FLUID-STRUCTURE INTERACTIONS


SLENDER STRUCTURES AND AXIAL FLOW


VOLUME 2




FLUID-STRUCTURE INTERACTIONS


SLENDER STRUCTURES AND AXIAL FLOW


VOLUME 2


MICHAEL P. PAIDOUSSIS
Department of Mechanical Engineering
McGill University, Montreal, Québec, Canada


ELSEVlER
ACADEMIC
PRESS


Amsterdam · Boston · Heidelberg · London . New Vork · Oxford
Paris · San Diego · San Francisco · Singapore · Sydney · Tokyo




This book is printed on acid-free paper


Elsevier Academic Press
84 Theobalďs Road, London WCIX 8RR, UK
http://www.elsevier.com
Elsevier Academic Press
525 B Street, Suite 1900, San Diego, Califomia 92101-4495, USA
http://www.elsevier.com


Copyright @ 2004, EIsevier Ltd, AU rights reserved


No part of this publication may be reproduced, stored in a retrieval system,
or transmitted in any fonn or by any means electronic, mechanical, photocopying,
recording or otherwise, without the prior written pennission of the publisher


Pennissions may be sought directly from Elsevier's Science & Technology Rights
Department in Oxford, UK: phone: (+44) (O) 1865843830, fax: (+44) 1865 853333,
e-mail: pennissions@elsevier,co,uk, Vou may also complete your request on-line via
the Elsevier homepage ( http://www.elsevier.com ). by selecting 'Customer Support'
and then 'Obtaining Pennissions'.


British Library Cataloguing in Publication Data
A catalogue record for this book is available fmm the British Library


Library or Congress Cataloguing in Publication Data
A catalogue record card for this book is available fmm the Library of Congress


ISBN 0-12-544361-7


Printed and bound in The Netherlands
04 05 06 07 08 9 8 7 6 5 4 3 2 I




Contents


Preface ....,......
Artwork Acknowledgements .


7 Thin Shells Containing or Immersed in Flow
7.1 Introductory Remarks . . . . . . . . . . . . . , . . . , , . .
7.2 General Dynamical Behaviour . , . . . . , , . . . . , . . .
7.2.1 The equations of motion of circular cylindrical shells
7.2.2 Shells containing quiescent fluid . . . . . . . . .
7.2.3 Shells with flow: Pai'doussis & Denise's analysis .
7.2.4 Experiments and comparison with theory .
7.2.5 Shells with flow: Galerkin-type solutions . . . .
7.2.6 Thin versus thick tubes conveying fluid. . . . .
7.2,7 Post-divergence flutter or dynamic divergence?
7.3 Refinements and Diversification ."...,'...,
7 ,3 ,1 Link between travelling and standing wave solutions
for shells with supported ends. . . . . .
7.3,2 Cantilevered shells and outflow model s .
7,3.3 Finite element methods . , . . . ,
7.3.4 Asymmetric support conditions . ,
7.3.5 Anisotropic shells containing flow
7.3.6 Other work ..,.........
7.4 Wave Propagation and Acoustic Coupling
7.4.1 Introductory comments . . , , . .
7.4.2 Wave propagation . . . . . , , . .
7.4.3 Complex wavenumbers and viscous fluids
7.4.4 Anisotropic shells . , . .
7.4.5 Concluding remark , . .
7.5 Viscous and Confinement Effects
7.5.1 Introductory comments .
7.5,2 Steady viscous effects on shells with supported ends .
7.5.3 Unsteady viscous effects: developing boundary layer
7,5.4 Unsteady viscous effects: general treatment . . . . .
7.5.5 Confinement and viscous effects on cantilevered shells
7.5.6 Concluding remarks . . . . , . . . . , . .
7.6 Swir1ing Flow and Rotofluidelasticity ..,,'.
7.6.1 Introductory remarks and literature review
7.6.2 Rotating shell conveying co-rotating fluid


X11l
XVll


573
573
574
575
582
584
594
601
605
610
612


612
621
627
639
646
649
652
652
654
665
667
669
670
670
671
676
681
687
696
696
696
699


v




CONTENTS


7,6.3 Tbe intense neutron generator system.
7.6.4 Boundary-layer effects for small Ro
7.7 Turbulence-lnduced Vibration ..'.....
7.7.1 General fonnulation . . . . . , . . , .
7.7.2 Response to an arbitrary random pressure field .
7.7.3 Response to a boundary-layer pressure field
7,7.4 Sample results . . . .
7.7.5 Concluding remarks .
7.8 Nonlinear Dynamics ....
7,8.1 Literature review "
7.8.2 Simply supported shells conveying fluid
7.8,3 Forced motions of fluid-conveying shells .
7,9 Pliable Shells . . , . . . , . . . , , . . , . .
7.9.1 Introduction,.......,.....
7.9.2 Pressure waves in horizontal systems.
7,9.3 Physiologica1 systems, .
7.9.4 Steady flow . . . . . . ,
7.9.5 Flow-induced oscil1ation
7.10 Applications.. . . . , . . . . .
7.10.1 General applications. , .
7,10.2 Internal-annular flow cases
7,10.3 Aeronautical app1ications ,
7.10.4 Shell-type Coriolis flow meter
7.11 Concluding remarks . . . . . . , , . ,
Solitary Cylindrical Structures in Axial Flow
8.1 Introduction.,..,.,..
8.2 Linear Equations of Motion . . . , , , . ,
8,2,1 General fonnulation ,..,...
8.2.2 Solitary cy1inder in unconfined flow
8.2.3 Cy1inder in a cluster or in confined flow
8.2.4 Boundary conditions .,.
8,2.5 Dimensionless parameters
8,2,6 Discussion ,....
8,2,7 Methods of solution
8.3 Linear Dynamics ......
8.3.1 Linear dynamical behaviour
8.3,2 Experiments and comparison with theory .
8.3.3 The mechanisms of instability
8,3.4 Thick boundary layers "."
8.3.5 Compressible and two-phase flows
8.3.6 Wave propagation . . . . . , . . .
8.3.7 Strings in axial flow . . . . , . . .
8.3,8 Deploying/extruding cy1inders and related problems .
8.4 Smal1-Amplitude Flow-Induced Vibration . . , . . . . . . .
8.4,1 Mechanism of excitation and prediction of vibration .
8.4.2 Chen & Wambsganss' analytical model
8.4.3 Further work and refinements . . . . . . . . , . . . .


706
708
710
711
713
717
720
721
723
724
731
741
745
745
746
750
752
760
781
781
782
782
783
786
787
787
789
789
792
794
796
798
800
800
802
802
815
829
834
836
848
850
856
858
860
872
878




CONTENTS


vii


8.4.4 Cantilevered cylinders. . . . . . . . . . . , . . . .
8.4.5 Closing remarks and acoustical1y excited vibration
8.5 Tapered Cylinders in Axial Flow . , . .
8.5.1 Tbe equations ofmotion .........,...
8,5.2 Transverse boundary conditions ........
8.5.3 Dimensionless equations and method of solution
8,5.4 The theoretical dynamical behaviour
8.5.5 Experiments .........
8.5.6 Closing remarks ......'...
8.6 Concurrent Internal and Extema1 Flow . . .
8.6.1 The theoretical model and solution method
8.6.2 Some theoretica1 results .........
8,6.3 Experiments and comparison with theory
8.6.4 Concluding comments
8.7 Parametric Resonances , .
8.7.1 Theoretica1 model
8.7.2 Experiments
8.7.3 Parametric resonances in two-phase flows
8.8 Complex Support Conditions . . . . .
8.8.1 Multispan cylinders and tubes
8.8.2 Loose supports .......
8.9 Towed and Self-Propelled Cylinders
8.9.1 Genera1 introduction .....
8.9.2 The Hawthorne-Pa'idoussis model
8.9.3 Dynamics oftowed flexible cylinders: theory
8,9.4 Experiments...."..
8.9.5 Rigid towed cylinders ,...,..
8.9.6 Mechanisms of instability . . . . . .
8.9.7 Acoustic streamers: Dowling's work
8.9,8 Conc1uding remarks for towed cy linders
8.9.9 Self-propelled/articulated cylinders
8.9.10 Locomotion of slender fish
8.10 Nonlinear Equations of Motion . . . . . .
8.10.1 Definitions and preliminaries . . .
8.10.2 The inviscid fluid dynamic forces
8.10.3 Hydrostatic and frictional forces
8.10.4 The equation of motion for inextensible centreline
8.10.5 Boundary conditions .......,......
8.10.6 Methods oť analysis . . . . . . . . . . . . . . .
8.10.7 Discussion: the equations used in Section 8.11.1
8.10.8 Equations for extensible centreline
8.10.9 Concluding remarks .......'
8.11 Nonlinear Dynamics ",...,....'
8.11.1 Cantilevered cylinders in axial flow
8.11.2 Cylinders with simply-supported ends
8.12 Articulated and Pendular Systems ..,...
8.12.1 Linear dynamics oť cantilevered system


882
883
885
885
889
890
891
894
897
898
898
901
903
907
908
908
912
918
922
923
924
926
926
930
933
937
949
951
953
957
957
963
971
971
974
977
980
982
983
986
986
988
988
989
999
1003
1003




vnl


CONTENTS


8.12.2 Chaotic dynamics of a pendular system. . . . . . . .
8.12.3 Osci11ation of a spherica1 pendulum in a flow-channe1
8.13 Practical Applications . . . . . . . . . . . . . . . . . . . . .
8.13.1 Nuc1ear reactor and other power-generating systems .
8.13.2 A sample practical case . . . .
8.13.3 Towed and propulsion systems . . . . . . . . . . . .


1005
1021
1024
1024
1027
1031


9 Clustered Cylinders in Axial Flow 1033
9.1 Introduction........ 1033
9.2 Hydrodynamic Coupling . . 1033
9.2.1 Genera1 discussion . 1033
9.2.2 Analytica1 methods for inviscid coupling . 1040
9.2.3 Analytica1 methods for viscous coupling 1049
9.3 Dynamics of Clustered Cylinders in F10w . 1054
9.3.1 The equations ofmotion ...... 1055
9.3.2 Solution of the equations of motion . 1060
9.3.3 The dynamica1 behaviour . . . . . . 1060
9.3.4 Experiments and comparison with theory . 1068
9.3.5 Clustered, flexibly interconnected cylinders 1081
9.3.6 Clustered cylinders simultaneously subjected to interna1 and
external flow. . . . . . . . . . . 1091
9.4 F1ow-Induced Vibrations . . . . . . . . 1097
9.4.1 Genera1 vibration characteristics 1097
9.4.2 The basic ana1ytica1 model . . . 1105
9.4.3 The excitation field and the response 1108
9.4.4 Comparison between theory and experiment 1113
9.4.5 Conc1uding remarks . 1126
9.5 Strings ofClusters. . . . . . . . . . . . . . . 1127
9.5.1 Motivation............... 1127
9.5.2 Modelling of c1uster-string dynamics . 1129
9.5.3 Dynamica1 behaviour . 1132
9.5.4 Conc1uding comments 1134
9.6 Summary and Conc1usion . 1134
10 Plates in Axial Flow 1137
10.1 Introduction.................... 113 7
10.2 General Ana1ysis ................. 1139
10.2.1 Infinite plate in incompressible fluid flow . 1139
10.2.2 Plate of finite length in incompressible flow 1145
10.2.3 Absolute and convective instabilities . . . . 1157
10.3 Representative Studies ............... 1161
10.3.1 Plates in contact with quiescent dense fluids 1161
10.3.2 Finite-width and finite-width-and-length plates in flow 1166
10.3.3 The effect of dissipation on stability 1172
10.3.4 Plate channe1s conveying fluid 1178
10.3.5 Para11el plate assemblies . 1181
10.3.6 Cantilevered plates in flow . . 1186




CONTENTS


IX


10.3.7 Nonlinear dynamics ..,...
10.3.8 Buckled plates in flow . , , . .
10.3.9 Travel1ing web, sheets and tapes
10.3.10 Spinning discs
10.4 Conc1uding Remarks . . . . . , , , . .


1195
1205
1207
1213
1220


11 Annular- and Leakage-Flow-Induced Instabilities 1221
11.1 Introduction........,...,..... 1221
11.1.1 General introduction . . . . . . . . . 1221
11.1.2 Preliminary discussion of mechanisms . 1223
11.2 Major Engineering Problem Cases , . . . . . . 1225
11,2.1 Nuclear reactor core barrel and thermal shield problems 1226
11.2.2 Feedwater sparger problems in BWRs . 1231
11.2.3 A final comment . , . . . . . 1233
11.3 Vibrations in Narrow Annuli . . . . , . . . . . 1233
11.3.1 Inertial and viscous coupling . . . . . . 1234
11.3.2 Vibration of fluid-coupled coaxial cylinders 1234
11.3.3 Numerical techniques ..,.....', 1237
11.3.4 Squeeze-film effects . . . , . . . . . . . 1245
11,3.5 Vibrations of fluid-coupled coaxial shells 1246
11.3.6 Forced vibrations . . . . . . 1248
11.4 I-D Leakage Flows . . . . , . . . , 1249
11.4.1 Mulcahy's I-d.o.f. analysis , 1249
11.4.2 Inada & Hayama's analysis 1256
11.4.3 The Porcher & de Langre 'pressure loss' model. 1264
11.4.4 Other work . . . . . 1268
11.5 Cylinders in Annular Flows . , , . . . . . . . . . . 1269
11.5.1 Hobsonetal. studies . . . . . . . . . . . . 1269
11.5.2 Other work on rigid centrebody dynamics . 1277
11.5.3 Flexible cy1inders in annular flow . . . . 1303
11.5.4 Numerical methods: prescribed motions . 1321
11.5.5 Numerical methods: dynamics 1338
11,6 Rotating Cylinders in Dense Fluid . . 1355
11.6.1 A mechanical analogy . . . . 1355
11,6.2 Rotor in fluid: linear analysis . 1359
11.6.3 Experiments .......,. 1364
11.6.4 A more general1inear theory . 1366
11.6.5 Non1inear effects . . . . , . . 1368
11.7 Coaxial Shells . . . . . . . , . . . . . 1371
II. 7.1 Coaxial shells with supported ends 13 71
11.7.2 Coaxial cantilevered shells . . . . . 1394
11.7.3 Wave propagation and simp1ified stability theory 1410
11.7.4 Practical experiences 1414
11.8 Miscellaneous Applications , . . . , . , . . . . . . . . . 1419


Epilogue 1421




x


CONTENTS


Appendices
L Matrices in Hybrid AnalyticaJ/Finite-Element Method of Lakis et a/. 1423
L.1 Matrices for a cy1indrica1 she11 in vacuo . . . , . . . . . 1423
L.2 Matrices associated with fluid flow in a cy1indrica1 shell . 1424
L.3 Matrices associated with fluid flow in a conica1 shell . 1424


M Anisotropic Shells 1428
N Effect of Viscosity in Rotof1uidelasticity 1430
O Nonlinear Motions of a Shell Conveying Fluid 1433
0.1 Theparticu1arso1ution, Fp . . . . . . . . 1433
0.2 The discretized equations ofmotion '" 1434


P Flutter of Collapsible Thbes Conveying Fluid 1437
P.l Bibliography.,. 1437
P.2 Analytical models . . . . . . . . . . . . . 1443


Q Viscous Force Coefficients for Cylinders in Axial Flow 1448
Q.l The longitudinal viscous coefficient Cy 1448
Q.2 The normal viscous coefficient CN . 1449
Q.3 The zero-flow normal coefficient CD . . 1451
Q.4 The base drag coefficient Cb ....., 1452
R Response to Arbitrary Force Field of a Cylinder in Axial F10w 1453
S The Equations of Motion of Articulated Cylinders in Axial Flow 1457
S.1 Self-propelled system . . , . 1457
S.2 Towed system . . . , . . . . . . . . . . . , . . . . . . . . . 1458
S.3 System with a gripped nose, . . . . . . , . . . . . . . . . . 1459
T The Nonlinear Equations of Motion of a Cylinder in Axial Flow 1460
T,I The inviscid hydrodynamic force FA (X, T) 1460
T.2 Tapered free-end boundary conditions . . . . . . . . . 1465
T.3 Axially extensible cylinders . . . . . . . . . . , . . . . 1468
T.4 Correction to equations for a fluid-conveying pipe with
supported ends ,.,..,......,........ 1473


U Added mass matrix by FEM 1474
V The Viscous Coupling Matrix 1477
W Spring Constants Used in the Calculations of Section 9.3.5 1480
X Flow-Induced Vibration of Clusters: Some Expressions in Detail 1482
X.1 Curve-fitted expressions for Kij,()(w) 1482
X.2 Expressions for Kln,l;(W) , . . . . . . . . . . . . . , . . . . . 1483




CONTENTS


Xl


Y Some Details for l-D Leakage Flow Analyses
Y.l Mu1cahy's (1988) l-d.o.fo analysis .
Y.2 Inada & Hayama's (l990a) analysis .. o
Z Coaxial Shells: Some Details
Z.1 Generalized forces; inviscid incompressible flows
Z,2 Numerical evaluation of the generalized inviscid forces
Z.3 The mean annular viscous pressure field o o .
Z.4 Model for cantilevered shell with annular flow
Z.5 Velocity profile and viscosity in annular flow .
Z,6 Solution of Navier-Stokes equations . . . . .


1484
1484
1485
1488
1488
1489
1490
1491
1493
1494


Q Recent Publications Related to Volume 1
Qol Fundamentals and related topics
Q.2 Pipes with supported ends o o . . o .
Q.3 Cantilevered pipes. , , . . . . . . .
Q.4 Pipes with pulsating flow; parametric resonances
Qo5 Risers and ocean mining systems
Q,6 Divers systems and applications
Q.7 Corrigenda to Volume 1


1497
1497
1498
1499
1500
1500
1501
1501


References .
Index. o . o


1503
1573




Preface


Apart from Chapters 1 and 2, Volume 1 is devoted to the dynamics of pipes conveying fluid,
in all its diverse variants. The extensive treatment allotted to that topic may be justified
part1y by the veritab1e cornucopia of interesting dynamical features it disp1ays, both linear
and non1inear, but mainly because it is a model prob1em (a paradigm in dynamics) in the
following sense: its study 1ays the foundations for (i) the physical understanding of the
phenomena for al1 systems invo1ving axial flow, and (ii) the mathematical mode11ing and
analysis of such systems.
A general Introduction to the aims and scope of this two-vo1ume set has been given in
Chapter 1 of Vo1ume 1 and wi1l not be repeated here; this is followed by Chapter 2 on
Concepts, Definitions and Methods, which app1y equal1y to the contents of both vo1umes.
Then, Chapters 3-6 deal with various aspects of the dynamics of pipes conveying fluid.
Frequent reference wi1l be made in this vo1ume (Vo1ume 2) to specific items treated in
Vo1ume 1. Indeed, the very fact that this vo1ume begins with Chapter 7 is indicative that the
two vo1umes constitute real1y one book, giving a coherent, integrated and comprehensive
treatment of the subject.
The material presented in this vo1ume cou1d have been ordered in many different ways.
The sequence adopted is as follows, First, the treatment ofaxial internal flow prob1ems
is comp1eted with Chapter 7, which deals with the dynamics of shells containing fluid;
however, the treatment is such that it app1ies to external axial flow over shells, as well.
Pure ly exte rnal flow prob1ems are treated next: solitary and clustered cy linders immersed
in axial flow in Chapters 8 and 9, and p1ates in contact with axial1y flowing fluid in
Chapter 10.
Hybrid cases invo1ving annular flow, in a sense both internal and external, are discussed
in Chapter 11. Final1y, the Epi10gue wraps up this two-vo1ume set. The reader is referred
to the introductory comments prefacing each chapter for a fuller discussion ofthe contents
therein.
A notab1e feature of the topics dealt with in this vo1ume is that, by their very physical
nature, they have wider and more direct application to engineering systems vis-o-vis those
treated in Chapters 3-6 of Vo1ume 1, as is amp1y made clear in the samp1e 'practical
experiences' presented in Section 1.1 of the Genera1 Overview section of Chapter 1,
and more fully in the comp1ete set of such practical experiences collected by the author
(PaYdoussis 1980) and by others (Naudascher & Rockwell1980a). In this vo1ume, refer
to Sections 7.9 and 7.10,8.4,8.9 and 8.13, 9.4,10.3.9 and 10.3.10, 11.2, 11.7.4 and 11.8,
for instance, for various engineering and bio1ogicaVphysiological app1ications,
The emphasis throughout is to give as clear a physical understanding of the system
dynamics and vibration as feasib1e, ln this respect, mathematics is used as a too1. More
space is allotted to ana1ytica1, ar more precise1y semi-ana1ytica1 methods, as compared to


xiii




xiv
PREFACE
numerical ones, because the former remain useful for generating benchmark solutions to
which numerical solutions may be compared in years to come,
The field itself is anything but static. New areas of interest emerge, just as interest is
flagging in some traditional engineering applications. Examples are found throughout,
but here let us mention the new bioengineering work in colIapsible tubes (Section 7.9)
and shelI-type Coriolis flow-meters (Section 7.10), towed acoustic streamers (Section 8.9),
workrelated to paper-making (Section 10.3.9), and disk drives (Section 10.3.10). The study
offluid-structure interactions in uůcro-electro-mechanical systems (MEMS) is largely stilI
in the future, but it has already begun [see, e.g. Lin et al. (1998)],
I owe a great deal to the many associates, colIeagues and friends who have read parts of
the book, checked and corrected it here and there, and contributed to the final form thereof.
Major contributors in this regard are: Prof Marco Amabili of U. Panna on several parts
of Chapter 7; Prof Chris Bertram of U. New South Wa1es on Section 7.9; Prof Aouni
A. Lakis of École Polytechnique (Montréal) on Sections 7.3,3 and 7.7; Prof Emmanuel
de Langre of École Polytechnique (Paris) on several sections of Chapters 10 and 11; and
Dr Christian Semler of McGi1l who, as in Volume 1, has read, commented upon and
corrected virtualIy the entire contents of this volume.
More liuůted, butjust as importantcontributions on specific items or sections of chapters
were made by the folIowing.
On Chapter 7: Prof. Jim Nemes of McGil1 on Section 7.3.5; Profs Charles W. Bert of
U, Oklahoma, Dr Paul Liang of Harley-Davidson and Prof. Henry Scarton of Rensselaer
Poly. on Section 7.4; Prof. John Dugundji of MIT, Dr Magnus Linde of the Aeronaut-
ical Research Institute of Sweden and Prof. Luca Cortelezzi of McGil1 on Section 7,6;
Prof. R.D. Kamm of MIT on Section 7.9; and Prof. Dave S. Weaver of McMaster U.,
Dr Vinhson N guyen, Dr Igor Zolotarev of the Academy of Sciences of the Czech Republic,
and Prof. R.B, Bhat of Concordia U, on other subtopics.
On Chapter 8: Profs Michael S. TriantafylIou of MIT and Stuart J. Price of McGil1
on Section 8,2.4; Dr Shoei-Sheng Chen of Argonne National Labs on Section 8,8;
Timothy Davies of ShelI Deepwater Development, Phil Wass of Dunlop Ltd, Dr Mark
A. Grosenbaugh of Woods Hole Oceanographic Institution, and Prof. S.K. Bhattacharyya
of lIT Madras on towed systems (Section 8.9); Jean-Louis Lopes, now at EADS Launch
Vehicles, on Section 8.10; and Yahya Modarres-Sadeghi on Sections 8.2 and 8.10.
On Chapter 9: Prof, Llewelyn R, Curling of U. Bahamas on Sections 9.2 and 9.4.
On Chapter 10: Prof. Earl H. DowelI of Duke U. on most sections of this chapter; Prof.
M,K. K wak of Dongguk U. (Korea) on Section 10,3.1; Dr Changqing Guo of N anhua U. on
Section 10.3.5 and 10.3.6; Dr Lixi Huang ofHong Kong Polytechnic U, on Section 10.3.6;
Prof. Peter Carpenter of U. Warwick on Section 10.3.7; Profs Antti Prauůla of Oulu U.
(Finland) and Peter Moretti and Y,B. Chang of Oklahoma State U. on Section 10.3 .9; Prof.
C. Dan Mote Jr ofU, Maryland on Sections 10.3.9 and 10.3.10; and Dr Morten H, Hansen
of RisSJ! National Lab (Denmark) on Section 10,3.10.
On Chapter 11: Prof. Abolghassem Mekanik ofBu-Ali U. (Iran), Prof. Woo-Gun Sim
ofHannam U. (Korea) and Prof, D. Mateescu ofMcGil1 on several sections ofthis chapter;
Dr Fuuůo Inada of CRlEPI (Japan) on Section 11.4; Dr Akira Yasuo of CRlEPI (Japan)
on Section 11.5.3; Dr Franois Bélanger of CAE on Section 11.5.4; Dr José Antunes of
Instituto Tecnológico e Nuclear of Portuga1 on Section 11.6; and Prof. René-Jean Gibert
of CEMIF (France) on Section 11.7.4,

PREFACE
xv
Many thanks also to Bill Mark for his expert and very willing help with several drawings
and photographs, to Kostas Karagiozis and Marika Asimakopulos for invaluable help with
hundreds of copies of joumal articles and bibliography, and to Yahya Modarres-Sadeghi
for help with proofreading the relabelled figures.
To all these contributors this book owes a great debt of gratitude, and I the most sincere
and heartfelt thanks.
Special thanks go to the many Ph.D., M.Eng, and Honours B.Eng. students, whose
these s I had the pleasure of supervising, as well as to the Post-Doctoral Fellows I was
priviledged to work with. Their names, too numerous (over 80) to list here, crop up in
the pertinent theses and co-authored papers cited throughout. Some of their contributions
have been pivotal, and this is reflected in this book as well. This reminds me of an essay
(booklet) that my father, who was an educator also, wrote: "Youth, the Spring oj Etemal
Renewal".t Indeed, I owe a great deal to the lively interactions with, and the youthful
enthusiasm and inventiveness of, all these then young collaborators.
My gratitude to my secretary, Mary Fiorilli, is unbounded, for without her virtuosity,
vigilence, dedication, helpfulness and loving care this book would not have materialized.
The loving support and constant encouragement by my wife V rissels (B pta r(is) has been
a sine qua non for the completion of this book, as my mother's exhortions to "be laconic"
have been useful, even if only partly heeded! For what little versatility in the use of English
this volume may display, I owe a great deal to my late first wife, Daisy.
Finally, I am grateful to my professors and supervisors, especially those who influenced
most my way of thinking and career path: Donald L. Mordell and A. Robert Edis in my
undergraduate studies at McGill, Sír William R. Hawthome PRS and T. Brooke Benjamín
PRS in my Ph.D. studies at Cambridge, and Ron I. Hodge at AECL, Chark River.
Acknowledgements are also due to the Natural Sciences and Engineering Research
Council of Canada, Fonds pour la Formation de Chercheurs et l' Aide a la Recherche
(FCAR) of Québec and McGill University for their support, the Department of Mechanical
Engineering of McGill for their forbearance, and to Academic PresslElsevier Science for
their help and encouragement.
I wish I could say that writing this two-volume set has been a breeze. The four-year
interval between the two volumes attests to the contrary. * It has been hard work. To quote
an unlikely source, Vidal Sassoon, "the only place where success comes before work is in
the dictionary". However, writing these two volumes has been immensely fulfilling.
Michael p, Paldoussis
McGill University
Montreal, Québec, Canada
t N eórT/s ws nnyIÍ ALwvías A vayevv1)uews, Publications of the Graduates' Society oť Abet School
(A/-L1Teuíov I:;XOAiís), No, 2; Cairo, Egypt, Dec. 1934,
tVery much contrary to the British popular-romance author, Dame Barbara Cartland, who boasted that "For
20 years, rve done a book a fortnight", before she died at 98 in 2000.
So, I cannot say, as John Tynda1l did in 1859, that "I think I shall never undertake to write a book again"
(F. Fahy 1985, Sound and Structural Vibration, London: Academic Press). Readers beware!

Artwork Acknowledgements


A number of figures used in this book have been reproduced from papers, reports or books,
appropriately relabelled, by kind permission of the publisher or the authors. In the great
majority of cases, permission was granted without any special requirement, other than
proper citation of the source, authors and other appropriate bibliographical information.
ln some cases, however, the publishers required specific statements. Rather than doing
so on each and every figure in this category, the following global statements are made in
the interests of brevity.
. Figures from papers t in theAlAA Journal and the Journal oJHydronautics, copyright
by the American Institute of Aeronautics and Astronautics, Inc" reprinted with
permlsslOn.
. Figures from the Annual Review oj Fluid Mechanics, copyright by Annual Reviews
www.annualreviews.org, reproduced with permission.
. Figures from NASA publications, reproduced by permission of the National
Aeronautics and Space Administration, USA.
. Figures from Nuclear Science and Engineering, copyright by the American Nuclear
Society, La Grange Park, lllinois, USA, reproduced with permission,
. Figures from ANL reports, copyright by Argonne National Laboratory, managed and
operated by the University of Chicago, U.S. Department of Energy, reprinted with
permission.
. Figures from Proceedings Asia-Pacific ConJerence'93, copyright 1993 IEEE,
reprinted by permission.
. Figures from Proceedings oj the Institution oj Mechanical Engineers and the Journal
oj Mechanical Engineering Science, reprinted by permission of the Council of the
Institution of Mechanical Engineers, UK.
. Figures from the Journal oj Fluid Mechanics, by permission of Cambridge
University Press.
. Figures from the Shock and Vibration Digest, by permission of Sage Publications
Ltd.
. Figures from A.A. Balkema publications, copyright by Swets & Zeit1inger, by kind
permission,
The il1ustrations on the front cover (from left, clockwise) are based on figures
appearing in:
(i) M. Amabili, F. Pellicano & M.P, Paldoussis (2000b), Journal oj Sound and
Vibration (EIsevier);


trn all cases, refer to the References, p, 1503 et seq, , for the complete title of the paper, the joumal or other
publication title, and other pertinent details,


xvii




xviii
ARTWORK ACKNOWLEDGEMENTS
(ii) Y. Watanabe, S, Suzuki, M. Sugihara & M. Sueoka (2002a), Journal oj Fluids and
Structures (Elsevier);
(iii) M. Amabili, F. Pellicano & M.P. Pai'doussis (2001), Journal oj Applied Mechanics
(ASME); those on the back cover are from Pa'idoussis (1979), Journal oj Sound
and Vibration (Elsevier) - aU with kind permission.
AU the publishers' cooperation is hereby gratefuUy acknowledged,

7
Thin Shells Containing or
Immersed in Flow


7.1 INTRODUCTORY REMARKS


The aeroelasticity of plates and shells, and particularly the flutter thereof, is a venerable
and well-researched subject area - see, for example, Dowelľs (1975) book on the very
subject and thereviews by Johns (1969), Parthan & Johns (1969), Komecki (197la, 1978,
1979), as well as the standard textbooks on aeroelasticity (Scanlan & Rosenbaum 1951;
Bisplinghoff etal. 1955; Fung 1958; Bisplinghoff & Ashley 1962; F6rsching 1974; BUlvol
1975; Librescu 1975; Dowell & Ilgamov 1988; Dowell et al. 1995),
The key word is panel flutter. A panel is a thin flat or curved plate supported on two
or all four sides and subjected to axial flow, generally on one side; the terrn is used not
only for plates, but also for membranes and shells. Clearly, such a system could represent
a portion of the skin of an aircraft or wing, or of the skin of a missile. As shown in
Section 7.3.6, such panels lose stability by divergence if the flow is subsonic and by flutter
if it is supersonic - refer also to Chapter 10. Since the panel is generally well supported at
its edges, structural nonlinearities associated with deforrnation-induced stretching of the
panel (cf. Sections 5.2.4 and 5.5,2 of Volume 1 t) limit the amplitude of both divergence
and flutter. However, whereas divergence may be ignored for practical purposes since it
is 'milď, flutter cannot be, because it can lead to fatigue failure. Hence, most work by far
has been devoted to flutter and, again by far, to supersonic flutter.*
In subsonic flows, traditional wisdom has it that any vibrations of interest are induced
by turbulence or other flow unsteadiness and are, therefore, forced rather than self-excited
(Section 1.2), although post-divergence flutter is possible, as discussed in Chapter 10. As
this book, arbitrarily and for purely practical reasons, is devoted mainly to incompressible
flows and to some extent to weakly compressible ones, a very large volume of the available
literature is beyond its scope and will be referred to only in passing; the interested reader
is referred to the references already cited,
The dynarnics of plates and shells is often treated together [see, e.g. Dowell (1975)],
which is especially attractive if the Fltigge-Kempner or the Donnell equations of motion of
the shell (to be discussedin Section 7.2.1) are used, in which shellmotions are described by
a single eighth-order differential equation, 'similar' to the fourth-order equation for a plate.
However, here a different approach is taken. Since the dynamics of thick-walled pipes


t AU sections cited beginning with a 1, 2, ' , , , 6 are associated with Chapters 1-6, which constitute Vo1ume 1
of this work (PaYdoussis 1998),
+The disintegration of the Second World War V-2 rockets in mid-flight was not due to navigation contro1
prob1ems, as initia11y thought, but to pane1s popping out due to panel flutter (Fnedmann 1998).


573




574
SLENDER STRUCTURES AND AXIAL FLOW
conveying fluid has been treated extensively and exhaustively in Chapters 3-6 (Volume 1),
it makes sense to proceed immediately with the treatment of thin-walled pipes (cylindrical
shells) conveying fluid, before embarking on problems involving externa1 axia1 flows over
cylinders and plates, which are covered in Chapters 8-10. This chapter deals with the
stability and flow-induced vibration of shells subjected to interna1 or external axial fluid
flow, a1though shells immersed in or containing quiescent fluid are also discussed briefly.
The dynamics of shells containing or surrounded by axia1 flow of incompressible fluid
was long thought to be uninteresting; the flow being subsonic, the system was expected
to be subject to only a mild (small amplitude) form of divergence. Nevertheless, a new
breath of life was blown into the subject, with the fortuitous discovery in 1969 that both
cantilevered and clamped-clamped shells, i.e. shells with both ends clamped, containing
low-speed flow do flutter (Pa'idoussis & Denise 1970, 1971, 1972),
The phenomenon was first observed in March 1969, in the course of experiments with
elastomer vertica1 tubu1ar cantilevers conveying fluid, which norma1ly lose stability by
flexural, beam-like flutter (Chapter 3, Section 3.5.6). On this particular occasion, how-
ever, the cantilevered pipe was unusua1ly thin-wa1led, and it was observed that, once
the flexural flutter developed, a shell-type oscillation in the second circumferentia1 mode
[n = 2; Figure 2.7(c)]t was superposed on the motion. The oscillation was accompanied
by a high-pitched sound (typica1ly 200-400 Hz). Subsequent experiment s with shorter
cantilevered pipes, which remain stable in beam-mode flexure up to high-flow velocities,
demonstrated that shell-mode flutter with n = 2 or 3 could develop on its own, as shown
in Figure 7.1. Even more momentous was the observation that a1so when both ends of the
pipe were clamped, the system developed shell-mode flutter at high enough flow,+
The first and simplest analytical model s to describe the phenomenon are presented in
Section 7.2, together with sample theoretical results. The experiments are a1so described
therein in greater detail, and theory and experiments are then compared.
More refined analytical models and more recent developments are presented in the
sections that follow, which deal with both the stability and flow-induced vibrations of the
system. The chapter ends with sections on physiological and engineering applications of
this work.
7.2 GENERAL DYNAMICAL BEHA VIOUR
The flutter phenomenon observed by Pa'idoussis & Denise (1970,1971), asjust described
in Section 7.1, is of the same general character as that studied before, in the 1950s and
1960s, in the case of external, usua1ly supersonic, axial flow over long cylindrical shells by
Bolotin (1956, 1964), Miles (1957,1958), Dowell (1966b), Dowell & Widna1l (1966a,b),
Olson & Fung (1966, 1967), Parthan & Johns (1969) and many others, during that period
and ever since.
t n is defined in Figures 2,7(c) and 7.3(a),
*The phenomenon was demonstrated to Prof. MJ. Lighthill who happened to visit the author's laboratory
in the summer of 1969, and who confinned that it had not been observed, or at least not reported, hitherto, He
suggested that understanding the underlying physical mechanism might be helpful in elucidating the generation
of Korotkoff sounds in haemodynamics - the characteristic sounds generated in the arterial system when
taking the blood pressure with a sphygmomanometer, rendered audible through a stethoscope placed over the
brachial artery in the antecubical space (Section 7,9), Although this line of investigation proved unfruitful, it
nevertheless provided an added impetus toward a systematic research effort into the stability of shells conveying
incompressible fluid.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
575
I
+ 11
1,/
-=-- - 1-- ":'___
--x:. _ _:..;r--
II
II
.,..
,- -
I J i \ \
I I, \ I
\ / I \ /
\ I , \ /
I
,
I
,
I
,
I
i
I
I
Figure 7.1 The observed flutter (n = 2) of an elastomer cantilevered cylindrical shell conveying
air (PaYdoussis & Denise 1972),
The main differences are that in this case (i) the flow was subsonic and essentially
incompressible, the Mach numbers in the experiment s being of the order of M = 0.05,
and (ii) the flutter amplitude was very large, at least for the elastomer pipes used in these
experiments, as may be appreciated from Figure 7.1. Another important and perplexing
feature of the observed phenomenon is that the conservative system (the shell with sup-
ported ends) apparently loses stability by flutter rather than divergence - cf. Chapters 3
and5.
7.2.1 The equations of motion of circular cylindrical shells
Unlike thick-walled pipes, the equations of motion of which in vacuo are different forms
of the beam equation(s) (Chapters 3-5), the equations of motion of a shell are much
more complex. Many investigators, e.g. Love, Fltigge, Naghdi, Timoshenko, Koiter,
Vlasov, Novozhilov, Donnell, Morley, Sanders and Kempner, have derived differential
equations for the static and dynamic analysis of shells. A nice discussion of these equations
and of the methods used to solve them, easily accessible to the non-expert, is given
by Forsberg (1965), who states that 'In a rather general way one can say that the wide
variety in the resulting equations arises basically from small differences in the formulation
of the strain-displacement relations, and the discrepancies occur only in terms which

576
SLENDER STRUCTURES AND AXIAL FLOW
numerically have 1ittle significance. As long as the 1imitations of thin shell theory are
observed, the various formulations give identical numerical results within engineering
accuracy.'t The interested reader is referred to Love (1927), Sanders (1959), Timoshenko
& Woinowsky-Krieger (1959), Fliigge (1960), Greenspon (1960), Novozhilov (1964),
Kraus (1967), Warburton (1973), Leissa (1973), Donnell (1976), Niordson (1985, 2000),
Soedel (1993) and many other books, reports and papers on the theory of shells. A nice,
easily accessible treatment is given in Markuš (1988), and a useful engineering summary
may be found in Blevins (1979). The discussion here is confined to linear, first-order shell
theory.
ln what follows, a few words will be said about the equations of Fliigge (1960), which
are used widely in this chapter, as well as about the simp1ified Donnell, Morley-Koiter,
Fliigge-Kempner and Sanders equations. The shell is assumed to be thin and to be made
of purely elastic, homogeneous and isotropic materia!.
The deformation of a thin circular cylindrical shell may be described by the displace-
ments ofthe 'middle' surface (half-way between the outer and inner wall, h apart, where h
is small compared to the mean radius, a): u(x, e, t), v(x, e, t) and w(x, e, t) in the axial,
circumferential and radial directions in a cylindrical-polar coordinate system (x, e, r), x
being the axial coordinate, as shown in Figure 7.2(a);+ t is the time. Fliigge's equations
for free motion of the shell are:
fO " 1 - v.. 1 + v /. / [ 1 - v.. '" 1 - v /.. ]
""'l(U,V,W)=u +-u +-v +vw +k -u -w +-w
2 2 2 2
a 2 u
= y at 2 '
fO l+v /. .. I-v" . [ 3 ,,3-v ". ]
""'2(U, v, w) = -u + v + -v + w + k -(1 - v)v --w
2 2 2 2
a 2 v
= y at 2 ' (7.1)
fO ( ) /. k [ 1 - V /.. '" 3 - v ". 't'74 2 .. ]
""'3 u, v, w = vu + v + w + u - u - v + v w + w + w
a 2 w
= -y at 2 '
where
k = /2 (hja)2, y = p s a 2 (1 - v 2 )j E,
()/ = aa( )jax, (r = a( )jae, V 2 = a 2 (a 2 jax 2 ) + a 2 jae 2 ,9
Ps being the density of the shell material; the last equation has been multip1ied by -1, to
achieve symmetry of the operator-matrix of the left-hand side,
(7.2)
tThis statement does not apply to the simplest form of the Donnell equations, as we shall see, Refer a!so to
Forsberg (1969).
*Note that this axis system and the directions of {u, v, w} violate the right-hand role; to satisfy it, r and w
for instance ought to be pointing inwards, A repercussion of this is the statement at the end of the paragraph
following equation (7.2).
'V'2 is a dimensionless form ofthe norma! Laplacian operator, with 8( )/8r = O.

TH.N SHELLS CONTAINING OR IMMERSED IN FLOW
577
(a)
(b)

Figure 7.2 (a) A cylindrical shell, showing the coordinate system and some key dimensions.
(b) An element of the shell showing the stress resultants N, M, and Q, following Fliigge's
sign convections,
The associated boundary conditions for a fulIy c1amped end of the shell are
u = v = w = 0,
w'=O,
(7.3)
For a free end, the normal force N x = ° and the bending moment Mx = O [see
Figure 7.2(b)], as welI as Kirchhoff's effective shearing stress resultants, Sx = Qx +
O/a)(aMxe/ae) = 0, Tx = Nxe - Mxe/a = ot (Fltigge 1960; Leissa 1973), which may
tDifferent sign conventions are used for the stress resultants by different authors, and one cannot switch
from ODe to the other without care.

578
SLENDER STRUCTURES AND AXIAL FLOW
be re-written in terms of displacement components as
N x = O : u' + \IV' + \lW - kw" = O,
Mx = O : w" + \lw" - \Iv. - u' = O,
S O ", (2 ) ,.. 3 - \I,. 1 - \I.. " O
x = : w + - \I W - v + u - u = ,
Tx = O : u' + v' + 3k( v' - w") = O.
For an axially unconstrained, simply supported end, we have
v = W = O,
(7.4a)
(7.4b)
(7.4c)
(7.4d)
(7.5)
as well as (7.4a) and (7.4b), i.e. N x = Mx = O.
To solve the equations of motion, one may generally take
{ u } { A(alax)cosne }
v = BsinnO 4>m(xIL)e iQt ,
w C cos ne
where 4>m (x I L) == 4>m (g) are the beam eigenfunctions with the appropriate boundary
conditions, and then obtain eigensolutions via the Galerkin or Rayleigh-Ritz methods
(Chapter 2). It is clear that each shell mode is associated with a circumferential wave-
number n and an axial wavenumber m. t The resulting vibration therefore involves given
combinations of n and m, as shown in Figure 7.3 for clamped-clamped ends.
A particularly interesting case is that of a shell with simply supported ends,* for which
4>m(g) = .J2sin(mng) - simple enough to result in an especially simple solution.
Substituting equation (7.6) into the equation of motion, for example, equation (7.1), the
radian natural frequencies, Q, may be determined from the roots of the following bicubic
equation in Q :
-6 -=4 -2
Q -k2 Q +k,Q -ko =0,
(7.6)
(7.7)
where Q = y'YQ are the dimensionless natural frequencies, and ko, k, and k2 are coef-
ficients depending on the shell theory utilized [e.g. see equations (7.28)]. It is noted that
sets of three different modes arise for each (n, m) pair: all three have the same number of
nodallines [see Figure 7.3(a)], but one is predominantly radial (generally the lowest), one
predominantly circumferential ('torsionaľ), and the third mainly axia1.
The lowest natura1 frequencies are not necessarily associated with the lowest n, i.e.
n = O, 1 or 2; the lowest frequency depends on the proportion of the total strain energy
associated with bending and stretching (extension ofthe circumference), as first elucidated
by Arnold & Warburton (1949). As shown for example in Figure 7.4, the minimum
strain energy in this case occurs for n = 7; the natural frequencies squared are of course
proportional to the strain energy. The value of n associated with the minimum natural
frequency depends mainly on hla and Lla,
tStrictiy, m is the half-wave number.
:j:Unlike beams, for which 'pinned' has been used almost throughout Volume 1 to denote 'simply supporteď,
this notation is not usua1 for shells and it is avoided in this chapter,
In this and similar figures, conventionally, continuous lines are drawn connecting the discrete points
corresponding to the only possible, integer values of n,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
579
Q9 -----
, ,
, ,
, I
, ,
, ,
, ,
-......--
n=O
n=1
n=2
n=3
(a)
 / ----------- '/ ,>
/ //,
/ '/ /
,./ ,//"
/ ....-_.._--...... / /
.................
/
/
'..............
,<
#...............
/'
........-
(b)
m= 1
m=2
m=3
Figure 7.3 Mode shapes of a clamped-damped cylindrical shell oscil1ating in (n, m) mode: (a) the
circumferential modal pattem associated with n circumferential waves; (b) the axial modal pattem
associated with m half-waves for the 'breathing modes', that is, for all m other than m = 1 (the
'beam modes'). 'Clamped-clampeď denotes clamped at both ends. The 'nodallines', equal to n,
are shown in (a) for n = 1, n = 2 and 3; they join the 2n nodes in each case.
>-.

Q)
í.i
.:::
.g
'"
'"
'"
Q)
-a
.8
'"
.:::
Q)
E
i5
Total energy
\
,
,
,
,
'...... ..,..",
......>............-
._._0
2
6
10
14
Circumferential wavenumber, n
Figure 7.4 The strain energy and its bending and stretching components with varying n for a
given circular cylindrical shell with simply supported ends [after Arnold & Warburton (1949) and
Blevins (1979)].

580
SLENDER STRUCTURES AND AXIAL FLOW
For very thin shells (k« 1), the equations of motion are sometimes simplified to
" 1 - v.. 1 + v /. / 02u
u +u +v +vw =y ot 2 '
1 + v /. .. 1 - v" . a 2 v
u + v + v + w = y at 2 '
/. 4 a 2 w
vu + v + w + kV w = -y ot 2 '
a fonn of the Donnell, or Donnell-Mushtari, shell equations (Donnell 1933, 1976;
Fliigge 1960, section 5.1.3). If additionally the in-plane inertia tenns, y(02ujot 2 )
and y(a 2 vjat 2 ) are neglected, one may then easily relate u and v to w, namely via
(Fliigge 1973)
(7,8)
V 4 v + (2 + v)w'" + w." = O,
V 4 u + vw'" - w/" = O,
(7,9)
and hence obtain a single equation of motion:
( a 2 w )
kV 8 w + (1 - v 2 )w"'/ + yV 4 - = O.
ot 2
(7,10)
This constitutes Donnell's 'sha11ow shell' theory, with equation (7.10) most commonly
referred to as the Donnell equation of motion.
In general, results obtained by different sets of shell equations are similar, since the
equations differ in tenns which have little influence, but the Donnell equations, particularly
(7,10), are an exception (Forsberg 1965): the simplifications introduced in this case limit
the range of applicabi1ity of the results, The use of the Donnell equations is generally
limited to cases where n is large (1 j n 2 « 1) and of course aj h large: say, n > 8 and
aj h > 500, However, as shown by Forsberg (1965) and Leissa (1973), they may be
applied to smaller n (say, n = 5) and aj h, provided that Ljma < 10 or 20, where mis
the axial wave number. U seful comparisons of the results obtainable with the Fliigge and
Donnell equations have been provided by Hoff (1955), Forsberg (1965) and Leissa (1973).
A slightly different form of equations (7.8), largely free from the limitations of the
Donnell equations has been developed by Mor1ey (1959) and rigorously established by
Koiter (Mangelsdorf 1971). These are the so-called Morley-Koiter equations; the first two
are identical to those in (7.8), but the third is
02W
vu/ + v' + w +k(V 2 + 1)2w = -y ot 2 ;
(7.11)
it differs from the corresponding one in (7.8) only in the tenn involving k. There is also
a single-equation version of these equations (Morley 1959), equivalent to but better than
Donnell's (7,10), namely
( a 2 w )
kV 4 (V 2 + 1)2w + (1- v 2 )w"'/ + yV 4 at 2 = O.
(7.12)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
581
Another set of equations which circumvents some of the limitations of the Donnell
equations, particularly with respect to the range of n to which they are applicable, are the
F1tigge-Kempner equations (Kempner 1955; Hoff 1955):
k[ V 8 w + 1  v 2 W llll + 2VW llllll + 6W IllI " + 2(4 - v)w ll ....
+ 2(2 - v)w ll " + 2w'..... + w.... ] + yV 4 ( a 2 w ) = O
at 2 '
(7,13a)
and two more equations relating u and v to w (not given here). This form of the equation
retains the same parameters as defined in (7.2). A more usua1 form, however, is the
following, established from (7. 13 a) by multiplying it by (P s h/y)V- 4 :
(D/a 4 )LF_K(W) + p s h(a 2 w/at 2 ) = O,
(7.13b)
where LF-K(W) = V- 4 [], with [] as in (7.13a)t and D = Eh 3 /[12(1 - v 2 )].
The reader should not be deceived by the length of the presentation devoted to the
Donnel, Morley-Koiter and Fltigge-Kempner equations. These are approximate equa-
tions, used sometimes for convenience at the co st of loss of some or a great deal of
accuracy. The F1tigge, or equivalently the Sanders, three-equation sets are the preferred
tool for the analysis of shell motions. The Sanders, a1so known as the Budiansky-Sanders,
shell equations (Budiansky & Sanders 1963) are presented in Section 7.3.3, and a nonlinear
form ofDonnell's equations in Section 7.8.
Before leaving this section, a short-hand notation introduced by Leissa (1973) is
presented, according to which equations (7.1), for instance, are written as
LF { : } = y Uw} ,
{ U } { LdU,V,W} }
where LF v == L2{U, v, w} ,
W L3{U, v, w} F
(7.14a)
in which the subscript F stands for F1iigge. Simi1arly, the Donnell equations (7.8) are
written as
L D {: }  y UJ
(7.14b)
and so on for the other sets of equations of motion,
FinaI1y, it is noted that, if the shell is subjected to a radia1 pressure perturbation
p* (x, e, t), with + p* being directed outwards, then the right -hand side of the third of
equations (7.1) and of equations (7.8) becomes
_y [ a 2 w _ L ] .
at 2 Psh
(7.15)
tln fact, even more usually, the equation is presented involving '\7*4 = a- 4 V 4 , which involves dimensional
derivative operators,

582
SLENDER STRUCTURES AND AXIAL FLOW
7.2.2 Shells containing quiescent fluid
Before considering shells containing flowing fluid, it is instructive to first treat the simpler
case of a shell filled with stagnant fluid. Specifically, let us consider free vibrations of a
thin circular cylindrical shell with simply supported ends, completely filled with quiescent,
inviscid and slightly compressible fluid - a problem first treated by Niordson (1953),
Berry & Reissner (1958) and Lindholm et aZ, (1962),
Irrotational motions of the fluid associated with small motions of the shell must satisfy
the Helmholtz equation,
1 a 2 <1>
V 2 <1> = "2---Z'
c at
(7.16)
<I> being the velocity potential and c the velocity of sound in the fluid; V 2 <I> is the Laplacian
of <I> (see Section 2.2.1). The impermeability boundary condition on the fluid-shell
interface, assuming no cavitation, is
 Ir=a =
aw
at '
(7.17)
w being the radial displacement. In accordance with equation (7.6) and with the discussion
leading to (7.7),
w(x, e, t) = C sin(mnx / L) cos ne J(t),
(7.18)
where J (t) is a harmonic function of t, of radian frequency Q. In view of (7.17), the form
of <I> for a solution by separation of variables should be
<I>(x, r, e, t) = Q(x)R(r) cos ne jet),
(7.19)
where jet) = dJ/dt. The function Q(x) depends on the conditions at the extremities of
the shell: at an open end, <I> = O; at a do sed end, a<l>/ax = O. Hence it is easy to verify
(Au-Yang 1984; Amabili 1997a) that
Q(x) = sin(mnx/L)
for both ends 'open',
(7.20a)
00
= L O'm cos(mnx/ L)
m=l
for both ends dosed,
(7 .20b)
00
= L O'm cos [(2m - l)nx/ L]
m=l
for an open end at x = O
and a closed end at x = L,
(7.20c)
where the O'm are coefficients depending on m. In (7.20a), 'open' means connected to
unconfined fluid, so that the pressure perturbation at an open end must vanish.
ln what follows we consider exclusively the shell to have open ends. This offers a
particularly attractive simplification, since Q (x) has the same form as the in vacuo axial
variation of w: 'the wet mode' shapes, i.e, the mode shapes in the presence of the fluid,
are the same as 'the dry modes', i.e. the modes in vacuo,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
583
Inserting (7.19) and (7.20a) into (7.16) leads to
d 2 R +  dR _ [ A2 _ ( Q ) 2 + n 2 ] R = O,
dr 2 r dr e r 2
(7.21)
where A = mJr I L; this is a form of Bessel's equation and hence admits solutions of
the form
([ 2 2 2 ] 1/2 )
R(r) = Amn1n A - Q le r
([ 2 2 2 ] 1/2 )
= AmnJ n Q le -A r
for Q/e ::: A
(7,22a)
for Q/e ::: A,
(7.22b)
where Amn is a coefficient to be determined, and where J n and ln are the Bessel and
modified Bessel functions of the first kind; those of the second kind have been eliminated,
as they violate the requirement that <I> be regular (finite) at r = O. From hereon, we shall
consider sufficiently small compressibi1ity (large e) and small enough frequencies for form
(7.22a) to apply throughout, for simplicity. Hence, equation (7.19) becomes
<I> = Amnln(A*rla)sinAxcosnej(t), (7.23)
where A * = [A 2 _ Q2/e 2 ]1/2a.
Substituting (7.23) and (7.18) into (7.17) gives Amn = C/[(A*la)I(A*)], where the
prime denotes the r-derivative. The perturbation pressure p*, exerted by the fluid on the
internal surface of the shell is given by the linearized form of equation (2.67a), i.e.
* a <I> [
p = -Pat .
r=a
(7.24)
Then, using the compact form of Donnell' s equations (7 .14b) and reca1ling that the fluid
loading appears as an additional term yp* I Psh on the right-hand side of equation (7.8),
we obtain
{} { U } { U }
L u  pa ln (A *) 
D  = Y _v w + Y Psh A*I(A*) _v w .
(7.25)
By defining the dimensionless virtual or added density of the fluid (see Section 2,2,2) by
pa ln (A *)
f3mn = h A *1' (A *) , (7.26)
Ps n
equation (7.25) may be re-written as
LD {  } = y {  } .
w -(1 + f3mn)W
Physically, this means that the contained fluid affects free motions of the shell primarily
by augmenting the inertialloading in the radial direction; the u- and v-components of
the motion are affected only through coupling with the w-components of the motion in
equations (7.8).
(7.27)

584
SLENDER STRUCTURES AND AXIAL FLOW
The characteristic equation for the eigenfrequencies in this case (simply supported,
'open' ends) has the same bicubic fonn as for the shell in vacuo, equation (7.7). The
coefficients ki for incompressible fluid are here given by (Amabili & Dalpiaz 1995)
k2 =  + 3 - v (n 2 + p) +  (n 2 + pf,
Ymn 2 Ymn
kl = 1 - v [ (3 + 2v)P + n 2 + Ymn ( n2 + P ) 2 + 3 = v k ( n2 + p/ ] , (7.28)
2n 1 v
ko= : [(1-V2)Á4+k(n2+PtJ.
where Ymn = 1 + f3mn and A = Aa.
ln equations (7.27) and (7.28), it is noted that (i) for thin shells, aj h "" 0(10 2 ) or
0(10 3 ), and (i i) as shown in Section 7.2.3, the Bessel functional in (7.26) is of O(ljn).
Hence, for air-filled metallic shells, for which pj Ps "" 0(10- 4 ), one obtains f3mn ""
0(10- 2 ), and the effect of the contained fluid on the vibration is quite small: the natural
frequencies in air and in vacuo are quite close. If the fluid is dense, however, for example,
water, so that pj Ps "" 0(10- 1 ), then 1 + f3mn "" 0(10 2 ), so that this effect can be quite
important.
This is demonstrated, for example, by some calculations and measurements by
Amabili & Dalpiaz (1995) for a water-filled steel shell (a = 175 mm, h = 1 mm, L =
664 mm, E = 206 GPa, Ps = 7.68 x 10 3 kgjm 3 , v = 0.3), shown in Figure 7.5. This
shell is simply supported and, so far as the fluid is concemed, 'open' at its extrernities,
which was achieved approximately in the experiments by closing the ends with thin flex-
ible plates. The shell was excited by an instrumented hammer and the signal was analyzed
by modal analysis software. Theoretical and experimental results agree remarkably well,
both in tenns of the eigenfrequencies (Figure 7.5) and the modal shapes (not shown here).
It is clear that the effect of the contained dense fluid is very important; had aj h been
higher, this effect would have been even more important.
7.2.3 Shells with flow: Pa'idoussis Br Denise's analysis
Despite the fact that the observed flutter was of very large amplitude, the first attempt to
model it (PaYdoussis & Denise 1972) was made via linear theory, which strictly applies to
small oscillations. t The shell is considered to be thin and the shell material to be purely
elastic, homogeneous and isotropic. The fluid-dynamic forces are derived from potential
flow theory: that is, it is assumed that the fluid is inviscid and non- heat -conducting. Further-
more, the fluid is assumed to be incompressible. In tenns of Fltigge' s shell equations (7.1),
the equations of motion take the fonn
(PU
LI (u, v, w) = Y----z,
at
[ a 2 w qr ]
L3(U, v, w) = -Y at2 - Psh '
a 2 v
L2(U, v, w) = Y----z,
at
(7.29)
tStrictly, nonlinear effects become generally non-negligible for amplitudes of the order of the shell
thickness - or larger,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
585
800
200
Experimental values
O Empty, m = I and 2
. Full vertical, m = I
O Full horizontal, m = I
/:::, Full horizontal, m = 2
700
600
g 500

g. 400
..g
]
'"
Z
100
Full, m = 1
O
3 4 5 6 7
Circumferential wavenumber, n
Figure 7.5 Theoretical (continuous lines) and experimental (symbols) natural frequencies versus
the circumferential wavenumber n, for m = 1 and 2, for an empty and water-filled shell with simply
supported 'open' ends (Amabili & Dalpiaz 1995).
where qr is the radial surface loading per unit area of the núddle surface, defined in tenns
of the internal and external pressures on the shell, Pi and Pe, by
qr = Pi - Pe.
(7.30)
Consider now the way in which the shell interacts with the fluid, (The suffixes i and e
indicating whether a quantity belongs to the internal or external region will temporarily be
omitted,) The steady potential flow can be taken as a steady state of undisturbed motion.
The undisturbed velocity V is then defined by the potential \11 (x, r, 8):
V = V\I1.
(7.31)

586
SLENDER STRUCTURES AND AXIAL FLOW
The disturbed state may be defined by the potential of the disturbances <I> (x , e, r, t), so that
W = u x + <1>. Hence, the velocity field for the perturbed state is given by
0<1> 18<1> 8<1>
V x = U + -, VB = --, V r = -. (7,32)
ox r oe or
Similarly, the pressure is defined by
p = PO + p*,
(7.33)
where PO corresponds to the steady potential flow and p* is the perturbation component.
It is assumed that the disturbances causing the deformations of the shell are sufficiently
smal1 for their squares and higher-order terms to be ignored, The governing equation for
the potential flow is the Laplace equation (see Section 2.2,1), V 2 W = O, which in terms
of the potential of the disturbances, <1>, may be expres sed as
1 o ( 8<1» 1 0 2 <1> 0 2 <1>
-- r- +--+--0
r or or r 2 oe 2 ox 2 - .
Consider next the boundary conditions, The condition of impermeability of the surface
of the shell may be expres sed mathematically as Ds(x, r, e, t)jDt = vrlr=a, where s =
a + w (x , (), t) is the instantaneous description of the shell surface and D jDt is the material
derivative [cf, equation (3.26)]; for hja « 1, a + !h ::::::: a. Thus, the condition
V r = 8<1> 1 = ( 8W + U OW )
or ot OX
r=a
must be satisfied at any point of the contact surface between the shell and the fluid,
The dynamic condition on this surface is given by Bernoulli's equation for disturbed
motion,
(7.34)
(7.35)
0<1> V 2 P
- + - + - = O,
ot 2 P
(7.36)
where V 2 = v; + v + v;, By 1inearizing this expression, it may be shown that the
perturbation pressures for the internal and external regions are, respectively,
* ( O<l>i O<l>i ) I * ( 8<1>e o<l>e ) I
Pi = -Pi fu + Uia;: r=a' Pe = -Pe at + U e OX r=a. (7.37)
We next look for a solution in the form of waves travelling along the shell:
u = A exp [i(hja + ne + Qt)],
v = B exp [i(A.xja + ne + Qt)], (7.38)
W = C exp [i(hja + ne + Qt)].
Here A, B, C are complex coefficients, and A. is complex also; t n is the circumferential
wavenumber. Similarly, the potential <I> may be expressed as
<I> = R(r) exp [i(A.xja + ne + Qt)],
(7.39)
tThe case of Re(Á) = o corresponds to a spatially decaying standing wave.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
587
where R (r) is a function to be detennined. t Substitution into the Laplace equation (7.34)
gives
d 2 R +  dR _ ( n 2 + 'A2 ) R = O,
dr 2 r dr r 2 a 2
(7.40)
which may be solved in terms of modified Bessel functions of first and second kind and
order n, yielding
R(r) = Dl In ('Ar/a) + D2Kn ('Ar/a).
The derivative of <I> with respect to r, taken on the sUĎace of the shell is
(7.41)
 Ir=a = R'(a)exp [i ('A + ne + Qt)] = [  + u : l
where equation (7.35) was utilized, and ( )' = d ( ) / dr. The potential <I> may then be
expressed as
(7.42)
R(r) [ aw aw ]
<I>(x, e, r, t) = R'(a) ar + U ax '
(7.43)
and similarly the pressure p* (at r = a) is
* R(a) [ a a ] 2
p (x,e,t)=-p- -+U- w.
R'(a) at ax
(7.44)
In addition, the pressure p* must remain finite, which means that R(r) = DIIn('Ar/a)
inside the shell, and R (r) = D2Kn ('Ar / a) outside the shell. The internal and external
pressures are then defined by
[ ] 2
* a a a
Pi =-Pi n+'AIn+l('A)/In('A) at +Ui ax w,
[ ] 2
* a a a
Pe = -Pe n _ 'AK n +1 ('A)/Kn('A) at + U e ax w,
(7.45)
at r = a - 0+ and r = a + 0+, respectively, for h / a « 1.+ They introduce three basic
new terms in the shell equations, corresponding to the three derivatives of w:
a 2 w/at 2 , U 2 (a 2 w/ax 2 ) and 2U(a 2 w/axat). These terms are associated, respectively,
with the inertia of the fluid, and the centrifugal and Coriolis forces of the moving fluid.
Thus, the fluid loading is wholly analogous to that acting on a tubular beam conveying
tThe form of (7.38) and (7.39), permitting solution by separation of variables, strietly holds for an infinite1y
long system, in which (u, v, w), <t> and p are implicitly eonsidered to be defined for x E (- 00, +(0); they are
spatia11y periodie (as if the system were periodica11y supported), with a prineipal period of 2L. Neverthe1ess,
this form of the solution remains fundamenta11y va1id for the problem at hand.
*Therelations Al(A) = n1n(A) +AIn+! (A), AK(A) = nKn(A) - AKn+1 (A) have been used to obtain (7.45)
in the form given.

588
SLENDER STRUCTURES AND AXIAL FLOW
fluid, as analyzed in Chapter 3 - see also Section 7,2.6. It is also noted that the first
part of these expressions, involving the Bessel functiona1s, is similar to others found in
Sections 2.2.2(e) and 2.2.3 in conjunction with added mass calculations.
We next rewrite the shell equations in dimensionless terms, utilizing the following
barred, dimensionless quantities:
x = x/a,
li = u/a,
v = v/a,
w = w/a,
t = tiTo,
(7.46)
Q = Q/Qo and U = U/Ua,
where the following reference quantities have been introduced:
[ E ] 1/2
Ua -
- Ps(1 - v 2 )
(7.47)
Substitution of expressions (7.45), (7.46) and (7.38) into equations (7,29) leads to three
linear, homogeneous equations in the unknowns A, Band C:
[ ps(l - V 2 ) ] 1/2
To =a ,
E
1 [ E ] 1/2
Qo =  Ps O - v 2 ) ,
A [A 2 + 1 ; v n 2 (1 + k) _ Q 2] + B [ 1 ; v An] _ iAC [v + k (A 2 _ 1 ; v n2)] = O,
[ 1 + V ] [ 2 1 - v 2 -2 ] . [ 3 - v 2 ]
A An + B n + A (1 + 3k) - Q - lC n + A nk = O,
(7.48)
AA [v T k (A 2 - 1 ; v n2) ] + B [n + 3 ; V A 2nk ] _ iC { 1 + k[(A 2 + n 2 )2
- - 2 - - 2
_2n2+1]_ Q2 _ f.1-i(Q+Ui A ) + f.1-e(Q+U e A) } =O,
n +Al n +1(A)/l n (A) n -AK n +1(A)/K n (A)
where
f.1-i = (a/h)(pi!ps) and f.1-e = (a/h)(Pe/ps).
(7.49)
For non-trivial solution, the associated determinant must be zero; this is the character-
istic equation. Being transcendental in A, this equation wil1 have an infinite number of
roots; hence, the complete solution is defined by the infinite series:
{  } = f {  } eXP[i(Ajx+ne+ Qt )].
w J=1 C J
The obvious difficulty lies in the fact that the number of boundary conditions needed
to determine the complex constants Aj, Bj and Cj is limited to eight, equations (7.3)
and (7.4a-d), whereas the number of these constants is infinite. Hence some simplifica-
tion is indicated; fortunately this is possible, without jeopardizing the essential va1idity
of the solution, as follows. The va1ues of A which contribute most to the modal shape
are those with small modulus; large real parts of A correspond to vibrations of shorter
(7.50)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
589
wavelength, and 1arger imaginary parts to vibrations with spatial1y increasing damping.
In practica1 tenns, the modu1us of the roots A} , j = 1, 2, . . . , 00, increases rapid1y with j,
and consequent1y the disp1acements may adequate1y be represented by the use of a finite
series solution, corresponding to a truncated set of A} (say, four or eight). We shal1 proceed
with the assumption that this is feasib1e and set up the solution procedure as follows (for
j = 1, 2, , . . , 8).
(i) For a particu1ar shell (k, fJ.i and fJ.e known) and given va1ues of n, U i and U e , the
determinanta1 equation associated with (7.48) is
:DI( Q ,A) =0.
(7.51)
For a trial value of Q (dose to one obtained at a previous step, say for U i - f). U i ),
the lowest A} are computed numerically from this characteristic equation, e.g. by
a secant method.
(ii) We se1ect the eight Aj with smallest modu1us and then express
Aj = a}C}, B} = fJ}Cj;
j = 1,2,...,8,
(7.52)
where the a} and fJ j are, in a sense, obtainab1e from the eigenvectors of the matrix
equation 1eading to (7.51).
(iii) We then substitute into the eight boundary conditions, which may now be written
as []{C} = {O}, where {C} = {CI, ..., C8}T. Again, for non-trivia1 solution the
determinant of this matrix must be zero, 1eading to the 'frequencyequation':
V 2 (Q ; AI, A2,..., A8) = O,
(7.53)
(iv) If equation (7.53) is satisfied, the trial Q is one of the eigenfrequencies; otherwise
step s (i)-(iii) are repeated unti1 it does.
A smaller-order scheme was a1so tried for the damped-c1amped shell, invo1ving the
four A} with the lowest modu1i. In this case, on1y the four boundary conditions in w, two
from equations (7.3) and (7.4b,c), are utilized in the equiva1ent of (7.53).
A1so, a set of calcu1ations were conducted in which the long-wave (small A)
approximations of the Besse1 functiona1s were used,
[ n+A ln+1(A) ] -1 '"  { 1-  [ A2 ]}
ln (A) - n 2 n(n + 1) ,
[ Kn+l(A) ] -l 1 { I [ A2 ]}
n - A Kn(A) = -;; 1 - 2 n(n _ 1) ,
obtained by expressing the Besse1 functions in truncated series fonn, the second one being
va1id for n ::: 2, When these are substituted into (7.48), it is dear that the effects of interna1
and external flows over the shell are dosely similar. It is therefore sufficient to study one;
in the calculations that follow, U e = O is taken throughout, and hence the dynamics
associated with internal flow on ly is examined.
It is also interesting that in the limit, A --+ O, the pressure tenn (7.45) becomes
(7.54)
a [ d a ] 2 a [ a a ] 2
P=-Pi- -+Ui- W-Pe- -+U e - w.
n dt dX n at dX
(7.55)

590
SLENDER STRUCTURES AND AXIAL FLOW
This ties in with the results on added mass presented in Chapter 2, showing that the effect
of the fluid is diminished with increasing n. lt a1so shows that for n = 1, this result is
identical to that obtained for beam theory in Chapter 3,
7.2.3(a) Clamped-clamped shells
Typical results for a clamped-clamped shell, for n = 2 and m = 1, 2 and 3, are shown
in Figure 7,6, ca1culated with (i) the eight-Á and (ii) the four-A approximations, (iii) with
and without the Bessel function approximations (7.54), and Civ) neglecting or not k with
respect to unity in equations (7.48). lt is seen that all these approximations give dosely
similar results, the maximum discrepancies being of the order of 10%. Of particular
importance is the closeness of the four- and eight-A solutions, establishing the validity of
the truncation of the infinite set of A into a small finite one. The neglect of k with respect
-........
3,6 '.
'. o
'\,
. o
\.
\0 o
O \
LLLi o o
 22 1 'i,i o \
   o \0
x 2.0 \
I o o o  ·
 1.8 .--Li  o o I Q \"
5< - --i'io o i o \0
 1.6 'i c  \
Ici.\ao
'\  .
tt \
}l \
3.4
3.2
'"
'"
 14

iJ 12
E
či
1.0
0,8
0,6
0.4
0.2
o
0.20
0.40
0.60
0.80
1.0
Dimensionless velocity, U i
Figure 7.6 Variation of the dimensionless frequency, Q, of a clamped-damped shell with dimen-
sionless flow velocity, Ui, for n = 2 and m = 1, 2 and 3; h / a = 2,27 x 10- 2 , L / a = 25,9, v = 0.50,
and U e =0; /Li = /Le =0,06: -.-, four boundary conditions, polynomials for In and Kn; +, four
boundary conditions, no extemal fluid; !:::", eight boundary conditions, full series In and Kn; O, four
boundary conditions not neglecting k with respect to unity (Pai'doussis & Denise 1972),

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
591
-
- ... . (Q)
uj=o
Figure 7.7 Argand diagram ofthe dimensionless frequency, st, as a function of U i for a c1amped-
clamped shel1; n = 2, and other data as in the last case of Figure 7.6. The loci on the real and
imaginary axes have been drawn s1ightly off the ax.es but parallel to them for clarity, - - - - -, m = 1;
- - -, m = 2; -, combined m = 1 and m = 2; - -, m = 3 (PaYdoussis & Denise 1972).
to 1 has the largest effect, which appears surprising until it is realized that k is of the same
order of magnitude as Q2 ; hence, this unnecessary simplification is not recommended. In
the absence of dissipation, the eigenfrequencies are wholly real.
It is seen in Figure 7.6 that the eigenfrequencies for each m diminish with increasing V i
and eventually vanish, their evolution being similar to that shown in Figure 3,10 for a thick-
walled pipe conveying fluid. This suggests that the system loses stability by divergence,
which is confirmed by the Argand diagram of Figure 7.7. It is seen that divergence at
Vi -=- V cd = 0.580 in the m -=- 1 mode is followed (according to linear theory) by
divergence in the m -= 2 mode at V i = 0.606; immediately after, at V i = V cf = 0.607,
the two mode loci coalesce on the Im( Q) -axis and give rise to coupled-mode flutter
('Pai'doussis-type coupled-mode flutter' as opposed to the Hami1tonian variety, as defined
in Section 3.4.1). The evident similarity in dynamical behaviour of tubular beams and
shells conveying fluid is discussed in Section 7,2.6.
The reader is referred to Sections 7.2.7, 7.3,6(b) and 7.8.2(d) for discussion on the
existence of coupled-mode flutter,
Calculations have also been conducted for n = 1 and n = 3 (Figure 7,8), and similar
dynamical behaviour for all n is found. It is of interest that, a1though the eigenfrequencies
for V i = O are ordered as Qn ,m == Q2 ,1 < QI ,I < Q 3,1 (see the discussion ofFigure 7.4),
the flow velocities for divergence are not similarly ordered; in fact, --u;;2 < 71'::;3 <
--U;;I.t Clearly, this order applies to the specific shell examined. It may at first sight
appear odd that mode loci cross, i.e. two different modes may have the same frequency for
the same Vi - see the discussion on 'mode veering', 'mode exchange' or 'role reversal' in
tThe notation ofUed for divergence and Uel for flutter in any (n, m)-combination is here (andelsewhere in
this chapter) used loosely, for simplicity, instead OfUd and Uf; clearly, for a given L/a, it is the lQwest such
value, corresponding to a particular (n, m), which is the critical one.

592

N
o
x
I
»
u
 3
6<
<l)

SLENDER STRUCTURES AND AXIAL FLOW
5
(1,2)
4
.-.
'.
'\.
\
.
\
.
 ._. (2,3)-'\
@'.. \
' (2,2)"">-'.
82 '\.
Q .-.-..... \ \
( 2 I ) . ·
, , \
\ \' \
\.  .

.
0,2 0,4 0,6
(I, I)
0,8 1.0
1.2
1.4
°
Dimensionless velocity, Vi
Figure 7.8 The dimensionless frequency, , versus V i for the first three circumferential modes,
n = 1,2,3, and axial modes m = 1,2 or 3 of a clamped-clamped shell (data as in the last case of
Figure 7.6): .:1, n = 1,., n = 2; o, n = 3 (PaYdoussis & Denise 1972).
Section 3,5.1. The difference here is that each n is independent; hence, different-n modes
may cross one another quite freely.
The modal shapes have also been obtained. An example is shown in Figure 7.9 for
n = m = 2, 'animated' by showing the shapes at different times over half a period. lt is
seen that only at V i = O and V i ::::: V cd does the system have normal modes, as discussed
in Section 3.4.1 (see Figure 3.13). Otherwise, the nodes and antinodes travel along the
shell; i.e., the modal shapes contain trave1ling wave components. It is also observed that
(i) vand w have the expected dependence on xl L in Figure 7.9(a)-(c), but so does u,
in view of equation (7.6), and (ii) there is a transition from second- to third-beam-mode
shape in vand w at Vi ::::: V cd. Similar results for (n, m) = (2, 1) may be found in Denise
(1971) and here in Figure 7,113. In that case, the v and w components of the displacement
change from first- to second-beam-mode form at V i ::::: V c d = 0.580,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
593
 :: f t (b) 1  j
;g t _x_x ';;fí
 - 1.50 --:
-3.00 1 1 I 1 I I I I ,
 :: f " -- ; ';-' j

 ::: f '" ' o--  _ _ _ _ 1 f '" ' j
i::: t'. t j
3,00 f 1
(c) (d)
S 150
x
 000
! ::: l 1
 ::  (C)I I I 1 I ,,-' I I 1
j o 002t  +:?:.
';' -100
"
o
..... -2,00
 I 60 I (c) II (d)
';:' 080
-;
@ 000 I
 u i -0.80
-1.60 .
0,00 0,10 0.20 0.30 O 40 050 0.60 0,70 0,80 0,90 100
] 000
:a
 -1.00
"
o
..... -2,00
o 10 0,20 0,30 0.40 O 50 060 0.70 0.80 O 90 I 00
x/L
Figure 7.9 Time-dependent second-axial-mode modal shapes (n = 2; nominal m = 2) over half
a period of oscillation for four different values of Vi of a clamped-clamped shell (data as in the
last case of Figure 7,6): (a) Vi = O; (b) V i = 0,5; (c) V i = 0.6; (d) V i just short of divergence
(Patdoussis & Denise 1972),

594
SLENDER STRUCTURES AND AXIAL FLOW
N
;::
I
3.
.,,850
. O 825
\
.\0: 750
\ 1 10 1.00
.0,700 \
"" 0,600 1.0 09
· 0.9 .'08
08 .0.5 0,70.:'" 0,4
0,6 07 \  0.65 't 0,7 0.5 06 03
. ,0.4 O. .' O'  4
0.2 0.4 .0,2 04....:....9.2 0,55 O, 04 '0,2
O O 0,7-. O I.Q 0.9 O 5 O ' 0.3 0.2 O O
0 ( 6. 0,65
110
0,7. ,
2
/
2
<S
I 1

O
O
3 4
(.Q)(X 10 2 )
5
6
Figure 7.10 Argand diagram of tbe dimensionless frequency, [2, as a function ofthe dimensionless
flow velocity, U i, for a cantilevered shell; hla = 2.27 x 10- 2 ; Lla = 12.9, v = 0,50, /Li = /Le =
0.06 and U e = O, showing the first two axial modes, m = 1 and 2, for n = 1,2,3, as well as the
n = 2, m = 3 mode: , n = 1;., n = 2; 0, n = 3 (PaYdoussis & Denise 1972).
7.2.3(bJ Canti/evered shells
For tbis set of boundary conditions, only the eight-)., solution is used, with no approxi-
mations in the Bessel functions or in tenns of k. Tbe eigenfrequencies for U i = U e = O
were compared to those byWarburton & Higgs (1970) and Lakis & PaIdoussis (1972a) in
which the Fltigge and Sanders shell equations are used, respectively, and different solution
metbods to that employed here, showing agreement to within 5%.
Typical results are shown in Figure 7.10 for n = 1, 2, 3 and m = 1, 2, plotted as Argand
diagrams, with U i varied and U e = O. It is seen that, as for a thick-walled pipe conveying
fluid (Section 3.5; Figures (3.27)-(3.29), for example), this inherent1y nonconservative
dynamica1 system loses stability by flutter, sequentia1ly (according to this linear theory)
in the n = 2, n = 3 and n = 1 modes, always in the m = 2 axial mode number. Thus, the
same axial fonn as for beam-mode flutter of thick-wa1led pipes (Sections 3.2.2 and 3.5.6)
is obtained, involving a 'dragging', lagging axial mode shape.
Some further results, on tbe effect of L/a and tLi on stability are discussed togetber
with the experimental results in Section 7.2.4.
7.2.4 Experiments and comparison with theory
Experiments were conducted witb cylindrical shells made of either silicone rubber
('silastic') or latex rubber (PaIdoussis & Denise 1972). The fonner, which were cast in
specia1ly prepared moulds (Apndix D, Volume 1) had the following physica1 and geo-
metrical data: E:::: 215 lb/in. (1.48 x 10 6 N/m 2 ), v :::: 0.47, Ps = 3,72 x 1O- 2 lb/in,3
(1.03 x 10 3 kg/m 3 ), a = 0.28 in. (7.11 mm), h = 0.06 in. (1.52 mm), so

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
595
that aj h = 4.7, and typically L = 2 - 10 in. (50.8-254 mm). The latter were made
from commercially available 'colostromy' (or 'Penrose') tub ing used in medical applica-
tions; thecorrespondingdatawere E::: 89,6x 10 4 Njm 2 , v::: 0.5, Ps = 0.85 x 10 3 kgjm 3 ,
a=7,85 mm,h=O,178 mm, so that ajh=44, and L=25.4-254 mm. Clearly, the
silicone rubber pipes are not truly thin shells.
Although some observations were made with water, in almost all experiments air was
used as the fluid. The apparatus consisted of a regulated and metered compressed-air
supply, piping and supports for the shells, sufficiently standard to require no detailed
description - beyond making the statement that adequate care was taken to ensure that
the flow entering the shell be straight and uniform (see also Figure 7.65). To help eliminate
pressure perturbations from the supply, the flow went through a number of pressure-
reducing valves, before reaching the shell at Iow pressure.
In these early experiments, observation of the dynamical behaviour with increasing flow
was made visually with the aid of a stroboscope, sometimes synchronized to the emitted
sound via a microphone and associated electronics, thus allowing either 'freezing' the
motion or slowing it down for easy observation.
7.2.4(a) Qua/itative observations
Cantilevered cylinders were tested first, as in this case visual observation of the free
end al10wed easy identification of the circumferential mode of vibration. The flow velo-
city was increased in small steps starting from zero, The following observations refer to
sheIls sufficiently short for flutter to occur first, rather than lateral beam-like oscil1ation
(cf. Section 7.1).
For velocities below the flutter threshold, the first vibrations to be observed corre-
sponded to n = 2 and were of very smaIl amplitude, the end cross-section deforming
between two extreme positions, the shape of which was quasi-elliptica1. With increasing
flow velocity the amplitude increased slightly and then, at a certain point, a mode change
occurred - the vibration now corresponding to n = 4; n = 3 could also be obtained by
disturbing the cylinder.
Upon increasing the flow velocity further, the flutter threshold was reached, where
large amplitude vibration corresponding to n = 2 developed spontaneously into a faidy
stable limit cyc1e, the amplitude being maximum in the region near the free end and
of diminished magnitude towards the c1amped end (Figures 7.1 and 7.11). The shape
of the free-end cross-section was irregular and became wobbly with increased flow; its
lips twisted altemately in and out, sometimes c10sing the opening almost completely at
the extremes of the cyc1e. The frequency of vibrations ranged between 100 and 800 Hz,
depending on the sheIl and its length, and was accompanied by a loud, shriIl sound.
When the flow velocity was then reduced, flutter persisted below the critical point for
its onset. The flutter described above developed spontaneously; it could be induced at a
lower velocity by pinching or disturbing the shell. Both these are attributes of nonlin-
ear behaviour; specifically, they suggest that the onset of flutter is via a subcritical Hopf
bifurcation [Sections 2.3 and 5.7; Figure 2.1l(d)]. Here, it ought to be remarked that,
generaIly, nonlinear effects are expected to come into play once the amplitude of motion
exceeds the sheIl thickness (see Section 7.8); therefore, for the relatively huge amp-
litudes pertaining to these experiments, and i1lustrated in Figure 7.11 for a similar sheIl to
those used in the experiments just described, the nonlinear effects should be that much
stronger - see also part (c) of this section.

596
SLENDER STRUCTURES AND AXIAL FLOW
Figure 7.11 Photographs of the large amplitude 'disturbance-induced' flutter of a specially cast
silastic shell (Lla = 9.8, hla = 0.094; a = 5.7 mm). The shell was cantilevered vertical,
as in Figure 7.1. The photographs were taken fmm above the free end, and the motion was
stroboscopically arrested, to capture the circumferential mode shape,
In these experiments, no flutter was observed for shells conveying water. This has little
to do with the vastly different f.ki = pia/Psht in water flow; see Figures 7.14 and 7.15.
A more likely reason is that internal pressurization effects, associated with the larger
pressure drop in water flow, become important - whereas they have been totally ignored
in the theory.
Clamped-clamped shells paradoxically lost stability by flutter. As we shall discuss fur-
ther [Section 7,2.7], it is not clear whether this is true flutter, as in theory, induced by
proximity of the divergence and flutter boundaries, or whether it is a 'dynamic diver-
gence', involving two collapsed forms differing by e = JT for n = 2 [Figure 7.3(a)]. The
maximum amplitude was downstream of the mid-point of the shell.
If the shell was long, lateral (n = 1) divergence occurred, as for thick-walled pipes
conveying fluid (Section 3.4.4). For very short shel1s, flutter was just preceded by, or
was coincident with, what appeared to be lateral buckling; however, observations in such
cases were difficult, as this generally resulted in bursting of the cylinder.
7.2.4(b) Comparison of experiments with theory
The observations just described may be considered to be in qualitative agreement with the-
ory, in the sen se that for both sets ofboundary conditions the observed flutter was associated
with the theoretically predicted circumferential wavenumber: n = 2. In respect to the axial
wavenumber, the observed flutter appeared to occur in a modal shape dominated by the
zero-flow mode components associated with m = 1 and m = 2. However, the nonlinear
character of the motion after the inception of flutter, and the then non-negligible frictional
forces tending to 'push' the region of maximum amplitude downstream, make comparison
between theoretical and experimental modal shapes problematica1.
Quantitative agreement between theory and experiment was assessed by comparing the
measured critical flow velocities for the on set af flutter with the corresponding theoretical
values (n = m = 2) in the case of cantilevered cylinders, and with the theoretical values
for divergence of the mode n = m = 2 in the case of clamped-clamped shells; as has
tThis is truly equal to twice the fluidJshell mass ratio, Pi7r:a2 / p,2n:ah, for a thin shel!.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
597
" 1.5
Io
."
Io
i3
u
o
"
> 1.0

.,
í3
.
o:
.,
8
i5
0,5
o
10
20
30
40
50
Lla
Figure 7.12 Variation of the theoretical dimensionless flow velocity for divergence, U cd, of a
clamped-clamped shell with Lla for n = 1,2,3 and m = 1,2, and also m = 3 for n = 2; hla =
2,27 x 10- 2 , v = 0.50, /l-i = /l-e = 0.06, and comparison with the experimentally observed
flutter. Theoretical results: b., n = 1; _, n = 2; 0, n = 3. D, Experimental results (Pai'doussis &
Denise 1972).
already been noted [Section 7.2.3(a)], this latter is essentiaIly coincident with the onset
of flutter - refer also to Section 7.2.7. Comparison of the associated frequencies would
be fruitless, for two reasons: (i) for c1amped-clamped sheIls, it is not certain whether the
phenomenon is one of dynamic divergence or trne flutter; (ii) for cantilevered sheIls it was
very clear that the Hopf bifurcation was strongly subcritica1. Comparison was confined to
the data obtained for the latex cylinders which were sufficiently thin for the theory to apply.
The experimental data are compared with theory in Figures 7,12 and 7,13, ln the case
of both ends c1amped, agreement between theory and experiment is rather good over
the whole range of Lja, the margin of discrepancy varying between zero and 16%. The
agreement for cantilevered sheIls is not nearly as good, being reasonable only for long
sheIls [i,e, for Lja > 16], where the margin of agreement is about 20%, The main reason
for the discrepancy, and the fact that the experimental U cf are lower than the theoretical
ones, is almost certainly due to the fact that flutter is associated with a subcritical Hopf
bifurcation, while the theory is linear. This ipso facto explains a great dea1. Entrainment
of extema1 fluid by the emerging fluid jet at the outlet may be a secondary cause. FinaIly,
for L j a ......, 5, the effects of imperfections in the transition between the flow-supplying
pipes and the sheIl are likely important. More fundamentaIly, however, the trave11ing wave
solution is most appropriate for fair1y long sheIls; its applicability deteriorates for L j a
smaIl;t see Sections 4.4 and 7.3.1.
tNevertheless, Dowell (1975; p. 26) states that 'For low supersonic and subsonic Mach numbers, a 'travelling
wave' ana1ysis gives results in reasonable agreement with more elaborate ana1yses which account for the effects
of finite panellength'.

598
SLENDER STRUCTURES AND AXIAL FLOW
1.0
0.8
1'1?

'g 0.6
i
o;
,
'E
U
0.4
o o o o o o o
o
o D o o 8 o o o o o D o o
o o o o
0.2
5 10 15 20 25 30 35
L/a
Figure 7.13 Variation of dimensionless critical flow velocity for flutter, U cf , of a cantilevered
shell conveying fluid; hla = 2.27 x 10- 2 , v = 0.50, /.Li = /.Le = 0.06. Theoretical results:
x, n = I; .ó., n = 2; o, n = 3. O, Experimental results (Paldoussis & Denise 1972).
The effect of Mi, defined in (7.49), on stability for a fixed va1ue of L/a is shown in
Figures 7.14 and 7.15. It is seen that U cd and U cf are diminished dramatically as Mi is
increased vis-a-vis Mi = 0.06 in the previous figures. Hence, by this reason alone, flow-
induced instabilities for water flow shou1d be easier to excite; but, as found experimental1y
and discussed in part (a) of this section, this is not the case.
A fina1 comment is that, in all this, the presence of acoustical coup1ing, prominent in
the canti1evered shell experiment s particu1ar1y, has been ignored in the theory, though
undoubted1y it must p1ay a role,
7.2.4(c) Some more refined experiments
A set of very extensive, careful, but unfortunate1y unpublished experiment s were conduc-
ted by Guernier (1978), with si1icone rubber shells (E = 1.32 x 10 6 -1.63 x 10 6 N/m2,
v =0.47, Ps =2.45 x 10 3 kg/m 3 , a =6,11 mm, h =0,53 mm, so that a/ h = 11.3).
Frequencies and amp1itudes were also measured, as well as the dynamics beyond the
onset of instabi1ity.
A samp1e set of resu1ts for cantilevered shells is shown in Figure 7.16. It is seen that U cf
for disturbance-induced flutter can easi1y be three times smal1er than U cf for self-excited
flutter! Furthermore, two types of disturbance-inducedjlutter may be identified:
(a) 'pure mode' flutter, inv01ving a single, c1ear1y identifiable mode (n = m = 2);
(b) 'combined mode' flutter, inv01ving several axial and circumferential modes.
In other experiments, the zone of pure-mode flutter, always below that for combined-mode
flutter, was found to extend up to L/a  30; in Figure 7.16 it stops at L/a  20.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
599
1.6
m 1
1.0
V ed
o
O
0,2
0.4
0,6
0.8
1.0
!Ii
Figure 7.14 The effect of /-li on U cd for clamped-clamped shdl conveying fluid;
hla = 2.27 x 10- 2 , Lla = 15, v = 0,50 (Pai'doussis 1974a).
10
Vcf
0.5
O
O
0.2
0.4
!Ii
0.6
Figure 7.15 The effect of /-li on U c ] for cantilevered shell conveying fluid; hla = 2.27 x 10- 2 ,
Lla = 15, v = 0,50 (Pai'doussis 1974a).

600
SLENDER STRUCTURES AND AXIAL FLOW
1.5
1.0

' 8 ' '"
\ , '"
o 1:::.... .6.
o' _ _ _ _ . _ _ '--__-__ __-:--_-_-_ -_ -_ -_ -:-_ -_ -_ -_ -_ -_ _ _
tJ
0,5
0,00
(a)
- - - - t - -fo' -+- - -+! - - f" - * 4- - - 4: - - - -1-- - - . '"
,
1 O 20 30 40
L/a
0,25
O
O
Q
(b)
tJ
Figure 7.16 (a) Different type s of self-excited and disturbance-induced flutter, for a canti-
levered shell. Disturbance-induced: ., 'pure mode' shell flutter; +, 'combined mode' shell flutter;
x, beam-like flutter, Self-excited: 0, shell flutter; D., combined beam-shell flutter. (b) The different
zones of flutter in an Q versus U diagram: the cross-hatched zone at the lowest U corresponds to
'pure-mode shell flutter'; the middle, lightly shaded zones to 'combined-mode shell flutter'; that at
the extreme right (darkly shaded) to 'combined beam- and shell-mode flutter', The arrows indicate
whether measurements were taken with the flow increasing or decreasing (Guemier 1978),
The self-excited flutter displayed random, perhaps chaotic, characteristics. It, also, was
of two distinct types. For small L/a, L/a < 10 for the system of Figure 7.16(a), shell-
mode flutter was observed. For L/a > 12 approximately, however, combined beam- and
shell-mode flutter was the normo
The zanes of distinctly different dynallŮcal behaviour are shown qualitatively in
Figure 7 .16(b), also showing the strong hysteresis effects that are possible, as well as
information on the dOllŮnant frequencies of osci1lation, af particular interest is the fact
that there exist zones in which it is impossible to excite self-sustained osci1lations (shown
as stable in the figure). The existence and extent of these various zanes depends on the
particular shell and on L/a,
Clearly, all these characteristics make it obvious that one really needs a nonlinear theory
before a meaningful comparison with these interesting experiments is undertaken - see
Section 7.8.2.
Finally, another set of experimental results, conducted in another apparatus, which
could have been presented here, may be found in Section 7.S.S(c).

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
601
7.2.5 Shells with flow: Galerkin-type solutions
The analysis in Section 7.2.3 may be tenned a 'travelling wave analysis' (Kornecki 1971a;
Dowell 1975, p. 26) because of the fonn of the solution, equations (7,38) and (7.39),
Similarly, the Galerkin-type solutions for the same problem, undertaken soon after by
Weaver and co-workers, may be called 'standing wave analysis' (Weaver & Dnny 1973;
Weaver & Myklatun 1973); they are discussed next. Of course, whatever itsfonn, the
solution itself must be approximately the same, with the modal shape of the oscillating
shell basically as in Figures 7.9 and 7.113.
In these Galerkin analyses, the Fliigge-Kempner shell equation is used, equa-
tion (7.13b), When a perturbation pressure, p*, acts on the shell due to an internal flow,
this equation takes the fonn
4 t' a 2 w *
(Dia )LF-K(W) + Psh = P ,
at
(7.56)
which is similar to the third of equations (7,29), The Galerkin solution takes the fonn
00
w(x, (), t) = L Cmťj)m(x) cos n(}e iS1t ,
m=l
(7.57)
where ťj)m = sin(mJrxl L) for shells with simply supported ends (Weaver & Dnny 1973),
and ťj)m = 1 - cos(2mJr x I L) approximately for clamped-clamped ones, in preference to
the clamped-clamped beam eigenfunctions, for convenience (Weaver & Myklatun 1973).
Only interna! flow is considered, and the perturbation pressure is detennined by potential
flow theory. The velocity potential has the fonn
<I>(x, r, (), t) = {1°O In (Xr)[A(X) cos Xx + B(X) sin Xx] dX} cos n(}e iS1t ,
(7.58)
instead of (7.39); X 2 is the separation constant in the Laplacian equation for <1>. It is
assumed that the shell, of length L, is connected to rigid tubes at either end, of the same
internal radius as the shell. t The radial surface perturbation is therefore detennined by
the impenneability condition: that is, such that <I> satisfies equation (7.42), with W as
given by (7.57), for O :::: x :::: L, and (a<l>lar)lr=a = O elsewhere. The constants A(X)
and B(X) are determined by means of Fourier integral theory - see Section 4.4.4 and
Hildebrand (1960). Thus, for the simply supported shell one obtains
! '" 1 00 In(Ar) mLC m { - -
<1>=: -  2 - 2 iQ[COSAX-(-l)mcOS A (L-x)]
m=l O A.I(Aa) [(mJr) - (AL) ]
- U X[(-I)m sin X(L - x) + sin Xx] J dX} cosnge iO ', (7.59)
tA flexible shell connected to semi-infinite rigid extensions is sometimes referred to as a bajJied shel/, by
analogy to a baffled plate (a flexible plate embedded in a rigid one of infinite extent),

602
SLENDER STRUCTURES AND AXIAL FLOW
where here ( )' = d( )/dr. This is then substituted into the first of equations (7.37), to
determine p*, and tbence into equation (7.56), Upon applying the Galerkin method (see
Sections 2.1.3 and 2.1.4), one eventually obtains
[F(U, n, m)]{C m } == {O};
(7.60)
because of the form of (7.59), F contains a number of improper integrals, independent of
U, which are computed numerically for various n and L/a. Then, the generally complex
eigenfrequencies are determined by setting the determinant of (7.60) to zero and, hence,
so are tbe critical flow velocities for divergence and flutter. The solution has not been given
in great detail here, since it is given in a more rigorous form in Section 7.3.
Typical results for the evolution of the eigenfrequencies with tlow for a rather short
(L / a == 2) simply supported steel she11 conveying water, analyzed by a two-mode Galerk:in
approximation, are shown in Figure 7,17. They are analogous to those shown in Figures 7.6
and 7.7, but with the following difference: coupled-mode tlutter in this case is oť the
Hamiltonian variety, which is consistent with the results for pipes conveying high-density
fluid (see Figure 3.12 for 'high' values of {3).
The dependence of the flow velocities for divergence and flutter on n, for the same
shell, is i1lustrated in Figure 7,18 (cf, Figure 7.4). The critical ftow velocity is therefore
associated with n = 5, as in Figure 7.17, and is given by ( UW- U)cd == 3.6; U w-u ==
(..Jfi/j(2)(a/ h)(L/a) U is Weaver & Unny's form of the dimensionless flow velocity,
and U is defined in (7.46). This corresponds to U c d == 0.051, whichis morethan ten times
30
10
2.0
3.0
4,0
5,0
20
::o
let
o
1.0
U w - u
Figure 7.17 The dimensionless eigenfrequencies Qw-u ofthe first and second mode (m = 1,2)
of a simpIy supported steeI shell conveying water (Lfa = 2, hfa = 0,01, /-Li = 12.7, for n = 5) as
functions ofthe dimensionIess flow veIocity, U w-u, via a two-mode Galerkin soIution (Weaver &
Unny 1973), Here Qw-u = (,J12fn 2 )(a/ h)(Lfa)2Q and Uw- u = (.Jfif;r2)(af h)(Lfa) U , in
which Q and U are defined in (7.46).

9
::>
I
I:?
6
3
O
THIN SHELLS CONTAINING OR IMMERSED IN FLOW
603
15
12
5
10
Circumferentia1 wavenumber, n
20
15
Figure 7.18 The dependence of ( U W-U)cd and ( UW- U)cf on n for a simply supported steel shell
with Lla = 2, hla = 0,01, conveying water (Weaver & Unny 1973); see caption to Figure 7.17
for detinition of U w- u,
smaller than the value of U cd in Figure 7.6 for the rubber-air system. However, Mi = 12.7
for the steel-water system, as compared with Mi = 0.06 for the rubber-air one; thus, with
the aid of Figure 7.14, this value of U cd is seen to be in the appropriate range,
Similar ca1culations for clamped-clamped shells, but only for divergence, have been
conducted by Weave.!:.. & Myklatun (1973), and some typical resu lts are shown in
Figure 7.19, giving (UW-M)cd versus hla for various values of Lla; UW-M is defined
in terms of U in the caption. This figure ge ner alizes the resu1ts in Figure 7.12 which are
but for a single value of hla. The values of (UW-M)cd in Figure 7,19 are about one order
of magnitude smaller than those for the rubber-air systems of Section 7.2.3, but again
this may be explained mainly by the difference in Mi, which for hla = 2 x 10- 2 in this
case is 85,
It is seen in Figure 7.19 that the system loses stability in the beam mode (n = 1) if LI a
is sufficiently large, provided that hla is not too smal!. Otherwise, stability is lost in a shell
mode (n 2': 2), the n involved being progressively higher as hla and Lla are diminished.
The cross-over point for beam (n = 1) versus shell (n 2': 2) in stability occurs roughly at
Lla = al h, with the former being the critical one for Lla > al h (Weaver 1972), This
is discussed further in Section 7.2.6.
Weaver & Myklatun (1973) also attempted to conduct experiments on shell ftutter, using
clamped-clamped Tygon tubes conveying water. t These tubes were rather thick-walled
(al h = 4), and hence can hardly be considered as thin shells. In all cases, the system
lost stability as a beam (n = 1), followed nevertheless immediately by an asymmetric
ftapping-type ftutter, in which 'the ftutter mode seemed to be a simple ftattening of the
tube in one direction'. It was also remarked that the same ftapping-type of ftutter can be
tExperiments with rubber tubes were also attempted, The commercial tubing used, however, was not straight
(see Chapter 3, particularly Section 3,5,6) and these experiments had to be abandoned,

604
SLENDER STRUCTURES AND AXIAL FLOW
0.03
0.02
0,04
Thickness ratio, hla
0,1
0,2
0.12
a
.g 0,09
O)
>
'"
()
<=
"
E!!
"
>
'6 006
Oj ,
g
.C
u
Figure 7.19 The critical divergence flow velocity, UW- M, for clamped-clamped shells as a
function of hla, for various Lla, showing the circumferential mode number n in which diver-
gence occurs; the shell properties are associated with a Tygon-type pipe conveying water (E =
5.87 x 10 6 N/m 2 , pi!ps = 1.7); UW- M = W2/n 2 )(pi!ps)]1/2 U , The generally unidentified
numbers in the figure give the value of n associated with divergence for the values of h / a and LI a
at that point (Weaver & Myklatun 1973),
produced 'by pinching the tube at flow velocities somewhat below the critical', that is,
below the critical for beam-type flutter, which echos some ofthe experimental observations
in Section 7.2.4(c). This observation is important, because it shows that there is some
similarity in the dynamics of water- and air-conveying systems. t
It was concluded that 'the theoretically predicted flutter mode s are not those observed
experimentally and (it appears) that the latter may even be an entirely different
phenomenon', which was later confirmed to be the case (Section 7.9). Hence, the exper-
iments were discontinued, and only beam-type flutter critical velocities were compared
with experiment, as described in Section 7.2.6(b).
Nevertheless, the predictions of this theory have been compared with Pa'idoussis &
Oenise's theory and experiments, as shown in Figure 7.20. It is clear that agreement
between the two theories is reasonable, and it is stated that for high enough L / a the
results are identica1. Neglect of in-plane inertia in the Fltigge-Kempner theory gener-
aBy results in overestimating stability, as also shown for cantilevered shells by Shayo &
tThe asymmetric flapping of the Tygon-water system may be related to (i) the larger materia! damping and
(ii) the much larger !.ti, as compared to the rubber-air system, ln terms of !.ti, it may be conjectured that the
nonlinear behaviour of the rubber-air system is 'shell driven', whereas the Tygon-water system is 'fluid driven',
in accordance with the corresponding dominant inertias,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
605
I 1.0
,
"
o
"iJ
:>-
"
"
5
I:!J o
" 0,5
:>-
'8
<;
,
.E
U
O 8 16 24 32 40
Length ratio, L1a
Figure 7.20 The critical divergence flow velocity, U c d, as a function of L / a: - - -, Pai:doussis &
Denise (1972) theory for clamped-clamped shells conveying air; -, Weaver & Dnny (1973) and
Weaver & Myklatun (1973) theory for simply supported and clamped-clamped shells, respectively;
0, PaIdoussis & Denise's experiments (Weaver & Myklatun 1973).
Ellen (1978; figure 1), It should be noted that, in addition to utilizing two different shell
theories, the fluid-dynamic models are also different; in particular, there is the ques-
tion of applicability of a trave1ling wave analysis for short shells, discussed further in
Section 7.3.1.
7.2.6 Thin versus thick tubes conveying fluid
7.2.6(a) Shells and tubular beams conveying fluid: transition
Perhaps the earliest work on the dynamics of shells conveying fluid was conducted
by Niordson (1953), and was motivated by vibrations in the penstock system of the
Aswan dam hydroelectric plant - involving tubing 1.8 m in diameter, 15 mm thick (so that
a/ h = 60, thus a thin shell!), 22.5 m between supports, conveying water at U = 7 m/s.
Niordson's fonnulation ofthe problem is essentially as in Section 7.2.3, in fact more gen-
eral, taking fluid compressibility into account. The only difference that compressibility
makes is that the ratio R(a)/ R'(a), which leads to the Bessel functional in the first of
equations (7.45), is replaced by
R(a)
R'(a)
a
n - iA*Jn+l (iA*)/J n (iA*) ,
(7.61)
where
A *2 = A 2(1 - M 2 ) - 2AM(Qa/c) - (Qa/c)2, (7.62)
A being the same as before - equations (7,38); M is the Mach number. Obviously, for
c -+ 00, M -+ O, equation (7.61) reduces to the fonn of the fraction in the first of
equations (7.45).
Although Niordson's fonnulation is complete and correct, he did not proceed to analyze
the general prob1em of a shell containing flow, studying instead a number of simpler special

606
SLENDER STRUCTURES AND AXIAL FLOW
cases: (i) the fluid-filled shell with U = O; (ii) beam-like motions (n = 1) of the shell for
U f=. O; (iii) the 'pipe string' (see Section 3.7.1) for U f=. O; (iv) the empty shel1 immersed
in quiescent liquid.
On1y item (ii) is discussed here. The shell equations are rep1aced by those of a beam,
and the perturbation pressure on the shell surface becomes
R(a) [ a a ] 2
p*(x, O, t) = -p----,- - + U - w(x, t) cosO.
R (a) at ax
The loading on the beam (n = 1) per unit 1ength is FA (x, t) = Jt 1T pa cos O dO, and
R(a)jR'(a)  a(1 + !A.*2fl,t which Niordson simp1y approximates as equal to a,
and hence arrives at the simp1est possib1e form of equation (4.58) of Section 4.4, name1y
FA = -pJra 2 [(ajat)+u (ajax)fw. In the process, heestablishes that thisapproximation
is valid on1y for motions oflarge wavelength, that is, for A. *  O, and hence for 1arge L j a.
He then proceeds to obtain one of the first derivations of the critical flow ve10city of pinned-
pinned pipes conveying fluid, name1y Ucd = (Mj E 1)1/2 LUcd = Jr (Section 3.4.1).
The transition from thin tube (shell) to thick tube (pipe) has been re-examined
by Pa'idoussis (1973b, 1975a) by calcu1ating the eigenfrequencies for n = 1 via the
Pa'idoussis & Denise theory and comparing them to those obtained by Eu1er-Bemoulli
beam theory in Chapter 3. The dimension1ess flow ve1ocity, frequency and mass ratio,
respectively u, (i) and {J, used extensively in Chapters 3-5, are re1ated to those used here
[Section 7.2.3, equations (7.46) and (7.47)] for very thin tubes through
u = (  ) 1/2 (  ) U , (i) = ( fki +2 ) 1/2 (  ) 2 Q ,
1 - v 2 a 1 - v 2 a
{J=.
2 + fki
(7.63)
A comparison of the eigenfrequencies for L j a = 15 is shown in Figure 7.21, where it is
seen that the shell-theory first-mode (m = 1) frequencies are lower than the beam-theory
ones by approximate1y 5%, and the second-mode frequencies by about 20%, which agrees
with Forsberg's (1969) results for U = O. Indeed, the evo1ution of the frequencies with
increasing flow ve10city as well as the critical U cd and U cf for beam- and shell-theory
resu1ts are also close.
As shown by Komecki (1971b), for 1arge enough Lja and n = 1, the shell equations
reduce to the beam equations. Also, for motions of long wave1ength, that is, for 1arge
Lja, the fluid-dynamic forces reduce to those in equation (3.1), as demonstrated in the
foregoing and in Section 4.4.3, and as first done by Niordson (1953); thus, potentia1 flow
theory converges to p1ug-flow, slender body theory. Hence, it is not surprising that for
1arge enough Lja, the eigenfrequencies obtained by the two theories converge. Indeed,
the main reason for considering Lja = 15 in Figure 7.21, rather than, say, Lja = 50, is
to make the differences clear1y visib1e.
As shown in Figure 7.22, the critical flow velocities for first- and second-mode
divergence with the two theories converge to within 5%, respectively, for L / a 2: 26 and 38,
for this particular set of parameters.:j:
tNiordson erroneously gives 1},,'2 instead of}" *2 in this expression, but in any event does not use it, simply
taking R(a)IR'(a):::: a,
+The demonstration of this convergence for large enough Lla and a fuH discussion of the coupled-mode
ftutter predicted by linear theory (see also Section 3.4.1) were the main contributions of the Pai'doussis

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
607
70
30
60
20
50 First mode
10
S 40 
l  O --1
30 I
-10 I
20 , \
, ,)
-20
10 - Shell theory \
- - - Beam theory \
-30 \
2 4 6 8 10 12 O 2 4 6 8 10 12
u u
Figure 7.21 The real and imaginary components of the first- and second-mode eigenfrequencies
of a relatively short thin-walled c1amped-clamped tube conveying fluid computed by beam and shell
theories (f3 = 0.0291, Lja = 15, aj h = 44, v = 0,5): -, shell theory; - - -, beam theory;
u = (Mj El)I/2U L, W = [(M + m)j EI]1/2QL 2 as in equations (3.71) and (3.73) of Chapter 3
(Paldoussis 1975a),
12
10
8
ucd 6
4
- Shell theory
2 - - Beam theory
O
O 5 10 15 20 25 30 35 40 45 50
L/a
Figure 7.22 The dimensionless critical flow velocities, Ucd, for first- and second-mode divergence
of thin-walled c1amped-clamped pipes conveying fluid, calculated by beam and shell theories
(aj h = 44, v = 0.5); (Paldoussis 1975a).
(1975a) paper; they sufficiently impressed the lnstitution of Mechanical Engineers (UK), presumably for
their novelty and quality, to award the author a George Stephenson medal for this work, What makes this tale
worth telling is the fact that the very same paper had been submitted two years earlier to another joumal (which
can here remain anonymous), and it was rejected for being insufficiently novel! Trus tale proves the author's
contention that the peer-review process inextricably contains elements of a lottery. Young researchers/authors
of rejected papers should therefore not be unduly dejected; their work may yet win a prize!

608
SLENDER STRUCTURES AND AXIAL FLOW
The conclusion may be reached by re-writing the perturbation pressure, with the aid of
equation (7.44), as
p* = -paI(n, Á)[(i1ji1t) + U(i1ji1x)]2 w ,
(7.64)
where I(n, Á) is a functional of appropriate Besse1 functions. Then, comparing to the
equivalent beam force per unit length, it is clear that the analogy in dynamical behaviour
must be very close (Pa1doussis 1987). This includes the energy transfer mechanisms dis-
cussed in Chapter 3. Thus, for example, it is no accident that even when n > 1, the axial
wavenumber m = 2 should be associated with flutter for cantilevered shelIs,
7.2.6(b) Beam-like divergence of thin shells
The frequency equation may be simplified drasticalIy for Q = O, n = 1 and Lja » 1,
leading to a simplified way for predicting U c d for n = 1, for shelIs with supported ends.
This is easiest done by utilizing the Fliigge-Kempner single equation of motion; this useful
work has been done by Weaver & Dnny (1973) and Weaver & Myklatun (1973), in the
latter of which, for clamped-clamped shelIs, it is found that
Uc d = [  (1- v 2 ) ( !!. ) (  ) 3 ( Ps )] 1/2 == [  (1- v 2 ) (  ) 3 ] 1/2, (7.65a)
4 F211 a L Pi 4 F211f-Li L
where
1 00 II (sajL) I-cos s d
F211 = s,
o I (saj L) S(47T 2 - s2)2
F211 being one of the integrals referred to beneath (7,60), and s = XL. Making use of
(7.63), equation (7.65a) may be rewritten in terms of the beam-theory dimensionless flow
velocity (equation (3.8», u, as
( 7Ta ) 1/2 1
Ucd = 4L .J F2l1 == 27T Fbi' (7 .65b)
Noting that u = 27T for a clamped-clamped 'thick' pipe, Weaver & Myklatun define the
ratio of the right-hand side to 27T as an 'end flow factor'; here we calI it the 'bluntness
factor', Fbi' This factor is graphed in Figure 7.23 and it is seen that, for sufficiently
slender shelIs, the beam approximation is quite adequate, and Fbi --+ 1. Similarly, for
pinned-pinned shelIs, an analogous FbJ is defined, but in (7.65b) we must write 7T Fbi;
for that case a1so, FbJ --+ 1 for large Lja.
The effect of bluntness on the divergence flow velocity has been considered by
Shayo & ElIen (1974) as welI. They also use the F1tigge-Kempner equation, but for
simply supported shelIs conveying compressible fluid. The fulI analysis for any n is given
in Section 7.3.1, but the resultforbeam-like motion (n = 1) is given here for completeness
of the present section, namely
U c d = 7Tb[(1 - v 2 )j f-L]1/2{ 1 - 0.945(1 - M 2 )1/2b + 2.467(7 + M 2 )b 2
} -1/2
- (1 - M 2 )1/2(13,348 + 5.301M 2 )b 3 + O(b 4 ) ,
(7.66a)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
609
3
Weaver & Unny (simply supported)

..... 2
B
<.)

'"
'"
"
B
<=
;:!

-- ---- ------ - - -- - - - - -:. - :.-:. - -:- -  - - -;;..-,:,- - ----.- -
---
'- -- -- --"'-- Shayo & Ellen (full)
o
O
5
10
Ua =b- 1
15
20
Figure 7.23 The 'bluntness factor' or 'end-flow factor', Fbf, for beam-mode divergence
of thin simply supported (Weaver & Dnny 1973) and clamped-clamped shelIs (Weaver &
Myklatun 1973), compared to Shayo & ElIen's (1974) 'fulI' expression of the bluntness factor
(Fbf )S-E, equation (7,67) (up to and including b 3 terms), and to its linearized versi on.
where b = a / L, IL = pa / (Psh) as in (7.49), and M is the Mach number. In terms ofthe
beam-theory u, this result is
Ucd = n{l- 0.945(1- M2)1/2b +... }-1/2,
where the { } quantity is the same as in (7.66a).
From equations (7 .66a,b) one may define a 'Shayo & Ellen bluntness factor',
(Fbf)S-E = (l- 0.945(1 - M2)1/2b +... }-1/2,
(7.66b)
(7.67)
the full form of which is also plotted in Figure 7.23 for M = O, as well as its linearized
versi on, {I + iO.945b}. It is seen that the full Shayo & Ellen bluntness factor converges
to unity from below, which is consistent with the results ofFigure 7.22: the blunter the shell,
the lower is the value of U cd. t The Weaver et al. bluntness factors, however, converge
from above,* suggesting the opposite effect of bluntness on stability. This discrepancy
has great1y perplexed the author, and many others no doubt, since the two sets of theories
are equivalent.
The answer to this conundrum becomes clear once the linearized version of the
Shayo & Ellen bluntness factor is computed; as seen in Figure 7.23, that approxima-
tion too converges from above. Thus, even for b '" 1/4 or 1/10 when one should expect
the b 2 , b 3 , etc. contributions to be negligible, their effect is anything but! The key to the
explanation lies in the fact that, unlike for a typical Taylor expansion, the coefficients of
tNevertheless, V ed increases with b because the bluntness factor is multiplied by b in equation (7,66a),
*Shayo & Ellen (1974) discussed this briefly, but not very clearly, In fact, by using the wrong word, they
rather added to the confusion, They state that 'the (Weaver & Unny) end flow factor , , , gives a misleading idea
of the stability condition since it approaches unity from below . , ,', instead of saying 'from above'!
Š As pointed out by Weaver (1998), the main purpose ofhis bluntness factors, which agree with the linearized
version of Shayo & Ellen's, was to show that they converge to unity for sufficiently slender shel1s, rather than
to yield approximate results for V cd,

610
0.30
::>:
I

l;j 0,24

'(3
o
<>
>
8 0.18
5
V
:.a
] 0.12
'.::
'c::
U
0,06
O
SLENDER STRUCTURES AND AXIAL FLOW
Unstable
'Full' theory
Stable
6
12
18
24
30
36
42
Length ratio, Va
Figure 7.24 Comparison of theory with experiment for beam-type divergence of clamped-
clamped tubes: -, 'fulI' Pltigge-Kempner shelI theory; - - -, beam theory; o, experiments
(hla = 0.25, E = 4.14 x 10 6 N/m 2 , v = 0.45, plps = 1.19); UW- M = [{l2/n 2 )(p;/ps))1/2 U
(Weaver & Myklatun 1973),
b 2 and b 3 are not of 0(1); hence, the approximation {I + 10.945b} is not correctto O(b)!
This is therefore a good example ofthe pitfalls ofhasty order-or-magnitude considerations.
ln fact, the mistake is easy to make since, physically, it makes sense for the fluid-
dynamic force per unit length for a blunt shell to be smaller than for a slender one - for
the same reasons as discussed in Section 4.4, This is undoubtedly trne here also,
However, two additional factors should be bome in mind. First, additional flexibility
is involved in shell theory, compared to beam theory - cf. the dual effect of bluntness
on (i) the shear deformation of Timoshenko beams and (ii) the fluid-dynamic forces, dis-
cussed in Section 4.4 (see Figure 4.22). Second, in the expression for U c d in (7.66a), the
bluntness factor is multiplied by b (or, equivalently, L is embedded in the detinition of Ucd),
As a result, the net e ffe ct is that U c d actually increases with b == a I L - cf. Figure 7.24,
where it is seen that U cd decreases with increasing Lla,
The important conc1usion from all of this discussion is that, unlike expression (7 ,65b),
equation (7.66a) may actually be used to obtain U cd approximately for beam-type diver-
gence of shells, Indeed, by comparison to the fuU theory, it is found that equation (7,66a)
gives accurate estimates of U c d for b == a I L as large as 0.5 - and LI a = 2 corresponds
to as short a shell as one might want to consider!
Finally, the experimentally determined beam-type divergence data of Weaver &
Myklatun, referred to in Section 7.2.5, can now be compared to both the 'fulľ F1iigge-
Kempner she11 theory and the simplitied theory, as shown in Figure 7.24. It is seen that
the beam approximation is quite valid for Lla > 30 (LID> 15). It is also seen that
agreement between the full (shell) theory and experiment is quite good throughout.
7.2.7 Post-divergence flutter or dynamic divergence?
The only important qualitative difference between beam and shell dynamics, apart from
the n associated with divergence or flutter, is that (i) for thick-walled tubes (beams) with

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
611
supported ends, the post-divergence coup1ed mode flutter predicted by linear theory does
not materialize in the experiments (Section 3.4.4) [and is shown to be impossib1e by
nonlinear theory (Section 5.5.2)], while (ii) for thin-walled tubes (shells) it does, for
n > 1. The question is why.
The work done by the fluid on a pipe conveying fluid in a period of oscillation, T, is
T f L 1 L f T
 W = - [ MU(ú} + UÚJW') dt - - [ M[ÚJ 2 - (Uw?] dx
10 O 2 10 O
(Pilldoussis & Issid 1974),t where ()' = d( )/dt, ()' = d( )/dx, and M here is the mass
of the fluid per unit 1ength. Assuming periodic motions, the second integra1 vanishes. For
systems with both ends supported, so does the first integra1, and hence, by this argument
a1one, no energy can flow to the pipe from the infinite energy reservoir of the flowing
fluid, thus engendering the paradox of amplified oscillations with no energy supp1y. How
this can be reconci1ed has been e1ucidated by Done & Simpson (1977) and is discussed in
Section 3.4.1. In Chapter 3, on1y the first integra1 in (7.68) has generally been considered,
assuming motions to be periodic, as in equations (3.5) and (3.11). However, considering
the full equation (7.68), it can be seen that, iffor some U > U e ! the oscillation is presumed
to be growing, then it is seen that it can be, that is,  W > O and energy can be gained,
despite w and perhaps a1so w' being zero at x = O and x = L.
Still, all this is academic, since it is known that beam-1ike flutter (n = 1) ofthick-wa11ed
pipes does not occur in experiments, nor in nonlinear stability ana1ysis. Yet, it does for
thin-walled tubes (n > 1). In Section 7.2.4(a) it was suggested that perhaps the observed
flutter is not trne flutter but rather a 'dynamic divergence'.:j: Unfortunate1y, no full-fledged
nonlinear theory is avai1ab1e for the she11 prob1em,9 which cou1d answer definitively the
question ofthe theoretica1 existence of coup1ed mode flutter as Ho1mes' (1977, 1978) did
for the thick-walled pipe prob1em. Hence, here we shall attempt to shed some light on
the issue by concentrating on the physica1 differences between the two prob1ems, which
are principa11y two: (i) as discussed in Section 7.5, un1ike for beam-like motions, there
is no reason why fluid frictiona1 forces wou1d cance1 out for she11 motions (n > 1);
(ii) shell-type limit-cycle motion impedes the flow by restricting the flow passage.
It is difficult to see what influence the pressurization and wa11-traction associated with
fluid viscous effects exert on post-divergence dynamics (here the reader is referred to
Section 7.5 for further discussion). The effect of flow restriction, however, is clearer: the
onset of divergence causes the collapse of the mid-span of the she11 into a form simi1ar
to that of one of the extreme free-end forms shown in Figures 7.1 and 7.11 for flutter of
a cantilevered she1!. C1ear1y, this severe1y restricts the fluid flow. The resulting build-up
of pressure behind the constriction forces the she11 to open up, whereupon fluide1astic
forces cause it to collapse again, but, because of inertia, into the other extreme form at
90° to the first (for n = 2); and so on. By this scenario, it wou1d appear that the 'dynamic
divergence' hypothesis for the observed flutter is indeed p1ausib1e. The fact that a clear
shell-mode divergence (predicted by theory), preceding flutter, is not observed reinforces
the argument.
(7.68)
tA typographical error in the second integral of the original form of (7,68) in PaYdoussis & lssid (1974)
has been corrected by Weaver (1974), who concludes that such 'gyroscopic conservative' systems are really
nonconservative,
"For example, refer to Brochard et a!, (1987) and Fukuyama et a!, (1999),
However, this deficiency is fast being redressed; see Section 7,8,

612
SLENDER STRUCTURES AND AXIAL FLOW
There are other pieces of negative evidence in favour of this hypothesis, First, the
mechanism just described applies to internal but certain1y not to external axial fiow over
the shell; the fiow in that case cannot be impeded by shell collapse, Nevertheless, it is well
known that there are no observations of fiutter of shel1s with supported ends in external
axial fiow, unless the fiow is supersonic (Dowell 1975); nonlinear theory with external
subsonic fiow agrees (Section 7.8).t
The second piece of negative evidence in support of the dynamic divergence hypo-
thesis is related to the effect of dissipation. It wi11 not have escaped the reader's attention
that, in all the calculations presented so far, dissipation has been neglected. Yet, as dis-
cussed in Section 5.5,2 in conjunction with Holmes' analysis of the thick-wal1ed pipe
problem, dissipation, even if small, profoundly alters the post-divergence dynamics; spe-
cifically, it eliminates the restabi1ization between divergence and fiutter that occurs for high
enough /Li - see, for example, Figure 7.17. Furthermore, this is a key element in mak:ing
coupled-mode fiutter impossible in the pipe problem.
As shown by Matsuzak:i & Fung (1977a) - see Section 7.3.6(b) - dissipation e1im-
inates post-divergence restabi1ization in shell modes as well. This is illustrated here
in Figure 7.25 ( = 0.01), which should be compared to Figure 7.17 (= 0).+ ln
Figure 7.17, the eigenfrequencies are wholly real for 0< Uw- u < (UW- U)cd, wholly
imaginary for ( UW- U)cd < Uw- u < 4,5 (not shown), and then wholly real again for
Uw- u > 4.5 approximately, implying restabilization - cf. Figure 3.11; then the first- and
second-mode (m = 1 and 2) 10ei collide on the 'R.e( Qw- u) axis and thereafter become
complex, one of them with Im (Q w-u) < O, thus giving rise to coupled-mode fiutter. With
damping, however, as seen in Figure 7.25, after the onset of divergence there is always a
mode with Im (Qw- u) < O; hence the system does notregain stability - cf. Figure 3.14.
This renders the occurrence of post-divergence fiutter less 1ikely.
Final1y, some convincing positive evidence in support of the dynamic divergence
hypothesis is presented in Section 7.8.2(d), based on non1inear theory.
Hence, on the basis of the currently available evidence, it would appear that the
coupled-mode fiutter predicted theoretically with internal or external fiow probably never
materia1izes; hence, the observed fiutter with internal fiow is most 1ikely a manifestation
of dynamic divergence.
7.3 REFINEMENTS AND DIVERSIFICA TION
7.3.1 Link between travelling and standing wave solutions
for shells with supported ends
The link between the Paidoussis & Denise travelling wave solutions and the Weaver et a!,
standing wave solutions described in Section 7.2 is provided in an excellent paper by
Shayo & Ellen (1974) for pipes with supported ends, in which (i) some important insights
and (ii) some very useful simplifications for limiting cases (e,g. for long shells) are given.
tThe same does not appear to be trne for fiat plates in axia1 fiow, and this is a mildly controversial question
(Chapter 10).
*The calculations in Figure 7.25 have been performed with the Donnell shell theory [equation (7.10)] rather
than Fliigge-Kempner's, but otherwise the same method as used by Weaver & Dnny (1973), It was confirmed
that the results are essentially coincident with those of Figure 7,17 for  = O; see Figure 7,26, to be discussed
in Section 7,3,1.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
613
25,
5
20
S 15
I
lď
llo
3
2
S
I
I O

-I
-2
-3
O
2
3
U w - u
4
5
6
Figure 7.25 The effect of dissipation on the dynamics of shells conveying fluid for the same
system as Figure 7.17 but with viscous damping added; I; = 0.01 (Amabili et a/. 1999a).
7,3.1 (a) Genera/ ana/ysis
Shayo & Ellen's (1974) analysis, though correct, is too compact and the notation
quaint. Hence, a more standard derivation is given here, sirnilar to Pa'idoussis,
Chan & Misra's (1984a) for the perturbation pressure, but reverting to Shayo & Ellen's
analysis subsequently, considering a simply supported shell conveying fluid, analyzed via
the Fliigge-Kempner equations.
The fluid is genera11y considered to be compressible and inviscid. Hence, small
perturbations in the velocity potential, <I> (x, r, e, t), must satisfy
1 ( a a ) 2
\72<1> - - - + u - <I> = O,
c 2 at ax
where c is the ve10city of sound. Expressing <I> as
<I>(x,r,e,t) = 1/fn(x,r)cosnee iQt ,
(7.69)
(7.70)

6]4
SLENDER STRUCTURES AND AXIAL FLOW
20
15
:>

IOJ
10
5
O
O 2 3 4 5 6
U w - u
Figure 7.26 The dimensionless eigenfrequencies of a simply supported shell conveying water
(Lja = 2, hja = 0.01, n = 5) - cf. Figure 7,17 - obtained by Donnell's sheU theory [equa-
tion (7.10)] via a two-mode Galerkin solution: -, with Weaver & Unny's (1973) tluid-dynamic
model; - - -, with Pai'doussis & Denise's (1972) tluid-dynamic model (Amabili et al. 1999a).
substituting in (7.69), and taking the Fourier transfonn of the resultant equation, leads to
a21fr; I a1fr; [ n 2 2 ] * 1 2 2 2 *
----a;:2 +  a;:- - r2 + a 1fr n - c 2 (-Q + 2QUa - U a )1fr n = O, (7.7])
in which
1fr;(a, r) = i: 1frn(X, r)e iax dx, (7.72)
a being the Fourier variable, and the asterisk denoting a transfonned quantity. To obtain
(7.71), use has been made ofthe fact that [a1frn(x, r)jax]* = -ia1fr;(a, r), and
lim 1frn(x, r) = O and lim [a1frn(x, r)jax] = O, (7.73)
x::I::oo x::I::oo
which is reasonable on physical grounds, particularly if (i) rigid extensions to the shell for
x < O and x > L and (ii) a small amount of dissipation are assumed.
Defining next the dimensionless parameters
QL
K=-
U'
equation (7,71) may be written as
a21fr; 1 a1fr; [ n 2 2 ] * O
-+--- -+A. 1fr = ,
ar 2 r ar r2 n
which adrnits solutions of the fonn
Ci = aL,
U
M=-,
c
A.L = [Ci 2 - M 2 (Ci _ K)2]1/2,
(7.74)
(7.75)
1fr;(a, r) = Cn(a)In(A.r) + Dn(a)K,,(A.r).
Since <I> must remain finite as r --+ O, clearly Dn(a) = O.
(7,76)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
615
If the radial displacement associated with a particular circumferential mode is
expressed as
00
w(x, O, t) == wn(x, O, t) = L C mn 1Jm(x) cos nO e iQt ,
m=l
(7,77)
then substituting (7.76) and (7.77) into the impermeability condition, equation (7.35),
leads to
*  iU(K - li) In(.A.r) *
1/f n (a, r) =  AL I' (Aa) 1J m (a)C mn
m=l n
iU(K -a) L oo _ *
= {En(a, r)1Jm(a)C mn } ,
AL
m=l
(7.78a)
(7.78b)
where En(ii, r) = In(Ar)/I(Aa) and I(z) = dln(z)/dz,
The perturbation pressure p may be expres sed as t
p(x, r, O, t) = p(x, r, O)e iQt ,
(7.79)
and the Fourier transform of p is
p*(ii, r, O) = L i: p(, r, O)eia d,
where  = x / L. + Then, substituting the Fourier transforms of (7.70) and (7.79) into
(7.37), we obtain
(7.80)
p*(ii, r, O) = -ip(U / L)(K - a) 1/f; (a, r) cos nO.
(7.81)
Next, substituting (7.78b) into (7.81), utilizing (7.79), and taking the inverse transform,
we obtain the surface perturbation pressure (at r = a):
00
p(,a,O,t) = L Qmn()cosnOeiQt,
m=l
(7.82a)
tThe asterisk on p, denoting the perturbation pressure, as in (7,33) and (7.37) for instance, has been dropped
since here the asterisk denotes a Fourier-transformed quantity.
:j:The inverse transform, not only for p but for all such quantities, is defined here in terms ofthe dimensionless
a = aL, for general use in what follows, as
1 1 00 '-, 1 1 00 '-,
p(, r, B) = 2 L p*(a, r, B)e- 1a , da = _ 2 p*(a, r, B)e-,a, da
n -00 n -00
- the latter form being so because of equation (7,80). Note that if the former form is used, an extra
L -I factor arises in equation (7 ,82b), as in Pai:doussis et a/, (l984a), This difference disappears after full
nondimensionalization of the final theoretical results,

616
SLENDER STRUCTURES AND AXIAL FLOW
P U 2 C 1 00 ( K - Ci ) 2 .
Qmn() = mn En(Ci, a)<p:n(Ci)e-fa dCi,
2n L -00 )"'L
where Qmn (O is the generalized pressure. t
Equations (7.82a,b) may be used in conjunction with any shell theory and for any
boundary conditions. However, it is for a simply supported shell and the Fltigge-Kempner
shell theory that a number of very useful simplifications can be effected, and hence it is
for that system that we proceed further. The Fliigge-Kempner equation (7.13b) with p on
the right-hand side (for the shell conveying fluid) may be re-written, after substitution of
equations (7.77) and (7.82a) and division by Psh, as
(7.82b)
00 1 00
(Qn - Q2) L Cmn<Pm () cos ne e iQt = h L Qmn () cos ne e iQt ,
m=1 Ps m=!
(7.83)
where Qmn are the in vacuo eigenfrequencies, given by
Q2 = Ek { 1 [ ( m2 + n2)4 _ 2vnt' _ 6nťn2
mn psa2(l - v2) ( m 2 + n 2 )2
_ 2(4 - v) m 2 n 4 + 1  v 2 nť + 2(2 _ v) m 2 n 2 + n 4 _ 2n 6 ]} , (7.84)
in which k = A (h j a)2 and m = mn a j L; it is noted that, if Qmn is divided by
{E j[p s a 2 (1- v 2 )]}1/2, one obtains the dimensionless frequencies Qm n defined by (7.46).
Transfonning <P:n (Ci) back into <Pm () in Qmn, cf. the penultimate footnote, multiplying
by <pq () and integrating over (-00, (0) -effectively over (O, 1) since this is the physically
meaningful domain of the problem - we obtain
2 2  pu 2 C mn 1 00 (K - Ci)2 _ _
(Qqn - Q )C qn -  h En(a, a)Hqm(a) da = 0,
2n Lps -00 )"'L
m=1
(7.85a)
q=1,2,...,00;
Hqm(Ci) = 11 <Pm()eia d 11 <pq()e-ia d. (7.85b)
Clearly, equation (7.85a) leads to a matrix equation of the form [HC} = {o}, where
{C} = {Cln, C2n, .,. CNn} if the summation is truncated at q, m = N. Then, det[] = °
is the characteristic equation from which the eigenfrequencies, Q, may be determined.
7.3. 1 (b) Simplification on the divergence boundary
On the divergence boundary, at U = Ucd, clearly Q = ° and hence also K = 0- see
(7.74). Furthermore, the off-diagonal terms, associated with the Galerkin solution are zero
t Approximate expressions for the genera1ized fluid forces, based on Galerkin discretization of the coupled
fluid-structure problem and bypassing the Fourier transform technique (and the numerical difficulties therein),
have recently been derived in an elegant ana1ysis by Langthjem & Olhoff (2003),

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
617
(cf. Section 3.4.1) and hence only the diagonal terms are of importance; thus q = m, and
equation (7.85a) simplifies to
2 pU;d 1 00 a In{a v' 1 - M2 a jL) -
Q - Hmm(Ci) da = O.
mn 2JT LPsh -00 (1 - M2)1/2 I(a v' 1 - M 2 aj L)
For the simply supported shell, m () = v'2 sin(mJT) and
_ 4m 2 JT2[1- (_l)m cos a]
Hmm(a) = ( 2 2 -2 ) 2
mJT -a
One may therefore write (noting the change in integration limits)
2 _ pU;d [ 8m2JT2 ] 1 00 ! a[1- (_l)m cos a] In(a v' l- M2 a jL » ) __
Q 2 da - O.
mn 2JTLp s h v' 1-M2 O (m 2 JT2-a)2 I(a v' I-M2ajL)
(7.86)
Using the shorthand notation
Qmn = m l°°{} da,
(7.87)
with the bracketed expression as in (7,86), as done by Shayo & Ellen, one can write
L p h ( 1 - M 2 ) 1/2Q2
U 2 _ s mn
cd - 4pmJT Qmn
(7,88a)
This is generalized to apply to n = O also, by introducing the factor £n, such that £0 = 2
for n = O and £n = 1 for n > O; then, in dimensionless terms we have
_ 2 £ ( I - M 2 ) 1/2 Q 2
U _ n mn
cd - 4/LbmJT Qmn
(7.88b)
in whichb = ajL and /L = paj(psh); U and Q aredefinedin (7.46). Equations (7.88a,b)
present a remarkable simplification to the problem of determining the critical ftow velocity
for simply supported shells, particularly if the expression for Qmn can be simplified. This
has been done by Shayo & E1len (1974), and this work is reproduced in outline in what
follows.
7.3.1 (c) Divergence of slender she/ls, b = aj L « 1
Slender shells signify large Lja. However, in view of the form of the argument of the
Besse1 functions in (7.86), it is more convenient to equivalently consider shells of very
small 'bluntness' b, and hence 'modified bluntness', b, defined by
b = (1 - M2)1/2b,
b=.
L
(7.89)

618
SLENDER STRUCTURES AND AXIAL FLOW
After 1engthy manipulations (Shayo & Ellen 1974; appendix), for m not 1arge - more
--1
precise1y for m "'-' O(b ) - one obtains
Qmn =
1 1- 2
--=--- + -bmn + O(b )
2bmn 8
- 2 2
bmn [ 4nFl- m n -2 2 -3
- l+-b- b +8m nnF2b
4n n 2n(n + 1)
(4 + 3n)m 4 n 4 -:-4 O -5 ]
+ 8n2(n + 1)2(n + 2) b + (b)
for n = O,
(7.90)
for n  1,
where
Fl(n) = (X> [ In(Z) -! ] dZ,
10 zI (z) n Z2
F2(n) = (OO [ In(Z) _.!.. + Z2 ]  dz.
10 zI(z) n 2n 2 (n + 1) z4
Hence, for I) « 1 and m "'-' 0(1), from (7.88b) and the appropriate asymptotic form of
(7.84), one obtains
U d = .!..(1 - M 2 )(1 - v 2 ) + 0( m 2 ), n = O, (7.92a)
fJ.,
U d = .!..(1 - v 2 ) m 2 + 0(m 3 ), n = 1, (7.92b)
fJ.,
(7.91)
U; d   {ne n2 _1)2 (  n + O (  ),
where k = l (h/a)2, fJ., = pa/(psh), and m == mnb as in equation (7,84). From (7,92a)
and (7.92b), it is c1ear that, for b -+ O, the n = 1 (beam mode) result determines the
most critical condition, and hence onlY n  1 needs be considered from here on. Next,
comparing (7 .92b) and (7 .92c) it is seen that the most critical mode is associated with m -+
00, which is inconsistent with the approximations employed in deriving the foregoing
asymptotic expansions. Hence, a different type of approximation is required, namely
I) « 1 and bm "-' 1, which is discussed in subsection (d).
Nevertheless, persisting with the approximation I) « 1, m "'-' 0(1)-1), it is noted that
equations (7.88a,b) and (7.90) may be used to obtain fj cd, provided that Fl and F2 are
known. These have been evaluated, as shown in Table 7.1.
n> 1,
(7.92c)
Table 7.1 Values of F, (n) and F2 (n) (Shayo & Ellen
1974).
n 1 2 3 4 5
Ft (n) -0,742 -0.210 -0.097 -0.055 -0.036
F2(n) 0.211 0.021 0.004 0.002 0.001

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
619
Unlike the general case of n > I, the case of n = I can quite ...!..eadily be finalized
here. From (7 ,92b), it is clear that the most critical, smallest value of U cd is obtained with
m = 1- similarly to the results obtained by beam theory in Chapter 3, Hence, taking
m = I, utilizing (7 .92b) and inserting the higher-order terms, one obtains equation (7.66a),
which has already been discussed in Section 7 .2.6(b).
7,3.1 (d) Short waves on long shel/s; b « I, bm '"" I
To find the most critical condition for shell-mode divergence (n > I), one still has b « I
but, instead oftaking m '"" I, one now imposes bm '"" 1. Following Shayo & E1len (1974),
the expression for Qmn is re-wriUen in the form
I i oo m xln(mx)
Qmn =  {l- (-I) cosmJl'x} (1 2)21' ( ) dx,
mJl' o - X n mx
where m = (1- M 2 )1/2 m == (1- M 2 )1/2(Jl'mb) == (1- M 2 )1/2(mJl'ajL).
Since m '"" b- 1 , the integrand oscillates very rapidly, and the main contribution comes
from the neighbourhood of x = I. Hence, expanding ln (mx) jl (mx) as a Taylor expansion
about x = I, the following approximation to Qmn is obtained:
Q = [ I + 2 ( 1 _ M 2 ) 1/2 I' ( m )  ] In(m)
mn Jn Jl' L 41(m)'
I' ( m ) _ { m [ foo 1(z)In(Z) d ( _I ) m ]
Jn - 10 (m2 _ z2)[I(z)]2 z +
m [ n2 ] [ ln(m) ] 2 } I(m)
- (-1) m 2 + I I(m) In(m) ,
(7.93)
Substituting into (7.88b), the following is obtained:
Q2 I' () [ 2 ] -1
U 2 = (1 - M 2 )1/2-.!!!:!!:..  n m 1+ -(1 - M 2 )1/2b!n(m)
cd J.L mIn (m) Jl'
(7.94)
A number of important conclusions may now be reached by examining the asymptotic
results for n = 1, equation (7.66a), and n > I, equation (7.94), discussed in greater detail
by Shayo & Ellen (1974) and Shayo (1977), as follows.
(i) The leading term in (7.94) corresponds to the stability condition of an infinite1y long
shell with periodic supports, or to that obtained by trave1ling wave analysis when
some damping is present or when J.L -+ O, This suggests that the trave1ling wave
analysis provides a reasonable approximation to the critical divergence condition
of finite length shells, provided that b = aj L is relatively small and the axial
wavelength is consistent with the finiteness of the shell,
(ii) The most critical condition must be determined by considering all n and m, thus
through use of both (7.66a) and (7.94). A comparison of the results of (7.94) with
Weaver & Unny's (1973) shows good agreement for bas large as 0.2 and hja as
small as 10- 2 .

620
SLENDER STRUCTURES AND AXIAL FLOW
Table 7.2 Values of U; d(/1,jk) == pU 2 a 3 1D, where {.t = pal(psh),k = -ť2(hla) and D =
Eh 3 1[12(1- v 2 )], for various thickness ratios, hla, and bluntness ratios b = al L, for v = 0,3 and
M = 0, The integers in brackets correspond to the (n, m) mode combination involved at in stability.
The first column gives the instability obtained fmm a travelling wave analysis.
hla pU 2 a 3 I D = ({.tI k)V 2 for mode (n, m)
Travelling waves b = 0,05 b = 0.1 b=0.2 b = 0.5
0.001 1.01 x 10 4 1.78 x 10 4 2,59 x 10 4 3.87 x 10 4 6.04 x 10 4
(2,1) (3,1) (4,1) (6,1) (9,1)
0.003 6,36 x 10 3 9.62 x 10 3 1.55 x 10 3
(3,1) (4,1) (7,1)
0,005 3,57 x 10 3 5.01 x 10 3 8,00 x 10 3
(3,1) (4,1) (6,1)
0,008 .. 2.87 x 10 3 4.38 x 10 3
(3,1) (5,1)
0,01 2.03 x 10 3 3,33 x 10 3
(3,1) (5,1)
0,03 301 893
(1,1) (4,1)
0.05 108 415
(1,1) (3,1 )
0.08 42.4 250
(1,1) (3,1)
0.1 27,2 183
(1,1) (2,1)
0,3 3,08 27.8
(1,1) (1,1) (1,1)
0.5 1.16 12,7 12.1
(1,1) (1,1) (1,1)
(iii) The transition from beam- to shell-mode divergence occurs when (aj L)(aj h) ::::: I,
as also found by Weaver (1972).
(iv) As hja is decreased, the critical conditions correspond to larger n - cf,
Figure 7.19.
Table 7.2 gives a sample set of the asymptotic results obtained with the aid of
equations (7,66a) and (7.94), which are in excellent agreement with the results of
Weaver & Dnny (1973). These results i1lustrate the conc1usions given in the foregoing.
Some discussion is also given by Shayo & Ellen for shells in an elastic shroud of
stiffness K, similar to an elastic foundation for beams. In the foregoing equations this
is accommodated by replacing Qk by Qk + K j(p s ha 3 ). !tis found that when n > I, the
variation of U;d with m is similar for K = O and K t= o. For n = I, however, U c d -+ 00
for K t= O, instead of U cd -+ O for K = O, in the limit of m -+ O. Moreover, for n = I,
the value of m associated with instability when K t= O can be large, and not m = I as is
typical for K = O (Table 7.2) - cf, Section 5.5.2(c),
!t is noted in the table, that in this version of the travel1ing wave solution, the critical
mode is always (n, m) = (2, I) - although in the Pai'doussis & Denise version, n > 2

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
621
is certainly possib1e (Figure 7.12), It is a1so noted that, when n = 2 is predicted by the
standing wave solution a1so (the shaded part of the tab1e), agreement between trave1ling
and standing wave solutions is quite good, provided b ::: 0.2 - a1though at the upper
1imits of hla the agreement is not rea11y so good; thus, for hla = 0.008 and b = 0,05, we
have 1.26 x 10 3 by the standing wave solution versus 1.24 x 10 3 by the travelling wave
solution, but for b = 0.1, this deteriorates to 2.22 x 10 3 versus 1.24 x 10 3 ,
Agreement between travel1ing and standing wave solutions is a1so good for canti1evered
shells, provided b is sma11, as shown in Figure 7.27, to be discussed in Section 7.3.2.
Fina11y, another kind of comparison between the Patdoussis & Denise and Weaver &
Dnny ana1yses has been done by Amabi1i et al. (1999a), using the same she11 theory
in both cases [Donnell's equation (7,10)] and the same solution method (a two-mode
Ga1erkin solution), but the two different fluid mechanics mode1s. A typica1 result is shown
in Figure 7.26 for the same system as in Figure 7.17 and n = 5, appropriate1y high for the
Donnell equation to be reasonab1y accurate. It is seen that the resu1ts are not too different
in terms of this comparison a1so, despite the shell being anything but slender (L/a = 2).
This shows that, with respect to fluid dynamics mode1ling, the two ana1ytica1 mode1s give
quite simí1ar results, even for b1unt shells.
7.3.2 Cantilevered shells and outflow models
Both in the beam-mode theory of Chapter 3 and in the travelling wave ana1ysis of
Section 7.2.3, a 'point function re1ationship' between the perturbation pressure and pipe
deflection was assumed, such that p ex [a/at + U(a/ox)fw for each axia110cation a10ng
the shell. On the other hand, in the Fourier-series Ga1erkin-type solutions, a more complex
integra1 re1ationship is invo1ved, for examp1e, equations (7 .82a,b). Indeed, an inseparab1e
part of the ana1ytica1 procedure is that the perturbation pressure far upstream and down-
stream needs to be specified, as in equations (7,73). This poses no prob1em for shells with
supported ends, as discussed in conjunction with (7.73), However, potentia11y at 1east, it
does for canti1evered shells. t
Specifically, the question arises as to what happens to the jet emerging from the free
end of the shell. It is known that such jets are subject to in stability, as e1ucidated for
instance by Batche10r & GiU (1962). * Thus, how does the jet dynamics coup1e to motions
of the she11? More precise1y, for the prob1em at hand, the question is: how does the
perturbation pressure at x = L decay, as it must do if some dissipation is assumed, beyond
the free end of the shell? This then, in an integral sen se, wou1d affect the perturbation
pressure at any point within the shell. This question has been addressed for the first time by
Shayo & E11en (1978). It has a1so been discussed, for n = 1 oscillations of short pipes, in
Section 4.4.4.
Shayo & E11en (1978), noting that canti1evered shells genera11y lose stability by flutter,
so that inertia1 effects are of importance, in this case use the F1tigge shell equations in
which in-plane inertias are not ignored - as they are in the Fltigge-Kempner theory, used
in their earlier paper. The same Fourier-transform analytica1 method is used to detennine
the perturbation pressure. Here the same analysis is presented, but with the fonnulation
tFor an alternative, approximate analysis, bypassing the use of the Fourier-transform technique and the need
of outflow models, see Langthjem & Olhoff (2003).
*Much newer work is also available; see, for instance, Crighton & Gaster (1976) and Corke & Kusek (1993).

622
SLENDER STRUCTURES AND AXIAL FLOW
and notation in Pai'doussis, Nguyen & Misra (1991), which treats a more general problem;
see Sections 7.5,5 and 11.7.2.
Shell motions are govemed by equations (7.29), which may be re-written as
LI (u, v, w) = O, L2(U, V, w) = O, L3(U, V, w) = O,
(7.95)
subject to boundary conditions (7.3) and (7.4a-d), the latter of which may be written as
Ri(U,V,W) =0, i=I,2,3,4.
(7.96)
ln this case, because of the complexity of equations (7,96), it is rather difficult to obtain
appropriate comparison functions, and it is therefore not convenient to use the ordinary
form of the Galerkin method. The extended Galerkin method is utilized instead, whereby
the equations of motion and boundary conditions are satisfied together in the mean, The
appropriate variational statement is
E(h/a) !1 21f 1 L
D£ = 2 [LIDU + L2DV + L3DW] <lx de
I-v O O
- 1.'" [ R,8u + (1 - v)R,8v - k R,8w + kR,8W'] dO} = 0,
where k = /2 (h / a)2, and the signs in the second integral are dictated by the sign
conventions in equations (7.4a-d) - cf. Figure 7.2: for N x , related to -u; Mx, related
to -w'; Sx, related to w; and Tx, related to -v. The upstream boundary conditions are
easily satisfied with an appropriate choice of admissible functions,t namely
(7.97)
{ U } 00 { aAm C OSne(a/8X) } ,
v = L Bm sin ne </lm (x )é"U ,
w n=1 Cm cos ne
(7.98)
where </lm (x) are the eigenfunctions of a cantilevered beam, equation (2.27).
Equation (7.97) may be obtained directly, at least for the empty shell, from the kinetic
and potential energies of the shell (Sharma 1974), application of variational procedures
on the time-integral of the Lagrangian, and integration by parts. Sharma (1974, 1979)
used this energy approach to obtain the natural frequencies of a cantilevered shell via the
Rayleigh-Ritz method, which is equivalent to what is used here.
Proceeding in the same manner as in Section 7.3,1, but considering incompressible
fluid flow, one can determine the perturbation pressure, with the following difference: in
view of the discussion at the beginning of this section, the domain of this problem is not
(O, L). Thus, what is required is an extension for </lm (x) which would describe the dynamic
behaviour of the jet emanating from the shell free end; this constitutes the so-called outjlow
model. Two such models are proposed by Shayo & Ellen (1978): the 'collector pipe model'
and the 'free flow model', described below; neither allows jet in stability to develop -
simply because its treatment would be too difficult to handle.
t Admissible rather than eomparison funetions, beeause they satisfy only the geometrie boundary eonditions,
at x = o.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
623
The perturbation pressure is given by
00
p(,a,e,t) = L Qmn()cosneeigt,
m=l
(7.99)
P U 2 c 1 00 ( K - a ) 2 I 'cib ) ,-
Q (  ) = mn n \"" { A.* ( a ) + r* « (i\ } e-Ia da
mn 'i 2n L -00 a I(ab) 'f'm m ""J ,
which is identica1 to (7 .82a,b) for M = O, except for the additional term r::Z (Ci); that is the
Fourier transform of r m (O, the extension to the corresponding beam eigenfunction, that
is, the outflow model, different forms for which are discussed next. In these equations,
K = QL/U,b = a/L and = x/L.
The collector pipe model supposes that the fluid issuing from the free end of the
canti1evered shell is 'swal1owed by a second tube in controlled motion but not mech-
anical1y connected to the canti1ever'. Thus, in dimension1ess terms, the domain of the
prob1em is expanded from <Pm (), valid for  = (O, 1), to <Pm () plus r m (), with
rm() = <Pm(1) [(I - )/(l -1)], 1 <  :S I;
(7.100)
identical to the so-cal1ed 'first-model' in Tab1e 4.1 for n = 1.
ln the free flow model, it is assumed that fluid partic1es downstream of the free end of
the shell travel with ve10city U in the x-direction and 'have a radial disp1acement which
is prescribed by what they had when they 1eft the pipe'; that is, for n > 1 the jet executes
varicose oscillations of the form
{ -iQL }
rm()=<Pm(l)exp --U(-1) ,
 > 1,
(7.101)
for time variations exp(iQt).t This is the on1y non-decaying solution for a circu1ar jet
with zero pressure condition on its boundary; all other solutions rapid1y decay, as pointed
out by Shayo & Ellen. Furthermore, the following germane point is made: for a stab1e
system and temporal1y decaying osci11ations, hence for Im(Q) > O, equation (7,101)
indicates an axial1y growing oscillation for  > 1 - cf. Figures 3.45 and 3.48
for beam-1ike motions; this is a reflection of the fact that 'the jet contains the time
history of the canti1ever exit in spatial form' - see also Shayo (1977; section 4,5 and
appendix A).
As discussed in Section 7.2.6(a), for very long shells one may write
pa [ a a ] 2
P = ----;;- at + U ax w(x, e, t),
(7.102)
in which any downstream end-effects are decoup1ed from shel1 motions. A thorough
ana1ysis has been done by Shayo & Ellen (1978) - see also Shayo (1977; section 4,5
and appendix B) - in which an asymptotic expansion of (7.99) is undertaken for
smal1 a/L, a definite1y non-trivial task, effectively 1eading to (7.102). This constitutes
tNote that Shayo & Ellen use exp( -iQt) and hence a positive exponent in equation (7,101), Here,
Tm(Q) < O for the fluttering (unstable) shell, so the exponent is negative for; > I,

624 SLENDER STRUCTURES AND AXIAL FLOW
Table 7.3 Values of the dimensionless flow velocity for flutter, 7J cf,
defined by (7.46) and (7.47), for various values of Lla and n, obtained
with different outflow models, for v := 0,5, hla := 0,0227 and f.J, := 0.06
(Shayo & Ellen 1978).
n L/a 'Collector pipe 'Free flow 'Long pipe
model' model' model'
5 1.70 1.66 1.40
10 1.23 1.25 1.20
15 0,94 0.96 0,93
20 0.75 0.76 0.74
2 5 0,86 0,87 0,83
10 0.53 0,54 0,52
15 0.39 0.40 0.39
20 0,39 0,39 0,39
3 5 0.57 0,58 0,56
10 0.54 0.53 0.53
15 0,45 0.45 0,44
20 0.48 0,47 0.48
Shayo & Ellen's long pipe model. The other two model s together with this one are
used to effect a comparative study of the importance of downstream flow conditions on
stability.
The critical flow velocities for flutter, U cf , in the n = 1, 2 and 3 modes, calculated
with the aid ofthese outflow models are compared in Table 7.3. These theoretical results
have been computed whi1e ignoring in-plane inertias for convenience, but the pattem in
Table 7.3 is similar if they are taken into account, although the values of U c f are then
uniforrnly lower. It is seen that (i) provided L j a  10, the effect of the outflow model is
quite negligíble; (i i) for n  2, this is trne down to Lja  5 or so, differences being ofthe
order of 5%; (iii) the inc1usion of a 'true' outflow model stabilizes the system vis-a-vis the
'long pipe model' . The physical reason for this last observation is not obvious; intuitively,
one might expect the opposite. The only explanation that can be proposed here is that, in
the 'collector pipe' and 'free flow' models, the explicit specification of fluid behaviour
beyond  = 1 renders the system effectively 1ess nonconservative.
Typical results for V c ! with varying Lja == b- 1 are shown in Figure 7.27, computed
with the aid of the 'co11ector pipe model'; these are compared to Pa'idoussis & Denise's
theoretical and experimental results, For the parameters in this figure, it is found that the
values of U c f calculated with this outflow model are indistinguishable from those obtained
with the 'long pipe model' (not shown), thus reinforcing the discussion on Table 7.3.
Furtherrnore, it is seen that travelling and standing wave solutions are quite close for
Lja  24 (b < 0,04), and reasonably close for Lja > 8 (b < 0,1), reinforcing the
conclusion reached in Section 7.3.1.
Additiona1 outflow model s have been proposed by Pai'doussis et al. (1986, 1991);
four models in a11: Models O, 1, 2 and 3, according to the order of the polynomial
describing them; see Figure 7.28 and Table 4.1 of Section 4.4.4. Model O simply takes
r m () = O, whi1e Model] is the same as Shayo & Ellen's 'co11ector pipe model', defined
by equation (7.100). In Model 2, continuity is additionally imposed to the slope at the

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
625
0.5
Unstable (flutter)
1.5
2
1.0
U cj 3
1 




-..;;:
'<::::


-..::::
......,
o
o
o
o
Stable
o
4
8
12 16
L/a
20
24
28
Figure 7.27 The dimensionless flow velocity for flutter, U cf , of a cantilevered shell conveying
fluid, as a function of Lja, for n = 1,2 and 3; v = 0.5, hja = 0.0227, I-t = 0.06. -, Shayo &
Ellen's theory; - - -, Paldoussis & Denise's (1972) theory; o, Pa'idoussis & Denise's experimental
data (Shayo & Ellen 1978).
shell outlet: r()I=1 = <p(l), leading to
{ ( - 1)2 } I { ( - 1)2 }
rm() = <Pm(l) 1 - (Z _ 1)2 + <Pm(l) (- 1) - (Z - 1) ,
1 <L::::Z
(7.103)
- cf. equation (4.68) of Section 4.4.5(b). Mode Z 3 imposes the additional condition that
r () I=l = O, thus leading to
rm() = <P m (l)3 { 2e - 3(Z + 1)2 + 6Z + Z2(Z- 3) }
(Z - 1)
+ <p(l) { e - (2Z + 1)2 + Z(Z + 2) _Z2 }
(Z - 1)2 '
1 <  S Z.
(7.104)
This, with Z = 3, is the model with which most of the calculations in Section 7.5.5 have
been conducted. Although the model is generally trouble-free, numerical difficulties are
sometimes experienced with it, as discussed by Pmdoussis ef aZ. (1991) and Nguyen ef al.
(l993a). These difficulties are related to the functional form of rm(), and specifically in
the determination of r::Z (Ci) for very large Z. In this respect, it is recalled that the appropriate
value of Z is selected such that increasing it further does not have a large effect on the critical
ftow velocity.

626
SLENDER STRUCTURES AND AXIAL FLOW
Consequently, Model 3 has since been abandoned and is supplanted by Model 4,
proposed by Nguyen et al, (1993a). Similarly to Model 3, the displacement and slope
at 1 vanish, but in this case as 1 ---+ 00. Thus, this outflow model should satisfy
r m (1) = 4>m(1), r:n(l) = 4>:n(l), rm(l) 11--+00 = O, r:n (l) 11--+00 = O. Intuitively then,
r m () = (A + B) exp(1-) is taken, which automatically satisfies the lasttwo boundary
conditions, and thus,
rm() = {[4>m(1) +4>:n(1)H -4>(1)}el-, 1 <   l.
(7.1 05)
Model 4 implies that 1 ---+ 00 should be used in the calculations. However, to test its
convergence with increasing l, ca1culations have been conducted with 'truncated versions'
of the model, involving finite values of l, for example, as in Figure 7.28 for 1 = 5. As
shown in Table 7.4, Model 4 is free of any numerical difficulties, displaying a smooth
convergence with increasing l, sensibly achieved with 1 = 10, and practically with 1 = 5.
Furtherrnore, by comparing with the results in Table 7.3 (interpolating for L/a = 12.9),
it is found that the values in Table 7.4 are quite close to those obtained with either the
Table 7.4 The dimensionless critical flow velocity for flutter, ff cf, of a
cantilevered shell (v = 0.5, Lla = 12.9, hla = 0.0227, fJ, = 0.06) with
Model 4 (Nguyen et aZ. 1993a).
n Z=3,O 1=5,0 1=10 I = 10 2 I = 10 3
1 0.967 0,988 0,980 0.982 0,982
2 0.419 0.426 0.423 0.424 0.424
3 0.465 0.473 0,470 0.469 0.469
4 0.562 0.541 0.556 0,554 0.554
5 0.611 0.614 0,611 0,616 0,616
5
 4

o. 3
c
o
 2
-e
€
 I
- - . . - Model O
- . - Model 1
- - - Model 2
- - . - Model 3
-Model 4
""0"""""". 0," ,
....a...: "
':,::--\
Model O
O
1
2
3
4
5
6
7
8

Figure 7.28 The so-cal1ed outflow models, describing the manner in which pressure perturbations
decay beyond the free end of the shell; m = 2. Model O corresponds to an abrupt disappearance of
i. - "J" .......-..-.... I, '/ '........-'-.........1-1"1............-""....
Models 2 and 3 are proposed by Pai:doussis et aZ, (1991), and Model 4 by Nguyen et aZ. (l993a).
In Models ° to 3, I = 3. In Model 4, Z -+ 10 or 100 should be used to approximate I -+ 00; a
truncated version for l = 5 is also shown, yielding acceptable results - cf. Table 7.4.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
627
'collector pipe model' or the 'free flow model' - albeit lower, for the reasons discussed
in conjunction with Table 7.3.
All these model s are empirical. A more physically based model is proposed by Ilgamov
et al. (1994), in which the jet dynamics is considered under certain assumptions. This
model has been developed for beam-like (n = 1) motions of vertical cantilevered pipes
with an end-nozzle, so that gravity and such nozzle-related effects as a vena contracta are
intimately woven into the model. However, this work may serve as a useful starting point
for a further improved model. The impetus for this, nevertheless, is not overwhelming, in
view of the results presented in Tables 7.3 and 7.4. Indeed, Model 4 may be considered to
be satisfactory in every way.
7.3.3 Finite element methods
The finite element method (FEM) has already been introduced in Chapter 6 of this book and
used for the analysis of curved pipes conveying fluid, The reader is referred to the stand-
ard references on FEM cited therein. For shells in vacuo, much work has been done with
FEM, utilizing flat or curved, rectangular or triangular finite elements. Examples of the
early work may be found in Percy et al. (1965), Zienkiewicz & Cheung (1968), Olson &
Lindberg (1971) and Gladwell & Vijay (1975); see also Gould (1985) and Bathe (1996).
Nowadays, computer packages may also be used for this task, for example, NASTRAN,
ABAQUS, ANSYS, MARC, ADINA, S AP, MSC/ADVANCED FEA, MECHANICA.
Typically, hundreds or thousands of finite elements are involved in the analysis, and
hence large matrices and heavy computational cost. Consequently, more recently, sev-
eral attempts have been made to develop 'macroelements', hybrid techniques and other
procedures, tak:ing advantage af geometrical symmetries for instance, to alleviate the com-
putational task (Zienkiewicz et al. 1983; Meirovitch & Baruch 1983; Oliveira et al. 1986;
Gonr;alves & Ramos 1997). One of the earliest such hybrid methods was developed by
Lak:is & Pai'doussis (1971, 1972a) and used for the analysis of empty and partly filled
shells, as well as shells conveying fluid (Lakis 1976; Selmane & Lakis 1997b); it will be
presented in outline here.
7. 3. 3 (a) Lakis' hybrid ana/ytical/finite-e/ement method
The finite element chosen is a cylindrical frustum (a cylindrical segment of the
shell - see Figure 7.29), rather than the more usual triangular or rectangular plate ele-
ments. Accordingly, no geometrical modelling of the structure is necessary. Moreover,
this allows the use of the shell equations in full for the determination of the displacement
functions, instead of the commonly used polynomial displacement functions. For vertical
cylindrical shells filled with stagnant fluid, the shell is divided into its empty and fluid-
filled parts, and each is subdivided into a number of finite e1ements, We shal1 first present
the analysis for an empty finite element and then, in part (b) of this section, modify it for
an element filled with stagnant or flowing fluid,
The displacement functions are determined by Sanders' theory for thin cylindrical shells
(Budiansky & Sanders 1963), forthe following reason: in Sanders' theory all strains vanish
for small rigid-body motions, which is not true for some ofthe other theories; accordingly,
the displacement functions based on Sanders' theory satisfy the convergence criterion af

628
SLENDER STRUCTURES AND AXIAL FLOW
the finite-element method stating that strains within the element should be zero when the
nodal displacements are generated by rigid-body motions.
Sanders' equations for thin, isotropic, circular cylindrical shells are t
u" + i(1 - v)u" + io + v)v" + vw' + k [o - v)u"
-i(1 - v)v" + io - v)w'''J = O,
I ( 1 ) ,. .. I 1 ) " . k [ 3 1 ) ,. 9 ( 1 ) "
2 + v u + v + 2( - v v + w + -S( - v u + S - v v
.. I (3 ) ", . . . J O
+v - 2 - v w - w =
,. k [ I ( I ) ,.. I (3 ) ". + ... ""
- u - v - w + -2 - v u + 2 - v v v - w
2 II' . . . . . ] - O
- w -w -,
(7.106)
where, similarly to Section 7.2.1, here (r = o( )joe and ( )' = ao( )jox, a being
the mean radius; k = /2 (hja)2. It is noticed that these equations differ from F1tigge's
equations (7.1), only in tenns multiplied by k. The strain-displacement relations are
, 1 ( ' ) I,. "
Bx = u, BO = -;; v + W, BxO = 2a (v + u), Kx = -w ,
Ke =  2 (v' - w"), K (J =  ( -w" + v' - u. )
a x a 2 4 4 '
(7.107)
where 8 x , B(J, 8x(J are the axial, circumferential and shear strains of the middle surface,
and Kx, KO, KxO are the axial and circumferential bending and modified twisting strains,
respectively.
Suppose we have a cylindrical finite element defined by two nodes i and j and the
corresponding circular nodal boundaries (Figure 7.29). The object now is to express u, v
and w within the finite element in tenns of displacements at the nodes i and j, thereby
defining the displacement functions.
For motions associated with the nth circumferential wavenumber, we may write
{ u(x, e) } [ COS ne
w(x, e) = O
v(x,8) O
O
cos ne
O
O ] { Un (x) } { Un (x) }
O wn(x) = T wn(x) .
sin ne vn(x) vn (x)
(7,108)
By substituting equation (7.108) into equation (7.106) and letting
un(x) = Ae Ax / a , vn(x) = Bé x / a , Wn(x) = Ce Ax / a ,
(7,109)
we obtain three simultaneous algebraic equations in A, Band C. For non-trivial solu-
tion, the determinant oť their coefficients must vanish, leading to a characteristic equation
which is a quartic in A 2 . Each root, Ap, yields a solution of equations (7.106), the com-
plete solution being obtained by the sum of all eight and involving the constants Ap, Bp
and Cp, p = 1,2,.,.,8. Finally, as Ap, Bp and Cp are not independent, we write
t Anisotropic shells are discussed in Section 7.3.5,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
629
v
h
ÚL n ,
U n ;
(dwn/dx)
iX=Xj=O
I x
-----L-
j
W nj
x=xJ=1
j
[ Un, }
w n .
Oj = (dwn/dx)j
V nj
I
..,.-- -.....
N-I
".- -......
N
N
......--......
N+ 1
Figure 7.29 Definition of the finite element used, a cylindrica1 frustum of length t, and the
displacement vector associated with node i, li; (Lakis & Pai:doussis 1971).
Ap = cxpC p , Bp = f3 p C p , so that U n , Vn, W n may be expres sed in tenns of only eight
constants. After considerable manipulation we obtain
{ U(X'O) }
w(x, O) = TRC,
v(x,O)
(7.110)
where R is given in Appendix L.l and C is a vector of eight constants which are lin-
ear combinations of the Cp, The eight C p are the only free constants, which must be
determined from eight boundary conditions, four at each edge of the finite element.
We now express the nodal displacement vectors as follows:
8i = {U nj , w nj ' (dwn/dx)i, v nj } T .
(7.111 )
Each 8i may be determined from equation (7.110), where x in R now has a definite value,
x = O or x = 1, as the case may be; hence, we obtain
{: } = AC ,
(7.112)
where A is given in Appendix L.I, its elements being determined from those of R. Final1y,
combining equations (7.110) and (7.112), we obtain
{ U(X,O) ! { 8' } { 8' }
w(x, O) = TRA -I 8/' = N 8/' '
v(x,O) J J
(7.113)
which defines the displacement functions.

630
SLENDER STRUCTURES AND AXIAL FLOW
The strains are related to the displacements through equations (7,107); accordingly, we
may now expres s f in terms of (;i and (; j, and after lengthy manipulation we obtain
f= [ ]QA-l ft} =BU},
(7. 114a)
where Q is given in Appendix L.l. The corresponding stresses may be related to the strains
by the elasticity matrix P,
{ (;i }
CT = Pf = PB (; j ,
where P is also given in Appendix L.l for isotropic shells.
The mass and stiffness matrices, m and k, respectively, for one finite element are given
by finite-element theory (Zienkiewicz & Cheung 1968; Zienkiewicz & Taylor 1989), as
follows:
(7. 114b)
k = [1.' r BTpB dA] .od m = [P'h 1.' 1.'" NT N dA l
(7.115)
where Ps is the density of the shell material, h its thickness and dA a surface element. As
N and B have been determined by equations (7,113) and (7.114a), and P is known, k and
m may now be eva1uated,
The determination of k and m is achieved ana1ytically by carrying out the necessary
matrix operations and integrations over x and e in equation (7,115), To do this, it has
been found necessary to introduce several intermediate matrices, eventually obtaining
expressions for the general terms k pq and m pq of k and m, respectively. Because of
the complexity of the manipulations, neither the intermediate steps nor the final result are
given here, The interested reader is referred to Lakis (1971) or Lakis & Paidoussis (1972a)
for details.
If the shell is empty, ideally in vacuo, the next step is to construct the global mass and
stiffness matrices, and then proceed to ana1yze the free vibrations of the system. This model
is ideal for the analysis of nonuniform shells, provided that nonuniformities are axially
symmetric, for example, for shells with discontinuously varying thickness. Each uniform
part of the shell is subdivided into a number, N, of finite elements - a number which, as
will be seen, is relatively small (of order 10). The global mass and stiffness matrices, M
and K, of order 4(N + 1), may then be constructed from m and k, by realizing that, for
free vibration, the sum of forces and moments at each node must be zero, F j + Fi+l = O,
while the displacement must be continuous, (; j = (;i+l. Hence, for free vibration, the
equation of motion may be written in the form
MA + KA = O,
(7.116)
where A = {(; 1 , (;2, . . . , (; N + I} T, (; N + 1 being the displacement vector associated with the
lower edge of the last finite element; the overdot denotes a time derivative.
In cases where the shell is not completely free, the kinematic boundary conditions must
be taken into consideration. Thus, for a shell simply supported atx = O and x = L, we must
have Vn = W n = O in the displacement vectors (;1 and (;N+l. To account forthis, appropriate
rows and columns of equation (7,116) are deleted, reducing the matrix equation to one

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
631
of order 4(N + 1) - 4 in the above example. In general, the order of the system will be
4(N + 1) - J, where J is the number of kinematic boundary conditions imposed.
The solution of equation (7.116) proceeds via standard matrix techniques, yielding the
natural frequencies, S1j, i = 1,2, . . . , 4(N + 1) - J, and the corresponding eigenvectors.
The matrice s obtained in the foregoing analysis hold for n ::: 2. A sinůlar development
has been done to treat the axisymmetric (n = O) and beam-like (n = 1) modes by
Lakis & Sinno (1992), Shells with mu1tiple, equispaced supports are treated by
PaYdoussis & Lakis (1972).
7.3.3(b) Shells fi/led with flowing or stagnant fluid
Consider the general case of a compressible fluid flow in the shell. Perturbations in the flow
velocity associated with shell motions are governed by equation (7,69). Then, proceeding
as in Section 7.2.3, expressing
8
<I> = L Rj(r)Sj(x, O, t)
j=1
- cf. equation (7.39) - and applying the impermeability condition (7.35), the
perturbation pressure may be obtained, as follows:
(7,117)
" . [ O Wj ° Wj 20 Wj ]
P.  - ""  Zj,. (,m ja.) :,2 + 2U. a:a, + U. :x 2 '
where i 2 = -1, while subscript u = i for internal flow and u = e for external flow, and
(7.118)
ai
Z.. -
J,I - n -imjaj [Jn+l(imjaj)/Jn(imjaj)]'
a e
z. -
J,e - n - imja e [Yn+l(imjae)/Yn(imja e )] '
In and Y n being the nth order Bessel functions of first and second kind, respectively,
(7,119a)
(7. 119b)
(m ja u )2 = ).](1 - M 2 ) - 2), jM(S1a u /c) - (S1a u /c)2,
W j (X, O, t) = Cj exp(i).jX /au) cos nO exp(iS1t),
and M = U u/ c is the Mach number - cf. equations (7.61) and (7.62).
Introducing next equation (7.113) into (7.118), and performing the matrix operations
required by the finite element method, the inertia, damping (Coriolis) and stiffness (centri-
fugal acceleration) matrice s associated with the flowing fluid, mf, cf and k f, respectively,
are obtained by evaluating the following integral:
(7.119c)
(7.119d)
i NTpu dA,
where Pu is the vector of nodal perturbation pressures. Specifically,
mf = (A-1)TSFA- 1 , cf = (A -I)TDFA -1, kf = (A -I)TGFA -I, (7.121a)
(7.120)

632
SLENDER STRUCTURES AND AXIAL FLOW
and so the total mass, damping and stiffness matrices for an element of the shell are
mT=m-mf, cT=c-cf, kT=k-kf;t
(7.121b)
the matrices S F, D F and G F are given in Appendix L.2.
Hence, the new globa1 matrices may now be formed, leading to
MX + CÁ + KA = O,
(7.122)
which can be solved by standard matrix-eigenvalue techniques - see Section 2.1.2.
7.3.3(c) Extensions of the hybrid method
Several extensions of the hybrid analytica1/finite-element method have been undertaken
by Lakis and co-workers. These are presented here in outline, with few detai1s of the
analysis; readers are referred to the papers cited.
(i) Cylindrical shells partially filled with stagnant fluid: free-suiface effects This exten-
sion takes into account motions of the free surface, which are coupled with motions of
the shell; see Figure 7.30(a). In general, one can write pj p = -ofjJjot + gx + f(t)
- cf. equation (2.67b) - in which f(t) is later included in ofjJjot. Taking 1J(r, O, t) as
the height of the free surface, since the pressure is constant or zero at the surface, one
obtains 1J = (ljg)[ofjJjot]x=o, and hence [(o2fjJjot 2 ) + g(ofjJjox)]x=o = O on the free
surface. One proceeds to calculate the kinetic and potentia1 energies of a fluid element in
the course of vibration, and hence the fluid mass and stiffness matrice s , The analysis is
more complex than in the foregoing; the reader is referred to Lakis & Neagu (1997). It
ought to be mentioned that, in this analysis, the velocity potentia1 is directly dependent
on the wall displacement and does not contain additional degrees of freedom representing
sloshing modes. For a complete analysis, including the sloshing modes, see Balendra et al.
(1982) or Morand & Ohayon (1995), for example.
(U) Open cylindrical shells A general analysis for nonisotropic open cylindrical shells
[see Figure 7.30(b)] either filled with stagnant fluid or subjected to external and internal
flow, neglecting free-surface effects, is presented in Selmane & Lakis (1997a,b). The ana-
lysis is initially sirnilar to that in part (a) of this section, for example, equations (7.108)
and (7.109), but in this case we have u(x,O) = Aexp(1JO)cos(mJTxjL),v(x,O) =
B exp(1JO) sin(mJTx j L), w(x, O) = C exp(1JO) sin(mJTxj L), where 1J is complex. Thus,
in this case, m (instead of n) is taken as given a priori, and the eight adrnissible va1ues of
1J are determined via the characteristic equation referred to helow (7.109). It is interesting,
however, that the solutions that eventually emerge involve modal shapes which are of the
same type as for a complete shel!. In this case also, following the classica1 finite-lement
procedures, the requisite mas s and stiffness matrices are obtained for both the shell and
the fluid.
tNote the different sign convention vis-a-vis Section 2,2.2: e,g., here mT = m - mf, instead of
mT =m+mf.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
633
H
(a)
-'
-,'
(b)
Undisturbed
fluid surface
x=O
--r:-
Sl
Sl
Fluid
Si
-' ;//'"
,
"""'\:
:
Figure 7.30 (a) Sketch of a vertical, partially filled shell, studied for free-surface effects by
Lakis & Neagu (1997); (b) an open (incomplete) cylindrical shell subtended by the angle (/JT,
subjected to both internal and external ftow (Selmane & Lakis 1997b).
(žii) Conical shells conveying fluid A general analysis has been developed for conical
shells in vacuo,t or containing stagnant or flowing fluid. In this case the equations of a
conical shell are utilized and proces sed in a similar way as in part (a) of this section; see
Lakis & Ouriche (1986) and Lakis ef al. (1992). Although detail s of the general analysis
are not given here, the method for obtaining the fluid inertia, Coriolis and centrifugal force
matrice s is outlined in Appendix L.3, since they are of more general utility than for this
particular analysis.
Some typical results from these analyses are presented in part (d), along with those for
cylindrical shells developed in parts (a) and (b) ofthis section.
tFor c1assical work, see, for example, Fliigge (1960), Saunders et a/, (1960), Komecki (1967),

634
SLENDER STRUCTURES AND AXIAL FLOW
24
20
---
16
I=O,O 
, iJ}r-O,125in 
 95in
A-A B-B
N-
g
g 12
OJ
='
<::r
OJ
....
....
\
\
8 \
\
\
q
,
4
o
O
4
8
n
12
16
Figure 7.31 Natural frequencies uf a ring-stiffened cylindrical shell first studied by Weingarten
(hla = 1.98 x 10- 2 , Lla = 1.77; m = 1): ---, Weingarten's (1965) theory; o, Weingarten's
experimental data; - - -, Sewall & Naumann's (1968) theory; -, Lakis & Pa'idoussis's (1972a)
hybrid FEM theory (Lakis & Pa'idoussis 1972a).
7.3.3(d) Typica/ results
Several calculations for shells in vacuo, previously analyzed by others, are presented in
Lakis & Pa'idoussis (1972a), with the aim of validating this model: uniform and axial1y
discontinuous cylindrical shells, under various boundary conditions. One example, shown
in Figure 7.31, is for a ring-stiffened she11 first studied by Weingarten (1965) and then
by Sewall & Naumann (1968). The theoretical results by the hybrid FEM theory have
been obtained with 23 finite elements (N = 23), one for each stiffened and unstiffened
segment. !tis seen that the earlier calculations tend to overestimate the natural frequencies,
particularly for high n - presuming, of course, that the experimental data are accurate.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
635
The hybrid FEM theory, on the other hand, is appreciably more successful. CaIculations
for m = 2 and 3, not presented here, show similar trends.
In another, typical example, a cylindrical shell with a single wall-thickness discontinu-
ity is analyzed with N = 20. The results (not given here) are in good agreement with
Warburton & Al-Najafi's (1969) theoretical and experimental data.
Figure 7.32 shows some results for the eigenfrequencies of a truncated conical shell
in vacuo, compared to Saunders et al. (1960) theoretical and experimental values, for
varying n and m = 1 and 2. The agreement with the Saunders et al. theoretical values is
quite good, and with their experiments even better.
Good agreement having been obtained in many other cases, the hybrid model may
be considered to have been validated; attention is now focussed on fluid-filled or
fluid-conveying shells.
A typical set of caIculations for a vertical, partially filled steel shell are shown in
Figure 7.33; biL is the fraction of the shell which is water-filled. Excellent agreement
with the experimental data by Lindholm, Kaňa & Abramson (1962) has been obtained.
It may be seen that the effect of added mass can be very important: for example, for
n = 3, the natural frequency of the fully filled shell (biL = 1) is about three times lower
than that of the empty shell (biL = O) - cf. Section 7.2.2. As expected, in view of
equation (7.ll9a), this effect is diminished with increasing n.
An example of the effect of a free surface on the natural frequencies of a liquid-filled
shell, as analyzed by Lakis & Neagu (1997), is shown in Figure 7.34. It is seen that, when
free-surface effects are tak:en into account, the natural frequencies are diminished. This is
related to the additional kinetic energy associated with free-surface waves; on the other
0,5
10) 0.4


j 0.3
]
c
. 0.2
Š
'i3
=
c
:z: 0,1
o
o
5
10
15
20
25
30
Circumferential mode number, n
Figure 7.32 Nondimensiona1 frequency Q = QRdPs(1- v 2 IE)]1/2 for a truncated conica1
shel1, in vacuo, with cone ang1e etc = 45.1° extending from Xo to XL (where X is measured a10ng
the cone; see Figure L.l) where the radii are Ro and RL, with v = w = O at the boundaries,
and xLlxo = 2.25, hl RL = 0.00112, RL = 0.228 m,8 = hl1(xo + xd = 1.098 x 10-3,
v = 0.30, E = 200 MPa and Ps = 7, 870kg/m 3 : --0--, Lakisetal. (1992)theory;-, Saunders
et a/. (1960) theory; 0, Saunders et al. (1960) experiment (Lakis et al. 1992).

636
SLENDER STRUCTURES AND AXIAL FLOW
2,0 o
N
g 1.0
.'\
1.5 \
,\
'-. \ '." n = 7
 .-"_._11 / 8
-...."'......... n = 6
i ., '\. -.-! ,/ 11
-.......... "'___..........__ n = 2
....... .. .-. 
"' 0-__ I
..........  /' a
f____ n=5 A
, 1"
n=4
;>,
u
5
50

 0,5
0.75
0,50
'-.
..........
.........
......... n =3
---.-. 1/'
-
0,25
o
0,25
0,50
biL
0.75
1.0
Figure 7.33 Comparison of theoretical natural frequencies of a partially liquid-filled cylindrical
shell with experiment for m = 1 : e-e, Lakis & Paldoussis' hybrid FEM theory with 12 finite
elements, Experimental values (Lindholm et at. 1962): ., n = 2;. n = 3; +, n = 4; O, n = 5;
Ll, n = 6; 0, n = 7 (Lakis & Paldoussis 1971),
hand, the additional potential energy due to the waves is negligible when compared to
the strain energy of the shell. [Further considerations on this point are given by Amabi1i
(1997b) and Amabi1i et al. (1998a).] Clearly, such effects are more important for low
filling ratios, Hl L, and for shallow shells (smaller LID). It is also seen in Figure 7.32
that, in this particular example, free-surface effects become more pronounced for the
higher axial wavenumbers, m. The literature on partially filled shells, with or without
free-surface effects taken into account, is quite rich, but is not be discussed further in
this book. The reader is referred to the literature: for example, Yamaki et al. (1984),
Mistry & Menezes (1995), Morand & Ohayon (1995), Mazúch et al, (1996), Gonrralves &
Ramos (1996) and Amabili et al. (1998a).
Typical results for shells containing flowing fluid are shown in Figures 7.35 and 7.36.
It is seen in Figure 7.35(a) that, with 20 finite elements, agreement with Paidoussis &
Denise's (1972) results is not very good. With 30 finite elements, however, the agreement
is much improved, and for U  0.5 it is quite good, as shown in Figure 7,35(b).
ln Figure 7.36, again reasonable but not very good agreement with Weaver & Unny's
(1973) theoretical results is obtained. A reasonable explanation for this, insofar as coupled-
mode flutter is concerned, is that in Weaver & Unny's theory in-plane inertia is neglected,
and it is known (Section 7.3.2) that this leads to nonconservative estimates of stability;

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
637
5000
4000
>'-.
\. ''Q.
\, '''"
\. '''"
'Q ,
"",.", "o.......
..,....0.,.,. ......,......
g 3000
»
u
<::
<!)
g.
J: 2000
---:;(>-
.-:::<1'___-_
- ---....'.,.-....-,- ----::}---: --z..-,-.. -"-.-.. ": ,'.-.. -.. ":
."""'.""':.:::,=..  2. Ck.'''''''
--
....0.......,....
......
1000
m 1
°
I
0,25
I
0,50
H/L
I
0,75
1.00
Figure 7.34 The influence of free-surface effects on the natural frequencies of a vertical clamped-
free shell (a I h = 100, LI a = 3) partially fiUed with water to various levels [Hl L is defined in
Figure 7.30(a)] for n = 2, For each m, the upper curve is according to Lakis & PaYdoussis' (1971)
analysis, with 20 finite elements, neglecting free-surface effects; the lower curve is according to
Lakis & Neagu (1997), taking these effects into account; for m = 1 the two curves are virtually
indistinguishable (Lakis & Neagu 1997).
0,005
Values of U 
0.4 q.3 0,2 02
, / 0,0
0,000 02
0,5 0.3 0,0
0.4
0,6
-0,005 Paidoussis 0,5
1& & Denise
:f '\.
-0,010
0,7 0,7
/" Lakis
-0,015 0,6
-0,020
0,02 0,03 0,02 0.03
(a) (.Q) (b) (.Q)
Figure 7.35 Argand diagram of the dimensionless frequency, Q, as a function of the dimensionless
flow velocity, U , for the n = m = 2 mode of a cantilevered cylindrical shell (hla = 2.27 x
10- 2 , Lla = 12.9, v = 0.5, JL = 0.06), comparingPaYdoussis& Denise's (1972)theory andLakis'
hybrid FEM theory (a) with 20 finite elements and (b) with 30 finite elements (Lakis et a/. 2000).

638
35
30
25
20
:o

101
15
10
5
O
O
SLENDER STRUCTURES AND AXIAL FLOW
-m=1
--- m = 2
'....... m=1
....m=2
} Lakis
} Weaver & Unny
._-....-=.:..:....... .
,
''''''''': = 2
"
"
"
'''''''' :::e /; d_mOde ..,.,
" '
"
" .
m= 1 " I
, :
\ I
\ !
\ I
\ !
I /
"
.'
2
3
U w - u
5
6
4
Figure 7.36 The dimensionless frequency Qw-u versus tbe dimensionless flow velocity, U w-u,
for a simply supported circular cylindrical sheIJ (Lja = 2, aj h = 100, pj Ps = 0,128, n = 5),
comparing Weaver & Unny's (1973) theory and Lakis' hybrid FEM theory.
120

 /I' 
:o .A .A
IJ .A
80

'"
::s
C"
"
<.!:: 60
'"
'"
'"
'E
o
';;;
s= 40
"
S
'6
20
o
8
28
12
16
20
24
4
Dimensionless frequency, U w - u
Figure 7.37 The dimensionless frequency Qw-u versus tbe dimensionless flow velocity, Qw-u,
defined in Figure 7.17, for a c1amped-damped open cylindrical shell simultaneously subjected to
internal and external flow (Selmane & Lakis 1997b).

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
639
clearly, however, this cannot explain the discrepancy in the critical value for divergence,
which is quite large and remains unexplained.
A typical result for an open shell simultaneously subjected to internal and external flow
is shown in Figure 7.37. The dynamical behaviour is similar to that of comp1ete circular
cylindrical shells conveying fluid (cf. Figure 7.8, for instance). The eigenfrequencies
of each mode, real in the absence of dissipation, decrease and eventually vanish with
increasing flow velocity. In this case the critical mode is associated with n = 5 and,
unusually, m = 2; the critical flow velocity, (UW -U)cd ::::: 8.5, is much higher than the
corresponding value for a comp1ete shell with simply supported ends (cf. Figure 7.36). This
may at first appear surprising, because the open shell with free edges is certainly more pli-
ab1e than a complete circular one; in this case, however, the longitudinal edges are clamped,
making the shell considerably stiffer. Furthennore, the fluid is wholly unconfined, and the
severity of the fluid-stmcture interaction is similar for flowing and stagnant fluids: the
greater the confinement, the larger is the effect, and vice versa (see Sections 7.5, 11.3.5
and 11.7).
Unfortunately, no ca1culations are available for conical shells conveying fluid, despite
the fact that the theory is available.
It may be concluded that the hybrid finite element method is a valuab1e and computa-
tionally efficient tool for the analysis of shells filled with or conveying fluid. Of course,
an enonnous effort was involved in the analytical evaluation of the matrices - some of
which are given in Appendix L. However, the task having now been done once and for all,
this method is a comp1ete and valuab1e addition to the repertory of tools for analyzing such
systems. Furthennore, this method allows (with ease) the analysis ofaxially symmetric
nonuniform systems.
Extension of this hybrid analytical/finite-e1ement method for anisotropic shells is
discussed briefly in Section 7.3.5 and for nonlinear oscillations in Section 7.8.
Other hybrid finite-element models, applicable also to orthotropic and fluid-conveying
cylindrical shells, have recently been fonnulated by Zhang et al. (2001, 2002) and
Kochupillai et al. (2002).
7.3.4 Asymmetric support conditions
In all of the foregoing, shells with supported ends have been considered to be under the
same support condition at both ends: either clamped or simply supported. The dynamics
in cases of mixed support conditions, for examp1e, shells clamped at the upstream end and
simply supported at the downstream one ('clamped-pinned' for short) have been studied
carefully, apparently for the first time, by Horáček & Zolotarev (1983, 1984) and Zolotarev
(1987 a,b).
In the earlier ofthese papers, the analysis is conducted with Voľmir's 'semi-membrane'
shell theory, which is similar to Donnelľs and Morley's, thus involving one equation - cf.
equations (7.10) and (7.12) - but with most of the higher x -derivatives absent. In Zolotarev
(1987 a,b), the fuller Goldenveizer -N ovozhilov equations (Leissa 1973; Markuš 1988) are
used, which are similar to Fliigge's (7.1), but differ in some of the tenns involving k;
furthennore, in this case the ends are flexibly supported, with distributed translational
springs constraining u, vand w and rotational ones constraining w' at the shell ends. The
fluid is considered to be slightly compressib1e and the shell is generally pre-stressed via
external pressurization. The perturbation pressure is expressed in travelling wave fonn,

640 SLENDER STRUCTURES AND AXIAL FLOW
6
3
m=2
N 4
:: 2
x
1&
II 2

X
O O 19:
I 3 4 5
- V ed I V ef 
V I -2
I
I
I I
I -4
I "
V h J. 1,
..,.
, ------- ft -6
Figure 7.38 The real and imaginary parts of the eigenfrequencies of a pinned--clamped shdl
conveying inviscid fluid, for n = 6 and m = 1 and 2; L/a = 6.7, -tz(h/a)2 /(1- v 2 ) = 1.7 x 10- 7 ,
pressurization pa/Eh = 5,2 x 10- 6 , f.L = pa/psh = 3.2; fj = U(Ps/E)I/2, š1 = rla(ps/E)1/2:
-, Re(š1); - - -, Im(š1); (Horáček & Zolotarev 1983).
and the method of solution is similar to Pai"doussis & Denise' s four-A. (Horáček & Zolotarev
1983, 1984)t and eight-A. (Zolotarev 1987a,b) approximations - see Section 7.2.3.
Calculations with c1amped-damped, pinned-pinned and cantilevered shells show qual-
itatively similar behaviour to that obtained in all of the foregoing. It is the c1amped-pinned
and pinned-clamped shells, however, that display unusual behaviour, on two counts, as
discussed below.
7.3.4(a) Paradoxical dynamics
First, the results obtained by Horáček and Zolotarev for both the c1amped-pinned and
pinned-clamped shells suggest that they are nonconservative systems for all values of U.
Thus, as shown for example in Figure 7.38 for a pinned-clamped system (for n = 6 and
m = 1 and 2), Im( S"2 ) > O for all O < U < U c d, implying the existence of ftow-induced
damping for this conventionally conservative system, The two branches of the m = 1
mode re-coalesce and give rise to coupled-mode ftutter; for slightly higher U , this mode
combines with the m = 2 mode on the real axis.
The second notable feature in Horáček's and Zolotarev's calculations ensues from the
fact that, if the system is c1amped-pinned (corresponding to a pinned-clamped one with
U replaced by - U) , the Re (S"2) remain the same but Im (S"2) changes sign: the Im (S"2)
versus U diagram is the mirror image across the U - axis of what is seen in Figure 7.38.
Thus, in this case the system is predicted to be unstable by ftutter, in both the m = 1
tWhen using the Vol'mir shell theory, since ( )1111 is the highest x-derivative, and small argument approxim-
ations to the Bessel functions are used similar to (7.54), only four A are involved, and hence no truncation of
the infinite set of A as in Section 7.2,3 is necessary,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
641
and m = 2 modes, and to remain so for all U in the range O < U < U c d! Here the
reader is referred to the discussion on cantilevered pipes sucking rather than discharging
fluid in Section 4.3. In that light, the sign reversal inIm( Q ) is expected- see also Bolotin
(1963),'
On the other hand, according to calculations;by means of the Paidoussis et al, (1984a)
standing wave Fourier-transform theory the pinned-clamped and clamped-pinned shells
conveying fluid behave as conservative systems (Paidoussis et al. 1993b), Thus, the
variation of Re( Q ) and Im (Q) with U obtained by that theory is similar to that shown in
Figure 7,21. Hence, two questions arise: (a) why is the dynamics as predicted by the two
theories different, and (b) which of them is correct? The answers are given in part (c) of
this section, after the extent of this apparent paradox is explored further.
If the expression for the perturbation pressure, equation (7,64), is considered with
w (x, O, t) = w (x, t) cos nO, and then the work  W done over a cycle of periodic
oscillations evaluated - cf. equation (7.68) and Section 3.2 - one readily obtains
ÓW  -p1ra'I(n, A)U f [ (  )' + U e: ) ("d ) J: dto
(7.123)
Clearly, in the case of shells with supported ends, no matter whether clamped or pinned
at one or both ends, one must have
W=O,
(7.124)
since o w j o t = O at both extremities of the shell, ln this light, the system would appear to
be conservative in the same sense as a clamped-pinned or pinned-clamped pipe conveying
fluid, of the type considered in Chapter 3: that is, gyroscopic conservative, but with the
Coriolis forces not doing any net work over a cycle of periodic oscillations. FQr sufficiently
high flow, the system loses stability by divergence; but certainly, by these considerations
alone, it should neither gain nor lose energy to the flow via oscillatory motions prior to that.
On the other hand, unlike the pipe, deflection of which is described by w (x, t)
alone, the shell deformation involves u, vand w; furthermore, as seen in Section 7.3.3,
equation (7,111), owjox is quite important on its own, and generally both u ¥= O and
owjox ¥= O at a pinned end. t Hence, things may not be as simple as they appear, despite
the unequivocal conclusion from equation (7,124). Here the reader is also referred to the
discussion at the end of Section 1 0.2.2(b).
7,3.4(b) C/ues for resolution of the paradox
Probably motivated by the same questions regarding o w j ox ¥= O at a pinned end, Zolotarev
(1987a,b) studied the dynamics of a shell conveying fluid, with its downstream end suppor-
ted by springs and the upstream end clamped, as shown in Figure 7.39(a). By setting some
of the spring constants equal to zero, while others equal to infinity, it is possible to repro-
duce clamped (ki = 00, i = 1, ...,4), simply supported (kl = k4 = O, k2 = k3 = (0),
free, and other intermediate boundary conditions (cf. Section 3.6.2). For these latter cases,
tThe standard axial-constraint boundary condition for a pinned (simply supported) end is that N x = 0-
see equation (7.4a). An altemative condition, however, is u = O; see equation (7.256b) and the discussion
following it.

642
SLENDER STRUCTURES AND AXIAL FLOW
. .
l
I
1ťl
u
(a)
140
120
100
,.-.,
N
!; 80
'-.
60
40
20
O
O
(b)
k 1 =0,5
-I
---- ó
-=.::-=-- ---7----- =-------- --::--, ....,
k 1 =0 ,
50 100 150 200 250
U (mis)
12
10
8
,.-.,
'"
6 


4
2
O
-2
300
Figure 7.39 (a) A cylindrical shell clamped at the upstream end and spring-supported at the
downstream end; k" kz and k3 are the spring stiffnesses in the axial, circumferential and radial
directions, respectively, while k4 is the rotational spring stiffness associated with w ' . (b) Variation
of the n = 6, m = 1 eigenfrequency with U for kz = k3 = k4 = 00 and various values of kl, for a
shellwithL = 1 m, a = 0.15m, h = 0.2mm, p, = 2800kg/m 3 , v = 0.34, E = 7,2x 1Q 1O N/m z ,
conveying incompressible fluid with p = 12 kg/m 3 andexternal uniform pressurization P = 500 Pa
(Zolotarev 1987b).
the boundary conditions (7.3) and (7 .4a-c) need to be appropriately modified. Thus, instead
of (7.4a),
u'+v(v'+w)+kIU=O at x=L (7.125)
is used, in which the kw" term has been neglected for a very thin shell (k « 1). t Therefore,
if k2 = k3 = k4 = 00, so that v, w and w' are all zero, u must satisfy u' - klU = O, and
thus u(L, t) i= O generally. The results of such ca1culations are summarized as follows.
(i) If k2 = k3 = k4 = 00, the effect ofthe axial spring kl is as shown in Figure 7.39(b),
indicating that, since damping is negative (8 < O), the system is unstable by
single-mode ftutter for al1 U > O, provided kl i= 00. Thus axial movement at
/
tThe other boundary cODlhtions, modified slightly vis-a-vis (7.4b-d) to correspond to the theory used are:
w" + v(w" - v') + k4W' = o; w lll + (2 - v) [w ' " - Vi'] - k3W = O; u' + (1 + 4k)v ' - 4kw " + k2V = O,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
643
the downstream end, that is u(L, t) :f= O, is sufficiently destabi1izing, according to
these results, to induce flutter.
(ii) If k2 = k3 = 00 and kl = O, calculations with k4 = O and k4 = 00 show that
the effect of rotation at the end affects the results very little, and the dynamics of
the system is essentially as in Figure 7 .39(b). Therefore, for a simply supported
downstream end (kl = k4 = O, k2 = k3 = (0), the result already obtained is
retrieved: the system is unstable by flutter for infinitesimally small and higher U.
(iii) If kl = k2 = k4 = Obut k3 :f= O, the system is subject to positive damping (and
it is stable) for O < U < Ucd, showing that v-motions at the downstream end are
stabilizing.
(iv) Calculations for the cantilevered system (al1 kj = O) show the system to be damped
at small U and to eventually lose stability by divergence, instead of flutter as
in other calculations (for example, by Páidoussis & Denise, Shayo & Ellen and
Lakis; see, for example, figures 7,10 and 7.35), which constitutes a weakness of
this theory. Nevertheless, importantly, the values of Im(Q) for U < Ucd are
10 to 20 times larger in this case than for the clamped-pinned or pinned-clamped
systems ofFigures 7.38 and 7.39.
Thus, the original supposition that a nonvanishing slope (a w / ax :f= O) at a pinned
end may have been responsible for the paradoxical dynamics proved not to be decisive;
a nonvanishing u, however, appears to be more promising. The problem is that according
to all the theories uti1ized to compute the results, the perturbation pressure - the ooly
possible source of energy input - is a function of w alone, as seen in (7,123), Hence,
whether u (L, t) is zero or not should make no difference,
One final item of interest that ought to be pointed out is that the flow-induced
damping in the Horáček and Zolotarev results is extremely small: { "'" 0(10- 3 ) in Fig-
ures 7.38 and 7.39; whereas for cantilevered shells (e.g., Figure 7.10), { "'" 0(10- 1 ) to
{ "'" 0(1),
7.3.4(c) Resolution of the paradox
The paradox remained intact well into the 1990s, and it was while writing the precursor
to this section that the author thought that it ought to be resolved, if at all possible!
It was decided to start with a system, the dynamics of which is well-established:
clamped-pinned and pinned-clamped pipes (rather than shells) conveying fluid, the gen-
eral dynamics of which is described in Chapter 3; by conventional wisdom, these are
conservative systems. The initial set of computed results, obtained with J.-L. Lopes' help
showed that (a) the pinned-clamped system is indeed conservative (stable for U < Ucd),
but (b) the clamped-pinned one behaves as a nonconservative system, with Im(w) < O
for all U < UCd, which is similar to Horáček & Zolotarev's results! At this point it was
believed that, unexpectedly, an important discovery had been made - and this in 1998!
More careful investigation, however, showed that the rogue finding for clamped-pinned
pipes was due to insufficient accuracy in the determination of the eigenvalues and eigen-
+n';M ;... .h!  "rr('t to 'only' six significant figures); calculations with nine
significant-figure accuracy, showed the clamped-pinned system to be indeed conservative
and to behave identically as the pinned-clamped one. Interestingly, correctness to four
significant figures is quite sufficient to obtain the correct results for the pinned-clamped

644
SLENDER STRUCTURES AND AXIAL FLOW
system, but even eight significant-figure accuracy is not good enough for the clamped-
pinned one. Evidently, the form of the eigenfunctions makes the numerics very sensitive
to precision in one case but not the other. t
These results for the pipe problem suggested that perhaps the origination of para-
doxical dynamics in the shell prob1em could also be numerical rather than physical.
Consequently, a thorough reexamination of the problem was undertaken by Wong (2000),
The analytical model s of Pa'idoussis & Denise (1972), Pa'idoussis, Chan & Misra (l984a)
and Zolotarev (l987b) were computerized anew, and calculations with all three were
conducted for the rubber-air, steel-water and aluminium-air systems analyzed before
in these papers, with asymmetric support conditions (Misra, Wong & Paldoussis 2001).
The results appeared to display the same discrepancies between the theories as discussed
heretofore. Then, it was notked that Re (Q ) is typically three or four orders of magnitude
larger than Im( Q ), which is of V(l0-5); it was then realized that in the Pa'idoussis et al.
(1993b) calculations this had 'reasonably' been equated to zero! !f, however, the results are
replotted in expanded sca1e, then all three theories predict a similar dynamical behaviour:
flow-induced damping, positive or negative, for U < U cd.
Figure 7.40(a,b) shows typical results for the evolution of the eigenfrequencies Q as
functions of U for a clamped-pinned rubber shell conveying air. The Re( Q ) vary with U in
a similar way as for a clamped-clamped shell (see Figure 7,25, for instance); divergence
occurs at U cd :::: 0.57. It is seen that the values of Im( Q) , although very small, are
nonzero and negative; furthermore, just as in Figures 7.38 and 7.39, the variation with U is
smooth and hence looks quite physical! Similar results have been obtained for a clamped-
pinned steel shell conveying water. Moreover, similar results are obtained whether the
calculations are done by the trave1ling wave or the standing wave method (Section 7.3.1);
also, significantly, whether u = O or N x = O is tak:en at the pinned end.
Next, in view of the results for the pipe system discussed in this subsection, the shell
calculations were repeated with tightened tolerances and increased accuracy (i) in the
calculation of the beam eigenvalues, Am, and hence the eigenfunctions in the standing
wave analysis, and (ii) in the axial wavenumbers Aj in the trave1ling wave analysis, also
reducing the step-size and tightening the program tolerances, which were found to be
the most important sources of inaccuracy in that analysis. In the former case, calculation
of the eigenvalues to eight-significant-figure accuracy gives Im( Q ) :::: -2,2 x 10- 5
for n = 2, m = 1, U = 0.3; but by increasing their accuracy to 18 significant
figures this is reduced to -0.7 x 10- 5 . The decrease in the magnitude of Im( Q ) is
not very rapid but it is monotonic, suggesting that eventually, for high enough accuracy,
Im( Q ) -+ O.
Similar results are shown in Figure 7 .40( c), obtained via the trave1ling wave ana-
lysis. Here it is seen that Im (Q) may be reduced from -5 x 10- 5 to -0.6 x 10- 5
approximately for U = 0,75 by tightening tolerances from 'simple' double precision
to V(l0-8). Moreover, as shown in Figure 7.25(d), for a pinned-clamped shell and U
changed to -U, that is, for the same physical system, Im( Q ) = O:f: 10- 8 at 1east;
evidently, for the travelling wave analysis, the pinned-clamped system is numerically
more robust. Further details of these calculations may be found in Wong (2000) and
Misra et al. (2001),
tThis was confinned by further calculations with the flow velocity changed fcom U to -U; it was always
the clamped-pinned system that was the numerically sensitive one, irrespective of flow direction.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
645
5
4
----.
----.---_.... r- m=2
---_. ;t
._--..
--

x
15
i
2
.........
"....""
'-.
\
\ .
;-
I
I
I
3
o
(a)
 0 r==E
'i _0,1 )- -. - - -:-- -:-- -  - - -: -f- - - - - - -  I
o
 -0,2
X -0.4
I -0,6
 -0,8 (c)
-1.0
..,...----..- '" ...,.... , . ......... ... D" ,. '"
. .  . - . =1--:::-...-......_....
- t( - x..-x_-:::-_;;-.:-:-__-.
'-x-._ oJ(
....
1& 3(d)----<>-

O 0.1 0,2 0.3 0.4 0.5 0.6 0.7 0.8 0,9
Dimensionless flow velocity, U
Figure 7.40 (a) and (b) Complex frequencies of a clamped-pinned rubber shell (clamped upstream
and simply supported downstream) conveying air (aj h = 44.4, Lja = 25,8, f-t = 0.06; n = 2),
obtained by the standing wave Fourier transform analysis. (c) Variation of Tm(?2) with ff for
n = m = 2, for different accuracies in the axial wavenumbers and program tolerances involved
in the travelling-wave analysis: x, double precision; Á, long double precision; D, 10- 6 tolerance;
+, 10- 8 tolerance, (d) Variation of Tm(?2) with U for the same system as in (c), but recast as a
pinned-clamped system with reversed flow direction: <>, m = 1; _, m = 2 (Wong 2000).
Hence, based on these results and the energy transfer arguments leading to (7.124), it
is conc1uded that clamped-pinned and pinned-clamped shells conveying fluid are con-
servative systems, just as clamped-clamped and pinned-pinned one are. t The very small
damping, positive or negative, obtained in the numerical results for arbitrarily small flow
velocity is but a numerical artefact. The same conclusion is reached by Amabili & Garziera
(2002a) in a somewhat different way.
Thus, ingloriously, the controversy on the nonconservativeness of asymmetrically
supported shells collapsed like an imploding soufflé. However, the story is sufficiently
interesting to be recounted. In this regard, it is the fact that the numerical departures from
Im(Q) = O look so real that is responsible (Figures 7.38-7.40): they neither fluctuate
tThus, the conc1usion in Pai'doussis et a/. (1993) is upheld, aJthough in that paper no exhaustive study had
been undertaken on just how c10se Im(Q) is to zero,

646
SLENDER STRUCTURES AND AXIAL FLOW
randomly about zero, nor do they change sign. Instead, the variation of Lm ( r.! ) with U is
sufficiently smooth t to mimic physically plausible dynamical behaviour.
Additionally, it should be recalled that, in all the foregoing, structural damping was
taken to be zero. If even a small amount of dissipation is taken into account, however,
the numerical damping is easily overwhelmed (Horáček & Zolotarev 1983), since it is
so small. In one case, for a clamped-pinned shell, Wong (2000) shows that the inclusion
of mechanical damping as small as 0.12% of the critical is sufficient to nuIlify the effect
of numerical flow-induced negative damping, even if insufficient care is taken with the
numerics. Hence, from the practical point of view, even if it were real, the numerically
generated damping would be of minor significance.:j:
Finally, a last caveat: in a full-fledged nonlinear context, a clamped-pinned shell may
not generally be a conservative system.
7.3.5 Anisotropic shells containing flow
Many pipes and shells used in engineering applications are anisotropic; specifically, they
are macroscopically orthotropic (see Appendix M), for example, braided hose used widely
in aerospace appIications, filament-wound reinforced plastic piping, and so on. It is natural
to wonder how the dynamics of such anisotropic shells conveying fluid differs from that
of the isotropic ones discussed so far. With the current interest in composite materials, this
topic cannot but receive ever growing attention.
Some work on orthotropic cylindrical shells containing stagnant fluid (Jain 1974;
Karimbaer et al. 1974; Muggeridge & Buckley 1979b) was followed by studies on the
dynamics of anisotropic shells conveying fluid, by Dong & Muggeridge (1975), Bert &
Chen (1975a), Librescu (1975), Buckley et al. (1976), Muggeridge & Buckley (1979a),
Holm-Christensen (1990), Selmane & Lakis (1997a-c), and on wave propagation by
Bert & Chen (1975b).
Different type s of anisotropy have been examined. Perhaps the most general case is
considered by Dong & Muggeridge (1975), Buckley et al. (1976) and Muggeridge &
Buckley (1979a), in which the shell consists of N homogeneous laminae, each of arbit-
rary thickness, elastic properties and orientation of its principal axes relative to the
shell axes - see inset to Figure 7.41 - and generally 'unbalanced', that is, uneven,
ply (lamina) layout with respect to the neutral axis of the shell. Less general cases,
involving 'balanced' lamina layout are considered by the other researchers cited in the
foregoing; the Fliigge-type equations of motion for such orthotropic shells are given in
Appendix M.
Muggeridge & Buckley (1979b) consider glass-epoxy shells, presenting calculations for
eight-ply (at :1:45 0 and other orientations) shells, with Young's moduli E 11 = 51.7 GN 1m 2
and E22 = 24.1 GN/m 2 , in the axial and circumferential direction, respectively, shear
tl n some cases not quite as smoothly as in Figure 7.40(c), but smoothly enough to make one think that it
is the departures from smoothness that are due to numerical problems, rather than the overall effect! See also
next footnote.
*In fact, the issue has not quite been laid to rest, as discovered in November 2002. In a fresh set of
calculations, Horáček & Zolotarev (2002) claim to have proven that the asymmetric boundary-condition system
is nonconservative! Moreover, they present results showing significantly larger Im(Q) for very small L/a,
This suggests that the source of nonconservativeness or, as this writer believes, fallacious modelling is related
to conditions at the shell boundaries which, in their model at least, is systemic rather than numerical.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
647
modulus G12 = 8.66 GNjm 2 , major Poisson ratio V12 = 0.25, Ps = 2 x 10 3 kgjm 3 and
Ps j P = 2. The dynamical behaviour of the system is found to be qualitatively similar to
that of an isotropic shell, but with some quantitative differences, of course.
Calculations in which the wrap angle y is varied are shown in Pigure 7.41. 1t is seen
that, for the parameters involved, divergence in the n = 2 and n = 3 modes is almost
coincident for an axial wrap (y = O); but, as the wrap angle is increased, the critical
velocity is diminished, the lowest value being reached at y :::: 60 0 . Thus, according to this
study, axial fibre-reinforcing is the best for enhancing stability of the shell.
Prom the fundamental point of view, however, a more 'objective' comparison of
orthotropic versus isotropic shells is required, as pointed out and implemented by
4,00
3,50
3.00
n= 1
'"
o
- 2,50
x
*"
I
S n=5
'G 2,00
o
Q)
;.
:;o; n=4
o
""
.. n-3
, 1.50
.E
u
n=2
1.00
0,50
0,00
O
10
20
30 40 50 60
Orientation, y (deg)
70
80
90
Figure 7.41 The dimensionless critical flow velocity for divergence U; d = Ued(E 11 I Ps )-1/2
versus the ply (lamina) orientation angle y for a simply supported, eight-ply orthotropic shell
(Lla = 10, hla = 0,01); y = ° corresponds to an axial wrap, and y = 90° to a hoop wrap of
the laminae; U * = 0.01 corresponds to U = 51 mis (Muggeridge & Buckley 1979a). In the
inset, (x, e, r) are the shell axes, and (1, 2, 3) are the principal axes of one of the laminae
(Buckley et al. 1976).

648
SLENDER STRUCTURES AND AXIAL FLOW
Ho1m-Christensen (1990) via the concept of 'comparab1e constitutions'. In this regard,
it is obvious that the critical ve10city increases as the stiffness of the shel1 is increased;
however, there are more ways and senses than one, in which an orthotropic shell can be
made stiffer. Hence, for a proper comparison of different orthotropic shells to an iso-
tropic one, an intrinsic parameter has to be emp10yed which wou1d remain invariant; the
parameter chosen is the stored e1astic energy. For an orthotropic shel1 with a stress-strain
re1ationship as in equation (M.1) and Q16 = Q26 = O, it is found that shells with equal
e1astic energy have to satisfy the following constraint relationships:
Q12 = 1 - (1 - VO)Q22,
- (3 - 2VO)Q22 - (1 - vO)2Q2 - 1
Q66 = -
2(1 + vo) Q22
(7.126)
in which Qij Qij/Ql1, and Vo is the Poisson ratio of the comparison material. In
addition, the stiffness matrix needs to be positive definite, which defines the admissib1e
interval for Q22:
(3 - 2vo) - ./ 5 - 4vo (3 - 2vo) + ./ 5 - 4vo
< Q22 <
2(1 - vO)2 2(1 - vo)2
(7.127)
The most important result obtained is shown in Figure 7.42, in which fj =
U(Ps/Qll)1/2[(1 - vo)/(2Q66)]1/2. It is seen that al1 three curves, corresponding to
different vo, have minima at Q22 = 1, which, it is recalled (Appendix M), corresponds to
1.0
0,8
"
I::;'"
0.6
0.4
0,2
0,0
0.0
1.4
1.2
-vo=O,3
_______o Vo = 0.4
- - - V o = 0.5
.
.
,
.
.
.
.
.
.
.
,
.
I
"
"
...
"
"
.'
"
_.--
...
I
I
,
/
'"
..
..
I
I
I
I
\
. . ...
'-......::...:;.::.... -
...
...
...
...,
......
10
3,0
4.0
Q22
5.0
70
8,0
2,0
6,0
Figure 7.42 The dimensionless critical flow velocity for divergence, Ucd, defined in the text, as
the orthotropy is changed by varying Q22, subject to the constraints in relations (7.126) and (7.127),
for a shell with Lla = 25,9, hla = 2.27 x 10- 2 and f.L = 0.06 (Holm-Christensen 1990),

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
649
an isotropic shell. Hence, surprisingly perhaps, it is shown that 'almost any conceivable
orthotropic material is superior to its isotropic counterparť! Actually, the highest critical
speeds are controlled by the limits in (7.127), and that is why only a part of the curves is
shown. It is also noted that, although this work is fundamentally important and interesting,
Muggeridge & Buckley's (1979a) comparison, though not 'fair', is more meaningful from
the practical viewpoint.
Bert & Chen (1975a) also studied the dynamics of anisotropic shells conveying fluid, as
a preamble to considering wave propagation in such shells and the dynamics of composite
shells containing swir1ing flow, discussed in Sections 7.4 and 7.6, respectively.
Finally, Selmane & Lakis (1997a,b) examine the dynamics of open cylindrical shells
immersed in stagnant or axially flowing fluid by means of their hybrid finite-element
method (Section 7.3.3), but the effect of orthotropy is discussed only for stagnant fluids.
The natural frequencies are compared with those obtained by Ramachandran (1979) via a
Rayleigh-Ritz procedure, A1though fair1y good agreement is obtained for isotropic shells
(within 10-15%), large differences (sometimes of the order of 100%) are inexplicably
found for anisotropic shells. The interested reader is also referred to the recent contributions
by Lakis & Selmane (2000), Toorani & Lakis (2001) and Zhang ct al. (2001, 2002).
7.3.6 Other work
No treatment of the dynamics of shells containing flowing fluid would be complete
without reference to the extensive work on compressible, mainly supersonic, external
fluid flows, for example, by Bolotin (1956, 1963), Leonard & Hedgepeth (1957), Miles
(1957, 1958), Stearman ét al. (1962), Librescu (1965, 1967, 1975), Dowell (1966b, 1970),
Dowell & Widnall (1966a,b), Brusilovskii ct al. (1966), Fung (1966), Johns (1966),
Olson & Fung (1966, 1967), Fahy & Pretlove (1967), Widnall & Dowell (1967), Parthan
& Johns (1969), Matsuzaki & Kobayashi (1971), Matsuzaki (1973), Kornecki (1971a,b,
1978, 1979), Horn ct al. (1974), and Mizoguchi & Komori (1978),t However, since this
subject is covered thoroughly in well-established and respected books, cited in Section 7,1,
it is not attempted to cover it here also, the emphasis in this book being heavily on incom-
pressible flows; if compressibility is taken into account at all, then the flow is always
subsonic. The serious student should nevertheless consult all these references, as well as
those cited in Section 7.8.1, if further research, even on incompressible fluid flows, is
contemplated,
ln most cases, the same researchers who have made key contributions on the aero-
elasticity of shells subjected to compressible, mainly supersonic external flows, consider
subsonic flows also, and in a few cases even incompressible flows - although, for the
reasons given in Section 7.1, devote much less attention to them. Examples are Kornecki's
(1974) and Matsuzaki & Fung's (1977a,b) work.
7.3,6(a) Proof that U e ! > Ued
Kornecki (1974) considers, in a generic way, the aeroelasticity of plates (flat panels) and
shells with internal or external, subsonic or supersonic flow. If E (w) is the structural
tResearch on this subject peaked in the 1960s and 1970s, with 1966 being truly prodigious, That is the main
reason for so many 'olď references being cited; they are the sellŮnal ones,

650
SLENDER STRUCTURES AND AXIAL FLOW
(self-adjoint, positive definite) operator and w the dimensionless defiection, one readily
obtains [cf, the foregoing in general and equations (7.13b) and (7.64) in particular]
E(w) - PshLQ2[w =r= /7M(w)] ::I: pU 2 [q(w) + 2iKr(W)] = O,
(7,128)
where /7 = Mjb = [paj(psh)]j(ajL) = pLjPsh, K = QLjU, M(w) is the effective
added mas s operator, while q (w) is related to stiffness and r (w) to the Coriolis forces.
The ::I: differentiates external from internal fiow, with the /ower sign (+ or -) here and
esewhere being associated with internal fiow. Expressing w = Ll a} iJ (), where the
fJ () are an orthononnal set satisfying
1 1 !m()[!}() =r= /7M(!})] d = Dm},
Emj ::I: pU 2 (qmj + 2iKjlmj) - P s hLQ2 Dmj = O,
Dmj being Kronecker's delta, application of Galerkin's method to equation (7.128) leads
to
where
Emj = 1 1 !m()Ej(iJ(» d,
(jmj = _ {OO a2P(li)Cmj(li)da,
Jr L 10
jlmj =  (OO P(a)Smj(a)da,
Jr L 10
Fe N _ _ In(Aa) 2 2 2 1/2
...)- AaI(Aa) ' Aa=[a -M (K-a)] (ajL),
Cmj(a) = 1 1 1 1 !m()f}(u) cos[a(u - )] du d,
Smj(a) = 1 1 1 1 !m()!j(U) sin[a(u - )] du d
(7.129)
(7. BOa)
(7. BOb)
(7. 130c)
(7. BOd)
(7. BOe)
(7.130f)
for a shell with interna! subsonic fiow, a being the Fourier transform variable -
cf, equations (7.74) and (7.82b). Different, but similar, expressions to (7.130b-7.130f)
apply to external fiow and fiow over fiat panels.
For non-trivial solution,
- . - 2
det[Lmj ::I:IQV m j - Q Dmj] = O,
(7. 13la)
- - 2 -
Lmj = (Emj::l: pU qmj)j(PshL), Vmj = (2/7UjL)jImj, (7.13 lb)
where, cf. Section 3.4,1, jlmj = -jljm and jljj = O. Hence, taking a two-mode
approximation,
2 1 - - -2 - - -2 2 - - -2 1/2
Ql,2 = Z{Lll + L22 + V 12 =r= [(L11 + L22 + V 12 ) - 4(L11 L 22 - L 12 )] }. (7.132)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
651
From this equation, it is realized that divergence, U = Ued, occurs at
- - - -2
/},,(Ued) = L11L22 - L 12 = O,
(7.133)
implying that, at that point, i 11 and i 22 > O [they cannot be negative, as argued below
equation (7 .134b )], On the other hand, flutter occurs when the radical in (7.132) vanishes;
that is, at Uef'
- - -2 2 - - -2
(L11 + L22 + V 12 ) - 4(L11 L 22 - L 12 ) = O,
(7. 134a)
or in another form
- - 2 -2 - - -2 -2
(Lu - L22) + V 12 [2(L11 + L22) + VnJ + 4L 12 = O,
(7 .134b)
Hence, it is easy to see that (7. 1 34a,b) can be satisfied only when both i 11 and i 22 are
negative, Now, since for internal flow the i ii (U) are continuously decreasing functions of
U and i ii (O) >:. O since the system is structuraly stable, at some flow ve10city U o < U cf
just one of the Li i functions must vanish, say LIl; thus,
- -2
/},,(Uo) = -L 12 S O.
Therefore, following the progression of Zi(U) from U = O toward Ued and Uo, it is clear
that Uo :::: Ued, and therefore
Ued S Uo < Uef. (7.135)
Hence, for subsonic internal or externa1 flow, the system loses stability by divergence and
it is a1so, theoretically at 1east, subject to post-divergence flutter. A more complete proof
may be found in Kornecki (1974), As already discussed, the observed flutter in internal
flow may be a manifestation of dynamic divergence (Section 7.2.7), while in external flow
flutter has never been observed,
Working in a similar way, one can prove that, for supersonic flow over a flat panel, it
is impossible for divergence to occur, and the system loses stability by flutter (Bolotin
1963; Dowell 1975), while for supersonic flow over a she1l no general conclusions may
be reached (Kornecki 1974), although flutter very clearly does occur.
7.3.6(b) Post-divergence stability range
As discussed in Section 3.4.1 (see Figure 3.12), pipes with supported ends and small f3
(small fluid-to-total mas s ratio) are predicted to lose stability by divergence and then to
be subject direct1y to post-divergence coupled-mode flutter, of the so-ca1led PaYdoussis-
type. On the other hand, pipes with larger f3, after losing stability by divergence, become
restabilized for a range of U, before they too become subject to coupled-mode flutter, in
this case via a Hamiltonian Hopf bifurcation, However, as shown in Section 5.5.2, if
dissipation is taken into account the restabilization is rendered impossible in all cases;
moreover, the nonoccurrence of restabilization has been shown to be one of the key
e1ements in proving the nonexistence of post-divergence flutter in pipes by nonlinear
theory.
Matsuzaki & Fung (1977 a) show the nonexistence of restabi1ization in the presence of
dissipation for incompressible ideal flow in shells. They utilize Morley's shell equation

652
SLENDER STRUCTURES AND AXIAL FLOW
(Section 7.2.1) and the standard Galerkin, Fourier transform type of solution as in the
foregoing. They show that for O < /l < /lc the system does not regain stability after
divergence, before the onset of coupled-mode flutter, where /l = /-Ljb = pLj Psh. For
/l > /lc' however, there is restabilization if dissipation is neglected - cf, the difference
between Figures 7,17 and 7,7, In contrast, if dissipation is taken into account the restabil-
ization zone vanishes, as it does for pipes. This matter is also discussed in Section 7.2.7;
see Figure 7.25.
ln a companion paper, Matsuzaki & Fung (1977b) consider the nonlinear dynamics of
two-dimensional channels, grosso modo modelling shells conveying fluid. A part of the
upper and lower plates is flexible, either straight or buckled, t The equation of motion is
remarkably similar to that utilized by Holmes (1977, 1978) for the nonlinear analysis of
pipes conveying fluid, equation (5.82), considered in greater detail also in Chapter 10:
the only nonlinearity taken into account is related to stretching of the plate due to lateral
deflection. The conc1usions reached are remarkably similar to Holmes' (see Section 5.5.2):
(i) the equilibrium points associated with first-mode divergence are stable, but not those
emerging from second-mode divergence; (ii) no steady-state post-divergence flutter is
found to exist, though transient flutter may arise for small amplitudes of motion. This
tends to reinforce the arguments in Section 7.2.7 that the observed flutter is most likely a
manifestation of dynamic divergence.
7.3.6(c) Nonuniform supports
The effect of circumferentially nonuniform supports - e.g. a simple support with a few
rivets - and also the effect of an elastic foundation have been treated by Amabili &
Garziera (2000a, 2002a,b) for shells in contact with quiescent or flowing fluid.
7.4 WAVE PROPAGATION AND ACOUSTIC COUPLlNG
7.4.1 Introductory comments
This is a venerable subject, with early work dating back to the second half of the 19th
century, by Helmholtz, Korteweg, Lord Rayleigh, Résal, Kundt, Lamb, and others; see,
for example, Lamb (1898) who considers the problem of 'the velocity of sound in a
tube, as affected by the elasticity of the walls', thus involving both wave propagation and
coupling between motions of the compressible fluid and the elastic tube wall. In general,
in the subsequent literature one may find work mainly concerned with wave propagation
in the fluid as affected by the deformability of the conduit,* or work concerned with
vibration and wave propagation in the tube (shell) as affected by the presence of internal
or external, quiescent or flowing fluid, Examples of the more recent, early literature are
papers by Thomson (1951), Morgan & Kiely (1954), Womersley (1955), Junger (1955),
tThis physical system in a way bridges the gap between internal and external flow problems; see also
Matsuzaki (1981), Section 7.9 and Chapters 10 and 11.
*With the current keen interest in bioengineering applications (hemodynamics, pulmonary and urinary flow
systems), one should distinguish between (i) systems of small defonnability, where defonnation is associated
with stretching and flexure of the conduit, and (ii) systems of large refonnability ('collapsible' tubing), where
defonnation also involves substantial changes in the fluid-filled cross-sectional area; these changes in area are
similar to changes in density, and hence to compressibility - refer to Section 7.9,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
653
Lin & Morgan (I 956b), Skalak (1956), Warburton (196I) and Dragos (1962),t Examples
of more recent papers on the subject are by Mizoguchi (1967), Rubinow & KeIIer (197 I),
HareI & Perulli (197I), Levin et al. (197I), Deng & Popelar (1972), Chen & Rosenberg
(1974), Bert & Chen (1975b), Berger & PaImer (1977), Firth (1977), Wilkinson (1978),
Mizoguchi & Komori (1978), Cummings & Brandt (1979), BuII & Norton (1980), FuIIer &
Fahy (1982), Junger & Garrelick (1984), Junger & Feit (1986), Horáček & ZoIotarev
(1981, 199Ia), ZoIotarev & Kruntcheva (1995), Liang & Scarton (1994, 1996), Bérot &
Peseux (1998) and Cirovic et al. (2002).
The field has expanded to cover many technologicaIly and/or scientificalIy important
subjects, such as propagation of surges in tubes, waterhammer, pressure waves in highly
deformable, coIlapsible tubes with bioengineering applications (see Section 7.9), and
the effect of a flexible waII on hydrodynamic stability (see Appendix C). Most of these
subjects per se are beyond the scope ofthis book. However, some further and more recent
references may be usefuI to readers who are interested in these subjects and are given here
without commentary. They are examples of the type of work that has been done, and do
not represent a complete bibliography, by any means.
(i) Unsteady fluid-structure interaction phenomena, pressure waves and acousticaI
effects, incIuding waterhammer: Streeter & Wylie (1974), Wylie & Streeter (1978),
Pa'jdoussis (1980), Gibert (1988), Moody (1990), Tijsseling (1996).
(ii) Large amplitude waves, surges, decompression transients in nucIear reactors:
Sarpkaya (1961), ForrestaI & Herrmann (1965), Olsen & Shapiro (1967), Kivity &
CoIIins (1974), Schumann (1979, 1980), Buggisch (1982), Cowley (1982).
(iii) Pressure waves in bioengineering systems: see Section 7.9 and Appendix P.
(iv) Effect of flexible boundary on hydrodynamic stability, compIiant coatings:
Benjamin (1960, 1963), LandahI (1962), Carpenter & Garrad (1985, 1986),
Gad-eI-Hak (1986, 1987, 2002), Yeo & Dowling (1987), Riley et al. (1988),
Carpenter & Morris (1990), Lucey et al. (1997), Peake (1997), Reutov &
Ribushkina (1998), Carpenter et al, (2000, 2001).
(v) Gravity-related pressure wave propagation: Patdoussis (1965a), Atta & Gibert
(1986), AHa et al. (1986), Fujita et al. (1992, 1996), Capodanno (1995), Bauer &
Komatsu (1994, 1998), Tanaka et al. (2000).
ln what foIIows, a smaII sample of this vast literature is presented - the part that is
physicalIy closest to the material in Sections 7.2 and 7.3, and which gives a coherent over-
view of the topic - based mainly on the work of Chen & Rosenberg (1974), Lin & Morgan
(1956b) and Horáček & ZoIotarev (1991a), involving propagation of smaII disturbances
in not very pIiable circular cylinders of essentiaIly infinite length. The fluid is considered
to be inviscid but compressible, since compressibiIity and hence acoustic coupIing makes
the resuIts particularly interesting and the treatment more reaIistic.
The effect of compressibility on the dynarnics of the sheIl is generaIly smaII, as shown
for instance (i) by Niordson (1953) for the effect on the eigenfrequencies at U = O,
(i i) by Chan (1984) on the effect of the eigenfrequencies for ° < U < U c d, and (iii) by
Horáček & ZoIotarev (198 I, 1984) an d Pa'jdoussis, Chan & Misra (19 84a) for the crit-
ical flow velocities for divergence, U cd, and coupled-mode flutter, U cf. For example,
t One of the reasons for citing some quite old references is that they are rife with insightful physical inter-
pretations of the dynamical behaviour. As in so many other topics, researchers in 'the old days', deprived of
computational pyrotechnics, were actually forced to think much harder!

654
SLENDER STRUCTURES AND AXIAL FLOW
for a clamped-clamped shell, U c d is lowered by only 0.7% when the Mach number is
increased from M = O to M = 0.5 (Pai'doussis et aJ, I 984a), Horáček & Zolotarev (1981,
1991a) find this increase to be only 0.1 % in one case, and 3.7% in another, purposely
chosen - the latter more substantial, but stilI small.
If, however, there is strong acoustic coupling, that is, if the frequencies of acoustic waves
and shell eigenfrequencies are close or coincident, the effect on both can be important.
Finally, before presenting the work on wave propagation and acoustic coupling, it is
recalled that the basics and concepts of wave propagation may be found in Section 3.7,
presented with the aid of the simpler problem of flexural waves (n = 1) in thick-walled
pipes, modelled as strings or beams, conveying fluid, t
7.4.2 Wave propagation
Consider an infinitely long, thin circular shell filled with generally flowing inviscid, com-
pressible fluid. Chen & Rosenberg (1974) utilize Fliigge's equations to describe motions
of the shell, while Horáček & Zolotarev (1991a) use Vlasov's semi-membrane, or half-
bending, theory (Novozhilov 1964), which is simpler since it involves only a single
equation (in tenns of the radial displacement), or a1ternatively and more genera11y the
three-equation set of Goldenveizer-Novozhilov.
ln general, one may write
{ U } { U cos ne }
v = v sinne e i (r2t-ax),
w wcosne
(7.136)
and for the velocity potential associated with fluid motions, cf. equation (7.69),
<I> = R(r) cos ne e i (r2t-ax), (7.137)
where a is the wavenumber and Q the frequency of the propagating disturbance; hence,
the phase velocity V p = Q/a. Note that the exponent in (7.136) and (7.137) may be
written in several equivalent ways; thus i(Qt - ax) = ia(Vpt -x) = i(2JT /Je)(Vpt - x),
where Je is the wavelength.
Solutions of equation (7.69) with (7.137) are of the fonn
R n (f3a) ( () () ) 2
P = -pa f3R(f3a) ar + U ()x w, (7.138)
13 = la 2 - (Q_aU)2/c 2 11/2, (7.139)
U being the flow velocity, p the fluid density, a the shell radius, and c the speed of sound;
Rn(f3a) = In(f3a)
Rn(f3a) = J n (f3a)
for 13 2 > O,
for 13 2 < O,
(7.140)
the latter being feasible only for c #- 00, J n and ln being the Bessel and modified Bessel
functions of the first kind. Correspondingly, if the flow were external instead of internal,
tRefer also to de Langre & Ouvrard (1999) and Doaré & de Langre (2002a),

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
655
Rn (fJa) = Kn (fJa) and Y n (fJa), Y n and Kn being the Bessel and modified Bessel functions
of the second kind. Then, substituting into equation (7.29), for instance, one obtains three
homogeneous transcendental equations, the determinant of which yields the dispersion
relationship between a and Q. In the proces s, the following dimensionless parameters
are defined:
K = aa,
_ [ Ps O - V 2 ) ] 1/2
Q = Qa ,
E
_ [ PsO_V 2 ) ] 1/2
u=u ,
E
_ Q _ [ Ps O - V 2 ) ] 1/2
v p - - - V p
K E
t [ E ] 1/2
l' = -;; Ps O _ v2)
_ x
x = -, (7.141)
a
c = c [ ps O ;- v 2 ) f/2
pa
j.L = -,
Psh
k =  ( !!. ) 2
12 a
where c is the dimensionless speed of sound in the fluid, and the other symbols as in
Section 7.2. Hence, the dispersion relationship takes the form
F(k, v, n, j.L, U, c, Q , K) = 0,
(7.142)
and the pertinent exponent in equations (7.136) and (7.137) becomes
eÍ(?h--KX) == eiK(vp-r-X).
(7.143)
Some typical results from Chen & Rosenberg (1974) forn = ° are shown in Figure7.43
for (a) the empty shell, and (b) the shell filled with quiescent fluid. It is clear, at a glance, that
wave propagation in both the empty and fluid-filled shell is generally strongly frequency-
dispersive; that is, the wave or phase velocity depends on the frequency, ln general, for
any n > 0, the empty shell has three frequencies for each K, corresponding to predomi-
nantly axia1, mainly circumferential, and predominantly radial motions; this is because
the dispersive relation, being free of the Bessel functions, is simply a cubic in Q 2 -
see equation (7.7). For n = 0, however, the circumferential equation of motion becomes
decoupled, since O' == 80j8e = 0, and hence purely circumfere ntia l ('torsional', T)
motions become possible, govemed by a linear relationship between Q and K; the u- and
w-components are coupled, yielding a biquadratic in Q , and hence the two other solutions
shown in Figure 7.43(a). _
For the fluid-filled shell, the dispersion relation is transcendental in Q and hence gives
rise to an innumerable number of modes, as shown in Figure 7.43(b); only the torsional
mode remains unaffected by the presence of the fluid, since viscous effects have been
neglected. The other modes are identified as Mnm, n being the circumferentia1 wavenumber
and m the order of the mode.
ln view of equation (7.139) for U = 0, the same Q is associated with both +a
and -a (and hence ::!::K), and only the positive values are shown in the figure. The pecu-
liar, 'segmented' form of the Q versus K relationship in Figure 7.43(b) will be clarified
shortly - in conjunction with Figure 7,50 et seq.
In Figure 7.43(a), excluding the T-mode which is purely circumferentia1, the other
low- Q mode is predominantly axia1 for K < I, whi1e the high- Q mode is predominantly

656
SLENDER STRUCTURES AND AXIAL FLOW
4.0
3,5
3.0
IOJ
 2,5
<.>
<=:
"
g.
"
<t:
 2.0
"
"2
Q
'"
<=:
"
8 1.5
1.0
0.5
o
2
3
Dimensionless wavenumber, K
(a)
4
4
Dimensionless wavenumber, K
(b)
Figure 7.43 Dimensionless eigenfrequencies Q versus the dimensionless wavenumber K for
n = ° in an (a) empty and (b) water-filled steel shell; hla = 0,01, J-L = 12,8 and c = 0.257; C r ,
defined in equation (7.144), is approximately equal to 0.3 (Chen & Rosenberg 1974).
radial (the only one with non-zem Q for K --+ O), For K > 1, however, the low- Q mode is
principally radial, while the high- Q one mainly axia1. Thus, a 'mode exchange' via 'mode
veering' takes place in the vicinity of K = 1 (cf. Chapter 3).
Figure 7,44 is helpful in clarifying the physical nature of the modes for the fluid-
filled shell in Figure 7.43(b). Thus, for example, M02 is a predominantly axial mode
for K < I, and radial for K > l, approximately. Similarly, Mm is predominantly axial
for 0.5 < K < 1.6, and radial otherwise. Thus, the slanted, nearly straight segments of
the curves in Figure 7.43(b), which can be thought of as the remnants of the straight-
line component of the u-dominated uppennost Q curve in Figure 7.43(a) at high K, are
associated with axial motion.
Similar results for the same system but with U = 0.03 are shown in Figure 7.45, The
diagram is no longer symmetric across K = O. Hence, the phase velocity of downstream-
trave1ling waves (K > O) is greater than for upstream-travelling waves (K < O); moreover,
the group velocity, v g = d Q /dK, which is the velocity of energy transport associated with
wave propagation, is also different.

657
THIN SHELLS CONTAINING OR IMMERSED IN FLOW
 2
Ó
.
 O
'"
.
P..
 -2
10
8
I
I
I
I
I
I
I MQ3
I)
6
4
-4
.,.-- ....
MOl /" (
I '
/ I
, M 02 I M03
I I
I I
I I
I I
I I
I ,
,
-6
-8
-10
O
Z
K
3
4
Figure 7.44 Amplitude ratio, Ii/UJ, with li and ID defined in equation (7,136), for the system of
Figure 7 .43(b), showing whether a mode is predominantly axial or radial (Chen & Rosenberg 1974),
Finally, in both Figures 7 .43(b) and 7.45 it is noted that, other than the T-mode,
only two modes (MOl and Moz) propagate at small Q . For each of the other modes,
there is a cut-off frequency, corresponding to the intercept with the ordinate; this is the
frequency below which the mode is non-propagating, in the sense that the mode cannot
propagate without attenuation.
These results, for n = O, are very similar to those obtained by Lin & Morgan (1956b),
who consider a number of limiting cases, as well as physical interpretation of the results,
using a simplified shdl theory for n = O (Lin & Morgan 1956a). They find that, gen-
erally, wave propagation is govemed principally by three parameters: the mass ratio
I-L = (pa)/(psh), the Poisson ratio v, and
c { KJlp } l/Z
C r = C s = E /[Ps(l - v 2 )] ,
(7.144)
which is the ratio of the speed of sound in the fluid, c, to that of longitudinal waves
in an infinite elastic solid, cs, with Kj being the bulk modulu s of elasticity of the
fluid - equivalent to c in (7.141). For small frequencies, Lin & Morgan obtain asymptotic

658
SLENDER STRUCTURES AND AXIAL FLOW
4.0
1.0
4,0
3,0
3,0
Q 2,0
2.0
1.0
-4
-3
-2
-1
o
K
2
3
4
Figure 7.45 The Q versus K diagram for the same system as Figure 7.43(b) and n = O, but with
flowing fluid, with U = 0.03 (Chen & Rosenberg 1974).
expressions for the two waves that propagate, MOl and M02 in Figure 7.43(b), and they
draw a number of insightful conc1usions, for example, on the effect of varying J-L, while
keeping C r constant, as follows. t
(i) We start with a very large J-L, which may be considered to be due to the fluid density,
p, being very large; hence, for a finite C r , for example, for C r '" 0(1), K j must
also be very large. In this case, wave propagation is govemed primarily by the
properties of the shdl and is independent of the stiffness of the fluid, since for very
large Kj the fluid is essentially incompressible. In the limit of infinite fluid density
(and hence K j), the phase velocity is that of longitudinal waves in an infinite elastic
medium.
(ii) As f..(, is reduced, p and Kj are diminished simultaneously to keep C r constant, and
a more coupled motion between the shell and fluid occurs in wave propagation,
giving rise, through the Poisson ratio effect, to radial motion as well.
tHere, it is noted that, in all of Chen & Rosenberg's (1974) ca1culations, a steel sheU containing water
is considered (f.t = 12,8), while in Horáček & Zolotarev's (1991a), discussed next, gas-filled metal shells
are considered (f.t = 0.129); hence, the effect of f.t on the dynamics is of interest.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
659
..32-
::..."
II 0.6
",,"
1.2
1.0
4 th mode I
I :
,
,
,
3,d mode'
\
'-.
0,8
..
0.4
.-.-'
0.2
- Fluid in an elastic shell
- Fluid in a rigid tube
- - - Fluid in an mfinitely flexible shell
, - . - Empty shell
- o, - Infinite solid
o
O
2
Q* = Qa/c s
3
4
Figure 7.46 The dimensionless phase velocity, v p = Vp/c s , versus the dimensi onI es s frequency,
Q* = Qa/c s , for wave propagation in fluid contained by an elastic shell, for n = O, t-t = 0,97,
k = 0.0013, C r = 0.40 and v = 0.3 (Lin & Morgan 1956b).
(iii) Further reductions in IL, with C r constant, may be interpreted as due to an increase
in the sheIl density, Ps, and a simultaneous increase in E; as E increases, so does
the phase velocity, For sufficiently smaIlIL, the propagation characteristics are
those of sound waves in an essentiaIly rigid sheIl.
The phase velocity as a function of the frequency, which is another way of presenting the
same information as in Figure 7.43, is shown in Figure 7.46 for IL = 0.97 and C r = 0.40.
Of special interest in this figure are the wave propagation velocities (i) in the fluid, (ii) in
a rigid sheIl, (iii) in an infinitely flexible sheIl (pressure-release walls), and also (iv) in an
empty sheIl and (v) in an infinite solid, which have been sketched in. Hence, for low Q*, it
is easy to see that the lowest ofthe two propagating modes (the 'Oth mode') corresponds
to plane-wave propagation in the fluid, in a somewhat less than totaIly rigid tube, while
the other (the '1st mode') corresponds to longitudinal sound waves propagating through
an empty tube.

660
SLENDER STRUCTURES AND AXIAL FLOW
At Q* == Qa/c s = O, for the Oth mode, vpo  c;[1- 2/1c;/(l- v 2 )] with v p = Vp/c s
as in (7.141). As Q* increases, the coup1ing between shell and fluid increases; it becomes
a maximum (and vpo reaches a minimum) as Q* approaches unity, the frequency ofhoop
vibrations of the empty sheU. For sufficiently 1arge Q*, vpo -+ c r , the wave velocity in
a rigid tube, For the 1st mode at Q* = O, v;!  1 - v 2 + 2v 2 /1c;, and it is essentia11y
a longitudinal wave propagating through the shell, with V p ! somewhat 1arger than in the
empty shell because of the fluid inertia. For high enough Q*, v p! -+ c r for this mode also.
AU the other modes have characteristics intermediate between the two limiting cases of
propagation in a rigid tube and in an infinitely flexib1e sheU over most of the frequency
range; a1so, they all have minimum cut-off frequencies.
Equivalent results for asymmetric modes, n > O, have been obtained by Chen &
Rosenberg (1974) but on1y those for n = 1 are presented, reproduced here in Figure 7.47.
It is seen that on1y one mode, Mli, propagates for Q < 0.6 approximately; it is
predominantly radial, whi1e the others are predominantly either axial or circumferential
for K not too 1arge, as seen in Figure 7.48. A1so, some calcu1ations for n = 2 have been
done by Bert & Chen (1975b) using the same theoretical model; see Figure 7.55.
The propagation characteristics for the jluid-conveying system and n = 1 are shown
in Figure 7.49. The absence of the pure torsiona1 mode in this figure and in Figure 7.47,
in contrast to Figures 7.43 and 7.45, is noted; for n > O, axial, radial and tangential
displacements are generally coup1ed [see the second of equations (7.1) with (). t= O], In
genera1, the propagation characteristics in Figures 7.47 and 7.49 for n = 1 are simi1ar to
those for n = O in Figures 7.43(b) and 7.45.
The peculiar quasi -discontinuous character of the frequency-wavenumber re1ationship
in the foregoing is c1arified once the acoustic modes of the system are superposed on the
figures, as was done by Lin & Morgan (l956b), Horáček & Zolotarev (1981, 1991a) and
Zolotarev & Kruntcheva (1995), for instance.
Figure 7,50 shows the same type of Q versus K diagrams as in the foregoing, for an
empty (/1 = O) and air-fiUed sheU (/1 = 0.129), but with the acoustic modes ofthe system
in a rigid sheU drawn in (for a finite c r , E -+ 00 implies Ps -+ 00, and hence /1 -+ O).
Q s! and Q s2 are the first (lowest) and second shdl modes in vacuum [cf. Figure 7.47(a)],
whi1e Q a!, Q a2, etc. are the acoustica1 mode s in a rigid shell.
For the fluid-fiUed system, it is seen in Figure 7.50 that the formerly continuous struc-
tural modes, cf. Figures 7.43(a) and 7.47(a), are chopped up, each 'discontinuity' being
associated with coincidence with one ofthe acoustic modes, Q a!, Q a2, etc. Thus, portions
ofthe predominantly radial (lowermost) mode in Figure 7.47(a) is seen in Figure 7.50(a)
to be captured successive1y by Qa !, Q a2, Q a3, and Q a4; simi1ar1y, portions of the acous-
tical modes get captured by the structural modes. On1y the sharp1y slanted parts of the
origina1 curves remain unaffected (heavy continuous lines) caUed 'structura1 modes' in the
figure, whi1e the 'acoustical modes' (heavy dashed lines) essentially coincide to those of
the acoustica1 waves in a rigid shel1- see, for examp1e, Howe (1998). t This 1atter near-
coincidence (to be henceforth simp1y caUed coincidence) is associated with the fact that
both /1 and Cr are small, implying a 1arge E, physically not too different from E -+ 00.
t One way to distinguish acoustical from structural modes, originally proposed by Warburton (1961) and
used in this work, is by comparing IUfI/ůl and IVfl/vl. where UfI and VfI are the motion-induced fluid velocities,
while ů and v the corresponding shell velocities, When these ratios are of 0(1) the mode is structural; when
they are of 0(100), they are acoustical- provided that the corresponding frequencies are not too close.

'"
:{
\
\
\
" M
.....- :i
-------
-----
M
:i
I;;:: I;;::
 f;;
.....
.... -
..............::s
.........
..........
.....,
_/
----

"".
00
I
N ON"'"
I I
/ pUU /!! SOqUl :!pnmdrnv
00
\O
\O
I
o
""

('<)
o
N

."
('<)
'"
N
."
'"
O
{j 'Á:JU:!nb:!.y SS:!IUO!SU:!IlI!O
"".
M
N >c
o

I
-O.... . ..... I>/)
1a.š  :;; e ..s: i)
",6.g
 - Q)
 I  E 
I:::: 1>/)--...1:::: '" 
,5 !ž, ::--I dC:!
_l
 ťJ . ,9 "8 
,- :<   U
 o '-'
....,-::., j-' I ,.
£8e
.g"c;j..cr-;:!
a t::  ''':: 'O "': 0'1
,_ .:= .&J r- -
- t='-'
p.. S 8°c;j
<r:: '..:: "2 s:.a 5'0
 .... =' '-
g.p..u....
QO Q) .:= .... 0 ....
 . u o
r:--: 8e
"'Q)I>/)Q)
Q,I -gea
"'" e "'- 
5'n 1;:3 I I '"
-- -- --"-]
r.. I;::, m I;::, I;::, ...
"".
:D,
'-' ---
-Oe,o
1a]
- 
- Q)
Q) '"
..co
"'
dC:!
S 
Q) Q)
..c
mU
'-'
---r-
\r)
C"'!
'-o
 II
1'->
::::-0
.... 
..s: 
e
m N
....-
II
:.a
.... .
Q)-
.Do
So
a II
Q)
 

1 .
>,-
g II
Q)
=' ::::
0"....
..s:
Q)=
..cQ)
ťJ
-o
t--
..,.-
.-=
t--'
t
5'n
._ m
r..
>c
M ....
CI)
1
CI)
1;
N 
'"
'"
CI)
--a
.8

CI)
E
i5
o
é:
"".
>c
....
CI)
M

CI)
1;

N 
CI)
--a
o
'U;
<::
CI)
E
i5
o



662
SLENDER STRUCTURES AND AXIAL FLOW
4,0
1.0
3,0
Q 2,0
-4
-3
-2
-1
°
2
3
4
K
Figure 7.49 The frequency-wavenumber diagram for the same system as in Figure 7.47 (b), n = 1,
but with flowing fluid, with U = 0,03 (Chen & Rosenberg 1974).
Only in the upper right-hand corner of Figures 7.50(a) and (b) do we see a significant
difference in the acoustical frequencies in the flexible as compared to the rigid shell,
A similar behaviour is displayed by the Q s2 mode shown in the upper left-hand portion
ofFigure 7.50(a) - cf. the middle curve in Figure 7.47(a).
ln the vicinity of intersection of structural modes in the empty shell and acoustical
modes, that is, near coincidence, the modes are truly coupled; these are the portions of the
curves drawn as heavy dots and called 'structural-acoustical modes' in the figure. Mode
'veering' or 'mode exchange' takes place in that neighbourhood. The reader is referred to
Chapter 3 for a discussion of this phenomenon, for example, Figures 3.29 and 3.77, where
it is shown that the modal content suffers alterations around these points (Figure 3.41);
here these alterations are even more profound: the structural mode becomes acoustical
and vice versa.
The reason why the fluid and shell solution curves do not actually cross at coincidence is
that, if they did, there would be two different rates of energy propagation in the two media,
corresponding to the different slopes, and hence different group velocities (= d Q jdK),
despite the two having an equal energy of vibration (Fuller & Fahy 1982).

\ ,.
\ '"
\ \
\, '
\ \
\ \
\\ '
\y., \
. . \
, . \
"f '
'\ \-
' \ N \ I r:
'\ leJ ....
\ \
\ '
\ \
, \
\ ......
,
\
\
\
 \
101 ,
\
I
\
\
,
\
,
,
,
,
,
,
,
\
,
\
,
\
\
\
\
I
':i \
IOJ I
I
<:<
""
o
'"
o
..".
o
N
o
101
<:>
\". \ \
\ \ \
\ \ \\
\ \
\ \ \
\ \ \
\ \ \
\ \ \
\ \ \
\ \ \
, \ \
\ \ \
, \ \
\ \ \
, ,\
, \ \
, , \
, , \
' \  , N \ -
lci 'Ici \ lci
\ \ \
, ' \
, ' \
, , ,
\ ' ,
, , \
I , '
\ -, \
'" Ie]  \
\ .:,. ---- \
I \ .
I I .:'
o
o
\
.\
,\
. :
,.\
\
,
,
,
,
,
,
,
, ..
'I ci
\
,
\
,
\
,
\
\
\
\
\
\
\
\
\
I 
:Ici
..".
o
N
o
""
o
'"
o
IOJ
. .
" 
, \
,
\
\
\
\
\
,
,
,
, M
, lci
- ,
Ie] ,
,
,
,
,

".J'..
\
\
\
I
I
I
..".
r<1
N.!':
""
o
o


..".
r<1
N
,
\
,
,
,
,
,
,
\
,
Id'
,
\
\
\
\
\
I
\
I
o

M
::t

"!
:::..
:::..
 +" 
,  + -
+ " :' :t-
( , +
@)
C  ":""+ 
+ =-" -
e.. + £1i' ci
M

\ \
\ \
\ \
\ \
\ \
\ \
\ \
\ \
\ \
\ \
\ \
\ \
\ -
\ ,Ici
\Iď \
, \
,
, \
, \
, \
, \
, \
, \
, '
, \
\ \
\ \
\ \
, \
\
.
"'
i
""
o
'"
o
..".
o
N
o
101

o5.S :"'.
. c : 
S' . 8:
uo
'" .... o :»
..s S v::
- -::I
o  «I o
E .S:a 1:j
V) o  c:d
.L ..c:: ::I....
'ta '" R a
'O 
a5«1t;
.... 1'.5
..s "O I "O
...c:: I '"
..". '" .. 
II c.g.9
o.. o 
::: S S o-
'-""'(1)_""""
3. o «I 
"" .g B'-
a ,5  
N ___ l::2..c::
N '" '"
II lci " :::3
:::. '
,,-.... - .. ::::
e-lci  
.- "'-'........ Q)
-d-g
II "" II S 
-=-- e ::i ta 
 ta tJ.g ?1
   .
':u8'@
Eooo
.B"'1:joáj
S I ,._..c::
- .... ---
::I """"
 .. .S:a o.g
0°.g"'«I
 II o
 :::t..c:: B 
 ....]:
 ..s -; CI:) o
 o u___""
.go2,.g
 '" M
ti) o '" «I....
S :3 ICI c;::, 
:oe  .;
 :@ICI 
'" '-" o
t  '" 
.g'O
ICI '-Ó 0-0
",-.oSN
00-
'u d 13 ct6
5 II '
6< .':- 5 
o  u ....
<t: ,«10
'" o .::r:
'" II I '-"
ti) I o
g I ,I.-g
'r;j  ::o S
5d8ta
.E II .g.g
Q ;>.g;g
o8
= o '... «I
"!J.
to- _ ti) «I
f II] B
& <:! . ě
?t;j
..".
r<1
N.!':
""
o
o

663

664
SLENDER STRUCTURES AND AXIAL FLOW
ji* 1.0
2,0
...... ' . :. '''' * '''''''',,,,,,,,'''''''''''''' ............
...........
I ,QsT ....................., ...".
",,"-?-""I'''''''''' ji* k..'''''''' ...,
'I .......,,........... ,........,.."
1.5 ....).............,.., ............
, I -* ............ ,I"
------ I Qs2 ............ ".." ",
. ...... ,....... ......
,,"', ............ . ,..' " o"
I I ..................... ,.' ,.'
.. ' ..........ij*....... ,...,,- ......
.  sl ...... .--:::...- , . , ' . . ,...::..:..:.....  , :...- , ' , ' . ...,'.. . : ,  , ,' : . : . : ., , ' . : , ,  , ':;;;{
H' j.........".,.,
I' ............. ,......... ,.......
. ........- ........ .....:....
..,.1'.... .,' ",.. ,.,.
. ............ .......:.....
'I ...' .""
0,5 , :.,..".,.,., ",." .,'
. ,,'
; ...,.."..,' :."..
, ..."'" .',.'.
.....,*
. ",.... Qal
o
4
8
K
12
16
Figure 7.51 Curves of Q* = QI.J'1=V2 versus K for n = O and hla = 1/300, li = 0.34,
U = O and C r = 0.066: - - -, predominantly axial mode; - - . - -, torsional mode; -,
predominantly radial mode; . . ., acoustical mode, The Qsk and Qak are defined in Figure 7.50
(Zolotarev & Kruntcheva 1995),
Comparing the different parts of Figure 7.50, it is noted that, not only the uncoupled
structural modes change with n, but also the acoustical modes (obtainable from the poten-
tial function solution but with boundary conditions appropriate for a rigid wall, hence
involving the zeros of the Bessel functionals). To help understand this, sketches of some
of these modes are presented in Figure 7 .50( d). These should be interpreted in a similar
manner to modal pattems in a membrane: the shaded and unshaded parts are in antiphase;
thus, in the (1,1) mode, when the left-side ha1f has negative acoustic pressure, the right-
side halfhas positive pressure. Obviously, acoustical and structural modes of1ike n couple
easiest.
One aspect of wave propagation that may escape attention in Figures 7.50(a) and (b)
is that, for sufficiently high K, the frequency of the structural modes suffers a step-like
reduction in Q as K is increased - seen in the upper right-hand comer of these figures.
This occurrence is shown accentuated in Figure 7,51 (heavy continuous lines) for n = O.
ln this figure, the other two shell modes ofFigure 7.43(a) are also shown, and the portions
corresponding to predominantly axial or radial motion are identified.
ln Figures 7.50(a)-(c) and 7.51, the structural-acoustical coupling is weak since J.L is
small (J.L = 0.129). For larger JL, achieved for example by decreasing h and raising p,
it becomes more difficult to distinguish between predominantly structural and acoustical
modes. An example for J.L = 3.125 for a shell pressurized extemally is presented in
Figure 7.52. The modes in the tluid-filled system are all drawn as heavy continuous lines
and are simply identified as modes of the strongly coupled tluid-shell system; they are
substantially different from both the empty-shell structural modes and the acoustical modes
in the rigid shel!.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
665
1.0
0,8
0,6
Q*
0.4
0,2
o
2
4
6
8
10
K
Figure7.52 Curvesof?2* = Q/.Jl=V2 versus K forn = 1 andh/a = 1/750, v = 0,34, no =
Poa/ Eh = -5.21 x 10- 6 , C r = 0.065. Thin lines are for f.L = O: -, structural modes; - --,
acoustical modes. Thick lines (-) are for f.L = 3,125 (Horáček & Zolotarev 1991a),
The propagation characteristics with flowing fluid are illustrated in Figure 7.53 for n = 4,
ln these figures, : H2 * instead Of::l:K is used to distinguish upstream (- Q *) from down-
stream (+Q*) travel1ing waves; in this regard, ::I: Q * should be thought of as (::I:v p )K, v p
being the phase velocity. In Figure 7.53(a), for U = O, it is seen that upstream and
downstream propagation is the same. In (b) and (c), however, the acoustical mode-loci
rotate counterclockwise, decreasing structural-acoustical interaction for downstream trav-
elling waves, while increasing it for upstream travelling ones. This counterclockwise
rotation is of course associated with downstream sound propagation being faster, and
hence associated with larger Q *, than upstream propagation.
7.4.3 Complex wavenumbers and viscous fluids
ln all the foregoing, only real wavenumbers have been considered. Generally, however,
Dne can have complex wavenumbers, representing attenuated wave propagation, and pure
imaginary wavenumbers, representing 'evanescenť modes, as discussed by Fuller & Fahy
(1982) and Fahy (1985), for instance - cf. Section 7.2.3. As already mentioned, below the
cut-off frequency the wavenumbers become complex; this is also trne if dissipation is taken
into account - see, for example, Scarton & Rouleau (1973). The analysis presuming the
wavenumbers to be real, for frequencies above the cut-off, is quite acceptable for inviscid
Dr nearly inviscid ftuids; however, this is not generally so for viscous ftuids.
A very meticulous and general analysis of wave propagation in cylinders of arbitrary
wall thickness containing quiescent viscous fluid and also considering thermoelastic and
thermoacoustic effects has been considered by Liang (1990) and Liang & Scarton (1994,
1996,2002). The shell is analyzed as an elastic annulus via the Navier-Cauchy equation,

1.0
r> ____ ___---- ---- ------
____t_
--
--
0,5
Qa
-------
--
--- - ---=--.
---- -------
ij ----
---------
Q* O
4
6
8
IC
-0,5
----
---- ---___-.;:0. _____
.   ---- -----------
-.....- --- - ----
--------- -- -
-------- - ------ -.'.: ::..---
---------------------- ..----
---
-1.0
(a)
1.0
--
---
-----
.................... ................ ""
--............ -........ 
-- _....-
--
---
-....-............ :-----
0,5
4
IC
6
8
Q* O
----
---- --::._-----=-== ---==.:.::-:---
-05
...--------
-1.0
(b)
------
-----
- - - - - --- ----- ......
----- ------
1.0
---
- '
.....-....,..- ... 
---
--
--
--
--
--
--
--
---
---
05


g* O
IC
4
6
8---
---------------------
------
--------
-05
----------
-1.0
(c)
Figure 7.53 The frequency (Q* = QjJf=V2) versus K diagrams for wave propagation in
a shell conveying fluid; n = 4, hja = lj300, v = 0.34, no = O, C r = 0_066: (a) U = O;
(b) U jJf=V2 = 0.04; (c) U j = 0.06, The nota ti on forthe various curves is as in Figure
7.50; - - -, asymptotes of the acoustical resonant frequencies (Horáček & Zolotarev 1991a),
------ -----
-- -
--- -------------- ----------:..

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
667
rather than thin shell equations, and the fluid by means of the thermoacoustic continuity,
Navier-Stokes, and thermodynamic energy equations. Solutions have to satisfy 12 radial
boundary conditions: three each for continuity of displacement and stress at the inner
surface; one each for continuity of temperature and heat flux on the inner surface; three
for the pressure-release conditions at the outer surface; and one for continuity of heat flux
to the surroundings! For each frequency and mode, one can express the 3-D deformation
as oCR, e, X) = u(R) exp(ine + KX), with K = aa, where a is the generally complex
wavenumber and a the outer radius, R = r/a and X = x/a. The phase velocity, V p =
J...(Q/27r), J... being the wavelength. In general, K = Kr + iKi and Kr < O; Kr == Re(K)
corresponds to the attenuation rate, and Ki == Lm (K) to the so-called propagation constant. t
Typical results are shown in Figure 7.54 for n = O, in terms of K and the dimensionless
frequency Q* = Qa / co, CO being the intrinsic sound speed in the fluid (e.g. Co = 1480 mis
in water). The dimensionless phase velocity, c; = Vp/co = Q* /Ki.
Figure 7.54(a) presents the uncoupled results: the solid lines for the empty shell, and
the dashed lines for the fluid column alone (with pressure-release boundary conditions).
It is seen in Figure 7 .54(b) that the coupled results are generally quite different from
either, which is an attribute of wave propagation with strong coupling. Here, the solid and
dashed line s indicate propagation dominated by solid and fluid effects, respectively; in the
dotted parts of the curves, solid and fluid contribute equally, that is, the coupling there is
particularly strong.
It is seen in Figure 7 .54(b) that the first coincidence occurs in the vicinity of Q* = 3. This
corresponds to the local minimum in the attenuation versus Q* curve in Figure 7 .54( c), and
this may be used to pin-point coincidence (Liang 1998; Scarton 1998), here at Q* = 3.35.
Propagation for n i= O has also been considered (Liang 1998; Liang & Scarton 2002),
but results are not presented here. In general, for n i= O, coincidence involves spiral wave
propagation, so that propagation in the lower wave-speed fluid (water) can catch up with
that in the higher wave-speed steel tube (Scarton .1998).
Thermoacoustic effects for this particular fluid-shell system are rather weak; for
example, the coincidence frequency is shifted by only 1% vis-o-vis the results of
Figure 7.54(b) and (c). Nevertheless, since such effects could be stronger for other fluid-
shell combinations, a mechanism for how thermal effects can influence wave propagation
has been proposed (Scarton 1998), as follows. The thin viscous, acoustic boundary layer,
which exists despite zero mean flow velocity, generates heat; this is readily conducted
into the inner surface of the steel wall, which has substantial1y higher thermal conduct-
ivity. Therein it acts like a heat source, causing expansion of the inner steel surface, thus
modifying wave propagation.
7.4.4 Anisotropic shells
ln the foregoing it was seen how the character of propagating modes changes with the
wavenumber, for example, from being predominant1y axial to radial and vice versa
(Figures 7.44 and 7.48). Hence, since anisotropic shells have inherent1y different properties
in different directions, the effect of anisotropy on wave propagation could be profound.
tThus, here the imaginary component of the complex wavenumber, Ki, corresponds to what in all the
foregoing had been the purely real wavenumber - because of the different exponential form adopted,

668
SLENDER STRUCTURES AND AXIAL FLOW
10
,;
,;
,;
,;
,;
,; ,;
8
---
*
CI ,;
,;
;.; ,;
<J ,;
= 6 ,;
CI>
5- ,;
,;
'" ,;
<I:: ,;
'" ,;
'"
CI>
'2 4
Si
'"
iJ "
s "
O "
"
2 "
"
"
"
"
"
"
10
,;
,;
,;
,;
,;
,;
,;
8
- -
, .'
_.... .0
'"
/ ///
. ,;
, ....
2
,;
,;
6
4
,;
,;
,;
,;
,;
"
"
(a)
o
O I 2 3 4 5 6
Dimensionless propagation constant, Kj
o
O I 23456
Dimensionless propagation constant, Kj
(b)
10- 3
'" ...
'" '"
CI> "r 10- 4
'2
o o
00 .
= ::I 10- 5
CI> c::
S CI>
.- 1:
Q os
10- 6
O 2 3 4 5 6
(c) Q*
Figure 7.54 (a) and (b) Dispersion curves, Q* versus Ki, for axisymmetric (n = O) wave
propagation in a water-filled relatively thick steel tube (radius ratio of 0,93). (a) Uncoupled
results: -, empty steel tube; - - -, water column alone. (b) Coupled results: -, solid-
dominated; - - -, liquid-dominated: ' . . ., very strongly coupled wave propagation, (c) Dimen-
sionless attenuation constant, Kr, versus Q* for the first solid-dominated propagation mode, the
local minimum pin-pointing coincidence (Liang 1998),
Bert & Chen (1975b) conducted an investigation in to the effect oť anisotropy on wave
propagation in essentially infinitely long shells, containing inviscid fluid, paralle1ling the
work oť Chen & Rosenberg (1974) on isotropic ones, discussed in Section 7.4.2. Love-type
first-approximation shell theory is used, but also simplifications corresponding to Morley's
and Donnelľs equations. Calculations are presented for a fibreglass-epoxy composite
shell (El1 = 52.3 GNjm 2 , E22 = 10.3 GNjm 2 , G12 = 6.88 GNjm 2 , Vl2 = 0.27,
Ps j P = 1.97, f.L = 51), and the results compared to those for an isotropic shell.
Wave propagation in the axisymmetric (n = O) and beam-like (n = 1) modes is
qualitatively similar to that presented in Figures 7.43-7.45, 7.47 and 7.49 for an isotropic
shell, Quantitative differences are quite significant, however. For example, amplitude

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
669
1.5
M 23
4,0
3,5
2,5
Q 2,0
1.0
0,5
----
0,0
-4 -3 -2 -1
°
2
3
4
K
Figure 7.55 Frequency-wavenumber diagram for shell conveying fluid with U = 0.03 and n = 2,
-, Glass-epoxy shell (IL = 51 and c = 36); - - -, steel shell (IL = 12,8, C = 0,257); [2, K, IL
and c are defined by equations (7.141), but, for the composite shell, instead of E, one uses A 11 [see
equation (M.5)] (Bert & Chen 1975b),
ratios for n = O are qualitatively similar to those in Figure 7.44, but (i) the intercept with
the ordinate for the MOl mode is at -7 approximately, instead of -1, and (ii) the M02 and
Mm curves are shifted to the right, to almost double the values of K,
The results for n = 2 presented in Figure 7.55 show similar trends: wave propagation
in isotropic and orthotropic shells is qualitatively similar, but quantitative differences
are quite significant, though perhaps falling short of being characterized as 'profounď,
as was initially supposed. In this regard, the differences displayed in the figure are not
wholly due to anisotropy, since the Young's moduli and density ofthe composite shell are
much inferior to those of the isotropic steel shell, as manifested by the difference in the
values of c.
An important result, not discussed in Section 7.4.2, is that, for n = 2, all modes are
non-propagating (evanescent) at low frequencies; that is, there are cut-off frequencies for
all of them - though for the M21 mode this occurs at Q = 0.01 and hence is hardly
visible.
Bert & Chen also give a number ofvery useful approximate, c1osed-form solutions, for
example, for the phase velocity, extending and refining Chen & Rosenberg's (1974) work.
7.4.5 Concluding remark
The work presented in Sections 7.4.2-7.4.4 gives but a glimpse of how very intricate and
interesting wave propagation in shells can be. The interested reader is also referred to
Section 7.9.2 for a discussion of wave propagation in pliable shells. However, further
consideration of this topic is beyond the scope of this discussion.

670 SLENDER STRUCTURES AND AXIAL FLOW
7.5 VISCOUS AND CONFINEMENT EFFECTS
7.5.1 Introductory comments
The subject of this section might have been entitled real fluid effects in the dynamics of
shells conveying fluids, or perhaps some real effects, the most prominent of which are
those associated with fluid viscosity, So far, the fluid has been assumed to be inviscid.
Yet, viscosity-induced pressure drop in a sufficiently long shell conveying fluid is gener-
ally not negligible. As the internal pressure is diminished vis-o-vis the external ambient
pressure, assuming them to be equal at inlet, steady compressive stresses in the shell wi1l
affect its dynamics; if the pressure drop is sufficiently large, buckling of the shell may be
induced, even at mode st flow velocities (cf. the discussion of Figure 4,11, Section 4.3,1).
These may be termed steady viscous effects. In addition, there wi1l genera11y be unsteady
viscous effects, associated with shelI motions, modifying the form of the perturbation
pressure given by equation (7.45), for instance, and introducing damping as welI; refer
to Section 2.2,3 for the effect of a viscous quiescent fluid on the dynamics of coaxia1
cylinders,
Also discussed in Section 7.5 is the effect of 'confinemenť. In many engineering
systems, shelIs are coaxially arranged with other shells or relatively rigid cylindrical con-
tainers. Examples may be found in nuclear or chemical reactors and in heat exchangerso
Coaxia1 shells conveying fluid in the annulus, exclusively or concurrently with flow within
the shell, are treated in Chapter 11. In that same chapter is given a brief review of the
extensive literature on the effects of confinement of a shell by outer shells and cylinders,
with stagnant fluid within the shell and in the annulus.
However, in this chapter, since some of the results to be presented are for shells con-
veying fluid and surrounded by quiescent, confined fluid, some comments on the effect
of this outer fluid are useful. t As discussed extensively in Sections 2.2.2 and 2.2.3,
for inviscid and viscous fluids, respectively, the main effect of the quiescent fluid is
to add to the effective mass of the shell. If the fluid is dense, and especia11y if it is
confined in a narrow annular space, the fluid added mass may be very large - sev-
eral times the shell mass - and this clearly has an important effect on shell dynamics.
Also, as discussed in Section 2.2.3, important damping effects are introduced; see
Figure 2.9.
The first attempt to account for viscous effects on the dynamics of shells conveying
fluid was made by Paoidoussis et al. (1985), modifying the Pa1doussis et al. (1984a) theory
and considering only steady viscous effects in shells with supported endso The theory was
later extended to cantilevered shells by Pa1doussis et al. (1991).
Unsteady effects were dealt with approximately, first for a developing boundary layer
(Pai'doussis & Mateescu 1987) and then for fully developed turbulent flow (El Chebair
et al, 1990). As wi1l be seen in what follows, unsteady viscous effects for internal
flow are rather weak, in contrast to annular flow (Chapter 11) where they can be
important; hence, further attempts to refine theory in this direction were confined to
coaxia1 shells subjected to annular flow, culminating in a successful CFD model (Nguyen
et al. 1994).
tThe interested reader ought to also consult Weppelink's (1979) work on confinement and end-effects,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
671
7.5.2 Steady viscous effects on shells with supported ends
ln the case of tubular beams (pipes) conveying fully developed turbulent flow, it has been
shown theoretically that viscous effects are negligible (Section 3.3.4), and this has been
confirmed by experiments (Sections 3.4.4 and 3.5.6). However, this is not necessarily the
case for cylindrical shells, and hence viscous effects cannot be neglected a priori. In a first
attempt to assess the importance ofviscous effects, the unsteady, shell-motion-dependent
forces are deterrnined as in the foregoing, while the viscous forces are determined separ-
ately in a time-averaged sense t (Pa'idoussis et al. 1985), Thus, only steady viscous effects
are taken into account;+ they lead to motion-independent stress resultants appearing in the
equations of motion of the shell, which are deterrnined in what follows,
Consider the she1l to contain a fully developed turbulent flow. Consequent1y, to over-
come viscous pressure drop, the pressure within the she1l will generally be higher than the
ambient, decreasing in the flow direction. Physically, two effects of viscosity are taken
into account in the averaged sense: surface traction, and pressurization/depressurization.
Hence, the shell is considered to be pre-stressed by the following so-called basic loads:
(i) a constant axial force per unit area (stress), Px' and (ii) an axially symmetric normal
pressure, Pr' which are assumed to have the following functional forms:
Px = B,
Pr = -(Cx + D),
(7.145)
where B, C, D are constants to be determined. The associated axial and hoop stress
resultants may be shown to be
N x = B (!L - x) - va (!LC + D),
where, as before, a and L are the shell radius and length, and v is the Poisson ratio.
Different forms of basic loads have been considered, for example, by Fltigge (1960),
Timoshenko (1961) and Brush & Alrnroth (1975), but not oftheform of (7,145), The equa-
tions of motion subject to this form of basic loads are derived ab initio by Chan (1984).
They are as follows:
Ne = -a(Cx + D),
(7.146)
LI (u, v, w) + qlu" + q2(V' + w) + q3(U" - w') = y(a 2 u/at 2 ),
L2(U, v, w) + ql v" + Q3(V" + w') = y(a 2 v/at 2 ), (7.147)
L3(U, v, w) - Ql w" - Q3(U' - v' + w") = -y[(a 2 w/at 2 ) - Qr/ Psh],
where Li (u, v, w), i = 1, 2, 3, are as in equations (7.1), Qr = (Pi - Po)lr=a is the
perturbation pressure associated with shell motions (Section 7.2.3), and
{Ql, Q2, Q3} = [(1 - v 2 )/ Eh]{ N x , apx' aPr},
(7.148)
where E, h and Ps are as in Section 7.2.1, and Ne = ap r has been used. Details of
how the basic loads enter the equations of motion are not given here. However, it may be
tA ful! analysis would involve the determination of the vibration-dependent viscous flow-field, necessitating
the iterative solution of the Navier-Stokes equations numerical!y, for arbitrary (i,e, not prescribed) shell
motions - see Section 11.7,2(c),
*Unsteady viscous effects are treated approximately in Section 7,5,3,

672
SLENDER STRUCTURES AND AXIAL FLOW
useful to il1ustrate how they do so, by considering the contribution of the Nf} term to the
equilibrium condition of forces in the circumferential direction. If the tota! stress resultant
is denoted by Nf} = Nf} + Nf}, where Nf} is the perturbation stress resultant, then the normal
force in the circumferential direction acting on one side of a stretched element of the shell
(Fltigge 1960) is Nf} (1 + Ef} )dx, where Ef} = (v' + w) fa is the circumferential strain; on the
other side ofthe element there wil1 be the same, augmented by (afaO)[Nf} (1 + Ef})dx]dO;
hence, the contribution of this term to equilibrium is
[Nf}(1 + Ef}) dx] dO = Ne dx dO + Nf} (v" + w") dx dO,
ao
to first order. The first term is the one normally appearing in the equilibrium equations of
non-pre-stressed shells, while the second gives rise to the extra terms involving q3 in the
second of equations (7.147), where it is noted that Nf} = aPr'
The quantities N x , P x and Pr appearing in (7.148) are determined from the time-mean
Navier-Stokes equations [e.g. Laufer (1953)],
..!.. ap = -(r uxUr ) +  ( r d U ) ,
P ax r dr r dr dr
..!.. ap = -(r u2 ) + ( u2 ),
p ar r dr r r f}
d 2
0= -( urUf} ) + -( urUf} ),
dr r
(7. 149a)
(7. 149b)
(7. 149c)
where U is the mean flow velocity, P the static pressure, (x, O, r) the cylindrical polar
coordinate system, (U x, Uf}, u r ) the corresponding fluctuating velocity components ofthe
turbulent flow, p the fluid density, and I} the kinematic viscosity; the overbar denotes the
time-mean.
Integrating equation (7 .149c), one has u r Uf} = CI f r 2 , where CI is a constant. At r = a,
c1early UrUf} = O on the shell wall. Hence, CI = O everywhere. Since all time-mean
quantities are independent of x (except the pressure) for fully developed flow, then oP far
from equation (7, 149b) must be independent of x, implying that ap fax is independent of
r; then, equations (7 . 149a, b) may be integrated with respect to r, yielding
r 2 ap dU
-- = -r (uxu r ) + qr- + C2(X),
2p ax dr
(7.150a)
P - I r 1 - -
- = -U; + -(u - u;)dr + C3(X),
Par
(7.150b)
where C2 and C3 are functions of x to be determined from the boundary conditions.
Considering now fully developed axisymmetric flow, dU fdr = O and UxU r = O at
r = O; hence, C2(X) = O. At r = a, since UxU r = O, equation (7. 150a) becomes
1 aPi _ 21}i dUi I
Pi ox - ---;; dr r=a'
(7.151)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
673
where the subscript i has been inserted to denote quantities associated with the internal
fluid. The shear velocity, UT' is defined next,
2 dUi I
U Ti = -1]i-
dr r=a
Twi
(7.152)
=
Pi
where Tw is the wall viscous shear stress - see Section 2.2.1(f). Substituting (7.152) into
(7.151) and integrating with respect to x yields
Pi 2 2
- = --UTix + C4(r);
Pi a
(7.153)
then, comparing this with (7 .150b) leads to
2 2
C3(X) = --UTix,
a
- I r 1 - -
C4(r) = -U;i + -(Ui - u;)dr + CS,
a r
(7.154)
where C S is a constant, easily found to be Pi (O, a) / Pi. Hence, the static pressure in the
internal fluid region may be written as
2Pi 2 2""' I r 1 2""' 2""'
Pi(X, r) = --UTix - PiUri + Pi -(U ei - ur)dr + Pi(O, a).
a a r
(7.155)
On the shell surface, this expression takes on the simp1ified form
2Pi 2
Pi(X, a) = - -UTix + Pi(O, a).
a
(7.156)
Hence, the radial basic load, Pr = Pi(X, a) - Po(x, a), where Po(x, a) - Pe is the
constant ambient pressure in the outer, quiescent fluid region, may be written as
- 2Pi 2
Pr = --UTix + Pi(O, a) - Pe.
a
(7.157)
The axial basic load is Px = Twi, and from equation (7.152) it may be written as
- U 2
Px = Pi Ti.
(7.158)
The only unknown in (7.157) and (7.158) is UTi. It may be determined empirically from
the well-known expression for pressure drop in a pipe,
I OPi I =  JiPi U
OX 2 Dhi I'
(7.159)
where Ji is the friction factor and Dhi is the hydrau1ic diameter (in this case equal to 2a).
Utilizing (7.152) and (7.151), the foregoing leads to
2 1 2
U Ti = "8 Ji U i .
(7,160)

674
SLENDER STRUCTURES AND AXIAL FLOW
Table 7.5 The lowest critical flow velocity for divergence, U c d, for the
Lgap rubber-air system (hla = 3/2), obtained with the inviscid version of
fue present theory (Paldoussis et a/. 1984a) abbreviated as P, C & M (84), for
n = 2, compared to independently obtained resu1ts by Paldoussis & Denise
(1972), [P&D (72)] Weaver & Myklatun (1973) [W&M (73)] (scaled fmm
Figure 7.20), Pham & Misra (1981) [Ph&M (81)] and Paldoussis & Mateescu
(1987) [P&M (87)] for hla = 00.
P&D (72)
W&M (73) Ph&M (81) P&M (87) P,C&M (84)
l+k=1
0,580
l+k;é:l
0,594
0.68
0,625
0,598
0,635
The factor Ji depends on surface roughness and the Reynolds number. It may be
obtained from the Moody diagram or the Colebrook equation [see, e.g., Murdock (1976)],
commonly used in hydraulics.
Examining equations (7.157) and (7.158), it now becomes clear how the functional
forms in equation (7.145) and (7.146) arise, leading to the evaluation of the constants
B, C and D. Hence, p x and pp N x and Ne , as well as ql , q2 and q3 have now been fully
detennined in terms of Pj, Uj, jj and the steady-state pressures Pj(O, a) and Pe,
Ca1culations have been conducted with the shell surrounded by a coaxial shell or rigid
cylinder and flow in the inner shell, in the annulus, or both, first with the fluid being
considered inviscid, incompressible or compressible (Pa'idoussis et a/. 1984a); then, the
flowing fluid was taken to be viscous but incompressible (Pa'idoussis et a/. 1985). In the
results to be presented here, however, (i) the fluid in the annulus is taken to be stagnant
throughout, and its effects on the dynamics are taken into account as in Section 2.2.2, or
altematively as in Section 11.7.1 but with Ua = O, and (ii) the outer cylinder is taken to
be rigid.
The perturbation pressures, related to shell motions, are determined as in
Section 7.3.I(a) - see also Chapter 11, where both incompressible and compressible
forms are given. The 'basic', steady-state loads are taken to be as just detennined in the
foregoing.
Ca1culations have been conducted with so-called '! -gap' and ' 1 -gap' systems, defined
as follows. If a is the shell radiu s and b the inner radius of the rigid outer cylinder, then
the !-gap system corresponds to (b - a)la = !' that is, bla = 3/2; and correspondingly
forthe I -gap system, bla = 11/10. The parameters forthe l -gap 'steel-water' system,
that is, for a steel shell conveying water, are: b = 100 mm, h = 0.50 mm, Ua = 5308,
alb = 10/11, Lib = 10, hla = 5.50 x 10- 3 , f.J., = 23.30, and v = 0.3 [refer to
equations (7.47) and (7.49)]; for the !-gap rubber-air system they are: b = 11.8 mm,
h = 0.178mm, Ua = 36.73, alb = 2/3, Lib = 17,3, hla = 22.7 x 10- 3 , f.J., = 0.06,
and v = 0.47.
This latter, rubber-air system was considered mainly for comparison with Pa'idoussis
& Denise's (1972) calculations for an unconfined shell (Section 7.2.3(a», and hence
indirectly with experiments. Typical results for inviscid flow are shown in Table 7.5.
The values of U c d obtained are within ::1:7% of those by others, utilizing other methods
(Sections 7.2 and 7.3). One significant difference vis-a-vis Pa'idoussis & Denise's results is

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
675
Table 7.6 Comparison of the critical flow velocity for diver-
gence, U c d, and coupled-mode flutter, U c f, neglecting and
accounting for steady viscous effects, for the /0 -gap steel-water
system (n = 3); from Pai'doussis et a/, (1985).
Divergence, U c d x 10 2
Coupled-mode flutter, U cf x 10 2
Inviscid
Viscous
Inviscid
Viscous
2.55
4,14
3,14
4.86
that coupled-mode flutter in this case (confined shell) is of the Hamiltonian type; that is, it
is preceded by restabilization of the first (m = 1) mode after divergence (cf. Figure 7,56),
rather than as in Figure 7.7. Another difference is that, in these results, coupled-mode
flutter is much doser to divergence (V cf = 0,641, V cd = 0.635) than according to
Pai'doussis & Denise's calculations,
The quantitative effect of confinement of the outer fluid by the rigid cylinder is very
small in this case, because of the lightness of the fluid (air) and the relatively wide annular
gap (b / a = 3/2). In general, the effect of confinement is to reduce the natural frequencies
because of the increased added mass of the confined fluid. However, it must be stressed
that the effect on Vcd is nit, since divergence is associated with Q = O and hence motion-
induced accelerations do not arise.
A typical result for the 1 -gap steel-water system conveying inviscid or viscous fluid
(taking into account only steady viscous effects) is shown in Figure 7,56, The critical values
are compared in Table 7.6. In inviscid theory, the static pressure of the inner and annular
fluids is equal and constant throughout the domain (O, L). In the viscous calculations,
however, these pressures are taken to be equal at x = L, while that of the inner flow varies
with x and V according to the associated pressure drop.
Similar calculations for the same system but with simply supported ends, analyzed by
the same method, have been presented by El Chebair (1988). The results are similar:
V cd = 2,00 x 10- 2 by inviscid theory, and V cd = 3,80 x 10- 2 by taking steady viscous
effects into account.
The dynamics according to viscous theory is broadly qualitatively similar to those by
inviscid theory, except that (i) los s of stability and coupled-mode flutter occur at higher
flows, (ii) the values of Q tend to remain high or to rise with increasing V , rather than
dipping towards zero monotonically, and (iii) there is no third-mode (m = 3) divergence.
The physical reason for these differences may be grasped in terms of the pressurization of
the inner fluid, required to overcome the frictional pressure drop, and the attendant static
pressure differences across the shell wall; this pressurization gives rise to a tensi1e hoop
prestress and, through the Poisson effect, to an axial tensile presstress, since the ends are
clamped, tending to stiffen and hence stabilize the shell (Fliigge 1960) vis-a-vis inviscid
theory where pressurization is totally absent. Also, it is recognized that, with increasing
V , both the steady (stabilizing) forces and the unsteady perturbation (destabilizing) forces
increase, which helps explain (ii) and (iii) above.
The magnitude of the pressurization-related stabilization effect may appear surprising.
However, for the physical system involved, it is noted that the extemal pressure necessary

676
SLENDER STRUCTURES AND AXIAL FLOW
2,0
Di )
m=3
1.5
--.
'"
o
x 1.0
IOJ m = 2
-"n"
''',..\...
0,5
B
0,0
°
3
4
5
6
7
 X 10 2
Figure 7.56 The dimensionless frequency, Q , versus the dimensionless flow velocity, V i, for
the ťo-gap steel-water system (defined in the text), n = 3 and m = 1,2 and 3, by inviscid and
viscous theory - in the latter taking into account only steady viscous effects - from PaYdoussis
et a/, (1984a, 1985): -, viscous theory; -- -, inviscid theory. A, divergence; B, restabilization;
C, coupled-mode flutter, for the viscous theory, A *, B*, C* denote the same for inviscid theory,
to collapse the shell in the absence of flow is 36.3 kPa. This compares to a value of pressur-
ization of 79,6 kPa at the upstream end to achieve zero pressurization at the downstream
end for U = 0.04; thus, the mean pressurization is quite large and of the same order as
the collapse pressure, but in the opposite direction,
The shell may also buckle due to sufficiently high compressive load, and indeed the
downstream part of the shell is subjected to such a load due to surface traction; however,
the magnitude of this traction is two orders of magnitude lower than that needed to collapse
the shell in this particular case; hence, it is of secondary importance here.
The conc1usion is that mean viscous effects, associated mainly with pressurization, are
genera1ly important. However, they stabilize the system as compared to predictions by
inviscid flow theory, and such predictions are therefore conservative,
7.5.3 Unsteady viscous effects: developing boundary layer
The first of the two analyses to be discussed here for assessing unsteady viscous effects
on the dynamics of the system explores the effects of a developing boundary layer in a

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
677
clamped-clamped shell ingesting incompressible viscous fluid (Pa'idoussis & Mateescu
1987), as shown in Figure 7.57.
The flow consists of a core flow, which may be treated as if it were inviscid (see
Section 2,2), and a thin, developing boundary layer, either laminar or turbulent. Steady
viscous effects are ignored, partly for simp1icity, but mainly so as to highlight more clearly
the unsteady viscous effects.
Fliigge's shell equations are utilized, but they are transformed into a single eighth-order
equation in the radial displacement w - however, not making the kind of accuracy-
limiting assumptions one associates with the Donnell, Morley-Koiter or Fli.igge-Kempner
equations (Section 7.2.1), This equation may be found in Pa'idoussis & Mateescu (1987),
and its full derivation in Mateescu (1984); here it is given in symbolic form as
£ [ w , -r x , -rf), P - Pe] = O,
(7.161)
where w = w / a, -r x and -rf) are the flow-induced axial and circumferential stress resultants,
and P - Pe == p(X, e, t) - Pe is the pressure difference across the shell wall, The task is to
express -r x , -re and P - Pe in terms of w . The solution, according to the Galerkin method,
proceeds in the normal way; here,
w (, e, -r) = L Cmf/>m() cos ne e AT ,
m
(7.162)
where  = x/a, -r = (Uo/a)t, in which Uo = {E/[Ps(1 - v 2 )]} 1/2, and f/>m() are the
dimensionless eigenfunctions of a clamped-clamped beam. These latter, for convenience,
are expressed as f/>m() = f/>mT() + f/>mH(), f/>mT being the trigonometric component,
- cos(fJm /1) + O'm sin(fJm / 1), and f/>mH the hyperbolic component, cosh(fJm / 1) -
O'm sinh(fJm / l); fJm are the dimensionless eigenvalues of the c1amped-clamped beam,
and 1 = L/a (cf. Section 2.1.3).
The perturbation velocity potentialfor inviscid flow may be expressed as
<I>(, r, e, -r) = aUof/>(, r)cosnee AT ,
(7,163)
where f/>(, r) is the so-called reduced potential and r = r/a. Proceeding then as in
Section 7.2.3 for instance, the unsteady pressure for inviscid flow in this notation takes
u-+ o P(x, r, O)
._._-
u-+
h
 x L
Figure 7.57 Geometry of the cylindrical shell conveying axial flow, showing a representative
point in the fluid, P (Pa'idoussis & Mateescu 1987).

678
SLENDER STRUCTURES AND AXIAL FLOW
the form
P - Poo = -pU L Cm {A 2 + 2 UA (a/a) + C8m/l)2 U2 }
m
x [l}m4>mH() + {m4>mT()] cos ne e ÁT ,
(7.164)
where l}m and {m are the values at l' == r/a = 1 of l}m(l') and {m(l'), defined by
l}m(r) = Jn({jm l' / 1)/ [nJn({jm/ I) - ({jm/ I)J n +1 ({jm/ l)],
{m(l') = In ({jm l' / 1)/ [nIn({jm/ I) + ({jm/ I)I n +1({jm/ l)],
(7.165)
in which, U = U/Ua and, as before, 1 = L/a; J n and In are the ordinary and modified
Bessel functions of the first kind, respectively, The effect of the stationary outer fluid is
neglected; hence, effectively Pe = Poo.
7.5,3(a) The unsteady viscous flow in the boundary layer
For unsteady viscous flow, it is recognized that the central core flow is quasi-inviscid, and
the viscous effects manifest themselves in the boundary conditions, applied at the edge
of the displacement thickness of the boundary layer, r = a - (i*(x), as wil1 be done in
determining P - Pe later.
Considering now the unsteady viscous flow in the boundary layer, it is assumed that
(i) the unsteady perturbation velocities are small because shell motions are presumed to
be small, and (ii) the radial and tangential components of the perturbation velocity are
small as compared to the axial component V x ' Hence, utilizing the usual assumptions of
the boundary-layer concept, the Navier-Stokes equations of the unsteady viscous flow in
a laminar boundary layer on the shell wall may be written in the linearized form
a V x a V x a Vxl a Vxl 1 a ( a V x )
-+U-=-+U-+Vj-- r-
at ax at ax r ar ar '
1 ap
O'" ---
- par'
aVe aVe a Vel a Vel 1 a ( aVe )
-+U-=-+U-+Vj-- r- ,
at ax at ax r ar ar
(7.166)
where Vxl and Vel are the values of V x and Ve at the interface between the boundary
layer and the core flow, which vary in time due to shell oscil1ations; v j is the kinematic
viscosity of the fluid. The radial component of the flow velocity at the boundary layer limit
is sensibly equal to the radial velocity of the shell; hence, there is no radial component of
the relative velocity with respect to the shell wall.
Assuming that the displacement thickness (i*(x) of the developing boundary layer is
small with respect to the shell radius a, the steady-flow velocity U at the boundary-layer
edge can be derived from the continuity equation; thus,
UI(X) = U {a 2 /[a - (i*(x)]2} ::= U(1 + 2(i* fa).
(7,167)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
679
y
r(J
x
x
o
V x
Figure 7.58 Diagram showing the transformation of coordinates and the angle f3 through which
the unsteady viscous flow oscillates in the boundary layer (Pai'doussis & Mateescu 1987).
The displacement thickness may be derived in various ways (Schlichting 1960;
White 1974). Here, because of the complexity of the overall problem at hand, it is des ir-
able to have a simple analytical expression for 8*(x); such an expression was derived in
Mateescu (1984), namely
8*(x)ja = ! {I + [4xjRe a - 1]j[l + (12xjRe a )1/2]} ,
(7.168)
where Re a denotes the Reynolds number based on the shell radius.
For small amplitude oscil1ations, the unsteady perturbation velocities Vxl - Ul and Vel
are negligible with respect to Ul; hence the magnitude of the fluid velocity VI relative
to the shell surface, considered at the edge of the boundary layer, can be assumed to be
invariant in time, since VI = [Vlí + VJi]1/2 :::: Ul. However, the direction of Vi varies in
time with an angle tJ, defined by tan tJ :::: Vel j Vxl.
The circumferential velocity Vel may be obtained approximately from the value of Ve
in the core flow near the wall (r = 1) t; that is, from equation (7.163)
Vel :::: Ve(, 1, e, T) = -Uon4J(, 1) sin ne e AT . (7.169)
At the shell wall, the direction of the flow velocity relative to the x-axis is
tan tJ = Ve(, 1, e, T)j Ul :::: -[n4J(, l)j U ] sin ne e AT ,
(7.170)
as shown in Figure 7.58, where V x Ir=a-8* :::: Ul, Ul :::: U, have been utilized,
Now, introducing the coordinate system (X, Y), as shown in Figure 7,58, Vx =
V x cos tJ + Ve sin tJ, which at r = a - 8* (x) becomes
VXI = Vxl cos tJ + Vel sin tJ;
(7.171)
hence, equations (7.166) may now be recast in the new coordinate frame and linearized
to give the remarkably simple resu1t
O:::: Vf! ( r aVX ) .
r ar ar
(7.172)
t Actually, the circumferential velocity at the boundary-Iayer edge should be calculated accounting also for
the displacement thickness of the boundary layer, cS*(x); an iterative process could be used in this respect.
However, the effects of the second iteration are expected to be small enough for this iterative procedure to be
considered an unnecessary refinement in this analysis,

680
SLENDER STRUCTURES AND AXIAL FLOW
Thus, the problem of unsteady flow in the boundary layer has been reduced to one of
quasi-steady flow, which may be treated by standard methods. This has been achieved
through the assumption that the flow velocity in the boundary layer is sensibly constant
in magnitude, but fluctuating in direction with respect to time, as characterized by the
angle f3.
Next, adapting Lighthi1l's approach for boundary layers on bodies of revolution to the
problem at hand (Mateescu 1984), one finds
Tx = 1PU 2 C[[I/(Re a x)]1/2,
Te = 1PU 2 C[[I/(Re a x)]1/2 [-ncp(, OI U ] sin ne e};r,
(7,173)
with C[ = 1/.J3. It is noted that Tx is associated with the steady surface traction on the
shel1; since here we are concemed with unsteady viscous effects, the only quantity of
interest is Te.
Proceeding in a similar manner for high flow velocities, in the case of a turbulent
boundary layer we have
Te = !pU 2 C t [1/(Re a x)]1/5 [-ncp(, OI U] sin ne e AT ,
with C t = 0.0592, as recommended by Schlichting (1960).
(7.174)
7.5.3(b) The unsteady f/ow in the centra/ core
The unsteady flow in the axially nonunifonn (because of boundary-layer growth) central
core is govemed by the same set of equations as for inviscid flow, with the important
difference that the impermeability boundary condition, equation (7.35), is now applied
at r = a - 8* (x) instead of at r = a. Although the manipulations in this case are more
complex (11::Z and ;:; being functions of  here), the same procedures as for inviscid flow
apply, leading to the following expression for the pressure:
P - Poo = -pUJ L Cm {).2 [11()CPmH() + ()CPmT()]
m
+ UA [211()cp:nH() + 2()cp:nT() + 11;;:()CPmH() + '()CPmT()]
+ U 2 [(f3ml 1)2{11::Z()CPmH() - ()CPmT()}
+ 11;;: ()cp:nH() + ' ()cp:nT()] } cos ne e AT , (7.175)
where
11 () = J n [(1 -l/)f3ml I] / { (f3ml I)J [(1 - '8*) f3m I I]} ,
;:; () = In [(1 - '8*)f3ml I] / { (f3ml l)I [(1 - -ť')f3ml I]} ;
(7,176)
the prime denotes the derivative with respect to the argument, I = Lla, and -ť' = 8* la.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
681
Table 7.7 Critical flow velocities for divergence,
U c d, according to inviscid theory and viscous the-
ory, accounting only for unsteady viscous effects
(Pai:doussis & Mateescu 1987).
System
Inviscid
Viscous
Air-rubber
Steel-water
0.598
0,064
0,612
0,065
7.5.3(c) Calculations and discussion
Ca1cu1ations have been conducted for both inviscid and viscous flow, and the results com-
pared. In the case of viscous flow, the eigenva1ues Ak are no 10nger pure1y imaginary;
they are comp1ex, due to the unsteady viscous effects, although the rea1 component (for
flows be10w the thresho1d of divergence) is quite sma11. Results for the second circum-
ferentia1 mode (n = 2) for both inviscid and viscous flow are presented in Figure 7.59,
showing the variation ofthe dimensionless eigenfrequencies Qk = 'Im(Ak) versus U for
two systems: the so-called rubber-air system (defined in Section 7.5.2) and a stee1-water
system (pj Ps = 0.128, Ua = 5387, v = 0.3, ť = Lja = 25.9, hja = 0,0227), so
named according to the materia1 of the shell and the fluid being conveyed. The ca1cu1ations
have been conducted with four terms in the Ga1erkin expansion, sufficient for convergence
with excellent accuracy.
The critica1 flow ve10cities are compared in Tab1e 7.7 (cf. Tab1e 7.5). From Figure 7.59
and Tab1e 7.7, it is quite clear that unsteady viscous effects are very weak, both in terms
of stability and of the eigenfrequencies of the system; indeed, for the stee1-water system,
the effect is virtua11y nonexistent. (Here it shou1d be mentioned that a1though U cd for the
rubber-air system is higher by a factor of 10, the dimensiona1 va1ue is actua11y 10wer.)
Severa1 of the results in this section have since been confirmed by Zhang et al.
(1999) by means of a finite-e1ement formu1ation using axisymmetric she11 element s
(cf. Section 7.3.3); the new results agree close1y with those here.
7.5.4 Unsteady viscous effects: general treatment
ln a sense, the treatment here is a genera1ization for flowing fluid, as deve10ped by
E1 Chebair et al, (1990), of what is presented in Section 2.2.3 for shells filled with quiescent
viscous fluid. Unlike the work in the preceding section, the flow here is fully deve1oped,
and the unsteady viscous effects (again the on1y viscous effects to be considered, and
for the same reason) are determined from the linearized Navier-Stokes equations. The
Reyno1ds number is supposed to be sufficiently 10w, however, for Reyno1ds stresses to be
neg1ected [Section 2.2.1(f)].
7.5.4(a) General formulation
The equations of motion of the shell are simi1ar to (7.147), but with the steady viscous
terms suppressed, ql = q2 = q3 = O, and unsteady viscous stresses added to a11 terms on

682
0.02
Icf
"r
'"
.0
<::
'"
6- 0,00
'"
<I::: 0,0
<::
'" (a)
b/)
<>
V>
V> 0.04
'"
í3
,::I
13
'"
S
es
0,03
SLENDER STRUCTURES AND AXIAL FLOW
0.06
0,04
0,02
0.15
0,01
0,00
0,00
(b)
Dimension1ess flow ve1ocity, U
Figure 7.59 The dimensi oni es s eigenfrequencies r2k = Re(r2k), k = 1,2,3,4, for n = 2,
as functions of fJ, (a) for the rubber-air system, and (b) for the steel-water system, defined in the
text: - - -, inviscid theory; -, theory taking into account unsteady viscous effects, in (b) the
results from this latter theory being sensibly coincident with those of inviscid theory (PaYdoussis &
Mateescu 1987).

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
683
the right-hand side:
{ ( 82U qx ) ( 82V qe )
{.cI, .c2, .c3} (U, V, w) = Y 8t2 - Psh ' Y 8t2 - Psh '
_y ( 8 2 W _  )}
8t 2 Psh '
(7.177)
where qx, qe and qr are the unsteady viscous stresses in the respective directions.
The equations describing the flow of an incompressible viscous fluid are the continuity
equation and the Navier-Stokes equation,
V.V = O,
P [8V j8t + (V.V)V] = -v p + f.LV 2 V,
(7.178)
(7.179)
respectively, where V is the ve1ocity, p the pressure, P the density and f.L the dynamic
viscosity of the fluid. .
The flow velocity may be expressed as
V = Vep + V x 1/f + U(r),
(7.180)
where U (r) is the steady viscous mean flow velocity in the axial direction, whi1e Vep and
V x 'ft are the unsteady inviscid and viscous parts of the velocity perturbation associated
with she11 motions. From equation (7.180), the ve10city components are given by
V x = U(r) + 8ep + 1/1e + 81/1e _  81/1r ,
8x r 8r r 8e
18ep 81/1r 81/1x 8ep 181/1x 81/1e
Ve=--+---, V r =-+----,
r 8e 8x 8r 8r r 8e 8x
where 1/1x, 1/1e and 1/1r are the components of 1/1 in the subscripted directions, Substituting
for V from equation (7,180) into equation (7.178) gives
(7.181)
v 2 ep = O,
(7.182)
and from equation (7.179), after linearization, one obtains
P [(8j8t)(Vep + V x 1/f) + {(Vep + V x 1/f). V} U + (U. V)(Vep + V x 1/f)]
= -V p' - V Po + f.LV 2 [Vep + V x 1/f] + f.LV 2 U, (7.183)
where it has been assumed that
p = Po + p',
(7.184)
with Po being the steady pressure and p' the unsteady perturbation in pressure.
Moreover, since the steady viscous flow must satisfy the Navier-Stokes equation a1so,
we have
P [(U. V)U] = -V Po + f.LV 2 U;
(7.185)

684
SLENDER STRUCTURES AND AXIAL FLOW
then, subtracting equation (7.185) from (7.183) gives
p [(a/at)(VeI> + V x 1fr) + {(Vel> + V x 1fr).V} U + (U,V)(VeI> + V x 1fr)]
= -V p' + MV 2 (V x 1fr). (7.186)
Equation (7.186), when written in the (x, O, r) co-ordinate system resu1ts in three equa-
tions, and four unknowns 1frx, 1fr(}, 1frr and p'. To solve this under-determined system of
equations, a constraint is introduced on the components of 1fr:
p [(a/at)(V x 1fr)] = MV 2 (V x 1fr);
(7.187)
then, subtracting equation (7.187) from equation (7.186) yie1ds
p [(a/at)VeI> + {(Vel> + V x 1fr). V} U + (U. V)(VeI> + V x 1fr)] = -V p'. (7.188)
The constraint on 1fr may be viewed as follows. Basica11y, the ve10city perturbation vector
V x 1fr is introduced to represent the viscous perturbations. Thus, it is possib1e to derive
these ve10city perturbations separately, as in equation (7.187), and to then inc1ude their
effect on the pressure perturbation in (7.188). In this sense, this device is sinů1ar to Chorin' s
(1973) operator-sp1itting scheme.
Equation (7.187) may be rewritten as V x [pa1fr lat - M v 2 1fr] = O, which is satisfied
by
pa1fr/at - MV 2 1fr = O.
The fluid stresses - cf. equations (2.64) and (2.65) - are given by
(7.189)
[ a V r a V x ]
-rrx = M ax + a;:- ,
-rr(} = M [ r  ( V(} ) +  a Vr ] ,
ar r r ao
aV r
-rrr = -p + 2M-.
ar
(7.190)
They mayeach be splitintotwocomponents, forexamp1e, -rrx = 'rrx(x, r)+-r:x(x, O, r, t),
where 'r rx etc. are the steady components responsib1e for pre-stressing the shell and
generating the steady loads, neg1ected in this ana1ysis, whi1e -r: x etc, are the unsteady fluid
stresses of interest here, given by
I [ a2e1> 1 a 2 1frx a 2 1fro 1fr(} 1 a1fro a 2 1fr(} 1 a1frr 1 a21frr ]
-rrx = M 2 aXor + -;. axao - ax2 - --;:z + -;: a;:- + or 2 + r 2 00 - -;: oOor '
I [ -2 ael> 2 a 2 e1> 1 a1frx a 2 1frx 1 a 2 1frx
-rrO = M --;2 ao + -;. aoar + -;:a;:- - ---a;:z + r 2 ao 2
1 a 2 1fr(} a 2 1frr 1 a 1frr ]
--;: oOox + axar - -;: ax '
I I ( a2 e1> 1 a1frx 1 a 2 1frx a21fro )
-rrr = -p + 2M ar2 - r2 ar; + -;: aoar - axar .
(7.191)
ln tenns of the foregoing, the unsteady fluid loads in (7.177) are given by
qx = -r:xlr=a'
q - -r ' I
() - r(} r=a'
qr = --r:r Ir=a'
(7.192)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
685
7.5.4(b) Boundary conditions
The velocity of the fluid at the wall must be equa1 to the velocity of the she1l; thus the
boundary conditions may be expres sed as
V x Ir=a
Du
--,
Dt
Ve Ir=a
Dv
--,
Dt
V r Ir=a
Dw
=-,
Dt
(7,193)
where the flow velocity components are as in equations (7,181), and D/Dt = o/at +
ua/ox.
It is understood that in viscous theory, the no-slip condition on the wall must be applied.
This requires that the mean flow U(r) at the wall be equal to zero. However, in applying
the no-slip condition in this analysis, difficulty is encountered in obtaining the centrifugal
forces which normally occur, say, in an inviscid treatment of the problem: these are the
forces which cause divergence (buckling). In the absence of the centrifugal forces, the
problem is not well defined physically (DowellI987),
To understand the physica1 nature of this difficulty, recall the unsteady pressure
derived by potential flow theory, in the form of equation (7.64): pf = - paI(n,).,)
x [a 2 w/at 2 + 2U e o 2 w/oxat + u;a 2 w/ox 2 ], where U e is the constantmean flow vel 0-
city. It is also recalled that divergence, a static instability, is caused by the centrifugal
forces, that is, by the last term in the expression, which should also be present in viscous
analysis. However, it is seen that by applying the no-slip boundary condition, U (r) = O
at r = a, the centrifugal (and Coriolis) forces vanish, This is the essence of the difficulty,
It should nevertheless be made clear that another facet of this difficulty is associated with
applying the boundary conditions at the mean location of the boundary (shell wall), as
if it were immobi1e, rather than the instantaneous surface position. (Here the reader is
referred to Chapter 11, for example, Section 1I,5,2(f), where it is seen that, by applying
the boundary conditions on the moving wall, stiffness terms are once again retrieved.)
This problem, of the appropriate boundary condition on a moving surface in viscous flow,
has been dealt with in different ways by different researchers, as described below.
Dowell (1971), in his paper on the generalized aerodynamic forces on a flexible plate
undergoing transient motion in shear flow, has found that application of the no-slip con-
dition at the wa11 resulted in overestimating the stabilizing effect of the boundary layer, To
overcome this problem, he applied the boundary condition at a distance zo from the fixed
reference frame, where U (zo) i= O. Ventres (1975), studying the effect of a boundary
layer on lifting surface aerodynaDŮcs with shear flow, has similarly applied the boundary
condition at some small distance zo i= O, away from the fixed reference surface, in order
to determine the pressure on the lifting surface; he then let zo -+ O and was able to obtain
an expression for the pressure which was independent of ZOo Pai'doussis & Mateescu
(1987) ha ve dealt with this difficulty by applying the boundary condition at the edge
of the boundary layer thickness, where the mean flow velocity is set equal to U e (see
Section 7.5.3).
The latter approach and Ventres' cannot be used in this analysis, since the flow here is
assumed to be fully developed. Instead, the effect of the mean flow velocity is incorporated
at the boundary, in two different ways.

686
SLENDER STRUCTURES AND AXIAL FLOW
(i) Method 1. In this scheme, a slip condition is permitted at the moving wall, and an
averaged flow velocity is presumed to act thereon. The averaging is done as follows:
l a 2rU(r)
U av = 2 dr,
O a
(7.194)
where U(r) = Umax(l- rja)l/s, and s is a constant depending on the Reynolds number
(Schlichting 1960).
(ii) Method 2. In this method, the boundary condition is applied at a distance 8 from the
wall, which is equivalent to the shell deformation in the radial direction. The associated
velocity is described in power-law form,
( 8 ) I/S
Ua = U max -;;
(7,195)
This method is similar to Dowell's (1971): the boundary condition is applied at different
distances, Zo, away fmm the fixed reference surface, thus obtaining a plot of the variation
of the dynamic pressure as a function of sUĎace deformation (ca1culated from nonlinear
theory). In this way the appropriate value of zo may be determined, which corresponds to
the correct dynamic pressure.
It is important to mention here that, there might be different and better methods to
handle this problem [see, e.g., Section 11.5.2(t)]. Mathematically speaking, the methods
utilized here are not very rigorous; indeed, they have been selected on physical grounds
and on the basis of previous work by others.
7.5.4(c) Method of solution
A travelling wave type of solution is utilized, much like that in Section 7.2.3, in
which w(x, (), t) = C n cosn() exp[i(Qt - kx)], and similarly for all other quantities:
u, v; '!/fx, '!/fr, '!/fo; -r: x ' -r;o' -r:r; and pf (El Chebair et al. 1990). This eventually leads to
four Bessel-type equations, with solutions of the form
fjJ(r) = CI In(kr),
'!/f X< r) = C3 I n(fJr),
1/Jr (r) = - 1/Jo (r) = CSln+l(,8r),
(7.196)
where (32 = (iwjv) + k 2 , and the modified Bessel functions are as in the foregoing, By
substituting (7.196) into (7.193) with U av or Ua, the three constants are determined.
Next, substituting the expressions for fjJ(x, (), r, t), 1/JX<x, (), r, t) etc. as defined above
into (7.191), and making use of the form of -r: x etc" also as defined above, the unsteady
fluid forces qx = 7ix(r) cosn() exp[i(Qt - kx)] and qO and qr, similarly formulated, are
found. Hence, equations (7,177) may be written in consistent form, which may then be
solved with the aid of the Galerkin method.
7,5.4(d) Numerical results
Solutions ha ve been obtained for the so-caI1ed /0 -gap and lÓO -gap steel-water systems,
defined in Sec ti on 7,5,2, and also with annular flow [see Sec ti on 11.7.1(c)], for shells with
simply supported ends, for simplicity.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
687
Table 7.8 Critical flow velocities for diver-
gence' U cd, obtained by the Fourier trans-
form and the travel1ing wave solutions for
the !b-gap steel-water system, with inviscid
flow (El Chebair et a/. 1990),
Method
Ucd
Fourier transfonn
Travelling wave
2.00 x 10- 2
1.94 x 10- 2
The inviscid version of the program was first tested against Weaver & Unny's (1973)
calculations, and fair agreementwas obtained: U c d = 5,10 x 10- 2 and U cf = 6.50 X 10- 2
by Weaver & Unny and U cd = 5.00 X 10- 2 and U cf = 6,80 X 10- 2 by El Chebair et a/.
(1988),
The travelling wave solution has also been compared to the Fourier transform method,
identical to Pai'doussis et al. (1984a); as may be seen in Table 7.8, the results are close
(see also Table 7.2 and Section 7.3.1),
The nature of the travelling wave solution may be discussed with the help ofFigure 7.60.
Two roots, denoted by  and Q -, are obtained t (here real, but generally complex
conjugate), corresponding to backward and forward travelling waves (see Section 7.4).
=+ -- - -
The values of Q and Q are equal for U = O, but different for U i= O. With increasing
U , Q- and Q+ approach each other and, for U = 1.94 x 10- 2 , both Q- and Q+ become
positive, so that no self-sustaining forward-going wave is any longer possible; this may
be considered to correspond to the onset of divergence. Eventually, the two roots coalesce
and give rise to coupled-mode flutter at U cf  2,2 X 10- 2 , If the exponent in the solution
is changed to exp[i(Qt + kx)], then the mirror-image results are obtained, flipped across
the Im (Q) -axis, with the same critical velocities.
The effect ofunsteady viscous forces on the dynamics may be assessed from Table 7.9
and Figure 7.61, It may be seen that the effect on stability is small, especially if the
U av approximation is used, tending to stabilize the system - similarly to the results
presented in Sec ti on 7.5.3. However, the viscous fluid also generates a damping effect, as
expected.
The conc1usion is that, for internal flow, unsteady viscous effects are small and, insofar
as stability is concerned, perhaps even negligible, This is the main reason why no further
effort for refinement has been expended. The case of annular flow in coaxial shells is
discussed in Section 11.7.l(c),
7.5.5 Confinement and viscous effects on cantilevered shells
This topic has been studied by Pa'idoussis et a/. (1991), presenting both an inviscid analysis
and one in which steady viscous effects are taken into account. As the essentials of this
analysis have already been presented in conjunction with other facets of the problem, it
t"Q = Qa/ {E/[p,(1 - v 2 )]} 1/2; U = U/ {E/[p,(1 _ v2)]} 1/2,

102
I&'

0,01
0,021 0,0
2 3
(Q) X 10 3
Va1ues of U i 
0,0 0,01 0,0194
-3 -2 -1 B
0,022
0,021
1
0,022
-10- 3
0,026
_10- 2
Figure 7.60 Argand diagram for the dimensionless frequency, Q, as a function of the dimension-
less flow velocity U; inviscid-flow solution with ei(Qt-kx) : 0, Q+; ., Q- (El Chebair et a/. 1990).
10- 3
 0,022
,-..
I 10- 4
'-"'

-
B / Va1ues OUi
I I 0.021_ 0,0194 0.01_ ,0,0_ I
-3 -2 -1 B ° 1 2 3
Q+) x 10 3
_10- 4
 0.022
-10- 3
Figure 7.61 Argand diagram for the dimensionless frequency g+ as a function of the dimension-
less U with and without unsteady viscous effects; solution with ei(Qt+kx) : ., inviscid; _, unsteady
viscous effects taken into account (El Chebair et a/. 1990).

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
689
Table 7.9 The critical flow velocity for divergence,
U c d, obtained with (i) unsteady viscous effects taken
into account and the boundary conditions applied via
Method 2 at different "8 = o/a, or with the average
velocity approximation of Method 1, and (ii) invis-
cid flow theory; for the l -gap system defined in
Section 7,5,2, but with simply supported ends,
With unsteady viscous forces
Method 2, "8 = 0,01
Method 2, "8 = 0,02
With unsteady viscous forces
Method 1 (using U a v)
Inviscid flow theory
2,3 x 10- 2
2,2 x 10- 2
1.95 x 10- 2
1.94 x 10- 2
wil1 not be repeated here, in the interests of brevity, The interested reader wil1 find it in
full in the original reference, as welI as in the sections to be cited in what follows,
The equations of motion are equations (7,147), and the steady viscous terms are
detefllÚned much the same way as in Section 7.5.2. The Fourier transform method is
used to detefllÚne the perturbation pressure, as outlined in Section 7.3.1, leading to
equations (7.99) of Section 7.3,2, involving also the use of an appropriate outflow model,
as discussed therein, ln the case of the canti1evered system, because of the more comp1ex
form of the boundary conditions, the extended Ga1erkin method is used to obtain solu-
tions - see Section 7.3.2 and equation (7.97).t Here, some of the results are presented
directly,
7,5.5(a) Inviscid results: effect of confinement
The inviscid model is tested first against Pa'idoussis & Denise's (1972) travelling wave
theory and experiments, as shown in Figure 7.62. EssentialIy, this figure is identical to
Figure 7.27, agreeing almost fulIy with Shayo & ElIen's (1978) results, as should be,
The effect of confinement is investigated in Figure 7,63, where the critical flow velocity
ofthe so-called 1 -gap steel-water system, defined in Section 7.5.2, is compared to that
of the 160 -gap system, in which the outer, rigid cylinder radius b is related to the inner
flexib1e shell radiu s a by alb = 100/101; in both cases, b = 0,1 m, h = 0,5 mm,
L = lm,ps = 7.8 x 10 3 kg/m 3 ,p = 10 3 kg/m3,E = 2.0 x I0 11 N/m 2 ,v = 0.3,
Uo = 5308 mis, and pa I (Psh) = 23.30; the stagnant fluid in the annulus is the same as
that flowing within: water.
ln the majority ofthe results in Figure 7.63, the system loses stability by flutter. However,
in some cases stability is lost by divergence, namely for n = 6-9 for the 1 -gap system and
n = 8, 9 for the 160 -gap system. In these latter cases, flutter, of the single or coup1ed mode
tConvergence of the method has been studied in detail (Pa'idoussis et a/, 1991), and eigenfrequencies of
quite satisfactory accuracy are obtained with (i) M = 6, that is, six admissib1e functions, (ii) an integration
step for the second of equations (7,99) of b.Ci = 2, and (iii) the range of li of [-00, 00] approximated by
[-200,200],

690
2,0
\.6
I:::;'''
tf.
'g 1.2


i<I 0.8
<>
.::
.;:::
U
SLENDER STRUCTURES AND AXIAL FLOW
. Tests }
Theory by Paidoussis & Denise (1972)
- Paidoussis et a1. (1991) theory (M = 6)
0.4
.
0.0
O 5 10 15 20 25 30 35
L/a
Figure 7.62 Comparison of the critical flow velocities for flutter of the cantilevered shell
studied by Pa'idoussis & Denise (1972) with the inviscid version of the Pai'doussis et al. theory
(Pai'doussis et al. 1991).
8
o 1/IO-gap system
6 <\ 1/100-gap system
N
:=:
x 4
I:::;'''
2
o
O
2
4
6
8
10
Círcumferentia1 mode, n
Figure 7.63 The dimensionless critical flow velocity, U c , obtained by inviscid flow theory for two
steel shells conveying water and surrounded by quiescent water in the annulus, to show the effect
of confinement. Parameters are given in the text; t-gap system signifies that (annular gap)/(shell
radius) = 1/ k, -, Flutter boundary; - - - , divergence boundary (Pai'doussis et al. 1991).

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
691
variety, folIows at higher U ; for example, for n = 7 and the l -gap system, divergence
occurs for U = 0.03427 and coupled-mode flutter for U = 0,03895, in both cases in the
m = 1 mode(s).t
It is seen that tbere is a very substantial reduction in tbe critical U c for each mode as tbe
annular gap si ze is diminished. Tbere are two main reasons for this. First, and this applies
to all the results, it is noted that the gap here is reduced by increasing a, while keeping
b constant; thus, a lower (b - a)/a implies a higher a and, since h remains constant, a
lower h/a* by appmximately 9%, Second, and this applies only to cases involving flutter,
a smalIer gap means a large increase in the added mass of the stagnant annular fluid. This
latter is mainly responsible for tbe 42% maximum reduction in U c, corresponding to the
lO-fold diminution in the annular gap. For flutter, the increase in added mass may be
thought of as an effective reduction in stiffness, hence causing a reduction in the critical
U c for each mode, as well as in tbe U; , tbe overall critical U c. Tbis effect is weakest
for n = 1.
7.5.5(b) Steady viscous effects
The intluence of steady viscous effects on tbe dynamics of the l -gap steel-water system
may be assessed fmm Table 7.10. It is seen that all modes are slightly stabilized by
the inclusion of viscous terms - even, albeit very slightly, the n = 1 mode, which
corresponds to beam-like motions oftbe shelI; it is recalIed that, according to beam theory
applicab1e to thicker cy1inders conveying fluid, the net effect of viscous forces is exactly
zem (Pai'doussis 1998). This resu1t for n = 1 constitutes a very important finding: even
by higher-order tbeory - shelI tbeory may be viewed as such for beam-1ike motions-
steady viscous effects on the dynamics of canti1evered tubes are minimal, if not nulI.
It is also seen in Tab1e 7.10 that stability according to inviscid theory is lost in the fourth
circumferential mode, n = 4, whi1e with tbe viscous forces taken into account in tbe n = 3
mode; tbe overall stabilizing effect due to the viscous terms is 15% on U ; - the asterisk
again being associated with the mode giving the smalIest U c.
Tbe physical explanation for tbis stabilizing influence of viscous effects is tbe same as for
clamped-c1amped shelIs (Pai'doussis et al. 1985): the steady 10ads due to viscosity induce
a hoop stres s and a tensi1e stress, both of which are 1argest at  = O and vanish at  = 1.
Both hoop and tensile stresses effective1y render tbe shelI stiffer, raising U c. However,
this effect is not as pmnounced as for shelIs with both ends supported. This 1ends support
to the physica1 explanation given for tbe strength of this effect in Section 7,5.2: it is the
Poisson-induced axial tension associated with the hoop pre-stress that causes most of the
stabi1ization for shelIs with supported ends; in the case of cantilevered shelIs, however,
the presence of a free end greatly diminishes this effect.
Further ca1cu1ations have been conducted on the effect of tbe shelllength on stability;
the shell radius is fixed and L varied, as shown for the /0 -gap system in Figure 7.64. It is
tNo differentiation is made in the notation 'coupled-mode flutter' between (i) cases where the two modes
involved are of the same mode-number (m) but one is from the right-hand plane of Q and the other from the
left-hand plane (e.g. Figures 3.9 and 3.14), and (ii) cases where modes of different m are involved.
*Thus, this is peculiar to the, in retrospect, unfortunate manner in which the parameters have been varied
(for computational convenience), This is the main reason for the lowering of the divergence boundary for the
16o -gap system in Figure 7,63 - misinterpreted in Pai'doussis et ul, (1991),

692
SLENDER STRUCTURES AND AXIAL FLOW
Table 7.10 The dimensionless critical ftow velocities for modes n = 1-7 for the 10-
gap steel-water cantilevered system subjected to interna1 ftow, according to inviscid
and viscous theory (i.e. inciuding steady viscous effects), with the axial mode number
(m) involved in each case shown beneath in brackets, Cases where stability is first lost
by divergence are marked by t, The accuracy of the computed U e is ::1:0,00002,
Theory Critica1 flow velocity, (je
n = 1 n=2 n=3 n=4 n=5 n=6 n=7
Inviscid 0.06899 0.03107 0.02772 0.02705 0.02990 0.02911 t 0.03427 t
(m = 2) (m = 2) (m = 3) (m = 4) (m = 5) (m = 1) (m = 1)
Viscous 0,06904 0,03261 0.03117 0,03356 0,03488 0.04808 0,11065
(m = 2) (m = 2) (m = 3) (m = 4) (m = 4) (m = 4) (m = 4)
recalled that for inviscid theory, U; decreases more or less monotonically with increasing
L / a; moreover, the mode associated with U; increases as L / a becomes smaller. It is seen
in the figure that the situation becomes more complicated when steady viscous effects
are taken into account. According to viscous theory, as L/a is increased, the steady-state
pressurization at the upstream end becomes higher (for inner and annular pressures equal
at the downstream end). This effect is very pronounced for shells with supported ends, but
as may be seen in Figure 7,64 it is less than dramatic here; nevertheless, the difference in
U ; between inviscid and viscous theory does increase with L/a: 10% for L/a = 5, and
52% for L/a = 30 - hardly negligible,
Finally, in view of the results obtained here and especially those in Sections 7.5.3
and 7.5.4, no attempt has been made to study unsteady viscous effects for cantilevered
shells conveying fluid.
7.5.5(cJ Experiments
The discussion oť this work (Nguyen et al. 1993b) could have been part of Section 7,2.4.
It is presented here mainly because the new apparatus in which the experiments have been
conducted involves an outer, containment cylinder - this apparatus being a variant of one
constructed for the study oť coaxial shells conveying fluid.
The apparatus is presented in Figure 7.65. For the experiments to be described, the inner
radiu s of the plexiglas cylinder was b = 63.35 mm, while the shell radius a = 24.79 mm,
so that this is a -gap system; the shell thickness was h = 1.37 mm - hence not truly
a thin shel!. The Silastic E silicone-rubber shell was cast in a special mould, as described
in Appendix D (Volume 1). The fluid was air.
The only type oť instability observed was flutter, which, in these tests, was always
associated with n = 2. Flutter was ťound to be quite violent and remained so even if
the flow was slightly reduced. It might be of interest to mention that the first shell tested
for instability was tom apart because of the intensity of the flutter and high stresses in
the shell, Once flutter occurred, the ťree-end cross-section of the shell became ova1 to
such an extent that one side oť the inner wall oť the shell touched the opposite one; the
cross-section appeared to have the shape of a bow-tie, with the axes oť symmetry rotated
by 90° half a cycle later.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
693
8
6
N
 n=4
x 4
*u
I:::, 3
3 2
2 3
2 2 2
O
O 5 10 15 20 25 30
Lfa
Figure 7.64 The most critical (for any n) value of U c , denoted by U; , for the to-gap steel-
water system, as a function of Lfa, with the circumferential mode involved in each case noted
near each data point: 0, inviscid flow theory; !:::", theory taking into account steady viscous effects
(Patdoussis et al. 1991)
Air out
Inner shell
(silicone rubber)
Outer cylinder
(plexiglass)
Table
Upstream
plenum
t
Airin
(compressed air supply)
Figure 7.65 Schematic of the 'new' test apparatus used for studying the dynamics of shells
conveying fluid (Nguyen et al. 1993b).

694
SLENDER STRUCTURES AND AXIAL FLOW
Table 7.11 The effect of annular fluid on the experimental
critical internal flow velocity, U ;, for different lengths of a
cantilevered silicone-rubber shell conveying aif.
Critical flow velocity, 17; (mis)
(b - a)/a
1.5
00
L/a = 8
63,7::1:: 2,7%
63,2 ::I:: 1.7%
L/a = 7
70,3 ::I:: 1.2%
70,1::1:: 1,0%
L/a = 6
78.0 ::I:: 0,6%
78,1::1:: 0,5%
Although only one annular gap is considered, the effect of the quiescent annular fluid
(air) on the stability of the shell could easily be examined by removing the plexiglas
cylinder, thus making the gap/radius ratio, g j a = (b - a) j a -+ 00. Table 7.11 shows
experimental resu1ts of the critical flow velocity U; with and without the plexiglas cylinder
installed; these results were obtained from test s on one shell. For three different lengths
of the shell tested, there were hardly any changes in Ir; as g j a was increased from 1.5 to
00, the differences being within possible experimental error, lndeed, this observation was
expected: for air as the annular fluid, the effect would be strong only for very small gja;
however, the violence and amplitude of the flutter precluded using very small g j a (e,g,
gja = 1/10).
Figure 7,66 shows the analytical and experimental results for the frequencies of the
shell (Lja = 8) at various flow velocities. The experimental resu1ts were obtained by
placing two fibre-optic displacement transducers at varying azimuthal or axiallocations;
then the cross-spectral density (CSD) and corresponding phase may be used to identify
the mode of vibration. The CSDs and the corresponding phase plots indicate that the
dominant frequencies excited are mostly associated with n = 2. In general, the measured
frequencies slightly increase with flow velocity; this may be due to pressurization, tending
to expand the shell radially, hence rendering it stiffer, On the other hand, there are slight
reductions in the predicted (by viscous theory) frequencies associated with all m -:F 1
(except for m = 2 and n = 3) - at least for the values of n considered. In spite of
this opposite trend, theory and experiment are in fairly good agreement; for example, for
n = 2 and m = 2, the discrepancy between theory and experiment varies from about 8%
at U = 29 rn/s to 24% at U = 64 rn/s.
The effect of L j a on the stability of the system is presented in Figure 7.67. Both theory
and experiment agree that (i) U ; is increased as Lja is reduced and Oi) the only type of
instability is flutter, which is associated with n = 2 for the range of Lja and the other
shell parameters for this set of experiments. Quantitatively, the agreement is considered
to be fair, and its extent appears to be a function of Lja. lt becomes better as Lja is
increased; predictions of U; differ from the test results by an amount varying from 27%
(based on the smaller test value) for Lja = 5, to 3.5% for Lja = 7.9. Nonlinear effects
undoubtedly play a role in this, since the amplitude of limit-cycle motion is several times
the shell thickness; refer to Section 7.8.
One important difference with the experiment s by Páidoussis & Denise (Section 7.2.4)
is that the experimental values of Ir; here are higher than the theoretical values, even
though these latter are for viscous theory. One possible reason could be that the shells

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
695
(a)
20 40 60
U (mis)
80
m=4
___ 60 m=3
N
::c m-2
'-'
>,
5 40
50

J,>., 20
O
O 20
(c)
80
80
60
40,
20
O
O
(b)
,
-m=3
80
,
m-2
. m=1
N 60
;;
>,
g 40

J,>., 20
... ..........
---
.-. ....
-. ..
m=2
m=1
O
O
20
40
U (mis)
60
80
-
00
m = I
40
U (mis)
60
80
Figure 7.66 Predicted and measured frequencies of flow-induced shell vibration as functions of
the internal flow velocity: (a) n = I, (b) n = 2, (c) n = 3; g/a = 1.5, L/a = 8, -, Theory
taking steady viscous effects into account; ., experiment (Nguyen et al. 1993b).
120 . . I
100 .
. .
. .
80 . . .
. .
--- --- . .
'" . .
S 60 ---
'-'

40 -
20
O I
4 5 6 7 8 9
L/a
Figure 7.67 Overall critical internal flow velocity for flutter, U;, as a function of length-to-radius
ratio ofthe shell, L/a, for g/a = 1.5, -, Theory taking steady viscous effects into account;.,
experiment (Nguyen et al. 1993b),

696
SLENDER STRUCTURES AND AXIAL FLOW
here are considerably thicker (a/ h = 18,1 rather than 44); however, this may also be a
function of the different apparatus used,
7.5.6 Concluding remarks
It has been shown in this section that motion-related unsteady viscous effects do not affect
in an important way the perturbation pressure acting on the wall of a shell conveying fluid. t
Hence, there is no overwhelming imperative for pursuing their determination by means
of more sophisticated theory: the perturbation pressure may be adequately calculated via
inviscid flow theory or by means of the approximate theories presented here.
ln contrast, steady viscous effects, manifesting themselves as mean pressurization and
surface traction pre-stresses on the shell, are genera1ly important and ought to be taken into
account, specially for shells with supported ends, as should be the effect of confinement
when the surrounding fluid is dense.
7.6 SWIRLlNG FLOW AND ROTOFLUIDELASTICITY
7.6.1 Introductory remarks and literature review
The study of the dynamics of shells containing swirling (helical) flow has been inspired by
practical problems all along. For example, swirling flow in dual spool aircraft jet-engines
is not uncommon; also in rotating drums used in the process industry, and in some nuclear
engineering applications.
Depending on the manner of generation of the rotation, the vortical flow in any cross-
section of the cy 1indrica1 system may be thought of as free vortex, e.g. in annular geometrie s
with a cylindrical inner shell or drum rotating, and the outer containment cy1inder or shell
stationary, as in Srinivasan (1971) for example; or it may be 1ike solid rotation ofthe fluid,
with the containing shell rotating and the fluid within also, with the tangential velocity
proportional to radial distance, for example, as in Lai & Chow (1973). A degenerate case
is that of a rotating fluid with no axial flow component, for example, as in Howard et aZ.
(1972), M,T, Landahl aptly coined the word roto.fluideZasticity to describe this field.+
This field is related to that of stability of rotating fluids, with applications ran-
ging from meteorology and astrophysics to every-day engineering (e.g. centrifuge and
vortex-type vacuum cleaner designs); see, for example, Rayleigh (1916), Howard &
Gupta (1962), Benjamin (1962), Greenspan (1968), Pedley (1968, 1969), Nagib et aZ.
(1971), Maslowe (1974), Gans (1975,1976), Hultgren (1976, 1978), Bark et aZ. (1978),
Leibovich & Stewartson (1983), Wang & Rusak (1996), Shtern & Hussain (1999).
Perhaps the first study of a rotating fluid in an elastic container is due to Hough (1896),
followed up by Suess (1971) and Bauer (1971). Kelvin's classica1 work (1910) on 'liquid
gyrostats' should also be mentioned - showing for the first time that oblate spheroidal
containers rotating on fixed supports are stable, but even slightly prolate ones are violently
unstable.
tThis is not generally true for annular flow; refer to Section 11.7,2,
* According to Maslowe (1999), a co-author in Howard et a/, (1972), Landahl referred to this field as 'A virgin
field, pregnant with possibilities' - which, by the way, is as apt today as it was in the 1960s and 1970s.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
697
The aeroelastic stability of a cylindrical shell subjected to inviscid helical air-flow on the
outer surface has been investigated by Srinivasan (1971), in the first of a series of studies
inspired by the jet -engine problem. The flow field is modelled generally as a compressible
free-vortex flow (tangential flow velocity Ve ex ljr), but ca1culations with an incompress-
ible fluid flow and a s1ender-body approximation are also conducted. A linearized version
ofWilson's (1963) nonlinear shell theory is used, somewhere between Donnelľs three-
equation set and F1tigge's (Section 7.2.1), containing a1so mean stress resultants in the
axia1 and hoop directions. A trave1ling wave solution is considered, of the form
{u, v, w} = {u(r), iJ(r), w(r)} exp [i(A. - rot + nO)].
(7.197)
The pressure on the surface of the shell in this case is
p I = - P DifJ I = - p ( (JifJ + U (JifJ + a V e * (JifJ ) I '
Dt (Jt (Jx r 2 (JO
r=a r=a r=a
V e * = Ve -aQ,
(7.198)
where ifJ is the perturbation velocity potential, V; is the relative tangentia1 velocity on the
surface of the shell, Ve being the fluid velocity on the surface and Q the angular velocity
of the shell. C1early, ifJ must satisfy the impermeability conditions
(JifJ I = Dw ,
(Jr Dt
r=a
(JifJ I - O
(Jr r=b - ,
(7.199)
where a and b are the radii of the general1y rotating inner shell and of the rigid outer
cylindrica1 tube, respectively (see in set to Figure 7,68),
Actually, a1though the problem is formulated for U 1= O, ca1culations are confined to
the case of U = O, a typical result being shown in Figure 7.68, with A. = O and also Q = O.
Hence, basically, the fluid is swir1ing in the manner shown in the inset to the figure, with
no rotation of the shell itself and no axial flow, Me = Ve j co is the Mach number related
to the rotating flow.
The results are characteristic of those normally obtained for the whir1ing of shafts -
see, for examp1e, Axisa & Antunes (1992) and Section 11.6. Thus, at Me = O, the fre-
quencies of forward and backward (retrograde) waves are equal. With increasing Me, the
frequency of the backward trave1ling wave diminishes and it eventual1y vanishes, at point
D, which corresponds to divergence; thereafter both waves are forward, moving with dif-
ferent wavespeeds. Eventually the frequencies of the two waves coalesce at point F, which
corresponds to c1assical coupled-mode flutter [cf. Figure 3.4(c)]. At this point, the hitherto
purely real frequencies become complex conjugate.
It is seen in the figure that the s1ender-body results overestimate stability considerably,
as compared to incompressible or compressible flow theory, the results of which are c1ose;
this latter observation is not too surprising, considering the smallness of Me,
Experimental verification of these findings was attempted by Dowell et al. (1974),
The experiment consisted of two cylinders with an annular space between them. In the
experiment, the outer cylinder rotated; fluid was ingested from the ambient air through
holes on the cylinder near one end, by applying suction to the annulus via a vacuum pump.
The inner cylinder, a thin shell, did not rotate; it was appropriately instrumented to monitor

698
SLENDER STRUCTURES AND AXIAL FLOW
15
5
Unstable
10

N
:r: °

-5
"
'-
"
-10
-15
°
0,01
0,02
0,03
MO
Figure 7.68 The frequencies of forward and backward propagating waves in an infinitely long
aluminum shell (al h = 400, (EI[Ps(1 - V 2 )]}1/2 = 5,28 x 10 3 mis, v = 0.33) subjected to an
annular rotating air-flow of the type shown in the inset figure (hla = 1.224, Psi P = 344) as
a function of the rotational flow Mach number, Mo, for n = 2: -/'<,.-, f (Hz) obtained by
compressible flow theory; -0-, by incompressible flow theory; -0-, by slender-body theory;
- - -, the imaginary component of the complex frequency after F, for slender-body theory (not
to scale). D: divergence; F: flutter (Srinivasan 1971),
its vibration as a function of the swirling mass flow rate or of the outer cylinder rotation.
For some combinations of radial in-fiow and r.p.m, of the outer cylinder, a vortex-like
fiow-field could be established, leading to large circumferential velocities near the inner
flexible cylinder, sufficient to induce fiutter.
Experiments were conducted with steel and aluminum shells (L:::: 2.44 m, a =
0.10-0.15 m, h = 0.30-0.81 mm), as well as a '!-scale model' involving PVC shells,
in relatively wide annuli (b / a :::: 3). The observed flutter corresponded to either n = 4
or n = 2, and m = 1 predorninantly. The fiutter mode involved axial standing waves, but
travel1ing circumferentially, Comparisons were made between the 'fiutter r.p.m.' and fiut-
ter frequencies, the fonner indicating the r,p.m, of the rotating outer cylinder at which
flutter of the inner shell occurs. Agreement with Srinivasan's theory is generally rather
poor, with discrepancies of the order of 2: 1 in some cases for the flutter r.p.m, and as
much as 10: 1 for the flutter frequency, the experimental values being high in both cases.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
699
Most of these discrepancies were blamed on the use of potential flow theory, and there
is no doubt that they must be at least partly related to real-flow effects, not the least of
which is the precise nature of the helical flow; t however, there were other, significant
differences between the experimental system and the idealized theoretical model. Fur-
thermore, unfortunately, the results were found to be very dependent on the inlet hole
geometry.
As a result, a great deal of effort was expended in refining the theory so that it could
accept more general axial and tangential velocity profiles, closer to those in experiments
(David & Srinivasan 1974). In paralIel, the flow-field has been examined experimentalIy
in great detail (McLean & DowelI 1975). Consequently, it was possible to achieve a
much improved agreement in flutter r.p.m.:j: Alas, the correlation between theoretical and
experimental flutter frequencies remains poor, for reasons that, to this writer' s knowledge,
stilI remain unresolved,
ln paralIel to this work, Lai & Chow (1973) studied the dynamics of a rotating cylindrical
shell containing axialIy flowing fluid which is also rotating, with the same angular velocity
(solid rotation). Further work on the same problem was done by Chen & Bert (1977a,b).
This analysis is discussed in greater detail in Section 7.6.2.
It is reiterated here that extensive work on the dynamics of rotating fluids per se
(i.e. without participation of the containing structure) have been conducted over the
years, which could be useful in the analysis of the dynamics of shells containing such
flows. For example, a very high-powered and extensive research effort was undertaken
by the Aeronautical Research Institute (FAA) of Sweden in the 1970s, on the so-called
'gas centrifuge projecť (Linde 1999); sample publications are by Howard et al. (1972),
Maslowe (1974), Gans (1975, 1976), Hultgren (1976, 1978) and Bark et al. (1978).
Other work, mainly on the n = 1 case, was done by Pai'doussis and by Skiba in
1967, discussed in Pai'doussis (1980), and also by Howard et al. (1972), presented in
Section 7.6.3.
FinalIy, before closing this section, it should be mentioned that a fair amount of work has
been done on rotating shelIs, free of any fluid interaction, starting with Bryan (1890). Some
more recent contributions are related to angular rotation rate-sensors, known as cylinder
gyroscopes; see, for example, Langdon (1982), Burdess (1986), Loveday & Rogers (1998)
and Kristiansen & Egeland (1999), to be discussed in Section 7.10.1.
7.6.2 Rotating shell conveying co-rotating fluid
The dynamics and stability of a shelI containing fluid in matched (solid-body) rotation
and generalIy also flowing axially was examined by Lai & Chow (1973), purportedly
inspired by fluid-structure interactions in 'the thrust chamber and the pipelines in the
liquid propelIant feed system of a spinning rockeť. In contrast to Srinivasan (1971),
Dowell et al. (1974) and related studies which are connected to a real system and the
t Actually, nonlinear effects may play an important role, but this was not explored at the time, Significantly,
the ratio of flutter r,p,m, to flutter frequency, in the same units, in several cases was near 1, i,  or t,
tPotentialIy at least; this depends on an unmeasured parameter Ve / br?, the ratio of the tangential velocity
cIose to the sheIl to the circumferential velocity of the outer rotating sheIl; if this is equal to 1.1, perfect
agreement results,

700
SLENDER STRUCTURES AND AXIAL FLOW
complications attendant thereto, the problem studied by Lai & Chow (1973) is very ideal-
ized and hence easier to describe in the brief treatment accorded to swirling flow in
this book.
The equation of motion is a modified form of (7.14b), based in a coordinate system
rotating with the shell,
{ U } { a2ujat2 }
L D v = y a 2 vjat 2 - Q 2 v + 2Q(awjat) ,
w -a 2 wjdt 2 + Q 2 w + 2Q(dvjdt) + pjPsh
(7.200)
where Q is the angular velocity of shell and fluid alike. The fluid velocities, V x , V r and Ve,
are related to the pressure p by
Dv x 1 ap
- ---,
Dt p ax
DV r V 2 1 dp
----2Qve- Q r=---,
Dt r p ar
Dve vrVe 1 1 ap
- + - +2Qv r = ----,
Dt r p r de
(7.201)
where
D d d d Vo d
Dt = at + V x ax + V r ar + -; ae '
(7.202)
assuming the fluid to be inviscid and incompressible. The continuity equation,
aV x 1 a 1 dVO
- + --(rv r ) + -- = o,
dX r ar r ae
(7.203)
must also be satisfied.
Considering travelling wave solutions, in view of the rotating coordinate frame, the
velocities are expressed as
{V x , Vn Vo} = {U, O, O} + {vx(r), vr(r), vo(r)} e ia ,
ex = wt - kx - ne,
(7,204a)
(7.204b)
where v x , v r , vo are related to the perturbation velocities and k here is the axial
wavenumber; similarly,
p = p + p*,
p* = pe ia ,
(7.204c)
in which p* is the perturbation pressure, while P is the mean pressure.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
701
Utilizing these equations, one eventually finds, for small perturbations,
2 1 ap
Q r = ---,
p ar
d 2  1 d  [ 2 ]
P P 2 2 n 
- + - - + k ()., - 1) - - P = O,
dr 2 r dr r 2
(7.205a)
(7.205b)
2Q
).,=
(w - kU)
(7.206)
Integration of (7.205a) gives the undisturbed pressure distribution,
per) = Pa - pQ2(a2 - r 2 ),
(7.207)
in which Pa is the undisturbed pressure at r = a. Equation (7.205b) is a Bessel equation,
adrnitting solutions of three different types, depending on the magnitude of )., 2 , The per-
turbation pressure is related to the perturbation radial displacement via the impermeabi1ity
boundary condition, equation (7.35),
V r Ir=a = i w (w - kU) e ia ,
(7.208)
where the shell displacements are taken to be
{ } { --- } ia
u, v, W = u, v, w e .
(7.209)
Hence, the solutions to equation (7.205b) may be written as
 pa(w - kU)2(l - ).,2)In(kr) _
P= w,
ka In-l (ka ../1 - ).,2 ) - n(1 + )")I n (ka ../1 - ).,2)
8p(n + l)Q 2 r n _
- w
- an-l[n(n + 1) + k 2 a 2 ] ,
pa(w - kU)2(1 - ).,2)J n (kr&-=!) _
- w
- ka&-=! Jn-l(ka&-=!) - n(1 + )")Jn(ka&-=!) ,
).,2 < 1
).,2 = 1
).,2 > 1,
(7.210)
where In and J n are as in equations (7.140).
Equations (7,210) and (7.207), evaluated at r = a, together with (7.209) are then
substituted into (7.200), giving three homogeneous equations in li, v and w , solutions of
which yield the desired frequencies w. Before doing this, it is convenient to introduce the

702
SLENDER STRUCTURES AND AXIAL FLOW
10
8
0.2
0.4
0,6
0,8
1.0
6
K
4
2
o
{J
Figure 7.69 The dimensionless axial wavenumber, K, versus the rotational frequency, n, for a
rotating empty elastomer shell (al h = 50, {EI[Ps(l - v 2 )]}1/2 = 36 mis, v = 0.49) for varying
circumferential wavenumber n (Lai & Chow 1973).
following dimensionless parameters:
- J p, (1 - ,2)
w = wa
E '
- Jp(l- v 2 )
V=V
E .'
Q  "a J p,(l - v 2 )
E '
(7.211)
K =ka.
Typical results are shown in Figure 7.69 for rotating empty elastomer she1!. It is seen
that, for a given K and a large enough Q, the empty she1lloses stability by flutter, not in the
n = 1 mode, but in one of the higher she1l modes (n > 1): for long enough wavelengths
(small K) in n = 2, but for larger K in progressively higher n.
_ With co-rotating and axia1ly flowing fluid, it is seen in Figure 7.70(a) that, at fixed
Q = 0.1 (1820 r,p.m.) and K = 10, the system first loses stability in the n = 3 mode at
{; = 0.089 (V = 3.3 mis) - at the 'nose' ofthe curve shown. When Q = O (not shown),
one obtains flutter in the n = 6 mode, at a higher critica1 {;, by about 11 %, Figure 7. 70(b)
shows the manner of variation of the critical n and K with {; for a fixed Q = 0.1. It is seen
that the most critica1 {; corresponds to n = 3, but for higher or lower K in other modes. The
important conc1usion is that fluid rotation destabilizes the system as compared to straight
axial flow, as rnight intuitive1y be expected.
In 1999, it was decided to extend this ana1ysis so as to take real fluid effects into
account. The first step was to reproduce Lai & Chow's results, whereupon some interesting,
if unexpected, difficulties were encountered (Cortelezzi et al. 2003; Pong 2000). It was
thought at first that the problem was numerical, but several different methods of solution

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
703
1.5
o
1.0
-
aJ 0.5
-0,5
O
0.02 0.04 006 0.08 0,10 0,12
0,14
(a)
n=3
25
20
15
K
10
5
O
0.1
02
0.3
(b)
tJ
Figure 7.70 (a) The dimensionless frequencies W of the same rotating shell as in Figure 7.69
containing co-rotating and axially flowing water with ť2 = 0.1 and axial wavenumber K = 10, for
different values of n. (b) The K versus [; diagrams for ť2 = 0.1 (Lai & Chow 1973),
yielded very similar results. The prob1em is real, and it is the foHowing. Referring to
equation (7.210), it has been found that (i) one generaHy needs both forms for p (for
A. 2 > 1 and A. 2 < 1) in orderto trace the curves in Figure 7.70(a) fuHy; (ii) the denominator
of the latter form is a functional of the ln Bessel functions, each of which crosses zero
an infinite number of times, and so does the denominator as a who1e. For examp1e, by
inspection, for n = O the denominator becomes zero at every value of ka.JI2=1 for
which LI (ka.JI2=1) = O. As a result, the perturbation pressure at these points is
indeterminate, and the solution becomes invalido

704
SLENDER STRUCTURES AND AXIAL FLOW
1.5
°
1.0
-
w 0.5
-0.5
°
0,02
0,04
0,06
0,08
0,10
0,12
0.14
u
Figure7.71 The eV versus ČJ curve for n = D shownin ??D(a), recalculated by Cortelezzi, Pong &
PaYdoussis (2003); islets of invalid (indetenninate) solution preclude the determination of the flutter
point.
A typica1 set of results for the n = Ocase ofFigure 7. 70( a) are shown in Figure 7.71, The
two straight lines delimit the domain containing the islets where a solution is not feasible.
Thus, it becomes impossible to properly 'close the curve' and hence determine the flutter
threshold. Generally, the width and orientation of the inva1id domain (no-solution band) is
parameter-dependent; thus, this band could disappear or cover most of the stability curve.
Severa1 attempts to overcome the problem failed, Of course, with the aid of a trusty
French curve, one may c10se the curve while ignoring the inva1id segments. t Fundamen-
tally, however, the qUfstion is this: does this mathematical difficulty have the physical
meaning that flutter in such cases is impossible? This remains an open question.
A dosely similar analysis to Lai & Chow's (1973) was conducted by Chen & Bert
(1977a,b) in which the shell is stationary but the fluid is rotating as in the foregoing. Thus,
the physica1 system is c10ser to engineering app1ications, but the use of inviscid flow theory
is 1ess justifiable than in Lai & Chow's system. The shell is taken to be either isotropic
or anisotropic (cf. Section 7.3.5). The analysis is c10sely similar to Lai & Chow's, and so
is the expression for the perturbation pressure, corresponding to (7,210), except that in
this case Ve = Qr + Ve exp(ia), and hence).. = 2Q/(w - kU - nQ). Typical results for
tlt has been suggested by an anonymous reviewer that these difficulties may a1tematively be circumvented
by recasting the fundamental eigenvalue problem from one involving shell displacements to one involving
pressures, Basica1ly, equations (7.200) and (7,210) may be expressed as Lw = p and p = Aw, so that the
eigenvalue problem may be stated as Lw = Aw == (ANI AD)W. Then, the altemative. equivalent eigenvalue
problem is SADP = ANP. This rnight indeed bypass the infinite-pressure problem at the offending frequencies
and al10w the closing of the curves, In this 1ight, these problematic solutions would then be viewed as being
associated with zero displacement amplitude rather than infinite pressure, The fundamental problem remains,
however; for zero amplitude solutions are as strange as infinite amplitude ones in the context of a linear
homogeneous system, where the amplitude ought to be arbitrary. Thus, the non-solution infinite-pressure islets
would simply be replaced by islets of zero shell amplitude,

40
35
30
25
20
15
10
5
4
3
\é 2
..:- I
..

 O
.. O
; 40
.. 35
 30
25
20
15
10
5
4
3
2
I
O
O
1 laN SHELLS CONTAINING OR IMMERSED IN FLOW
705
- E-glass/epoxy
- - Steel
Unstable
-'-'-
0.1
0,13
0,16
0,22
0,25
0,19
Stable
- E-glass/epoxy
-,- Steel
n=7,'
,V
n=8(
// '-
! ( '-- __ -c::- n =10
\ '......... n = 6' _
'''"'''""'-.--':':-::::'':.- .---.
- ,
0,06 0,08 0,10 0.12 0,14 0.16 0.18 0,20 0,22
Axial flow velocity, U CB
Figure 7.72 The axial wavenumber versus axial flow velocity U CB diagrams for a steel shell and
an E-glass/epoxy shell for different n, when (a) C B = O and (b) QCB = 0.2 (Chen & Bert 1977a).
QCB = O and 0.2 t are shown in Figure 7,72; here
( p ) 112 ( - ) 1/2
QC B = --.!!... Qa, VC B =..!!!...- V,
An All
where AI1 is asinequation (M.5). Forahomogeneous isotropic shell, All = Ehj(l- v 2 ),
so that V C B = U and QC B = f.-t 1/2 Q , the nonsubscripted quantities being as defined by
equation (7.46), It is clear that rotation of the fluid reduces the critical axial flow velocity
by approximately 40% and 20% for the steel and epoxy shells for values of Qaj V of
0,98 and 0.35, respectively.* An attempt to reproduce these results encountered the same
(7,212)
tln the text associated with Figures 3 and 4 of Chen & Bert (1977a; errata), QCB = 0,05 is erroneously
given.
+Figure 5 in Chen & Bert (1977a) gives the wrong impression, since the values of [lal U given appear to
unaccountably be two orders of magnitude smaller than they ought to be.

706
SLENDER STRUCTURES AND AXIAL FLOW
difficulty asin the case ofLai & Chow's results; in view ofthe similarity in the expressions
for p in the two cases (Cortelezzi et a/, 2003), this is not surprising.
ln all the foregoing, unusually, analysis of the dynamics was done by manipulation
of the dimensiona/ equations, and dimensionless quantities were introduced toward the
end and in the results presented. Hence, one of the key dimensionless parameters for the
dynamics of rotating fluids was not explicitly used, that is, the Rossby number, defined by
U
Ro = -. (7.213)
Qa
ln terms of the Rossby number, the results in Figures 7.68 and 7.69 correspond to Ro --+ O,
but Ro is of 0(1) at the critical point in Figure 7.70(b),
7.6.3 The intense neutron generator system
Before presenting any further theory, it is of interest to review one documented practical
case in which swirling flow has led to fluidelastic instabilities - other than the turbo-
machinery/aeromechanics problem which motivated the work by Srinivasan, Dowell et a/,
described in the foregoing - in model tests for the so-called lntense Neutron Generator
system, No doubt there are many other such cases, which nevertheless remain unreported
because of commercial and possible litigation considerations.
The intense neutron generator (ING) is a device for producing thermal (relatively low
energy) neutrons by means of a spallation reaction, designed by Atomic Energy of Canada
(Bartholomew & Tunnicliffe 1966). A high-energy proton beam from a linear accelerator
collides with the nuclei of a dense heavy-element target, thereby generating thermal neu-
trons, as well as heat and electromagnetic radiation. The ING target must have a low
fission cross-section, that is, low heat production for each useful neutron emitted, and
low thermal-neutron absorption cross-section, Furthermore, the cooling of all structural
components must be simple, These requirements are satisfied by a Pb-Bi (lead-bismuth)
eutectic acting both as target and primary coolant in a loop containing the target area
and a heat exchanger. The design should also satisfy the additional requirement that the
target be 'open' to the proton beam, the possibility of cooling a window of any kind being
impractical because of the high ("" 38 MW) heat production in the small space around the
target.
These requirements are met in the design shown diagrammatically in Figure 7.73(a).
The liquid Pb-Bi enters the target tube as a swirling annular flow with a gaseous core,
thus allowing the proton beam to reach the target relatively unimpeded. Somewhere below
the top of the tube, as the swirl is diminished by dissipation, an annular hydraulic jump
occurs, filling the tube and transforming the annular flow to a swirling pipe flow, thus
presenting a target in the central part of the tube cross-section.
The device was never built, but was extensively studied. Of interest here are the hydraulic
simulation experiments, performed in 1967 and described briefly by PaYdoussis (1980;
Case 39). Water was used as the fluid. It passed through a 'vaned swirl heaď [Fig-
ure 7. 73(b)] which controlled the annular flow (film thickness and swirl angle), for example
by moving the central tube up or down. The maximum swir1 angle was 45° or less, meas-
ured from the vertical, Typical axial flow velocities were 4.3 mis in the filled region and
12 mis in the annular region. In this apparatus, swirl increased with increasing flow rate.
The swirling annularflow entered the aluminum tube below it [Figure 7.73(c)] and therein

m
o
2
4
6
8
(a)
THlN SHELLS CONTAINING OR IMMERSED IN FLOW 707
:1:::::"' Proton beam tube Vacuum line
- Vacuum line Clamp
Swirl-head
Vanes cm
l:O
20
30 Metres
O
Hydraulic jump
0.5
Water film
F illing line
Swirling Pb-Bi flow
@
15,2 cm
(b) (c) epoxy
joint
Figure 7.73 (a) Diagrammatic view of the lntense Neutron Generator ( 8.5 m long). (b) The
vaned swirl head, and (c) the 50.8 mm diameter test tube used in the hydraulic simulation
experiments (PaYdoussis 1980).
took place the hydraulic jump, its position located by an ultrasonic ftaw detector. Three
different aluminum tubes of D = 50.8, 76.2 and 101.6 mm, a1l3.35 m long and 1.24 mm
thick, were used.
Violent vibrations were recorded. For the 50.8 and 76.2 mm tubes, ftexural whirling
vibrations in their first and, at higher ftow, in their second modes developed, with a
maximum amplitude of 25 mm; third-mode vibrations were also observed for the smallest
tube. In the 50.8 mm tube, the hydraulic jump occurred at its upper end, so that the tube was
filled with water all along; it cracked at the lower support by fatigue. Vibration started in
the first mode, increasing in amplitude with ftow, reaching a plateau, and then decreasing,
before the next mode would appear. For the 101.6 mm tube, shell-mode vibrations in the
second circumferential mode (n = 2) were observed, involving several axial wavelengths,
of 1.5 mm maximum amplitude; n = 3 vibrations were also obtained.
The mechanism for the 50.8 and 76.2 mm tube vibration is akin to whirling of shafts.
For a given ftow rate, there exists in the tube a band of swirl frequencies, decreasing down

708
SLENDER STRUCTURES AND AXIAL FLOW
the tube as the swirl ve10city is decreased by friction. As flow increases, so do es the swir1
ve10city and hence frequency, and eventually at the top of the tube, it coincides with the
first-mode flexura1 frequency, precipitating whirling, As flow increases further, the swirl at
the top increases further and the resonant driving swirl area moves down, unti1 maximum
amp1itude is produced when the resonant swirl reaches the tube midpoint. Thus, unlike
the whirling shaft, the resonance peak is not sharp; indeed, it is very broad, as observed
in the experiments. At higher flow, and higher swir1, second-mode resonance is reached,
et seq. For examp1e, for the 50.8 mm tube, the maximum swir1 ve10city at the wall was
12,80 mis, giving Q = 503.9 radJs or 80 Hz, which may excite, in turn, each of the three
10west beam modes of 13, 36 and 71 Hz (calcu1ated taking into account the interna1 fluid
mas s distribution), but not the fourth at 118 Hz. Simi1ar1y, for the 76,2 mm tube, the
calcu1ated maximum swirl was 48 Hz, sufficient to excite the first two flexura1 modes of
17 and 46 Hz, as observed.
The mechanism for the 101.6 mm tube vibration appeared at the time to be physically
more obscure. The calcu1ated maximum swirl frequency of 27 Hz shou1d have been suf-
ficient to excite the first flexura1 mode at 20 Hz, but this did not occur. t Instead, shell-mode
vibrations (n = 2 and 3) occurred, with frequencies in the range of 150-200 Hz, depend-
ing on the 10cation of the hydrau1ic jump, However, in view of the dynamica1 behaviour
discussed in Section 7.6.2, the dynamics must be simi1ar to that in Figure 7.70, in this
case the critica1 circumferentia1 mode being n = 2 rather than n = 3. It ought to be noted
a1so that the calcu1ated n = 2 frequencies (assuming simp1y supported ends and com-
p1ete1y waterfilled tubes for simplicity) were dosely spaced: 119, 120, 122, 129, 142, . . . ,
312 Hz for m = 1, 2, 3, 4, 5, . . , , 10, agreeing with the observation of 'severa1 axia1
wave1engths' .
7.6.4 Boundary-Iayer effects for small Ro
Here we follow dosely the ana1ysis by Howard et a/. (1972) in which, by means of a
matched asymptotic expansion, an approximate expression is obtained for the perturbation
pressure acting on the shell, taking into account the boundary 1ayer.
We start with equations (7.201). Dividing by Q 2 a and defining
A ú)
ú)= -,
Q
A r 1 + A i (wt-kx-ne )
r=-= 1Je ,
a
K =ka,
A p*
p=,
pa "
{u,v,w} { A A A } i(wt-kx-ne)
= u, v, w e ,
Qa
(7.214)
where r is the dimension1ess deformed shell radiu s and P the corresponding pressure, we
obtain the dimension1ess linearized perturbation equations for fluid flow in the essential1y
inviscid care
i(w - K RO)V r - 2ve = -dPJdr,
i(w - K Ro)ve + 2v r = (in/f)p,
i(W-KRo)v x =iKP,
(7.215a)
(7.215b)
(7.215c)
tThis was attributed, partly, to disruption of the ftow by a 'ftow expander' at the tube entrance,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
709
where Ro is the Rossby number, defined by (7.213). Considering now the mean axial flow
velocity to be small compared to the rotational one, that is, Ro --+ O, and eliminating the
velocity components one by one leads to
 i (  dp 2n  )
V r = 4 _ w 2 -w dr + T p ,
A 1 ( dp wn  )
Ve = 4 _ w2 2 dr - T P ,
(7.216)
 K A
V x = -;:p,
W
Substitution of these into the dimensionless fonn oť the continuity equation (7.203),
1 d A A love oV x
-; d/ rvr ) + -; oe + OX = O,
(7.217)
leads to a Bessel equation for the pressure p, namely
d 2 P 1 d P ( 2 n 2 ) 
dr 2 + -; dr + O' - r 2 P = o,
2 _ ( 4-W 2 ) 2
O' - 2 K,
W
(7.218)
with solution
p = AJn(O'r) + BYn(O'r).
(7.219)
This much is essentially the same as in Section 7.6.2, but with the new, dimensionless
fonnulation; 0'2 here corresponds to K 2 (A 2 - 1) there.
We next consider the situation in the boundary layer. The equivalent of equa-
tions (7.215a,b,c) for viscous flow and Ro = O is
.A 2 A dp E ,,2
lWV r - Va = -----;- + v V r ,
dr
.A 2 A ln E ,,2
1WVa + V r = 7 P + v Ve,
r
d 2 1 d n 2
V 2 = _ + __ _ _ _ K 2
dr 2 r dr r 2
(7.220)
.   .  E ,,2 
lWV x = lKp + v V x ;
where
v
E=-
Qa 2
(7.221)
is the Ekman number - the equivalent of the inverse Reynolds or Stokes number
[see Section 2.2.2(c)] for rotational flow. Solution to this 'inner' (here physically outer)
boundary layer flow is sought, assuming p = PO + E 1 / 2 Pl + ... and {v r , Ve, v x } =
{vro, veo, vxo} + E 1 / 2 {Vrl, Val, vxd + ..., considered to be functions of the stretched
variable  = (1 - r)j E l / 2 . This solution is then matched to that of the 'outer' (here
physically inner) inviscid core, as shown in Appendix N.
The final expression for the wall pressure perturbation, Pw, is
A (4 - w 2 )fi
Pw = 2 2 2 '
2n O' J n -l (O') O' (n + K) 1 / 2
n+-- + E
W Jn(O') K 2 (iw)1/2
(7.222)

710
SLENDER STRUCTURES AND AXIAL FLOW
where ry = w ja, the dimension1ess radial disp1acement amp1itude. It is noticed that the
wal1 pressure perturbation is smaller than it wou1d be in inviscid flow (E = O); moreover,
Pw is not in phase with ry by virtue of the (iw) 1/2 term.
It was initial1y thought that inc1usion of viscous effects, thus rendering the model more
realistic, might solve the difficu1ties in obtaining a valid solution throughout the domain
of interest as detailed in Section 7,6,2, at 1east for this case of U = O, Some calculations
are presented in Pong (2000). Unfortunately, however, the same difficu1ties are generally
encountered, again particular1y near the nose ofthe curve which wou1d define the thresho1d
of flutter; for this reason these resu1ts are not presented here.
7.7 TURBULENCE-INDUCED VIBRA TlON
All the material presented so far in this chapter deals with the dynamics of a shell conveying
fluid, that is, with the effect of a mean flow on the free vibration characteristics of the shell
and possib1e loss of stability. However, if the flow is turbu1ent, there are smal1 random
fluctuations in flow ve10city and in the associated pressure fie1d, which can excite the shell
at any flow ve1ocity, and certain1y at ve10cities inferior to those which may be associated
with fluide1astic instabilities. The pressure fluctuations everywhere in the fluid, and in
particu1ar on the shell wal1, have a wide frequency spectrum. The shell is excited by them
by acting as a mu1tip1e narrow-band filter, each pas s frequency being associated with one
of its eigenfrequencies. This prob1em is considered in this section, after a brief literature
review.
Methods for determining the response of continuous structures (e.g. beams, p1ates and
shells) to distributed forces random1y varying with time have been deve10ped in tandem
with their principal application: that of turbu1ence-excited response due to an axially
flowing fluid. Ear1y contributions in this area (Eringen 1957; Powell1958; Dyer 1959;
Thomson 1962; Tack & Lambert 1962; El Baroudi et al. 1963) dea1 principal1y with beams
and p1ates, and the first such work specifical1y on cylindrical shells excited by flow-induced
noise is due to Cottis & Jasonides (1964), Cottis (1968) and C1inch (1967, 1970). Examp1es
of more recent work on cy1indrical shells are by Levin et al. (1971), Lakis & Pai'doussis
(1972b), Lakis & Doré (1974), E1ishakoff (1975), Au-Yang & Connelly (1977), Bo10tin
(1979), Horáček (1985, 1986) and Liu & Gorman (1996).
Here, an abbreviated treatment of the analysis by Lakis & Pai'doussis (1972b) is given,
based on the hybrid analytical/finite-element method presented in Section 7.3.3, thus
being app1icab1e to shells with axial variations in their properties, provided they remain
axial1y symmetric. The treatment of the excitation pressure field, however, may be used
in conjunction with any other type of analysis, t
One of the key assumptions common1y made in analyses of turbu1ence-induced vibra-
tion is that the fluctuating pressure field induces the vibration, but that the vibration is
sufficiently smal1 not to significantly affect the excitation field; thus, there is no dosed
feedback loop, and the problem may be regarded as one of forced rather than self-
excited vibration - extraneously induced excitation (EIE) in the Naudascher & Rockwell
dassification; see Section 1.2.
tTurbulence-induced vibration analysis by means of c1assical continuum theory is il\ustrated in Section 9.4,
for instance,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
711
The task here is twofold: (i) given a random pressure field, how can the response of the
shell be predicted, and (ii) how does the pressure field, in particular a turbulence-induced
pressure field, need to be characterized for the purposes of such analysis.
7.7.1 General formulation
It is assumed that the shell has been subdivided into a number of cylindrical frusta as in
Figure 7.29, govemed by the equation of the discretized system,
My + cý + Ky = F,
(7.223)
where y is a displacement vector, M, C and K are the global mass, damping and stiffness
matrices, respectively, and F is a vector of the extemal forces - cf. equation (7,122), This
implies that the continuous excitation pressure field has been discretized; this is done by
the method developed by Mercer & Hammond (1968), as shown in Figure 7,74.
Consider the pressure field acting on an area of the shell Se, surrounding the node e at
coordinate Ze, as shown in Figure 7.74(a), where Se extends axially from Z and Z. It is
always possible to approximate the distributed pressure over the area Se by two mutually
perpendicular forces per unit length, fRn and fCn, for each circumferential wavenumber
n, with a resultant
F(x, e, t) = L fRn(x, t) cos ne + L fcn(x, t) sin ne,
n
n
Xo
o o o
rn
--jJ--
=J_--
fRn
, ,
N-l ťB ' I
N N-l
N+l N
(a)
I I
N-l ťB ' I
N N-l
N+ 1 N
(b)
Figure 7.74 (a) Transformation of the continuous pressure field to a discrete force field.
(b) The equivalent discrete force field acting at the node e, involving fRn, fCn and .Mr (Lakis &
PaYdoussis 1 972b).

712
SLENDER STRUCTURES AND AXIAL FLOW
where fRn and fCn are at a distance xo from the origin of the shell as shown in
Figure 7.74(a); they are approximately given by
(2:n:
fRn(X,t) =a Jo p(x,e,t)cosede,
(2:n:
Jcn(x, t) = a Jo p(x, e, t) sine de,
(7.224)
(7.225)
where p(x, e, t) is the instantaneous pressure on the suďace. These two forces, fRn and
fCn, acting at point A are transformed to two forces and one moment, .Mr, acting at the
node e, as shown in Figure 7.74(b). The extemal force vector associated with the nth
circumferential wavenumber at node e can now be written in the following form:
! ["
{F(t)}e = 0, f I fRn(X;, t) dx;,
[" ) T
fP Jcn(x p , t) dx p ,
1':
{ J (X) -1})fRn(X}, t) dx},
JI',
J
(7,226)
in conformance with the structure of the nodal displacement vector, ; = (u n ;'
w n ;' (dwn/dx);, vn; }T, equation (7.111); here I;' = li = l = I, I; = Ij = I = I;
and le = I j, the peculiar indicial notation having been introduced for convenience in
subsequent manipulations.
The eigenvalue nroblem associated with the homogeneous part of (7.223) is of order
4(N + 1) - J, where N is the number of finite element s and J the number of kinematic
constraints imposed, thus leading to 4(N + 1) - J eigenfrequencies Q; and eigenvectors
;. Next, we proceed to obtain a decoupled form of equation (7,223) by the methods
out1ined in Section 2.1.2, leading to
.Mz + Di + Sz =  TF = Ff,
(7.227)
where  = [l, 2, ' . . , 4(N+l)-J] is the modal matrix, .M the genera1ized mass mat-
rix and S the spectral matrix, both diagonal, whereas D =  T C  is generally nondiagona1.
However, mean flow effects are considered to be small t and are neglected* henceforth,
while damping in the shell is taken to be hysteretic; hence C becomes proportiona1 to K,
thereby making D also diagonal. Therefore, equation (7.227) is decoupled and may be
rewritten as
Zr + 2rQrŽr + QZr = F:I.M r ,
r = 1,2,.." 4(N + 1) - J.
(7.228)
tPor a steel cylindrical shell with Lla = 26 and hla = 0,023 conveying air-flow, fJ = 0,20 corresponds to
U  1006 mis; for this magnitude of flow velocity. which is really beyond the range we shall be considering
here, the natural frequencies of the shell (clamped at both ends) are found to diminish by only 3% as a result
of the flow. Accordingly, for metal tubes conveying fluid with flow velocity in the 'normal' engineering range,
these effects may be considered to be secondary and wi11 not be taken into account in this analysis,
*Other than added mass effects, which are taken into account.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
713
Upon solving equations (7.228), the response in tenns of the original coordinates is
found by
4(N+l)-J
Yq = L <l>qrZr.
r=l
(7.229)
This then is the instantaneous response due to an arbitrary force vector F; it represents
the response at the nodal points, that is, at specific values of the coordinates x and B,
namely x = Xi, i = 1, 2, . . . , N + 1. and B = Bo, where Bo is defined consistently with
F. Moreover, the response obtained above is associated with a specific n; by repeating the
analysis for a sufficient number of n, the total response for any point on the nodal circles
may be obtained by superposition, in accordance with equation (7.108) summed over n.
7.7.2 Response to an arbitrary random pressure field
The spatial cross-correlation function of the response is defined by
VrYij (Xb x p ; Bo; T) = Yi(Xb Bo, t)Yj(x p , Bo, t + T),
(7.230)
where Yi and Y j are two elements of the nodal displacement vector (it is recalled that Y
is a generic notation for U n , Vn, W n and dwn/dx); Xk and x p are the axiallocations ofthe
two nodes involved, where indices k and p are related to indices i and j, such that (for an
unconstrained shell) to each k correspond four sequential values of i [in accordance with
the definition ofthe nodal displacement vector, equation (7.111)]; T is the time delay and
the overbar denotes a time average,
Assuming that we are dealing with an ergodic proces s, using the correlation theorem
we may write
1 f oo ,
VrYi/Xk, x p ; Bo; T) = lim - Yt(Xb Bo, Q, T)Yj(x p , Bo, Q, T) e'r2'C dQ,
T-+004:rrT -00
(7.231)
where the Y s are the finite Fourier transfonns of the ys, such that
Yj(Xk, Bo, Q, T) = f T Yi(Xb Bo, t) e- ir2t dt;
-T
the asterisk denotes the complex conjugate. Now, for T = O, and i = j = q, and hence
k = p = g (where q and g must be consistent), equations (7.230) and (7.231) may be
written in the fonn
Y(Xg, Bo, t) = VrYij(x g ; Bo; O)
= lim  [00 Y;(x g , Bo, Q, T)Yq(x g , Bo, Q, T) dQ,
T-+oo 2:rrT 10
(7.232)
where the fact that Y; Y q /2T is an even function of Q has been used.

714
SLENDER STRUCTURES AND AXIAL FLOW


.QI
.Q2
.Q
.Q3
Figure 7.75 The magnification factor as a function of frequency for several modes (Lakis &
PaIdoussis 1 972b) ,
Introducing the Fourier transforms of equations (7,228) and (7.229) into (7.232), we
obtain
4(N+l)-J 4(N+1)-J
Y(Xg, 80, t) = L L
r=l s=l
4>qr 4>qs
Q;Q.Mr.Ms
1 1 00
x lim 2 T ei(t/Js-t/Jr)IHr(Q)IIHs(Q)I.;F(Q, T).YF*(Q, T)dQ,
T -+00 7f o
(7,233)
where Qr is the rth natural frequency, .Mr is the rth element of the generalized mass
matrix, 4> qr is the (q, r)th element of the modal matrix, CPr and CPs are phase lags between
the force and the response, Hr(Q) is the magnification factor defined by
H,(Q) = {[l- (  )T + [2" J TI/2,
(7,234)
and F(Q, T) is the finite Fourier transform of the force vector.
Figure 7,75 shows IHr(Q)1 plotted against Q for a lightly damped multi-degree-of-
freedom system. We see that IHr(Q)1 displays pronounced peaks in the neighbourhood of
the natural frequencies ofthe system. The products IHr(Q)IIHs(Q)1 for r i= s are seen to
be small in comparison to the products for r = s. In addition, the terms in equation (7.233)
with r i= s may be negative or positive depending on the sign of the product 4>qr4>qs,
while terms with r = s are always positive. Therefore, ignoring the contribution of the
cross-product terms to the mean-square response, equation (7.233) may be written as
4(N+l)-J 4>2 1
Y(Xg, Bo, t) = L 4 qr 2 lim -
r=l Q 2 .M r T-+oo 27fT
x 1 00 IHr(Q)12.;F(Q, T).;F*(Q, T) dQ.
(7,235)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
715
If, however, the natura1 frequencies are very dosely spaced, constituting a near-continuum,
then special techniques may have to be used for obtaining the response, for example, by
means of 'statistica1 energy analysis' (Lyon & Maidanik 1964; Lyon 1973; Lyon & De J ong
1995; Keane & Price 1997).
We next reformulate the force fie1d in equation (7.235) in terms of cross-correlations.
We first define the cross-correlation spectra1 density of a force f by
Wf(Xi, Xj; r.!) = lim [ .!..F(Xi' r.!, T)F*(xj, r.!, T) ] ,
T-+oo T
(7,236)
where F is the Fourier transform of f. Then, substituting the Fourier transform of
equation (7.226) into (7.235), expanding and making use of (7.236), we obtain
4(N+l)-J <p2 (IX!
Y(Xg, eo, t) =  27rr.!rM} Jo IHr(r.!) 1 2
N+l N+l I" I"
x {  !; <Pir <Pur ' i u W fR (r.!; xi, Xu) dXi dxu
N+l N+l
+ L L <Pir <Pkr
i=l k=l
I" I
{I { (Xk -lk) W fR (r.!; Xi, Xk) dXi dXk
JI; J/
N+l N+l I" I"
+ L L <Pir<Pvr ti 1 v WfRC(r.!;Xi,Xv)dxidxv
i=l v=l I; I
N+l N+l l'! I"
+ L L <Pjr <Pur {J {u (Xj -lj)WfR(r.!; Xj, Xu) dXj dxu
JI' JI!
j=l u=l J u
N+l N+l
+ L L <Pjr <Pkr
j=l k=l
N+l N+l
+ L L <Pjr <Pvr
j=l v=l
N+l N+l
+ L L <P pr <Pur
p=l u=l
N+l N+l
+ L L <P pr <Pkr
p=l k=l
N+l N+l
+ L L <P pr <Pvr
p=l v=l
I" I
{J { (Xj-lj)(xk-1dWfR(r.!;Xj,xddXjdXk
Jlj J/
I" I"
{ J 1 v (Xj -lj)WfRc(r.!; Xj, xv) dxj dx v
Jlj I
I" I"
{p (u WfcR(r.!; X p , Xu) dx p dx u
JI J/
I" I"
1 p 1 k (Xk -lk)WfcR(r.!; Xk, X p ) dx p dxk
I I
I" I"
1 p (v Wfc(r.!; X p , xv) dx p dxv } dr.!,
I J/
(7.237)

716
SLENDER STRUCTURES AND AXIAL FLOW
where W f R and W Je are ťorms of (7,236) involving the ťorces fR and fc, respectively,
and WfRC and WfcR representmixed terms involving both fR and fe, In this equation the
indices i and u are associated with radial forces, pand v with circumťerential ťorces, and
j and k with moments.
The next step is to ťormulate the cross-correlation spectral densities oť the ťorce field
in terms oť those of the pressure field, The narrow-band space-time correlation ťunction
oť a ťorce field f (x, t) may be obtained experimentally, in principle, by multiplying
electronically f (x s , t) and f (Xk, t + .) after passing them through identical narrow-band-
pass filters with centra! ťrequency (2, and taking a long time average; thus,
1frfQ.(x s , Xk;.) = fg,(xs, t)fg,(xk. t + .),
Now, the cross-correlation spectral density is the Fourier transťorm oťthe cross-correlation
function; moreover, it is complex, its real part being an even ťunction and its imaginary part
being an odd function, Accordingly, the real component of the cross-correlation spectral
density is equal to the narrow-band space-time correlation function when . = O, that is,
Re[Wf , (x s Xk; (2)] = fg, (x s , t)fQ(xk. t),
(7,238)
and the imaginary part may be obtained by setting . = Jr /2(2.
Now, if the ťorce field is derived from a homogeneous pressure field, that is, one
independent oť location, the cross-correlation spectral density oť both the pressure and
the ťorce fields may be expressed in terms oť distances oť separation  = Ixs - Xk I and
'f/ = la(Oi - Oj)l, instead oťthe coordinates themselves. Thus, we have
PQ(X s , Oi, t)PQ(Xk, Oj, t) = PQ(X, O, t)PQ(X +, O + 'f/, t),
(7,239)
and
fQ(xs, t)fQ.(xk. t) = fQ(x, t)fQ(x +, t).
(7.240)
In this analysis the pressure field is related to the components of the ťorce field fR and
fc through equations (7.224) and (7.225). Accordingly, the cross-correlation spectral
densities employed in equation (7.237) may now be written as follows:
Re[W fR (, (2)] = fRQ. (X, t)fRQ. (x + , t)
= a 2 21C 21C PQ(X, O, t)PQ(X +, 0+ 'f/, t)
x cos O cos (O + 'f/)dO d (O + 'f/),
Re[WJe(' (2)] = feQ.(x, t)fcQ.(x +, t)
= a 2 21C 21C PQ(X, O, t)PQ(X +, 0+ 'f/, t)
x sin O sin(O + 'f/)dO d(O + 'f/),
(7.241)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
717
Re[WfRC(' Q)] = !RQ(x, t)!cQ(x +, t)
[2n [2n
= a 2 10 10 pg,(x, e, t)pdx +, e + 1'}, t)
x cos e sin(e + 1'})de d(e + '1),
and WfcR(' Q) = WfRC(' Q).
The displacement power spectral density is, of course, a real quantity. The pressure
cross-correlation spectral density is complex; however, its imaginary part vanishes as a
result of the double area integration. Thus, the Re[W f] and, hence, the response of the
shell [through equation (7,237)] wi11 have been determined once the quantity
Re[pg,(x, e, t)pg,(x +, e + 1'}, t + T)] = pg,(x, e, t)pg,(x +, e + 1'}, t)
is specified.
We introduce the norma1ized spatial cross-correlation function for the pressure field
,I, ( t ) _ p(x,e,t)p(x+,e+'1,t+T)
'P P <;, '1, T - ,
p2(x, e, t)
(7.242)
where p2(x, e, t) == p2(t) is the mean-square of the pressure fluctuations. It wi11
be assumed that we can expres s 1/Ip(, '1, O) = 1/Ip(, O, 0)1/I p (0, 1'}, O), The quantities
1/Ip(, O, O) and 1/I p (0, 1'}, O), which are, respectively, the axial and circumferential spatial
correlation functions, may be determined experimentally with relative ease; by a proced-
ure such as that out1ined at the beginning of this section we may obtain these values of
1/1 p for specific frequency bands, which we denote by 1/1 PQ' Accordingly, equation (7.242)
may be re-written in the form
pg,(x, e, t)pg,(x +, e + '1, t) = 1/IPQ(' O, 0)1/IPQ(0, '1, O) p(t) .
(7.243)
Equations (7.243) and (7.241) expres s the spectral cross-correlations of the force field
in terms of the pertinent correlations of the pressure field. Thus, equations (7.243), (7.241)
and (7.237) together express the response of the shell to an arbitrary homogeneous random
pressure field.
7.7.3 Response to a boundary-Iayer pressure field
ln the case of subsonic boundary-layer pressure fluctuations, the streamwise and lat-
eral spatial correlation functions have been examined theoretically and experimenta1ly
by severa1 investigators, for example, Corcos & Liepmann (1956), Wi11marth (1958),
Smith & Lambert (1960), Weyers (1960); Bakewell et al. (1962), El Baroudi et aZ. (1963),
Bakewell (1964), Wi11marth & Roos (1965), C1inch (1969). Reviews of the subject are
given by Wi11marth (1975) and Horáček (1989). In particular, since this wi11 be adopted in
the ana1ysis that follows, it is noted that Bakewell and co-workers measured, and derived
expressions for, the axial and circumferential correlation functions, in experiments with
air flowing in a cy1indrical pipe, The experiments covered a range of Reynolds numbers
from loS to 3 x 10 5 ,

718
SLENDER STRUCTURES AND AXIAL FLOW
We denote the axial and circumferential correlation functions per unit bandwidth with
centre frequency w by 1/1pw(' O, O) and 1/rpw(O, 1], O), respectively; avoiding f which has
been used to denote force, the subscript w is now used instead of Q to indicate that the centre
frequency is in Hz and not in rad/s (we shall be using expressions with numerical constants,
so the distinction is important). Prom his experiments, Bakewell (1964) detennined the
following approximate expressions for the (real) spatial correlations:
1/1pw(' O, O) :: e-bIS1 cos(aSg),
1/1pw(O, 1],0):: (1 +cS;)-1[2-e-dSrl,
(7,244a)
(7 .244b)
where Sg = w/ U conv and SI'J = 1]w/ Ut, are the axial and circumferential Strouhal num-
bers, and a, b, c and dare constants to be detennined (here a is not the radius); Uconvand
Ut,are, respectively, the convection and the centreline velocities. In equations (7.244a,b),
there should have been added the subscript Re for Reynolds number; but, at least in the
range investigated by Bakewell, the effect of Reynolds number is small and does not
appear explicitIy in the expressions.
The values of the constants in these two expressions depend on the fluid. Por turbulent
air-flow, the values of a, b, c and d to be used in equations (7.244a,b) with Strouhal
numbers based on the convective or centreline velocity are given by
a = 8,7266, b = 1.0,
c = 20, d = 100,
for Sg = !;w/U conv ,
for SI'J = 1]W / Ut;
(7.245)
It may be expected, nevertheless, that these constants would be approximately the same
for different fluid s at the same Strouhal number, at least for sufficiently high Reynolds
numbers. This is supported by Clinch's (1967) measurements in water.
Bakewell also obtained measurements of th e mean-square pressure per unit band-
width (i.e. the power spectral density), p1(t), which are reproduced in Pigure 7,76
plotted against Strouhal number 2aw / UL a being the shell radiu s again from here on.
Por the purposes of this analysis, the following expression for the curve of best ftt is
utilized:
p1(t) = 2k2 p 2 aUl exp( - 2k law/ UiJ,
(7.246)
where kl = 0,25 and k2 = 2 x 10-6; P is the fluid density. Clinch's (1967) measurements
in water flow have been added to the ftgure, and they are found to agree quite well with
Bakewelľs in air, especially at high frequencies. At low frequencies, it is suspected that
the measurements are contaminated acoustically, and that is why the high readings at small
2wa / Ut, have not been included for the lines of best ftt.
Upon substituting the experimentally based relations (7 .244a, b )-(7.246) into equations
(7,243) and thence into equation (7,241), we obtain expressions for the cross-correlation
spectral densities of the force fteld arising from boundary-1ayer pressure fluctuations.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
719
,-,
'S""
",'3

I 
N
10- 10
0,01
0.1
1
2wa/U f
10
100
Figure 7.76 Bakewelľs (1964) and Clinch's (1967) measurements of the mean-square pressure
per unit band-width, in air- and water-flow, respectively, compared with the expression used by
Lakis & Pai:doussis (1972b). Bakewelľs data points and Re are: 0,1.0 x 105; .,1.5 x 105;
/:::",2.0 x 105; A,2.5 x 105; D, 3.0 x 10 5 . Clinch's data points and mean flow velocities are:
+,6,78 mis (Re  1.03 x 10 6 ); x, 11.43 mis; *, 13.21 mis; -, Bakewelľs line of best fit;
.. ., Clinch's line of best fit; - - -, equation (7.246). Clinch's mean flow velocities have been
converted to centreline velocities by mu1tiplication by 1.14.
These expressions may then be substituted into equation (7.237) to obtain the following
expression for the mean-square response of the shell:
4(N+I)-J 2
- " 2 a
y(t) =  <Pqr 16 24M2
r=1 7r w r r
[ N+I N+I N+I N+l
X {;!; <Pir <Pur !f, I + 2 {; {; <Pir <Pkr!fff I
N+IN+l N+IN+l ]
+ ?; {; <Pjr <Pkr !fn M I + ?; ?; <P pr <PvrIf;v I ,
(7.247)

720
SLENDER STRUCTURES AND AXIAL FLOW
where the rs, which are complicated functions of the lj and ofthe constants a, b, c, d of
equation (7.244a,b), involving integrals over w, too long to give here, are given in detail
in Lakis & Pai"doussis (l972b).
7.7.4 Sample results
Calculations have been conducted for a simply supported steel shell conveying air; the
shellparametersare: a = 103.6 mm, L = 470,9 mm, h = 1.19 mm, E = 2.07x 105 MPa,
Ps = 7,83 X 10 3 kg/m 3 , v = 0.3, Šr = 10- 5 for all r - see equation (7.228); the fluid
parameters are P = 1.20 kg/m 3 , Ut, = 7.32 mis (24 ft/s), corresponding to Re = 105. For
this shell, the lowest natural frequency is associated with n = 3 (435 Hz, approximately),
as compared to those for n = 2 (750 Hz), n = 4 (470 Hz) and n = 5 (680 Hz) - see
Figures 7.4 and 7.5, and Lakis & Pai'doussis (1972b, figure 6), Convergence for the free-
vibration characteristics is sensibly achieved with N = 5, that is, with five hybrid finite
elements. This same number has been used for the calculation of the response.
The maximum r.m.s. displacements as functions of n are shown in Figure 7. 77(a). Since
the excitation spectrum is highest at the lowest frequencies, as shown in Figure 7.76, it is
not surprising that the response is highest for n = 3, and next highest for n = 4. It is also
clear that the maximum response occurs in W n , that is, in the radial displacement, with
tangential and axial response being roughly one and two orders smaller.
The response in the n = 3 mode with increasing centreline flow velocity Ut,is shown in
Figure 7.77(b), covering the range of24-240 ft/s (1 ft/s  0.305 mis) for two values ofthe
10- 8
V>
3
II
,
"'"
10- 5
g 10- 9
s
I
10- 10
10- 11
2
3
4
5
10- 10
O
10- 9
(a)
n
(b)
120
Uf (ft/s)
240
Figure 7.77 Maximum turbulence-induced r.m.s. displacements of a simply supported steel shell
conveying air (parameters given in the text) in inches (1 in. = 25.4 mm): (a) as functions of n, for
Re = 10 5 and l;r = 10-5; (b) for n = 3 and varying centreline flow velocity Ut-in ft/s (1 ft/s 
0.305 mis) and two different l;r; (Lakis & Pai'doussis 1 972b) ,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
721
structural damping ratio. It is seen that the r.m.s. response increases proportionately with
Ul 2 approximately. Jf the total response (for all n) were desired, as would be the case
in most app1ications, the calcu1ation would have to be repeated for all n (or a sufficient
number) and summed, as discussed in Section 7.7.2.
The second set of calculations undertaken is for a shell first studied by C1inch (1970),
both theoretically and experimentally. It is a long, slender and thin stainless steel cy1indri-
cal shell (a = 76.2 mm, L = 6096 mm, h = 0.64 mm, E = 1.97 X 10 5 MPa, Ps =
8.01 X 10 3 kg/m 3 ) conveying water with flow velocities in the range of 6.30-13.2 mis.
C1inch obtained experimental data of the mean-square radial displacement of the shell in
frequency bands corresponding to the third-octave filters he used. Then, by summation, he
obtained the response in the frequency range of 100-1000 Hz, approximately; the response
in only this frequency range is given by C1inch (1970), and it is compared with his own
theory, which deals with only the high-frequency response of cy1indrical shells. Moreover,
the experimental values of the mean-square radial displacements given by C1inch are mean
values of measurements taken at severallocations on the shell.
This shell has been analyzed by the theory given in the foregoing by subdividing the
shell into eight identical finite elements; the boundary conditions have been taken to be
those of simple support. The natural frequencies of the shell, calculated first, show that
many of the natural frequencies of the shell for n = 2 and n = 3 are below 100 Hz,
indicating that the high-frequency response as calculated and measured by C1inch would
differ appreciably from the total response.
Calculations of the response are confined to n = 2-6 (from which the approximate
'totaľ response was obtained by summation). The values of this 'total' response of the
shell at its axial mid-point are shown in Figure 7.78. Also shown are values of the high-
frequency response, as calculated by this theory, obtained by taking into account only
the modes whose natural frequencies are in the range 93-1000 Hz, as well as C1inch's
experimental results.
It is immediately evident from Figure 7.78 that the response at the high frequency range
is but a small part of the total. The second point of interest is that agreement between
this theory and experiment, in the frequency range of 100-1000 Hz, approximately, is
quite good, which lends confidence in the theory. However, as discussed by Lakis &
Pa'idoussis (1972b), there are a number of weaknesses in this comparison, notably taking
r = 2 X 10- 2 for all r, as given by C1inch, although the measurement was made but for
one mode; in this connection, it should here be pointed out that the difficulties in obtaining
re1iable experimental values of  for shells are formidable. The results in Figure 7.78 show
that root-mean-square response depends on UL 6 approximately.
7.7.5 Concluding remarks
A theory has been presented, capable of predicting the response of a cy1indrical shell
to an arbitrary as well as a turbulence-induced random pressure field. The comparisons
with experimental measurements, which are rather scarce (for reasons to become clear
at the end of these remarks), show that the theory is sound. However, more important is
that this shows that this type of theory is sound; that is, the foregoing can also be used
in conjunction with a continuum analysis of the she1l or another type of finite-element
analysis.
A number of assumptions have been made, which may be worth recounting here:
(i) mean flow effects, other than added mass effects, are neg1igible; (ii) the damping matrix

722
SLENDER STRUCTURES AND AXIAL FLOW
20,000 20,000
10,000 10,000
3,000 3,000
 
g "'o::
 1,000 1,000 o
o -O
-
x X
1"'$ 1"'$
300 300
100 100
t
30 30
150 200 300 400500600800
Uf (in/s)
Figure 7.78 Turbulence-induced radial displacement at x = L/2 for a shell conveying water,
studied earlier by Clinch (1970), as a function of the centreline velocity (1 in, = 25.4 mm). - - o - -,
Clinch's experimental results for high-frequency response; - - -, theoretical results obtained
by Lakis & Paidoussis' theory (with n = 2-6) for high-frequency response (93-1000 Hz); -,
'total' response obtained by the same theory (with n = 2-6) considering all frequency components
(Lakis & Paidoussis 1972b).
is linearly related to the stiffness matrix; (iii) the pressure field is stationary, ergodic and
homogeneous; (iv) coupling between the various n is neglected; (v) the cross-product terms
in equation (7,233) are neglected; (vi) the only source of excitation is due to boundary-
layer pressure fluctuations (so that far-field effects are neglected); (vii) pressure loss,
pressurization and compressibi1ity effects are ignored; and (viii) Bakewelľs correlations
apply, Most of these assumptions are quite reasonable and should apply in many cases.
Concerning the Bakewell-based mode1ling of the turbulent wall-pressure field, it should
be reasonable so long as the Reynolds number is not small; in any case, the form of the
correlation should be the same for different Reynolds numbers, but the constants may
need to be adjusted.
Perhaps the only major limitation of this analysis concerns assumption (vi), Pressure
perturbations not locally generated in the turbulent flow within the shell, but generated
upstream or downstream and convected by the fluid to the shell, are both ubiquitous and can
be quite large. This so-calledfar-field noíse can easily be one or more orders ofmagnitude
larger than the boundary-Iayer related pressure fluctuations - see also discussion on this
topic in Chapters 8 and 9. Unfortunately, this far-field noise is not easily predictable, being
a function of the installation or laboratory apparatus. Even if all the upstream piping is
specified and known, prediction is very difficult - though not impossible ran example may
be found in Gibert (1988; section 19.5)], Usua1ly, however, the amplitude and frequency
spectra of the far-field contribution need to be measured in sítu.
If one wishes to have an apparatus which generates only turbu1ence-induced pressure
fluctuations, copious precautions need to be taken, and this is the reason for the statement

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
723
that experimental data of this sort are rather scarce, as stated at the beginning of this
discussion. UsualIy, some contamination from the far-field remains in the 'turbulence-
induced spectrum' , representing facility or apparatus specificity. This is the reason why,
if both the excitation field and the response are measured on the same apparatus, and the
excitation used in the theoretical model, one generalIy gets a better agreement between
theory and experiment than otherwise,
As is discussed in Chapter 8 (Section 8.4), far-field noise constitutes a much smalIer
component of the total excitation field in two-phase flow, since acoustic disturbances do
not 'travel welI' in most two-phase flow regimes. Studies in this area for internal flow
have been initiated (Lakis et al. 1978; Lakis 1978; Mohamed 1980), but have not been
published fully.
FinalIy, it is remarked that the maximum r.m,s. ampIitudes in the two sample calculations
of Section 7.7.4 are very small: w rms / h  10- 3 (Figure 7.76, for relatively low velocity
air flow) and w rms / h  60 x 10- 3 (Figure 7.77, for high velocity water flow). Such
smalI amplitudes are mainly of concern for high-cycIe fatigue considerations; refer to
CrandalI & Mark (1963, chapter 3) for the probablistic estimation of the mean Iifetime
to failure due to random vibration.
7.8 NONLlNEAR DYNAMICS
Other than some discussion in Section 7.3.6(b) and in the description of experimental
behaviour in Sections 7.2.4 and 7,5.5(c), the theoretical work treated so far in this chapter
has been in tenns of linear theory; yet, it is known (Section 7.8.1) that amplitudes of radial
defonnation of the order of the shelI thickness may result in important structural nonlinear
effects!
There has been a great deal of work on the nonlinear dynamics of shelIs in vacuo
and shelIs subjected to external, generally supersonic, flow - to be reviewed briefly in
Section 7.8.1, However, nonlinear work on shells filIed with dense fluid or conveying fluid
is much rarer, Readers interested mainly in fluid effects may jump over the review on
shelIs in vacuo, and go directly to Section 7.8.1(b,c).
Before the literature review, three important characteristics of nonlinear shelI dynamics
ought to be introduced. The first, common to alI forced nonlinear systems, is the sof ten ing
or hardening response, characterized by the backbone of the resonance curve leaning to
the left or the right, respectively. The interesting thing about shelIs is that both softening
(most frequently) and hardening nonlinear behaviour have been reported, and the two may
indeed exist in different parts of the parameter space (Amabili et al, 1998b; DowelI 1998;
Evensen 1999; PelIicano et al. 2002).
The second characteristic of interest is the existence of 'companion modes' . For linear
vibration, if a shell is excited periodicalIy, then the standing wave generated is azimuthalIy
fixed vis-a-vis the point of application of the force; this is referred to as the driven mode.
However, in experiments with large amplitude vibrations, circumferentialIy travelIing
waves have been observed in the response of the shelI, ln the theory, trus can be reproduced
by the addition of another standing wave mode with circumferential orientation Jr / (2n)
vis-a-vis that of the driven mode; this is calIed the companion mode. It has the same
frequency and mode shape as the driven mode, and it arises because of the axial symmetry
af the shelI via nonlinear coupling.

724
SLENDER STRUCTURES AND AXIAL FLOW
The third item of interest is that, a priori, one cannot ignore coupling between dif-
ferent circumferential modes; thus, the analysis can generally not be conducted for each
n independently as in linear theory. Of particular interest is that, even if one analyzes
oscillations in a particular n, nonlinearity generates a net circumferential contraction for
large amplitude oscillations; that is, inwards maximum deflections are larger than out-
wards ones. This suggests the inclusion ofaxisymmetric modes (n = O) in the assumed
mode expansion as first done by Evensen (1967), even if one is primarily interested in
deformation in some other n > O.
7.8.1 Literature review
This review is subdivided into three parts: (a) shells in vacuo, (b) shells in contact with
dense fluids and (c) shells in contact with flowing fluid. It draws heavily from the recent
literature reviews in Amabi1i et a/. (1998b, 1999c) and Dowelľs (1998) discussion, as well
as in Amabi1i & Pa'idoussis' (2003) review paper, wherein fuller versions may be found.
7.8. 1 (a) Shells in vacuo
The first study on this subject may be attributed to Reissner (1955), who considered
a single half-wave lobe of the vibrating shell, analyzing it as a simply supported open
shell by means of Donnelľs shallow shell theory, Donnelľs nonlinear shallow-shell the-
ory introduces a stress function, in order to combine the three equations of equilibrium
involving the shell displacements in the radial, circumferential and axial directions into two
equations involving only the radial displacement and the stres s function, Chu (1961) con-
tinued with Reissner's work, extending the analysis to closed cylindrical shells; he found
that non1inearity in this case always 1eads to hardening type characteristics, whereas for
the open shells Reissner found it could be either hardening or softening, depending on
the geometry. However, Evensen (1963, 1967) shows that in both these studies continuity
of in-plane circumferential displacements is not satisfied; more importantly, he notes that
experiments suggest that the nonlinearity should be of the softening type; Olson (1965),
in experiments with a thin seam1ess copper shell, found a small degree of softening, of
the order of 1 % in frequency for an amplitude equal to 2.5h, h being the shell thickness,
Evensen (1967) attempted to reconcile this quantitatively small but qualitatively import-
ant discrepancy between theory and experiment, by using a different form of the assumed
mode expansion, involving both companion modes and an axisymmetric (n = O) term to
account for nonlinear circumferential contraction.
Indeed, Evensen (1963, 1964, 1966, 1967) has made several pioneering contributions
in the study of nonlinear dynamics of rings t and cylindrical shells, both theoretical and
experimental. He notes that for rings the nonlinearity is of softening type and that this is
frequently true for cylindrical shells as well. Dowell (1967b) confirmed Evensen's (1966)
results for a ring using a somewhat different, but complementary, approach. Dowell &
Ventres (1968):1: then studied the cylindrical shell using the approach they had previously
developed for the ring and emphasized that, as the length-to-radius ratio, Lja, of the
tA ring is a shell, the cross-sectional deformation of which involves no axial variation,
*Some algebraic errors in Dowell & Ventres' analysis (1968) were noted by Atluri (1972). These have now
been corrected (Dowell & Ventres 1998) and it is shown that the results of Dowell 's approach are not essentially
changed,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
725
cylindrical shell becomes large, the shell behaves like a ring, with softening nonlinearity;
whi1e, in the linůt of small L/a, the shell behaves like a flat rectangular plate (i.e. a shell
of zero curvature or infinite radius), and one should expect a hardening nonlinearity.
The principal difference of the Evensen and Dowell analyses lies in the different assump-
tions made for the axia1 variation of the axisymmetric mode. Dowell & Ventres (1968)
show that Evensen's and Dowell's results are essentially the same for a ring and in the
ring linůt of a shell (Dowell1998). For cylindrical shells of intennediate L/a, however,
Dowell's and Evensen's results do not necessarily agree; thus Evensen's modal expansion
may give a softening nonlinearity and Dowell & Ventres' a hardening one, as found by
Varadan et aZ. (1989), for example, and further discussed by Prathap (1978), Evensen
(1978a,b), Dowell (1978, 1998) and Amabili et aZ. (1998b, 1999b),
The controversy has lasted for several years, and has only recently been resolved.
However, it seems to be true that whether one gets a softening Or hardening nonlinearity
depends on (i) geometry, and (ii) the type and number ofaxisymmetric tenns added in the
assumed mode expansion. This latter item has recently been convincingly demonstrated
by Amabili et aZ. (1999b), by an example in which, if one such axisymmetric mode is
used [m = 1 in equation (7 .254b) of Section 7.8.2], a hardening non1inearity is obtained,
but with two or three tenns (m truncated at 3 or 5) the nonlinearity is definitely softening,
Item (i) has been investigated by Pellicano et aZ. (2002).
Moving now to other work, Matsuzaki & Kobayashi's (1969a,b) work and Leissa's
(1973) review on large amplitude vibrations of circular cylindrical shells should be men-
tioned, as should Ginsberg's (1973, 1974) and Chen & Babcock's (1975) papers, which
constitute important fundamental contributions in this area.
Ginsberg (1973, 1974) uses a different approach in his study offree and forced vibrations
of circular cylindrica1 shells, avoiding the assumptions of mode shapes and employing
instead an asymptotic analysis to deal with the nonlinear boundary conditions for a simply
supported shell. The solution is reached by us ing the energy of the system to obtain the
Lagrange equations, which are then solved via the harmonie balance method, Ginsberg
uses the more accurate Flfigge-Lur'e-Byme shell theory, instead of Donnell's sha1low
shell theory, considering both the driven and the companion modes. Both softening or
hardening nonlinearities are found, depending on system parameters.
Chen & Babcock (1975) use the perturbation method to obtain solutions to the non1in-
ear Donnell equations, again without selecting a particular deflection solution, They solve
the c1assical simply supported case with companion mode participation, and examine
the appearance of a travel1ing wave. The solution involves a sophisticated mode expan-
si on, including 'boundary-layer' -type (in the singular-perturbation theory sen se) tenns in
order to satisfy the boundary conditions, They also present experimenta1 results in good
agreement with their theory, showing a softening nonlinearity.
Fu & Chia (1993) find that for the larninated shells they consider, softening or hardening
nonlinearity depends on the h / a ratio. Ganapathi & Varadan (1995, 1996) consider large-
amplitude vibrations ofunifonn and lanůnated shells by means ofNovozhi1ov's non1inear
theory and a finite element fonnulation. Also, Selmane & Lakis (1997c) and Lakis et aZ,
(1998) study open and closed orthotropic shells, using Sanders' nonlinear theory and their
hybrid finite element method (Section 7.3.3).
Chiba (1993a) studied experimental1y large amplitude vibrations of cantilevered circu-
lar cylindrical shells made of polyester sheet. Almost all responses display a softening
nonlinearity. It is found that for modes with the same axial wavenumber, the weakest

726
SLENDER STRUCTURES AND AXIAL FLOW
degree of softening nonlinearity can be attributed to the mode having the minimum nat-
ural frequency; a1so, shorter shelIs have a larger softening non1inearity than longer ones.
Travel1ing wave modes were also observed,
Some of the classical and modem methods of non1inear dynamics - see Chapter 5 and
Appendix F - have been used here also. Nayfeh & Raouf (1987) examine the problem by
means of plane strain theory of shelIs and a perturbation ana1ysis, more suitable for rings
than for supported shelIs of finite length. They investigate the response when the frequency
of the axisymmetric mode is approximately twice that of the asymmetric mode (2: 1 internal
resonance). Raouf & Nayfeh (1990) and Nayfeh et a/. (1991) examine the response of
infinitely long shelIs, uti1izing the method of multiple scales and perturbation analysis,
while retaining both the driven and companion modes. They find amp1itude-modulated and
chaotic solutions. Popov et a/. (1998), Foale et a/. (1998) and Laing et al. (1999) present
interesting studies of different methods for obtaining low-dimensional models from the
non1inear equations of motion of a shalIow cy1indrical shelI panel under periodic axia1 or
trans verse forcing.
Amabi1i et a/. (1998b, 1999b, 2000a) study the dynamics of simply supported shelIs by
synthesizing the characteristics ofEvensen's and DowelI & Ventres' solutions, satisfying
tulIy the radial displacement and bending moment boundary conditions and continuity of
circumferential displacement, while satisfying the axial constraint 'on the average'; both
driven and companion modes are considered. The results agree welI with Ganapathi &
Varadan's (1996), Chen & Babcock's (1975), and also with Olson's (1965) experiments.
Figure 7.79 illustrates this last comparison for a shell with L = 0.39 m, L/a = 1.92,
ajh = 1816, E = 1.24 x 105 MPa, Ps = 8,93 x 10 3 kg/m 3 , v = 0.365, for modal


h::
0,5
1.5
1.0
o o
o
0,990 0.992 0,994 0.996 0.998
1.002 1.004 1.006 1.008 1.01
W/W nm
Figure 7.79 Response-frequency curves and backbone curves for the driven mode for the shell
in vacuo originally studied by Olson (1965), for n = 10, m = 1: o, Olson's (1965) experiments;
- - - Evensen' s (1967) backbone curve, essentially coincident with that of Ganapathi & Varadan
(1996); -, Amabili et at. (1998b) ca1culations.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
727
excitation fnm = 0.002 h2Pswm' W nm = 2n x 91.71 rad/s, and damping ratio nm =
0,0008; it is seen that agreement is quite satisfactory.
A great deal of excellent work on nonlinear dynaITÚcs of shells has been published in
Russian, whích is not cíted in the foregoing, Interested readers are encouraged to refer to
the extensive review by Kubenko & Koval' chuk (1998), wherein most of this literature,
as well as some in the foregoing, is discussed - a total of 223 references,
Several studies have also been conducted on the effect of initial imperfections, for
example, by Kubenko et al, (1982) and Watawala & Nash (1983).
Before ending this section, Yamaki's (1984) excellent book on the nonlinear stability of
circular cylindrical shells should be mentioned, a topic of interest to readers of this chapter.
7,8.1 (b) Shells in contact with dense fluid
Most of the work on this subject is inspired by applications, see also Section 7.10, namely
containers, reservoirs, silos, and so on. A great deal of work on the linear dynamics
of such structures, rectangular or cylindrical, has been undertaken, especially on the
resonant frequencies of partly or comp1ete1y liquid-filled containers with an open top
or hermetícally sea1ed, the effect of flexibility or rigidity of the bottom, coupling with
sloshing of the liquid, and so on, Also some work has been done on shells immersed in
liquid, The readeris referred to Berry & Reissner (1958), Abramson et al, (1966), Rapoport
(1968), Lakis & Pai'doussis (1971), Gupta & Hutchison (1989), Fe1ippa & Ohayon (1990),
Morand & Ohayon (1989, 1995), Chiba (1992, 1993d), Gon9alves & Ramos (1996), Liu
et al. (1998), Amabíli et al. (1998a,b), for instance, and to reference s cíted therein, For
an excellent review of sloshing dynamics, see Ibrahim et al. (2001).
In most cases, the applications are such that motions must be small. However, in some
cases, for examp1e, for earthquake-induced motions or blow-down situations in nuclear
reactors, large transient motions are possib1e; although, motions need not be truly large for
nonlinear effects to manifest themselves. Moreover, in sloshing situations, large motions
of the fluid arise, and hence hydrodynamic nonlinearities. Here, some of this literature,
considerably 1ess extensive than for shells in vacuo, is reviewed briefly, strictly for steady-
state rather than transient oscíllations,
ln liquid-filled shells, one has two oscíllators: the fluid and the she11, with the fluid
frequencíes generally being much higher, unless there is a free surface and hence gravity
waves, ln this latter case, strong coupling between the shell and the liquid modes is
possib1e. Stíll, one generally distinguishes between (i) 'sloshing modes', which are modes
characteristíc oť the liquid free surface, where the amplitude of the free surface waves is
larger than the wall displacement and (ii) 'bulging modes' that are characteristic of the
oscíllating tank wall excíting the liquid, generally with much smaller free-surface motions.
Chu & Kafia (1967) develop a theory for nonlinear vibrations of partially filled tanks,
where the nonlinearity is associated with free-surface waves, while linear she11 theory
is employed, Kafia (1966) also perfonned experiments on large-amplitude vibrations of
partially filled shells.
Ramachandran (1979) studies the large-amplitude vibrations of circular cylindrical
shells with circumferential1y varying thickness and immersed in a quiescent, inviscid
and incompressible fluid. Donnell's shell theory is used, together with a simplified ver-
sion of Evensen's mode expansion and a Rayleigh-Ritz solution procedure, Numerical
results for free vibration show a softening nonlinearity both in vacuo and in water.

728
SLENDER STRUCTURES AND AXIAL FLOW
Boyarshina (1984, 1988) examines theoretically the nonlinear free and forced vibrations
and stability of a circu1ar cylindrical tank partially filled with a liquid having a free surface,
ln this case, nonlinearity is attributed to the interaction of free surface waves and elastie
flexural vibrations of the shell. In partieu1ar, three characteristie cases are investigated:
(i) the natural frequency, Ws, of sloshing of the liquid free surface equal to the natura1
frequency of shell vibration, W n ; (ii) W n = 2w s ; and (iii) W n = 3w s .
Gonalves & Batista (1988) consider simp1y supported circu1ar cylindrica1 shells filled
with inviscid incompressib1e fluid, using Sanders' nonlinear shell theory and a novel
mode expansion that inc1udes two terms in the radial direction (the asymmetric and the
axisymmetric ones) and 10 terms to describe the in-plane disp1acements. The companion
mode is not considered, and no boundary conditions are imposed on the fluid at the
ends of the shell. Both free vibrations and undamped response to harmonie excitation are
investigated. It is found that the presence of a dense fluid gives stronger softening results
than for the same shel1 in vacuo,
Sivak & Te1a10v (1991) conducted experiments on a vertical circu1ar cylindrical shel1
made of titanium alloy, partially filled and partially submerged in water; experimental
boundary conditions simu1ate a c1amped shell, Many experiments indicate softening non-
linearity; however, the comp1ete1y filled shell shows a hardening nonlinearity, an effect
increasing with n, possib1y due to the steady stresses associated with the weight of the
water,
Chiba (1993b) studied experimental1y 1arge amplitude vibrations of two vertical can-
ti1evered circu1ar cy1indrical she11s made of po1yester sheet, partial1y filled with water to
different 1evels. It was observed that, for bu1ging modes of the same axia1 wavenumber,
the weakest degree of softening nonlinearity can be attributed to the mode having the min-
imum natural frequency, as in empty shells, It was a1so found that shorter tanks (shells)
have a 1arger softening nonlinearity than taller ones, as do tanks with a 10wer liquid 1evel.
Trave1ling wave modes and coupling between two bu1ging modes (and between two slosh-
ing modes) were also observed. The effect of a thin film floating on the free surface and
the behaviour of the free surface was also investigated, Large-amplitude vibrations of two
vertieal circu1ar cylindrical shells with c1amped ends, partially filled to different heights
were studied by Chiba (1993c). In this case also, the response generally disp1ays a sof ten-
ing nonlinearity. Again, the nonlinearity is 1arger for partial1y filled shells, as compared
to empty and comp1etely filled ones; this may be attributed to 1arge-amplitude free-
sUrface waves, when the free surface is away from a nodal circ1e in the shell mode being
excited.
Lakis & Laveau (1991) and Se1mane & Lakis (1997d) use their hybrid finite element
method (Section 7.3.3) to study the nonlinear dynamics of open and c10sed cylindrica1
shells in contact with inviscid incompressible fluid, neg1ecting free-surface effects. Both
softening- and hardening-type nonlinearity is found for c10sed circu1ar cylindrical shells,
empty, filled, and submerged in quiescent fluid, depending on the geometry. Significantly,
the influence of the nonlinearities associated with the fluid are found to be negligible - in
contrast to Chiba's findings, presumably because in Lakis et al. free-surface effects are
neg1ected.
A great deal of work has also been done on linear and non1inear dynamics of tanks with
a flexib1e bottom and on tank-like damping/attenuation deviees invo1ving sloshing; see,
for examp1e, Welt & Modi (1992), Ikeda et al. (1998), Chiba (1992, 1996), Chiba & Abe
(1999) and the references cited therein.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
729
Amabili et al. (1998b) study the nonlinear free and forced vibrations of simply supported
shells filled with and/or surrounded by liquid, by means of Donnell's nonlinear shallow
shell theory and a solution method a1ready discussed at the end of the review of shells
in vacuo. Solutions are obtained (i) numerically and (ii) by a perturbation analysis using
the method ofnormalforms (Appendix F.3). Typical results are shown in Figure 7.80(a)
for a water-filled simply supported shell originally studied numerically by Gonalves &
Batista (1988), with L = 0.5 m, Lja = 0.6, aj h = 200, E = 2.06 x 10 5 MPa,
Ps = 7.576 x 10 3 kg/m 3 , v = 0.3, and P = 10 3 kglm 3 , neglecting companion-mode
participation, for modal forcÍng fnm = 0.003 Eh 2 j[a 2 (1 - v 2 )], W nm = 2n x 157 rad/s,
and no damping (nm = O). Agreement between the numerical and the perturbation
solutions is reasonably good only up to Anm = 0.4 h, where Anm is the amplitude of the
driven mode. The discrepancy for larger amplitudes is most probably due to the fact that
the perturbation analysis has been carried out to second order only.
Whatis not evident in Figure 7.80(a) is that a partofthe response curve is unstable, This
is shown in Figure 7,80(b) and (c), where results with companion mode participation, that
is, Bnm (t) i= O, are also shown, in this case with nm = 0.014. The response curve being
unstable could mean one of two things: either that a stable periodic solution does exist
but a higher-order analysis is required to find it, or that no periodic solution is possible
and the response is quasiperiodic, amplitude-modulated - which is the case here - or
chaotic. Nevertheless, with companion mode participation, most of the response curve is
stable; only in the vicinity of wjw nm = 0.95 is there an amplitude-modulated region. The
maximum amplitude is smaller as a result of the damping.
Nonlinear vibrations of water-filled shells have been studied by Amabili et al. (1999b,
2000a) by means of refined models; an experimental study of this system has recently been
conducted by Amabili et al. (2002). A problem of internal resonances in a water-filled
circular cylindrical shell is treated in Amabili et al. (2000c) and Pellicano et a/. (2000) by
means of both numerical and analytical (normal forms) methods,
Additional work, particularly from Eastern Europe, is reviewed by Kubenko &
Koval'chuk: (2000).
7.8.1 (c) She/ls with flow
Nonlinear studies of the dynamics of shells subjected to internal or external flow are few.
The first to appear deal with external, mostly supersonic flows, and were probably inspired
by nůssile applications (Olson 1965; Olson & Fung 1967; Evensen & Olson 1967, 1968;
Horn et al. 1974).
ln Olson & Fung (1967) shell motions are described by means of Donnell's nonlinear
shallow shell theory and the aerodynamics by linear piston theory or potential flow theory,
obtaining solutions via a two-mode expansion, without companion mode participation
but including axisymmetric modes. Evensen & Olson (1967, 1968) use piston theory and
a four-mode expansion, incorporating the companion mode. t Limit-cyde solutions are
obtained via Galerkin discretization and the harmonie balance method, and their stability
is exanůned numerically. Two types of linůt-cycle flutter are obtained: a standing-wave
flutter and a circumferentially travelling wave flutter; typical1y, trave1ling wave flutter
tHowever. similarly to Evensen's (1967), these mode expansions are not moment-free at the ends of the
shell, as they should be for c1assical simply supported shells, and the homogeneous solution for the stress
function is neglected,

730
SLENDER STRUCTURES AND AXIAL FLOW
1.0
0.9
0.8
0,7
0.6
 0,5
I..J
0.4
0.3
0,2
O,]
O
0,7 0,8 0,9 1.0 1.1 1.2 1.3
(a)
1.5
0.5
..../
Bnm (t) *" O
r Bnm (t) = O
Bnm (t) *" O
,
] O

I..J
(b)
O
0,60 0.65 0.70 0.75 0,80 0.85 0.90 0,95 ],05
1.0

Ir:c." 0,5
O
0,60 0.65 0.70 0,75 0.80 0,85 0.90 0,95 1.05
(c) w/w nm
Figure 7.80 The frequency-response curves for a shell filled with fluid originally studied by
GOll!;:alves & Batista (1988) for n = 10,m = 1: (a) Ani == AIO,l without companion mode
participation and zero damping: - - -, Gonalves & Batista's (1988) numerica1 solution; -,
Amabili et al. (1 998b ) second-order perturbation solution. (b) An 1 and (c) Bn 1 with companion
mode participation, and with damping: -, stable solution; . . ., unstable solution; in (b) the curve
without companion mode participation (Bnm(t) = O) is also shown (Amabili et a/. 1998b),

THlN SHELLS CONTAINING OR IMMERSED IN FLOW
731
arises through instability of the standing wave limit cyde, The onset of flutter can be
subcritica1. Olsson (1978) reexamines the problem in the presence of a temperature field
causing variations in the material properties.
The reader interested in the dynamics of shells subjected to compressible axia1 flow is
referred to Dowell (1975; chapter 3) and Dowell & Ilgamov (1988).
Lakis & Laveau (1991) and Selmane & Lakis (1997d) are probably the first to consider
nonlinear effects on the vibrations of shells with intemal flow. Lakis & Laveau uti1ize
the hybrid finite-element method of Section 7.3.3, considering only fluid-related non-
linearities in a potential flow framework; specifica1ly, the Bernoulli equation is expanded
to second-order and a linear boundary condition is used to satisfy the impermeability con-
dition at the fluid-shell interface. Results show that this kind of fluid nonlinearity can be
neglected for vibration amplitudes ofphysical significance, confirming Evensen & Olson's
(1967,1968) findings. In contrast, Selmane & Lakis (1997d) consider a1so structural non-
linearities, by using the nonlinear Sanders-Koiter shell theory and a more conventiona1
finite element scheme. Theoretical results are obtained only for circumferentially open
cylindrical shells conveying fluid, although the formulation is quite general and could a1so
have been applied to dosed shells. For open shells, the effect of vibration amplitude on
vibration frequencies is studied, and it is found that nonlinearity produces either harden-
ing or softening behaviour, depending on the circumferentia1 wavenumber n, for both
quiescent and flowing fluid.
Recently, Amabili et al, (1999a) have initiated a systematic reconsideration of the
dynamics of cylindrical shells conveying fluid by nonlinear theory, in the first of a series
of studies on the subject. This work is discussed in the next section.
7.8.2 Simply supported shells conveying fluid
Amabili et al. (1999a) reexamine the nonlinear dynamics of simply supported shells
conveying fluid by means of the nonlinear Donnell sha1low shell theory and linear potentia1
flow models,
7,8.2(a) Shell equations and boundary conditions
For a shell with the nomenc1ature and axes as in Figure 7.2(a), but with w in this case
being inward, Donnelľs shallow she1l equations may be written as (Chu 1961; Evensen
1967; Dowelll975; Yamaki 1984)
4 h ' h .. f 1 a 2 F
DV w + c w + Ps w = - p + -;; ax 2
( a2F a 2 w a 2 F a 2 w a 2 F a 2 w )
+ a 2 ae 2 ax 2 - 2 aaxae aaxae + ax 2 a 2 ae 2 '
(7.248)
1 4 1 a 2 w ( a2w ) 2 a2w a 2 w
Eh V F = - -;; ax2 + aaxae - ax 2 a 2 ae 2 ' (7,249)
where D = Eh 3 /[12(1 - v 2 )] is the flexural rigidity, E is Young's modulus, v the
Poisson ratio, h the she1l thickness, a the mean shell radius, Ps the mass density of the

732
SLENDER STRUCTURES AND AXIAL FLOW
shell, c the damping coefficient, and f and p are the radial pressures on the surface of
the shell due to extemal forces and the contained flowing fluid, respectively; here f = O.
The overdot denotes a time derivative. F is the Airy potential stress function, related to
in-plane stresses, via which the three shell equations are reduced to two. The biharmonic
operator here is defined as V 4 = [a 2 /ax 2 + a 2 /(a 2 ae 2 )]2.
These equations are based on Donnell's shallow-shell theory, with only the dominant
nonlinear terrns retained. t Results should be accurate for very thin shells, hla « I, and
only for modes of high circumferential wavenumber n; specifically, l/n 2 « I must be
satisfied, so that n  5 is required in order to have fairly good accuracy. The in-plane
stres s resultants are given by (DowellI975)
1 a 2 F
N ---
x - a2 ae 2 '
a 2 F
No = ax 2 '
1 a 2 F
Nxo = --;; ax ae '
(7.250)
and they are related to the shell displacements by
2 N x vw 1 ( aw ) 2 v ( aw ) 2 au vav
(1 - v )- = -- + - - + - - + - +--
Eh a 2 ax 2 aae ax a ae '
2 No w v ( aw ) 2 1 ( aw ) 2 au 1 av
(I - v ) Eh = -  +"2  + 2 aae + v ax + -;; ae '
2 Nxo [ 1 awaw 1 au av ]
(I-v)- =2(I-v) ---+--+- .
Eh a ax ae a ae ax
(7.251)
(7.252)
(7.253)
For motions in the nth circumferential wavenumber, the deforrnation w may be
expres sed in terrns of the linear eigenmodes as basis,
N) N2
w(x, e, t) = L L [Anm(t) cos(ne) + Bnm(t) sin(ne)] sin(Am X )
n=i m=i
N3
+ L AOm(t) sin(Am X ),
m=l
(7 .254a)
where Am = mj( / L, t is the time, and Anm (t), Bnm (t) and Aom (t) are unknown functions
of t; thus, axisymmetric modes are included which, as discussed in Section 7.8.1, are
important. On the other hand, nonlinear coupling among asymmetric modes of different n
can be neglected, for simplicity.:j: ln the following analysis, the sums in equation (7.254a)
are truncated at Ni = 2 and N2 = 5; it is also noted that axisymmetric terrns having an
even m value can be eliminated, as they do not contribute to shell contraction (half the
shell moves inwards and the other half outwards). Hence, the following mode expansion
tFor example, nonlinearities in curvature strains are neglected,
:J:The effect of such coupling has been studied in Amabili et al. (2oooa) and Pellicano et a/. (2002) and
found to be much less significant than that of the axisymmetric modes; in the last paper, models with more than
20 degrees of freedom have been used, to examine the convergence of the solution.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
733
is used:
2
w(x, O, t) = L [Anm(t) cos(nO) + Bnm(t) sin(nO)] sin()"m x )
m=l
3
+ L AO,(2m-1)(t) sin()"2m-l X ).
m=l
(7 .254b)
This may be considered as a more sophisticated version of the expansion used by Amabili
ct a/. (1998b) with which the results in Figures 7,79 and 7.80 have been obtained, where the
sums are truncated at NI = 1 and N2 = 3, since only modes with m = I are investigated
in that study and an artificia1 constraint is introduced to reduce the number of degrees of
freedom. Equation (7 .254b) satisfies the boundary conditions
w = O and Mx = _D{(o2wjox 2 ) + v[o2wj(a 2 00 2 )]} = O at x = O, L,
(7.255)
where Mx is the bending moment per unit length [Figure 7.2(b)]. The other boundary
conditions depend on whether we have a 'c1assicaľ simply supported shell of finite length
(here denoted as Case 1) or an infinitely long, periodically supported shell with restrained
axial displacement at the supports (Case 2). They are as follows:
Case 1: N x = O and v = O }
Case 2: u = O and v = O
at x = O, L;
(7.256a)
(7 .256b )
moreover, u, vand w must be continuous in O. The conditions imposed in Case 2 are
welljustified by the reciprocal constraint between the part ofthe shell under consideration
and extensions thereof outside (O, L). Case 2 also approximates a shell with thick rings
at the ends; by means of this formulation, it is also possible to study a shell subjected to
axisymmetric prestress, that is, N x = Ň x at x = O, L.
The expansion used for w satisfies the boundary conditions given by equation (7.255);
moreover, it satisfies exactly the continuity of circumferential displacement, that is,
1 21f o V
- dO = V(21T) - v(O) = o.
o 00
(7.256c)
The boundary conditions for either of the in-plane displacements, equations (7.256a,b),
cannot be satisfied for all O at the ends of the shell; instead, they are satisfied only 'on the
average', Specifically, the following conditions are imposed:
Case 1:
[21f
10 N x a dO = O,
(7.257a)
Case 2:
1 21f l L ou 1 21f
- dx adO = [u(L, O) - u(O, O)] adO = O;
O O ox O
(7 .257b)

734
SLENDER STRUCTURES AND AXIAL FLOW
and for both cases,
1 27r l L Nxe dx a dO = O. (7.258)
Equation (7.257a) assures a zero axial force N x 'on the average', while equation (7.257b)
states that the axial displacement u is zero 'on the average' at x = O, L; equation (7.258)
is satisfied when u and w are continuous in O on the average and v = O on the average at
x = O, L.
As a consequence of these in-plane constraints, mean stress resultants N x , N e and N x e
are induced - see Amabili et aJ. (l998b, 1999a,c). They may be expressed in terms of
w, giving for Case 1
N x = O, N x e = O,
(1- v 2 ) Ne =  [27r [L [ (V 2 -1) w + 1- v 2 ( aw ) 2 ] dxadO,
Eh braL 10 10 a 2 aao
(7,259a)
and for Case 2,
2 N x 1 1 27r l L [ VW 1 ( aw ) 2 v ( aw ) 2 ]
(1 - v ) - = - -- + - - + - - dx a dO,
Eh braL o o a 2 ax 2 aao
2 Ne 1 1 27r l L [ w v ( oW ) 2 1 ( aw ) 2 ]
(1 - v )- = - -- + - - + - - dx a dO,
Eh braL O O a 2 ax 2 aoB
N xe = O.
In what follows, only Case 1 is considered,
(7 ,259b)
7.8.2(b) The fluid-dynamic model
The fluid-dynamic model employed is wholly linear, based on the observation that although
structural nonlinearities arise for amplitudes of ťJ(h), such amplitudes are hardly large
enough to invalidate a linear treatment of the fluid dynamics (Evensen & Olson 1967,
1968; Gonr;alves & Batista 1988; Lakis & Laveau 1991). Furthermore, in agreement with
the work in Section 7.5, unsteady viscous effects are neglected. Steady viscous effects are
a1so neglected, but this can only be justified for sufficiently short shells.
Two models, already discussed in Sections 7.2.3 and 7.2.5, are employed to determine
the perturbation pressure on the shell wall, by potential flow theory: Pai:doussis & Denise's
(1972) and Weaver & Unny's (1973). Without further ado they are given by
L In(mna/L) ( a a ) 2
P Ir-a = P - + u - w
- mnI(mna/L) at ax
(7.260)
for the Pa'jdoussis & Denise model, cf, equation (7.45), and
_ L 1 00 m[1 - (_I)m e -iaL] In(aa)
P I r - a - P - c os ( nB )
- 2 2 2 2 L 2 I , ( )
-00 m n - a a n aa
x [A(t) + 2iaU Á(t) - a 2 U 2 A(t)] e iax da
(7.261)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
735
for the Weaver & Vnny model, where ex is the Fourier transfonn variable and the deflection
fonn w(x, t) = A(t) sin(ml'xjL) cos (nO) has been assumed for simplicity,
It is important to state that solutions using the Weaver & Vnny model with the full
modal expansion of equation (7 .254b) are impossible, since the axisymmetric modes in
that theory imply an infinite kinetic energy of the fluid, because of the presence of semi-
infinite rigid bafftes (extensions) on either side of the shell in this model. Hence, for
the Weaver & Vnny model, solutions are possible only with AOl = A03 = A05 = o.
This, however, fundamentally alters the nonlinear behaviour of the shell; besides, a zero
axisymmetric contraction is virtually unattainable in practice.
7.8.2(c) Reduction to a finite-dimensional model
By substituting equation (7 .254b) into (7,249), a partial differential equation for the stress
function F is obtained. Its solution, in tenns of a homogeneous and a particular solution
may be written as
F = Fh + Fp,
Fp = CI(t) cos(nO) sin(nxjL) +.,. + C33(t) sin(2nO) cos(3nxjL),
where the full equation is given in Appendix 0.1, and
(7.262)
(7.263)
1- 2 2 I 2 { - I l L 1 21f [ a2Fp ] 1 -
Fh = -Nxa O + -x No - - ---z adOdx - NxoxaO
2 2 2naL o o ax
1- 2 2 I 2 { - 2Eh [ A03(t) A05(t) ] } -
=-Nxa O +-x No+- AOl(t)+-+- -NxoxaO;
2 2 an 3 5
(7.264 )
N x , No and N x O are the in-plane stress resultants generated at the ends of the shell,
as a consequence of the in-plane constraints on the average - equations (7,259a) for
Case I. Equation (7.264) is not the most general homogeneous solution, but it is chosen
in order to satisfy the boundary conditions on the average. In fact, it satisfies equa-
tions (7.250) on the average as a consequence of the contributions of Fp (i) to No being
(2naL)-1 fOL fi 1f [a 2 F p jax 2 ]adOdx and (ii) to N x and Nx o being zero, For Case I,
which is the ooly case studied here, simple calculations yield
N o = Eh { - [ AOl(t) + A03(t) + A05(t) ]
na 3 5
+ ;;2 [Al (t) + B;l (t) + A2(t) + B;2(t)]} ,
(7.265)
and it is recalled that N x = N xO = O. At this point, all the tenns involved in the equation
of motion, equation (7.248), have been evaluated.
Following the Galerkin method, seven coupled second-order non1inear ordinary
differential equations for the variables AnI (t), Bnl (t), A n 2(t), B n 2(t), AOl (t), A03(t) and
A05(t) are generally obtained, by successively weighting the single original equation

736
SLENDER STRUCTURES AND AXIAL FLOW
with suitable functions Zs, s = 1,..., 7, and integrating over the sheU middle surface.
The weighting functions Zs are defined as
Zs(X, (J) =
cos(n(J) sin(7l'x/L)
sin(n(J) sin(7l'x/L)
cos(n(J) sin(27l'x/L)
sin (ne) sin(27l' x / L)
sin(7l' x / L)
sin(37l' x / L)
sin(57l'x/ L)
for s = 1
for s = 2
for s = 3
for s = 4
for s = 5
for s = 6
for s = 7.
(7.266)
The Galerk.in projection of the equation of motion (7.248), in this case, may be effected
by tak.ing the appropriate inner product, (a, b),
{271: {L
([eq,(7.248)] , zs} = 10 10 [eq.(7,248)] zs(x, e) dx a de,
(7.267)
which has been done using Mathematica (Wolfram 1996); the resulting equations of
motion are given in Appendix 0.2.
Numerical results are presented here for a steel-water system originaUy studied by
Weaver & Unny (1973): L/a = 2, hla = 0.01, E = 2,06 x 10 5 MPa, Ps = 7.85 x
10 3 kg/m 3 , v = 0.3 and p = 10 3 kg/m 3 ; the results are presented in terms of Weaver &
Unny's form ofthe dimensionless U and Q, namely
V == U w-u = U / [(7l'2/ L )(D/P s h) 1/2] ,
Q == Q w-u = w/ [(7l'/L)2(D/Psh)1/2],
(7.268)
where D is as in equation (7.248) and w is the dimensional radian frequency,
7.8.2(d) Línear dynamics
Calculations of the eigenfrequencies according to both the Weaver & Vnny and
Paldoussis & Denise fJ.uid-dynamical model s for n = 5 and with no dissipation ({ = O)
have already been presented in Figure 7.26, from which it can be seen that the dynamics as
obtained by the two models is dosely similar, lt is noted that n = 5 is the circumferential
wavenumber associated with the lowest critical fJ.ow velocity for this system.
AIso, calculations with some dissipation induded ({ = 0.01) are presented In
Figure 7.25, showing that damping eliminates post-divergence restabilization.
7,8.2(e) Nonlinear dynamics
Numerical solutions have been obtained for the same system, principaUy by using AUTO
(Doedel & Kernéves 1986). Some results are presented first for a simplified modal
composition, with no companion mode participation: Bn 1 (t) = B n 2 (t) = O,
The corresponding bifurcation diagrams, calculated with the Patdoussis & Denise
model, are shown in Figure 7.81. AU solutions found, for all V, are nonoscillatory

1 h.N SHELLS CONTAINING OR IMMERSED IN FLOW
737
20
10
 O

-.;
-10
-20
(a)
10
5
 O


-5
-10
(b)
15
10
.:::
 5
O
-5
O
(c)
3
-----
--
- -
2
---
- - -
------
- - - -
...
3
....
....
- -
3
-
2
3
V
5
6
Figure 7.81 (a) and (b) Nonoscillatory dimensionless modal amplitudes AnI / h and A n 2/ h versus
the dimensionless flow velocity V, using Pa'idoussis & Denise's fluid-mechanical model, for  =
0,01 and without companion mode participation; (c) the corresponding dimensionless displacement
for a point on the shell at x = L/4: -, stable branches; - - -, unstable branches ofthe solution
(Amabili et al. 1999a).
(stationary), that is, they correspond to fixed points, thus reinforcing the conclusion in
Section 7,2.7 that the observed flutter of shells with supported ends is really a dynamic
divergence,
ln the bifurcation diagram of Figure 7,81(a), it is seen that a subcritical pitchfork
bifurcation is obtained, associated with the first mode (branch 1) at V = 3.33 - indeed,
strongly subcritical: the range of V over which the system may diverge, given a large
enoughdisturbance,extendsover1.31 < V < 3.33!AtV = 1.91, thereappears asecond
stable solution [branch 2, in Figure 7.81(b)], associated with divergence in the second
mode. Beyond this value of V, the system could easily undergo jumps, as a result of the
coexistence of four stable and some unstable solutions.
Figure 7.81 (c) shows the dimensionless displacement, rather than the individual modal
amplitudes. It is seen that the system is no longer symmetric with respect to the initial equi-
librium, as a consequence ofaxisymmetric contraction; it is clear that inwards (positive)
axisymmetric displacements are larger than outwards ones.
Taking into account the symmetry of the system, static solutions with no companion
mode participation can have the first and second longitudinal mode s (i) in-phase in e, the

738
SLENDER STRUCTURES AND AXIAL FLOW
case just examined, or (ii) orthogonal in () [Le. with the modes rotated through 7T / (2n)
relative to each other], to be considered next. In this case, we take A n 2(t) = BnI (t) = o,t
and shell defonnation in vol ves AnI (t) and B n 2(t), as well as the three axisymmetric
modes. The dynamics, in tenns of the modal amplitud es AnI, B n 2 and AOI is shown in
Figure 7.82. As expected, the dynamical behaviour, beyond the subcriticalloss of stability
by divergence, is different from that in Figure 7.81. In particular, it is seen that, for high
enough V, both branches 1 and 2 become unstable, and no stable solutions exist - that is,
--" ...
20 3 -:: ....
-
.". ...
I "
10  4
- - ,.- - ... ... 2
 - . -
.. O - - - - - -
- - ...
-..: - ;JI. - - -
- -
-10 '-,.... ...
.....-
.....-
.. .......... ....
-20 ........... -
(a)
3 ." - -
" -
10 .- 2
,,"
5 d;,
. .. -
 . -
-
'" O - - - -
"
 . .. -
. - ..
-5 .
-10 "'III '..
'- -
- - -
(b)
, .
30 .
# .
. .
# ,
25 # .
3 .f #
. . #
20 . 1 .
# #.
 . . .
.
<=> 15 . . .'
-..: #
10 " .,. 2 ...,..
.... . 4 _ .- ..-'"
5 .. .. - -
O ... ... ,.. .. .. :.
O 2 3 4 5 6
(c) V
Figure 7.82 Nonoscillatory dimensionless amplitudes versus V without companion mode parti-
cipation, taking An2(t) = Bnl (t) = O, and for  = 0.01: -, stable branches; - - -, unstable
branches. Amp1itude of (a) the first longitudinal mode, AnI! h; (b) second longitudinal mode,
B n 2/ h; (c) first axisymmetric mode AoI! h (Amabi1i et a/, 1999a).
tThis case does not involve companion mode participation, A trne companion mode should have the identical
form to the driven mode in every way, except for being azimuthally orthogonaI; thus, for trne companion mode
participation, both AnI and Bnl, or both A n 2 and B n 2, should be present.

THlN SHELLS CONTAINING OR IMMERSED IN FLOW
739
20
o
10
o Bn1/h
v
-10
-20
20
10
O An/h
-10
-20
Figure 7.83 Behaviour of the system starting from a point on the coupled-mode divergence
solution branch at V = 4,93 when slowly decreasing the dimensionless flow velocity V, with
Pai:doussis & Denise's model; generalized coordinates An 1 and Bn 1 versus V (Amabili et a/, 1999a).
none with this restricted mode composition, Of special interest is the folding of branch 1
[Figure 7.82(a)], leading to the emergence ofbranch 3 at V = 1.84. That branch also is
'unstable'; t however, if all the mode components are taken into account, it becomes stable
and is associated with travel1ing wave behaviour, as discussed next.
When all the generalized coordinates are retained in the discretized equations of motion,
equation (0.4) et seq., there is no preferential azimutha1 coordinate to specify the orient a-
tion of shell defonnation. The solution can be imaged in 3-D plots, involving the amplitude
of the driven and companion modes versus V for the two pairs of variables {An 1, Bn I} and
{An2, B n 2}, In particular, a section of this 3-D plot for zero amplitude of the companion
mode (B n 1 = B n 2 = O) gives the 2-D plots already obtained in Figure 7.81; obviously,
in view of the symmetry of the equations and of the physical system, the same 2-D plot
is obtained for a section at zero amplitude of the driven mode (AnI = A n 2 = O), There-
fore, the nonoscillatory solutions can be imagined as surfaces generated by rotation of
the corresponding curves in Figures 7.81 and 7.82. Generally, stable branches in these
figures generate stable surfaces here; however, unstable branches in these figures can also
generate stable surfaces since, although a stationary solution in a given plane may not be
stable, it may be stable in a rotating configuration.
Figure 7,83 shows a 3-D plot of AnI and Bnl evolving with varying V, These results
are obtained by numerica1 integration, starting from one of the fixed points obtained at
V = 4.93 and then slowly reducing the flow velocity, All the seven generalized coordinates
are nonzero for a large range of flow velocities. It is interesting to observe that, for
V > 1.84, the solution belongs to an axisymmetric surface, rotating he1icoidally around
its axis; this axisymmetric surface is obtained by rotation of branch 3 in Figure 7.82
around the V axis. For 1.31 < V < 1.84 first-mode divergence is obtained, but now
without rotation, At V = 1.31, the system regains the stable undefonned configuration.
Figure 7.84 shows the shape of the shell in post-divergence configurations for the same
tThe meaning of 'unstable' should be nuanced a linie, ln fact, each point on branch 3 is stable on its own!
However, if V is incremented, tbe next stable point cannot be found on tbe same curve, since its azimutbal
location has changed: it lies elsewhere on a helicoidally developing surface, as shown in Figure 7.83. Hence,
in tbat specialized sense, tbe section oftbe suďace which is branch 3 in tbe plane (Figure 7.82) is unstable_

740
SLENDER STRUCTURES AND AXIAL FLOW
(a)
0.5
(b)
0,5
(c)
0,25
05
Figure 7.84 Shell shape for coupled-mode divergence at different flow velocities: (a) V = 2;
(b) V = 3; (c) V = 4 (Amabili et al. 1999a).
branch (coupled-mode divergence) at V = 2,3 and 4. One should note the twist of the
shell, the axisymmetric contraction, and the increasing post-divergence amplitudes with V.
Recently, experiments have been conducted with c1amped-damped elastomer shells
conveying air and aluminium shells conveying water (Karagiozis et al. 2003). The
dynamical behaviour, particularly the subcriticality of and hysteresis in the observed
divergence, are in excellent agreement with theoretical predictions in the foregoing.
For other results, the interested reader is referred to Amabili et al, (1999a); also, to
Amabili et al. (1999b, 2000a) and Pellicano et al. (2002) for an exploration of other
mode expansions at zero flow - in particular showing (i) the importance of inc1uding a
sufficiently large number ofaxisymmetric modes in the mode expansion, and (ii) the less
than absolutely essential nonlinear coupling of modes of different n (by considering modes
with 2n). It is of interest that, generally, more than two longitudinal modes are necessary
for predicting with accuracy the critical velocity for divergence. Calculations with as many
as six longitudinal modes, however, show that the results of Figures 7.81 and 7.82 remain
qualitatively unchanged.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
741
20
10

S O
O;
"<:
-10
-20
10
5

S O
,..,
"
"<:
-5
-10
O



-'-
-o.
... .-
----.---
, .--
'----------
2
3
V
4
5
6
Figure 7.85 Nonoscillatory dimensionless modal amplitudes An I / h and A n 2/ h versus the dimen-
sionless velocity, V, of an externaZ axial flow - cf. Figure 7,81(a) and (b); -, stable branches;
- - -, unstable branches (Amabili et aZ, 2001),
7.8.2(f) Externa! f/ow
The study of nonlinear dynamics of cylindrical shells in contact with internally flow-
ing fluid discussed in subsection (d) has been extended to the case of externaZ axial,
incompressible flow (Amabili et a[, 2001), using PaYdoussis & Denise's fluid-dynamic
model. As seen in Figure 7.85, the dynamical behaviour of this system is quite similar to
that obtained for internal flow, both qualitatively and quantitatively.
The case of nonlinear supersonic flutter has been reinvestigated (Amabili & Pellicano
2001) also, by means of a refined shell model.
7.8.3 Forced motions of fluid-conveying shells
The response of a shell conveying fluid and subjected to a harmonie excitation in the
spectral neighbourhood of one of its natural frequencies has been investigated by Amabili
et aZ, (2000b), and some interesting results have been obtained.
The mathematical model is the same as in Section 7.8.2, but here I i= o. Specifically,
1= ln cos(ne)sin(Jrx/L)cos(Wft),
- 2
ln = Ihwn1m1[2/(JrL»
(7.269)
are defined, ln being the modal amplitude of the excitation and j its dimensionless
counterpart; Wf is the forcing frequency, and W n 1 and ml are defined in Appendix 0.2.
The fluid-structure interaction model is that of PaYdoussis & Denise, as redefined in
Section 7.8.2.
Calculations have been conducted, both by the use of AUTO and by numerical integra-
tion, for the steel-water system described just before equations (7.268) in whieh Vand Q,

742
SLENDER STRUCTURES AND AXIAL FLOW
1.5
c
0.5
 O
"
"'<:
-0,5
-1
-1.5
0,15 0,20 0,25 0,30 0.35
(a) f
1.5
0.5
 O
"
"'<:
-0.5
-1
-1.5
0.1 0.15
(b)
2
1.5
1.0
0.5
 O P

"'<:
-0,5
-1
-1.5
-2
1
(c)
0,2 0,25 0.3 0,35 0.4
f
C
1.05
1.1
1.15
Q/Q51
Figure 7.86 (a) Bifurcation diagram without companion mode participation for Q f / Qs, 1
1,07446 and V = 2, the other parameters being defined in the text, showing the evolution of
An 1 / h with increasing j, where: P stands for simple periodic motion, M amplitude-modulated
(quasiperiodic) motion, and C chaotic motion, (b) The same with companion mode participation,
(c) Bifurcation diagram with companion mode participation for j = 0,3 and V = 2, showing the
evolution of An 1/ h with increasing Q f / Qs, 1 (Amabili et a/. 2000b).

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
743
2
 O 
 N

 
-1
100 200 300 -0,8 0 100 200 300
(a) fWnt (b) fWnl
2
0,8
0.4
 
S 
O S O
 
 N
' , -0,4
-1
-0,8
-2
(c) -2 -1 O 2 (d) -0.8 -0.6 -0.4 -0.2 O 0,2 0.4 0,6
An/ h A n2 /h
10° -0.6
10- / -0,8
 10- 2 
 -1
 !
3-
E-< 10- 3 .:; - 1.2


10- 4 -1.4
-1.6
O 2 3 4 S (f) -0,8 -0.4 O 0.4 0,8
(e) w/wnl An/ h
Figure 7.87 Response of the system without companion mode participation for 1 = 0.3,
rltl rl5, I = 1.07446 and V = 2: (a, b) time traces of Ant/h nd An2/h; (c, d) phase portraits
of AnI! h and A n 2/ h; (e) power spectrum for Anl/ h; (f) Poincaré map for {Anl/ h, Anl/(hw5,1);
(Amabili et a/. 2000b),
the dimensionless flow velocity and frequency, are defined; Q f is the dimensionless
forcing frequency. As always, it is understood that, for the shell frequencies, Q == Qnm (V).
The first set of results is the bifurcation diagram in Figure 7.86(a), for V = 2 and mode
(n, m) = (5,1), with QJlQS,1 = 1.07446 and nl = 0.01, without companion mode
participation. From Figure 7.81, it is clear that the system at V = 2 is subcritically stable.
Hence, given enough excitation, it may develop divergence; or, at somewhat smaller
amplitudes, its response would be affected by the unstable fixed points (saddle points)
shown as dashed line s in Figure 7.81. Jt is seen [Figure 7.86(a)] that for small enough
1, the response is periodic, whi1e for j > 0.1444, it becomes (i) amplitude modulated,

744 SLENDER STRUCTURES AND AXIAL FLOW
2 0,8
0.4
 O 
.. ..
 
-1
-0,6
-2
-1
(a) O 400 800 1200 (b) O 400 800 1200
tW n1 tWnl
2 0,8
0,4
,-., ,-.,
.. 
S S
 O ;? O
. ..
'
-1 -0.4
2 -0,8
-2 -1 O 2 -0,8 -0.4 O 0.4 0,8
(c) An/ h (d) Bn/ h
10° 10°
10- 1 10- 1
 10- 2 ,-.,

 10- 3 .. 10- 3
3- 
f-< f:;'
 
 10- 4 
10- 5 10- 5
10- 6
O 2 3 4 5 O 2 3 4 5
(e) WIW n l (f) wlw nl
-0.2 0.5
-0.4 0.4
-0,6 0.3
0,2
--:;:: -0,8 ,-.,
.. 0.1
S S
 -1.0 :::s O
. -1.2 .:{ -0.1
-0,2
-1.4 -0.3
-1.6 -0.4
-1.8 -0,5
-1.5 -1 -0.5 O 0.5 1.5 -1 -0,8 -0,6 -0.4-0.2 O 0,2 0.4 0,6 0.8 1.0
(g) Anl 1h (h) Bn/ h
Figure 7.88 Response of the system with companion mode participation for the same parameters
as Figure 7.87, showing selected time traces, phase portraits, power spectra and Poincaré maps
(Amabili et a/, 2000b),

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
745
(ii) regains periodicity, but with period 5T, T being the excitation period, (iii) chaotic,
(iv) periodic, with period mT, (v) chaotic again, (vi) 9T periodic and (vii) chaotic once
more, At j = 0.3359, the forcing amplitude is sufficient to cause a jump, so that the system
then oscillates periodically (with AnI! h :::: 12, not shown) around the bifurcated stable
fixed point - see Figure 7.81 - with period T, Figure 7 .86(b) shows that with companion
mode participation the periodic windows vanish. Interestingly, the companion mode is not
excited until j = 0.2057. The response for j > 0.15 approximately is predominantly
amplitude modulated, while for j > 0.27, it becomes chaotic. At j = 0.3364 the
system again jumps to a large-amplitude periodic orbit about the bifurcated statically
stable equilibrium point.
Similar bifurcation diagrams have been obtained for a fixed j = 0.3, while Q f / Q5,l
is varied (Amabi1i ct aZ. 2000b). A typical case is shown here in Figure 7.86(c) for
[2 f / [25,1 = (1, 1.15). Of special interest is the dynamics beyond the 9T orbit; this latter
has already been encountered in the previous results. This orbit suddenly loses stability at
[2 f / [25, 1 = 1.0643, giving rise to a chaotic region. This sudden change of orbit is often
called a blue sky catastrophe [see, e.g. Thompson & Stewart (1986)].
Figures 7,87 and 7.88 show typical time traces, phase-plane plots, PSDs and Poincaré
maps for the same parameters but, respectively, without and with companion mode par-
ticipation, such that the motion is amplitude modulated in the former and chaotic in the
latter. The PSD and especially the Poincaré map of Figure 7.87 make it absolutely clear
that the motion is indeed amplitude-modulated, that is, quasiperiodic. Here the interested
reader is referred to Chapter 5 and Appendix F for a detailed discussion of similar res-
ults for pipes conveying fluid, Similarly, the phase-plane plots and Poincaré maps of
Figure 7,88 show clearly that the motion is chaotic. In this regime the chaotic behaviour
is 'weak', in the sense that the characteristics of the quasiperiodic motion are still visible
in Figure 7.88: the peaks in the spectra are still clearly discernible; also, the fundamental
frequency is one-ninth of the excitation frequency in both cases.
Thus, for the first time, chaotic oscillation has been shown to be possible for the dynam-
ics of shells conveying fluid. This makes the likelihood high that similar behaviour is
possible in the case of free motions of cantilevered shells conveying fluid,
7.9 PLlABLE SHELLS
7.9.1 Introduction
Pliable shells are tubular membranes of relatively low rigidity, such that, in the absence
of internal pressurization or external supports, they may collapse or at least substantially
deform easily, for example, under the action of gravity if horizontal, or via small external
pressurization, Such shells are aptly also called collapsible tubes. t Virtually all fluid-
conveying conduits in mammals are af this type: arteries, veins, pulmonary and urinary
passages, which are not only very pliable but are capable of collapse.* However, there
t Tubes here should not be eonfused with the pipes eonsidered in Volume 1, whieh are strietly subjeet to n = 1
deformation. Collapsible tubes are generally noncircular pliable shells, the deformation of which manifests
itself basica11y inextensiona11y and involves substantia1 cross-sectional area changes.
*Collapse can occur not only as a result of extema11oads, but a1so by interna1 fluid-dynamiea1 action; see
Section 7.9.4.

746
SLENDER STRUCTURES AND AXIAL FLOW
are some engineering applications a1so, for example, the pliable containers used for the
transport of oil and fresh water by sea discussed in Chapter 8,
The physiological systems are especia11y interesting and have been and stilI are being
studied intensively, The basic characteristic that makes these and alI such pliable systems
unique is that the cross-sectional area of the shelI depends on the loca1 internal pressure.
Hence, intuitive understanding of flow characteristics based on elementary hydraulics, for
engineers at least, can genera11y no longer be relied upon. In this respect, the flow is more
akin to compressible flow, with the triad of {flow velocity, pressure, cross-sectional area}
replacing {flow velocity, pressure, density}, as discussed in Sections 7.9.2 and 7,9.4.
The literature on the statics and dynamics of pliable shelIs conveying fluid, directly or
indirectly related to physiological studies, is truly vast: one could readily come up with a
reference list of 100 or 200 important papers on the subject. Indeed, this topic could easi1y
be the subject of a separate book. t In this section, only some ofthe most important aspects
are discussed, without much detail, but hopefulIy conveying the savour of this fascinating
subject.
The study of pliable shelIs (colIapsible tubes) has developed separately from that
of shelIs as treated in the foregoing, and only recently has cross-linking become feas-
ible, albeit in a limited way, There are good reasons for this, For one thing, as already
mentioned, the shelI shape depends on the interna1 pressure and hence on the flow rate.
For another, the amp1itude of motions or deformations in physiologica1 systems can be
very large: of the order of the radius; yet, important nonlinearities occur in shelI dynamics,
even for amplitudes of the order of the shelI thickness (Section 7.8)!
In the sections that folIow, a nonphysiologica1 system is considered first, to introduce
the reader to variable-area wave propagation (Section 7.9.2), This is folIowed by a brief
1iterature review (Section 7.9.3), and a treatment of steady and unsteady flow in p1iable
shelIs mode1ling physiologica1 systems and the interesting phenomena related to such
systems.
7.9.2 Pressure waves in horizontal systems
By way of conveying the flavour of pliable systems, let us first consider a p1iable, col-
lapsible shelI, treated as an inextensible membrane, filIed with stagnant liquid and lying
on a horizontal surface.
The expressions for the stresses and the shape of the tube (shell) have been obtained by
Hawthorne (1961), The pressure heads at the top and the bottom of the tube are defined by
h* and H, respectively, as shown in Figure 7 .89( a), Then, considering the circumferentia1
tension in the wan, Ts, to be constant, one may express
Ts/ Rs = Pr],
(7.270)
and from geometrica1 considerations
dz dr]
ds = RsdO = - = -, (7.271)
cos O sin O
where the meaning of the symbols is obvious from the figure; P is the liquid density. After
some manipulation and integration between the top of the tube (O = o, r] = h) and the
tlndeed, such books are already available, for example, by McDonald (1974), Pedley (1980) and Nichols &
O'Rourke (1998),
*The pressure at the top is therefore ph; here h should not be confused with the shell thickness, which in
the section is denoted by h s '

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
747
h
Ts
H
TJ
y
Zl
Z
Ts
(a)
a = 0.88
(b)
Figure 7.89 (a) Definition diagram for a horizontal liquid-filled pliable tube. (b) Different
cross-sectional shapes of the inextensible tube for different fractional fillings ex, defined by
equation (7.273); (Patdoussis 1965a).
bottom (O = Jr, rJ = H), one obtains Ts/ P = !(H 2 - h 2 ) (Hawthome 1961; PaYdoussis
1965a), and then the coordinates of every point of the tube are related via
; = (1 _ k 2 sin2 rjJ) 1/2 ,
 = E(k, rjJ) - E - (1 - !k2) [F(k, rjJ) - K],
k 2 =1-(h/H)2, rjJ= !(Jr-O),
where K and E are the complete elliptic integrals ofthe first and second kind, and F(k, t/J)
and E(k, rjJ) are the corresponding incomplete e11iptic integrais - see, for example,
(7.272)

748
SLENDER STRUCTURES AND AXIAL FLOW
Hancock (1958), Dwight (1961). Typical shapes are shown in Figure 7 .89(b) for different
'fractional fillings', a, defined by
A (2 - k 2 )K - 2E
a= =Jr
(Jr /4)D2 (K - E)2
in which A is the cross-sectional area of the tube and D is its diameter when the cross-
section is circular. The same type of analysis has been done for floating tubes, for example,
fresh-water-fil1ed tubular containers floating in sea water (Mikolajczak 1957; Hawthorne
1961; Szyszkowski & Glockner 1987a,b). Similar pressure-area variations are found in
physiological systems, but neither the tubes in that case are necessarily horizontal, nor is
the pressure hydrostatic,
To obtain the velocity of1ong waves in the tube under consideration (Pa1doussis 1965a),
it is first noted that
(7.273)
p = Po + pg(H - y),
(7.274)
po being the atmospheric pressure, The continuity and momentum equations are
aA a(uA)
- + - = O,
at ax
au + u au + .!. ap = O,
at ax p ax
where u is the perturbation ve1ocity, p the fluid density, x the axial coordinate along
the length of the tube, and t the time. Then, considering waves of sma11 amp1itude,
equations (7,275) may be linearized and combined to give
a 2 u ( 1 ap ) a2u ( A d P ) a2u
at 2 = p alnA ax 2 = p dA ax 2 '
(7.275)
from which one readily obtains the ve10city of very 10ng waves to be
Co = ( : Y/2,
(7.276a)
or, using (7.273) and (7.274),
Co [ H / D ] 1/2
gD = 2+(  -) ,
in which the subscript O is for the long-wave limit. If one considers the area to be the
equivalent of density in compressible flow, one can appreciate the analogy to compressible
flow, since then co = (dp/dp)I/2, the velocity of sound waves.
ln the limit of very small values of the internal pressure (H / D < 0.4, corresponding
to a < 0.6),k 2 I,E:::::: l,a :::::: 2(H/D) - (4/Jr)(H/D)2, and equation (7,276b)
simplifies to
(7.276b)
:::::: {[ Jr - 2(H/D) ] H } 1/2;
gD Jr - 4(H/ D) D
(7,276c)

THIN SHELLS CONTAINING OR IMMERSED IN FLOW 749
1.6
1.4
1.2
 1.0
o
"
 0,8
'G
o
O)
;>-
" 0.6
;>-
$
0.4
0.2
O
O 0.1 0,2 0,3 0.4 0.5 06 0.7
(a) Interna! pressure, HID
18
17
16
15
14
13
 12
11
o
" 10
 9
'G
o 8
O)
;>-
" 7
;>-
$ 6
5
4
3
2
1
O
0,5 1.0 1.5 2,0 2.5 3,0 3.5 4,0
(b) Interna! pressure, Hl Do
Figure 7.90 Velocity of propagation of very long waves in the tube: (a) for small internal pressure
and inextensible tubes; (b) for larger values of pressure, showing the effect of extensibility of the
tube (Pai'doussis 1965a).
for even smal1er H / D, the right -hand side tends to y' H / D , and hence Co = y' g H , which
is recognized as the wave velocity of long gravity waves in shal10w channels with a free
surface (ti dal waves) - see, for example, Lighthi1l (1978).
The wave propagation velocity, as given by equation (7 .276b), is shown in Figure 7.90(a)
as a function of H / D. As expected, as the tension in the tube wal1 increases with increasing
H / D, the wave velocity increasingly deviates from that of free-surface waves.

750
SLENDER STRUCTURES AND AXIAL FLOW
The analysis may be extended to deal with extensible tubes, by taking into account
the circumferential strain es of the tube wall. The corresponding wave velocity, COe, is
found to be
C5e (1 + es)[d(HI Do)/d(k 2 )] - (Hl Do) [des Id(k 2 )]
-=a
gDo (1 + e s )[dald(k 2 )] + 2a[des!d(k 2 )]
where es = [gpD1;14E y h s ](HI DO)2k 2 , Ey being Young's modulus, hs the tube wall
thickness, and Do the unextended tube diameter. In the limit of a nearly circular tube,
k 2 ---+ O, a ---+ 1, E:::- JT and k 2 (HI D) ---+ 2, one obtains
(7.277a)
C5e _ Eyhs _ 
gDo - gpD5 2 Do
If Eyhs!gpDf; » HIDo, this expression reduces to COe = (EyhslpDo)1/2, which is
the wave velocity obtained long ago by Résal for long axisymmetric waves in thin elastic
circular tubes (Lamb 1898).t Typical values of coe l.JgDo for varying Eyhs!(gpD1;)
are shown in Figure 7 .90(b). It is seen that the wave velocity decreases with decreasing
tube stiffness; indeed, for small enough tube stiffness, the wave velocity curve reaches a
maximum with increasing Hl Do, and then goes down if Hl Do is increased further, in
accordance with (7 .277b).
The analysis may also be extended to approximately deal with waves of finite
wavelength, by taking into account shear deformation along the sides of the tube.
Experiments have been conducted as well, on the propagation of long and finite-
wavelength waves in horizontal water-filled polythene and latex tubes. Typical results
are shown in Figure 7.91, showing reasonably good agreement with theory.
Although there is some intrinsic interest in this analysis and experiments, the main
purpose here is (i) to introduce 'variable-area' systems and, by examining one aspect of
their dynamics, namely wave propagation, to illustrate the analogy to gas-dynamic and
free-surface hydrodynamic systems; and (ii) to present an analysis in which there exists
an analytical relationship between internal pressure and cross-sectional area, which is
generally not the case for physiological systems.
Finally, it should be noted that the extension of this analysis to cases of flowing fluid is
straightforward, but more complex than for non-pliable tubes, for in this case the viscous
pressure drop implies a continuous change in the cross-sectional area along the tube.
Before c10sing this section, it should be mentioned that a great deal of work has
been done on wave propagation in compliant and collapsible tubes; see, for instance,
Womersley (1955), Kamm & Shapiro (1979), Bonis & Ribreau (1981), Moodie et al.
(1982), Cowley (1983), Tait & Moodie (1984), Shimizu (1985), Ropchan & Swaters
(1993), Peake (1997), and the references cited therein.
(7.277b)
7.9.3 Physiological systems
Veins, arteries, pulmonary passages, the urethra and ureter are the premier examples of
physiological conduits which may be considered as pliable shells or collapsible tubes. An
tl n the physiologicalliterature this is known as the Moens-Korteweg equation; see McDona1d (1974).

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
751
8
6
0.4 0,8 1.2 1.6 2,0
Interna! pressure, H/D o
2.4
7
 5
\:)

c:; 4
o

 3

2
(a)
2,O
\:)

'C:;
o
 1.0
O,)
>

o
O
0.4
0,8 1.2 1.6
Wavenumber,2n:D o /A
2.0
(b)
Figure 7.91 Comparison of theory and experiment for the wave velocity. (a) For long waves:
-, inextensible tube theory; - - -, o, extensible tube theory and experiment for polythene
tubes; - - -, 6, extensible tube theory and experiment for latex tubes. (b) For waves of finite
wavelength, A, in polythene tubes (Pa'idoussis 1965a).
excellent introduction to such 'physiologic examples' is provided by Shapiro (1977a) in
one of two landmark papers of interest here. t
Veins are very pliable and hence exhibit large area changes in response to small changes
in transmural pressure - which is the difference between internal and external pressure,
p - Pe. Hence, if P - Pe < O, veins are prone to collapse (buckling), They display the
characteristic ofjlow limitation, discussed in Section 7,9.4, whereby the flow rate becomes
independent of the inlet-to-outlet internal pressure difference, depending instead only on
tThe other one is Shapiro (1977b), These two papers, Bertram (1999) rightly observes, 'have resisted all
attempts at what ISI would call obliteration by incorporation' - which is a handsome and deserved tribute to
their quality and pertinence,

752
SLENDER STRUCTURES AND AXIAL FLOW
the inlet transmural pressure. The same phenomenon also manifests itself in pulmonary
passages, for example, in the trachea, as expiratory flow limitation - a characteristic of
high exertion rates in healthy lungs and of emphysema in diseased ones. Arteries are less
interesting in this context, since P - Pe > O nonnally, and they remain inflated.
Nearly collapsed tubes are also subject to self-excited oscillations and flutter, with cor-
responding flow-rate oscillations and sometimes sound ernission, for example, wheezing
and borborygmi. One may find a wealth of fascinating infonnation on the behaviour of
collapsible physiological systems,t as well as in associated diagnostic devices (e.g. the
Korotkoff sounds) and therapeutic devices in Shapiro (1977a).
The veins, arteries, urethra and trachea are surrounded by or attached to large masses of
tissue; they are not necessarily straight or unifonn along their length and, especially in air-
flow passages, they are not even axisymmetric, Hence, realistic analysis of their steady and
unsteady behaviour can 'become forbiddingly complex' (Shapiro 1977a). Consequently,
idealized models have been used to study the essentials of their behaviour, the princi pal
one among them being the 'Starling resistor' (Section 7.9.4).
The literature on this topic goes back to the early 1890s and even earlier. * U seful reviews
are provided by Kamm (1987), Skalak: et al. (1989), Kamm & Pedley (1989b) and Grotberg
(1994) for instance, and useful collections of papers may be found in special editions, for
example, by Kamm & Pedley (1989a), Pedley et al. (1992), Friedman (1992), Ellington &
Pedley (1995). An especially clear brief summary of the dynamics of collapsible tubes, to
which this section owes a great deal (in some aspects following it very closely), is provided
by Bertram (1995a). Perhaps, in these days of computerized literature retrieval, it is as
useful to give authors' names as to cite references, The main contributors are considered
to be: W,A. Conrad, A,H, Shapiro, T.J. Pedley, R.D, Kamm, C.D. Bertram, J,B. Grotberg
and M. Heil; other important contributors are Y. Matsuzak:i, C. Ribreau, M. Bonis, O.E.
Jensen, M.E. McClurken, G.c. Oates and D.J. Griffiths.
7.9.4 Steady flow
Selectively, two inter-related aspects of steady flow through collapsible tubes are discussed
here, namely flow limitation and elastic jumps. To do this effectively, it is best to first
describe the Starling resistor, via which these phenomena are easily understood.
7.9.4(a) The Starling resistor
The Starling resistor (Knowlton & Starling 1912) has become a classical tool for the study
of collapsible tube behaviour, both theoretical and experimental (Kamm 1987). A general
schematic is shown in Figure 7.92, Basica11y, it consists of a segment of collapsible
tubing, attached at either end to a rigid pipe of circular cross-section and enclosed in a
large chamber of constant but adjustable pressure, Pe. The upstream and downstream ends
of the system are connected to reservoirs of adjustable pressure, Pu and Pd, respectively,
through val ves of hydraulic resistance ku and kd.
t An example is that flow speeds in the trachea during coughing, for the purpose of dislodging phlegm and
other objects, can approach the speed of sound!
*For instance, Young (1808) studied bl00d pressure pulses at the beginning of the 19th century; also, perhaps
the first derivation of equation (7.276a) was made by Weber (1866), in conjunction with work motivated by
the problem of the arterial pulse wave.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
753
The three pressures of interest are the in1et and out1et pressures, Pl and pz, and the
external pressure Pe, which define the corresponding two transmural pressures: Pl-e =
Pl - pe and PZ-e = PZ - Pe. The transmural pressure elsewhere is Px-e = Px - Pe,
and is often simply denoted as Ptm. The other quantity of interest is the volumetric ftow
rate Q.
For simplicity, consider first the situation when Q = O and hence Pl = pz, whi1e Pe is
varied; in this case, the same value of Ptm applies for alllocations x. Figure 7.93 shows
the shapes of the central segment of the collapsib1e tube (near its end, the shape evolves to
match that of the rigid pipes). For Ptm > O, the tube is fully inftated and its cross-section
is circular as in (a). At slight1y negative Ptm, the tube can either remain circular or begin to
oval as in (b), depending on its stiffness. However, as Ptm becomes sufficiently negative,
the tube collapses (buckles) progressively into shapes (b)-(e) if the tube is sufficient1y
thin-walled, or as in (f) if the tube is small and thick-walled (Bertram 1995a).
Experimental pressure/ftow-area results for different relatively thick (nominally
12.7mm inside diameter and 2.1 or 3.2mm wall thickness) silicone rubber tubes
(Bertram 1987, 1995a) are shown in Figure 7.94(a). In segment AB of the curves, the
tubes are fully inftated and a == AI Ao > 1, Ao being the ftow area corresponding to the
unstretched circular tube - cf. equation (7.273). The tube remains inftated in segment BC,
but its cross-section begins to look 1ike Figure 7 .93(b). Thereafter, for segment CD, very
small increases in Pe (decreases in Ptm) result in precipitous reductions in area, from shape
(b) to (d) of Figure 7.93. After the tube has collapsed, the resistance of the uncollapsed
end-lobes to further ftattening is high, and relatively large changes in Ptm are necessary to
Pe
Pu
Pd
Pl\
jP2
--
--
--
--
ku
kd
Figure 7.92 Diagrammatic view of the Starling Resistor. The tube outside the chamber pressurized
to Pe is rigid, but the tube within is very pliable.
008ZZ0
(a)
(b)
(c)
(d)
(e)
(f)
Figure 7.93 (a)-(e) Generic shapes associated with the collapse of a thin pliable tube, as the
transmural pressure Plm becomes increasingly negative; (f) completely dosed tube, achievable
only by small, thick-walled biological vessels (Bertram 1995a).

754
o
'" D
E
s:S:;
II

-5
-10 E
0.0
(a)
SLENDER STRUCTURES AND AXIAL FLOW
st
5
a
o
O
-5
-10
-15
-20
-25
- Experimental
_,_,_ _[jJ=a- 3 / 2
_ _ _ _[jJ= a- 3 / 2 _1
-30
-00
0,5
a
1.0
(b)
Figure 7.94 (a) The relationship between P and ex for relatively thick elastomer tubes of two
different thicknesses: -, thinner; - - -, thicker (Bertram 1995a); (b) the same for much
thinner tubes, also showing typical shapes and analytical relationships (Shapiro 1977a,b),
produce further area reductions (segment DE of the curves). Another experimental curve,
this one from Shapiro (1977a,b) for very thin tubes (probably colostomy or Penrose tubing
- see Section 7.2.4), is shown in Figure 7 ,94(b), together with the following two analytical
approximations for the collapsingfcollapsed area-pressure relationships:
-p = -Ptm/Kp = a- n ,
-p = a- n - 1,
(7.278a)
(7 .278b)
in which n = 3/2 typical1y; Kp is a parameter proportional to the bending stiffness of the
tube, t such that P is dimensionless. The latter form, equation (7 .278b), is preferred since
it gives the correct value a = 1 as P --+ O, whi1e being reasonably c10se to (7.278a) for
a --+ O. (For the tubes of Figure 7,94(a) such analytical expressions are more difficult,
because of the sharp 'comers'.) Analogous pressure-area relations have been discussed
in Section 7.9,2, but in that case analytica1 description is feasible.
The experimental curves in Figure 7.94(a) and (b) and/or analytical expressions related
thereto define the so-called tube law, which supplants all of the structura1 side of the
problem, that is, what was hitherto described by means of shell equations. Clearly, the
tForexample, Cancelli & Pedley (1985) give Kp = E(2hf Do)3 /[12(1- v 2 )]. where E is Young's modulus,
h is the waJl thickness. Do the diameter corresponding to Ao, and v the Poisson ratio.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
755
N
I

P e -2
Pe
(a)
Q
(b)
Q
(c)
Q
Figure 7.95 The relationship between PI-2 and Q. In (a) each curve is for a constant Pe, in (b)
for a constant Pe-2, and in (c) for a constant PI-e(= -Pe-I), with arrows indicating increasing
values of the parameter involved (Bertram 1995a),
tube law is highly nonlinear, and it is due to this nonlinearity that most of the interesting
phenomena in collapsible tube dynamics arise,
With flow, the local internal pressure varies, and hence the shape of the tube varies
with x. As succinctly put by Bertram (1995a), 'The small pressure differences associated
with the fluid flow within the tube suffice to alter the cross-sectional shape considerably.
The shape of the conduit and the nature of the flow thus acquire an unusually high degree
of coupling; viscous and inertial pressure losses act in concert to cause a uniform tube
conveying a flow to become progressively more collapsed towards the downstream enď.
7,9.4(b) Flow limitation and elastic jumps
As stated at the beginning of this section, the fact that the area varies with transmural
pressure complicates things; for example, the flow-rate does not only depend on Pl-2, but
also on Pe, which can affect the flow area. Possible relationships between Pl-2, Pl-e, P2-e
and Q, shown in Figure 7.95, have been constructed and discussed by Bertram (1995a)
for different experimental 'protocols', t
ln Figure 7.95(a), for each constant Pe, Pl-2 is allowed to find its own level whi1e the
flow rate Q is varied (by varying Pu and/or Pd). At the origin of the graph, the tube is
fully collapsed and there is no flow. As Q is increased, Pl-2 increases sharply; the tube
'resistance' Pl-2/ Q is high. At the peak of each ofthe three curves, Pl-e becomes positive,
and thereafter the tube becomes inflated with increasing Q, starting from the upstream
end; the resistance falls, and the incremental resistance dpl-2/dQ is now negative, This
continues until the tube is fully inflated, at the right-end foot of each of the curves; the
resistance at that point reaches a minimum, whi1e dpl-2/dQ becomes positive again.
ln Figure 7.95(b), Pe-2 is kept fixed and positive for each curve (Le. P2-e < O); thus,
to change Q, Pe needs to be adjusted so that P2-e takes on the desired value. Again,
initially the tube is fully collapsed, and Pl-2 rises sharply with increasing Q. However,
in this protocol, as the tube begins to become less collapsed, a plateau (or near-plateau)
is reached which corresponds to pressure-drop limitation: Pl-2 depends only (or mainly)
on Pe-2; with increasing Pl the flow rate increases, but the pressure drop Pl-2 remains
sensibly constant. This is very interesting of course since it is impossible for rigid tubes. t
t An early, seminal study on this subject is Brower & Noordergraaf's (1973).
*Though interesting to fluid mechanicians, this is not so fascinating to biomechanicians, not nearly as much
as flow limitation, which has physi010gical repercussions (Bertram 1995a),

756
SLENDER STRUCTURES AND AXIAL FLOW
The curves in Figure 7.95(c) correspond to fixed values of PI-e. Thus, Pu is varied
while Pe is adjusted to achieve a constant Pe-I. It is seen that, for each value of Pe-I,
there is a segment of the curve where Q remains almost constant, despite large increases
in pressure drop, PI-2! This is the .flow-rate limitation phenomenon, ln fact, the flow
rate decreases somewhat with increasing PI-2; this, for example, corresponds to the so-
called 'negative effort dependency' in maximal forced expiration (Grotberg 1994), ln an
appealing analogy, flow limitation is likened to flow over a waterfall, where the flow rate
is independent of the level of the downstream reservoir.
Two mechanisms may be proposed for flow limitation. The first is the wave-speed
mechanism (Shapiro 1977b; Wilson et a/. 1986).
Consider flow with a local stagnation pressure P == Px, so that
I U 2
P = P + 'IP ,
(7.279)
where, strictly, also P and U should have subscripts x. The volumetric flow may therefore
be written as
Q = AU = A [(P - p) r/ 2 = A {[(P - Pe) _ (p _ pe)]} 1/2
Furthennore, in this context, the wave velocity given by (7.276a) may be written as
(7.280)
c 2 = [ A d(p - pe) ] 1/2
P dA
Suppose now that P - Pe has a particular local value that we may regard as fixed. Hence,
as P - Pe is varied, U is also varied via equation (7.280), and so is A via the tube law,
equations (7.278a,b). Next, examination of equation (7.280) shows that, as (p - Pe) --+
(P - Pe), U = O and hence Q = O; it is also noted that, as -(p - Pe) --+ 00, then
A --+ O. This suggests the existence of a maximum Q, Qmax. To detennine this, we take
the derivative of Q with respect to P - Pe, noting that A is a function of P - Pe also, After
simp1ification by means of (7.281), the result is
dQ A ( u2 ) A 2
d(p - Pe) = pU 1 - c2 = pU (1- S ).
(7.281)
(7.282)
This indicates that Qmax arises at the location (value of area or transmural pressure) for
which U = c. By analogy to gas dynamics, ujc is equivalent to the Mach number, M;
or to the Froude number, F, in hydrau1ics. Shapiro calls u j c the 'speed index', S. t If we
denote by Co the wave velocity in a circular tube,:j: we can set u = c in (7,280) and, using
(7 ,278b), obtain
Qmax = Ac = ťi(2-n)/2 = (1 _ p)(n-2)/(2n). (7.283)
Aoco Aoco
Thus, the maximum flow rate depends solely upon the local transmural pressure. In prac-
tice, this means that the downstream pressure where P is a minimum yields the maximum
flow rate, irrespective of what the upstream pressure is.
tperhaps the time is now ripe - in fact, it has been so for a long time - to call S the Shapiro number,
*Note that the subscript O here has a different meaning from that in Section 7,9.2,

-
-1 - -
---
-2

 -3
'"
'"
£
-4
(a)
THIN SHELLS CONTAINING OR IMMERSED IN FLOW
757
o
o
. --.......... -- ----.... ---.... ..-"
I
"
..
"
"
"
I'
,.
.'
I'
.'
.','
ff, I
__-;.-;.-;.-;.-;.ooo:;:»:
' "
--
0,8
0,6
- -....
,,'/
'/,'
,

",
'"
.ii:
0.4
0,2
-5
-6
O 0.5
Distance along the tube, g = x/L
O
O 0.5
Distance a10ng tbe tube,  = x/L
(b)
Figure 7.96 (a) The bold line shows Px-e = Px - Pe in a tube with minimum undisturbed area
at  = 0,5, conveying inviscid fluid at the wave-speed 1imited Q. The flow becomes supercritical
at  = 0.5, the choke point, and retums to subcritica1ity at  = 0.875 in this case, where the energy
loss at the elastic jump is such that it matches the given value P2-e. Dashed lines correspond to
subcritical flows, (b) The corresponding dimensionless area, Ci as a function of  (Wilson et a/.
1986; Bertram 1995a),
Figure 7.96 shows various situations for the transmural pressure Px-e and area ratio
along the tube for different upstream ftow conditions, following Wilson et a1. (1986) and
Bertram (1995a). The bold lines correspond to critical conditions (S = I) being reached at
 = x / L = 0.5, where it has been presumed that a minimum-area condition pre-existed.
Beyond  = 0.5, the ftow is supercritical [Qmax/ A() > e()], returning to subcriticality
via an elastie jump, similar in nature to a hydraulic jump or a gas-dynamic shock. The
analogy to a hydraulic jump is closest when viewed as a jump from a minimum area to a
larger area as in Figure 7.96(b).
We next consider the viseous-fiow limitation meehanism. For laminar viscous flow, we
can write
d ( 1 2 ) U
dx p+ '2 PU = -af.l. A '
(7,284)
in which it is once more recalled that p == Px - Pe, and p, U and A are all functions of
x; f.l. is the dynamic viscosity, while a is a numerical constant [see, e.g. Massey (1979)].
Noting that Q = U A is constant, this equation may be integrated to give
dp ( I _ U 2 ) = -aJlR;
dx e 2 A 2

758
SLENDER STRUCTURES AND AXIAL FLOW
this shows that, if S = u / c < 1, nonnal subcritical behaviour pertains, and d P / dx < O; if
S > 1, however, this indicates anomalous conditions (dp/dx > O) since the wave-speed
ftow-1imitation condition has been surpassed.
Now, for simp1icity, the inertial tenns in (7.284) are suppressed (Poiseui11e ftow t ),
yie1ding
dp Q
dx = -a/L A2 ;
integrating from fixed conditions at x = O (pl and AI constant) to variab1e (p2, A2) at
x = L, we obtain
1 l PI
Q=- A 2 dp,
a/LL P2
written so that the integra1 be positive. This integra1 may be eva1uated by using the tube
1aw - specifically, equation (7 .278b) - yie1ding
(7.285)
Q = const [ (1 _ pd n - 2 )/n _ (1 _ P2)(n-2)/n ] ,
a/LL
where it is recalled that P = p / Kp. For n = 3/2, this gives
3A6K P [ 1 1 ]
Q = a/LL (1 - Pl)I/3 - (1 - P2)1/3 .
lt is seen that as P2 is diminished, Q reaches a constant value; here it is presumed that
Pl is at 1east slightly negative - see Figure 7.94. lndeed, taking P2 --+ -00 gives the
maximum ftow rate
3A6K P [ 1 ]
Qmax = a/LL (1 - PI>I/3 .
(7,286)
Thus, ftow limitation can also arise by pure1y viscous considerations,
As computed by Bertram and shown in Figure 7.97, whatever the downstream trans-
mural pressure (at  = 1), the collapsed tube (hence for Px - Pe < O) deve10ps the same
upstream transmural pressure Pl - Pe and the same Qmax, corresponding to that given by
equation (7.286). The on1y difference is that the collapsed tube 1ength increases as Pe-2
decreases; before collapse, al1 the curves are essentially coincident.:j:
As Wi1son et a1. have pointed out, the two mechanisms are in competition, in a sense;
which of the two actual1y causes ftow limitation depends on system parameters. For 1arger
tubes, the wave-speed limit is reached first and hence is the control1ing mechanism, as
shown in Figure 7.98(a); for smal1er tubes, it is seen in Figure 7 .98(b) that the viscous limit
is reached first (lower curve) ar concurrently with the wave-speed limit (upper curve).
This is a great1y abbreviated, one-dimensional treatment of this system, The interested
reader must refer to the reference s cited here, and others cited therein, for fuller and
tIt is of interest that J,-L.M, Poiseuille (1799-1869) was both physician and physicist and his work on
laminar viscous flows was inspired by his interest in haemodynamic flows,
tS ee also Bertram (2000a) for an appealing, very simple, purely resistive model, showing flow limitation
and affording other insights.

10
O
<:i;
I -10


'"
'"
 -20
""
1
 -30
-40
-50
O 0,5
Distance along the tube, 
Figure 7.97 The variation of Px - Pe with  for various va1ues of p2-e == P2 - Pe for laminar
viscous flow. With increasing  the collapsed section of the tube tapers down, and the same values
of Q and pl-e resultJor all values oj P2-e, indieating flow limitation by the viscous mechanism
(Bertram 1995a).
Q(l/s)
.i?_i!___.___.__ _ _ ____ __1___ _. __.___. ____
Pl =2 ,
0.3 I
.
I
.
0,2 /
Viscous limit
...-.:.----------.-------- --;--
Pl - 1 0.1 .
Wave-speed limit /"/
-"
-"
-4
(a)
Wave.speed -..:
limit I
.
/
/
.
/'
,"
-4
(b)
-2
o
P (cm H 2 0)
2
4
.I
Q (mVs) Viscous limit
------------------------
Pl =2
-2
4
Viscous limit
Pll
o
P (cm H 2 0)
2
Figure 7.98 Relations between wave-speed and viscous flow-limiting mechanisms, in geometrie-
ally similar tubes with the same specific compliance (i.e. the same tube-Iaw curve applies), showing
system behaviour for different va1ues of Pf. (a) For the larger tube (Amax = 0.2 cm 2 ) flow lim-
itation occurs via the wave-speed mechanism, the viscous-flow limits for both PI = I and 2 cm
H20 being much higher. (b) For the smaller tube (A max = 0.002 cm 2 ) the flow is limited via the
viscous mechanism for Pl = I cm H20 and concurrently by viscous and wave-speed limitation
mechanisms for Pl = 2 cm H20; (Wilson et al. 1986).

760
SLENDER STRUCTURES AND AXIAL FLOW
200
Pe-I (kPa): 8 30
«> 31
O 32
L,. 33
... 34
035
O 36
040
lil 50
150
';'

N
I

li 100


"
'"
'"
'"
...
p..
50
13
o
o
100
200 300
Flow-rate, Q (ml/s)
400
500
Figure 7.99 Flow limitation in a relatively thick-walled collapsible tube conveying water, The
results for nine different constant Pe-l values show the pressure drop versus the flow rate, All points
were approached by increasing Pe, until the required value of Pe-I was reached. The symbols are
annotated with the corresponding value of Pu. Five cases clearly show the transition to tube collapse
and flow limitation via in increase in Pu of only 1 kPa - see slanted lines (Bertram & Castles 1999),
more sophisticated accounts and analyses. Examples of this literature are: Webster et al.
(1985), Cancelli & Pedley (1985), Lowe & Pedley (1995), Luo & Pedley (1995), Heil
& Pedley (1995, 1996), Heil (1996), Alstr0m et al. (1999), Bertram & Castles (1999),
Bertram (2000a) and Bertram & Elliou (2003). In some of this work, the analysis is
extended to 2-D and in some, notab1y Heil & Pedley (1995, 1996) and Heil (1996),
shell theory and viscous flow theory are used together for the fust time in much more
sophisticated studies of this topic.
Figure 7.99 shows experimental results demonstrating flow limitation, from a very nice
study by Bertram & Castles (1999). In each experiment, similarly to Figure 7, 95( c), Pe-l
was kept constant, while Pu was varied. It is seen that a change of Pu of only 1 kPa, from
the near-horizontal cluster of data for the yet uncollapsed tube, is sufficient to precipitate
tube collapse, flow limitation and negative effort dependence, in the nearly verticallines
of data points, ln each case, Pl-2 varies widely, while Q remains sensibly unchanged.
7.9.5 Flow-induced oscillation
7.9.5(a) The physícaJ characterístícs of the osciJJatíon
In the previous section, 'steady flow' implied that, other than wave propagation, the system
could be studied mainly by statics, in the sen se that, for given parameter values, a steady

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
761
operating point has been presumed to exist. Here, ftow-induced oscil1ations of collapsible
tubes are discussed, which generally entrain ftow oscil1ations as well, and hence unsteady
jlow is generally involved.
An enormous amount of work has been done on this topic, both theoretical and
experimenta1, aimed at understanding the mechanism of ftow-induced oscil1ations in
physiological systems and in models of such systems (e.g. related to Starling-resistor-type
experiments). Appendix P contains a long but nonexhaustive list of important contribu-
tions on this topic over the past 30 years - an effort which is continuing, as it cannot
be said that the problem is fully resolved yet. The motivation had initially been wholly
practical and clinical (to understand and predict the mechanism for physiological oscil-
lations) but, with the discovery of a wealth of interesting dynamical behaviour, a healthy
curiosity-driven element has been injected in the quest over the years.
From the very beginning, it was realized that ftow-induced oscil1ation is related to
collapse of the tube; indeed, the oscil1ation is strongest at the 'throať, that is, at the
minimum tube area, clo se to the downstream end in a Starling-resistor-type experiment. It
has since become clear, however, that collapse is a necessary but not sufficient condition
for the onset of oscil1ation.
Initially, it was thought that osci11ation is only possible for parameters corresponding to
the negative-slope segments of Figure 7.95(a) (Conrad 1969). However, oscil1ations also
arise in the regions ofFigure 7.95(b) and (c) associated with pressure-drop and ftow-rate
limitation, observed as soon as the relevant experiments (exploring the pertinent parts of
the parameter space) were performed, by Brower (1970) and Bonis (1979), t respectively. *
Recent experimental and numerical work [e.g. Ohba et al. (1989), Yamane & Orita (1992),
Matsuzaki & Seike (1996)] poses new challenges to Conraďs belief that oscil1ations can
only occur on the negatively sloping parts of the curves in Figure 7 .95( a). It now appears
certain that oscil1ations can also occur in the pressure-drop and ftow-limitation regions of
Figure 7.95(b) and (c), as discussed by Bertram (2000a), wherein other work supporting
this statement may be found.
The Reynolds number range for the onset of oscil1ations quoted in Bertram (1995a) is
300-5000, thus straddling the laminar-turbulent divide. However, no upper limit is known
or postulated; the range covered in Bertram's experiments extends to Re = 50,000 -
pro bab ly the highest ever documented (Bertram 2000b).
Typical results are shown in Figures 7.100-7.102 from the work of Bertram et al. (1990,
1991), obtained in a Starling-resistor-type apparatus using water-ftow in relatively thick-
walled silicone rubber tubes (E = 3.8 MPa, hla = 0.30).9 These results help to define
these ftow-induced oscil1ations and some of their salient features.
ln Figure 7.100, it is seen that oscil1ations occur over most of the 'pressure-drop-limiteď
parts of the curves for Pe-2 = 100, 80, 60 and 40 kPa; the frequency is marked close to
each data point. Also shown on the same figure are the dynamic orbits of two oscil1atory
operating points (star and circled black dot), for (i) a highly asymmetric low-frequency
tThere is a wealth of important and useful results in Bonis' (1979) thesis, which unfortunately remain
unpublished, A version of some of the results of interest here may be found in Bonis & Ribreau (1978),
+In fact, the large-amplitude periodic oscillations to be described in what follows [see, e,g, Figure 7 ,102(c)-
(t), (i), (j)] almost never materialize if the flow-limitation protocol is followed in reaching collapse, but rather
noise-like fluctuations [e,g, Figure 7,102(a), (b) and (h)] (Bertram & Castles 1999),
šSimilar curves for thinner tubes and steady jlow generated by Bonis & Ribreau (1978) have been found to
be in good agreement with Jensen & Pedley's (1989) theoretical predictions.

120


I
I
I 100


Q)

'"
Q)
....
i:>..
80
200
180
,1Pe2 R RZ R
100()Oe
80 [J o .
60&6.6.
40  'V T
20  <> .
! '
" ' !
I  II
  H
   ..
',1ft
I I' II
: I :
I 1'1
IIJllt
 I  I "
   "
\)-
 I II
\ I III
I III
1,11
I I II,
   } I
,  II
I I I
,"
,"
,"
,"
,,,
,"
,,,
,"
,"
,"
,"
,"
,"
,"
,"
,"
,"
,"
,"
,"
,"
,"
,II
,"
,"
,"
,"
,"
,"
,"
,"
,"
,II
."
,"
,,,
,,,
,"
. :::
11.7:::
,"
,"
,"
,,,
,,,
,"
,"
,"
,,,
,"
,"
,,,
,,,
,"
,"
,"
."
,,,
'"
,"
,"
,"
,"
,"
.
"
'.3,11::
, l:
..' ..
\{
400 
500
160
'"'"
,',' ..
, I \ \ 
I I \ \I
I I  ,\
I' \ 1\
 I \ ,
" .
,,
:: '''', I \
, .
tU 
u \
  I \
  \
III \
III \
, "
.
. .
. "
, "
:\
,',
,',
,',
,',
,',
,',
,',
,',
,',
"
"
"
22.3/irreg. '
nf
140
60
40
20
o
O
100 200 300
Mean flow-rate (mlIs)
Figure 7.100 Equilibrium curves for a collapsible tube with L / D = 17.4 for five different settings
ofthe mean Pe-2, Pe-2 = Pe - P2, and variable kd: half-open symbols indicate high kd, open ones
a mediumkd (e.g., o, <», and a filled symbol (e.g.,.) a low kdo ForPe_2 = 20 the tube was circular,
and no oscillations were found; for higher Pe-2, however, oscillations at the frequencies shown (in
Hz) were widely present. For the operating points associated with the high-frequency oscillation at
68 Hz (Pe-2 = 100 kPa) and the low-frequency (LV) oscillation at 4.1 Hz (Pe-2 = 40 kPa) the
dynamic orbits of the osci1lation are also shown, drawn, respectively, as full and dashed 1ines [from
figures 2 and 3 of Bertram et a/. (1990)].

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
763
'LU-type' oscillation at 4,1 Hz, with a large positive Pl-2 and a much smaller negative
component, and (ii) a high-frequency, more symmetric oscillation at 68 Hz. In both cases,
the amplitudes of oscillation are quite enormous, both in terms of pressure and of fiow-rate
(and also in terms of flow-area), It is clear, therefore, that these are not perturbation-like
oscillations, but large-amplitude, strongly nonlinear ones.
Forsteadyfiow, ithad been foundthat Pl-2, Pe-2 and Q are a sufficient setofparameters
'to telescope all possible operating point curves onto a common surface in the three-
dimensional variable space' (Bertram et aJ. 1991); see, for example, Figure 7,95. For
unsteady behaviour, however, other parameters, for example, the downstream resistance
kd and the tube length L, become equally important. Hence, a more meaningful and
practical way ofpresenting the results is via 'control-space diagram s , ,as in Figure 7,101
for example, In these diagrams, each for a different L / D and kd combination, the domains
of different types of oscillation are demarcated: L, I or H for low-, intermediate-, or
high-frequency oscillation - the first being further subdivided into LU, involving a brief
tube collapse within the period, and LD, involving collapse lasting about half the period,
or just over. Further, MC denotes multiple-collapse-type osci11ations, and nf noise-like
osci11ations; UN denotes unattainable region s t of the parameter space (at least in the
protocol used to reach an operating point in the region), from which the system diverges
towards other points; c and o denote stable, nonoscillatory operating points, with the tube
either collapsed (c) or not (o, for open).
The waveforms, associated with these various types of oscillation, are illustrated in
Figure 7,102 in terms of P2(t); similar waveforms are available for Pl (t), Q(t) and A(t).
It is clear that in most cases the oscillations are highly nonlinear, sometimes of intriguing
waveform, and in a few cases probably chaotic. No wonder then that so much effort
has been devoted to the understanding of their inception and of the transition from one
type to another. Similar results though less extensive, have been obtained, for example, by
Griffiths (1977), Bonis & Ribreau (1978), Ohba et aJ. (1984, 1985), some with thin-walled,
Penrose-type tubing - see Appendix P.
An important characteristic of these oscillations is that they can be boro as fully fiedged
large-amplitude ones, as shown for example in Figure 7.103(a), rather than growing and
dying away gradually as a parameter is varied. They also disappear equally precipitously.
Figure 7,103(b), on the other hand, is an example of an oscillation growing gradually as
a parameter is varied. Figure 7.I03(c) gives a possible notional topology for the initiation
and cessation of the oscillations (Bertram I 995a,b ). Thus, Figure 7.1 03( a) corresponds
to the sudden onset of oscillation, shown diagrammatically in the middle of (c) as Pe is
increased (the downward arrow), while 7,103(b) corresponds to the gradual onset shown
on the right of (c) as Pe is diminished. * Thus, Figure 7 .103( c) is a generalization of
Figure 2.ll(d), showing a subcritical Hopfbifurcation, an unstable 'inner' attractor and a
stable 'outer' one. In this case, however, the stable equilibrium is not constant: it changes
as the control parameter (Pe) is varied, evolving on a smooth S-curve which may be
thought of as a fiow-distorted tube law of the type shown in Figure 7.94, Starting with
a low Pe and increasing it causes the area A to decrease, until the fold, at which point
there is no longer a stable nonoscillatory operating point, so that the system jumps to
tThe nomenc1ature is influenced by the experimental system and hence the researchers involved, Thus, 'unat-
tainable' in Bertram's experiments probably corresponds to 'aperiodic instability' in Bonis' (Bertram 2000b).
*However, Figure 7.103(c) does not deal with the transition from LU to H, i.e. from osci1lations represented
at the left-hand (sudden-onset) end of the osci1latory path (...) to those at the right-hand (gradual-onset) end,

764
SLENDER STRUCTURES AND AXIAL FLOW
200
(a)
(b)
o

'"

 100 .
I

"'-
200
(d)
(e)


 100
I

"'-
.. . 14 . . .
c nf
.... .. .. .... ..
LU .
. ....-: :MC: J / UN
:::::í-=t=: -t ..
.. .. o ..
. . c . l . nf . . H
.... .. :__.t_:
.: LD UN
:-==-=;==:-..-..--+_..
.. "LV" O" .. .. ..
(c)
.
.
j .
.. .. nf..
c ......
._*=i: -: _. UN.
o
200
100
(f)
c :
-*-. UN
*-=.:::::::--+-..-..
o 100 200 O 100 200 O 100 200
200
(g) (h) (i)

 j'
N 100 . .
"'-
I C . nf

"'- .-.
.-* UN
..-*=*::::::..-..-..-..
O 100 200 O 100 200
Pu (kPa) Pu (kPa)
Figure 7.101 Control-space diagrams: (a-c) LI D = 4 and, respectively, low, medium and high
kd; (d-f) LID = 8,2 and, left to right, low -+ high kd; (g-i) LID = 17.4, same variation of
kd' The crosses indicate the main operating points investigated. The dynamical behaviour of the
system is identified as: c, steady flow; nf, noise-like fluctuation; L (LU or LD), low-frequency
oscillation; I, H: intermediate and high-frequency osci11ation; MC, multiple collapse oscillation;
UN, unattainable region; o, open tube (Bertram et al. 1991),
the only stable solution branch available, which represents an oscillatory state (shown as
filled circles), just as in a subcritical Hopfbifurcation. If Pe is then reduced, the oscillation
follows the filled-circ1e path and eventually ceases suddenly, the systemjumping up to the
stable nonoscillatory upper branch with considerable hysteresis, via what is often called
a 'blue-sky catastrophe' (Thompson & Stewart 1986) - cf. Sec ti on 7.8.
Another important characteristic of these osci11ations is that they are not necessarily
of the standing wave type. The low-frequency (L) oscillations in long collapsible tubes,
for instance, originally and very aptly termed 'milking oscillations', involve the periodic

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
765
:: . (a)
o 0.4 0.8
80
: (g)
O 0.4 0,8 1.2
(b)  :: I (h)
 50 O 0.4 0,8
(c) f O  (i)
'" O ] 2 3
£]0: 1 (j)
:  )
O 04 0,8
Time (s)

 -50 ] 2
S
]OO 
 00
- ]00 0 ] 2 3
7: nrmmlflfIDl (e)
O ] 2 3
_]OO  I (f)
]000 ] 2 3
Time (s)
Figure 7.102 Various waveforms from the oscillations in Figures 7,100 and 7,10 1 in terms of P2
versus time: (a) nfwith Pu = 201 and Pe-2 = 160 kPa; (b) nfwith same Pu but Pe-2 = 140 kPa;
(c) H oscillation at 68 Hz with same pu but Pe-2 = 100 kPa; (d) I oscillation at 11.1 Hz with
Pu = 135 and Pe-2 = 60 kPa; (e) '2 out of 3' multip1e-collapse (MC) oscil1ation at 4.06 Hz with
Pu = 102 and Pe-2 = 44 kPa; (f) LV oscillation at 4.08 Hz with Pu = 166 and Pe-2 = 44 kPa;
(g) oscillation at 14.1 Hz with aberrantly high frequency bursts superimposed, c10se to the L-to-c
boundary, with Pu = 13 and Pe-2 = 33 kPa; (h) irregu1ar or nf oscillation c10se to the I-to-nf
boundary, with Pu = 102 and Pe-2 = 80 kPa; (i) LD osci11ation at 4.17 Hz at the same operating
point as for (d) but approached from lower rather than higher Pe-2; (j) multip1e collapse (MC)
cyc1e at 4.97 Hz with Pu = 134 and Pe-2 = 60 kPa; (k) irregu1ar osci11ation for Pu = 135 and
Pe-2 = 80 kPa. In (a, b, c, d, f, k), LI D = 17.4; in (e, g, j), LI D = 8,2; in (k) LI D = 4.0. In al1
cases, a low kd was used, except for (g) where it was medium; (Bertram et a/. 1991),
formation of a constriction at mid-length, its propagation towards, and eventual disappear-
ance at the downstream end, whereupon the same cyc1e is regenerated. These oscil1ations
are c1early associated with 'choking' (by analogy with gas dynamics), which implies that
upstream wave-propagation is no longer possible. Choking or critical wave-speed, how-
ever, is not necessarily a prerequisite of flow-induced oscil1ation (Bertram & Pedley 1982).
In contrast, the high-frequency (H) oscil1ations, discussed in the foregoing, tend to be
stationary (i.e. of standing-wave type), near the downstream end.
The spectrum of observed oscil1ations is enriched if tapered tubes are also considered,
with the wal1-thickness increasing in the streamwise direction - see, for example,
Kamm et al. (1993) and Bertram & Chen (2000) with air- and water-flow, respectively, It
is of interest that for such tubes there can be as many as six flow-limited flow rates for the
same upstream transmural pressure, with correspondingly different forms of oscil1ation.
The flow-induced oscil1ations discussed so far represent but a smal1 subset of a wider
spectrum, the nomenclature varying with the beholder's perspective and with the type of
analytical model used. The milking-type oscil1ations, for example, are also termed long-
wavelength osci1lations, thus revealing their kinship to long-wavelength osci1lations in

766
SLENDER STRUCTURES AND AXIAL FLOW
p, 3500 l
2000
O
:mm
2
3
4
5
6
(a)
3200
0,5
1.0
1.5
Time (s)
2,0
2,5
3.0
pz
2400
0.0
(b)
D
,". 1
A L :.. .........
\ ..
" ..
... .. .. - ...
- - - - - - - - .,---
Pe
(c)
Figure 7.103 Waveforms recorded during transitions across regional boundaries. (a) The open
(stable)-to-LUtransition (LI D = 17.4, Pu = 201 kPa, smallkd); duringthe6 s shown, Pe increased
from 172.3 to 172.5 kPa. (b) The nf-to-H transition (LID = 17.4, pu = 166 kPa, medium kd);
during the 3 s shown, Pe decreased from 166.4 to 165.7 kPa (Bertram et a/. 1991). (c) Notional
topology of the initiation and cessation of oscillation as the control parameter Pe is varied: -,
stable fixed points; - - -, unstable fixed points; .., locus of stable limit-cycle solution; the arrows
show transitions from one stable state to another (Bertram 1995a).
hydraulics and gas dynamics (Kamm & Shapiro 1979), and to continuity-type oscil1ations
in two-phase fiows. On the other hand, the high-frequency oscil1ations are of a different
type and are sometimes called flutter, a1though LaRose & Grotberg (1997) reserve this
nomenc1ature to situations where tube inertia is an important element in the oscil1ation -
being explicit1y taken in to account in theoretical analyses or simply playing an important
role in experiments. For liquid-filled tubes, wa1l inertia is not very important, as shown
theoretically by Luo & Pedley (1998). For air-filled tubes, however, it c1early is, These
questions are further discussed in what follows and in Appendix P. However, before doing
so, the important question of how these oscil1ations, or indeed the system of a partially
collapsed she1l, relates to all of the foregoing in this chapter (Sections 7.1-7.8) is briefiy
de1iberated,
An early attempt to explore this issue was made by Weaver & Pa'idoussis (1977),
motivated by the question of post-divergence fiutter of shells with supported ends -
see Section 7.2.7. Experiments were conducted both with collapsible colostomy (Penrose)
thin-walled tubes and thicker-walled, normally circular, silicone-rubber tubes which were
rendered fiat over a certain length by constricting the tube at the 1 L or i L point by means

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
767
250
200
Classical flutter
N 150
e:;
;...
u
=
OJ
g.
Q)
....
 100
I
\ I
, I
, -'
I
J
J
J
J
50
:,; F1'PP;"'
I
I
I
I
J
J
Stable
O
4 X 10- 4
6 X 10- 4 8 X 10- 4
Flow rate (m 3 /s)
10 X 10- 4
Figure 7.104 Modes and frequency of oscillation of a collapsible Penrose tube conveying air-flow
(D = 12 mm, L = 152 mm, h = 0,28 mm, E = 2,59 MPa, Ps = 1121 kg/m 3 , P = 1,206 kg/m 3 );
(Weaver & Pai'doussis 1977).
of adjustable flat plates, in both cases conveying air. In the case of the collapsible tubes, the
qualitative behaviour is shown in Figure 7.104, Whi1e the tube is incompletely inflated, a
self-excited flutter mode called 'flapping' develops; with increasing flow, there is a sharp
transition to classical flutter; for the artificially flattened tube, this transition is gradual.
From these observations, it is clear that collapsible tube self-excited osci1lations (at least
those of the high-frequency, standing-wave type) resemble 'flapping' and they are quite
distinct from classical flutter.
The prediction of flapping was pursued by means of two 'parallel-plate' analytical
models (Weaver & Paldoussis 1977) involving: (i) two finite-Iength infinite-width plates
with flow in-between, and standing wave osci1lations; (ii) an infinite-Iength, finite-width
version, and travelling wave oscillations. In both cases, osci1lations develop at sufficiently
high flows. t Agreement between the experimental critical flow velocities for flapping
and those of the standing wave analysis is reasonably good, but only for small values of
interplate gap, b (for b / D ::'S 0.1); itdeteriorates rapidly with increasing b. The significance
of this work is that it is among the first - the other being by Matsuzaki & Fung (1977b) -
to use a parallel-plate or channel model in the study of the dynarnics of collapsible tubes,
later exploited successfully by many others, for example, Grotberg & Reiss (1984), LaRose
& Grotberg (1997) and Luo & Pedley (1998).
t Some weaknesses in the model itself have been pointed out by LaRose & Grotberg (1997).

768
SLENDER STRUCTURES AND AXIAL FLOW
7.9.5(b) Ana/ytica/ mode/s
A large number of analytical models have been proposed, which are very nicely classified
and briefly discussed by Luo & Pedley (1998) and Pedley & Luo (1998), Here, the same
classification is adopted and somewhat extended.
The different models may be grouped into (i) lumped-parameter models, (ii) l-D (one-
dimensional) models, and (iii) 2-D parallel-plate/channel models. Furthermore, one may
distinguish models taking into account wall inertia and those neglecting it.
(i) Lumped-parameter mode/s, ln the lumped-parameter (zero-dimensional) models, a
single axiallocation of the tube is considered, say the 'throať, where the tube is narrowest,
and all relevant variables are related to that location and are therefore functions of time
only: Q(t), A(t), Pl-2(t) etc. Such models result in ODEs and are therefore simplest to
handle numerically. t
The first to be proposed and simplest of the lumped-parameter models is Conraďs
(1969), for oscillations associated with the negative-slope segments of the curves in
Figure 7.95(a), By considering the system characteristics and noting that the collapsible
tube behaves similarly to a flow(charge)-controlled nonlinear resistance (QNLR) cir-
cuit in electronics, an inductance-capacitance-resistance (L, C, R) model is heuristically
proposed:
LQ+RQ+(1/C) f Qdt+Pl-2=0,
where Pl-2 = -rQ +  y Q3, -r = OPl-2/0Q, and y is a measure of the strength
of nonlinearity. Then, by defining v = 2[ y / r - R] 1/2 Q, í = t / (LC) 1/2, € = (r - R)
x (C / L) 1/2, this may be transformed into a Van der Pol equation (Andronov et a/. 1966;
Hagedom 1981),
(7.287)
d 2 v 2 dv
- - 5(1- V )- + v = O,
dí 2 dí
(7.288)
Accordingly, for 5 « 1 one obtains sinusoidal oscillations, while for 5 > 1 relaxation-
type osci1lations. However, there are inherent weaknesses in this model. For one thing,
the downstream network and its effect on the oscillation is not part of the model,
though it would be easy to include it - indeed, this is one of the strengths of lumped
parameter models. As found by Conrad et a/, (1978), for instance, the downstream net-
work plays an important role in the oscil1ation, showing that waves are transmitted to and
from points beyond the point of collapse of the tube, suggesting that 'the one-dimensional
hydraulic jump analogy, with its super- to subsonic transitions (which are the basis of the
model), is not appropriate for describing oscillations in collapsible tubes.'*
A more fluid-mechanically based model is Bertram & Pedley's (1982). The great virtue
of this model is that the 1- D continuum is considered to start with, and then the equations
are 'projecteď onto a single axiallocation, in a notionally similar way as one performs a
Galerkin projection, ending up with ODEs. Thus, continuity, an energy equation upstream
of the point of maximal collapse (the throat), and a momentum equation downstream are
tThis is specially true if chaotic behaviour is explored.
*Nevertheless, there is no doubt that supercritica1 flow does occur in collapsible tubes at substantial Reynolds
numbers (Bertram 2001).

TH.N SHELLS CONTAINING OR IMMERSED IN FLOW
769
considered, taking into account unsteady head loss at the throat (and partial pressure recov-
ery), as well as the mechanical properties ofthe system upstream and downstream. Some
detail s are given in Appendix P.2. Eventually, one obtains three autonomous differential
equations of the form (Armitstead et al. 1996)
Ji = f (free parameters; A, Ul, U2),
Ůl = ft (free parameters; A, Ul, U2),
(7.289)
Ů2 = h (free parameters; A, Ul, U2),
where A is the cross-sectional area at the throat, and U I and U2 are the fluid velocities
upstream and downstream. Equaůons (7.289) are suitable for numerical soluůon.
Limit-cycle oscillations are predicted, and some of the available experimental obser-
vations are found to agree qualitatively well with theoretical predicůons. The principal
conclusions are that (a) choking, which is ignored, is not a necessary prerequisite for self-
excited oscil1aůons and (b) the most crucial fluid-mechanical factors involved in these
oscillations, in the absence of choking, are the jet head-loss and pressure recovery at the
throat, as well as the physical properties of the system upstream and downstream of the
collapsible segment of the tube. In particular, 'the importance of the jet head loss . , .
cannot be over-emphasized. If there were no head loss there would be no re-opening and
collapse would continue to zero area. On the other hand . . . if there were no pressure
recovery [f(A) = O] there would be no oscil1ations and all equilibria would be stable.
The pressure recovery is the only mechanism whereby the instability can derive energy
from the flow' (Bertram & Pedley 1982),t Consideration of the results from this model
and how they compare with observation have been instrumental in formulaůng the l-D
model by Cancelli & Pedley (1985).
The fundamental weakness oflumped-parametermodels is that, sui generis, they cannot
incorporate wave propagaůon; they therefore cannot even distinguish between subcritical
and supercritical flow,
(ii) One-dimensional models. Kamm & Shapiro (1979) analyze l-D unsteady flow in
collapsible tubes subjected to extemal pressurization or body forces, taking frictional
effects into account. It is a classical, benchmark study, in which the linearized theory of
wave propagation is thoroughly examined for varying S, both subcritical and supercritical;
large amplitude nonlinear motions are also considered. Damped oscillaůons are success-
fully modelled, but self-excited oscillations of the type arising in Starling resistors are not
examined,
Other l-D theories which do consider such oscillations, for example, by Brower &
Scholten, Griffiths and Bonis & Ribreau (Appendix P), are based on the experimental
ob servati on that breakdown of steady flow and inception of oscillation in the collapsible
tube occurs only if the flow becomes supercritical, thus presuming that criticality or
choking is not only necessary but sufficient for flow-induced oscil1ations, which is not
trne. These models are capable of predicting the onset of the oscil1ations, especially for
thin-walled tubes and water flow,+ as seen in Table 7.12 for instance. However, these
tHowever, the model of Hayashi et a/, (1998) predicts limit-cycle-type oscillations, despite omitting
separation, So, it is the head loss that is important, rather than its putative source (the jet itself),
*It is recalled that wall inertia is generally neglected in these models, For air-flow, agreement between theory
and experiment is not as good (Sakurai & Ohba 1986), ln air flow, however, wall inertia is usually accounted
for - see, e,g" Walsh's work discussed a little further on,

770
SLENDER STRUCTURES AND AXIAL FLOW
Table 7.12 Agreement between theoretical and experimental
thresholds of oscillation, both by Ohba et al. (1985), for thin-
walled tubes conveying water, in terms ofthe critical flow-rate
Qc (l/min); tube dimensions are in mm,
Type of tube Diameter Thickness Qc (theor) Qc (exp)
Natural rubber 18 3,5 12,6 11.5
12 2,5 5,0 4,5
6 1.2 4.3 1.2
4 0,9 0,14 0.42
3 0.8 0.097 0,24
Penrose 12 0.15 0.33 0.24
6 0,15 0.06 0.18
models have not been successful in modelling the ensuing osci11ations. The main reasons
for this are that they do not incorporate energy loss or pressure recovery, and they do not
take into account the mechanical properties of the upstream and downstream rigid pipes.
A model that does, and represents a milestone in the analytical model1ing of flow-
induced vibrations in co11apsible tubes, is Cancelli & Pedley's (1985), which marries the
essentials of I-D wave-propagation and the important aspects of the Bertram & Pedley
(1982) lumped-parameter model. This model thus provides a link between I-D model s
which predict choking (but had previously ignored the end constraints of a finite-Iength
tube) and oscillations. Further, it includes the effects of longitudinal tension which re-
presents an important restoring force, t particularly with regard to sub- to supercritical
transition, and affects also the steady transmural-pressure to area relationship. The essen-
tial feature of the model is that flow separation is assumed at the throat (with attendant
pressure loss and some recovery), with the throat moving up- and downstream in response
to wave propagation and to prevailing pressure gradients. To give a flavour of the model,
the principal equations as modified by Jensen (1992) are given here, and in more detail in
Appendix P.2, as fo11ows [a11 quantities here are dimensionless].
(i) The modified tube law:
P - Pe = P(a) - !(o2ajox 2 ),
where P(a) = 1 - a- 3 / 2 for a < 1 [cf. equation (7.278b)] and P(a) = k(a - I)
if a ::: I, a being the area ratio as in Section 7.9.4. The o2ajox 2 term represents
the effect of longitudinal tension.
(ii) The mass and momentum conservation equations:
(7.290)
oa a
- + -(ua) = O,
ot ox
ou ou ap ou
- + u- = -- - (x - I)u-,
ot OX OX OX
(7.291)
tlt is recalled that flexural (bending) rigidity is often ignored.

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
771
where u(x, t) is the cross-sectionally averaged 10ngitudina1 velocity, and the last
tenn represents dissipation due to flow separation; thus, X = I for fully attached
flow, and generally O < X < I.
After considering steady-flow - with some very nice theoretical results on flow limita-
tion - the existence of self-excited, long-wave oscilIations is demonstrated. A discussion,
in physical tenns, is presented by Cancelli & Pedley, but here an abbreviated version
by Jensen (1990) is given instead: (i) as the separation point moves downstream, an
increasing proportion of the flow along the tube becomes attached, and this results in
a high-pressure wave moving downstream and a low-pressure wave moving upstream;
(ii) the fonner is reftected at the junction with the downstream rigid tube, resulting in
an area increase in the downstream half of the collapsible tube; (iii) this causes the con-
striction to diminish and the separation point to move upstream; (iv) the reverse sequence
then begins, with a low-pressure wave moving downstream, and ultimately causing the
separation point to move downstream again, It is important to realize that the move-
ments of the throat and the corresponding flow-velocity maximum and pressure minimum
[0.58 < x / L < 0.85 approximately in Bertram et al. (1994; figure 13)] are rather modest,
as shown in Figure 7,105. Comparison with experiments is difficult, since some of the
parameters required by the model have generally not been measured; stilI, by making
some reasonable assumptions and using some of the more recent and complete data, it
may be said that the model captures the essentials of long-wave oscilIations quite well
(Bertram et aJ, 1994). Hence, the basic model has been developed and further exploited,
by Jensen & Pedley (1989), Jensen (1990, 1992) and Bertram et aJ. (1994),
At a given point in the parameter space, unsteady behaviour may arise (i) if no steady
solution exists or (ii) if an existing steady solution is unstable. The first case is stud-
ied by Jensen & Pedley (1989), and the second, associated with small time-dependent
perturbations, by Jensen (1990) by means of powerful c1assical and modem dynamics
approaches,
Jensen (1990, 1992), using the modified Cancel1i & Pedley equations, finds by linear
analysis that the tube may be unstable in at least three different fonns of oscilIation, with
frequencies in distinct bands - which broadly correspond to the L, I and H oscilIations in
o
a(x, t)
o
x
Figure 7.105 The oscillating tube area versus axiallocation x and time t, for Pe-2 = 30,6 and
Q = 1 (both dimensionless; see Appendix P.2); (Jensen 1992).

772
SLENDER STRUCTURES AND AXIAL FLOW
34
32
30
P
28
26
24
0.5 1.0 15 2.0
(b) Q
30
20
\ ,;
\J=
,
p
10
Mode 2
I
o
(a)
5
10
Q
40
20
P2 O
-20
-40
-60
(c) O
O
P2
(d)
Figure 7.106 (a) The neutral stability curves of modes 2 and 3 in a region of the (P, Q)-space,
defined in Appendix P.2; (b) a part of the diagram, blown up, in the neighbourhood of a double-
Hopf double degeneracy; (c) Mode-Iocked quasiperiodic motion, in which the modal frequencies are
rationally related, for P = 30.6 and Q = 1.45; and (d) quasiperiodic motion where the frequencies
are incommensurate, for P = 30.6 and Q = 1.477 (Jen sen 1992).
the Bertram et al. experiments (Bertram et al. 1994).t For a partieular set of parameters,
the neutra1 stability (threshold) eurves for modes 2 and 3 are shown in Figure 7. 106(a,b) ,
in both eases associated with a Hopf bifureation, A weakly nonlinear analysis is then
employed, using eentre manifold and nonnal form teehniques (see Appendix F). lt is
shown that, for the range of parameters eovered in the figure, the Hopf bifureation for
mode 3 is suberitiea1 for 1.29 < Qc < 1.55 and supereritieal otherwise, while mode 2 is
suberitiea1 everywhere. The intersection of the two eurves eorresponds to a double- Hopf
double degeneracy [cf. Section 5.6.2(b,e)], in the vicinity of which eomplex dynamiea1
behaviour may be expeeted. A four-dimensional manifold is obtained, represented by
a set of eoupled amplitude equations, via which eodimension-2 bifureation diagrams
sirnilar to those in Figure 5.27 are obtained. Sinee both oseil1ations arise subcritieally,
weakly nonlinear mode interaetions are unstable, henee unobservable, and therefore the
stable fully nonlinear interaetions are examined numeriea11y (Jensen 1992), as shown in
Figure 7 ,106( e) and (d).
Keeping the dimensionless P == pe-2 eonstant at P = 30.6, it is found that the
system is stable for Q < 0.89; at Q = 0.89, a supereritieal Hopf bifureation leads to
mode 3 oseil1ation - initially harmonie, but with inereasing Q inereasingly distorted.
tThis suggests that, after all, these osci\Jations may not be distinctly different in terms of the underlying
mechanism,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
773
As Q is augmented further, a secondary Hopf bifurcation occurs, and mode 3 oscillation
becomes unstable, leading to a series of quasiperiodic states, involving two frequencies:
W3, associated with mode 3, and W2 arising from 'the growing influence' of mode 2.
Figure 7.1 06( c) for Q = 1.45 shows such a quasiperiodic, mode-Iocked oscillation where
W2/ W3 = p / q, in which pand q are integers, such that p / q is c10se to 2/3 (a strong
resonance); this oscillation repeats itself every 27T3 == 27(2Jr/W3)' For Q = 1.477,
Figure 7.l06(d), pand q are incommensurate, and the oscillation displays the bursting
characteristics that may signal the onset of chaotic oscillation reached via the intermittency
route [see, e.g. Sections 5.8.3(c) and 8.12.2(c)].
The mathematical simplicity of the model is more apparent than real, and numerical
difficulties prevent the exhaustive exploration of the nonlinear dynamics of the system.
The fact that PDEs are involved does not help, Hence, for a more extensive exploration
of the bifurcation behaviour of the system, Armitstead ef al. (1996) have reverted to the
Bertram & Pedley (1982) lumped-parameter model, involving ODE s (cf. the discussion
in Section 5.6, Volume I, p. 316), and obtained very interesting results, beyond the scope
of this brief review.
Another type of I-D model, for flow-induced oscillations in the trachea, has been
proposed by Walsh ef al. (1991). This interesting model involves annular flow about
an inner cylindrical membrane with rigid extensions and an outer rigid cylindrical con-
tainer; this artifice allows the membrane to collapse (balloon) outwards, so that it is in
tension in the collapsed state. t It is noted that in most models the tube area depends
only on the transmura1 pressure (tube law) - though modified sometimes to inc1ude
the effect of tension; thus, wa11 inertia is neglected and hence no trne aeroelastic flutter
is possible, Moreover, in most models, axial motions are neglected (the tube is axi-
ally 'tethered'), though in the physiological system axial motions are dominant. Here,
the axial inertia is taken into account, and later (Walsh 1995) trans verse inertia also.
Furthermore, a systematic scaling of the finite-deformation equations is used to obtain
a rational I-D model in which membrane curvature effects and variable axial tension
are taken into account, such that axial and normal wall variables are coupled elasti-
cally. The system is completed by a I-D fluid-dynamic model, ignoring separation
and pressure losses. Despite this last omission, flutter does occur, at subcritical flow
velocities. The mechanism is intimately related to elastic coupling in the two direc-
tions (axial and trans verse) and to the presence of wall inertia, preferably both axial and
transverse.
It is interesting that such a radica1ly different model to, say, Cancelli & Pedley' s predicts
flow-induced oscillations (flutter). However, this flutter is believed to be distinct from that
in Starling resistors.
(iii) 2-D parallel-plafe/channel models. According to Pedley & Luo (1998), both the
solid mechanics (the modified tube law) and the fluid mechanics (e.g. in the Cancelli &
Pedley model) habitually employed in l-D models are 'extremely crude', with severe
embedded weaknesses. It would be ruce, of course, to develop a 3-D model. However, the
fluid mechanics of such a model would be much more complex than that in the Cancelli &
Pedley model, for instance, and the solid mechanics, in view of the large shell deforma-
tions, much more difficult than what is presented in Section 7.8, not to mention numerical
tA membrane cannot support a compressive hoop stress, which would arise if the membrane collapsed
inwards due to internal flow,

774
SLENDER STRUCTURES AND AXIAL FLOW
m{'t
uc:=::> zLx
b
I
 1
(a)
Symmetric mode
--------------------------
(b)
Antisymmetric mode
Figure 7.107 (a) The parallel-plate model of Grotberg and co-workers; (b) detinition of the
symmetric and antisymmetric modes of deformation; (£Tom Grotberg & Reiss 1984).
difficulties in the solution. Hence, a compromise for the way forward is the development
of 2-D models. Such model s typically involve a channel of rectangular cross-section, with
either only one of the faces flexible [e.g. Webster et a/. (1985)], or two opposing faces,
the other two being considered to be infinitely far apart [e.g. Weaver & Pai'doussis (1977),
Grotberg & Reiss (1984)].
There are numerous 2-D models (Appendix P). In particular, in conjunction with work
related to pulmonary flow dynamics and the origin ofwheezing, Grotberg and co-workers
have developed several variants and refinements of one such model, starting with Grotberg
& Davis (1980). Here, the Grotbert & Reiss (1984) version is outlined, to give a flavour of
all such 2-D models. The model is fundarnental1y similar to Weaver & Pai'doussis' (1977),
but includes the effect of fluid friction and is developed and exploited much more fully.
Consider a 2-D channel, such as shown in Figure 7,107, the two paral1el flexible
walls ofwhich can general1y move symmetrically (varicose motion) or antisymmetrical1y
(sinuous motion), The pertinent equations for the fluid are
'V 2 c1> = O,
a ci> I 2 P
-+-('VcI» +!cI>+-=O,
at 2 p
(7.292)
(7.293)
that is, the continuity equation and the unsteady Bemoulli equation, modified by the
inclusion of a Darcy friction term (wherein ci> may be thought of as the spatial integ-
ral of the fluid velocity). The equation for the plates is the l-D von Karman equation

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
775
1.0
11* = IO
/
0,5
Q O
s
-0,5
-1.0
" JJ * --+- 00
Figure 7.108 The relationship between tbe dimensionless frequeney Q and dimensionless flow
velocity S = U j[Vj(pb 3 )]1/2 for k = 0.3, for forward travelling waves (up per braneh) and
baekward travelling on es (lower braneh, up to Ss) for two values of t-t *, The eritieal flow veloeity
for divergence ('stasis') is Ss, for both values of t-t*. The threshold flow velocity for flutter is S f'
defined as shown in the ease of t-t * = 10; for t-t *  00, S f is essentially coincident with Ss
(Grotberg & Davis 1980),
[cf. equation (10.115)]; in dimensionless form (see Appendix P.2) it is: t
a 4 w a 2 w a 2 w aw
'D ax 4 + [w - T ax 2 + Psh at 2 + Pshg-at + P - Pe = O, (7.294)
where'D = Eh 3 j[12(1 - v 2 )], w = Z - b is the deflection ofthe plate, [ is the elastance
oť the walls themselves or due to the springs, T = (bj6'Ah2)J+A(awjax)2dx is the
deformation-induced tension - cf. Section 5.5.2; 'A is the wavelength of motion, and g
is the coefficient of damping in the wall. The system is subject to symmetry or antisym-
metry conditions on the centre1ine and to the usual impermeabi1ity boundary condition on
the wall,
Consider solutions of the form {<I>-,--w} ex exp{i [k(xjb) - Q r]} where k = 2nbj'A is
the dimensionless wavenumber, and Q and r are the dimensionless frequency and time.
Jypical results when tension and damping effects are ignored (T = O, f = g = O) are
shown in Figure 7.108 for a very light fluid (J-l..* = Pshjpb -+ (0)* and a heavier one
(J-l..* = 10), This figure is similar to some presented in Section 7.6 (Figures 7.68 and 7.70,
for instance), and the interpretation of the results is similar: (i) the point of crossing of
the S-axis signifies that upstream wave propagation is thereafter no longer possible, and
the 'stasis' for velocity Ss corresponds to what has heretofore been called the critical flow
velocity for divergence, which is independent of J-l.. *; (ii) the point of coalescence of the two
solution branches corrsponds to where Q becomes complex and it is the critical value for
co up led-mode flutter, S f. By conscting several such curves for different wavenumbers,
the most critical values of S s and S f may be determined. It is seen in Figure 7.108 that,
t AU equations are given in dimensionless fonn in Grotberg & Reiss (1984) and most subsequent papers,
but the nondimensiona1ization scheme is not aU that obvious - see Appendix P,2, For instance, the elastance
of the wa1ls is £, and the 10ng-wave propagation velocity in the channel is defined as (£bj p)I/2 and used
for nondimensiona1ization; by comparison with equation (7.276a), taking A = bx (width), it is seen that
£ = dpjdb, and hence has dimensions ofN/m 3 ,
tNote that Il * here is the inverse of Il, defined in (7.49) and used extensively in the early parts of this chapter.

776
SLENDER STRUCTURES AND AXIAL FLOW
as f..l* -+ 00, Ss becomes sensibly coincident with Sf, whereas for f..l* = 10, there is
a sizable gap between them (cf, Figures 7,7 and 7.17). This has also been convincingly
demonstrated by Webster et aZ. (1985).
It is interesting that, if wan damping is included, coupled-mode flutter 'does not occur'
according to Grotberg & Reiss (1982, 1984).t What is certainly true is that, whether
with or without structural damping, the system Zoses stability by divergence, whereas
in the experiments it is flutter that is observed. This, Grotberg & Reiss consider to be
'anomalous' , al though as we have considered at length in Section 7.2.7 there are reasonable
interpretations for it, notably that the observed flutter may in fact be a dynamic divergence.
Nevertheless, this prompted the authors to include fluid damping in their model, which
had felicitous effects, as may be seen in Figure 7,109, where Sf and the corresponding
frequency, Q f, are plotted versus the ratio of fluid to structural damping y. For vanishing
fluid damping and nonzero structural damping (y -+ O), the frequency of coupled-mode
flutter vanishes, while a substantial fluid damping causes the system to 'lose stability
by flutter'. This is a very interesting result, although it is not a unique occurrence: if a
sufficient degree of nonconservativeness is injected into an inherently conservative system
(in this case via fluid damping), then it may behave as a nonconservative one, losing
stability by flutter rather than divergence.:!:
Much further work has been conducted by Grotberg and co-workers which will
not be discussed here; for instance, taking into account the nonlinear terms in
equations (7.292)-(7,294), the nonlinear dynamics is explored, finding that flutter arises
via a supercritical Hopf bifurcation and that some of the linear instabilities are singular
when examined in the nonlinear regime (Grotberg & Reiss 1984; Grotberg & Shee 1985).
A major question that shou1d be discussed, however, is how do these theoretical results
relate to the behaviour described in conjunction with 1-D models. The superficial differ-
ence is that here wan inertia is taken into account, while in Cancelli & Pedley's model,
for instance, it is not. A more profound difference is that here the solid mechanics is
fundamentally linear, except for the modification arising from the deformation-induced
tension, whereas in the 1-D theories the tube law introduces strong nonuniformity and
nonlinearity in the area-pressure relationships.
The reconciliation of the findings of Cancelli & Pedley, Jensen and others, on the one
hand, and Grotberg and associates, on the other, has begun with two important studies:
LaRose & Grotberg's (1997) and Luo & Pedley's (1998). Both are too involved to present
in detail here, but the physica1 systems are easy to grasp.
LaRose & Grotberg (1997) abandon their assumption of uniform flow through the chan-
nel, and consider developing viscous flow, modelled by the Orr-Sommerfeld equation.
Assuming a travelling wave solution as before, the resultant eigenvalue problem for the
fluid-structure interaction is solved both numerically and, for the long-wave limit, analy-
tically. A long-wave instability is found, for the first time in this type of channel flow
model, which, it is reasoned, must correspond to the long-wave flow-induced oscillation
found in the Cancelli & Ped1ey 1-D model, for instance; it is independent of the mass
ratio f..l * and hence of wan inertia. Moreover, the finite-wavelength instability of the type
tBy following their analysis, however, what is really shown is that a Hamiltonian Hopf bifurcation does
not occur, characterized by Im(Q) = O at the threshold [see Figure 3.4(c)); while a so-called Paldoussis
coupled-mode flutter [Figure 3.4(d»), modified by the inclusion of damping, is feasible - see Figures 7.7
and 3.14.
*See, for example, Section 11.5,3 - specifically Table 11.9,

2.25
THIN SHELLS CONTAINING OR IMMERSED IN FLOW
777
2,00
1.75
SI
1.50
1.25
1.00
1.50
1.25
1.00
{JI 0,75
0,50
0.25
o
-2
-I
o
logy
2
Figure 7.109 The critical dimensionless flow velocity for flutter, S f, where S = U /(£b/ p)I/2,
and corresponding frequency, Q f' for symmetric and antisymmetric plate motions, as a function
of the ratio of fluid to structural damping, y = f / g [corrected version of the figure in Grotberg &
Shee (1985)].
discussed in the foregoing continues to exist in this new model. For large f..l *, typical of
airflow, the critical flow velocity for the long-wave instability, Slw, is higher than the flutter
threshold, Si' For smaller f..l*, however, f..l* :S 100, Si can be higher than Slw' Which
is higher also depends on wall damping to which Slw is insensitive, channel height, and
other parameters,
Comparison with experiment is difficult, again because some of the pertinent quantities
are missing from the experimental data and have to be estimated. Table 7.13 gives some
selected data from a more extensive comparison in the original paper. In this table, for
the theoretical results we give those for Poiseuille rather than developing flow, and for
the experimental values of U i we give those at the centre of the collapsed channel [see
Figure 7.93 (c)-(e)], computed numerically - both of which lead to a more favourable
agreement. It is seen that agreement in U i is reasonable only for the smaller channel
heights, but is generally of the correct order of magnitude; in the experiments, b corres-
ponds to the central, parallel-wall segment of the col1apsed tube. Surprisingly perhaps,
the flutter frequencies agree quite well,

778
SLENDER STRUCTURES AND AXIAL FLOW
Table 7.13 Comparison of theoretical and experimental flut-
ter speeds, S f, and corresponding frequencies [from LaRose &
Grotberg (1997)].
Channel height, Uf (mis) f (Hz)
b (mm)
Experiment Theory Experiment Theory
0,81 39.5 6,2 292 307
0,70 10,0 5.4 300 308
0,37 4.7 3.5 325 362
An important study, also considering a nonuniform velocity profile, but an inviscid
shear flow, in a channel undergoing symmetric (varicose) travelling waves is pres-
ented by Huang (1998). It is recalled that the pressure perturbation on the wall has
two components: one in phase with the displacement, and the other in phase with
transverse velocity of the wall. For potential flow, only the first works towards destabiliz-
ing the system (the Bernoulli effect); for shear flow, however, the sign of the pressure
is reversed ('reversal of the Bernoulli effecť) and it is the second component that is asso-
ciated with destabilization - specifically by producing wave drag and hence irreversible
transfer of energy from the flow to the waves. t It is reasoned that this is a plausible mech-
anism for generating the flow-induced oscillations observed in the experiments. This
mechanism is convincingly tied to the dynamics of flexible walls subjected to uniform and
shear flows and to the work of Benjamin, Landah1 and others [see, e,g., Appendix C and
Section 10.3.3]. A useful critique of LaRose & Grotberg's theory is given, to which this
work has many similarities (but also differences). The interested reader is also referred to
Huang (2001a).
Luo & Pedley (1998) and Pedley & Luo (1998) present a new parallel channel model
of their own, in which the channel is nonuniform, as shown in Figure 7.llO(a); only the
portion CD of the upper plate is flexible, modelled as a membrane, the trans verse inertia
of which is taken into account, as is a linear axial tension in it, but dissipation in the wall
is neglected. The inlet flow is a steady Poiseuille flow. Typical instantaneous streamlines,
computed at dimensionless times t = 4.80 and 27.34 are shown in Figure 7.11 O(b,c), and
many more in the original papers.
A time history of the wall displacement at xw/b = 8,5 and an associated 3-D plot of
the deformation is shown in Figure 7.111 (a,b) for a value of the axial tension allowing the
development of long-wave instabi1ity. This indeed is shown to occur for f..t * = O (dashed
line), corresponding to a dense fluid and a thin membrane. For a less dense fluid (but likely
not aS light as air), f..t* = 0,1, it is seen that, superimposed on the long-wave oscillation,
a higher-frequency flutter develops, which c1early requires a nonzero wall inertia for it
to arise.
For a higher axial tension, the long-wave oscillation does not develop, as shown in
Figure 7,lll(c). Here, the dotted line trace for f..t* = O eventually returns to Yw = O,
while that for f..t * = 0.1 shows flutter once more, For the same value of tension but an
tMoreover, streamline pattems show the existence of Kelvin's 'cats' eyes' in the wider channel segments,
thus rendering the effective flow passage narrower,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
779
B 
A
(a)
C D
______ y = h(x, t) 
E
F
(b)
(c) O
5
10
15
20
25
30
35
40
Figure 7.110 (a) Sketch of the 2-D model of Luo & Pedley. (b,c) Instantaneous streamlines
(lighter lines) and energy dissipation contours (darker lines) for t = 4,80 and 27.34 (Luo & Pedley
1998).
even lighter fluid or heavier wall (f.L * = 1), it is seen in Figure 7 .lll(d) that, again, only
flutter arises, but with considerably lower frequency and highly nonlinear waveform. The
osci11ation grows fast and nonlinearly, so much so that the automatic mesh generator used
in the solution can no longer follow it.
Clearly, therefore, Luo & Ped1ey also find that the two forms of osci11ation - the
low frequency, long-wave osci11ation and the higher frequency, shorter wave1ength flutter
generally coexist, but in some cases only the latter arises. t
With the aid of this most recent work, the major of the hitherto model-specific behaviour
traits of collapsible tube osci11ations have been reconciled to some extent, though adequate
agreement between theory and experiments remains prob1ematic. This reminds this writer
ofthe discussion in Section 5.8.1 (Volume 1, pp, 365-366) on 'a case study into some of
the pitfalls of analytical mode1ling of nonlinear systems when trying to match experiment-
ally observed behaviour' . By taking this or that effect (here, tension, wall inertia et seq., of
the many possible ones) into account in the model, and beholding an improvement in the
agreement with experiment, one can easi1y conc1ude that one has homed in on the major
effect; however, the improvement may have only been fortuitous, and a second improve-
ment may thus make things worse in the multidimensional parameter space involved. It is
only when a suitably robust model has been assemb1ed, that all the pieces fall in to place,
We have yet some way to go for that!
7.9,5(c) Chaotic oscillations
Some of the experimental time traces of flow-induced vibration, in Bertram et al. (1990,
1991) for instance, show aperiodic, chaotic-looking osci11ations - see, for example,
tThis analysis has recently been extended (Luo & Pedley 2000), and multiple steady solutions are found for
a constant upstream transmural pressure, some stable and some unstable when the flow is limited, depending
on which parameters are allowed to vary, The important conclusion is reached that there is no definite link
between flow limitation and self-excited oscillations,

1.0
0,8
0.6
Y w
0.4
t( t 2
0,2
O
O 10 20 30 40
(3)
1.2
0,8
Y w
0.4
O
(b)
1.0
0,8
0.6
Yw
0,4
0,2
O
(c) O
1.0
0,8
0,6
Y w
0.4
0.2
O
O
(d)
30
10
20
30
10
20
30
40
50
60
Figure 7.111 (a) Time history of the wall displacement Yw/h for xw/h = 8.5 and low enough
axialtension to pennitlong-wave oscillation: - --, J1.-* = O; -, J1.-* = 0,1. (b) 3-D displacement
corresponding to one period in (a) for J1.-* = 0.1. (c) Time history of Yw/h for a higher tension,
for which long-wave oscillation does not develop, but flutter does for nonvanishing wall inertia:
.,'" J1.-* = 0;-, J1.-* = 0.1. (d) the same as in (c) butforalarger J1.-*, J1.-* = 1 insteadofO.l (Luo
& Pedley 1998).

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
781
Figure 7.102. However, showing that such vibration is indeed chaotic is difficult, espe-
cially for experimenta1 systems - see, e,g., Section 5.8.1. t However, despite the authors'
disclaimers of total success and fresh doubts (Bertram 2001), it may be said that Bertram
et a/. (1991) and Bertram (1995b) present sufficient evidence in terms of reconstructed
phase plots, retum maps, spectra and time traces, to provide a strong indication, but not
definitive proof, that some of the observed oscillations were indeed chaotic.
More quantitative measures are obtained by Wagner & Persson (1995) for experimental
data of arterial blood pressure and renal blood flow: fractal dimensions of 3.3 ::!:: 0.37
(equivocal) and 3.6::!:: 0,23 (much clearer), andLyapunov exponents (cf. Section 5.8,1) of
0.1 and 0.04, which are indicative of weak chaos.
Jensen (1992) has explored the nonlinear dynamics of the system analytically and
numerically, especially in the neighbourhood of subcritical double-Hopf double degen-
eracies (as discussed in (b) just before), The time traces and the vector field are certainly
suggestive of chaos through intermittency, but this could not be proved quantitatively.
Finally, fully quantitative measures of chaos (fractal dimension, Lyapunov exponents)
have recently been obtained for a collapsed tube, driven by a pulsatile head applied to
an operating point where self-excited oscillation is otherwise periodic (Bertram & She
2000); see also She & Bertram (1996).
This writer believes that chaotic oscillations do exist in many parts of the parameter
space of the problem at hand. However, perhaps this exploration is best left till after a
model capturing all the essential features of linear and nonlinear behaviour and agreeing
well with experiment has evolved,
7.10 APPLlCATlONS
Cylindrical shells are widely used in science and engineering: all thin-walled piping and
cylindrical containment structures (e.g. for power generation, aeronautical and process
industry applications) are thin shells subjected to fluid loading or flow excitation.
7.10.1 General applications
Much of what has been presented in the foregoing was inspired by concrete engineering
applications, or conducted to solve a real problem. Examples are: (i) the work on swirling
flow (Section 7.6) for aircraftjet-engine and centrifuge applications, and the Intense Neu-
tron Generator system; (ii) all of the work on turbulence-induced vibration (Section 7.7);
(iii) the work on the dynamics of fluid-filled cylindrical containers (Section 7.8.1(b».
Furthennore, all of the work in Section 7,9 is related to real physiological systems. Here,
we shall present some further 'general' applications, and then consider some specific ones
in the following subsections.
Further to the comments in Section 7.9.2 on inflatable membranes, it should be men-
tioned that such systems are used for the transport of lighter than sea water cargos
[e.g. Hawthorne (1961); see also Chapter 8], floats, pollution-control barriers and air-
springs (Charrier & Hannoyer 1974), flood-control devices and dams [e.g. Anwar (1967),
Namias (1985), Hsieh et a/. (1989), Wauer & Plaut (1991), and many other papers referred
to therein], and air-cushion devices (Dowell 1980).
It is recalled that one of the earliest studies on the dynamics of shells conveying fluid
has been Niordson's (1953), in conjunction with the Aswan hydroelectric plant penstocks.
tl n particular, see 'Moon's parable', Volume I, pp, 356-357,

782
SLENDER STRUCTURES AND AXIAL FLOW
Interest in this has continued, especially on resonances associated with draft-tube vortices,
the turbine blade-passing frequency or dynamic imbalance, waterhammer, and so on [e,g.
Shiraki et a/. (1975), Bhave et a/. (1987), Ruud (1991), Moore (1991), Helwig (1993)].
Flexible shell segment s are sometimes inserted in piping to reduce not only vibration
but also acoustic noise transmission; see, for example, Chapnik & Currie (2000).
Furthermore, the statics and dynamics of deeply submerged underwater pipelines
involves mainly pressure-related but also flow-induced shell-type responses; see, for
example, Haughton & Ogden (1979), K yriakides & Chang (1990), Mikkelsen & Tvergaard
(1997),
AIso, since the dynamics of rotating shells has been discussed in Section 7.6, it is
appropriate to mention another application, this one without fluid interaction: the Coriolis-
based or 'vibrating cylinder' gyroscope (Langdon 1982; Burdess 1986; Loveday & Rogers
1998; Kristiansen & Egeland 1999). This gyroscope involves a cy1indrical shell forced
to oscillate in the n = 2, m = I mode, say; if a rotation about the axis is superposed,
then with the aid of a simple diagram, it is easy to see that Corio1is forces at 45° to the
antinodes are generated - which are essentially 1inearly related to the rotation rate,
7.10.2 Internal-annular flow eases
ln several engineering systems, a cylindrical shell or tube is simultaneously subjected to
internal flow and to annular flow, confined on the outside by a stiffer cylindrical strnc-
ture. Indeed, flow-induced vibration problems have been experienced in such systems.
Examples are heat shields used in after-bumers of certain types of aircraft jet-engines
(Ziada et a/. 1988), thermal shields in nuc1ear reactors (Pai'doussis 1980; Wald 1983), and
other systems (de Langre 1999; Gibert 1999).
If the velocity of the two flows is simi1ar, instabilities and flow-induced vibrations are
much more likely to be due to the annular flow (Chapter II). For that reason, these concrete
flow-induced experiences are discussed in Section 11.7.4.
7.10.3 Aeronautieal applieations
A number of very interesting practical experiences (' cases' ) of flow - induced vibration and
flutter of shells conveying fluid are reported by Dowell (1980).
The first relates to shell flutter of the main shaft of a Pratt & Whitney jet engine, This, in
fact, is the case that motivated most of the very interesting work discussed in Section 7.6
by Srinivasan, Dowell and their associates, This is a case of swir1-induced shell flutter,
which in the end was suppressed by reducing the strength of the vortical flow about the
shaft (DowellI980, Case I).
Another interesting case (Dowell 1980, Case IV) involves flutter of the conical exten-
sion on re-entry missi1e exit-cones. The extension, made of 'rigidized c1oth' or other
1ight-weight material, is very flexible; it helps improve performance at high speeds and
high altitudes. Incidents of flutter of the conical extension have been reported, The exten-
sion may be modelled as a conical shell or even a conical membrane, and it is noted
that, qua1itatively, flutter of cylindrical and conical shells is simi1ar. Hence, it has been
possible to obtain 'reasonable correlation with flight experiments'. In the end, however,
the problem was solved by using stiffer re-entry cone extensions.
Such extensions are designed to operate in an underexpanded condition, that is, with the
static pressure greater than the ambient one. Under off-design conditions, however, over-
expansion can occur, resulting in collapsing-pressure loading which helps to precipitate

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
783
(a)
(b)
Figure 7.112 Flutter of rocket nozzle conical extensions: (a) for a 0.356 mm thick, fibre-reinforced
plastic extension with area ratio AR=lOO; (b) for a 0,241 mm thick one, with AR=54 (Tuovila &
Land 1968),
flutter. An experimental study of the problem is reported by Tuovila & Land (1968); see
also Bert & Egle (1969). The nozzle throat was 76 mm and the exit 305 mm in diameter
at a 22.5° expansion half-angle. The extension was made of glass-fibre reinforced cloth
moulded with polyester resin, with an exit diameter of either 559 or 762 mm at the same
angle. In the tests, buckling of the extension occurred, and was followed by 'erratic oscil-
lation' (flutter), as dramatically shown in Figure 7,112; in the thinner extension, probably
in the n = 9 mode.
7.10.4 Shell-type Coriolis flow meter
Other than the biomechanical systems, most engineering applications related to this
chapter involve 'problems', real or potential. It is therefore refreshing to have at least
one application, which is wholly benign (see Volume 1, p. 3, third paragraph). This is
the shell-type Coriolis flow meter; t the ordinary, beam-type Coriolis flow meter is briefly
discussed in Section 4.7.1.
The basis of operation of the shell-type meter may be appreciated by looking at
Figures 3.13 and 7.9 (n = 2, nominal m = 2), and more particularly Figure 7.113 (n = 2,
nominal m = 1). It is seen that for U = O the mode shape is stationary (the mode is c1as-
sical- Volume 1, footnote on p. 95); hence the phase difference in I w I between any two
axially separated points is zero. This is no longer trne for U > O; a phase difference /14>
tThe author is grateful to Studzinski (1999) for some useful information on this,

784
2,50
1.25
_o
" -
:e X 0.00
" '-"
 
- 1.25
-2,50
1.50
,9 O' 0,75
"00
'X 0,00
gfI'-"
.3 :: -0,75
-1.50
1.20
_ (a)
' 0,60

1  0,00
.g -0,60
U
-1.20
2.50
1.25
-:'0
"'-
 X 0,00
';
-1.25
-2.50
1.50
tU  0,75
8
. X 0,00
01),-"
<=
.3 :: -0,75
-1.50
1.20
  0,60
O) o
 X 0.00
8,-"
;:J '"
e -0,60
iJ
SLENDER STRUCTURES AND AXIAL FLOW
(a)
(b)
(b)
(a)
(c)
- 1.20
0.0 0,1 0,2 0.3 0,4 0.5 0,6 0.7 0.8 0.9 1.0 0,0 0.1 0.2 0.3 0.4 0,5 0.6 0.7 0,8 0,9 1.0
xlL xlL
Figure 7.113 Time-dependent first-axial-mode modal shapes (n 2; nominal m = I) over
half a period of oscillation for the same clamped-clamped system as in Figure 7.9: (a) Vi = O;
(b) V i = 0.4; (c) V i = 0.5; (d) V i just short of divergence,

THIN SHELLS CONTAINING OR IMMERSED IN FLOW
785
(b)
Figure 7.114 The mode shape (amp1itude exaggerated) of a c1amped-c1amped shell oscil1ating
in the n = 2, m = 1 mode for (a) U = O, (b) O < U < Ucd (DMC/FMC 1998).
1.0 - -
-
- -
, -
0,8 - -
-
- -
- -
- ,
0,6 -


"<:]
0.4
0,2
25 50 75 100 125 150 175 200 225
6
5
 4

or,
o
3
x
-
"<:] 2
5
10
15
20 25
qm (kg/s)
30
35
40
45
Figure 7.115 (a) The phase difference /),fjJ for three shells conveying fluid (p = 10 3 kg/m 3 , Ps =
4.5 x 10 3 kg/mJ, E = 110 GN/m2, a = 10 mm, h = 0,5 mm, L = 0.1,0.15 and 0,2 m) versus
the mass-flow rate qm. (b) /),t = /),fjJ / úJ for the same three shells (Kutin & Bajsié 1999).

786
SLENDER STRUCTURES AND AXIAL FLOW
exists, as is also shown more graphically in Figure 7.114, Moreover, this phase difference
increases with U, almost linearly (Kutin & Bajsié 1999), as shown in Figure 7,l15(a). The
flow rate for divergence, Ucd, for each of the three shells in the figure is approximately
where the curve goes off the figure in each case. If one considers flow rates considerably
lower than Ucd and plots ó.t = b..cp / Q-recalling that Q varies with U -the variation b..t
versus flow-rate is linear [Figure 7.115(b)]; moreover, the three curves sensibly collapse
into one.
This basic idea has been exploited, and flow meters measuring b..cp and hence the flow
rate are now commercially available (DMC/FMC 1998),t As in the beam-type meter, an
exciter is used, in this case to excite the shell in the n = 2, m = 1 mode, located at
x/L = 0,5; electromagnetic sensors measure the vibration at x/L  0.25 and 0.75, and
hence b..cp is computed, Other designs are under development; see, for example, Amabili
& Garziera (2000b).:j: The main advantage of the shell-type flow meter vis-a-vis the beam-
type one is that the flow is straight through, with no bends; hence, no appreciable pressure
drop is incurred. Moreover, cleaning, if necessary, is easy.
7.11 CONCLUDING REMARKS
It is abundantly clear that an enormous amount of work has been done on the dynamics of
shells containing or immersed in axial flow - even though not everything has been touched
upon in the extensive treatment accorded to the subject in this chapter, for example, the
dynamics of toroidal shells conveying fluid (Zhu 1995)!
A number of topics have been well covered and may be said to be well understood,
for example the linear dynamics of shells conveying straight flow. However, the physical
behaviour of the system is strongly nonlinear, and a great deal remains to be done on
the nonlinear dynamics of the system - and deserves to be done, especially now that
computational facilities make it much more feasible.
Other topics, such as the dynamics of shells containing swirling flow, have been
abandoned 'in mid-stream', so to speak, probably because of shifts in the priorities of
research-funding agencies. Others are still vigorously being pursued, for example, the
biomechanics work, and steady progress in understanding and prediction capabilities may
be expected in the decade to come.
t Gratifyingly, the street address of Direct Measurement Coorporation (DMC) is 4040 Coriolis Way!
*In conjunction with Krohne Messtechnik.

8
Solitary Cylindrical Structures ln
Axial Flow


8.1 INTRODUCTION


Unlike vibrations of cylindrical structures induced by transverse flow ('cross-flow'), the
study of vibration due to axial flow is a relatively new phenomenon, beginning in eamest
in the 1950s, The reason for this is easily understood when considering that, in the former
case, even moderate flow velocities may cause large amplitudes of transverse vibration, for
example, vortex-induced oscillation (Naudascher & Rockwell1994; Zdravkovich 1997,
2003), and hence, severe, immediate problems. Hence, cross-flow-induced vibration has
received the lion's share of designers' and researchers' attention. In axial flow, however,
the transverse vibrations in industrial structures are of generally small amplitude: typically
10- 3 < AI D < 10- 2 , A being the r.m.s. amplitude of vibration and D the cylinder dia-
meter. Accordingly, such vibrations would ordinarily be of litt1e concem, were it not for
the peculiar geometry in certain applications, rendering vibration of even such small amp-
litude worrisome, namely (i) in nuclear reactor fuel-element bundles, monitoring tubes
and control rods (for systems where the mean flow is nominally axial) and (i i) in the case of
heat exchanger tubes, in regions where the fl.ow is mainly axial. In both cases, the problem
arises from clo se spacing, either between the cylindrical elements and 'spacers' on adja-
cent elements, or between the cylindrical elements and intermediate imperfect supports;
thus, even small-amplitude vibration results in impacting and, hence, fretting and wear-
which, in time, might cause the rupture of the (tubular) cylindrical elements, with serious
consequences [see, e.g. Ko (1997)]. Thus, unlike cross-flow-induced vibration problems,
low-amplitude axial-flow-induced vibration problems surface only in the long tenn, caus-
ing, apart from fretting, high-cycle fatigue and stress-corrosion cracking, for instance.
Nevertheless, these all too easily and conveniently neglected problems eventually become
of major concem as long-service systems, such as power plants, grow older.
Another important reason for interest in axial-flow-induced vibration is that some sys-
tems are inherently very flexible because of their material make-up, manner of support,
or length, so that they are prone not only to larger-amplitude vibrations but also to flu-
idelastic instabilities; examples are Dracone barges, towed acoustic arrays (Section 8.9),
and pendular systems (Section 8.12). In this respect, it is important to stress that although
much of what is presented in this chapter is specific to cylindrical structures, the basics
apply equally to all one-dimensional structures that may be modelled as strings or beams:
for example, in paper making and tape winding (Pramila & Ákesson 1987), deployment
of string for medical implants (Frei et al, 2000), etc; refer also to Chapter 10,


787




788
SLENDER STRUCTURES AND AXIAL FLOW
Conceming cylindrical structures in axial flow, one may distinguish unconfined flow
and confined flow, for example by an outer containment conduit. Most engineering
applications are c1early of the second type, although in many cases confinement is not
tight enough to affect the dynamics substantially, However, the dividing line between
axial (essentially unconfined) flow and annular flow is arbitrary, and hence, some of the
work involving confined flows is presented in this chapter, whi1e some is reserved for
Chapter 11. Indeed, this provides an additional very important reason for interest in axial-
flow-induced vibration: its analytical modelling provides the framework for modelling
annular and leakage flow oscil1ations, which are a major hazard (Pai'doussis 1980; Cases
43-52; Au-Yang 1999,2001).
Most of the early interest in the low-amplitude vibration of cylindrical structures in
axial flow was directly related to the power generating industry and more specifically to
nuc1ear power plants: in the United States (Burgreen et al. 1958; Shields 1960; Quinn
1962, 1965; Pavlica & MarshallI966), France (SOGREAH 1962), Sweden (Rostrom &
Andersson 1964a,b; Rostrom 1964), and Canada (Pai'doussis 1965b, 1966c). These first
studies to appear in the literature had one or more of the following aims: (i) measurement
of the amplitude of vibration of particular cylindrical structure configurations, modelling
nuc1ear reactor components and flow conditions; (ii) understanding the causes of vibration;
(iii) development of means for predicting the vibration amplitudes, Although from the
designer's point ofview, item (iii) was the most important, it became evident that the lack of
understanding of the underlying mechanism causing the vibration rendered its prediction,
with any degree of confidence, unsatisfactory. Thus, the need for more fundamental and
systematic as opposed to ad hoe studies became evident.
The dynamics of a cylinder in axial flow were studied by Pai'doussis (1966a,b) both ana-
lytically and experimentally and the equation of motion was derived for the first time. It was
shown that the effect ofaxial flow is to damp free oscil1ations at small and moderate flow
velocities and to reduce the natural frequencies of oscil1ation. However, at sufficiently
high flow velocities, cylinders in axial flow become subject to fluidelastic instabilities,
namely to buekling (divergence) and to oscillatory instabilities (flutter), These fluidelastic
instabilities were shown to be of little praetieal concem (PaIdoussis 1966c), as the crit-
ical flow velocities necessary for their inception - for cylindrica1 structures of nonna1
engineering application - are much too high (typically > 50 mis for water flows).
Thus, the vibrations of practical interest occur with flow velocities smaller than the
critica1; they have been ca1led subcritieal vibrations, to differentiate them from those
associated with oscil1atory instabilities. The subcritical vibrations are stochastic in nature,
with the dominant frequency component corresponding to the first-mode of the cylinders;
as mentioned earlier, the amplitude is nonnally very small, but can be larger for long,
slender cylindrical structures.
Having obtained the equation of motion, at least some of the parameters influencing
the vibrations of a cylinder in axia1 flow could be identified, thus enabling the fonnulation
of a semi-empirica1 expression for the prediction of amplitude of subcritical vibration
(Pai'doussis 1966c, 1969) which proved to be one of the most successful (Section 8.4).
This and other semi-empirical theories for subcritical vibration suffered from a common
weak:ness: in the course of their fonnulation, the forces causing the vibration were incom-
pletely specified. Accordingly, a great deal of effort has been expended in c1arifying the
mechanism of subcritica1 vibration excitation, as discussed in Section 8.4. This, indir-
ectly, provided further impetus for the study of the dynamies and stability, as opposed to

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
789
small-amplitude ('subcritica1') flow-induced vibrations, of cy1indrical structures in axial
flow, which constitutes much of what is presented in Chapters 8 and 9.
In what follows, after deriving the linear equations of motion in Section 8.2, the linear
dynamics is considered first (Section 8.3), and then low-amplitude subcritical vibrations
(Section 8.4). After that, a number of more particular topics, for example, nonlinear
dynamics, the dynamics of towed and articulated systems, are discussed in the rest of the
chapter.
Before closing this Introduction, it ought to be recalled that there exists a very rich
literature on the fluid mechanics and dynamics of generally noncylindrical axisymmetric,
usually rigid bodies ofrevolution, mainly in potential flow, going back to Munk (1924),
von Kánnán (1930) and Laitone (1947), for examp1e; see also the classical work cited
in Section 2.2.1(a)-(e). Work on the subject has continued to be done ever since, more
recently focussing on computational aspects; see, for example, Zedan & Dalton (1980)
and Wu cf al. (1989) and reference s cited therein.
8.2 LlNEAR EQUATIONS OF MOTION
8.2.1 General formulation
The equations of motion have been derived by Paí'doussis (1966a)t and again later in a
more general, corrected form by Pai'doussis (1973a, 1 974b), as presented here, The reader
is encouraged to review the preambulatory comments of Section 3.3.1, as they apply here
equally, and to refer to the derivations in Section 8.10.
The system under consideration consists of a flexib1e cylindrical body of circular cross-
section, immersed in an incompressib1e fluid of density p flowing with uniform velocity
U parallel to the x-axis, which coincides with the position of rest of the cylinder axis.
The cylinder is considered to be either fixed at the upstream end and free at the other,
as in Figure 8.1, or supported at both ends. Except for a short tapering piece by which
the cylinder is terminated at a free end (which wi1l be considered with the boundary
conditions), the cylinder has uniform cross-sectional area A, mass per unit 1ength m and
y
u
..
x
.,
u
..
Figure 8.1 A canti1evered cylinder in axial flow (Pai:doussis 1966a).
tFollowing dosely Hawthome's (1961) fonnulation for a towed cylinder (Section 8,9),

790
SLENDER STRUCTURES AND AXIAL FLOW
flexural rigidity E I. The x - and y-axes lie in a horizontal or vertical plane wherein all
motions of the cylinder are supposed to be confined,
We consider smalllateral motions of the cylinder about its position of rest, during which
the angle of incidence i and o i j o x remain sufficiently small so that (a) no separation occurs
in cross-flow, and (b) the fluid forces on each element of the cylinder may be assumed
to be the same as those acting on a corresponding element of a long straight cylinder of
the same cross-sectional area and inclination. Furthermore, the fluid is supposed to be
contained by boundaries sufficiently distant from the cylinder to have negligible influence
on its motion; later, we shall consider the case when either neighbouring structures or the
containing channel are not far away,
We suppose that the cylinder is given a smalllateral displacement y(x, t) from the
straight position, The resultant relative velocity between the cylinder and the fluid flowing
past it is given by (Lighthi1ll960a)
oy ay
v(x,t)=-+U-.
ot ox
Near the cylinder, this lateral flow is identical to the 2-D potential flow that would result
from the motion ofthe cylinder through fluid atrest with velocity v(x, t). We suppose that
this flow has momentum Mv per unit length of cylinder, where M is the added (virtual)
mass of the fluid per unit length, which is equal to pA for a circular cylinder in unconfined
flow, provided that the wavelength of motion is large in comparison with the diameter of
the cylinder [cf. Sections 7.2.6(a) and 2.2.2(a)J.t The rate of change of this momentum
per unit length is [(ajot) + U(ojax)](Mv) and gives rise to an equal and opposite lateral
force on the cylinder (Lighthil1l960a), that is, the inviscid force per unit length according
to slender body theory,
(8,1)
FA = (  + U ) [ M ( oy + U OY )] ,
ot OX ot OX
(8.2)
cf. equation (2,84).
In addition to the inviscid force, there are viscous friction and pressure forces acting
on the cylinder. Unlike the case of intemal flow in a pipe (Section 3,3), these two sets of
forces are generally not intimately related and do not cancel out in the final equation of
motion; indeed, in unconfined flow, the frictional forces on the cylinder neither control
nor substantially influence the mean pressure along the cylinder.
Although the viscous forces cannot be neglected,* they do not dominate the dynamics
in high Reynolds number flows either. Hence, it is possible, approximately at least, to
simply add them on to the inviscid forces, in the same way as the hydrostatic and gravity
forces [see also Hawthome (1961)J. This avoids embarking on a full viscous-flow analysis
of the problem, In this derivation, from hereon, the cylinder is supposed to be vertical, so
that x points in the direction of gravity.
The forces and moments acting on an element, ox, of the cylinder undergoing small
oscil1ations, y(x, t), are shown in Figure 8.2(a), where FN and FL are the viscous forces
tlt is implicitly assumed here that the displacement thickness of the boundary layer remains small in
comparison with the radius of the cylinder.
*The equations of motion would then be identical to those for a pipe conveying fluid, though the dynamical
behaviour is not (Pai:doussis 1987),

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
791
(a)
m e:;) ox
F;, OX
y
(b)
Figure 8.2 (a) An element 8x of the cylinder showing forces and moments acting on it; (b) an
equivalent rigid element surrounded by fluid (Pa'idoussis 1973a).
per unit length in the normal and longitudinal direction, respectively, and Fpx and Fpy
are the hydrostatic forces in the x- and y-direction, respectively; Q is the lateral shear
force, .M the bending moment, T the axial tension, and m the mas s of the cylinder per
unit length. Hence, the equations of motion in the x- and y-directions may be written as
aT ay
_ a +mg + h + (FN + FA) - - Fpx = 0,
x ax
aQ ay a ( Oy ) a2y
--FN-FA+F +h-+- T- -m-=O,
ox Py ax ax ox ot 2
(8.3)
(8.4)
where the fact that y and its derivatives are small has been utilized; the inertial forces
in the axial direction are of second order of magnitude for smalllateral motions of large
wavelength and have been neglected, Similarly, the effects of angular acceleration of the
element are neglected and, by elementary Euler-Bemoulli beam theory, we have
a.M a ( 02y ) * o ( 03 y )
Q = -  = - ax EI ox2 - E OX I ox 2 ot '
(8.5)
in which E is Young's modulu s and E* the corresponding viscoelastic constant, assuming
dissipation in the cylinder to be ofthe Kelvin-Voigt type (Section 3.3,2),
The next step is to determine the forces F Px and F Py' which are the resultants of the
mean pressure p acting on the outer surface of the element. Although they may be found
by integration, it is more convenient to determine them as follows. Consider the element
ofFigure 8.2(a) momentarily frozen and immersed in fluid on all sides as in Figure 8.2(b);
then, there wil1 be additional forces, pA and pA + [o(pA)fax]8x, on the other faces of
the element. However, the resultant ofthese forces and ofthe forces FPx8x, and Fpy8x is

792
SLENDER STRUCTURES AND AXIAL FLOW
known: it is the buoyancy force. Accordingly, assuming p == p(x) is a linear function of
x (which is reasonable, as it applies to both a hydrostatic pressure distribution and to one
modified by skin-friction pressure drop), one may write
[ a(pA) ] . [ a { ay }] J'{
-----a.;- - Fpx 8x1 + Fpy - ax (pA) ax 8xj = - :t;ťpndA
= - fIJ v p d(vol) = - ap A8xi.
vol ax
Hence,
-Fpx = - : A + axA) , Fpy = a: (PA : ). (8.6)
Thus, it has been shown that Fpy is equivalent to a tensile force, acting at the cross-section
of the cylinder, of magnitude pA, Note that, if aAjax = O, Fpx = O; for convenience,
account of this will be taken only later.
Substituting now equations (8.6) into (8.3) and (8.4), one obtains
a ( ap ) ay
-(T + pA) + mg - -A + FL + (FN + FA)- = O,
ax ax ax
a Q ay a [ a y ] a2y
- - FN -FA+h- +- (T+pA)- -m- =0.
ax ax ax ax at 2
(8.7)
(8.8)
As wil1 be shown, if y '" o (E), then F N and FA are of the same order, so that
(FA + FN}(ayjaX) '" O(E 2 );
neglecting these tenns in equation (8.7) and integrating it from x to L, upon combining it
with equatios (8,7), (8.8) and (8.5), one obtains
( * a ) a4y [ ap
E - + E I  + FA - (T + pA)L + mg(L - x) - -A(L - x)
at ax ax
l L ] a2y ( ap ) ay a 2 y
+ hdx -+FN+ mg--A -+m-=O,
x ax 2 ax ax at 2
(8.9)
where the axial uniformity of the cylinder has been invoked.
ln this equation, FA is as in equation (8.2), but the tenns (T + pA)L, apjax, FN and
FL have yet to be determined in explicit formo This is done first for the case of a solitary
cylinder in unconfined axial flow.
8.2.2 Solitary cylinder in unconfined flow
When the cylinder is isolated in a faidy large flow channel, the pressure drop due to flow
may be considered to be negligible, and
ap
-=pg,
ax
(8,10)

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
793
The inviscid hydrodynamic force is given by equation (8.2), where M = pA. (The more
genera1 case where M > pA is considered in Section 8.2.3.)
The viscous forces acting on long inc1ined cy1inders have been discussed by Tay10r
(l952a) in connection with fish propu1sion. For turbu1ent boundary 1ayers, Tay10r noted
that viscous forces will depend on the exact nature of surface roughness, Where it may
be assumed that roughness consists of a number of protuburences pointing equal1y in all
directions, he proposes that
FN = 1pDU 2 (Cf sin i + CDp sin 2 i),
FL = 1PDU 2 Cj cos i,
(8.11 )
where D is the cy1inder diameter, CDp and Cf are the coefficients associated with fonn
and friction drag, respective1y for a cylinder in cross-flow, and
i = tan- 1 (ay/ih) +tan- 1 [(ay/at)/U].
(8.12)
Hoerner (1965) compi1ed data on the viscous forces on cylinders inc1ined at ang1e i to
the flow and obtained expressions for the forces acting in the x- and y-directions: Fx =
1PDU 2 (CDp sin 3 i + Cf) and Fy = 1pDU 2 CDp sin 2 i cos i. Upon being transfonned to
the directions used here, that is, nonna1 and a10ng the cy 1inder axis, these reduce identically
to equations (8.11). It is of interest that Cf is given by
1 . [ (Drag force h ]
Cf = Jr lm ,
i--+O 1pU2 Jr DL
the Jr factor appearing since the skin friction coefficient is based on the surface area. It
is recognized that, at best, equations (8.11) give mean va1ues since, strictly speaking,
the coefficients vary from point to point with the deve10ping boundary 1ayer. Typically,
Cf = 0,01 to 0.03 for ReD ::: 10 4 (see Appendix Q).
For small motions, sin i « 1, equations (8.11) may be 1inearized and, using (8.12), the
viscous forces may adequately be represented by
1 ( a y ay ) 1 ay
FN = -pDUCN - + U - + -pDCD-,
2 at ox 2 at
1 2
FL = -pDU CT,
2
(8.13)
where the more genera1 fonnu1ation, with different nonnal and tangential coefficients, C N
and CT, has been adopted, The second tenn in FN represents a 1inear mean contribution
ofthe quadratic viscous forces at zero flow velocity, 2PDCDpl(ay/at)l(ay/at); this has
been retained, since all other viscous tenns vanish at U = ° [cf. Chen & Wambsganss
(1972)]; CD is not dimensionless.
Consider next the tenn (T + PA)L in equation (8.9), first for the case where the cyl-
inder is either free to slide axially or is completely free at x = L. Recalling that the pA
tenns arise from the pressure on the sides of the cylinder, and considering a slice of
the cylinder at x = L, one obtains TL = - PbA, where Pb is the base pressure; hence

794
SLENDER STRUCTURES AND AXIAL FLOW
(T + pA)L = (PL - Pb)A. Since Pb < PL generally, this is something like form drag,
more accurately referred to as base drag, which may be expressed as
1
(T + pA)L = -pD2U2Cb.
2
(8,14)
On the other hand, if the total length of the cylinder remains constant between fixed
supports, the situation is different. The fixed length requirement implies that at x = L,
there will be a compressive load -!pDU 2 CT(!L) - mg(!L), where the existence of
the second term depends on whether the length is fixed while the cylinder is horizonta1 or
not. There will a1so be a compressive load due to radia1 contraction; for a slender tubular
cy1inder this is equal to -2v(pA), where p is the mean va1ue of P at x =!L - cf.
Section 3.3,2, Finally, there may also be an externally imposed uniform tension T. Hence,
in this case,
(T+pA)L = - [!PDU 2 CT +mg - (apjax)A] (!L)+(1-2V)(PA)+T, (8.15)
Combining equations (8,14) and (8.15), and substituting into (8,9) along with
equations (8,2), (8.10) and (8. 13),t one obtains the equation of sma1llateral motions:
( a ) a4y ( a a ) 2 {
E*-+E I+pA -+U- y- 8[T+(1-2v)(]JA)]
at ax at ax
[ 1 2 ] [ I ] 1 2 2 } a2y
+ 2 pDU CT + (m - pA)g (1 - z8)L - x + 2 pD u (1 - 8)Cb ax 2
1 ( a y ay ) 1 ay ay a 2 y
+ -pDUCN - + u - + -pDCD- + (m - pA)g- + m- = O,
2 at ax 2 at ax at 2
(8.16)
where 8 = o signifies that the downstream end is free to slide axially, and 8 = 1 if the
supports do not allow net axial extension.
This equation is more general than that derived in Pai'doussis (1966a), as gravity and
pressurization effects have been taken into account. * Moreover, the frictional forces have
been taken into account in a systematic way, which eliminates an error in the original
derivation, arising from the absence of the term Fdayjax) in equation (8.8), which
effectively amounts to taking FN and FL to act in the y- and x- directions, respectively.
This erroneous interpretation of the frictional forces has unfortunately found its way
into others' work as well, for example, Ortloff & Ives (1969), Pao (1970) and Chen &
Wambsganss (1972), until fina1ly corrected in Pai'doussis (1970, 1973a),
8.2.3 Cylinder in a cluster or in confined flow
Here is presented an equation of motion for a cylinder in confined flow - confined by the
presence of adjacent cylinders or proximity to the flow-containing channel. The analysis is
approximate in many ways. One aspect that is stressed here is that the adjacent cylinders,
tIt is more proper to introduce equations (8,11) into (8,9) and then linearize, but this leads to the same result.
*These effects may be negligible in the case of systems with large E I, and L not too large,

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
795
IU
Figure 8.3 A flexible cylinder in a cluster of cylinders (PaYdoussis 1973a).
if any, are considered immobile; the case of a cluster of flexible cylinders is treated in
Chapter 9, while the rigorous treatment of truly annular flow is reserved for Chapter 11.
To fix ideas, however, we consider the case of a cluster of j cylinders, Figure 8.3, but
with only one of them free to oscillate, and then only in the {x, y}-plane,
ln this case, the pressure drop is clearly not negligible, but is closely related to friction
forces exerted on the cylinders and on the flow channel. To obtain an insight into the char-
acteristics of such a system one can look at the 'internaľ flow in the channel. Considering
a fluid element 8x of overall flow area Ach, one has, approximately,
ap
-Ach- - Ff + pAchg = o,
ax
(8.17)
where Ff is the total frictional force, the assumption having been made that the pressure
across any cross-section is uniform, Now Ff may be expres sed in terms of the shear forces
on all surfaces; for uniform conditions, denoting the cylinder surface area by Sj == S,
Ff  hj +Fo' <X (s+s",),
from which it can be concluded that equation (8.17) may be written as
ap ( Stot )
-Ach ax = h S - pAchg,
h = pDU2Cf'
(8.18)
with Stot being the total surface area per unit length.
Next, consider the term FA in equation (8.2). In accordance with the work in
Section 2,2,2 and Chapter 9 - see also Chen & Wambsganss (1972), Clasen (1972),
Paldoussis (l973a) - equation (8.2) holds trne, but now with M = XpA, in which

796
SLENDER STRUCTURES AND AXIAL FLOW
x = [(Dh + D2)/(Dh - D2)] > 1 increases with increasing confinement; Dch is the
diameter of the flow-channel.
ln conjunction with the viscous hydrodynamic forces, F Lhas already been expres sed
in terms of the pressure drop. Again, in the absence of better experimenta1 data, FN can
be taken to be re1ated to F L in a simi1ar manner as for an iso1ated cylinder: that is,
FN = h {[(ay/at) + u(ay/ax)] fU} + 1pDCD (ay/at).
(8.19)
Final1y, the term (T + pA)L wi11 c1ear1y have a simi1ar form as for an isolated cy1inder;
uti1izing equations (8.18) gives
(T + pA)L = - [1PDU2Cf (1 + D/Dh) + (m - pA)gJ (!L) + (1- 2v)(pA) + T
(8.20)
instead of equation (8.15), and equation (8.14) remains unaltered; Dh = 4Ach/ Stot is the
hydraulic diameter.
Therefore, by substituting equations (8.2), (8.14) and (8.18)-(8.20) into (8.9), after some
manipu1ation the equation of smalllatera1 motions of a cy1inder in a c1uster is obtained as
( a ) a4y ( a a ) 2 { _
E*- + E I---:i + xpA - + u- y - o [T + (1- 2v)(pA)]
at ax at ax
+ [PDU2Cf (1 +  ) + (m - PA)g] [(1- 1o)L -x]
1 2 2 } a2y 1 ( a y ay ) 1 ay
+ -pD u (1 - O)Cb - + -pDUCf - + u - + -pDCD-
2 ax 2 2 at ax 2 at
[ 1 2 ( D )] ay a 2 y
+ (m-pA)g+-pDU Cf - -+m=O.
2 Dh ax at
(8.21 )
Note that in the limit, as D/Dh --* O, and if CN = CT == Cf, this equation reduces
exactly to that of an iso1ated cy1inder, as it shou1d. Therefore, this equation may be
uti1ized, henceforth, as the general equation ofmotion, although equation (8.16) wi11 also
be used whenever CN i= CT.
8.2.4 Boundary conditions
It is convenient to use the following set of generalized boundary conditions:
a 3 y
E 1- + k oy = O
ax 3 '
a 3 y
EI3 - kLY = O,
ax
a 2 y ay
EI - - CO- = O
ax 2 ax
a 2 y ay
EI- +q- = O
ax 2 ax
at x = O,
(8.22)
at x = L,
in which ko and kL are trans1ational spring constants, and Co and q rotational ones, for
x = O and x = L, respective1y - cf. Section 3.6; from these, al1 the standard boundary
conditions may be obtained according1y as ko, kL, Co and CL are either zero or infinite.

SOLlTARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
797
ln the case of a free downstream end, however, a more general boundary condition
is considered, to treat cases of an ogival free end, as shown in Figure 8.1 for instance.
It is assumed that the cross-sectional area tapers smoothly from A to zero in a distance
sufficiently short, so that y and the lateral velocity v may be considered constant. This
requirement aHows the forces acting on the tapered end to be lumped and considered
in appropriate boundary conditions (Hawthome 1961). Equating the lateral shear and
inertial forces to the rate of change of lateral momentum over the tapered end, say for
L -l < x < L, and neglecting frictional and gravity effects for now, we obtain
j L aQ l L ( a a ) l L a2y
-dx - f - + u- [M(x)v] dx - m(x)-dx = O,
L-l ax L-l at ax L-l at 2
where v is given in (8.1). The parameter f, which is equal to unity for slender-body,
inviscid-flow theory, has been introduced because the theoreticallateral force at the free
end may not be fuHy realized as a result of: (i) the lateral flow not being truly 2-D, since
the fluid has opportunity to pas s around rather than over the tapered end (Munk 1924);
and (ii) boundary-layer effects (Hawthome 1961). Accordingly, f will normaHy be less
than unity, [Munk (1924) shows that the momentum of the lateral (inviscid) flow about
an inc1ined rigid body of revolution is proportional to Ky - Kx, where Ky and Kx are,
respectively, the lateral and axial virtual masses of the fluid for the whole body, For
ellipsoids of revolution, Lamb (1957) has calculated that Kxl Ky = 1 for LI D = 1, and
Kxl K y = 0.022 for LI D = 9.47. Thus, for elongated bodies, Kx may be neglected in
comparison with K y' For the flexible body under consideration, however, we are dealing
with distributed virtual masses along the length, Over the cylindrical portion of the body
the axial virtual mass is, of course, identically zero, and the lateral virtual mas s per unit
length is M. For the tapered end-piece, on the other hand, the lumped, lateral virtual mass
Kye which equals !Z_IM(x) dx is not necessarily much greater than the corresponding
axial virtual mass Kxe since liD is not always large. Accordingly, it is c1ear that f, which
may now be assumed proportional to 1 - Kxel Kye may be considerably less than unity,
quite apart from boundary-layer effects. See also Paidoussis & Yu (1976).]
We now write M(x) = XpA(x) and m(x) = PbA(X) and, since y and v are considered
constant over the tapered end, equation (8,23) leads to
(8.23)
f L aQ j L dA a 2 y j L
-dx-fxpUv - d dx-(Pb+fxp)----z A(x)dx=O,
L-{ ax L-l x at L-l
which yields
a 3 y ( a y ay ) a2y
EI-+fMU -+U- -(m+fM)x - =0
ax 3 at ax e at 2 '
(8.24a)
where
l i L
xe = - A(x) dx,
A L-l
Furthermore, to first order, it may be assumed that the bending moment at the free end is
zero, so that
M = XpA.
a 2 y
'2=0.
ax
(8.24b)

798
SLENDER STRUCTURES AND AXIAL FLOW
It is understood, of course, that 1/ L « 1, so that the boundary conditions may be
considered to apply at x = L.
If gravity and friction effects are taken into account, a more careful derivation
(Lopes et al. 1999a, 2002) yields
03y ( Oy Oy ) [ ( 02y a2y ) 02y ]
EI-+fMU -+U- - fM -+U- +m- Xe
ox3 ot OX ot 2 axot ot 2
oy 1 2 D oy 1 ( OY OY ) _
- (m - pA)g-X e - -pDU Cr--Xe - -pDUCN - + U- Xe = O,
OX 2 Dh OX 2 ot OX
(8.25)
where
1 [ L
Xe = - D(x) dx.
D L-l
An interesting connection between the principal term in (8.24a) and (8.25), namely
fMU[(oy/ot) + U (oy/ox)] == M*Uv, and the lift on a low-aspect ratio (slender)
wing has been pointed out by Triantafyllou (1998); here v is the lateral velocity and
M* = f M. This lift, on a wing of chord C and span s, with aspect ratio AR == s /c « 1,
is L=!pU 2 ACL, in which A=cs. Taking CL :::::: (dCL/d1/1)1/1 :::::: (dCL/d1/1) (v/U)
and reca1ling that dCL/d1/1 :::::: !7l'(AR) from slender wing theory,t it is seen that
L =  ps2 U V = M* U v, where M* is the added mass of the wing. This is identical, there-
fore, to the expres si on for the cylinder-tail boundary condition obtained by slender-body
theory. Furthermore, slender wing theory predicts (Katz & Plotkin 1991, section 8.2,2)
that there is no lift if the wing form is not tapered (e.g. as it is for a delta wing); this is
compared to (8.23) and to the equation preceding (8.24a), and here again it is seen that a
rigid 'end-piece' has to be tapered in order to generate any lift.
8.2.5 Dimensionless parameters
ln order to render the equation of motion and the boundary conditions nondimensional,
the following dimensionless terms can be defined:
=x/L, TJ=y/L, r:={EI/(m+pA)}1 / 2 t /L 2 .
(8.26)
Substitution of these terms into equation (8.21) yields the dimensionless equation of
motion,
0 5 TJ 04TJ { 2 [ 1 2 ] [ 1 ]
a o40r: + o4 + XU -8[r+(1-2v)IT]- "2€CfU (1+h)+y (1-28)-
1 2 } 02TJ 1 / 2 o2TJ [ 1 2 ] OTJ
- -(1- 8)cb U - + 2xf3 u- + -€CfU (1 + h) + Y -
2 02 oor: 2 o
[ 1 1 12 ] OTJ 02TJ
+ -€Cff3 1/2 u + -€cf3 I - + [1 + (X - 1)f3] - = O
2 2 or: or: 2
(8.27a)
tS ee Jones (1945) and Hoemer (1985, p, 17,2) and refer to Katz & Plotkin (1991; Section 8.2).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
799
in the general case, For the case of a solitary cy1inder when CN i= CT generally,
equation (8.16) gives
( o ) 04ry { 2 [ 1 2 ] [ 1 ]
a o. + 1 0;4 + u - o [1 + (1 - 2v)n] - 2SCTU + y (1 - 2 0 ) -;
1 2 } 02ry 1/2 02ry [ 1 2 ] Ory
- - (1 - O)Cb U - + 2,8 u - + -SCNU + Y -
2 0;2 0;0. 2 o;
+ [ !SCN,81/2 u + !SC,81/2 ] ory + 02ry = O.
2 2 (Ji or 2
In these equations, the following dimensionless parameters have been used:
[ I ] 1/2 E*
a= E(pA+m) L2 '
S= L , u = ( PA ) I/2 UL,
D EI
(8.27b)
pA
,8 = pA + m '
_ pAL 2
n- ,
EI
(m - pA)gL 3
y= EI
1'L 2
1=-
EI'
D
h=-
Dh'
4
cf = -Cf,
7r
(8.28a)
4
CN=-CN,
7r
4
CT = -CT,
7r
4
Cb = -Cb,
7r
4 ( PA ) I/2
C = -; EI LCD; t
o and X have been defined with equation (8.16) and following equation (8,18). In the
solutions to be obtained, the dimensionless complex frequency (J) crops up; it is related to
the dimensional radian frequency Q by
( pA +m ) I/2
(J) = QL 2 .
EI
Similarly, the standard boundary conditions become
03ry 02ry * ory
0;3 + Kory = 02 - KO o = O
03ry a 2 ry *ary
K ry +K O atl:c=I' ,
a3 - 1 = a2 I o; = s
for a tapered free end, the special boundary conditions at  = 1 are either
03ry ( 1 ) ( 1/2 ary 2 ary )
- + xl - -x SCN ,8 u- + u -
0;3 2 e a. a;
[ 02ry a 2 ry ]
- Xe [1 + (xl - 1),8] or 2 + xI,81/2u o;a.
( 1 2 ) ory a 2 ry
- Xe 2 SCTU h + Y O = 02 = O,
(8.28b)
at ; = O,
(8.29a)
(8.29b)
tNote the correction in the definition of c - the 4/ Jr factor - vis-a-vis Pa'idoussis (l973b), undetected for
some time, as c = O had normally been used in calculations; corrected, but not discussed, in Pa'idoussis & Suss
(l977a) and subsequent work.

800
SLENDER STRUCTURES AND AXIAL FLOW
corresponding to (8.25), or the simplified fonn used in the earlier studies,
a 3 1/ a1/ a1/ a 2 1/ a 2 1/
- + xlu 2 - + xlf3 1/2 u- - {l + (xl - 1)f3} x - = - = O
a3 a a-r e a-r 2 a2 '
(8.29c)
where
koL3
KO=-,
E/
Xe
Xe =-,
L
kLL 3
Kl =:. E/'
Xe
X e =-.
L
COL2
K*--
0- E/ '
* cLL 2
K 1 =-,
E/
(8.30)
8.2.6 Discussion
If gravity, tensioning, pressurization, and dissipative effects are absent, the simple form
of equation (8,16) may be written as
a 4 w a 2 w a 2 w a 2 w
E/- 4 + MU 2 - 2 +2MU- + Fv + (M+m)- =0,
ax ax axat at 2
(8.31)
whereby the similarity to the equation of motion of a pipe conveying fluid, say equa-
tion (3.1), becomes evident. Indeed, the only difference is the presence of the viscous
forces for extemal flow, represented by Fv in (8.31). These forces render the system
nonconservative, even if both ends are supported and internal dissipation is neglected,
They nevertheless do not radically alter the theoretical dynamical behaviour of the system
vis-u-vis that with internal flow (Pai:doussis 1987), especially in the case of a cylinder
with supported ends - although in experiments the predicted post-divergence flutter does
materialize in this case. In the case of cantilevered cylinders, the analogy is less clo se,
because of real flow effects at the free end.
Examining equation (8.21) or (8.27a,b), it is clear that all parameters are easily meas-
urable, with the exception of the viscous coefficients cf (or CN and CT), Cb and c. Some
comments on plausible ranges ofvalues for the physical systems considered later are given
in Appendix Q.
Final1y, it should be said that the equations derived here apply to the simplest systems
under consideration in this chapter. Other forms, applying for example to tapered cylinders
(cones) or towed systems, are derived in the pertinent sections,
8.2.7 Methods of solution
The methods of solution are quite similar to those of Section 3.3.6, and hence, will be
given here in highly abbreviated fonn.
8.2.7(a) Systems with classica! boundary conditions
For cylinders with ideally clamped, pinned or free-end boundary conditions (in the latter
case with Xe = X e = I = O), the Galerk.in method may be used.
Considering a solution ofthe fonn 1/(, -r) = Ll q,r()qr(-r) as in (3.85), with q,r()
the appropriate comparison functions, and following the Galerk.in procedure, one would

S0U1ARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
801
eventua1ly obtain the equation
[M]ij + [C]4 + [K]q = O,
(8.32)
where q = {ql, Q2, . . . , qN}T; in the case of equation (8.27a), but more genera1ly with CN
and CT rather than cf, the elements of the matrice s are
M sr = [l + (X - 1),8]8sr,
C sr = [aJ...; + !€CN,81/2 u + !€c,81/2] osr + 2X,81/2ub sr ,
Ksr = J...;Osr + {xu 2 - o [r + (l - 2v)n] - !(1 - O)Cb u2 } C sr
+ [! (€CN + €CTh) u 2 + Y] [b sr - (l - O)Csr + d sr ]
(8.33)
- cf. equations (3.86)-(3.89); b sr , C sr and d sr are defined in (3.87), and osr is Kronecker's
delta.
ln the case of KO = KO = Kl = Kr = O, the comparison functions CPr() are the eigen-
functions of a beam with the appropriate boundary conditions, J... r are the corresponding
eigenva1ues, and b sr , C sr and d sr are tabulated in Table 3.1. If, however, any of the K
are nonzero, then the comparison functions have to be determined first (see Section 3.6,
for instance); a1tematively, the extended Galerkin or the Rayleigh-Ritz method could be
utilized (see the next subsection or Section 7.3.2).
8.2. 7(b) Canti/evered system with tapered free end
The solution in this case is more complicated because one of the boundary conditions is
time-dependent (cf. Section 2.1.4).
ln the origina1 ana1ysis (Pai'doussis 1966a), a power series solution is used, 11(, T) =
(Lo Arr) exp(iú>T). This is substituted in both the equation of motion and the four
boundary conditions. Collecting like powers of  from the equation of motion, and truncat-
ing at r = n, one obtains n linear equations in Ar, but it is possible to obtain a recurrence
relationship, so that they can a11 be expressed in terms of A2 and A3. Then, the two bound-
ary conditions at  = 1 lead to a 2 x 2 detenninant from which ú) may be detennined
for given parameter va1ues. The method is quite efficient, although 40-60 terms may be
necessary in the expansion to achieve convergence. (Application of this method predated
symbolic manipulation computer packages; but now that they are available, determining
the higher Ar in terms of A2 and A3 would be quite easy.)
A more 'modem' method, which reduces the solution to an eigenvalue problem is
given next, namely the extended Ga1erkin method. As the cantilevered system is con-
sidered, o = O in this case. Expressing the solution as TJ(, T) = Ll CPr()qr(T), with
CPr() the eigenfunctions of a cantilevered beam, it is observed that a11 boundary condi-
tions are satisfied, except (8.29b) or (8.29c), Hence, the boundary-va1ue problem may be
expressed as
F(TJ(, T» + o( - l)B(TJ(, T» = O,
(8.34)
where F (11) is the equation of motion, B (11) is boundary condition (8,29b), and o ( - 1) is
Dirac's delta function. Applying the Ga1erkin procedure to equation (8.34), for CN =1= CT,

802
SLENDER STRUCTURES AND AXIAL FLOW
leads to an equation of the form of (8.32), but with
M sr = [1 + (X - 1),8] 8sr + [1 + (fx - 1),8] Xe4>s (l)4>rO)
C sr = [a).,; + !SCN ,81/2u + !sc,81/2J 8sr + 2X,81/2ub sr
- (xf - !XeSCN) ,81/2 u 4>sO)4>r(l) + Xexf,81/2 u 4>s(l)4>;O),
Ksr = )";8sr + xu 2 c sr - !CbU2Csr + [!(SCN + sCTh)u 2 + yJ b sr
+ [!SCTu2(l + h) + yJ (d sr - c sr )
+ [! (SCN Xe + sCThXe) u 2 - xfu 2 + Y Xe ] 4>s(l)4>;O),
(8.35)
ln the results to be presented, unless Xe is specified, the simpler boundary conditions
(8.29c) are used instead of (8.29b), with attendant modifications in (8.35).
8.3 LlNEAR DYNAMICS
ln this section the dynamics of cylinders in axial flow is discussed, presenting results
obtained with the theoretical model of Section 8,2, followed by a description of experi-
ments on the same, in which the general character of the dynamical behaviour is stressed;
nonlinear aspects are covered in Section 8.11. A discussion of the mechanisms of instabil-
ity is presented in Section 8.3.3. The treatment of a number of special topics then follows,
for example, the dynamics of strings in axial flow.
8.3.1 Linear dynamical behaviour
8.3. 1 (a) Basic dynamics for cylinders with supported ends
The dimensionless complex frequency w == Re(w) +iIm(w) for a typical pinned-pinned
(simply supported) system, t plotted as an Argand diagram with the dimensionless flow
velocity u as parameter, is shown in Figure 8.4+ for the lowestthree modes ofthe system. It
is evident that small flow velocities damp free motions ofthe cylinder [Im(w) > O] - see
discussion in Section 2.3, For sufficiently high flow velocities, the frequencies of all three
modes successively become purely imaginary, bifurcating on the Im(w)-axis; for the
sake of clarity, the evolution of the mode loei on that axis is shown schematically in the
diagrams on the left-hand side of the figure. The frequency associated with one branch
of each mode eventual1y vanishes altogether, indicating the onset of static divergence
(buckling) via a pitchfork bifurcation,
The first divergence occurs at u = 3,137, in the first mode, followed at u = 6.312 by
divergence in the second mode. However, at s1ightly higher flow velocity, u :::: 6.48, the
tThe rather low value of f3 in this figure is typical of natural uranium-oxide-filled cylindrical cans of the type
used as fuel elements in the CANDU-type nuc1ear reactors [Figure g,135(b)]; the dynamics of such systems
was one of the prime motivators for this work,
*The unusual scales arise from plotting the square roots of Re(w) and Im(w) on a linear scale, for
convenience of presentation,

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
803
negative branches of the first- and second-mode loci coalesce and leave the axis at a point
where Im(w) < 0, indicating the onset of coupled-mode flutter. The other branches of
these modes behave similarly, leaving the axis where Im(w) > O, Finally, at u é::: 9.45
there is divergence in the third mode, Here the reader is referred to Sections 3.4 and 5.5.
In particular, it is stressed yet again that linear theory is capable of predicting only the
point of first loss of stability, in this case Ucd = 3.137. Whether the other bifurcations
(higher-mode divergence and coupled-mode flutter) materialize has to be ascertained by
nonlinear theory and tested by experiment (Sections 8,3.2 and 8.11),
The dynamics of a system with different system parameters (higher f3 and lower BCf) is
illustrated in Figure 8.5. In this case also, first-mode divergence occurs for u é::: 3.14. t lt is
followed by coupled-mode flutter via recoalescence of the two bifurcated branches of the
first mode, rather than be coalescence of first- and second- mode branches as in Figure 8,4.
Moreover, the role of the positive-Im(w) branch of the coupled mode flutter is assumed
by the second mode for u > 6.5. This, therefore, is another instance of 'mode veering' or
'mode exchange' discussed at length in Volume 1.
It is of interest to compare the dynamical behaviour il1ustrated in Figures 8.4 and 8,5 to
that predicted in Paldoussis (1966a) in which, it is recalled, the viscous forces are resolved
erroneously. The system loses stability in its first mode in essentially the same manner and
at the same value of u according to both sets of calculations. However, the post-divergence
characteristics are different: in PaYdoussis (l966b), coupled-mode flutter is replaced by
single-mode flutter involving, in turn, the second and then the third mode; quantitatively,
the critical flow velocity for flutter, ucf, is lower in Paldoussis (1 966b ) by about 10% for
the system of Figure 8.4.
Next, the dynamical behaviour of a system in confined flow, as described by equa-
tion (8.21), is il1ustrated in Figure 8.6 - basically for the same parameters as in Figure 8.4,
but with X = 4 and h = 1.5, reflecting the confinement. lt is noted that, qualitatively, the
dynamics is broadly similar, Of particular interest are: (i) the dramatic reduction of Ucd
from 3,14 to 1.57 approximately; and (ii) the increase in the flow-induced damping.+
The onset of coupled-mode flutter is also lower, at ucf é::: 3.14 rather than Ucf é::: 6.31.
lnterestingly, the critical values for divergence scale with X as u é::: Jr / "jX, and similarly
for coupled-mode flutter (cf. Section 3.4) - refer to equation (8.36) and the discussion
around it. Moreover, in this case of higher confinement, there is an additional coupled-
mode flutter, involving the third- and fourth-mode loci (the fourth-mode locus has been
suppressed in the figure, for clarity).
8.3, 1 (b) Basic dynamics for canti/evered cy/inders
The dynamics of a cantilevered cylinder with a fairly well-stream1ined free end (f = 0,8)
is illustrated in Figure 8,7. Free motions are damped at small u. At sufficiently high u,
however, the cylinder first loses stability by divergence in its first mode, at u é::: 2,04, The
system is then restabilized (note that the first-mode locus on the negative Im(w)-axis
doub1es back to stability), and then loses stability by single-mode (second-mode) flutter at
u é::: 5.16; it is restabilized at u é::: 8,6, but, before that, it loses stability at u é::: 8.17 in the
third mode. The important qualitative difference in the dynamics of this system compared
tDivergence, and hence Ucd, is independent of p, but genera11y not of sCJ.
*Here it is recalled that  = Im(w)/Re(w); a1though Im(w) is lowerin this case, Re(w) decreases with u
much faster.

52
3,5
4 2
---
6445
3 2
6.344
2 2
6.312 12
---
3.14
s
 O

6312
_1 2
9.46
3,5
6.344
_2 2
9.5
45
6.437 _3 2
6.25 _4 2
_52
O
60
96
6.45
I st and 2nd mode
6297 9.445
Values or u

9.375 9
7
"
1
'"
8
3 2
6
5
O
1st mode
2nd mode
3rd mode
..!:!

c

6.5
1st and 2nd mode (coupled-mode flutter)
6,75
1 2
10 2
4 2 52 6 2
911.-(m)
2 2
3 2
Figure 8,4 Argand diagram for the complex frequencies, úJ, of the lowest three modes of a
solitary pinned-pinned cylinder in unconfined axial flow, as functions of u, for f3 = 0,1, 8Cf = I,
/) = X = I, c = ex = h = y = rr = r = ° (Pai:doussis 1973a),
4 2
5 6 3 2
4
6,25 2 2
325
6,281 "
,2 1
31406 '"
 O

1st mode "
_1 2 
3.25 6.281 
_2 2
625
4
5 6 _3 2
-4 2 52
O 1 2 2 2 3 2 4 2 6 2 7 2 8 2
911.-(m)
Figure 8.5 Argand diagram for úJ, similar to that of Figure 8.4, but for f3 = 0.48, 8Cf = 0,25,
/) = X = 1, c = ex = h = y = rr = r = ° (Pai:doussis 1973a),

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
805
6,28
52
4,75
4 2
32

3.14  2 2
1 st and 2nd mode
175
o
o
"
2 
'"
4,6875 4.5 4
3 3
O O
2nd mode 3rd mode
"

:5
1 2 3,1418
1.53
1.5
1.578
1 st mode
1.75
1 st and 2nd mode
(coupled-mode flutter)
4.75
3rd and 4th mode
(coupled-mode flutter)
"_ 5
52 6 2
;w"-(w)
10 2
Figure 8.6 Argand diagram (cf. Figure 8.4) for a pinned-pinned cylinder in confined axial flow,
showing the complex ev as a function of u, for f3 = 0.1, BCf = 0,25, X = 4, h = 1.5,/) = I,
C = ex = y = IT = r = O (Pai'doussis 1973a).
to that of a pipe conveying fluid is remarked here: in the internal flow system (Section 3.5)
there is no divergence prior to flutter.
The results obtained by Pai'doussis (1966b) in this case are qualitatively similar, though
the critical velocities are higher for divergence (Ucd :: 2.6), and lower for flutter (ucf ::::
4.78 for the second mode, and similarly for the third mode).
The Argand diagram for another system, with a higher BC f, is shown in Figure 8.8. The
main point of interest here is that, after divergence, the system is subject to coupled-mode
fiutter [of the Pai'doussis variety, see Figure 3.4(d)], rather than single-mode flutter - so
that, at onset, the frequency of oscillation is zero. Only the first mode locus is shown in
the figure; the higher modes have been suppressed for c1arity.
Calculations for a cantilevered cylinder with f = 1 generate complex frequency dia-
grams similar to those of Figure 8.4 for a cylinder with supported ends! This is not
unreasonable, however, as in this case the only nonconservative forces acting on the sys-
tem are the viscous hydrodynamic ones, which is also trne for cylinders with supported
ends. This is further discussed in Section 8.3.3.
Finally, Figure 8.9 shows in an encapsulated form the evolution of the dynamics of a
system as f is varied - in terms of eigenvalue diagrams - illustrating several mode-
veering, mode-exchange occurrences (cf. the discussion around Figure 3.29); see also
Figure 8.64. In Figure 8.9(a), it is seen that for f = 0.7 flutter occurs in the second mode

60
525
50
8,81
4,0
s
 5
O O
5.25
-10
2,5
8,5
45 8,5
-2 O 30 5,5
4.0
-3,0
O 10 20 30 40 50 60 70
(w)
6
Figure 8.7 Argand diagram of the complex frequencies, w, of the lowest three modes of a soI-
itary cantilevered cylinder with a tapered free end in unconfined axial flow, as functions of u, for
fJ = 0,5, ECf = 1,8 = O, X = I, f = 0.8, Xe = 0.01, Cb = Y = O (Pai'doussis 1973a).
12
04-- 10 u= 618
8
6
u = 2,98
4
 u = 2.18
 2 u = 1.47
u = 7,77 u =0.
-4 -2 O u=249 2 4 6 8
u = 7.90 e----+ -2
U=7.48
(w)
Figure 8.8 The complex frequencies, w, of a solitary cantilevered cylinder with a tapered free
end in unconfined axial flow, as functions of u, for fJ = 0.47, f = 0.8, ECf = 2.50, Cb = 0.2,
Xe = 0.00667, Xe = 0.00785, y = O.

SOLITARY CYLINDRICAL STRUcrURES IN AXIAL FLOW
807
100
50
100

 50
o
-40
-20
o
20
40
(c)f=O.4
20
u = 15.47
100
100
14,77


 50

50
15,54
o
-40
o
-20
(A)
o
-40
-20
o
20
o
20,59
u = 16,37
-40
-20
o
20
(A)
Figure 8.9 Argand diagram s in tenns of the eigenvalues A = iw, showing the effect of f on
the evolution of the loci and on stability for a cantilevered system with f3 = 0.5, X = 1, h = O,
sCf = 1, Xe = 0,00667, Xe = 0.00785, Cb = Y = O (Lopes et al. 1999b).
at U c f2 = 5.48 and in the third at U c f3 = 8.56, In Figure 8.9(b) for f = 0.5, flutter occurs
in the third mode fint, at U c f3 = 9.17, then in the second at U c f2 = 15,18 and the third
again at uf3 = 15.24. In Figure 8,9(c) for f = 0.4, flutter occurs in the third mode first
at U c f3 = 9,77, then in the second (U c f2 = 15.54) and the fourth (U c f4 = 15.47). Last,
for f = 0.1, we have U c f3 = 16,37 and U c f2 = 20.59. Ť
8.3. 1 (c) Effect of system parameters; cylinders with supported ends
The effect of system parameters on the stabili ty of the pinned-pinned system is summarized
in Tab1e 8.1. The so-cal1ed 'standard case' is for f3 = 0.1, BCf = 0.25, 8 = X = 1, C =
Y = ex = n = r = h = O; whi1e, for the other entries, on1y the parameter changed
vis-a-vis the standard case is given. Bere Ucdl and U c d2 represent the critica1 ve10cities
for first- and second-mode divergence, whi1e ucf is the critica1 ve10city for coup1ed-mode
flutter.
tThe high values of critical u, for example, ucf = 15.54, are probably not very accurate, as convergence (in
tenns of sufficiency of the number of modes used in the Galerkin expansion) was not tested in these specific
calculations.

808
SLENDER STRUCTURES AND AXIAL FLOW
Table 8.1 Effect of parameters on stability of pinned-pinned
cylinders in axial flow (Pai'doussis 1973a),
Parameter
ucf
Ucdl
U c d2
Standard case
£cf = 1
£c = 0.25
a = 0.003
{3 = 0.48
y = 10
rr = 5, 1) = 0.3
r = 10
8=0
h = 1.5, X = 4
3.143
3.137
3,143
3,143
3.143
3.107
3.444
4.455
3.246
1.572
6.28
6.31
6,28
6.28
6.28
6.30
6.44
7.04
6.49
3.142
6.41
6.48
6.41
6,52
6.28
6.44
6.55
7.11
6.60
3,143
Within the range of parameters covered in the table, it may be said that the effect on
Ucdl - that is, on the principal instability - is relatively smal1, Ucdl :::: Jr being a
reasonable approximation for solitary cylinders in unconfined flow, when n and r are
smal1. The second critical flow velocity for divergence may similarly be approximated by
U c d2 :::: 2Jr, and the onset of coupled-mode flutter may be considered almost coincident
with that. Indeed, for the first divergence, simply denoted here by Ucd, al1 the results in
Table 8,1 are found to satisfy
V;d = XUd - [r + (1 - 2v)ll] 8:::: Jr2
(8,36)
- cf. equation (8.27a). For X = I and 8 = I, this relationship is similar to that in
equation (3.100) for internal flow - but note the change in the sign for the pressurization
effect. AIso note that, for internal flow, v;d = Jr2 exactly, whereas here this is only
approximately true, because of the presence of the viscous terms.
It is a1so noted that, if 8 = O, the whole of the cylinder is under tension due to the
distributed longitudinal viscous drag, while for 8 = I only the upstream half is, the other
half being under compression. As a result, Ucd and ucf are al1 higher for 8 = O.
Qualitatively similar results may be found for clamped-clamped cylinders, though in
this case Vcd :::: 2Jr, None are given here in the interests of brevity.
8.3. 1 (d) Effect af system parameters; cantilevered cylinders
Tbe overal1 effect of f on stability of a typical cantilevered system is shown in Figure 8.10.
Tbe effect is profound: for sufficiently smal1 f and Cb = 1 - f, that is, for a sufficiently
blunt end, for example f = 0.3 and Cb = 0.7, there are no instabilities at al1 for u :::: 16.
A threshold value of f for such behaviour depends on the values of the other parameters,
but it always exists, For a slightly larger f, say f = 0.35 in this case, the first-mode
divergence and second-mode flutter are missed out, the first instability encountered being
third-mode flutter; there is also a higher-mode divergence within this flutter boundary.
For higher f, for example f = 0.6, the dynamical behaviour with increasing u is much
as shown in Figure 8.7 t for a different set of parameters: (i) the system loses stability by
tPor this system, cantilevered or with both ends supported, SDme Df the higher-mode instabilities do
materialize, unlike the interna! flow system; hence this discussion is not entirely academic,

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
809
16
14 -
12
]O
u 8 Stable
6 -
A
4
2
o
O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
f
Figure 8.10 The dynamical behaviour of a cantilevered system (tJ = 0.47, sCf = 0,5, (CN =
CT == cf), 8 = O, X = 1, Cb = 1 - f, Xe = 0.00667, Xe = 0.00785, y = 1.9, C = O) as afunction
of f: - , divergence boundaries; - , flutter boundaries. The significance of A, B, C is made
clear in Section 8.11.1(a); (Semler et al. 2002),
divergence; (ii) it regains stability; (iii) it becomes subject to second-mode flutter; (iv) it is
restabilized; (v) it develops third-mode flutter; and (vi) this is followed by more complex
dynamical behaviour. For stilI higher f, for example, f = 0.85, there is no restabilization
after the initialloss of stability.
Figure 8.II(a) shows the effect of tJ on stability for a system with f = 0.7 and otherwise
the same parameters, other than tJ, as in Figure 8,10. The shaded regions are stable. It
is recalled that, sinůlarly to f, the possible range for tJ is O < tJ < 1 - although,
realistica1ly, it is from '"'-'10- 3 for a light-gas flow over a heavy cylinder to '"'-' 0.7 for a
dense fluid flowing over a hollow cylinder; for tJ > 0.7 approximate1y, the cylinder would
begin to behave as a shell. Consider the case of tJ = 0.49 first. The system remains stable
up to u = 2,1, losing stability by divergence; it is restabilized at u = 5.2, and then loses
stability by flutter at u ::::= 5.6; after that, it never regains stability, being subject to different
types of flutter and divergence. For larger tJ, there may be a third range of stability. For
tJ  0,38, on the other hand, the system never regains stability after divergence, going
directly into second- and higher-mode flutter. The horizonta1line at u = 9,6 represents
the onset of the second zone of divergence, and u = 12.8 marks its cessation. Note

810
SLENDER STRUCTURES AND AXIAL FLOW
16
14
12
10 
'" .. ,.., .., ...." .... ../' ..,....,

" 8
6
A
4
2
o
(a)
10
8
'c>
"
Stable
4
0,0 0.2 0.4 0,6 0,8 1.0
(b) tJ
Figure 8.11 (a) The dynamical behaviour ofthe same system as in Figure 8.10, but with J fixed
at J = 0.7 and f3 variable; the shaded areas are zones of stability, (b) The dynamical behaviour
of essentially the same system but with 8cf = 1 and y = O, for several values of J, as fol-
lows: -, J = 0.5; - - -, J = 0.6;-, - '-, J = 0.7; "', J = 0.8;-.. - "-, J = 0,9
(Semler et al. 2002).
that static instabilities are independent of {3. The perplexing figure-of-eight behaviour is
clarified in Figure 8.1l(b), where it is seen how it evolves with varying f, initially as
a single instabi1ity-restabi1ization loop for f = 0.5 and 0.6 to a figure-of-eight closed
loop for f = 0,8 in this case of £cf = 1 (whereas £cf = 0,5 in Figure 8,ll(a), and the
figure-of-eight behaviour arises for f = 0.7).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
811
The effect of f together with 8CN and 8CT, for CN ICT = !, I and 2, is shown in
Figure 8.12 for divergence and Figure 8.13 for flutter. In Figure 8.12, in addition to the first
divergence zone, one can see a second and the beginnings of a third one; cf. Figure 8.II(a).
On the other hand, only the first (second-mode) flutter zone is shown in Figure 8.13. In
Figure 8.13( a,b) it would appear that flutter abruptly disappears for high enough eCN when
f is high. This is not trne, The results in Figure 8,13 have been obtained using AUTO
(Doedel & Keméves 1986), which can only detect single- mode flutter, that is, flutter via a
Hopf bifurcation. For high ec N and f, however, the flutter is of the coupled-mode variety,
such as shown in Figure 8.8. In one case, for f = 0.8 in Figure 8. 13(a), the values of ucf
for coupled-mode flutter have been added (dotted line), and are seen to follow smoothly
those for single-mode flutter. This is trne for all cases, but the ca1culations are tedious and
the curves for other f have not been similarly extended,
It is dear from Figures 8.12 and 8.13 that, provided eCN is sufficiently small, say
eCN < 0.2, its effect on both Ucd and ucf is quite small, as to a lesser extent is the effect
af varying C NI CT. This, however, is not trne for very small f and for larger values of ec N ,
when both eCN and CN ICT influence the dynamics quite significantly.
ln general terms, increasing eCN stabilizes the system in both divergence and flutter.
For high enough 8CN and low enough f, divergence may be suppressed altogether, for
example, for eCN = 2 and f = 0.5 in Figure 8.12(b,c). This is also trne for flutter. AIso,
again generally, increasing CN ICT stabilizes the system for divergence [e.g., compare Ucd
fOUCN = 2 and f = 0.7 in Figure 8.12(a--c)); this stabilization becomes more significant
athighervalues of eCN, for example, for eCN = 6. On the other hand, increasing CN ICT is
destabi1izing forflutter [e.g., compare U c ! fOUCN = I and f = 0.6 in Figure 8. 13(a,b,c)).
However, there are exceptions to these genera1 observations, for example, because the
instability curves become do sed in some cases, as in Figures 8.12(b,c) and 8. 13(b,c).
8.3. 1 (e) Divergence of pinned-free cylinders
The dynamics of a cylinder pinned at the upstream end and free downstream is investig-
ated by Triantafyllou & Chryssostomidis (1984), as an idealization of a towed-cy1inder
system - see Section 8.8. The simple-support, rather than clamped, upstream boundary
condition allows the analytical determination of the critical flow velocity for divergence,
using Airy functions (Abramowitz & Stegun 1970),
E1iminating the time-dependent terms from equation (8.27b) and taking y = Cb
0,8 = O and CN = CT = cf, the equation is simp1ified to
rl"" + u 2 [ I - !eCf(1 - )] rl" + !ecfU 2 TJ' = O,
(8,37)
where O' = dO/d, Similarly, the shearboundary condition (8,29c), for X = I, Xe -+ O,
becomes
TJ'" + fu 2 TJ' = O
at  = I,
(8.38)
where it has been presumed that, in this case also, the cylinder is generally terminated by
a tapered, ogival end. The other boundary conditions are
1](0) = 1]'(0) = 1]"(1) = O.
(8.39)

812
SLENDER STRUCTURES AND AXIAL FLOW
""
-
""
'<t
:!

o
00
p:J n
.£

CI)

o
o
o
.,)
00
8
ci
II
I
r-:
8
ci
II


IN
-8
II 
.., 
r- 
- ...
(1)
II a
><
ó
II II
"O f-.
r-:
"": :<:
o \.)
113;
<:Q.':";
] II
 f-.

..... :<:
o \.)
r.IJ ,-.,
(1)
=---
":I;"
 II
o f-.
'ij 
> :<:
!3 \.)
'+'<';;;'
':'
;::t"s
S6
:<: II
 >--
b/) II
.6 \.)

'O

(1)

00
""
G-
'"
'<t

o

,3,

00
""
G-
'"
'<t

o


00
""
,,:<:.
'"
'<t


....
 oč
2::



SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
813
....:: Q)
o..
I 
- '"
.-
'"
II '"
;::I
00 o
-,,"O
<J :03
,..;p..
, .
II ,
\C ,
>< d
Ó .9
II 'i;j

?' oo,E
'<1'", r-::o
'<T .....
ci o..
o
II :I:
N o:j
<:Q.. o:j
"O '> --'
C'I
 ... 
Q)
,-., tj
 . ;::I
o '+-.,q:: ......
..... . <:::
o I -
'I>
'" Ň 
Q)
;::I e
'<I' -a II
;:- Q)
I-. tZ:I
'" \.) '-'
?' ;::I --- >.
o =
'" '  o
N ;:- -IN
...
.B II
..: II I-.
B \.)
6 1-.---
o '5 \.) 
q:: --- \.)
11)  .
o <J 
N "O ____ o
o
S e, II
..o -'1;" '+-.,
I:i
 o II !3
 I-. ""
'" <J I:i
... --- 
.B  o
<J -=
'-, ---- '"
"! \.) cIj ..:
- ::s '-' B
I:i !3 ....
?' o ..... ;::I
.. q::
'" o
\.) II
00 '"
o 01) >..
I:i II
' \.)
;:- \()
'<I'
o ..... 00
o t-
.... o
u o
 ci
Q) II
o 11) "
\C :!: o \C ,-., 15
 - 00 '<I' N o  1><
t-
J:J n  'O
-' 'O
QQ o
o
f ci
SSm II
 "
><

814
SLENDER STRUCTURES AND AXIAL FLOW
Ingeniously introducing the new variables
=A2/3[£*(1-)-1], A=U/£*, £*=!£Cf' (8.40)
equation (8.37) is transformed into (d4rJ/d4) - (d2rJ/d2) - (drJ/dn = O, which, after
integrating once and substituting w = rJ', gives w" -  w' = C, where C is some arbitrary
constant. The solution of this equation may be written exact1y in terms of Airy functions,
as follows:
w(n = CIAi(n + C2Bi(n + C3Ci(n. (8.41)
The solution in terms of rJ(n involves an additional constant of integration, Co. Sub-
stitution of (8.41) into (8.38) and (8.39) gives a system of four homogeneous algebraic
equations for Co, CI, C2 and C3, from the vanishing determinant of which the critical
value of A, Acd, may be obtained.
A particularly simple solution for the case of £  00 (£*  (0) corresponding to very
long cylinders is also found, namely,
A;' ( ->-;n  _>-;3 (I - f) [1 - {"II Ai(,) ds ] . (8.42)
Solutions for the general case are given in Section 8,3,2(g), but for the simplified case of
£*  00 in Figure 8.14. It is noted that for £*  00 these results should also apply for
cantilevered (clamped-free) cylinders. It is seen that for f < 0.75, there is no divergence
at all. The results in this figure should be compared to those in Figure 8.12(b) for £CN very
large, For £CN = 10(£* = 5) and f = 1, one has Acd = Ucd/£* :::: 1,1 and dUcd/d£* :::: 1;
hence, Acd :::: 1,1 is in accord with the analytical prediction of Acd = 1.028 for f = 1 in
Figure 8.14 for £*  00.
3.0
1.0
2,5
2,0
*

'13
:s 1.5
II
'13
<-<
0.5
0,0
0,25
0.50
0,75
1.00
f
Figure 8.14 The critical flow speed for divergence. in terms of Acd = Ucd / e*. of an infin-
ítely long pínned-free cylínder, where e* = teC[, as a function of f, as obtained analytícally by
Triantafyllou & Chryssostomidís (1984).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
815
8.3.1 (f) Other work
Similar work to the foregoing has been done by others, e.g. Pao (1970), Grigoryev
cf al. (1979), Prokhorovich cf al. (1982) and Vendhan cf al. (1997), the latter by means of
the finite element method. Also, a great deal of1inear work was done by Lopes cf al. (1999b)
and Augu (1999) as a preamble to the study ofthe non1inear dynamics ofthe system, Refer
also to Sections 8.3.5 and 8.3.6.
8.3.2 Experiments and comparison with theory
8.3.2(a) Low-u experiments and flow-induced damping
Experiments with metal cy1inders (so1id brass rods and hollow steel cy1inders) in water
flow were conducted by Chen & Wambsganss (1972), aiming to show how the fun-
damental natural frequency and moda1 damping vary with flow. The cy1inders were
excited electromagnetica1ly, and the vibration was sensed optically. Typical results for
clamped-clamped cy1inders are shown in Figure Q,2, while for cantilevered cy1inders
with a blunt end (f = O) in Figure 8,15, in both cases showing excellent agreement with
their theory - s1ightly different from that presented here.
It should be remarked that, although the maximum flow velocities in these figures are
high (of the order of 30 mis), the dimensionless flow ve10cities are but a fraction of those
necessary for fluidelastic instabi1ities to materia1ize, For example, for a so1id steel cy1inder
with LI D = 20 in water flow, taking Ucd ::::: 2 one obtains Ucd ::::: 1180 ft/s ::::: 360 mis
(Pai'doussis 1966b), which is quite enormous and impractica1. For a very slender, hollow
cy1inder with LID = 100 and thickness-to-diameter ratio of 1/50, one obtains Ucd :::::
29 mis, which is stilI very large but feasible, although such a thin-walled cylinder would
be excited by the flow in both its beam and shell modes. For these reasons, the experimental
dynamics all the way to divergence and flutter has been explored with elastomer cy1inders
in water flow; the values of Ucd are then 50-100 times smaller, thus e1iminating the need
for a high-velocity, high-pressure flow system.
8,3,2(b) Apparatus for Paidoussis (1966b) experiments
The first such experiments were conducted by Pai:doussis (1966b), The apparatus consisted
of a water-circulating rig with a section in which the various test cy1inders could be mounted
horizontally, Flow straighteners were placed ahead of the test-section in order to e1iminate
secondary flows, and appropriate devices were used for smoothing the pump de1ivery
pressure. The supported ends of the flexible cy1inders were anchored on slender struts,
great care being taken to avoid flow separation ahead of the test-section. The test-section
with a cantilevered cylinder in position is shown diagrammatically in Figure 8.16, Later,
some experiments were conducted in another rig where the test-section was made from
a Plexiglas block of square cross-section in which was machined a 4 in. (:::::102 mm)
diameter flow passage. This arrangement afforded observation of the cylinder inside with
minimum visual distortion, and the photographs of Figures 8.17 and 8,18 were tak:en in
this test-section.
For some of the ear1y experiments, the flexible cylinders were made by cutting a rubber
tube into small segments which were subsequently rotated with respect to adjacent ones

10,5
'N 10,0
t:S
<.o::::
 9.5

6-
<1)
....
 9,0
Test 2-A
8,5
O
20 40 60 O
(a)
0,25
0,20
"-.1>
.g"' 0.15
f::!
OJ)
<=
t
Oj 0,10
"O
o
::E
0,05
20 40 60
(b)
00
Test 2-B
20 40
60 80
Test 2-C
o
20 40 60 80 100
o
20 40 60 80
Mean axial flow velocity, U (ft!s)
o
20 40 60 80 100
Figure 8.15 (a) The fundamental natural frequency and (b) the corresponding modal damping of
three different metal cylinders in axisymmetrically confined axial flow (h = 1, 1, 1 fmm left to
right panels). In (b) the damping was evaluated by: 0, the bandwidth method; 1::::., the magnification
factor method (constant force); ., the magnification factor method (constant displacement); -,
theory with CN for best fit (Chen & Wambsganss 1972),
"-
Fl 
cylinder
I 43 in, I
JE. 'J «
 :
support
-c+
Glass
tube
8i in,
'I
Flow
straightener
Figure 8.16 Schematic diagram of the experimental apparatus used by Pa'idoussis (1966b).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
817
and glued together, in such a way as to eliminate the permanent bow which is found in
commercially available rubber tubing. The outside surface of the composite cylinder was
then ground to a uniform diameter, For most experiments, however, the cylinders were
cast from liquid silicone rubber ('Si1astic'), which hardens into an elastic solid with the
aid of a catalyst; refer to Appendix D.
For experiments with c1amped-free cylinders, tapered plastic end-pieces of the same
specific gravity as the cylinder were glued to the free end. For pinned-pinned cylinders,
very short tapered ends were glued to the flexible cylinder and were connected to the fixed
struts on either side of the cylinder by thin, narrow, polythene strips; the unsupported
length of strip was just long enough to allow flexure. The strips were mounted in the
vertical plane, so that the preferred motion of the cylinder was in the horizontal plane, The
downstream strut could be moved axially before the flow was turned on, so as to apply
tension to the flexible cylinder if desired; once the flow was turned on, however, no further
motion of this support was possible.
ln most experiments with c1amped-free cylinders a thin metal strip was embedded in
the cylinder along its length, in the vertical plane of symmetry, effectively limiting all
motion to the horizontal plane and eliminating sag (positive or negative). Cylinders with
both ends pinned were not fitted with a metal strip, so that the development of in stability
would not be impeded by excessive resistance to axial extension.
Typical parameters for these cylinders are: D = 16.6 mm, L = 391 mm, m =
0.25 kg/m, EI = 6.39 x 10- 3 N m 2 ; or D = 16.7 mm, L = 330 mm, m
0.25 kg/m, EI = 7.68 x 10- 3 N m 2 .
For the flow velocities associated with fluidelastic instabilities, the flow was turbulent
(the Reynolds number based on channel diameter was::: 5 x 10 4 ), and hence the assumption
of a flat velocity profile over most of the test-section is quite reasonable.
8,3.2(c) Genera! observatíons
Considering pinned-pinned cylinders first, it was observed that, at small flow velocities,
disturbances induced by small flow irregularities were damped quickly (cf. Figure Q.2).
If the cylinder was flexible and long, it sagged slightly at its mid-point under its own
weight. Increasing the flow velocity in such cases exaggerated this sag and slowly shif-
ted it to the horizontal plane, which was the preferred plane of motion, There was
no distinct threshold for divergence instability (buckling). With increasing flow, the
point of maximum lateral displacement moved downstream, as axial drag tended to
put the downstream half of the cylinder in compression; eventually the cylinder became
J-shaped, extending beyond the downstream support. No osci11atory instabilities (flutter)
developed.
On the other hand, if the cylinder was more resistant to axial extension, or ifaxial tension
was applied at its ends, the behaviour of the system was much c10ser to that predicted by
theory. In this case, for a sufficiently high flow velocity, a small bow developed just
downstream of the mid-point of the cylinder, increasing in amplitude with flow, until it
reached the maximum allowable by the extensibility of the cylinder; see Figure 8,17 (a).
Contrary to the predictions of linear theory, divergence did not develop precipitously
with flow, presumably because of imperfections and nonlinear effects, notably due to an
increase in tension with increasing deflection.

818
SLENDER STRUCTURES AND AXIAL FLOW
(a)
(b)
Figure 8.17 Photographs of flexible cylinders in axia1 flow in divergence (buckling): (a) a pinned-
pinned cylinder; (b) a cantilevered one, Flow is from left to right.
(a)
(b)
Figure 8.18 Photographs of flexible cy1inders in axia1 flow undergoing second-mode flutter: (a)
a pinned-pinned cylinder; (b) a cantilevered one (Paldoussis 1966b),
Increasing the flow further, eventual1y caused spontaneous oscillation to develop, which
was in form very much like second-mode vibration of the cylinder in sti1l fluid, that is,
second-mode flutter developed [Figure 8.18(a)]; however, a trave1ling wave compon-
ent in the motion of the cylinder could be detected in the direction of flow, which is
a requirement for the cylinder to absorb energy from the fluid stream, 11S predicted by
theory (Section 8.3.3), Further increase in the flow velocity caused both the frequency

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
819
and amplitude of osci11ation to increase. In some cases, for even higher flow velocities,
third-mode flutter was observed.
Transition from divergence to second-mode flutter involved a gradual shift of the pos-
ition of maximum bow downstream, until a hump in the opposite direction appeared in
the upstream ha1f of the cylinder. The S-shaped cylinder sometimes flip-flopped (snapped
through) between two mirror-image configurations, apparently aperiodically, as captured
on ciné-film (Pai'doussis 1 966b ). The transition from this quasi-static condition to a fully
unstable one necessitated a usually small, but finite, increase in the flow velocity. This
behaviour is in qualitative agreement with theory, which predicts that, at its onset, flutter
occurs with vanishing frequency (see, e.g. Figure 8.5).
Considering clamped-free cylinders next, it was found that, at small flow velocities,
small random vibrations were always damped in this case also (cf. Figure 8.15). At
sufficiently high flow velocities, however, the system became unstable, provided that:
(i) the free end tapered smoothly; and (ii) the cylinder was not too long; or, in terms
of our theoretical parameters, provided that f was not too small nor stoo large. If these
conditions were met, as the flow velocity increased the system first buckled [Figure 8.17(b)]
and then spontaneously deve10ped second-mode flutter [Figure 8.18(b)]. At sti11 higher
flow velocities, third-mode flutter was observed. In the course offlutter, the free end sloped
backward to the direction of its motion for the greater part of the cyc1e. This 'dragging'
motion is necessary to allow the cylinder to absorb energy from the fluid stream; see
Sections 8.3.3 and 3.2.2.
Divergence developed slowly with increasing flow velocity, and it was not always pos-
sible to determine the point of neutral stability accurately. The point of onset of osci11atory
instabilities, on the other hand, was usually very c1early defined. According to linearized
theory, once the threshold of instability is crossed, the amplitude of motion should increase
without limit. In fact, because ofnonlinear effects, the maximum displacement in buckling
and the amp1itude of amplified oscillations were limited to usually one to three cylinder
diameters; see Section 8.11. If the flow velocity was reduced below the point where oscil-
latory in stability first occurred, the osci11ation persisted with reduced amplitude; this also
is an attribute of nonlinear behaviour.
Transition from divergence to second-mode flutter involved a gradual retum of the
cylinder to its point of rest along the x-axis, before further increase in the flow velocity
resulted in unstable oscillation; this agrees with the theoretical results in Figure 8.7.
Increasing the flow velocity further at this point caused the amplitude and frequency of
osci11ation to increase, until the motion became erratic and the frequency was gradually
reduced to níl; the cylinder then appeared to be buckled in its second mode. At this point,
a small increase in the velocity precipitated third-mode flutter of the system.
These ob servati on s show that the general behaviour of the system with increasing flow
velocity is in substantial agreement with theory [Section 8.3.1(b)].
If the cylinder did not contain a metal strip which effectively limited its motion to
the horizontal plane only, instability developed in one particular plane and, in the case
of divergence, remained in this plane; amplified osci11ation, on the other hand, soon
degenerated into a 3-D motion, in which the plane of osci11ation rotated slowly while the
free end described a quasi-circular path.
Similar observations were made with a cylinder pinned at the upstream end and free at
the other; see Section 8.9.4(a) and Figure 8.87. The specific gravity of this cylinder was
very nearly that ofwater (f3 = 0,50), so that gravity and buoyancy forces balanced almost

820
SLENDER STRUCTURES AND AXIAL FLOW
exactly, making experiments in the horizontal plane feasible. The various instabilities
occurred at lower flow velocities than for a similar cylinder with the upstream end clamped,
and it was possible to ob serve fourth-mode flutter.
A ciné-film and video have been made, showing the development of these instabilities.
One remarkable sequence that was fortuitously captured on film occurred because flutter
on one occasion was large enough to cause impact of the free end on the flow-containing
channel. Eventually, while the camera was running, the glued end-piece came off and was
swept downstream, whereupon the cylinder was instantly stabilized and remained straight
thereafter! This is a graphic illustration that a system with a blunt end is stable.
8.3.2(d) More recent experiments
A new experimental programme was initiated at McGill more recently (Adamovic 1999;
Pa'idoussis et al. 2002), mainly to collect data for comparison with the results of nonlinear
theory (Section 8.11); however, similar information as that just discussed, on the general
behaviour of the system, was perforce gathered. Rather than repeat these general obser-
vations, the main similarities and differences are remarked upon briefly, in point form, as
follows; further discussion may be found in Section 8.11. '
(i) The apparatus was similar, but the test-section was larger (Dch = 203 mm), and
so were the elastomer cylinders used in most experiment s (D = 25.4 mm, L =
520 mm). More importantly, the test-section was vertical (see Figure 9.17) and so
were the cylinders, which were generally not fitted with a metal strip, as this was
not necessary in this case; so, free motions were not restricted to a plane.
(i i) The general observations were similar to those already made in the foregoing. In
this case, however, flutter always involved 3-D whirling (orbita}) motion - sim-
ilarly to the horizontal system without a metal strip, as just discussed. Figure 8.19
(a)
(b)
(c)
(d)
Figure 8.19 Photographs oť a vertical flexible cylinder (a) in divergence at relatively low flow
velocity, (b) in divergence at higher flow velocity, (c) in second-mode orbital (3-D) flutter, and (d) in
third-mode orbita! flutter.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
821
shows one cy1inder with a well stream1ined end, subject to (a,b) divergence and
(c,d) second- and third-mode flutter. These photographs were taken from the side
of the test-section without a flat surface; so, the cy1inder diameter looks larger than
it really is,
(iii) There were some qua1itative differences in the transition from one instabi1ity to
another, however; for example, the transition from second- to third-mode flutter
was not usually preceded by a quasi-stationary (vanishing frequency) state, but it
involved erratic, perhaps quasiperiodic, oscillations (Section 8.11).
(iv) Experiments were conducted with one hollow cy1inder fitted with a metal strip,
physically different from the others, but with very similar dimensionless para-
meters. The dynamics of the two systems, with and without the metal strip, was
essentially the same, despite motion being planar in one case and whir1ing in the
other. Indeed, apart from this 2-D/3-D difference, the dynamics in the two cases
was qualitatively quite similar in every way.
Additional experiment s were conducted with a vertical, blade-fitted cylinder, with three
holes drilled in the end-piece, such that the fl1\id could go laterally through, rather than
around, as the cy1inder oscillated. With one end-piece, both divergence and flutter were
e1iminated in this way; with another, only divergence was, while flutter still occurred, but
at higher flow velocity. t
8.3.2(e) Ouantitative comparison with theory: pinned-pinned cylinders
A number of tests were conducted with cy1inders pinned at both ends and subjected to
varying uniform axial tension, As described before, the tension is app1ied by moving
the supports apart; its value was detennined by measuring the extension of the cy1inder
(Pai'doussis 1966b). Results obtained with two cy1inders, one considerably stiffer than the
other, are shown in Figure 8.20. Agreement in Ucd and ucf between theory and experiment
is reasonable but not very good - generally within 25%, It is reca1led that divergence is
very sensitive to imperfections, and that its onset is difficult to pinpoint because of the
imperfections and the additional deflection-induced tensioning. This latter, but in a more
comp1icated way, also affects the occurrence of coupled-mode flutter.
Since coupled-mode flutter arises with vanishing frequency at its onset, the experimental
va1ues of flutter frequency were compared with the theoretical ones at flow velocities
1 and 5% higher than the threshold values of ucf' Again, and not unexpectedly, agreement
is not very good.
8.3,2(f) Ouantitative comparison with theory: canti/evered cy/inders
ln the Pai:doussis (1966b) experiment s the thresholds of divergence and flutter were
estimated, the latter more precisely than the former, mostly by visua1 observation.
ln the newer experiments, however, in the vertical system (Pai'doussis ef a/. 2002),
motions could be measured by a noncontacting optical motion follower. Despite this, pin-
pointing the onset of divergence proved difficult. Figure 8,2l(a) gives a diagram of the
type used to determine Ucd, showing the average dimensionless displacement at the free
end of a cylinder, 1}*(l) = y(L)j D, as a function of u. It is clear that divergence does not
tThis offers an attractive possibility of stabilizing the system without increasing substantially the axial drag,

822
20
rolO
O
16
12
u 8
4
O
SLENDER STRUCTURES AND AXIAL FLOW
.. ..
u  1 05 ucf
u  1.01 ucf
..
20 40
160
Figure 8.20 The effect of extemally applied tension on stability of pinned-pinned cylinders in
axial flow; comparison of experiments with theory; -, theory (tJ = 0.48,8cf = 0,25,8 =
X = I, C = a = h = y = O); o,!:::", experiment for first-mode buckling; .,., experiment for
coupled-mode flutter (Pa'idoussis 1966b, 1973a).
2,0 0.40
.
l.5 0.30
 J
'"
 1.0 g 0,20
*
II::-
*

0,5 010
00 3 00 2 3 4 5 6 7 8
(a) u (b) u
Figure 8.21 (a) The dimensionless mean displacement of the free-end tip of the cylinder, r;* (I) =
y(L)/ D, as a function of increasing dimensionless flow velocity, u, showing the development of
divergence. (b) The r.m,s, amplitude ofvibration at  = 0.39, [1)*(0.39)]rms, versus u, showing the
onset of second-mode flutter (Pa'idoussis et a/. 2(02).
develop abruptly; instead, two zones may be discemed: (i) for u < 2 approximately, the
rate of increase of 17*(1) with u is moderate, and this may be identified with the growth
(exaggeration) of initial imperfections; (ii) for 2 < u < 2.5 approximately, there is a
steeper increase in 17*(1), which may be identified with the development of divergence.
The results in this figure are relatively 'c1ean', whereas in other cases the rougher but

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
823
more reliable criterion of 1]*(1) surpassing a threshold, say of 1.0, had to be used to
pin-point Ucd.
ln contrast, the onset of flutter in the second mode (and later the third mode) was
easy: visually or by monitoring the r.m,s, amplitude, as well as looking at the PSD of the
vibration signal. Figure 8,21(b) shows a typica1 diagram of the r,m.s. amplitude versus u,
used to pin-point flutter, It is seen that here there is little uncertainty as to the value of
ucf (U c f2). The reduction in amplitude, after a high of 0.37 D is reached, reflects gradual
changes in modal shape with increasing u.
Figure 8.22 shows the PSDs of the vibration of the cylinder, with the same sensor
trained at the same location. Figure 8.21(a,b) shows typical spectra of the turbulence-
induced vibration for pre-divergence flow velocities, u < Ucd. The dominant vibration
frequencies at JI :::: 0.6, fz :::: 2.9 and h :::: 8,0 Hz may be identified with the first-,
second- and third-mode eigenfrequencies of the system. Here, it should be noted that the
eigenfrequencies vary with u, but they remain identifiable, more or less at the va1ues cited.
ln Figure 8.22(a) and also 8.22(b) if the location of sensing the vibration is taken
into account, first-mode vibration at f1 dominates. This is not so for post-divergence
flow velocities, u > Ucd, as shown in Figure 8.22(c) and (d), where fz becomes more
prominent. Nevertheless, a component at JI remains identifiable in most cases. This
suggests that, in its buckled state, the cylinder has nonvanishing first-mode frequency,
o
-20
-40
(a)
u= 1.11 ft/s
(b)
u = 4,38 ft/s
-80
o
(c)
u = 5.46 ft/s
(d)
u = 6.59 ft/s
--. -20

O' -40
t:IJ
 -60
-80
u = 11.05 ft/s
U= 14,77 ft/s
-40
-60
-80
O
5
10
15
20 25 O 5
Frequency (Hz)
10
15
20
25
Figure 8.22 PSDs of vibration at  = 0.39: (a, b) at pre-divergence flow velocities; (c, d) at
post-divergence flow velocities; (e) in the course of second-mode flutter; (f) in the course of third-
mode flutter. (a) U = 0.34 mis (Ul ft/s), u = 0.53; (b) u = 2.10; (c) u = 2.61; (d) u = 3.16;
(e) u = 5.29; (f) u = 7,08 (Paldoussis et al. 2002).

824
SLENDER STRUCTURES AND AXIAL FLOW
that is. nonzero stiffness. This statement. heretical from a linear dynamics perspective.
is peďectly reasonable in nonlinear terms. Once second-mode flutter develops, at u =
U c f2, h is clearly dominant, as shown in Figure 8.22(e). Similarly, once third-mode
flutter is established at u = U c f3. the dominant frequency is h. as seen in Figure 8.22(f),
(i) Effect oj shape oJJree end. A number of tests were conducted in which a cylinder
was successively fitted with several tapered end-pieces, ranging from well-streamlined to
blunt shapes. At the end of each test, the end-piece was removed and a new one glued in
its place. The sequence was repeated several times, to check repeatability and to overcome
the effect of misalignment to some extent (Pa1doussis 1966b).
The critical flow velocities for divergence, Ucd, and flutter, ucf, are compared with theor-
etical values in Figure 8.23(a). The experimental results in each case cover a considerable
f=O.4
7
f=O.5
6
I
f=O,6
f=O,7
f=O,8
'- f= 0,9
""
::t
5
4
ABCDE FOR
t166&it
 !     
4
l!
::t
I
f=O.4
3
I
f=O.5
f=O,6
1-0,7
1-0.8
f=O,9
2
Figure 8.23 (a) The effect of shape of the tapered end-piece on stability of canti1evered cy1inders;
comparison of experiments in the rig of Figure 8.16 with theory ({3 = 0.47, SCN = SCT = 0,25,
X = 0.01); experimenta1 points from Pai'doussis (l966b).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
825
range in u, indicating relatively poor repeatability (small misalignments may have con-
siderable effect on stability); this is especially trne for Ucd, the determination of which
involved an element of subjective judgement. The arrows indicate that the system remained
stable up to the maximum flow velocity of about 6 mis attainable with the apparatus, corres-
ponding to u ::::: 14. Under such circumstances, 'stable' means that no distinct divergence
or oscil1ation developed; nevertheless, there were damped, apparently random motions of
irregular frequency and amplitude.
For an ideally slender end, f -+ 1 and Cb -+ O; for blunter ends, f < 1 and Cb > O.
According to theory, decreasing f stabilizes the system for divergence, while increasing
Cb destabilizes it. t These conflicting effects may have contributed to the poor agreement
between theory and experiment for divergence, since for most theoretical ca1culations
Cb = 1 - f was taken arbitrarily. The main reasons, however, are: (i) the sensitivity of
Ucd to imperfections, and (ii) the lack of refinement in dealing with the free-end boundary
conditions. Nevertheless, it is c1ear that, for a sufficiently blunt end, no divergence occurs.
For flutter, theory predicts that decreasing f and increasing Cb both have a stabilizing
effect, as observed. Qualitatively, agreement with theory is considerably better in this
case, Again, for a sufficiently blunt end, flutter is neither observed nor predicted in this
flow velocity range.
The effect of the free-end shape has also been studied by Hannoyer & Pai'doussis (1978);
see also Sections 8.3.4 and 8.6.1. For ca1culations with e = 20, they find that, whereas
decreasing (I / L)e == I / D from 2 to 1 does not have a very important effect, decreasing
it further does: thus, for I / D = 0.5 the system is unconditionally stable; I here is the
length Df the end-piece, so that, 1/ D < 0.5 corresponds to a hemispherical end-shape or
blunter.
Similar experiments were conducted in the vertical system (Pa'idoussis et al. 2002),
The results are compared with theory in Figure 8.23(b). In addition to Ucd, experimental
values for both second- and third-mode critical flow velocities for flutter, U c f2 and U c f3, are
shown; theoretical values for only Ucd and U c f2 are shown, since transition from second- to
third-mode flutter is generally nonlinear, as discussed in Section 8,11 - in the sense that
the system dDes not develop third-mode flutter from a stable equilibrium state. Agreement
with theDry for Ucd and U c f2 is quite reasonable in this case; perplexingly, however,
theory generally overestimates stability for divergence here, contrary to the results in
Figure 8.23(a). For the cases where the system does theoretically develop third-mode
flutter from stable equilibrium (for low values of j), the theoretical values (U c f3 > 9) are
cDnsiderably higher than the experimental ones (U c f3 = 7-7.5).
Some further experiments were conducted with one end-piece [#15 in Figure 8,23(b)]
the end of which was lopped off progressively more radically, resulting in a flat end of
increasingly larger area, The experimental results are given in Table 8.2, where it is seen
that taking off a small amount (5 or 10 mm) results in a reduced Ucd and an increased ucf.
Evidently this cDuld be assDciated with f being Dnly slightly reduced but Cb becoming
substantially higher as a result Df the flat end. The effect of decreased f and higher Cb is
stabilizing for flutter, as already remarked; hence, ucf increases. Eventually, however, as
the end is cut enough, the system is stabilized in both divergence (Ucd) and second- and
third-mode flutter (U c f2 and u c f3).
tFor instance, for a system E:Cf = 0,5 and f = 0.8, we find: Ucd = 1.93 for Cb = O; Ucd = 1.75 for
Ch = 1; Ucd = 1.65 for Cb = 2, For flutter, however, ucf = 7.51 for Cb = O, and ucf > 10 for Cb = 1.

826
SLENDER STRUCTURES AND AXIAL FLOW
9
8
'"
'v
"
7
I I I I
6
8
7
N
;p 6
J:
1-0,6
5
4
UULJ6060D
#115 #15 #2 #91 #35 #92 #90
OD
4
3
J 2
o
Figure 8.23 (b) The effect of shape of the tapered end on stability of cantilevered cylinders;
comparison with theory of more recent experiments in the apparatus with a vertica1 test-section
(/3 = 0.47, cCN = cCT = 0.25, X = 0.01); (Pai'doussis et al. 2002),
(ii) Effect ofthe mass ratio {3, Experiments in which f3 was varied were conducted in the
horizontal system with two hollow cylinders successively filled with various substances,
such as air, water, sand, lead shot and mercury (Pai'doussis 1966b). One cylinder was cast
and had a very smooth suďace finish, while the other was constructed from small segments
[Section 8.3 ,2(b)] and was fairly rough. The two tapered end-pieces were almost identica1.
Both cylinders had steel strips in their central planes. Even so, it was found impossible
to do successful experiments with f3 < 0.3, because the steel strip could not support the
weight of the cylinder without twisting.
The experimental results are compared with a single set of theoretical results for cIarity,
in Figure 8.24. Overall, agreement with theory is reasonable, especially for ucf.

Table 8.2 Dimensionless critical flow velocities for a can-
tilevered cylinder with end-piece #15, the end of which was
lopped off, progressively more. The asterisk denotes irregular
flutter about the bucked state,
End-shape "cd "cf2 "cf3
Intact (37 mm) 2.0--2.2 5.88 7.22
5 mm lopped off 1.6-1.8 6.01 7,62
10 mm " 1.7-1.8 6,73* 7,86
13 mm 1,8-2,0
15mm 2,2-2.4
20 mm
8
6
"t;- ....
::I
f 0,8
4 f 0,7
f=O,6
2
20
15 ....
....
"t;-
a .
.
10
S
4
.... ....
. .... ...
1! 2
::I
f=0.8
f=0.7
O f=0.6
0,1 0.2 0,3 0.4 0.5 0,6 0,7
P
Figure 8.24 The effect of the mass ratio {3 on stability of two cantilevered cy1inders fil1ed with
different substances; comparison of experiments with theory: -, theory (eCj = 0.2, X = I, Xe =
0,02, Cb = 0.1); ., experiments with the smoother cylinder; ., experiments with the rougher
cylinder (PaYdoussis 1966b, 1973a),

828
SLENDER STRUCTURES AND AXIAL FLOW
Table 8.3 The effect of slendemess, s,
on stability of a cantilevered cylinder
(Pai'doussis 1966b).
s Ucd ucf
16,3 2,01 4,58
22,9 3,08 4,54
25,9 4,94
39.5 5,85
45
(iii) Effect 01 surface roughness and slenderness. Experiments were conducted with cyl-
inders artificially roughened, in one case by rubbing with coarse sandpaper, and in another
by gluing cotton thread helically around the cylinder (Pai'doussis 1966b). A stabilizing
effect was obtained, mainly on flutter: with the thread, ucf was increased from 5,25 to
6,9-7.5. When rubber rings (1 in. inside and i in. outside diameters) were fitted over the
cylinder (D = 0.575 in.) at equally spaced intervals, no instability occurred at all.
Experiments were also conducted with one long cylinder (8 = LID = 46), progress-
ively reducing its slendemess by cutting pieces offthe free end and then replacing the same
end-piece. As seen in Table 8.3, when 8 = 25,9 (actually for all 8 > 24) no divergence
developed, while at 8 > 40 there was no second-mode flutter, at least to the maximum
attainable flow. At 8 = 39.5, the flutter amplitude and frequency were erratic, the latter
vanishing occasionally while the cylinder retained an S-shape.
Pai'doussis (1966b) tried to explain - only partly successfully - these observations in
terms of the theory as it stands. As seen in Figures 8.12 and 8.13, increasing 8 stabilizes
the system and may eliminate the instabilities a1together. However, a fuller explanation
should take into account the thickening boundary layer (when the length is large or rings
are present) and the resulting 'insulating effecť of the cylinder from the mean flow. This
is discussed more fully in Section 8.3.4, and in the section on conically tapered cylinders
(Section 8.5).
8.3,2(g} Quantitative comparison with theory: pinned-free cylinders
Some calculations have been conducted by Triantafyllou & Chyssostomidis (1984) for
tbe critical flow velocity for divergence, the first instability encountered as u is increased;
see Section 8.3.l(e).
Figure 8.25 gives Ucd 18* versus 8*, where 8* = 18Cf' for various values of I. Similarly
to the cantilevered system, for large I, the instability exists even for infinitely large 8; for
small enough I, however, in this case I < 0.75, the curve does not extend to infinity,
making a loop instead. Thus, for I = 0.70 and 8* = 1.50, there is no divergence.
Here, again invoking the boundary-layer insulating effect (Section 8.3.4), it may be said
that, in reality, this would be true at even smaller 8*. Also shown in the figure are some
experimental data from Pai'doussis (1966b), in reasonably good agreement with theory.
ln some unpublished work by the author (1965), some experiments with pinned-free
cylinders were conducted, yielding Ucd = 3.00 and 3.75 for 8 ::::= 22, depending on the
manner of pinning the upstream support; see also Figure 8.87.

829
SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
*

1!
'"
II


12.5
10,0
\
\
\
\
\
\
\
\
\
\
\
\
\
\
" \
,"- \
" ...........---.---.
'-'-,
7,5
5,0
2,5
0,0
°
2
3
e*
Figure 8.25 The critical flow velocity for divergence, Ucd, of pinned-free cylinders versus e* =
!ecr -, f = I; --, f = 0.8; -. -, f = 0.7; 0, experimental datafrom PaIdoussis (1966b) for
f :c:: 0.7 (Triantafyllou & Chryssostomidis 1984),
8.3.3 The mechanisms of instability
8.3.3(a) Divergence of cylinders with supported ends
The mechanism underlying divergence in cylinders with supported ends may be understood
by considering the static equilibrium of a cylinder, assuming it to have momentarily taken
an arbitrary bowed shape y (x). Eliminating the time-dependent terms in (8.16) and putting
g = O (horizontal system) and 8 = 1, we have
d 4 y [ 1 ] d2y
El dx4 +pAU 2 1-2cT(L-x)/D dx 2
- d 2 y 1 2 dy
-[T+(1-2v)pA] dx 2 +2(pAU /D)CN dx =0,
having used CT = (4/JT)CT and CN = (4/JT)CN. If T = pA = O, the dominant
term leading to divergence is pAU 2 (d 2 y /dx 2 ) = MU 2 (d 2 y /dx 2 ), M being the added
mass. By analogy to internal flow - see Section 3.2.1, Lee & Mote (1997a,b) - this
term represents a centrifugal force, which tends to exaggerate the assumed initial bow;
eventually, for sufficiently large U, it overcomes the flexural restoring force, resulting in
divergence.
(8.43)

830
SLENDER STRUCTURES AND AXIAL FLOW
This simple picture is modified by the presence of the viscous terms, the effect of
which is not readily assessable. Extemally applied tension and pressurization, however,
clearly stabilize the system, as expected; the latter becomes physically obvious if one
considers that the bowed cylinder offers a greater overall surface area on the side of the
bowed cylinder flung out than on the inner side, thus leading to a restoring force. These
observations agree with the results of Table 8.1, as they should.
8.3.3(bJ Divergence of cylinders with a free downstream end
For cylinders with a free downstream end, more particularly cantilevered cylinders, the
main destabilizing effect is associated with the tapered free end. Here the reader is referred
to the discussion following equation (8.25) and the analogy with a low-aspect ratio wing.
The destabilizing force is associated with the shear boundary condition (8.24a) which,
after elimination of the time-dependent terms, becomes
d 3 d
EI  + fpAU 2 -.!.. = o.
dx dx
(8.44)
Clearly, if f = O, that is, in the absence of a tapered end, no lift is generated at
x = L and hence divergence is impossible t; refer also to discussion at the end of
Section 8.2.4.
A special set of calculations have been conducted, in which viscous forces were
neglected, with just this side-force (lift) at x = L taken into account. The resultant
Ucd is reasonably close (for high enough f) to that given when considering the forces
on the rest of the cylinder as well: for example, for f = 0.8, for end-force only,
one obtains Ucd  1.78, and taking the forces on the rest of the body into account
Ucd  2.03; indeed, the forces on the rest of the cylinder tend to stabilize the system for
divergence.
One interesting observation that has been made in the discussion of experimental results
is that the effect of form (base) drag on divergence is destabilizing. This may be clarified
by analogy to a taught string, The work done W in displacing the string a small distance
y(x) from equilibrium - cf. Morse (1948, p. 30) - is
l i L d ( dY ) 1 [ d y ] L l i L ( dy ) 2
W = -- y- T(x)- dx = -- T(x)y- + - T(x) - dx.
2 O dx dx 2 dx O 2 O dx
If the string, or cylinder, is supported at both ends, the first term vanishes, while the second
is always positive. If, however, there is a free end, and if y and dy /dx have the same sign,
then the first term may dominate, which would mean a negative-stiffness, destabilizing
effect. As seen in Figure 8.17(b), y and dy/dx do have the same sign. Hence, setting
T (L) = ! P D 2 U 2 C b shows that increasing the base drag may indeed be destabilizing; cf.
the results in the penultimate footnote [in Section 8.3.2(f), p. 825].
tMore convincingly perhaps, the reader is referred to the equation just before (8,24a); it is clear
that, if A(x) = const for L - I ::S x ::S L, the last two tenns in that equation vanish, leading to
EI(d 3 Yldx 3 ) = 0,

SOLlTARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
831
8.3.3(cJ F/utter in genera/
Using equation (8.16) again, the rate of work done by the fluid on a cylinder with any
boundary condition, in the course of free periodic motion, is
dW = _ fL Elyy'''' dx _ fL yM(y + 2U.y' + U 2 y")dx + fL Y{B - Fx}y" dx
dt Jo Jo Jo
- !CN f\MU j D)y (y + Uy') dx - !c* f\Mj D)y2 dx
2 Jo 2 Jo
-l L Hm - pA)gy'dx,
where c* = (4jJr)CD with dimensions of velocity, M = pA, and primes and overdots
denote x and t-derivatives, respectively; here {B - Fx} stands for the long expression in
curly brackets in (8.16). Over a period of oscillation, T, the work done /:). W is
l T 1 l T
/:).W = - LHEly'" + MU(y + Uy')}] dt + -(1- 0)MU 2 Cb [yy'] dt
o 2 o
1 l T 1 L 1 l T 1 L
- -cN(MU j D) (y2 + U yy') dx dt + -cT(MU 2 j D) yy' dx dt
2 O O 2 O O
1 l T 1 L
- -c*(MjD) y2dxdt.
2 o o
(8.45)
To obtain this equation, extensive use of integration by parts has been made, while recal1ing
that the oscillation is periodic. For instance,
fT fL fT L fT fL
/:). WT* = Jo Jo yT*y" dx dt = T* Jo [yy']o dt - T* Jo Jo ly' dx dt,
where T* == B = o [f + (1 - 2v)pA] + !o - 0)MU2cb. The last integrand may be
written as ![(ojot)(y')2] and, since y,2It=T = y,2It=o for all x, the second integral
is zem. 50 is the first integral, for a system with both ends supported, since y(L) =
j>(0) = O. However, for a cantilevered system (o = O), we do have a contribution, namely
1 MU 2 rT, , d
2: CbJO YLh t.
Proceeding in a similar way with all the terms, for a cy1inder supported at both ends
('supported-supporteď), we have
1 fT fL
/:). W s - s = - 'icN(MU j D) Jo Jo (y2 + Uyy,) dx dt
_ !(MjD) fT f\c*y2 -cr U2 yy')dxdt.
2 Jo J o
(8.46)

832
SLENDER STRUCTURES AND AXIAL FLOW
For a cylinder supported upstream and free downstream ('s- f'), we have
 Ws-f = - (1 - f)MU {T [y2 + Uyy'h dt + MU2Cb {T [yy']L dt
10 2 10
1 {T (L
- ZCN(MU / D) 10 10 (i + Uyy') dx dt
1 {T (L
- Z(M/ D) 10 10 (c*y2 - cTU 2 yy') dx dt,
(8.47)
to obtain which we have made use of equation (8.24a) or (8.25) for Ely"'lx=L,
while neglecting the small terms involving Xe and xe. Expressions (8.46) and (8.47)
are a little different from those given before (Pa'idoussis 1 966a, 1973a), which are
incomplete.
If  W < O, oscil1ations are damped by the flow, while for  W > O oscil1ations are
amplified, i.e. the system wil1 be unstable by flutter.
8.3.3(d) Flutter of cylinders with supported ends
It is clear from equation (8.46) that, in the absence of viscous effects,  W s - s = O, which
is identical to the result obtained for internal flow in pipes, equation (3.6). At the time of
writing the original papers on the subject (Pai'doussis 1966a,b, 1973a),t it was not known
that coupled-mode flutter can occur, theoretically at least, for pipes supported at both ends
(see Section 3.4.1), and it was concluded that flutter must be due to viscous effects. As
shown by Done & Simpson (1977), however, coupled-mode flutter is in principle possible
even if C N = CT = O, that is, in the total absence of viscous effects, as discussed in detail
in Section 3.4.1. However, the existence of coupled-mode flutter has to be ultimately
assessed by non1inear tneory and experimental observation.
ln the case ofinternal flow it has been shown by non1inear theory (Holmes 1977, 1978)
that coupled-mode flutter does not actually occur, as discussed in Section 5.5.2 and as
confirmed by its nonexistence in experiments. In the case of external axial flow, however,
post-divergence flutter has been observed in experiments (Section 8.3.2(c)). Hence, it is
meaningful to continue this discussion.
As seen in Table 8.1, stability conditions are virtually independent of SCN, provided it
is reasonably small. Similarly to internal flow, the onset of flutter evidently corresponds
to a change in the character of the mode or modes involved: for u < Ucd oscil1ations are
damped, while for u > Ucd they may be amp1ified; cf. Figure 8.5 for external flow and
Figure 3.14 for internal flow: the main difference is that for u < Ucd, Lm (w) = O for
internal flow. (Indeed, one of the key elements for the nonexistence of post-divergence
flutter for internal flow is that, if dissipation is added, a branch of the solution remains
unstable between divergence and the presumed flutter; this is not trne for external flow,
where there is restabilization prior to the onset of coupled-mode flutter - see Figure 8.4;
but not always - see Figures 8.10 and 8.11.)
tS ee nevertheless 'Comment added in proof' in Paldoussis (1973a, p, 382),

SOLllARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
833
Inspecting equation (8.46) more closely, we observe that the condition for flutter
(f). Ws-s> O) requires that
-(CN - CT )U21L yy' dx - (CNU + c*) l L y2 dx > O,
the overbar denoting the long time average. The second term (including the negative sign)
is always negative. Hence, for inequality (8.48) to be satisfied, presuming that CN > CT
(smooth cylinders),t we must have Jo L yy' dx < O, which implies that different parts of
the body must vibrate in quadrature; this is an effect that can only be produced by the
action of flow. [In the absence of flow, the normal modes for both ends pinned are of the
form y = A sin(wt + <tJ) sin(mrxj L), where <tJ is independent 01 x, and the integral in
the first term of (8.48) is zero.] However, if the solution is represented as the sum of a
standing and a travelling wave, it is easy to see that the first term of (8.48) depends only
on the amplitude of the travelling wave component; furthermore, the phase velocity of
this component must be in the direction of flow for the first term of (8.48) (including the
negative sign) to be positive. This characteristic of cylinder flutter is corroborated by the
experiments - as seen in Figure 8.18(a) and more clearly in the available ciné films.
It is clear that Ll Ws-s> O is possible thanks to the presence of terms involving
U 2 (ay j ax) in the equation of motions; such terms are absent in the case of a pipe conveying
fluid, where f). W s - s = O, as noted by Holmes (1977). See also Section 10.3.7,
ln conclusion, energy transfer is generally dependent on (i) the mechanism elucidated
by Done & Simpson (Section 3.4.1) and Oi) the balance of the work done by the various
frictional forces, as is clear from (8.46) and (8.48).
(8.48)
8.3.3(eJ Cylínder wíth only the upstream end supported
ln this case, it is clear that, unless 1 = 1, the dominant term controlling Ll Ws-j in (8.47)
is the first. Indeed, apart from the (1 - f) factor, this term is identical to Ll W for internal
flow, equation (3.11). Hence, the same observations and conclusi ons may be reac hed albeit
modified by viscous effects; for example, the requirement that y'(L, t)HL, t) < O over
a cycle must be satisfied for energy transfer from the fluid to the cylinder, which means
that over most of the cycle the free end of the cylinder slopes backwards to the direction
of motion, exactly as observed in experiments.
One important difference vis-a- vis the internal flow system is the existence of the second
term in equation (8.47), especially since (i) it works in opposition to the first term, and
(ii) we nonnally peg Cb on I: Cb = 1 - I. This is discussed at the end of this subsection.
Equation (8.47) also shows that if 1 = 1, the inviscid first term vanishes, that is, the
work done by the nonviscous nonconservative forces vanishes. This has already been
remarked in the foregoing: the main source of energy transfer from the fluid to the solid
ceases to exist, though energy exchange may still occur via the viscous forces; indeed,
apart from the viscous forces, the system becomes like a conservative one. Significantly,
this also coincides with the suppression of single-mode flutter and its replacement by
coupled-mode flutter [see Figure 8,13 and discussion in Section 8.3.1(d); also compare
with the dynamics ofpipes conveying fluid (Section 3.4.1)].
tl n tenns of Argand diagrams, there are no fundamental differences in the linear dynafiŮcs for C N = CT and
CN 1= CT.

834
SLENDER STRUCTURES AND AXIAL FLOW
Table 8.4 The effect of Cb and sc N on flutter for f = O and
f3 = 0.47.
Cb ucf(scf = O) ucf(scf = 0.25) ucf(scf = 1.0)
.0 9,13 9.72 14.9
1 28,1 36.9 *
2
* If u exists in this case, it must be very high; see comments in the text.
The last item of interest is what happens when f -+ O. According to equation (8.47),
for f = O the system would then be more strongly nonconservative than for any other f;
yet, it has been found that flutter with a blunt end does not occur.
The system does indeed become more strongly nonconservative as f is diminished;
for example, 'Im(w), which is a measure of flow-induced damping, becomes larger. In
fact, despite what might appear to be trne from Figure 8,13, flutter is possible for f = O,
provided that, artificially, Cb = O. This is made clear by the results in Table 8.4. Even
for Cb = 1, flutter is possible, but at flow velocities beyond the range of the experiments
reportedhere. t For sCf = 1 and Cb = 1, the critical value could not be pin-pointed, beyond
stating that ucf > 40, if it exists at all, since results for such high u are very sensitive to
the number of comparison functions, N, used in the calculation; in fact, calculations with
increasing N suggest that perhaps ucf -+ 00.
For Cb = 2, however, no flutter is possible with f = O. As may be seen from equation
(8.47) the first two tenns then cancel each other out. Although the possibility of flutter
via the viscous terms involving CN, CT and C exists, it was found not to materialize in the
calculations.
Hence, it may be concluded that, for very blunt ends, (i) divergence disappears because
f becomes too small, and (ii) flutter disappears because Cb becomes too large.
8.3.4 Thick boundary layers
The expression used for the inviscid fluid-dynamic force or lift, equation (8.2), which has
been used in all the foregoing, has effectively been derived while neglecting completely
the presence of a boundary layer on the cylinder. This is reasonable only in cases where the
boundary-Iayer thickness is small compared to the cylinder radius. Indeed, as discussed in
the case of conical bodies in axial flow (Section 8.5), where boundary layers are particularly
thick, theory neglecting their presence fails utterly to predict the observed dynamical
behaviour, Here, it is attempted to take the boundary layer into account, approximately
and in a simple manner, following Hannoyer & Pa'idoussis (1978), that is, without engaging
in full-scale 3-D unsteady boundary-Iayer calculations.
A simple way of proceeding is to suppose the cylinder radius to be augmented by
the boundary-Iayer displacement thickness, (h, and then replace A (in M = pA) in
equation (8.2) by the augmented area, A *, This, however, would be basically wrong, as it
would imply that the displacement thickness associated with the cross-flow is identical to
that associated with the axial flow. Indeed, it would be unreasonable to suppose that, in
tThe maximum attainable in the early experiments (horizontal system) was u :::: 14, while in the more recent
set u :::: 8,

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
835
v
f
oy
of
(a)
(i)
(ii)
j _v oy
ox
(b)
(i)
(ii)
Figure 8.26 (a) Sketches of (i) the boundary layer associated with the axial flow, and (ii) the
cross-flow assullŮng that the axia! boundary layer is 'glued to the beam.' (b) Interpretation of
equation (8.49) in terms of superposition of two flows involving (i) velocity ay / ar and (ii) velocity
-u(ay/ax); (Hannoyer & Pa'idoussis 1978).
the course of lateral motions of the beam, the boundary layer moves as if it were glued to
the cy1inder [see Figure 8.26(a)]. It will be assumed instead that there is no boundary layer
associated with lateral motions (involving ay/at); hence, since the lift FA calculated by
simply replacing A by A * is overestimated by -[(a / at) + u(a / ax)][p(A* - A)(ay / at)],
the corrected expression for FA becomes
( a a ) { ( ay * a y ) }
FA=- -+U- p A-+AU- .
at ax at ax
(8.49)
This expression may be interpreted quite simply, upon considering the cross-flow to be
due to superposition of two potentia1 flows [Figure 8.26(b)]: one resulting from motion
of the cy linder a1one, involving area A moving at a velocity a y / a t, and the other from
motion of the fluid with velocity -U(ay/ax) past the contour of the cylinder and the
axial boundary layer, involving area A *; hence, equation (8.49) states that the totallift is
due to the sum of the two flows. C1early, this result is a vast oversimp1ification of the real
situation, but should be more accurate than just taking A * throughout.
The above expression may be refined further by noting that, as its thickness increases,
the boundary layer becomes 1ess sensitive to the motion of the cy1inder, which in tum
becomes insulated fcom the axial flow. We may then imagine that the cy1inder moves
within its boundary layer as if the axia1 velocity were reduced in the neighbourhood of the
cylinder, The corrective factor by which U would have to be multip1ied to take this into
account should be c10se to unity when 8. is negligible, and close to zero when 8. is very

836
SLENDER STRUCTURES AND AXIAL FLOW
large. The ratio AI A * appears to be a good and convenient measure of this effect; hence,
defining thereduced velocity U* = U(AI A *) and substituting U by U* in equation (8.49),
we obtain the following expression for the lift, derived by simple reasoning to tak:e into
account approximately the effect of the boundary layer:
( a * a ) { ( ay ay ) }
FA=- -+U - pA -+U- ,
at ax at ax
(8.50)
where it is noted that U* == U*(x).
For the calculations to be presented, it is assumed that the upstream cylindrical support
(cf. Figure 8.16) is xo 10ng and the boundary layer is turbulent from its leading edge.
Hence, the displacement thickness is approximated by the linear expression 15 1 (x) ::::
O.2(xo + X)CT, which presumes an almost constant rate of growth of the momentum
thickness, dÓ2/dx  !(CTIJT), and a shape vector 151/152  1.3 [cf. Schlichting (1968,
chapter XXI)], so that
U* [ D ] 2 [ 1-a(xIL) ] 2
U = D +O.4(XO +X)CT:::: a '
where a = O.4CTla, a = 1 + O.4(xOIL)CT.
The main thrust of Hannoyer & Pai'doussis' work is for cylinders subjected simul-
taneously to internal and external flow; so, a slightly different set of dimensionless
parameters has been used. Thus, (i) the ratio of internal to external cross-sectional areas
is required,"8 = (Ai! A)1(2, (ii) instead of Xe and Xe (see Section 8.2.4), liL is used, and
(iii) y* = [(1 - t) I c 3 ] [(pAl E /)g L 3] instead of y, along with other parameters - see
Section 8.6.
Some calculations have been done on the effect of skin friction and boundary-Iayer
thickness which, of course, are interrelated. However, analytically at Ieast, they may be
separated, It is found that increasing CN and CT but keeping the boundary-Iayer thickness
constant has a small stabilizing effect; similarly, increasing the length of the support Xo, so
that cO = xol L increases from zero to 10, raises the values of Ucd and ucf by roughly 5%.
If, however, the boundary-Iayer thickness is allowed to increase with CT, the stabilizing
effect is very significant, as shown in Figure 8.27. Some experimental data are also shown
in the figure: those on the left from Hannoyer (1977), and those marked 'with thread'
from earlier experiments, in which a fine cotton thread was wrapped helically around the
cylinder. The agreement is reasonably good.
Figure 8.27 also helps explain the observations summarized in Table 8.3. If boundary-
layer effects are neglected, then the theoretical cessation of divergence and flutter occurs
for c much larger than in the experimental results in the table. With boundary-Iayer effects
accounted for, however, the theoretical results in Figure 8,27 are in better quantitative
agreement with the experimental results of Table 8.3 - although it must be noted that the
results in the figure have been obtained by varying C N = CT, rather than c.
(8.51 )
8.3.5 Compressible and two-phase flows
The theoretical work into the dynamics of the system already described in Sections 8.2
and 8.3 has been extended to cover compressible flow (Pai'doussis & Ostoja-Starzewski

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
837
]0
8
Flutter
6
-------------.----
'1?
::
'B
::
4
Divergence
.
2
o
0,2
0.4
eCN= eCT
0,6
0,8
Figure 8.27 Comparative influence of skin-friction and boundary-layer thickness on the critical
flow velocities for divergence (lower curves) and flutter (upper curves) for 8 = 0.5,8 = 20,80 =
xol L = 10, li L = 0,075, y* = 0.0024: - - -, for negligibly thin boundary layer; -, with
boundary -layer thickness allowed to vary with CT; ., O, Hannoyer (1977) experiments; I, Paldoussis
(1966b) experiments; (Hannoyer & Pa'idoussis 1978),
1981). The initial motivation for this was to explain why the dynamics of the system
in experiments with single-phase (water) flow was so radically different from that in
simulated two-phase (air-water) mixtures, Specifically, whereas divergence and flutter
occur for single-phase flow (Section 8.3.2), they do not in two-phase flow, the cylinder
displaying random vibration response of ever increasing amplitude instead. The contrast
is c1early depicted in Figure 8,28 for a cantílevered cylinder in annular flow. t In contrast,
theory predicts the same kind of behaviour, whether in single-phase or in homogeneous
incompressible two-phase flow - that is, in equivalent single-phase flow with properties,
e.g. density and viscosity, averaged according to the void fraction [see Section 2.2.2( v)( c);
Wa1lis (1969)]. It is c1ear, therefore, that the existing model neglects something essentia1 in
two-phase flow which is responsible for this dramatic difference in the observed dynamics.
The question is: what? Since slip and spatio-tempora1 inhomogeneity between the two
phases are difficult to incorporate in a fluid-structure interaction model, a more convenient
culprit was sought; the first candidate was compressibi1ity.
It is well known that two-phase flow is anything but incompressible. Here, it is reca1led
that the sonic speed in two-phase Ipixtures can easi1y be one-tenth that in either the
liquid or the gas; and as low as one-fiftieth at low pressures (see Figure 8.29). This is
because the two-phase mixture has essentia11y the massiveness of the liquid phase and the
compressibi1ity of the gas,
tThese experiments are described in Chapter ] ] ,

838
SLENDER STRUCTURES AND AXIAL FLOW
4,0
 2,0
S
'"
.:.9

''::= O
 10
;; (a)

20
30
40
' l
1.0
00
(b)
I
100
I I
200 300
Frequency (Hz)
I
400
I
500
Figure 8.28 Typical vibration spectra of a cantilevered cylinder in highly confined axial flow,
the vibration being sensed by an embedded accelerometer: (a) flutter in single-phase (water) flow
at U = 4.26 mis; (b) broad-band random vibration of the same cylinder in simulated two-phase
(air-water) flow (mass flux 2700 kg/m 2 s, quality  15%). The FFf analyzer filter bandwidth was
set at 0,16 Hz in (a) and 1.6 Hz in (b); (Paldoussis & Pettigrew 1979),
Hence, a study into the dynamics of simply supported (pinned-pinned) cylinders in
compressible flow was undertaken (Pa'idoussis & Ostoja-Starzewski 1981), which stands
on its own merits, that is, irrespective of whether its predictions answer the question on
the perplexing dynamics in two-phase flows or not.
8.3.5(a) Dynamics in compressible f10w
The equation of motion is the same as before, namely equation (8.21) or (8.27a), sub-
ject to the standard boundary conditions for a pinned-pinned beam; however, in al1 the
calculations here, ex = y = 8 = O and CN = CT = cf have been taken for simplicity.
The inviscid hydrodynamic forces are obtained with the aid of the velocity potential
<I> = Ux +</J*,
</J* being the perturbation potential, satisfying
V2</J* _  a 2 </J* = O,
c 2 at 2
(8.52)
(8.53)
subject to the boundary conditions
a</J* I
- - O for - 00 < x < 00,
ar r=R
a</J* I { [ ay + U ay ] cos e for O  x  L
- = at ax
ar r=a O for x < O and x > L,
(8,54a)
(8.54b)

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
839
10 4
10 3
Pressure
(lb/in 2 abs)
2000
1000
500
100
 10 2

"CI
<=
;:I
o
'"
4-0
o
"CI
.,
., 10 1
"'"
</l
14.7
1.0
10°
10- 1
0,0 0,2 0.4 0,6 0.8 1.0
Void fraction, a
Figure 8.29 Variation of sonic speed with void fraction and pressure in a mixture af steam and
water; 1 ft = 0,3048 m, 11b/in 2 = 6,897 kPa (Davies 1965).
a being the cylinder radius and R that of the surrounding flow-channe1; c is the speed of
sound.
The perturbation pressure is given by
p* = _p Df/J* = _p ( af/J* + u af/J* ) ,
Dt at ax
(8.55)
and the lateral force on the cylinder is
{27r
FA = 10 p*a cos O dO.
(8.56)
Three different solutions have been obtained according to (i) a slender-body approxim-
ation, (ii) a 3-D approximation, and (iii) an integral transform, generalized force approach.
The reader is referred to Section 4.4 for similar analyses relating to internal flow.

840
SLENDER STRUCTURES AND AXIAL FLOW
(i) Slender-body approximation. According to this approximation the forces on each
element of the cylinder may be determined as if each element were part of an infinitely
long rigidly deformed cylinder; hence, the axial mode shape does not influence the loeal
perturbation pressure, and one may tak:e 4>* = 4>*(r, e, t).
Assuming separable solutions,
4>*(r, e, t) = B(r)E>(e) exp(iQt),
(8.57)
and recognizing that E> (e) = cos e in view of (8.54b), with the aid of the foregoing
equations one eventually obtains
FA = -pA B(a) (  + U  ) 2 y.
aB'(a) at ax
The expression for B(a)j[aB'(a)] may be found by substituting (8.57) into (8,53), fmm
which, after changing variables from r to r* = )..r, where
(8.58)
)..a = Qajc,
(8.59)
one obtains a Bessel equation, and hence B(r*) = ClH}()..r) + c2HI()..r), in which H}
and HI are Hankel functions of order 1, of the first and second kind, respectively. Finally
with the aid oi the boundary conditions one obtains
-B(a) H}()..a)+bHI()..a)
X=-=- ,
aB'(a) )..aHA()..a) - H}()..a) + b[)..aH6()..a) - HI()..a)]
(8,60)
where
)"RHA()"R) - H}()"R)
b=
HI()"R) - )" RH 6()"R) ,
HA and H6 being the corresponding zero-order Hankel functions; it is recalled that Hj () =
J1<) + iYj().
Therefore, the inviscid dimensionless force according to the slender-body approxima-
tion is
_ { 2 a217 1/2 a217 a217 }
fA - X u a2 + 2f3 u aa-r + f3 a-r 2 ' (8,61)
with the dimensionless quantities defined as in equation (8.28a). This expression is the
same as thatused to obtainequation (8.21), but X is no longersimply [(R2+a 2 )j(R2-a 2 )];
indeed X is a function of Q.
Then, solution to equation (8.21) proceeds by Galerkin's method, with comparison
functions 4>r() = J2sin(rJT).
(ži) 3-D approximation. In this case, equations (8.52) and (8.54a,b)-(8.56) still hold, but
4>* = 4>*(x, r, e, t) and it is governed by
( ) 2
2 * 1 a a *
v 4> - - - + U - 4> = O.
c 2 at ax
(8.62)

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
841
Writing <{J*(x,r,e,t) = 4J*(x,r,e)exp(iQt) and y(x,t) = y(x)exp(iQt) and intro-
ducing a combination of Ga1i1ean and Lorenz transformations, x' = x, r' = r(1 -
M 2 )1/2, e' = e, t' t + xMj[e(1 - M2)], M = U je being the Mach number, <{J*
may be written as
A.* ( x r' e t ) = e iQť e ikMxT ( x r' e )
ty , , , , , ,
(8.63)
where
1 a ( ,aT ) 1 a 2 T a 2 T 2
-- r- +--+-+k T=O
r' ar' ar' r,2 ae 2 ax 2
is a He1mholtz equation, with k, in reduced frequency form, given by
Qa M
ka = - 2 .
U 1-M
(8,64)
Severa1 attempts to obtain genera1 solutions of the coup1ed prob1em fai1ed, and an
approximate solution in which the same axia1 forms fory(x) and T(x, r', e) are assumed
has been pursued, name1y
y(x) = sin(knx), T(x, r', e) = sin(k n x)B(r')8(e); k n = n:rrjL, n = 1,2,...
(8.65)
Taking 8(e) = cose again, B(r') is governed by a Besse1 equation and hence 1eads to
the same functiona1 form for X as equation (8.60), but the argument in this case is
An a = [(ka)2 - (k n a)2]1/2 = [(Qaje)2 _ (n:rrj26')2]1/2
(8.66)
instead of Aa, with ka and kna defined in (8.64) and (8,65), and 6' = Lj D. The second
form for Ana is usefu1 to show immediate1y that Ana "7 Aa of equation (8.59) as 6' -+ 00,
and hence the 3-D solution reduces to the slender-body form,
(iii) lntegral-transform, generalized-foree solution. This solution is dosely simi1ar to
those used in Section 4.4.4 for interna1 flow in pipes and Section 7.3.1 for flow in shells,
Hence, there is no need to give it here in detail. On1y the fina1 form for fA is given, name1y
fA(, i) =  J OO (K - a)2 F(ii) 1 f t Am1/fm(Oeia d ) e-ia daé vT , (8.67)
2:rr -00 10
m=l
where 1/fm (O arethecomparisonfunctions used, such that 1J(, i) = [y()j L] exp(iQt) =
L=l A m 1/fm(O exp(iWi), W as in (8.28b), a = aL, a being the Fourier transform
variab1e, and
_ 1 [ K (Rf-t )11 (af-t) - I (Rf-t )K1 (af-t) ]
F(a) - -
- af-t I (af-t)K (Rf-t) - K (af-t)I (Rf-t) ,
f-t={a 2 -[(Q-aU)jef}1/2 or f-tL=[a2_M2(a_K)2]1/2,
(8,68)
in which K = QLjU.

842
SLENDER STRUCTURES AND AXIAL FLOW
Table 8.S The effect of the analytical model (method of solution) on
the critical flow velocity for divergence, "cd, of a pinned-pinned cy linder
in axial flow, for tJ = 0.1,8 = 20, X = 2.125, cf = 0.025, h = 1.5
(Pa1doussis & Ostoja-Starzewski 1981).
Analytical model used Ucd (úl (u = 2)
8 =20 8=5 8 =20 8=5
Slender-body model 2.33 2.19 4,72 + 0.068i 3,79 + O,Ol7i
3-D model 2.34 2.37 4.81 + 0.069i 5.01 + O,020i
(iv) Some typical results. The dynamical behaviour of the system as obtained by the fore-
going analyses for compressible ftow shows that the dynamics is the same as in Section 8.3
for incompressible flow: for sufficiently high flow velocity the system loses stability by
divergence, and at higher flow velocity by coupled-mode flutter. All calculations have
been conducted with two-mode Galerkin approximations; accordingly, the second-mode
eigenfrequencies and coupled-mode flutter thresholds cannot be very accurate, At the
time, this was done to reduce computational costs with the Fourier transform method (see
the next footnote), and was adopted throughout.
Before comparing compressible to incompressible flow, results by the slender-body
and 3-D analyses are compared first, in Table 8.5 for a system with Rla = 5/3, It is
seen that slender body theory underestimates stability, slightly for a slender system with
B = 20, but more seriously for a nonslender one (B = 5). The reasons for this have been
discussed in Section 4.4.9: slender-body theory overestimates the fluid-dynamic forces by
not taking into account that for nonslender systems there is a less-than-idea1 virtual mass
per unit length, The reduction in the slender body Ucd with decreasing B is likely due to
the smaller axial tension as BCf is diminished. The effect is overwhelmed by the reduction
in the fluid-dynamic forces with diminishing B, as discussed in the foregoing, when 3-D
theory is used, resulting in a slight increase in Ucd.
A typical Argand diagram for a system in incompressible flow (M = O) and in com-
pressible flow (M = 0.9) is shown in Figure 8.30. Here it should be clarified that the
differences between incompressible and compressible ftow results become more marked
as M becomes larger; this is thereason forusing M = 0.9, notthat it is believed thatthe ana-
lytical model would truly be applicable for such high M. It is concluded from Figure 8.30
that the effect of compressibility is very weak for these system parameters. It becomes
more important, however, as the slendemess B is reduced, as seen in Table 8.6, although
stili weak. (Here, it should be recalled that the dimensional U and Q would be much
higher for the shorter cylinder, irrespective ofthe small variations in u and ú) in Table 8.6.)
Calculations by means of the Fourier transform model yield slightly larger values of Ucd
and ucf, as shown in Table 8.7, most probably because of inaccuracies in the integration
for the evaluation of the generalized forces. In this table, the results for unconfined flow
(R  00, h = O) and confined flow (Rla = 513, h = 1.5) are also compared -
cf. Figures 8.4 and 8.6. Confinement does not enhance the effect of compressibility:
Ucd = 3,54 for M = O, and Ucd = 3.44 for M = O,9,t
t Here it is recalled that, because of limited computational resources at the time (1981), the method was less
precisely implemented than in the work of Chapters 4 and 7 (done in 1984 and the 1990s),

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
843
3 2 52 6
3
2 2 2.5 4 2
4.5 v.J= ufu 1\
4 .....---
1 2 3 2 4,7
4,65 4.65
4,5 4 3,5 3
2.31 2.25 2 4,7 2 1
 O O O

1?- 4n 2
-e _3 2
4 ----\'6'
4,5
-2 2 2,5 _4 2
3 o M=O
-3 2 -52 . M= 0,9
3.5
O 1 2 2 2 3 2 4 2 52 6 2 7 2
(w)
Figure 8.30 Argand diagram for the dimensionless complex frequencies, w, of the lowest two
modes of a pinned-pinned system (f3 = 0.1, e = 20, cf = 0.025, X = 2.125, h = 1.5) for two
different Mach numbers, M, as obtained by the 3-D approximation. The points on the verticallines
inside the frame of the figure are points on the Tm(w)-axis, drawn-in this way for compactness
(Pai'doussis & Ostoja-Starzewski 1981).
Table 8.6 The effect of compressibility for slender and
nonslender cylinders, according to the 3-D model for the
same parameters as in Figure 8.30.
M
ucd (VI (u = 2)
1':=20 1':=5 1':=20 1':=5
2.34 2.37 4.81 + 0.069i 5.01 + 0.020i
2.33 2.23 4.55 + O,068i 3,84 + 0.016i
o
0.9
Table 8.7 Comparison between unconfined and confined
t1ow, and between the 3-D and Fourier-transform analyt-
ical models for M = O and e = 20 (PaIdoussis &
Ostoja-Starzewski 1981).
Analytical model
ucd
ucf
3-D model (h = 1.5)
3-D model (h = O)
Fourier-transform model (h = O)
2.34
3.40
3.54
4,70
7,04
"-'7.2

844
SLENDER STRUCTURES AND AXIAL FLOW
The conc1usions from this work may be summarized as follows:
1. The effect of compressibi1ity on the dynamics of the system has been shown to be
rather weak for slender cylinders. Perhaps this could have been foreseen on physical
grounds; nevertheless, the quantitative differences between Mach numbers M = O
and M = 0.9 are truly surprisingly small.
2. For nonslender cylinders, the effect of compressibility becomes more significant (for
B = L/2a = 5) and should be even more important for smaller B (say, B = 2).
3. Increased confinement, although severely destabilizing the system, leads to no
significant enhancement of the effect of compressibility.
4. The effect of compressibility is obviously too small to account for the differences in
dynamical behaviour of reasonably slender cylinders in incompressible single-phase
flow and compressible two-phase flow - the initial motivation of this work.
8.3.5(bJ Random variations in the two-phase added mass
Having conc1uded that compressibility is not the main reason for the observed dynam-
kal behaviour of cylinders in two-phase flow, it was thought that the root mu st lie in
the inhomogeneity of two-phase flow and in the observed random variations in the void
fraction.
Carlucci (1977, 1980) and Carlucci & Brown (1983) studied the added mass t in sim-
ulated two-phase flow, namely air-water mixtures. They deduced the added mass from
measurements of the frequency of osci11ation of a single cylinder in annular two-phase
flow via ftp/fo = m/(mh + m), where ftp and fo are the frequencies in the two-phase
mixture and in vacuum (in air), m the mass of the cylinder per unit length, and mh 'the
hydrodynamic' or added mass, heretofore denoted by M; hence,
mh = m [Uo/f t p )2 -1],
(8.69)
The unexpected thing about the results is that mh is not proportional to the homogen-
eous mixture density, as seen in Figure 8.31. The homogeneous mixture density is P =
(1 - a)pz + apg, where Pl is the density of the liquid and Pg that of the gas phase, and
the void fraction a = Vg/CVz + V g ), the ratio of the volumetric flow rate of the gas to
that of liquid plus gas. Although the precise reason for this discrepancy is sti11 not fully
understood, it is now attributed to the nonuniform void distribution across the ftow area
(Pettigrew & Taylor 1993; Pettigrew et a1. 1995),
Another peculiarity ofaxial two-phase flow is the strange damping characteristics of
a cylinder in such flow (Carlucci & Brown 1983). The total damping ratio, T, may be
wri tten as
T = s + v + f + tp, (8.70)
where s, v, f are, respectively, the structural, viscous, and ftow-dependent damping
ratios adapted from homogeneous two-phase flow and semi-empirical considerations, and
tp is the two-phase component. Typical measured values are: for s = 0.2-0,5% of
tHere and elsewhere in this book, the simple inviscid interpretation of added mass is used, More sophisticated
interpretations are available, especially important at large oscillation amplitudes, which include 'viscous inertia'
contributions, associated with surface viscous effects and vorticity in the flow field - see, for example, Sarpkaya
(2001), Pan & Chew (2002),

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
845
 100 '
t 90 
"" ..,
 <hO
.., 80 t 0 0 "
i:
Ó 70 O o,  Proportional to mixture
!  O O . densityPm=apg+(I-a)pl
'" 60 : ;\O '

 50  i: O;g o
 40

 fCbO
 30 80 0o .
» O "'"&(\ -o
:I: 20
10 Oo (b0 O O '.
o
O 0°0 
O 10 30 40 50 60 70 80 90 100
Void ftaction, a (%)
Figure 8.31 Variation of the two-phase hydrodynaIIŮc (added) mass, mh, with void fraction,
relative to the hydrodynamic mas s in liquid flow, mh,l; mas s fluxes of 500-5000 kg/m -2s-1 ;
various vibrating cylinders, with in-air first-mode frequencies of 19-72 Hz and the same diameter,
D = 25.4 mm; flowtubes withintemal diameter 31.5 mm (Dch/ D = 1.24) and40mm (Dch/ D =
1.57); (Carlucci & Brown 1983),
3

f;.
J..J>
Ó
 2
00
"

"'"

oj
-&1
6

20 40 60 80 100
Void fraction, a (%)
Figure 8.32 Variation of two-phase damping ratio with void fraction for Dch/ D = 1.24, The
various symbol s correspond to different mass fluxes in the range of 0-5000 kg/ (m 2 s). Results for
DCh/ D = 1.57 display a les s prominent peak at ex :::: 35% of l;tp < 2% (Carlucci & Brown 1983),
critical (Le. 0.002-0.005), v = 0.1-0.3%, f = 0.1 - 0.5% (Carlucci & Brown 1983),
while typical values of tp are shown in Figure 8.32. They are seen (i) to be one order
of magnitude larger than the rest, and (ii) to have a perplexing peak at a :::: 40%. Again,
the precise reason is not fully understood even today, although this damping is clearly

846
SLENDER STRUCTURES AND AXIAL FLOW
dependent on the flow regime (Wallis 1969; Collier 1981), that is, on whether the flow is
annular, slug, chum, bubbly, and so on (Pettigrew et a/. 1995).
An investigation by Ch, Klein and the author was undertaken in 1980, with the joint
objectives of explaining (i) the unexpected results of Figure 8.31, and perhaps 8.32 also,
and (ii) the different behaviour of a cylinder in two-phase flow in Figure 8.28(b). The
working hypothesis was that the key difference between real two-phase flows and ideal
homogeneous flow models is related to the observed narrow-band random fluctuations in
void fraction and hence added mass. t This work was never published, but details may be
found in Klein (1981).
Initially, an ambitious analytical approach was launched, based on a fully probabilistic
micromechanics model of the fluid-structure interaction, with the assistance of Professor
D.R. Axelrad (Klein 1981). Specifically, two coupled generalized Langevin equations
for the motion of fluid partic1es are formulated, coupled via the forcing function to the
solid, which is modelled as a string. With appropriate probability distributions, one can
obtain the evolution of, say, the linear momentum-distribution in the form of a set of
Fokker-Planck equations, or that of the density fluctuations in terms of the Chapman-
Kolmogorov evolution equation. However, to carry this analysis all the way to a useful
engineering result appeared too long and arduous, and it was abandoned in favour of a
much simpler approach.
A one-degree-of-freedom modal approximation of the vibrating cylinder is adopted,
namely [m + mh + mh(t)]x + eX + kx = ° or f(t); mh is the mean value ofthe added
mass, Dividing by m + m h, we obtain
[1 + a(t)] x + 2wni + wx = {ft) ,
(8.71)
in which a(t) is the time-dependent void fraction. Once a(t) is specified, this equation is
integrated numericall y, with parameters corresponding to so-called 'realistic' conditions in
which fn = w n /27r = 15-60 Hz,  = 0,005-0.1, perturbationfrequencies fp = 5-25 Hz,
and perturbation amplitudes for a(t) of ap = ::!::O,OI to ::!::0.25. Then, as in Carlucci's
experiments, the frequency and damping in the response are measured and compared to
the unperturbed (input) values. Taking (m + mh )/m = (5/4)2, the added-mass 'defect'
in Figure 8.31 at 50% void fraction, which this analysis should hopefully reproduce,
translates to a response-frequency increase of ""6%,
Various ea1culation schemes with digitally generated deterministic and pseudo-random
inputs for a(t) in both the time domain and in the frequency domain (using FFf tech-
niques) were generated - for example, using pseudo-random Gaussian distributions via
the Monte-Carlo algorithm - and the response obtained by direct integration. For the max-
imum 'realistic' perturbations of ap = ::!::0.25, an increase in the oscillation frequency
of only 0.3% was obtained; for the unrealistic ap = ::!::0.75, this increases to 3.3%, still
far short of the desired 6%. Moreover, the effect on  was almost negligib1e. The main
problem in these simulations (and hence in the results) was the rather poor characteristics
of the a (t) input that could be achieved; for example, its FFT consisted of a number of
tSimilar but much more sophisticated analyses exist in aeroelasticity on the effect of turbulence-related
random flow fluctuations, For example, on the fluid-dynamic stiffness and its influence on flutter; see, e.g.,
lbrahim et a/, (1990), Poirel & Price (200 I),

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
847
[]]ID
Analog
computer
[Xg
x (t)
a (t)
f(t)
Function
generator
Oscilloscope
Signal
analyser
-,
I
lM
Keyboard
Figure 8.33 Schematic block diagram of the numerical experiment to test the effect of random
void-fraction variations on the two-phase added mass and damping; et(t) is the shaped random
input, f(t) the force input, and x(t) the system response, N-S is a noise source output; L-P and
H-P stand for low- and high-pass (Klein 1981).
Table 8.8 The effect pseudo-random narrow-band per-
turbations on the frequency of oscillation f and on the
damping ratio (measured at the forcing frequency).
o etrms resp fresp/ fo
0.02 ° 0.020 1.000
0.10 0,022 1.002
0,05 ° 0,050 0.998
0,10 0,052 0.998
0,32 0,059 1,032
distinct peaks in the desired frequency band, with relatively 1itt1e background noise, rather
than a quasi-continuum.
A more successful set of ca1culations was conducted with the aid of simulations on an
analog computer t and FFT analysis via a digital signal analyzer. The block diagram for
this scheme is shown in Figure 8,33,
Some typical results are summarized in Table 8,8, The damping was measured at the
half-power point. The results are promising. For instance, for Cl rms = 0.32, just a little
above the 'realistic' range, the added mass is 3.2% less than the homogeneous value,
of the same order as the 6% in Car1ucci's measurements. Also, there is an increase in
damping, ranging from 3 to 18% in the results in the tab1e, still short of the "-'100%
in the measurements,:j: These damping results have been obtained with a forcing fre-
quency },3 times the natura1 frequency, and the response PSD contains peaks at both the
response and the natural frequency; the va1ues of tp just cited correspond to the forced
vibration peak.
tThe analog computer [see, e,g, Kom & Kom (1956)] provided an analog output acceptable by the FFf
analyzer, whereas a digital signal did not; recall this was 1981.
*MoSl of the increase in damping in the experiments must be due to additional interphase dissipative
mechanisms, ignored in both the homogeneous flow model and in this one.

848
SLENDER STRUCTURES AND AXIAL FLOW
ln conclusion, the working hypothesis that the two-phase added mass 'defecť and,
partly at least, damping increase are related to the random fluctuations in the void frac-
tion yields results in qualitative agreement with observation, which in itself is a positive
result. Furthermore, quantitatively, the results on added mass are of the correct order of
magnitude. Thus, the hypothesis appears to be fundamenta1ly sound: although it cannot
be claimed that it is the only mechanism involved, the assumed mechanism must play an
important role.
Having concluded this, the reasons why fluidelastic instabilities do not materialize in
two-phase flow have not yet been clarified. Here again, unsteadiness in the flow must play
a role, t However, the main reason for the suppression of flutter may be the large increase
in damping, Taking e ten times larger than the value of e ::: 0.02 given for the single-phase
flow in Appendix Q, so that se ::: 3, shows that no flutter can arise; indeed the same result
is obtained with se = 1 - cf. Table 8.3 and the discussion thereof in Section 8.3.2(f).
Hence, this aspect of the dynamical behaviour of the system in two-phase flow can be
explained, partly qua1itatively and partly quantitatively.
8.3.6 Wave propagation
A modest amount of work has been done on flexura1 wave propagation in a cylinder in
general1y compressible flow (Pa'idoussis & Ostoja-Starzewski 1981).
Usual1y, wave propagation as opposed to vibration, is of interest if the system is quite
long (cf. Section 3.7). Here, for simplicity, the cylinder is considered to be infinitely long
and free of al1lateral constraints, so that
y(x, t) = exp(iknx) exp(iQt).
(8.72)
For qualitative comparison of the results to those in Section 8,3.5(a), the wavelength is
fixed to correspond to an eigenfunction of a pinned-pinned cylinder of length L, so that
k n = n7rjL, n = 1,2,3,.... Similarly, insteadoftheform ofequation (8.63), thevelocity
potential f/J* should now be written as
f/J* = e ikMx e::l:: iknx B(r')8(B) e iQt
(8.73)
which perfectly satisfies field equation (8.62).
Proceeding as in 8.3.5(a)(ii), it is found that
( a a ) 2
FA = xpA at + u ax y,
where X is as in equation (8.60), with the argument ofthe Hankel functions given by (8.66).
To render the results to be obtained as clear as possible, and also because exact solutions
are then possible, al1 dissipative forces are put to zero. Hence, substituting the above in
. dimensionless form into the equation of motion yields
K: + xu2K + 2f31/2XuWKn - [1 - f3(1 + X)] w 2 = O,
(8.74)
tlt has fortuitously been observed that when a lot of air tTapped somewhere in the water tunnel started
circulating (until it was eliminated by the deaeration system), the norma! divergence became akin to dynamic
divergence which looks like an unsteady, random vibration.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
849
where Kn = k n L = mr, and where it is recalled that X is a function of w. For given u and
M, this equation is easily solvable.
Typical solutions, in Argand diagram form with u as parameter, are shown in
Figure 8.34(a,b), corresponding to solutions 1J(, T) = exp(iwT + iKn) and exp(iwT-
iKn), respectively, with Kn = Jr. Both positive and negative branches of the solution, w+
and w-, are shown in each case. It is noted that, since all coefficients of equatiOll (8.74)
are real, the ws may be either wholly real or complex conjugate pairs.
Considering Figure 8.34(a) first, where the waves have been presumed to travel
upstream, it is seen that the solution gives both upstream-travel1ing (w+ branch) and
downstream-travel1ing waves (w- branch). The latter generally have a higher phase velo-
city, and only at u = O are the two equal, so that a standing wave could then be obtained
if both waves are present.
Mathematically, the general solution may be written as
1J(, T) = AI exp(iw+T + iKn) + A2 exp(iw-T + iKn).
(8.75)
4 2 (a)
2 2
S 0 2 ,25 2,25 2 O
 O
2
_2 2 2,5
o M=O
. M = 0.9
-4 2
4 2 (b) 4
3
2 2
S VaIues OfUr2,5
O I 2 2,25 20
 O
_:rr;2 2,25
:rr;2
_2 2 2,5
_4 2 _2 2
_4 2 O 4 2
(w)
Figure 8.34 Argand diagrams of the complex frequencies, w, of the lowest mode of a laterally
unconstrained infinitely long flexible cylinder in bounded axial flow, demonstrating the propaga-
tion characteristics of flexural waves, for solutions of the fonu (a) TJ (, T) = exp(iwT + in) and
(b) TJ(, T) = exp(iwT - in) : o, M = O;., M = 0,9. System parameters: f3 = 0,1, cf = O,
h = 1.5; (Paldoussis & Ostoja-Starzewski 1981).

850
SLENDER STRUCTURES AND AXIAL FLOW
Writing w- -Wi and w+ = W2, let us consider inteďering forward and backward
waves of equal amp1itude and wavenumber Kn = 7r, ln this case, equation (8.75) may be
written in the fonn
TJ(, T) = 2 cos [!(Wi + (2)T] exp [i7r + i!(Zih - wdT].
This is a wave travelling with frequency !(W2 - Wi), and with amp1itude modulated at
a frequency i(W2 + Wi). Since W2 - Wi < O always for u > O in Figure 8.34(a), the
wavefonn travels downstream.
It is noted that for u :::::: 2,17, W2 vanishes and there exists only a downstream-
travelling wave
TJ(, T) = 2 cos (!w i T) exp(i7r - iiwi T) .
For u > 2.17, W2 itselfbecomes negative, so that waves can no longer propagate upstream,
but we have two waves both travelling downstream, with different phase velocities, (Of
course, all this app1ies only to waves of wavenumber 7r; waves of wavenumber 27r, etc.,
may still propagate ustream until higher cut-off flow velocities are reached.) For higher
flow stilI, u :::::: 2.41, Wi and W2 become equal, the loci of w- and w+ coalesce and
bifurcate, the ws becoming complex conjugate, as shown in Figure 8,34(a). This signifies
that for u higher than u :::::: 2.41 waves become amp1ified, which is directly equivalent to
an in stability threshold.
Considering Figure 8.34(b) next, it is noted that it is the mirror image ofFigure 8.34(a),
Since w+ now corresponds to a downstream-travelling wave and w- to one travelling
upstream, it is evident that exactly the same infonnation is contained in each of the two
figures.
The interested reader is referred to the work of Dugundji et a!, (1963) on subsonic
flutter of panels and to Section 3.7 for wave propagation in pipes conveying fluid. By
comparing the behaviour of finite and infinitely long panels, in terms of standing and
travelling wave analysis, respectively, very interesting results have been obtained, some
bearing remarkable similarity to those discussed here.
ln terms of general conc1usions, first it may be seen in Figure 8,34(a,b) that the effect
of compressibility is truly very small. Secondly, the point of disappearance of upstream
wave propagation, u :::::: 2,17, is reasonably c10se (recall dissipation is neglected here) to the
divergence threshold in Table 8.5 (Ucd :::::: 2.33). This is similar to the results presented in
Chapter 7 on shells conveying fluid. On the other hand, the critical flow velocity for ampli-
fied wave propagation (u :::::: 2.41) is much smaller than the flutter threshold (u :::::: 4.7) for a
finite-length cylinder. Here the reader is referred to de Langre & Ouvrard (1999) and Doaré
& de Langre (2002a) on convective and absolute instabilities in fluid-conveying pipes.
(8.76)
8.3.7 Strings in axial flow
If the cylinder is sufficiently long and slender, flow-induced tension effects become very
important, while the flexural rigidity loses its dominance as the principal restoring force.
Thus, the cylinder may be modelled as a taught string, rather than a beam, which also
results in a mathematically simpler system - see, for example, Sections 3.7 and 5,5.4.
Towed arrays of hydrophones housed in very long, neutrally buoyant cylinders are the
premier example making such systems of practical, as well as academic, interest. In some

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
851
y
.. u
---
'r:To
x
Figure 8.35 The pinned-free cylinder in axial flow considered by Triantafyllou &
Chryssostomidis (1985), with a drogue added which causes a mean tension at the free end, To.
cases, the existence of a towrope is neglected, and the system is modelled as a cylindrical
string supported at its upstream end and free downstream. The dynamics of this latter
system is discussed here, while that of towed systems in Section 8.9.
Analytical studies of the generic system shown in Figure 8.35 have been conduc-
ted by Ortloff & Ives (1969), Pao (1970), Lee (1981), Lee & Kennedy (1985), and
Kennedy (1987), all of which are unfortunately based on the Pai'doussis (1966a) fonn
of the equations of motion, in which the viscous-frictional forces on the cylinder
are taken into account inconsistently (see Section 8.2.2), rather than on the correc-
ted fonn of the equations as in Pai"doussis (1970, 1973b) and as given in Section 8.2.
The dynamics according to the uncorrected and corrected equations is not dramatic-
ally different, provided that the cylinder is relatively short (and the ftexural rigidity is
taken into account). This, however, is not trne for the very long cylinders considered
here, precisely because the viscous-frictional effects are then so very important. Hence,
although some of these studies are quite admirable, their usefulness is ultimately rather
limited, t
Ortloff & Ives (1969) use Pai'doussis' (1966a) equations ofmotion setting the ftexural
rigidity (bending stiffness) to zero. They obtain solutions by means of approximate ana-
lytical methods and conc1ude that the system is always unstable. This analysis, however,
contains a mathematical error, which is corrected by Lee (1981), showing the system to be
stable if c = LI D is sufficiently large. Furthennore, it is noted that the approximate meth-
ods used in both studies introduce a simplification which, implicitly, is valid only for large
cylinder/displaced-ftuid mass ratio - which is not satisfied by the near-neutrally buoyant
practical systems inspiring this work (Pao 1970; Triantafyllou & Chryssostomidis 1985).
This difficulty is circumvented by Lee & Kennedy (1985) who, instead of an analytical
solution, utilize a numerical, finite-difference method. Finally, Kennedy (1987) considers
the response of the system to transverse motions of the 'towpoinť and to cross-ftow
excitation.
The definitive study on this system has been done by Triantafyllou & Chryssostomidis
(1985), using the corrected form of the equations of motion in a remarkable analysis.
tlt is interesting to note that, although the corrected equations were first published in 1970 and again in
1973, papers based on the uncorrected ones continued to appear until 1987, at least, and to be submitted
for publication up until 2000! Nevertheless, the author grasps this opportunity to apologize to the scientific
community for inadvertently being responsible for leading so many astray,

852
SLENDER STRUCTURES AND AXIAL FLOW
8.3.7(a) Triantafyllou & Chryssostomidis' ana/ysis
The equation of motion and boundary conditions are derived via Hamilton's principle.
The fonner corresponds to equation (8,16), with EI = O, Cb = O, Cd = O and no gravity
effects, but with an 'extemally imposed' tension, To, at the free end, for example, due to
a drogue t as in Figure 8.35:
[ 1 ] a2y a 2 y
MU 2 - To - -pDU 2 Cf(L - x) - + 2M U -
2 ax 2 axat
1 ( a y ay ) a2y
+ "2pDUCf ar + U ax + (m + M) at 2 = O.
(8.77)
The boundary conditions correspond to a simple support at the upstream end and to
equation (8.25) at the free end, with the tenns involving Xe and xe neglected, or
y = O at x = O,
To(ayjax) - f MU [(ayjat) + U(ayjax)] = O at x = L.
(8.78)
Since E I = O, a different nondimensionalization scheme is necessary, slightly adapted
here to match that in Section 8.2.5 as much as possible. Introducing
To
P = MU2 '
=xjL, 1]=yjL, r:=(UjL)t,
e* =  (Cf) (  ) = !eCJ,
(8.79a)
m 1-,8
ji, = M = {3'
(8.79b)
equations (8.77) and (8.78) become
* a 2 1] a 2 1] a 2 1] * ( a1] a1] )
[1-p-e(1-)]-+2-+0+ji,)-+e -+- =0,
a2 a ar: ar: 2 ar: a
[ a1] a1] ]
(p - f)- - f - = o.
a ar: =l
(8.80a)
1](0, r:) = O,
(8.80b)
Stability is investigated by considering
1](, r:) = Re [Y()eST].
(8.81)
Furthennore, defining
z=l-p-e*(1-) and A=sje*,
equations (8.80a) and (8.80b) may be written as
zy" + (2A + 1)y I + [O + ji,)A 2 + A]Y = O, (8.83a)
Y = O at z = Zl == 1 - P - e*, (p - f)y I - fAY = O at Z = Z2 == 1 - p,
(8.83b)
(8.82)
tOf course, the drag on a drogue would be flow-velocity dependent, but here a constant tension is
considered.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
853
where () = d( )jdz. A c1osed-form solution of (8.83a) may be obtainedin terms ofBessel
functions, as fol1ows:
Y(z) = A1 zV/2Lv(2,8*,JZ) + A2 zV/2J v (2,8* ,JZ),
(8.84)
where v = -2A, ,8* = [(1 + JL)A 2 + A]1/2, and A1 and A2 are arbitrary constants.
Equation (8.84), however, cannot be used directly to find the eigenvalues A ofthe system,
since the order of the Bessel functions itself depends on A. In Ortloff & Ives (1969) and
Lee (1981), it is assumed that A is smal1, so that v  O, which circumvents the prob1em;
but, as first pointed out by Pao (1970), this is valid only for JL large, which is inconsistent
with the physical system under consideration, as already mentioned,
Therefore, Triantafyl10u & Chryssostomidis (1985) proceed with a different, ingenious
approach. Equation (8.83a) is transformed into a self-adjoint form by 1etting
Y(z) = v(z)z-\
(8.85)
leading to
(zv')' = v { A 2[1 -  + JL)z] _ A} ;
the boundary conditions, equations (8.83a), become
(8.86)
v=O atz=zl, (p-f)V'=VAp(l-f)j(l-p) atZ=Z2.
(8.87)
Multiplying (8.86) by v*, the complex conjugate of v, and integrating from Zl to Z2 gives
l Z2 (zv')v* dz = l Z21v,2 { A2[1- (1 + JL)Z] - A } dz,
Zl Zl Z
(8.88)
Integrating the left-hand side of this equation by parts, one obtains
I 1 2 AP (1 - f) j Z 2 I ' 1 2 d
V2 f - z v z,
p- Zl
where V2 = V(Z2), Hence, equation (8,88) may be written in the form
MA 2 + BA + K = O,
(8.89)
l Z2 2 [ 1 - (1 + JL)Z ]
M = - lvi dz,
Zl z
j Z 2
K = - zlv ' 1 2 dz.
Zl
j Z 2 1 f
B = Iv 2 1dz + IV212p-=--,
Zl p- f
Stability is discussed with the aid of equation (8,89) [and also solution (8.84)] by pure
logical reasoning, without resorting to ca1culations; the argumentation, however, is too
complex to give in fun, and it is given here in outline only. It is first argued that if p > 1
(so that z ::S I - p < O), Re(A) < O and the system is stable, provided that M, Band

854
SLENDER STRUCTURES AND AXIAL FLOW
p
u
o
O
t:*
Figure 8.36 Stability diagram for a pinned-free cy1inder in axial flow in terms of the modified
slendemess e* and the tension parameter p, Stable regions are denoted by s, and the unstable one
by u (Triantafyllou & Chryssostomidis 1985).
K have the same sign; indeed, all three are shown to be positive. Then, it is shown that
the necessary condition of instability is p < 1 - £* - see Figure 8.36. It remains to
investigate stability for 1 - £* < p < 1. It is shown that the system is stable (motions
are bounded) if Re(v) ::: O which, in view of the definition of v below (8.84), means
Re()...) ::: O. Hence, the remarkably simple stability diagram in Figure 8.36 is obtained.
It is nevertheless noted that in the range 1 - £* < p < 1 the total tension, T (x) =
To + !pDU 2 Cj(L - x), becomes equal to MU 2 somewhere a10ng the string - cf.
Section 8.9.7. Hence, from thereon there is no tensile restoring force, and the neglected
flexural forces, even if vanishingly small, become important and may have a destabilizing
effect on the string.
The instability in the analysis and in Figure 8.36 is oscillatory (flutter). Divergence
does not arise for the string, since K in (8.89) cannot change sign without violating the
above-mentioned zero-tension limit; but see Sections 8.3.l(e) and 8.3.2(g) for divergence
of the pinned-free cylinder system, modelled as a beam.
The interested reader is also referred to Anderson (1973), Yadykin (1982), Yadykin
cf al. (1987) for other analyses of this system.
8,3.7(b) Experiments by Hansen & Ni
Experiments with very long cylinders, hence essentially strings, in axial flow are discussed
in Hansen & Ni (1976, 1979) and Ni & Hansen (1978).
The experiments involved (i) a polypropylene rope (cable) through which a water-filled
Tygon tube was threaded and (ii) a paraffin-oil-filled Tygon tube - thus a rough and a
smooth cylinder, respectively - in both cases achieving near-neutral buoyancy in water
flow of controlled temperature. The cylinders were 15.9 mm in diameter and 7.92 m 10ng,
resulting ine = LID:::: 500. F10w ve10cities in a transparenttest section, 190.5 mm in dia-
meter, were in the range of 4.6-8.3 mis (hence fully turbulent flow). Defonnationlmotion
was sensed optically. Some experiments were with tensioned cylinders supported at both
ends, but most with the downstream end free and fitted with a hemispherical, conica1 or
paraboloidal end-piece.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
855
These are very difficult experiments, for two reasons mainly: (a) maintaining near-
neutra1 buoyancy over such a length; and (b) e1iminating locked-in stresses, which are
known to become 'unlocked' and cause static deformation with increasing flow [see dis-
cussion of Bishop & Fawzy's (1976) experiments in Section 3.5.6 (Volume 1, p. 145)].t
Hence, the results should be treated with caution.
In the experiments with both ends supported, three regimes have been identified with
increasing flow: (i) with the who1e cab1e under tension; (ii) with zero tension at the down-
stream end due to the drag-induced compressive force for x :::: !L [equation (8,15)];
(iii) with zero tension farther upstream. Condition (ii) occurred at U := 5.2 mis in these
experiments. Moreover, although divergence is not discussed, the r.m.s. vibration amp-
litudes versus distance along the cable are suggestive of divergence at U := 5.6 mis and
higher (Hansen & Ni 1976, figure 9).
Experiments were a1so conducted with both the polypropylene cab1e ('cable' for short
from hereon) and the Tygon tube ('cy1inder' for short) contilevered. The time-averaged
deflection displayed a long wave1ength and a maximum of about 1.5D at 0.75L (cab1e)
and at '" 1.0 L (cylinder), attributed to s1ight deviations from neutra1 buoyancy.
Furthermore, the cylinder developed a static divergence over its most-downstream por-
tion at about 4.60-5.12 mis. 'No comparab1e behaviour was observed' for the cable 'under
any circumstances' , and this was attributed to the larger tension therein due to its rougher
surface. The critica1 u for the cylinder was found to be Ucd := 30; but the authors suggest
that ooly the length over which buckling occurred should be used (L := 1 m) in ca1cu-
lating Ucd,+ whereby Ucd := 3.5 is obtained; this is of the same order of magnitude as in
Figures 8.23(a), 8.24 and 8.27 for shorter systems.
This observation is apparently contrary to the Pai"doussis et 01.. and Triantafyllou &
Chryssostomidis' theoretical predictions in Figures 8.12(b) and 8.25 that for f ::s 0.7
there should be no divergence for B > 50 or so. On the other hand, if one considers the
greatest upstream part of the cylinder 'rigidified' by the large flow-induced tension acting
on it, and only a small downstream part constituting 'the flexible cy1inder', then these
observations may not be unreasonable. Furthermore, the boundary-layer thickness will
not grow indefinitely with x in practice, especially on a flexib1e cy1inder; it will be shed
periodica1ly after a certain axiallocation and start growing again. Hence, the insulating
effect discussed in Section 8.3.4 is app1icable only up to a point.
Both cable and cy1inder are subject to flow-induced vibration with maximum r.m.s.
amplitudes of '" 0.5D. The PSDs of the vibration signa1 are suggestive of turbulence-
induced vibration. For the (smooth) cylinder the vibration was of similar amp1itude al1
along the 1ength. For the rough cab1e, however, it was restricted to a relatively short length
tThe manufacturing-related bow in the tubes was 'eliminated' by stretching them while full of hot water
(refer to Volume 1 also); however, in this author's experience, a complete eradication ofboth bow and locked-in
stresses is very difficult to achieve. See Pai:doussis' 1980 discussion of the Ni & Hansen (1978) paper and the
authors' response.
* A very simi1ar set of observations and a corresponding elegant analysis have been presented by Doaré &
de Langre (2002b) for long hanging pipes conveying fluid, where the tension is due to gravity but never-
theless has the same x-variation as the drag-induced tension here, They find that for pipes longer than
L' ::: 4 [E I / (m + M)] 1/3, or approximately twice the length for which a standing cantilevered pipe would
buckle under its own weight (Section 3.5,2), the dimensional critica1 flow velocity for flutter, Vef, remains
sensibly constant. Hence, for pipes with L > L', Vef may be calcu1ated by: (i) the ana1yses presented in
Volume 1 for the appropriate value of y; (ii) the asymptotic methods developed by Doaré & de Langre; or
(iii) simp1y by taking Vef = V;f' the value obtained with L = L',

856
SLENDER STRUCTURES AND AXIAL FLOW
near the free end, because of the 1arge tension on the upstream part; this 'active 1ength' of
the cab1e diminished with increasing flow (to "-'0.7 m at U > 7.5 mis),
8.3.8 Deploying/extruding cylinders and related problems
A great deal of work has been done on travelling chains, strings and beams, starting
with Aitken (1878), part1y discussed in Vo1ume 1 because their dynamica1 behaviour
resemb1es that of pipes conveying fluid; see a1so Tabarrok et a/. (1974), Mansfie1d &
Simmonds (1987), Wickert & Mote (1989), Wickert (1992), Downer & Park (1993),
Sugimoto (1996), Behdinan et a/. (1997), Behdinan & Tabarrok (1997), Lee & Mote
(1997a,b), Chang et a/. (1997), Zhu & Ni (2000), Zhu et al.. (2000, 2001) and the
references cited therein. This area of work has applications to extrusion processes and
more recently to dep10yment of appendages in space, for examp1e unfo1ding antennae, or
wire for tethering sate11ites (e.g., Tsuchiya 1983; Kalayciog1u & Misra 1991; Be1etskii &
Levine 1993; Misra & Modi 1982; Ka1antzis et a/. 1996; Tabarrok & Behdinan 2000),
and for texti1e manufacturing processes invo1ving yam (e.g., Shih 1975; Perkins & Mote
1987; Zhu et al.. 1998),t The work on buckling flows, e,g, in capil1ary jets, incomp1ete1y
cited in Volume 1, is a1so re1ated (Stockman & Bejan 1982; Anderson & Bejan 1983;
Kimura & Bejan 1983; Bejan 1982, 1987), as is the work on high-speed trains cited in
Sections 4.7,7(c) and 11.8.
8.3.8(a) Deploying cylinders in dense fluid
ln most of the foregoing, either there is no surrounding fluid or, if there is, it is ignored. In
some cases, however, the sUITounding fluid is tak:en into account, for examp1e, by Taleb &
Misra (1981) who consider the dynamics of a cylinder being dep10yed or extruded in a
dense fluid.
The equations of motion are simi1ar to those in Section 8.2.1, but with some important
differences, as follows: (i) the flow ve10city U is rep1aced by the rate of dep10yment,
L; (ii) the inertia1 force of the beam itse1f is m [(a/at) + L(a/ax)]2 y, whi1e that of the
fluid is taken to be of the same fonu as (8,2); (iii) FN is the same, but it is 1inearized
into [cN(M/D)L + CN(M/D)] [(ay/at) + L(ay/ax)]; (iv) h = CT(M/D)L2,
but pointing towards x = O; (v) acce1erated dep10yment may occur, so that l i= O;
(vi) a fonu d.rag is considered at the 1eading edge, so that T (L) = !CT ML 2 is inc1uded.
It is assumed that the cy1inder is dep10yed horizontal1y and, this being a canti1evered
system, gravitationa1 and pressurization effects are not operative. Hence, the equivalent
of equations (8.7) and (8.8), after 1inearization, are
aT ..
- - F L - mL = O
ax '
aQ ay ( a, a ) 2 a ( ay )
--FN-FL--(M+m) -+L- y+- T- =0,
ax ax at ax ax 8x
(8.90)
tS ome of these papers have rather exotíc but descriptive titles involving, for example, 'ballooning strings'
and 'the inverse spaghetti problem'; of course, anyone who has suckedlslurped spaghetti into his mouth can
understand what the 'spaghetti problem' is, and hence guess what the inverse problem might be!

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
857
where
T(x) = -1(Mj D)t2cT(L - x) - mL(L - x) + T(L).
ln fact, the second of equations (8.90) is incorrect. Although in a viscous fluid some fluid
mass is dragged along the deploying beam, it is considerably smaller than M; altematively,
if M is used, its effective axial velocity will be much smaller than L (Mote 2(02). Hence,
tbe use of equation (8,2) for the inviscid forces is inappropriate, It would have been better
to write -m [(ajat) + U(ajax)f y - M(a 2 yjat 2 ) for the fourth tenn of the second of
equations (8.90); cf. Section 10.3.9. However, as we expect that the error affects the results
only quantitatively, we proceed with presentation of this work as published. Eventually
the following equation of motion is obtained:
( a ) a4y [ '2 1M . 2 l' 2 .. ]
E+E* at I ox 4 + (m+M)L +"2CT D L (L-x)-'iCTML +mL(L-x)
o2y ,a 2 y [ 1, ,.. ] ay
x ox 2 + 2(m + M)L oxat + '2 (cNL + cN)(MjD)L + ML ax
1 M ' _ ay a 2 y
+ '2 D (cNL + CN) at + (m + M) ot 2 = O. (8,91)
The systemis then partly nondimensionalizedby defining T = {E I j[(m + M)LÓ]} 1/2 t =
at, r = Lj Lo, s = Loj D, tJ = Mj(M +m) andcN = CN j(aLo), where Lois theoriginal
length. Solutions are obtained via Galerkin's method. Ca1culations are then conducted
with E* = O, tJ = 0.5, SCN = SCT = 1, CT = 1, SCN = 0,1, and different rates of
deployment ř = drjdr.
Figure 8.37 shows typical time-traces for the second mode at uniform rates of deploy-
ment, ř. As seen in Figure 8.37(a), at a small ř, ř = 0,1, the system remains stable; this
is so for the first and higher modes, Note the reduction in frequency as L is increased,
eventually leading to overdamped oscillation. For ř = 0,5, there is no noticeable damping
of motion, as seen in Figure 8.37 (b), Fina1ly, in Figure 8.37 (c) for ř = 1.0, it is seen that
the system becomes unstable before the cylinder reaches 20 times its originallength; the
first two modes show oscillatory motions changing to a static divergence, while the third
and fourth (not shown) display a flutter instability.
Taleb & Misra note that a critical flow velocity of u = 2.6 for a cylinder in axial
flow translates to ř = 3,67 for the deploying cylinder, the ratio (3.67/2.6) being close to
[(m + M)j M]1/2 = tJ-1/2 = ,.Ji. This is considerably higher than the ř = 1 which has
been found to lead to instability. However, the problem is not the same; for example, see
item (ii) in the paragraph prior to equation (8.90) and the discussion following it. Indeed,
tbe deploying cylinder may become unstable in the absence of any surrounding fluid; see,
for example, Kalaycioglu & Misra (1991).
Most interestingly, it is found that accelerated deployment, that is, with ř =
Vo exp( -h) and k < O, leads to greater stability than for unifonn deployment (k = O)!
This must be related to the acceleration-induced tension on the cylinder.
8.3.3(bJ Other, related problems
The dynamics of flexible rocket vehicles and missiles is similar to that of some of the
systems considered in this chapter. Tbe vehicle is often modelled as a free-free beam

858
SLENDER STRUCTURES AND AXIAL FLOW
o
-1
1 o 50 100 150
'O
S
u
OJ
'" O
<o-
o
 -1
J O 10 20 30 40
2
-2
Unstable
10 20
r
30
O
Figure 8.37 Typical time-traces of the second-mode vibration of a deploying cylinder in dense
fluid: (a) for ř = 0.1; (b) for ř = 0,5; (c) for ř = 1.0 (Taleb & Misra 1981),
(with the rigid-body modes taken into account), subjected to a purely axial flow or one at
a smal1 angle of attack; see, for example, O'Keefe (1965), Platus (1989), Elyada (1989),
Wu et a/. (1989), Karpel et a/. (1998), Dotson et a/.. (2000), Sugiyama & Oshima (2002).
Spin of the vehicle, to promote stability, is sometimes considered, as is the effect of the
end thrust. The equivalent flow velocities of interest are nonnal1y very high, supersonic
or hypersonic, and the vehic1e aerodynamics is complicated by the presence of addi-
tionallifting, stabilizing and control surfaces. Hence, this topic is beyond the scope of
this book.
lt should nevertheless be mentioned that divergence and flutter instabilities are
encountered. Also, interestingly, it is found that, in part of the parameter space, structural
damping may be destabilizing (cf. Sections 3.5.3 and 3.5.5),
8.4 SMALL-AMPLlTUDE FLOW-INDUCED VIBRA TION
The generic dependence of vibration amplitude versus flow velocity is shown in
Figure 8.38. 'Fluidelastic instability' therein stands for the first flow-induced divergence
or flutter instability encountered, as discussed in Section 8.3; the 'response to flow peri-
odicity' is associated with parametric resonances (Section 8.7); final1y, the 'nonresonant

SOLlTARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
859
Y nns
Fluidelastic instability
Response
to flow
"periodicity"
Nonresonant l
buffeting
response I
u
Figure 8.38 Idealized response of a cylinder in axial ftow (Pai'doussis 1981),
buffeting response' is what is of interest here. t Buffeting, or 'turbulence-induced vibra-
tions', are the sma11-amplitude (A) vibrations, typically of order 10- 3 < Aj D < 10- 2
mentioned in Section 8,1; they are sometimes called 'sub-critical vibrations', precisely
because they occur at flow velocities inferior to those related to fluidelastic instabilities
(Pai'doussis 1966c),
A great deal of work has been donej: towards understanding the mechanism of excitation
of these vibrations and developing tools for its prediction - particularly in conjunction
with nuclear fuel vibration. Nuclear fuel 'rods' or 'elements' are tubular cylinders, effect-
ively slender cans, containing fissile material, They are very closely spaced, and even
very small vibrations cause fretting wear, in some designs by impact of opposite-facing
spacers attached on the rods, or in others at the imperfect (to allow for thermal expansion
and for ease of assembly) intermediate supports, along the span. This work goes back to
the 1950s when, it is suspected, it was not universally appreciated that 'if there is flow,
there is vibration' . Still, the wear rate strongly depends on vibration, and wear is a serious
concern. If the cladding, that is the wa11 of the containing tubular cylinder, becomes too
tmn, the tube may crack, releasing radioactive material into the primary coolant circuit.
However, this work is by no means uniquely of interest for nuclear applications; it is also
of interest for heat exchanger vibration (in regions where the flow is primarily axial) and
a11 other systems involving nomina1ly axial flow. Reviews may be found in Chen (1974),
Pa'idoussis (1974c, 1981), Blevins (1979) and Au-Yang (2001); an extended, ana1ytical
review is given in Pa'idoussis (1974b).
tThe same diagram characterizes response of a cylinder in cross-flow, but flow periodicity then refers to
vortex-induced vibration (Paidoussis 1981),
*The author's bookshelves a10ne contain a piJe 0.75 m high of published papers and reports, let alone
conference proceedings and book chapters. Mercifully for the readers, this account is briefer.

860 SLENDER STRUCTURES AND AXIAL FLOW
8.4.1 Mechanism of excitation and prediction of vibration
ln most engineering applications, the dimensionless flow velocity u, defined in equa-
tions (8,27a), is less than 0.5,t and it is known (Section 8.3) that the main effect of ideally
uniform, steady and straight axial flow in that range is to damp free motions, However,
real flows, especially in industrial environments and particularly for clustered cylinders,
are not ideally uniform, steady and purely axial, and these departures from the ideal could
well cause vibration (Pai'doussis 1966c).*
For instance, swirl, unsteady trans verse flow components, or unsteady separated flow
upstream of the cylindrical structure, may give rise to unsteady lateral forces on the
cylinders, Convection, boi1ing, or chaoges io flow regime in two-phase flows, gener-
ally not axially symmetric, may equally produce lateral forces acting on the cylinders.
Small-scale turbulence within the boundary layer around the cylinders generates random
pressure perturbations around them; these are not spatially uniform at any given instant,
and when iotegrated around the circumference of a cylinder give rise to a random
lateralload,
AU of these presumed sources of excitation arise as the fluid flows over the cylindrical
structures. In addition there wil1 be 'system' disturbances acting on the cylinders, for
example, far-field disturbances (not necessarily axiaUy symmetric) propagated through
the fluid; also, mechanical excitation transmitted through the apparatus and supports,
which may have a fluid-mechanical origin farther upstream or downstream.
AU the foregoing may be considered to be sources of energy which could excite and
sustain vibratioo as a classicalforced vibration, However, in principle, the vibratioos could
be parametrically excited, or may iodeed be self-excited. Thus, is the excitation of the EIE,
I1E or MIE type (Section 1.2) in the Naudascher & Rockwell (1994) classification? This
question exercised the research cornmunity in the 1960s and 1970s. It is now accepted that
the vibration is generated by extraneously induced excitation (EIE) and that it is therefore
a forced vibration. Nevertheless, a few words on the other hypotheses are justified; for a
more complete account, see Pai'doussis (l974b,c).
8.4. 1 (a) Self-excited vibration models
The first such model is proposed by Quinn (1962). It is govemed by an equation of motion
of the cylinder,
EI(a 4 y/ax 4 ) + pAU 2 [(a 2 y/ax 2 ) + (d 2 YO/dx 2 )J
+ 1pDCv lay/atl (ay/at) + (m + pA)(a 2 y/at 2 ) = O,
(8.92)
similar but simpler than equation (8.16), and a momentum equation for l-D flow,
p(dU /dt) + 1pfU 2 {Dh[l + f3e(e + yo + y)]}-l = -ap/ax,
(8.93)
tConsider a typical tubular cylinder which might be used for a nuclear fuel element (D = 15 mm, L =
0.60 m, wan thickness 0.4 mm, E = 6.87 x 10 10 Pa). Neglecting the unreliable rigidity contributed by the
contained fuel, a ftow velocity of 10 mis corresponds to u  0.43.
+Thus, 'axial ftow may be regarded as something of a Trojan horse; although apparently beneficent, damping
free motions, it contains within it the very forces which excite sub-critical vibration' (Pai:doussis I 966c ).

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
861
where YO represents a presumed permanent cylinder bow, f the friction coefficient
of channel flow, Dh the hydraulic diameter, {Je a coefficient of eccentricity, e the
hydraulic eccentricity of the cylinder supports; other symbols are as in Section 8.2.
Thus, broadly speaking, deformation of the cylinders affects the flow velocity and, in
that sense, this model is similar in spirit to the Rousselet & Herrmann and the Bajaj,
Sethna & Lundgren models for the dynamics of pipes conveying fluid discussed in
Section 5.2.8.
Since the damping term is retained in nonlinear form, it is clear that amplitude inform-
ation may be obtained by approximate solution of these equations. Assuming a first-mode
deflection, Quinn reduces the system to an equivalent one of one-degree-of-freedom,
and obtains a criterion for existence of self-excited vibration depending entirely on the
eccentricity of the supports, the permanent bow and the time-dependent bow. He sim-
ilarly obtains an expression for the amplitude of vibration which depends on the same
parameters. In fact, the amplitude obtained is 1inearly proportional to the permanent
bow of the cylinder. Herein lies the main difficulty in using Quinn's theory for estimat-
ing vibration amplitude, since the permanent bow in the cylinders is not known ahead
of time.
Quinn tested his model mainly by comparing trends in experimental data with trends
indicated by his approximate solutions; on this basis the model was found to be in agree-
ment with experiments. In terms of quantitative comparison, however, the best that could
be said was that 'this comparison may be said to not refute the model.'
Today there are plentiful experimental data, allowing the objective evaluation of
Quinn's hypothesis. Two key characteristics of self-excitation are the existence of a
threshold parameter and a constant or nearly-constant limit-cycle amplitude. However,
there is no clear evidence of a critical or threshold value for the onset of vibration. t
Moreover, there is no evidence of limit-cycle oscillation. On the contrary, all indications
are that the vibration is random, with the r,m.s, displacement amplitude obeying a nor-
mal (Gaussian) probability distribution and the autocorrelation having the characteristic
damped-sinusoid form (Rostrom 1964; Wambsganss & Chen 1971; Harris & Holland
1971). Hence, the fundamental hypothesis for the model is unsound, and this model has
been abandoned.
8.4. 1 (b) Parametric excitation model
Y.N. Chen (1 970a,b), observing the poor performance of the early forced vibration
models - see subsection (c) - rejects them completely, arguing as follows: 'The
natural frequency of the fuel rods usually used is very low, so they can only be
excited to vibration in resonance by the large eddies, the size of which is com-
parable with the path width of the flow along the rod assembly. The diameter of
the rod will be small compared with this eddy size, so that the phase shift of the
dynamic eddy force along the periphery of the circular cross-section of the rod will
be small too. No significant resultant dynamic force perpendicular to the rod axis can
tpa'idoussis (1974b,c) theorizes that what Quinn supposed to be a 'critical flow velocity' may,
in fact, correspond to the point where truly Bow-induced vibration becomes of comparable mag-
nitude to vibration transmitted mechanically or caused by other background noise (which remains
sensibly constant with Bow velocity); thus, Bow-induced vibration would appear to 'emerge' at a
particular Bow.

862
SLENDER STRUCTURES AND AXIAL FLOW
therefore be exerted by the eddy.'t:I: He then goes on to propose a parametric excitation
model.
Chen use s a simplified form of the equation of motion,
EI(a 4 yjax 4 ) + (pAU 2 + !CTPU 2 LD)(a 2 yjax 2 )
+ !CNpD(ayjao + (m + pA)(a 2 yjat 2 ) = O,
(8.94)
cf. equations (8.16) and (8.43), but note that the C N and CT terms are not exactly the same
(indeed, C N here has dimensions of velocity). Chen then postulates that the observed vibra-
tion is parametric, arising from velocity fluctuations which affect the effective 'centrifugal'
(or 'compressive') force represented by the second term ofthe equation. He sets
u = U + ucost,
u = a2S U ,
(8.95)
where U is the mean flow, and u is the amplitude of the velocity fluctuations; a
is a proportionality factor,  the incident turbulence level, and S = f Dhj U the
Strouhal number, f being the frequency and Dh the hydraulic diameter, Thus, the
basic hypothesis is that the excitation is associated with the fluctuating effective tension,
(pAU 2 + !CTPU 2 LD)(2Uu), obtained once (8.95) is substituted in (8.94).
Equation (8,94) is discretized and, using (8.95), transformed into a Mathieu-Hill
equation (Bolotin 1964, Nayfeh 1981),
d2Bn
-----z + (on - £n cos -r) Bn = O,
dT
(8.96)
o. = ( Q; )2 " [1- ( , )'] (  )',
_ ( Qn ) ( uu )
£n - 2 2 '
 -
U e
where Bn(T) is related to qn(T), the nth generalized coordinate, by qn(T) =
exp(-T)Bn(T), 2=!CNPDj[(pA + m)Q], T=Qt, nf is the in-flow nth eigen-
frequency, while Qn is the corresponding value in stagnant fluid; U ; =
[(mrjL)2EI]j[!CTPDL + pA] is the critical flow velocity for divergence. Refer to
Pa'idoussis (197 4b) for an accessible treatment of the missing steps.
It is known [cf. Section 4.5 or Bolotin (1964)] that parametric resonances arise in the
vicinity of  j nf = 2, 1, , etc. for Sn --+ O, For Sn > O there are alternate regions of
stability and instability, Chen argues that, in this particular case, the secondary resonance
(QjQnf :: 1) is the main one (cf. Section 8.7.3), rather than the principal resonance.
Moreover, he faidy arbitrarily assumes the amplitude of vibration, y*, to be proportional
tThis is now known to be wrong; see, for examp1e, Gonnan (1971), Chen & Wambsganss (1972),
Gafnon & Pai:doussis (1994a,b),
Dr Y.N, Chen has been a great proponent of vortices and vortex shedding as mechanisms for flow-induced
vibrations of a11 kinds, An eminent researcher, and an urbane and cu1tured man, he neverthe1ess came close to
a physical altercation with Professor D.S. Weaver, to gain contr01 of the microphone during a question period
at the 1982 Keswick Conference (thereafter known to all as 'the batt1e for the mike'), to support his thesis that
fluide1astic instability of cylinder arrays in cross-flow is really a fonn of vortex shedding (Pai:doussis 1982). The
quotation in the text above is the only known instance ofDr Y.N. Chen repudiating a vortex-related mechanism!

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
10- 1
8'

8
"
 10- 1
Oj
<:i
, 10- 2
"

 10- 3
8
:':; 10- 2

"
.S
&10- 3
><

J
g 10- 1
"
O)
S
'i::
 10- 2

10- 3
10- 4
10- 5
(a)
10- 3 10- 2 10- 1
Predicted Ynns (cm)
(c)
10- 3 10- 2 10- 1
Predicted Y max ID
(b)
863
10- 4 10- 3
Predictedy*IDh
10- 2
Figure 8.39 Small, 'subcritical' vibration amplitude of solitary cylinders and arrays in axial flow,
Experimental values (ordinate) plotted as functions of calculated ones (abscissa), according to
the empirical or semi-empirical expressions of (a) Reavis (1969), (b) Pa.i'doussis (1969), and (c)
Y.N, Chen (1970a,b): -, line of 100% agreement; \ \ \ \ \, experimental data (Pa.i'doussis 1981).
to E: n ; specifically that fy* I U = kE: n , where k is arbitrary. Hence, eventua1ly he finds the
following remarkably simple expression for y*:
-ť- = [1- (UIU e )2r 1 ( f3U ) 2,
Dh U e
(8.97)
where n = 1 in U e , f3 oe (2ka) 1 /2 , and it is recalled that  is the incident turbulence
intensity level; empirica1ly, Chen proposes f3 = 1, 1, and 2, for 'quiet', average, and
particularly 'noisy' flow conditions, respectively.
Theory and experimental data from many sources have been compared in a plot of y* I Dh
versus f3 U I U e (Y.N. Chen 1970b). It is clear from the plot that [1 - (U I U e )2] ::: 1 to a
very good approximation, and hence, in view of (8.97), a simpler notation for the axes is
possible (Pai'doussis 1981), which is used here in Figure 8.39(c), It is seen that prediction
within one order of magnitude is achieved,
Does this mean that the parametric excitation hypothesis is sound? Alas, no! By the time
various assumptions have been introduced, notably that the system responds at Q ::: Qnj
(secondary-resonance assumption), that fy* I U = kE: n and f3 oe , this is really a semi-
ernpirical theory, not offering a definitive test of whether the excitation mechanism is
related to a parametric resonance or not. Indeed, these assumptions, reinterpreted to state
that: (i) the cylinder responds at its (first-mode) natural frequency; (ii) the amplitude
is proportional to E: n U, hence to U 2 , and also to the incident turbulence level , this
arnounts to a forced-vibration model in disguise, Indeed, this is the main reason for its
success!
Nevertheless, equation (8.96) does provide a simple means of prediction; in this regard
the only pararneter not readily available is CT, and for that Chen recommends 0.002 <
CT < 0.003. In retrospect, the main interest in this model, nowadays, is tutorial, showing
that, even with the wrong precepts, it is possible to devise a model giving reasonable and
useful predictions.

864
SLENDER STRUCTURES AND AXIAL FLOW
8.4. 1 (c) Forced vibration models
A complete forced-vibration model would take into consideration all the forces which
could directly excite lateral motion of the cylinder. However, since most of these forces
cannot be characterized easily, and hence are not known a priori for a given system,
the forced-vibration models invariably take into account only those which are known,
namely, only the forces arising from pressure perturbations in the turbulent boundary layer
surrounding the cylindrical structures. Statistical correlations ofwall-pressure fluctuations
of turbulent boundary layers are available, and these have been utilized to obtain estimates
of vibration amplitude.
Let L[y(x, t)] = O be the equation of small free lateral motions of the cylinder in axial
flow, for example, equation (8.21); then, if p(x, t) is the instantaneous pressure on the
wall of the cylinder, the equation of motion for forced vibration due to this pressure field
is given by
L[y(x,t)] = 1 2 7Tp(x,e,t)COse!Dde, (8.98)
Theories based on the forced vibration model have been developed by several research-
ers (Reavis 1967, 1969; Kanazawa & Boresi 1970; Chen & Wambsganss 1970, 1972;
Gorman 1971; Ohlmer etal. 1972; Lin 1984), They differ from one another in the form of
the equation of motion and the pressure-fluctuation correlations used. Reavis (1967, 1969),
who was the first to propose such a model, and Gorman (1971), who extended Reavis'
work concentrating mainly on air-water mixtures simulating two-phase flows, both use
essentially the damped beam equation in sti11 fluid as the equation defining L [y(x, t)].
Thus they do not take into account the terms explicitly dependent on the mean flow, This
has been done by Chen & Wambsganss (1970, 1972), approximately. A version of this
analysis is given in Section 8.4.2 in fair detail.
Conceming the pressure-fluctuation correlations used, Reavis uses the Bakewell
et al. (1962) measurements of turbulent wall-pressure fluctuations for flow in a pipe.
Gorman made his own measurements of wall-pressure fluctuations, also on the flow tube.
On the other hand, Chen & Wambsganss use measurements by Bakewell (1968) on wall-
pressure fluctuations on a body of revolution, measurements by Clinch (1969), as well as
other sources. More recent work on the characteristics of the excitation field are given in
Section 8.4.3(a).
Some of the assumptions commonly made in these theories are as follows:
(i) all far-field pressure fluctuation components are neglected, as well as those arising
from turbulence induced by supports, grids, etc., from boiling and flow-régime
changes;
(ii) no correlation is assumed between the pressure fields on adjacent cylinders
(in the case of multi-cylinder structures);
(iii) the motion of a cylinder is assumed to have no effect on the pressure field, nor on
the motion of adjacent cylinders;
(iv) the proces s is assumed to be ergodic and the pressure field homogeneous;
(v) the cylinder is lightly damped, and response in the first-mode of the cylinder is
considered to be dominant.
Of these assumptions, a few words ought to be said on (i i) and (iii), which effectively
mean that no distinction is made between the response of a solitary cylinder and one in a

SOLI'iARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
865
c1uster. This is the usual assumption, and this is why, when it comes to experimental data,
they are al1 indiscriminately mixed together, irrespective of cluster geometry, with those
from solitary cylinders. A more sophisticated theory, however, is given in Section 9.4.
The final product of each theory is an expression for the root-mean-square of the
displacement produced by the boundary-layer-induced pressure field. We shal1 now con-
sider the measure of agreement between the vibration amplitude thus predicted and
experimentally measured values.
Reavis (1969), using a much simplified but similar analysis to that in Section 8.4.2
and utilizing for the excitation field the Bakewell et al. (1962) data and wall-pressure
correlations in a pipe, in the absence of anything more appropriate at the time, obtained
the following expression for the r.m.s. displacement of the cylinder:
C1/tD1/t D h 1/tLDN I / 2 pU1)I/2
Yrms = 3/2 1/2 '
mf l I
(8.99)
where C is a factor ranging from 2 to 200 approximately, depending on Dh/ L and on
whether the system is of low- or high-noise level, 1/tD, 1/t D h and 1/tL are semi-empirical
pressure-field correlation functions [of 0(1) for smal1 f, and depending on Strouhal
numbers based on the subscripted quantities], N is the number of cylinders in the cluster,
1) the kinematic viscosity, JI the first-mode natural frequency, and I the corresponding
damping factor; the other symbol s are as in Section 8.2 (Reavis 1969; Pai'doussis 197 4b). t
The original expression did not involve C.
Reavis compared the theory to some of the available vibration data. While the trends
agree, theory underestimates the observed amplitudes by one to two-and-half orders of
magnitude. As a result, Reavis empirical1y modified the theory, incorporating C to achieve
agreement on the average. The result is shown in Figure 8.39(a).:I: Agreement is now within
one order ofmagnitude, no better or worse than by Y.N. Chen's model in Figure 8,39(c).
Gorman's (1969, 1971) theory is similar to Reavis', minus the empirical adjustments;
it also differs in that Gorman obtained his own fluctuating pressure correlations and PSDs
experimentally, in both water and air-water mixtures (simulating two-phase flows). In
all cases, the near-field pressure fluctuations were measured on the flow-channel wall,
reasoning that the pressure field must be clo sely similar to that acting on the cylindrical
structure within. The final expression for the vibration amplitude, slightly re-written, is
[ - ] 1/2
1/tD1/tLD p(t)
Ynns = 4 3/2 f 3/2 ;-1/2 '
mJr I" I
(8.100)
where most of the symbol s have the same meaning as in (8.99); p(t) is the measured
time-mean per unit bandwidth of the differential-pressure-fluctuation field (from two dia-
metrically opposite points) in the vicinity of the first-mode eigenfrequency, wherein is
inc1uded the dependence of Yrms on U, p, simulated quality and other fluid properties. It
is noted that the functional dependence of Ynns on D, m, JI and I is identical to that in
Reavis' expression.
tThus, m is the mass per unit length, while in the original formulation Reavis used the total beam mass, and
then L appears in the numerator,
tS ee discussion in subsection (d) that follows, on 'maximum' versus r.m.s, displacement amplitudes.

866
SLENDER STRUCTURES AND AXIAL FLOW
Some results are shown in Figure 8.40(a), t for air-water mixtures, showing remarkably
good agreement between theory and experiment. We see that the comparison of amp1itudes
is made on linear scales for the first time. Good agreement has also been obtained for
single-phase water flows; an example is shown in Figure 8.40(b). Similarly good agree-
ment has been obtained for cy1inders in clustered arrangements, in air-water mixtures by
Gorman (1975a) - see Figure 8.40(c) - and by Cedo1in et al. (1971) for experiments in
nitrogen-water mixtures, using a modified form of Gorman's theory.
Finally, Chen & Wambsganss (1972) conducted their own experiment s in water flow and
compared the measured amplitudes to those oftheir theory (Section 8.4.2). Typical results
are shown in Figure 8.41, where good agreement is found. ff one disregards the rightmost
panel s in the figure [which correspond to a cy1inder in a relatively tight annular passage,
Dh/ D = O,5(Dch/ D = 1.5), vis-o-vis the rest], agreement is excellent, especial1y at the
higher flow rates.
The question arises as to why Chen & Wambsganss, Gorman and Cedolin et al. obtained
fair to excellent agreement with experiment, while Reavis did not, in spite of the similarity
of all the theories used.
We shall first compare Gorman's and Reavis' work, as they use essentially identical
analytical models. First of all, Gorman obtained his pressure fluctuation correlations and
vibration data from the same experimental rig, so that any 'system' disturbances present*
would be automatically incorporated in the excitation field used in the theory. This is
supported by the fact that certain 'pecu1iarities' evidently exist in Gorman's measured
pressure correlations for single-phase flows (Chen & Wambsganss 1970) as compared
to others' measurements. This could explain the good agreement between theory and
experiment obtained by Gorman for single-phase flows. In connection with Gorman's and
Cedo1in's et al. excellent results for two-phase flows, the above would also apply, but
there is another, more fundamental, reason in this case. Two-phase flows have notoriously
short memories, that is, internal damping in two-phase flows is quite high. Accordingly,
pressure fluctuations arising upstream of the cylindrical structure - other than plane
pressure waves - do not survive to be felt there. Hence, in two-phase flows, the near-field
essentially represents the total pressure field, so far as vibration of cy1inders is concerned.
We next compare Chen & Wambsganss' and Reavis' work. There are substantial dif-
ferences in the theoretical models, but their effect has been reasoned to be rather minor
(Pa'idoussis 1974c); therefore, the difference in agreement between theory and experi-
ment in the two cases must 1ie with the experiments. Chen & Wambsganss obtained their
experimental points from a well-designed circulation system with minimum 'system' dis-
turbances, so that the boundary-layer-induced pressure field was probably a substantial if
not the main component af the excitation field. Furthermore, most importantly, in their
theoretical ca1culations they used PSDs of the excitation field measured in the same exper-
imental rig. This contrasts with the experimental data utilized by Reavis which were fram
various saurces, in some af which system disturbances were probably cansiderable. It
may be reasoned that if pressure-fluctuatian PSDs and correlations had been available ta
Reavis for each set af the experimental data used and, better still, fmm the same apparatus
tlt is noted lhat a simulated quality of 40% corresponds to a void fraction a > 90% at the pressure and
temperature in this experiment - cf. Wallis (1969).
*Other than (plane) pressure waves which were electronically suppressed as is normal for near-tield pressure
correlation measurements. Possible sources of far-tield noise are pump noise, cavitation, large changes in flow
area, other discontinuities and large-scale separation upstream (Blake 1986; Gibert 1988).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
867
1.2
0, predicted
1.0 O, measured
 0,8
'"
o
X 0.6
J 0.4
o
0,2
O
O 10 20 30 40
(a) Simulated quality (%)
12 o, predicted
0, measured
1.0 o
 o
0,8 a
o;-
O
- o
X 0,6
J O
0.4 V Pump running "
" " D
0,2 'p .
ump not runmng
00 5 10 15 20 25 30 35 40
(b) U (ft/s)
2.4 0,06
 , predicted
2,0 -0-, measured 0,05
 . 4 6 6 6
1.6 0,04 I
'O o o
X 1.2 0.03 J
J o .
0.8 0,02
0.4 0.01
O O
O 16 32 48 64 80 96
(c) Simulated quality (%)
Figure 8.40 Comparison between Gorman's theoretical and experimental values of mid-span
vibration amplitude of a solitary cylinder: (a) for flows of air-water mixtures (Gorman 1971), mas s
flow rate 3100 kg/h; (b) for water flow (Gorman 1969). (c) Comparison in the case ofthe central
flexible cylinder in a duster of cylinders, in air-water mixture flow (mass flow rate 18100 kg/h);
(Gorman 1975a).

868
SLENDER STRUCTURES AND AXIAL FLOW
2.0
o
1.0 /
g 0.8
'" 0,6 o 0#0
o 0,5
X 0.4 ° o<ff
 0,3 00
000
'" o o o
0,2
Test I-A Test I-B Test I-C
0,1
10 100 10 100 10 100
(a)
 10
'"
o
X

100
10
100
10
100
10
(b)
U (ft/s)
Figure 8.41 Comparison between Chen & Wambsganss' theoretical and experimental values of
vibration amplitude. (a) Mid-span amplitude for clamped-clamped cylinders; (b) amplitude at
I; = 0.906 for cantilevered cylinders (Chen & Wambsganss 1972).
as used for vibration measurements, then the agreement between theory and experiment
could have been comparable to that achieved by the other researchers.
8.4. 1 (d) Empirica/ expressions
Empirical expressions for predicting small-amplitude flow-induced vibration have been
proposed by Burgreen et al. (1958), Basile et al. (1968)t and Paldoussis (1965b, 1966c,
1969). The latter is stilI being used* - main1y for convenience, but also for cross-checking
the results from more sophisticated models (Blevins 1990; Au- Yang 1999,2001).
The basis for using empirical expressions for predicting the vibration amplitude is
no different, in principle, than the analogous case in heat transfer where, for instance,
knowing that the Nusselt number may depend on the Reynolds and Pradtl numbers we
may construct an empirical expression from which the heat transfer coefficient may be
ca1culated, In the case of vibration induced by axial flow, knowing that the amplitude
depends on u, f3, s, and the frictional coefficients (and, hence, the Reynolds number), etc.,
one may proceed in the same way,
tBasíle, Fauré & Ohlmer's is a very va1uable EURATOM study, with an extensive experimental component,
well worth consulting even today,
*Ever since receiving Den Hartog's (1970) gracious sea1 of approva1,

SOLITARY CYLINDRICAL STRUcrURES IN AXIAL FLOW
869
Paidoussis (1965b, 1966c) conducted many experiments with a single cylinder in
a channel, as well as with a 19-cylinder bundle, varying the length L, diameter D,
hydraulic diameter Dh, flexural rigidity E I, mass per unit length m, the first-mode
eigenvalue, al == AIL, and flow velocity. To these were added the experimental data
of Burgreen et al. (1958), SOGREAH (1962), Quinn (1962), Rotstrom (1964) and
Rostrom & Andersson (1964a,b), the last group involving hot and cold water, two-phase
flows and superheated steam. All these were put together (by systematic trial and error,
and with the benefit of physical understanding), and the following correlation has been
obtained:
Ymax = a-4 [ u1.6t:1.8ReO.25 ] [( Dh ) O.4 ] [  ] [ 5 x 1O -4K ]
Dli + u 2 D 1 + 4,8 ,
(8,101)
where u, t: and ,8 are as in Section 8.2, Re = U Dh/V, al = AIL is the first-mode
eigenvalue of the system, and K is an empirical factor; K = 1 for quiet, laboratory-type
conditions, while K = 5 for realistic, industrial conditions. The amplitude Ymax represents
a 'maximum' mid-span amplitude; since the vibration is random, this is meaningless, but
it may be taken to mean 2Yrms to 3Yrms, approximately.t
The range of the parameters covered in constructing the empirical correlation is given
in Table 8.9.* The degree of 'success' achievable, for the same data as those used in con-
strncting it, is shown in Figure 8,39(b), showing agreement within one order of magnitude.
Subsequent testing with new data showed a similar degree of agreement, even for multi-
span bundles of cylinders (Pavlica & MarshalI1966). It is of interest to look at Figure 8.42,
showing the original form of the data of Figure 8.39(b). It is noted that the line of best fit
for small amplitudes flattens out to a nearly horizontallevel. This is commonly attributed
to far-field and mechanically transmitted excitation, effects eventually overshadowed by
'trne' flow-induced vibration at higher flows; and this is why this part ofthe data has been
excised in Figure 8.39(b) [see also, e.g., Figure 8.41 and Mulcahy et al. (1980)],
Table 8.9 Range of parameters covered in
constructing empirical expression (8,101);
[fmm Pai"doussis (l965b)).
Parameter
Range of va]ues
u 2
Re
t:
2,1 x 10- 3 -8,0 x 10-)
2,6 x 10 4 -7 x 105
54,3-190
4.9 x 10- 4 -6,2 x 10- 1
1.45-4.56
f3
a)
t A more eo]ourful, practical solution is offered by Reavis: Ymax is " , , the maximum disp]aeement from
equilibrium to be expected if one were to sean through an oseillograph record of rod vibration about 5 ft in
length. This length of record corresponds to the author's experienee that a data sample is about two arms'
length long'. This fundamentally unsatisfactory answer is nevertheless a reasonable one from the practical
point of view!
*To peg these dimensional]y, it is noted that D = 9.5-]5.9 mm.

870
SLENDER STRUCTURES AND AXIAL FLOW
o Burgreen et a/, (1958)
10 - 3
C::::J Quinn (1962)
D SOGREAH (1962)
E8čI Rostrom & Andersson (1964)
EJ Paidoussis & Sharp (1967)
Empirical expression
10-\
,
,
,
I
I
,/
I
,
,
I
I
,
I
.. --'
8 2
" 10-
J
10- 4
14 13 12 Il
-4 [ u 16 el 8 Re 025 ] [ D j 04 ] [ {32/3 ] [ _ ]
aj 2 ---.!!.. - 5 x 10 4 X K
I+u D 1+4{3
Figure 8.42 Agreement between measured (ordinate; from various sources, as shown in the
legend) and predicted (abscissa) amplitudes of vibration according to the Pai'doussis empirical
expression (Pai'doussis 1974b).
Further experiments have been conducted to obtain an assessment of the effect of large
departures from axial flow (i) by putting a cylinder in cross-flow or an inclined plate
upstream of the test-cylinder, (ii) swirl, with the cylinder off the centreline, which yielded
maxima of eight and 100 times larger vibration levels than without them. t These effects
are not inc1uded in the empirical expression.
Figure 8.43(a) shows typical acceleration spectra as functions of the mean flow velocity,
demonstrating that most of the response is at ft  40 Hz. Then, the flow-straightener was
removed from the circulation 100p, and a 15.9 mm cylinder was mounted in cross-flow,
four test-section diameters upstream ofthe test-cylinder; as seen in Figure 8.43(b), a very
substantial increase in vibration amplitude has been produced, mainly due to the swirl (in
the absence of a flow-straightener),
8.4, 1 (e) Oiscussion and conclusion
The cumulative evidence at this point is compel1ing that the small-amplitude vibration
under discussion is a forced vibration due to correlated random pressure fluctuations on
the surface of the cylinder. These pressure fluctuations are fairly wide-band - see, for
t See a1so Fomichev et a/. (1973) for some results involving self-generated swirl due to asymmetric entry
into the flow channel containing the cylinder; refer to Jonsson & Sparrow (1966),

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
871
1.4
1.2 120
""'
 1.0
"
,g 0,8

.!:! 0.6
"
u
u
< 0.4
0,2
O
(a)
1.4
1.2 120
3 1.0
"
, 0,8

.!:! 0.6
"
u
u
< 0.4
0,2
O
(b)
Figure 8.43 The effect of increasing disturbance on vibration amplitude: (a) at minimum disturb-
ance level; (b) with the flow straightener removed and a cylinder in cross-flow inserted upstream
(Pai"doussis 1966c).
example, Figures 7.76, 8.45 and 9.48. The cylinder extracts energy in the vicinity of its
natural frequencies, most1y the first (Figure 8.43), acting as a band-suppression filter.
As seen in Figures 8.40 and 8.41 for instance, the forced vibration model, the most
sophisticated version of which is given in Section 8.4.2 [modified in Section 8.4.3(b)],
is quite sound: if one knows the pressure field accurately enough, one can predict the
resultant vibration quite well. Even under 'realistic' flow conditions, Ohlmer ct al. (1972)
have shown that the vibration amplitude can be predicted fairly well, provided that the
actual pressure field is used in the theoretical model.
The problem is that, at the moment, there exists no means for predicting fully the
characteristics of the pressure field in a yet unbuilt system. Clearly, the boundary-Iayer-
induced pressure field is but one component of the excitation field, often a very small
component - with the possible exception of two-phase flows. If one needs meas-
ure the actual pressure-fluctuation characteristics in a particular system in order to

872
SLENDER STRUCTURES AND AXIAL FLOW
predict the vibration amplitude, one might as well measure in situ the actual vibration
amplitude.
The importance of the far-field component of excitation has been condusively demon-
strated by a remarkab1e series of experiments by Kadlec & Ohlmer (1971), ln these
experiments, the same cylindrical structure was mounted in the test-section of four dif-
ferent flow loops, with several additional modifications. The pressure field was found to
vary quite considerably from one flow arrangement to another, under nominally identical
conditions; in fact, the r.m.s. values of the pressure fluctuations were found to be 7-30
times higher than corresponding values in 'ideaľ flow channels. The spectral densities
were also quite different. The r.m.s. vibration amplitudes were found to vary by as much
as 300 and 400%. Similar supporting evidence on the importance of far field noise - or,
otherwise put, the incident noise at the inlet of the test-section - was obtained by Card
using an organic coolant and varying the temperature. Card found that variations in r.m.s.
pressure amp1itude of 100% could lead to similarly large variations in vibration amplitude
(Pai'doussis 1974b).
Hence, some degree of empiricism must remain in the predictive tools for this vibration.
See also discussion in Section 8.4.3.
8.4.2 Chen Br Wambsganss' analytical model
A slightly modified form of this model is given here, so as (i) to correct for the, in this case
fairly inconsequential, error in the equation of motion, as per Pai'doussis (1966a, 1973a)
and as discussed at the end of Section 8.2.2, and (ii) to bring things doser in form to
all of the foregoing in this chapter (Pai'doussis 1974b,c). An excellent discussion of the
essentials of turbulent wal1-pressure fluctuations, with an extensive bibliography is given
by Blake (1986, chapter 8).
The main assumptions underpinning the model have already been presented and dis-
cussed in Section 8.4.I(c). According to assumption (ii), we consider a single cylinder
in axial flow. The random fluctuating pressure field, p(x, (J, t), on the surface of the cyl-
inder is translated to an equivalent lateral force field, f(x, t), by integrating around the
cylinder, as shown in Figure 8.44(a); the cylinder is now subject to a distributed random
f(x, t)
(a)

f(x, t)
(b)
x
Figure 8.44 Translation of (a) instantaneous random pressure field into (b) an equivalent random
force field acting on the cylinder (Pa1doussis 1974b).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
873
force f(x, t) as in Figure 8.44(b), and its motion may be ana1yzed by standard means.
Hence, the equation of motion is the dimension1ess form of equation (8.98),
L [1] (, r)] = (L 3 jEl) 1 27r p (x, O, t) cos O i D dO,
= (L 3 jEI) 1 7r 6p(x,O,t)cosO iDdO,
where L [1](, r)] is the left-hand side of equation (8.27b), in which, assuming a horizonta1
cy1inder, y = O; 6p is the pressure difference from two diametrally opposed positions
on the circumference. The use of 6p instead of P is important, in that most ofthe far-field
contributions, travelling down the test-section as 1-D plane waves are thereby cancelled.
The genera1 boundary conditions of equations (8.29a) are assumed to app1y.
The disp1acement 1](, r) is expanded in terms of eigenfunctions CPn () as follows:
(8.102)
1](, r) = L CPn()qn(r).
(8.103)
n
Since classical norma1 modes do not exist in this case, the eigenfunctions of the following
sub-system are used:
cp"" + {u 2 [1 - iSCT (1 - i8) - i(1 - 8)Cb] - 8r} cp" - ACP = O,
(8.104a)
with
cp'" + KOCP = cp" - KOCP' = O at  = O,
cp'" - KICP = cp" + Ktcp' = O at  = 1,
where A is the eigenva1ue to be detennined, and the primes denote differentiation with
respect to  ' In general, this eigenva1ue problem is not self-adjoint. According1y, to be able
to decouple the equations of motion we must a1so define the adjoint eigenva1ue problem,
which of course will yie1d the same eigenva1ues but a different set of eigenfunctions,
1/1n()' The CPn() and 1/1n() form a biorthogona1 set (see Section 2,1.6); that is, each of
the eigenfunctions CPn is orthogonal to al1 eigenfunctions 1/1m, for n =I- m.
Substituting now equation (8.103) into (8.27b) while making use of (8.104a), multip1y-
ing by 1/1m and integrating from  = O to 1, we obtain
(8,104b)
.. " . 213 1 / 2 " b ' l f3 1/2 ( ) .
qm +a amnqn + u  mnqn + '2 SCNu +SC qm
n
n
+ i SC NU 2 Lbmnqn + i SC TU 2 Ldmnqn + Amqm = Qm,
n n
(8.105)
m = 1,2,3, ..., where
a mn = 11 CP"1/1m d / 11 CPm1/1m d,
d mn = 11 cP 1/1m d / 11 CPm 1/1m d,
b mn = 11 CP1/1md / 11 CPm1/1md,
Qm(r)= 11 Q(, r)1/1md/ 11 CPm1/1md.

874
SLENDER STRUCTURES AND AXIAL FLOW
As may be seen, equation (8,105) is not decoupled. Making the usual assumptions for
small damping, the off-diagonal terms associated with internal and Coriolis-type damping
are neglected; similarly, for the fifth and sixth terms in the equation. t Accordingly, we
obtain the following decoupled set of equations:
.. [ 2 R1/2 b 1 R1/2 ( )] .
qm + aa mm + fJ U mm + 2 fJ SCNU + SC qm
+ [A m + !SCNu 2 b mm + !UTu2dmm] qm = Qm,
(8.106)
which may be re-written in the standard form as
.. 2 ' 2 Q
qm + l;mwmqm + wmqm = m'
(8,107)
We define the correlation function of the generalized force field by
1 l T
RQnQmCrO) = lim _ 2 Qn(T)Qm(r + To)dT
T---+oo T -T
1 tt
= MnM m 1010 Vrn(S)Vrm (s + so)RQQ(S' S + sO, TO) d s des + so),
(8,108)
where
I l T
RQQ(s, S + sO, TO) = lim 2T Q(s, T)Q(S + so, T + To)dT,
T---+oo -T
M n = 11 <PnVrn d s .
Note that Q (s, T) is the force field and Qm (r) is the genera1ized force field, as defined in
equation (8,105). Now, the cross-correlation spectral density is the Fourier transform of
the corre1ation function,
<f?QQ(s, s', w) = ! 1 00 RQQ(S, s', To)éJt"o dTO,
Jr -00
where, for simplicity, we write s' == s + SO from here on, and
,-fo, ( ) _ 1 1 00 R ( ) iúno d '
'i'QnQm W - - QnQm TO e TO,
Jr -00
(8.109)
and the Fourier transform of equation (8.108) may be written as
1 1 1 1
, , ,
<f?QnQm(w) = MnM m 1 o Vrn(S)Vrm(S )<f?QQ(S,S ,w)dsd s .
(8.110)
t A more general, but a1so more complex method is given in Appendix R.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
875
We may similarly write down expressions for the cross-correlation function and cross-
correlation spectral density ofthe displacement from equation (8,103), namely
RlIlI(' /, TO) = L L Rqnqm(TO)4>n()4>m(f),
n m
<I>1I11(' f, w) = L L <I>qnqm(w)4>n()4>m(/)'
(8.111)
n m
where
Rqnqm(TO) = lim _ 2 1 j T qn(T)qm(T + To)dT,
T-+oo T -T
and <I>qnqm (w) is the Fourier transform of Rqnqm (TO).
From linear random vibration theory, we know that
<I>qnqm(w) = IHn(w)Hm(w)I<I>QnQm(w),
(8.112)
where Hn(w) = [(w - ( 2 ) + 2{nwnwirl. Hence, substituting equation (8.112)
into (8,111) and using (8.110), we obtain the cross-correlation spectral density of the
displacement
<I>1I11(' f, w) = L L IHn(w)Hm(w)l 4>n()4>m(/)
MnM m
n m
X 1111 <I>QQ(, /, w)1frn()1frm(f)d d/.
(8.113)
We now presume that we can write
<I>QQ(, /, w) = AQQ(w)\I1QQ(, f, w),
(8.114)
which will be justified a posteriori by equation (8,119). AQQ(w) is the power spectral
density of the force field and \I1QQ is a spatial correlation function. Substituting this into
equation (8.113) and letting o = O ( = /), we obtain the power spectral density of the
displacement,
<I>1I11('' w) = AQQ(w) L L IHn(w)Hm(w)IJ;m(w)4>n()4>m(),
n m
(8.115)
where
J;m(w) = Mnm 1111 \I1QQ(' f, w)1frn()1frm(/)d df
(8.116)
is the so-called cross-acceptance, ff n = m, J;m is the joint acceptance which, like the
receptance (Section 4.6), is a measure of the ability of the forcing field to excite the nth
mode of the structure. The cross-acceptance, on the other hand, is a measure of the
contribution due to coupling between different modes,

876
SLENDER STRUCTURES AND AXIAL FLOW
Finally, the mean-square displacement is given by
lP(O = 1 00 <I>1]1]('' w)dw.
The solution, as given by equations (8,115) and (8.117) may be simplified by assum-
ing that damping is sufficiently small for coupling between the modes to be negligible;
the doub1e summation in equation (8.115) may then be reduced to a sing1e summation.
Moreover, if we assume that the response is predominantly in the first mode, we can
consider on1y the 1eading term in the series. Hence,
(8.117)
<I>1]1]('' w)  AQQ(w)Hl(w)J1 1 (w)<pi().
(8.118)
Next, we have to corre1ate the pressure field to the force fie1d. Since the pressure field is
assumed to be homogeneous, pressure corre1ations wil1 depend on the separation between
measurement points rather than absolute positions. Thus, the cross-corre1ation spectral
density wil1in genera1 be <I>pp(xo, eo, Q), wherexo = IXI -x21 andeo = lel - e21. Using
Corcos' (1963) model, one may write
<I> pp (xo, eo, Q) = App(Q)'¥1 (QXO/ Ve) '¥2(QaeO/ Ve),
(8.119)
where A pp (Q) is the power spectral density of the wal1-pressure fluctuations, '¥ I and '¥2
are spatial corre1ation functions in the axia1 and circumferentia1 directions, respective1y,
Ve is the convection velocity (cf. Section 7,7.3), a is the radiu s of the cylinder, and Q is
the radian frequency.
Different forms of the functions in (8.119) may be used, depending on specific applic-
ations and available measurements, Here, we give those by Chen & Wambsganss (1972)
[see a1so Wambsganss & Za1eski (1970)], a1so given in Chen (1987):
'¥1(QxO/VC> = exp(-O.lI Qx o/V e l),
'¥2(Qaeo/Vc) = exp(-0.55I Qae o/V e l),
Ve! V = 0.6 + 0.4 exp [-2.2(Q8* / V)] == s,
(8.120)
where 8* is the boundary-1ayer displacement thickness.
The power-spectral density of the pressure field in dimensionless form is written as
<I> ( f ) _ App(f) '" { 0.272 x 10- 5 / SO,25,
pp - p 2V3D h - 22.75 x 1O- 5 /S 3 ,
S<5
S>5
(8,121)
in which S = QDh/ V and f = Q/2Jr - see Figure 8.45(a). An alternative, more
recent compilation is given in Figure 8.45(b); but see also Section 8.4,3(a). With App(f)
specified, AQQ(w) and '¥QQ(, ', w) may be computed - both needed for determining
172() - namely,
16 1/2- 2 2 3
AQQ(w) = 2 f3 8X e u <I>pp(w),
Jr
'¥QQ(,f,w) =exp(-O,lal -fj)cos( -'),
(8,122)

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
877

'S
"'<:I,

'1:
;::j
10- 6
/0-7
0.1
(a)
10- 3
/0-4
 10- 5
'S
"'<:I,
;:o:: 10- 6
'-,
'1:
 /0-7
10- 8
/0-4
(b)

''''''-10
1 /O
S fDh/U
10- 2 /O-I
k*(mm- I )
Figure 8.45 (a) Dimensionless PSDs or measured pressure fluctuations on a cylinder and the
ap pro xi mat ion used in the model (Chen & Wambsganss 1972). (b) An altemative compilation by
Mu1cahy et al. (1982); k* = JI U is a wavenumber.
where
2 = V { Jr _ v[1 - exp( -2Jrv)] }
X 2(1 + v 2 ) 1 + v 2 '
j31/2w QL j31/2w
v = O,275(QDjV e ) = 0.275 -, (j = - = -.
esu Ve sV
- Dh
(j =-,
L
(8.123)
It is with the aid of this model that the results shown in Figure 8.41 have been computed.
The interested reader is also referred to Wambsganss & Chen (1971), wherein one may

878
SLENDER STRUCTURES AND AXIAL FLOW
also find an approximate expression
0.018K D1. 5 Dh,5 U 2 cpl (x)
Yrms(x) = 15 '
LO.5 JI' (m + M)ŠO,5
(8.124)
where the symbols are as before, M is the added mass per unit length [see Section 2.2.2( a)],
CPI is the first-mode eigenfunction and K = 7.85 N s2.5 /m 5 ,5 or 2.56 x 10- 3 lb s2 5/ft5,5;
care must be exercised in the units,t The similarity of this result with that of Gonnan's
and Reavis', equations (8,100) and (8.99), is now evident.
8.4.3 Further work and refinements
What has been presented in Sections 8.4.1 and 8.4.2 constitutes the basics of small-
amp1itude subcritical vibration. Here, some further work on the characterization of the
excitation field and on the mechanism itself is discussed briefiy.:j:
8.4.3(a) The excitation pressure field
It should be noted that most of the vast amount of work conducted on turbulent fiows
focuses on the velocity field (of interest to fiuid mechanicians) and very little on the
wall-pressure fiuctuations, especially in complex geometrie s such as clustered cylinders
(of interest to us here). Further to the studies discussed in the foregoing to characterize
wall-pressure fiuctuations in fairly unencumbered, 'open' channels, some work has been
conducted for particular geometries, for example, in annuli and cylinder clusters.
A careful and valuable study of the turbulent pressure and velocity fields in concentric
annular air-fiow has been conducted by Wilson & Jones (1983), providing particularly
useful data on the wall-pressure PSDs at low values of D Dh / U, which is the range of direct
excitation for practical engineering structures. It is shown that the PSD is fiat ('horizontal')
all the way down to DDh/ U max = 0.05, and that it 'breaks' at DDh/ U max  10. Thus,
A pp (f) should be fiat rather than sloping, and furthennore slightly lower than shown
in Figure 8.45(a) for small J Dh/ U, Hence, some amount of far-field excitation remains
in the Chen & Wambsganss (1972) model as applied and, as discussed at the end of
Section 8.4,1 (c), this contributes to the superior agreement between theory and experiment.
Mulcahy et oZ, (1980) have studied the decay of upstream disturbances produced by
different kinds of 'grids' [Figure 8.46( a)] with distance in annular geometries, The decay of
turbulence intensity (Tu) is slow, even for a rathernarrow annular passage (Dh / D = 2.71),
as shown in Figure 8.46(b), not decaying fully until beyond the test section exit. For a
narrower annulus (Dh / D = 0.85), however, this occurs at x  ! L.
The effect on vibration is ordered in the same way as in Figure 8.46(b): the slower
the decay in 1\1, the higher is the vibration amplitude, as shown in Figure 8.47 - cf.
Pai'doussis' (1969) experiments.
tPor example, in the British units, m and M must be given in slugs/ft, that is, lb s2 /ft 2 , and the rest in ft.
Note also that the value of K in SI units given by Chen (1987) is incorrect - as also found by Au-Yang (2001,
section 10.3); that given here has been corrected,
*Much of this work was done thanks to a very ambitious and extensive research programme at the Argonne
National Laboratory, administered by General Electric Company (San José, CA) and funded by the US
Department of Energy,

A
.
c
(a)
SOLITARY CYLINDRICAL STRUcrURES IN AXIAL FLOW
879
B
W
\2V
0.3

' 0,2

.6
"
5
"S
 0,1
f-
D
Grid position
/" Test rod
O 10
O
-5
20
30
(b)
Axial position, xlDh
Figure 8.46 (a) Different 'grids' located just upstream of the test-section, and (b) their effect on
turbulence intensity within the test-section containing a centrally located test-cylinder, for U =
8 mis water flow; Dch = 50,8 mm, D = 13.7 mm, Dhl D = 2,71. The measurements were made
at a radial distance r, such that (2r - D) I Dh = 0.5 : o, no grid; O, grid A; o, grid B; V', grid D
(Mulcahy et al. 1980),
400
390
380
50
S
 40
J
30
20
10
O 2 4 6 8 10 12
U (mis)
Figure 8.47 Vibration amplitude of a cylinder in annular flow versus mean flow velocity, with
various 'grids' just upstream of the test-section - see Figure 8.46(a): <>, no grid; O, grid A;
o, grid B; V', grid D (Mulcahy et a/. 1980).

880
SLENDER STRUCTURES AND AXIAL FLOW
Extensive measurements on wal1-pressure fluctuations within a seven-rod c1uster have
been made by Lin et al. (1981). The wall-pressure PSDs obtained are qualitatively similar
to that in Figure 8.45(b), for instance. It has been found that the PSDs near the entrance of
the c1uster (Dh = 8,6 mm) are quite c10se to those for annular flow with Dh = 37.1 mm,
for the 10w-frequency range (f < 1000 Hz), and lower than those for annular flow with
Dh = 11.7 mm. Within the c1uster, the PSDs are again similar to those for annular flow
with Dh = 37.1 mm for f < 200 Hz, but higher than those for Dh = 11.7 mm. These
results illustrate the dual effect of increased confinement: on turbulence production and
on turbulence decay, depending also on frequency. Generalizations are difficult to make.
Some additional measurements of the wall-pressure field in c1ustered cylinders
(Curling & paldoussis 1 992a,b ) are discussed in Section 9.4.
8.4.3(b) The excitation mechanism
The influence of the longitudinal and circumferential scales of turbulence (lx and le) on
the vibration are discussed by Lin (1984) - specifically the ratio of lx to the vibration
wavelength (2L), and of le to the cylinder circumference; in particular, the limiting cases
when both tend to zero or one of them to infinity - cf. Fung (1955). In real engineering
systems, other than aero- or hydronautical ones, the scale of turbulence is of the order of
the cylinder diameter or less.
As first observed by Ohlmer et al. (1972), the pressure field in typical engineering
systems is not axial1y homogeneous. Similar observations have been made by Mulcahy
et al. (1980) for annular geometries, as discussed in subsection (a) - see Figure 8.46(b).
Ohlmer et al. (1972) deal with this complicating feature of the excitation field by seg-
menting the cylinder in the analysis and considering the PSD of the pressure field to be
different in each segment, thereby obtaining superior agreement with the observed vibra-
tion amplitudes. This, of course, necessitates the characterization of the pressure field at
more than one point.
Lin (1987a) considers a modification of the excitation function in equation (8.119),
replacing App(Q) by App(Q, x) == CPpp(Q) exp(-bxj L), in which b = b(Dh, U) is an
empirica1 factor. Typical resu1ts are given in Figure 8.48. As can be seen, taking axia1
variation of the PSD into account great1y improves prediction. For the case of a narrower
annulus, however, the agreement is not as good. Unfortunately, this research has not been
pursued to the point of characterizing b for any arbitrary system geometry.
Fung (1983) considers another effect of nonideal flow conditions: the departure from
axisymmetry in the pressure field. The resultant modification to the theoretical model
of Section 8.2 involves the introduction of an azimuthal time delay, It is shown that the
predicted amplitude decreases as the time delay is increased; no experimental verification,
however, could be made.
Fina1ly, Lin & Jendrzejczyk (1984) conducted experiments on the flow-induced
response of a cylinder mounted eccentrical1y in a cylindrical flow channel, and found
that the small-amplitude vibration response increases with eccentricity, as shown in
Figure 8.49(a) - cf. the end of Section 8.4.l(d). Here Dh = Dch - D = 12,7 mm, D =
12,7 mm, and the eccentricity, e, is just the radial distance of the test-cylinder from the
channel centre. Non-negligible changes of the first-mode natural frequency and damping
factor with flow and eccentricity were a1so observed, Physica1ly, turbulence generation
is increased as the cylinder comes c10ser to the wal1, but so is the damping. Moreover,

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
881
S
-3
J 60
40 ?::w
20
O
O 2 4 6 8 10 12 14
U (mis)
200
Analytical
180
160
140
120
100
Figure 8.48 The effect of an axially nonuniform pressure field on the vibration of the central
cylinder in annular flow (Dh = 37.1 mm; Dh/ D = 2,71): V, experimental data; o, analytical
predictions considering the pressure field uniform; L, predictions taking the nonuniformity of the
pressure field into account (Lin 1987a).
o
10- 2 ..*
,. '"
.. '"
..
S o '"
.
 o '"
;:;
" . o
S " o
" o
u
oj 10- 3 o o
 '" o
'i3 o
 .
o
'"' "
o o o
 o
;:; "
 . o
10- 4 e
,



10°
10 1
U (mis)
10 2
t{}

(a)
Figure 8.49 (a) The response of a cy1inder placed eccentrically in a cylindrical channel: o, e / Dh =
O; L, e / Dh = ; D, e / Dh = ;., e / Dh = ft; (b) some typical cross-sectional pattems of the
vibration at different times for e/ Dh =  (four on top) and e/ Dh = ft (four at bottom); (Lin &
Jendrzejczyk 1984).

882
SLENDER STRUCTURES AND AXIAL FLOW
secondary flow is almost certainly generated, resulting in appreciable cross-flow over the
cylinder, which enhances the vibration. Interestingly, the vibration is generally orbita!
(whirling), especially at the larger eccentricities - Figure 8.49(b).
8.4.3(c) Multispan cylinders and vibro-impact
ln shell-and-tube heat exchangers the tubes are supported at several positions along their
length by baffte plates - see Figure 3,76 and the related discussion on modes and frequen-
cies. The same applies to some designs of nuc1ear fuel c1usters. Typically, the supports
are loose, to allow for thermal expansion and ease of assembly/disassembly, and hence
the support state is intermittent, thereby resulting in interesting nonlinear fluid-stmcture
interactions. These are discussed in Section 8.8.
ln the small-amplitude vibration of concem here, the uncertainties associated with the
actual forcing field far outweigh possible refinements of the type just mentioned, and
hence generally the analysis may be done as if each span were independent.
However, there are additional mechanisms of damping present in multispan cylin-
ders due to (i) dissipation and energy transfer to the intermediate supports and adjacent
structures as a result of vibro-impact, rubbing, etc., and (ii) the so-called squeeze-film
damping; see, for example, Rogers & Pick (1976), Shin et az.. (1977, 1978), Ko &
Rogers (1981), Pettigrew et aZ.. (1986a,b), Au-Yang (1986b), Axisa et aZ. (1988), Han &
Rogers (2001) and the references cited therein. Much of this work deals with cross-flow-
induced vibration, but the elements ofinterestc1early apply to axial-flow-induced vibration
equally.
8.4.4 Cantilevered cylinders
Most of the foregoing has been on cylinders with both ends supported; an exception
is Figure 8.41(b), Some very interesting experiments on the effect of trailing-end geo-
metry on the vibration of cantilevered cylinders have been conducted by Wambsganss &
Jendrzejczyk (1979).
Some of the results are surnmarized in Table 8.10. It is seen that the r.m.s, amplitude and
its rate of growth with flow are strongly dependent on the trailing-end shape. Moreover,
most ofthe ordering is counter-intuitive: some ofmost well-stream1ined ends generate the
highest amplitudes (H, E and F), but not always so (e.g" trailing-end G), These results are
qualitatively similar to Brepson & Léon's (1974) and Blake's (1986) ordering of trailing-
edge geometry for plate vibration, even though in that case the vibration is believed to
be vortex-induced, ln addition to the results given, information on the damping ratio is
also provided, but this does not help to understand the perplexing ordering in Table 8.10;
the near-wake stmcture and how it couples to motion of the cantilever must be quite
complex.
Some experiments with cantilevered cylinders with a blunt trailing end (probably similar
to A or B in the foregoing) in air-water mixtures simulating two-phase flow are presented
by Harris & Holland (1971), Some high-powered analysis (for that time) of the experi-
mental data gives strong support to the vibration being random (e.g., the autocorrelation
function is a decaying cosine wave and the PDF is Gaussian - cf. Section 5.8.1) and
hence excited by turbulence. The flow could be directed from the fixed end towards the
free end or vice versa, The latter was found to give higher amplitudes. In both cases the

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW 883
Table 8.10 Empirical relationship for the displacement response of a cantilevered cylinder
in axial flow as a function of trailing-edge geometry, Yrms/ D = aU b with U in mis; and
the response at U = 16,8 mis (Wambsganss & Jendrzejczyk 1979).
Trailing-end geometry Growth rate constants Displacement response
ax10 3 b Yrrns/ D Nonnalized
J [CJ 0.50 1.36 0.024 0,80
K C 1.41 0.97 0.024 0.80
G [:::> 2.63 0,84 0.028 0,93
A I 1.51 1.06 0.030 1.0
B ) 1.51 1.14 0.042 1.4
r> 1.26 1.26 0.046 1.5
H [> 2.60 1.02 0.046 1.5
C ) 1.10 1.34 0,050 1.7
D ) 1.89 1.29 0,074 2,5
E ) 1.01 1.58 0.084 2,8
F -=> 1.20 1.60 0,112 3.7
amplitude increases with void fraction almost linearly for ex. ::: 30%, as in Gorman's results
[Figure 8.40(a)].t
8.4.5 Closing remarks and acoustically excited vibration
The small-amplitude vibration of cylindrical structures in axial fl.ow has been shown to
be a forced random vibration, excited by turbulence-related random pressure fl.uctuations
on the suďace of the cylinders. However, apart from two-phase fl.ows, this is but one -
though sometimes the main - excitation source, far-field 'noise' being the significant
other.
The theoretical model for predicting the amplitude of the random vibration given in
Section 8.4.2 is sound and gives good results in all cases where the assumptions made
tNote that a vaid fraction af 30% corresponds to a simulated quality af < 10% (Wallis 1969),

884
SLENDER STRUCTURES AND AXIAL FLOW
in Section 8.4.1(c) are reasonab1y valid - in particu1ar, assumption (i) that the far-fie1d
components of the excitation fie1d are small. The excitation fie1d is generally not homo-
geneous [assumption (iv)], however, and the procedures proposed by Oh1mer et al.. and
Lin in Section 8.4,3(b) shou1d be used whenever feasib1e. If assumption (i) is not satisfied,
or if it is not known whether it is, then empirica1 expression (8.10 1), or equivalently (8.97),
shou1d be used for an approximate estimate,
The flow-induced vibration is a narrow-banded one, centred about the first-mode eigen-
frequency. This is simple for single-span cylinders, but not so clear-cut for mu1tispan
ones - see Section 8.8; however, a sing1e-span calcu1ation using the lowest system
eigenfrequency and mode shape gives an acceptable estimate.
The vibration amp1itude is sma1!, and hence the main concems in design are (i) fatigue
and (ii) wear at intermediate supports or with adjacent structures, if any. The vibra-
tion being random, the cumu1ative fatigue methodo10gy should be used (Crandall &
Mark 1963); see a1so Au-Yang (1999, 2001). In most of the problems that have arisen
(Section 8.13), however, the damage is re1ated to vibro-impact and other tribo10gica1
interactions, resu1ting in impacting or fretting wear (Ko 1993, 1997).
Fina11y, before closing this section, a few words shou1d be said about acoustic-
ally induced vibration. Simi1ar1y to pipes (Section 3.7.3) and shells conveying fluid
(Section 7.4.2), cy1inders in an acoustic medium may experience amplified vibration
when there is coincidence between the phase ve10city of pressure or acoustic waves and
structura1 waves,
This is studied by Hine (1973a), both theoretically and experimentally, in quiescent
fluid. In the experiments, a cylinder was centrally suspended in an air-filled channe1 (U =
O) and subjected to travel1ing acoustica1 waves generated by a 10udspeak:er. It is found
theoretica11y that vibrations may be excited for n = 1, where the azimutha1 dependence of
the acoustica1 ve10city potentia1 is cos ne (cf. Chapter 7). The maximum response occurs at
(dual) coincidence, when the frequencies and wave1engths ofthe acoustica1 and structura1
(flexura1 waves) are equa1.
Large acce1eration amplitudes were observed, of the order of 20g at frequencies of
the order of 1000 Hz for slender glass or a1uminium cylinders [e.g., L/ D "" 0(10 2 ),
D = 12.7 mm], in very good agreement with theoretica1 va1ues. Simi1ar calcu1ations and
experiments were conducted with multispan and clustered cylinders. The displacement
amplitudes Ad, however, are rather small [in the foregoing examp1e, Ad/ D "" 0(10- 4 )]
and the flexura1 mode excited rather high. Neverthe1ess, in rea1 systems they cou1d be
considerab1y 1arger, since the vibration amplitude is linear1y proportional to the pressure
amplitude of the acoustica1 mode. Thus, in the foregoing example, the sound pressure
leve1 was 109 dB (re 2 x 10- 5 Pa), whereas b10wers in a gas-cooled reactor have typically
been found to generate leve1s of 170 dB; hence, at dua1 coincidence, this gives Ad/D ""
0(5 x 10- 3 ), which begins to be interesting. This work has been extended to tak:e axial
flow into account (Hine 1973b).
The foregoing does not imp1y that all far-fie1d vibration excitation is via acoustica1-
structural coincidence, but serve s to show that acoustically propagated pressure
disturbances can indeed excite vibration and contribute to the observed vibration
amplitudes.
Further work on vibration excitation by sound, with acoustica1 waves impinging on the
cylinder transversely, considering a1so scattering and radiadion of the sound waves, has
been done by Yousri & Fahy (1976) and Lin & Chen (1977).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
885
8.5 T APERED CVLlNDERS IN AXIAL FLOW
The study of the dynamics of conical beams goes back to Kirchhoff and 1879; a definitive
analysis may be found in Wrinch (1922). Although considerable work has been done on
vortex shedding from tapered cylinders (truncated cones) and their dynamics in cross-
ftow [see, e.g, Zdravkovich (2003, Section 22,7)], the dynamics in axial ftow has not been
examined until relatively recently (Hannoyer 1977; Hannoyer & Pai:doussis 1979a,b). It
is a summary of this work that is given here.
The initial motivation for that study came fmm the following paradoxical finding, Uni-
form cylindrical cantilevers in axial ftow are less stable, the more streamlined the free end
(as discussed in Section 8.3); for example, an elongated ellipsoidal or conical end renders
the cylinder less stable than a hemispherical one. Yet, a conical cantilever, with a cone
ang1e small enough to preclude separation in axial flow - which may be considered as
the ultimate in an elongated 'enď - was found experimentally to be free of ftuidelastic
instabilities!
Hannoyer & Pai:doussis (1979a,b) consider the dynamics of hollow conically tapered
cylindrical cantilevers subjected to internal or external axial flow, or to both simultan-
eously. The hollow cantilevers could be of a variety of external-internal shapes, for
example, cylindrical-conical, or conical-conical; or they could be solid, as opposed to
hollow, conical cantilevers. The dynamics ofthese systems subjected to internal flow only,
and externally immersed in stagnant fluid, is discussed in Section 4,2, while the case of
simultaneous internal and external flows in Section 8.6. In this section, the system subjec-
ted to only external ftow is discussed. However, to avoid repetition, and as the internal flow
aspects have been thoroughly covered, the equations of motion for concurrent internal and
externa1 flows are derived here.
8.5.1 The equations of motion
Consider a nonuniform tapered cylinder of internal flow area Ai (x) and externa1 area
Ae(x), with corresponding diameters Di(X) and De(x), subjected simultaneously to
internal and essentially unconfined external flow of velocities Vi (x) and Ve, respectively,
and corresponding densities Pi and Pe, as shown in Figure 8.50(a).
An element of the cantilever, which is assumed to be hanging vertica1ly, is shown in
Figure 8.51. A ba1ance of forces in the x- and y-direction and a moment balance yield
aT aw a ( aw )
ax + Fit + Fet - (Fin + Fen) ax + ax Q + mg = O,
a Q aw a ( aw ) a2w
--a; + Fin + Fen + (F it + Fet) + ax T = m at 2 '
a [( * a ) a2w ] a.Mf
Q+- E -+E 1- +-=0,
ax at ax 2 ax
(8,125a)
(8.125b)
(8.125c)
where w (x, t) is the deflection of the canti1ever, E* is the viscoelastic damping coefficient
[cf. equation (3.22)], and .M f is the moment exerted by the ftow.
The forces due to interna1 flow, Fin and Fit, are given by equations (4.10a,b), and those
due to the external fluid, Fen and Fet. by equations (4.lla,b) for Ve = O.

886
y
SLENDER STRUCTURES AND AXIAL FLOW
 U e
r
Tapered cantilever
(a)
(b)
Figure 8.50 (a) Diagram of the flexible cantilever, generally tapered both externally and internally,
and generally subjected concurrently to internal and external flow; (b) the ogival end-piece at the
end of the cantilever (Hannoyer & PaYdoussis 1979a).
I
I
I
I
I
I
I
I
I
I
I
I
I
I
.
.
I
I
I
I
.
I
I
I
.
.
.
I
.
.
T
x
y
Figure 8.51 An element of the tapered cantilever showing the forces and moments acting on it.
Note that Q + (a Q / ax )8x etc. on the lower surface of the element have been omitted for clarity.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
887
With U e =F O, it is first noted that, a forteriori for a conical cantilever, the existence of
the boundary layer should not be neglected. Therefore, adopting the simplified approach
of Section 8.3.4, the inviscid fluid-dynarnic force is given by
( a * a ) { ( aw aw ) }
FA = - at + U e ax PeAe at + U e ax '
(8,126)
cf, equation (8.50), where u; = Ue(Ae/ A;), A; being the externallocal diameter of the
cantilever plus the annular boundary-Iayer displacement thickness.
Consider the hydrostatic and gravity forces next. Utilizing equation (8,6) and noting
that dpe/d.x  Peg, we have for the resultants in the x- and y-direction:
a a ( aw )
Fx = -(AePe) - PeAeg and Fy = - AePe- .
ax ax ax
(8,127)
Next, adopting equations (8.13) for the frictional forces in the normal and tangential
directions, we have
1 ( aw aw ) 1 aw
qen = "2 C fnPeDeUe at + U e ax + "2 MeCD at'
qet = CftPeDeU;,
(8.128)
where C fn, C ft are the frictional coefficients (C fn  C ft), Me is the dynarnic viscosity
of the fluid, and CD an empirical coefficient depending on the Reynolds and Strouhal
numbers.
Hence, one may write, correctly to first-order, the following relationships from which
Fen and Fet may be deterrnined:
aw aw a
Fet - Fenf); = qet - qen ax - PeAeg + ax (AePe),
aw ( a * a ) { ( aw aw ) }
Fen + Fet ax = - at + U e ax PeAe at + U e ax
aw a { aw }
+ qen + qetf); + ax AePef); .
(8.129a)
(8.129b)
One cannot presume that there are no moments induced by the externa! and interna!
flows. Careful study (Hannoyer 1977) has shown that, in general,
_ a 2 .Mf  PiAi dAi  (  + Ui ) 2 w
ax 2 27r dx ax at ax
PeAe dAe a ( a a ) 2
+--- -+U e - w,
27r dx ax at ax
(8,130)
in which .Mf is the moment due to both external and internal flows; cf. equation (4.12),

888
SLENDER STRUCTURES AND AXIAL FLOW
We next consider the axial boundary conditions - cf. equations (4,13), (4.15)-(4.20).
Substituting equations (4,lOa) and (8. 129a) into (8, 125a) and integrating from x to L gives
T + AePe - AiPi = T(L) + Pi(AU)iUi[ + LX [(PeAe - Pi, Ai - pA)g - qet]dx,
(8.131)
where pA is the mass of the cantilever per unit length, and A = Ae - Aj, To define
T(L) == T +AePe - AiPi Ix=L' the downstream boundary conditions have to be specified.
Two cases may be distinguished: one where the downstream end is free, or at 1east free to
slide axially, and the other where it is fixed.
For a free-to-slide axially or totally free end, the beam is considered to be tenninated
byan 'end-piece' oflength l, as shown in Figure 8.50(b), such that l « L. This end-piece
is subjected to the same kind of forces as the rest of the cantilever, plus a base drag.
N eglecting the effect of frictional forces over l, it may be shown that
T(L) = Db + {L (pA + PiAi - PeAe)gdx,
lL-l
(8.132)
and the base drag, Db, may be approximated by the quadratic form
Db = C tePe(AeU;)L + C tiPi(AiUhL + C tX(PiPeAiAeUlU;)If2, (8.133)
where the coefficients may be detennined experimentally; Cte and Cti are related to
external and internal flow, while Ctx is present when both flows coexist (and interact),
The following empirical formulae were established by Hannoyer (1977): Cte = 1.35/[9+
(2l/ De)3), Cti = 0.02/[1.25 + (2l / De)1/2], Ctx = 0.05[0.2+ (2l / D e )]/[O.l + (2l/ De)],
where De = De(L). Here and e1sewhere []L == []L-l, and strict1y De(L) == De(L -l).
ln the case of fixed ends, one may write
T(L) = To + [T' + AeP - AiP;]L,
(8.134)
where To is the external1y imposed tension, and the second term representing the fluid-
induced effects may be found by imposing the condition that the axial strain ex satisfy
fOL ex <lx = O, yielding
I I I {L 1
[T + AePe - Ai P;JL 10 Ae _ Aj dx
= Pi(AU)i (L Ui(L) - Ui(X) dx _ {L foLqet dx dx.
10 (Ae - Aih 10 (Ae - Aih
(8.135)

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
889
FinaIly, utilizing equations (8. 125a,c), (8.130) and (8.131), equation (8.125b) may be
re-wriUen as the equation of motion, in the form
( * O ) { 02 ( 02W )} piAi dAi o { 02 W 02w 202W }
E+E - - 1- ---- -+2U,-+U,-
ot ox 2 ox 2 2n dx ox ot 2 'oxot I ox 2
PeAe dAe o { 02w 02w 2 02w } { 02w 02w 2 02w }
- --- -+2U e -+U - +PiAi -+ 2U i-+ U ,-
2n dx OX ot 2 oxot e ox 2 ot 2 oxot I ox 2
{ 02w * 02w *02w dAe ( OW Ow )}
+ Pe Ae ot 2 + Ae(U e + U e ) oX ot + AeU e U e OX2 + U e dx ar + U e OX
- {T(L) + PiAiUi [Ui - Ui(L)]+ lX [(Pe - p)gA e + (p - pi)gAi - qed dX} :
1 ( OW OW ) 1 OW
+ 2Pe D e U e C fn ar + U e OX + 2 JLeCD ar
OW 02w
- (PeAe - PiAi - pA)g + pA ot 2 = O,
(8,136)
where T(L) is given by either (8.132) and (8.133) or (8,134) and (8,135),
8.5.2 Transverse boundary conditions
For a supported end, clearly the standard beam boundary conditions would apply. For
a free end the situation is complicated by the possible presence of the special end-
piece. Due to the small length of this end-piece, one may take the bending moment
induced by it to be negligible (02w / ox 2 = O at x = L), but not necessarily the shear
force.
Proceeding as in Section 8.2.4, the shear boundary condition in this case is
[( o ) 03w ] {L 02w
E* ot + E I ox 3 L-I + 1 L-I [(Pe - P )A e + (p - p;)AiJg ox 2 dx
l L 02w l L ( O O ) 2
- pA-dx - PiAi - + Ui- wdx
L-l ot 2 L-I ot OX
{L ( O * O ) [ ( ow ow )]
- f lL-1 Pe ot + U e ox Ae ar + Ue dx = O,
(8.137)
where the factor f is introduced to account for 3-D effects at the free end which reduce
the lift obtained by inviscid slender-body theory (Section 8,2.4); here []L-I == [ ]L for
l« L.

890
SLENDER STRUCTURES AND AXIAL FLOW
8.5.3 Dimensionless equations and method of solution
Before solving the system, the equation of motion and the boundary conditions are
rendered dimensionless by defining the following parameters:
 = x/L, 1J = w/L, 'l' = [EI/(pA + PeAe + PiAi)]ot/L2,
-2
8 = [Ai/Aeh=o, a e = Ae()/Ae(O), ai = Ai()/Ai(O), (8.138a)
£ = L/ De(O), Vd = [I/(pA + PeAe + pjAj)]oE* / L 2,
e = T(L)L 2 / EI(O), n = To L 2 / EI(O),
E ] 1/2 * * 1/2
U e = [PeAe/ 1 s=oUeL, U e = ue[Ae/ Ae]' Uj = [PiAj/ E/]s=OUi(O)L,
 3
Yj = (pi/p -1)8 , Y = [pAe/E1h=o gL ,
Ye = 1 + pe! P,
y* = (1_ť)y/£3,
C n = 2C fn/n,
Ct = 2C ft/n,
(8. 138b)
1/2
C v = (2/n)[Pe A e/ E/]s=o(f.LeCD/ Pe)£,
x: = l/ De(L),
Se = [1/ lAe(L)] f L Ae(x) dx;
L-l
(8.138c)
here X: = £ Xe, for Xe defined as in (8.30). In the ca1culations, the viscoelastic damping
model (characterized by E* and Vd) has been replaced by an ad hoc viscohysteretic
model, more appropriate for the materials of the tubular beams used in the experiments.
Accordingly, Vd is replaced by vd[1 + (Vd/f.Ld)levlr l ; hence, the values of f.Ld in the
figure captions in what follows. t AIso, since for values of U e of interest (for stability
considerations) C v « C n U e , in all ca1culations C v = O is taken [see Appendix Q(c)].
ln cases where the external or interna I forms are conical, it is more convenient to use the
parameters, a e and ai, or the cone angles f3e and f3j, in preference to a e and aj, defined by
a e = (1 - ae)2, aj = (1 - aj)2,
f3e = tan-l{[De(O) - De(L)]jL}:::: a e /£,
f3j = tan- l {[Di (O) - Dj(L)]/ L} :::: 8ai/£.
(8.139)
The physical significance of a e , £ and f3e is c1arified in Figure 8.52, The same pattern of
course holds for aj and f3j - see Figure 4.3.
The method of solution is a modified Galerkin scheme described in detail in Hannoyer
(1972, 1977) and briefly in Section 8.6.1. In essence, separation of variables is achieved
by seeking solutions of the form 1J(, 'l') = \11() exp(iev'l'), where ev is a dimensionless
tNote that tbis is a peculiar, albeit convenient model; tbus, a purely viscoelastic or hysteretic model is
retrieved by taking /-td -+ 00 or Vd -+ 00. respectively, rather than zem, Nevertheless, for tbe materials of
interest /-td and Vd have reasonable values.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
891
-- L !--, I
. r-:-::t- Pe a = De(O)-De(L)
e<L) E- =:E:f- =--- e L De(O) Lc
f = De(O)
(a)
L
{- m _ _  :,
(b)
/a,y a,l a,o
1- - ---------- - s=5
(c)
f=3 f=4 f=5 f=6
a,   fl a, %1 a,l l
f------- -+ i
(d)
Figure 8.52 (a) Definition diagram for <¥e, f3e and s, (b) Diagram showing how s may be varied
while keeping <¥e constant, and (c) how <¥e may be varied while s is kept constant. (d) How both <¥e
and s vary together for a constant f3e cone (tan 1e = ť2), corresponding to a system shortened by
cutting pieces off the free end tiannoyer & Pai'doussis 1979a).
circular complex frequency, The function \lI() is expanded in terms of Fourier-series
comparison functions with periodicity of 2, satisfying the boundary conditions at both
 = O and  = 1. Substituting into the equation of motion and the boundary conditions,
one obtains a set of homogeneous equations, leading to a vanishing determinant from
which the values of ev may be determined.
8.5.4 The theoretical dynamical behaviour
The ca1culations presented here are confined to extemal flow only and to solid rather than
hollow cantilevers. The parameters used have been chosen to be close to those pertinent to
the experiments of Section 8.5.5. Cantilevered truncated cones are considered, terminated
by a short ogival end-piece; assuming that separation does not take place and that the end-
piece itself is approximately conical and of length I, then it may be shown (Hannoyer &
Pai'doussis 1978) that
f ::::: 41 2 /{41 2 + [De(L) - Di(L)]2}.
(8.140)
Certain assumptions have been made with regard to the boundary layer in the ca1cula-
tions. Supposing that the beam is supported at the upstream end by a cylindrical support of

892
SLENDER STRUCTURES AND AXIAL FLOW
52
r 4 2 - With b,l.
_.- Without b,l.
7.38 3 2
S Mode 3
 2 2 5

i+- 1 2 1
! O
4,6 2.7(9 O
7,38 3 / -e
4 -2 2
10 8 -3 2
-4 2 2 2 3 2
1 2 4 2 52 6 2 7 2 8 2
(w)
Figure 8.53 The complex eigenfrequencies of the three lowest modes of a vertical cantilevered
conical beam in external flow with and without boundary layer (b.l.) taken into account; 8 = 20,
80 = 10, '8 = 0, Cl e = 0.5, Ye = 1.9, y* = 0.0024, C n = Ct = 0.008, C v = 0, /-td = 0.08, Vd =
0,015, X: = 1.5 Cf = 0.9); (Hannoyer & PaIdoussis 1979a),
length xo and that the boundary layer is turbulent from its leading edge, the displacement
thickness is approximated by (h (x)  0.2(x + xO)Cft; this linearized expression has been
obtained presuming that the rate of growth oť momentum thickness is constant, that is,
d82/dx = !Cft!Jr and that 81/82  1.3 (Schlichting 1960). With these assumptions, one
obtains
c:i = [ D e + O.4e + xo)C ft r  c  a r '
where (1 = 1 + O.4£oC ft, a = O.4C ft /(1, with £0 = xo/ De(O).
The locí oť the three lowest modes of a typical conical cantilever (a e = 0.5) are shown
in Figure 8.53. Also shown as chain-dotted lines are the loci for the same system, but
neglecting the effect oť the boundary layer. In both cases, the system loses stability first
by divergence, in its first mode, and then by flutter in its second and higher modes. This
qualitatively agrees with the behaviour oť cy lindrical cantilevers; however, the critical flow
velocity oť the equivalent cylindrical beam ťor divergence is lower, at U e  2,3, while
those for flutter are higher, at U e  5.65 and 9.15. The most important effect shown in
Figure 8.53 is that of the boundary layer, indicating a very strong stabilizing effect - the
critical flow velocities being higher by over 20%.
For a given length, increasing the cone fraction a e independently of the slendemess £
means making the cone sharper, a e = O corresponding to a cylinder and a e = 1 to a full
cone [Figure 8.52(c)). On the other hand, changing £ while keeping a e constant means
stretching the shape lengthwise while the cross-sectional areas at x = O and L remain
(8.141)

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
893
]O
8
-:::::.:::-..;:::-::.:::=....'"':::-.:..-- .
---=:::....
......-=:t.
Divergence
6
U ec
4
2 '_'_'_'_'_'_'_'_'_'_'_'_'_'_'
°
°
0,2
0,4
0,6
0,8
1.0
a e
Figure 8.54 The effect of CX e on stability for a conica1 cantilever in extemal flow: - , with the
boundary layer taken into account; -'-, - - -, without the boundary layer; other parameters
as in Figure 8.53 (Hannoyer & Pai"doussis 1979a).
fixed [Figure 8.52(b)]. The effect of changing e while keeping CX e fixed is not great, at
least in the range 10 < e < 20 and O < ex e < 0.6, where a 50% increase in e typically
generates a 5-10% increase in the critical flow velocity for divergence. Actually, the
stabilizing effect of e itself is not large for quasicy1indrical cantilevers; the mode st increase
referred to above is mainly the result of a thicker boundary layer, especially near the
free end.
The effect of changing ex e while keeping e constant is i1lustrated in Figure 8.54. The
most important aspect of this figure is that for sufficiently sharp cones (i.e, cones of larger
cone angles and hence, for a fixed length, more 'complete' cones), no in stability can occur.
Thus, for ex e > 0.85 (85% of full cone) the system does not become unstable iť U e < 10
(full1ines in the figure).t This is contrasted to a half-cone ofthe same length (ex e = 0,5)
which loses stability at U e  2.5. That this is due to the effect of the thickening boundary
layer is evident by comparing to the results for zero boundary-layer thickness (dashed
1ines). As in Figure 8.53, it is seen that the boundary layer has a strong stabilizing effect
on all the modes shown. Moreover, if the boundary layer were neglected, one would find
that even very tapered cantilevers (ex e  0.9) would be subject to instabilities, which the
experiments of Section 8.5.5 show to be incorrect.
Finally, considering the effect of shape oť the free end, it is noted that, in the results
discussed so far, relatively elongated tapered ends have been used (X: = 1.5, f = 0.9).
tThere is a flutter instability zone almost coinciding with the uppennost divergence zone which is not shown
in this figure, for cIarity.

894
SLENDER STRUCTURES AND AXIAL FLOW
It is found that, the less streamlined the end, the higher is the critical flow velocity - just
as for cy1indrical cantilevers. Thus, for cantilevers with e c:::: 15, no first-mode divergence
is possible at alI if X: < 0.5 (f < 0.5) appmximately. This effect seems to be insensitive
to a e for 0< a e < 0.6; for sharper cones, a e > 0,6, however, the boundary-Iayer effect
dominates (Figure 8.54), leading to the disappearance of divergence irrespective of free-
end shape.
8.5.5 Experiments
Experiments have been conducted with specialIy cast elastomer conical cy1inders
(Appendix D) in the water tunnel described in Sections 8.3,2(d), 8,6.3 and 8,11. No
experiments have been done with conical systems supported at both ends.
8.5.5(a) Generalobservations
A set of experiments was conducted with fulI, untruncated cones. No instabilities were
found to occur, up to the maximum flow attainable, in contrast to similar cyUndrical
cantilevers. This was considered rather surprising at the time, since one may suppose
that a conical body is clo se to the ultimate in a stream1ined 'enď - the 'enď essentialIy
extending fmm the point of support to the free end - and should, therefore, be susceptible
to instability at relatively low flow velocities.
It was confinned that separation was not possible for the smalI cone angles involved
« 2°). It was observed, however, that a1though no instabilities developed, there was
apparently random vibration (Section 8.4) of the cantilever, especialIy at the downstream
end, the ampUtude increasing with flow velocity. It was thought that these flow-induced
vibrations may actuaUy inhibit the estab1ishment of instabilities. However, experiments
with stiffer types of rubber showed that this was not the cause either. This paradoxical
behaviour of conical cantilevers was alI the more surprising upon considering that the then
existing theory predicted that instabilities should indeed occur.
It was at this point that it was decided to look into the effect of the boundary layer, the
existence of which had been ignored theretofore. It is recognized that at the downstream
end of conical bodies, even relatively thin boundary layers are quite thick in relation to
the loca1 diameter. The modified theory shows that the boundary layer exerts a strong
stabilizing effect on the system (Section 8.5.4) - indeed, sufficiently strong to preclude
instabilities of fulI or almost fuU cones (Le. for a e -+ 1). This is in clear qua1itative
agreement with observation.
With this matter clarified, systematic experiments were conducted, fmm then on
only with truncated conica1 cantilevers for which a e < 0.8, where instabilities are pos-
sible. It was found that this system is subject to a succession of instabilities with
increasing flow, starting with divergence in the first mode, followed by flutter with second-
mode modal shape, and then by higher-mode instabilities with more complex modal
shapes.
It was found that the onset of in stability was sometimes difficult to pinpoint accur-
ately, especiaUy for high a e (in the vicinity of a e = 0.7) because of the presence of
the smal!, random tlow-induced vibration, which occasional1y masked the initiation of
in stability. In such cases the possible experimental error in the va1ues of U ec was of the order
of  10%.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
895
12
a.
05 0.6 0.7 0,8 0,9
Mode 4 '. [X.  0,75
....... ... .
-....
20 "''''
8 
................""'...
"
U ec 6 15 Mode 3
W c
4 -- 10
"\ -,
__----"// Mode 1
-:_._.- .......w = O
2 [Xe= 1.5 5
Xe = 0.75
O 25 O
15 20 15 20 25
(a) E (b) E
Figure 8.55 Theoretical and experimental critica1 flow velocities and the corresponding frequen-
cies for a conical cantilever truncated at various lengths, subjected to axial flow (8' = O, /3e =
0,04, fO = 10, Ye = 1.9, y* = 0.0024, C n = Cf = 0.008, X: = 0.75 or 1.5, C v = f..Ld = Vd = O):
-.-, o, D, divergence (X: = 1.5); - - -,!:::", divergence (X: = 0,75); -, . ,. , *, flutter
(X: = 1.5); - - -,., flutter (X: = 0.75); (Hannoyer & Pai'doussis 1979b).
8.5.5(b) Quantitative comparison with theory
The experimental and theoretical critical fiow velocities for the various instabi1ities to
which the system is subjected are plotted in Figure 8.55 for different lengths of a Con-
ical cantilever of constant Cone angle ({Je = 0.035 rad = 2°). An almost full cone was
used to start with; then, it was shortened progressively by cutting pieces off the free end
[cf. Figure 8,52(d)]. For each length, the free end was ground to a well-streamlined ogival
shape, such that X: = 1.5; subsequently, for the same length, the experiment was repeated
with the free end groundinto aless pointed shape (X: = 0.75),t before reducing the length
further.
ln comparing theory with experiment, it is recalled that if the boundary-layer effect is
neglected, then the theoretical stability curves do not close at high a e , and instabilities
would be predicted up to a e -+ 1 (i.e, even for nearly full cones). It is clearly seen that
both experiment and theory (in which the boundary layer is taken into account) agree that
no instabilities may occur for a e > 0.75 approximately.
The sequence of instabilities predicted by theory with increasing fiow starts with first-
mode divergence, followed by second-mode fiutter; at higher flows, there is a zone of
essentially coincident third-mode divergence and flutter, and then fourth-mode fiutter.
This sequence was also observed experimentally; moreover, the modal shapes involved
were found to be very close to those computed by theory.
Quantitatively, the experimental critical fiow velocities are generally considerably
higher than the theoretical ones, sometimes by as much as 30-40%. The critical flutter
tTbese are tbe triangular experimental points in Figure 8.55, which should be compared to tbe dashed
tbeoretical curves.

896
SLENDER STRUCTURES AND AXIAL FLOW
frequencies also are not c10se to the theoretical ones; t thus, agreement between theory and
experiment is not as c10se as for cylindrica1 cantilevers.:j: Several reasons may be invoked.
One is the difficulty of deciding the exact point of onset of divergence in the experiments
because of the presence of the tlow-induced vibration referred to earlier. Another, the
main one, is that a number of qua1itatively sound but quantitatively untested assumptions
have been made in the theory in connection with quantifying the insulating effect of the
boundary layer (Section 8.3.4); in view of the strong overal1 effect on stability, these may
have important consequences on the theoretical predictions.
Figure 8,56 shows the effect of altering the cone fraction a e on stability, The theoretical
results are for a constant slendemess, e = 17; the experimental points however. correspond
to cantilevers covering the range 11 < e < 21; it was verified theoretical1y that varying e
over that range while keeping a e constant produces effects of less than 10%, and hence
comparison of al1 these points with the theoretical curves (e = 17) is meaningful. In
the figure. the longer cantilevers (larger e) are indicated by dark symbols, while the
2
U ec
o
o
0.2
0.4
0.6
0.8
1.0
a e
Figure 8.56 Comparison of theoretical and experimental results on the effect of 'sharpness', ať,
of truncated conical cantilevers on their stability. Theoretical results Oines) are for 8 = 17 with
the boundary-layer effect taken into account or without (b.l. = O). Experimental points are for
tubular beams of different materials and with"8 = 0,80 = 10, Ye = 1.9, y* = 0,0024, C v = ILd =
Vd = O, C n = Ct = 0,008, and X: = 0.75 or 1.5. The different materials correspond to different
types of Silastic silicone rubber: D, ., type A; V, type B; ., type D; o, ., type E (Hannoyer &
Pa'idoussis 1979b).
tThe direction of the arrows on the critical frequency curves identifies to which branch of the critical flow
velocity curves they correspond.
*The vertical dotted lines in Figure 8.55(b) for large E represent a range of frequencies; the vibration in these
cases was too erratic for a definite frequency value to be established.

S(\I.EARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
897
shorter ones by open symbols. The form of the in stability zones, as weIl as the kind of
agreement between theory and experiment are similar to those of Figure 8.55. Although
the lowest four regions of instability are quite distinct, stiIl quantitative agreement with
theory is not very good; besides, no clear difference between the longer and the shorter
cantilevers may be noticed. Nevertheless, the patterns of experimental in stability zones do
correspond to the theoretical ones; moreover, it is seen that theory ignoring the boundary
layer (b.l. = O) shows a complete lack of correspondence with experiment, whereas with
the boundary-Iayer effect accounted for, albeit approximately, agreement with experiment
is fair.
It is noted that in Figures 8.55 and 8.56 the theoretical results have been computed
with the internal dissipation constants of the canti1ever taken to be zero (JI = Vd = O).
Trus is because their effect on stability of canti1evers in externa1 flow has been shown
to be a1most negligible ( < 1 %) - at least in terms of the critical velocity; however, the
critical frequencies can be significantly different, above or below the values predicted with
dissipative effects neglected.
8.5.6 Closing remarks
The main contribution made in investigating the dynamics of conica1 cantilevers lies in
the uncovering of the importance of boundary-Iayer effects on the stability of all types of
cylindrical and quasicylindrical structures in axia1 flow.
The basic reason underlying the initia1ly perplexing observation that fully, or nearly
fuIly, conica1 cantilevers do not become unstable in axia1 flow, has been identified: namely
that it is associated with the thickness of the boundary layer, wruch near the free end of
the tapered cantilever is quite substantial when compared to the beam cross-sectional
radius. t If the boundary layer is thick enough, it may be said that the structure becomes
'insulateď from the mean flow, and hence no fluidelastÍc instabilities develop, The theory
taking into account the boundary layer agrees with experiment that (i) no instabilities are
possible for almost fuIly conical cantilevers (a e > 0.75), and (ii) instabilities are possible
for truncated conical cantilevers (a e < 0.75), albeit at substantiaIly higher flow velocities
than would be predicted if the existence of the boundary layers were ignored. Indeed,
the aforementioned may be viewed as the principal point of interest in this theoretical
model.
Considering a specific truncated conical cantilever, it has been found that the tip of the
free end must be streamlined sufficiently for instabilities to occur. If this is done, then the
system is subject to a sequence of divergence and flutter instabilities with increasing flow,
substantially the same in both theory and experiment - as for cy1indrical cantilevers.
Clearly, the degree of quantitative agreement between theory and experiment in the
case of external flow leaves room for improvement and this may usefuIly be the subject of
further research. In developing the theoretical model, a number of quantitatively untested,
a1though basicaIly sound, assumptions were made in order to allow taking the boundary-
layer effect into account in a tractable manner. Testing these assumptions and refining the
model in this area would be worthwrule.
t A more direct 'proof' was attempted by sucking the boundary layer into the interior of the cone and thence
out of the system, This proved unsuccessful, since the internal flow thereby generated was substantial and its
effect could not be ignored.

898
SLENDER STRUCTURES AND AXlAL FLOW
8.6 CONCURRENT INTERNAL AND EXTERNAL FLOW
Cylindrical structures simultaneously subjected to internal and external axial flow are
not as rare in engineering as it might seem. One example is the dril1-string shown in
Figure 3.18; the internal fluid flow that is used to turn the dril1-bit serve s a1so to carry away
the material dril1ed out, flowing upwards and around the dril1-string. Another example is
the heat-exchanger tubes carrying the primary fluid internal1y and the secondary one
externally; in many cases, at least over substantial segments of the tubes, the external flow
is axial, rather than a cross-flow (see Section 8.13).
In tenns of the small-amplitude vibration of Section 8.4, the effect of the two flows
may reasonably be expected to be merely additive. In tenns of stability, however, hence
more at the fundamental than the practicallevel, it is not evident that the effect of the two
concurrent flows may be deduced from that of each separately.
This topic has received relatively scant attention. The first such study may be by Cesari
& Curioni (1971), who consider theoretical1y the static stability of a tubular beam with
internal and external flow, for various support conditions. More general and extensive
studies have been made by Hannoyer & Pai'doussis (1978). For specific studies of the
dri11-pipe problem, see Grigoriev (1978) and Luu (1983), and for a silo water-mixing unit
in the pulp and paper industry, see Wang & Bloom (1999).
Here, in the interests of brevity, only some of Hannoyer & Pai'doussis' (1978)
work is presented. Only cylindrical tubular beams, both internally and extemally, are
considered.
8.6.1 The theoretical model and solution method
The theoretical model is identical to that developed in Section 8.5, and the equations
of motion and boundary conditions for a cylindrical tubular beam are obtainable from
equations (8.136) and (8.137) by imposing dAifdx = dAe/dx = O and Vi(L) = Vi =
const. Hence, the equation of motion is given by
( * a ) ( a 4 w ) ( a a ) 2
E + E - 1- + Pi Ai - + Vi - w
ar ax 4 ar ax
( a * a ) ( aw aw ) aw
+ P A - + V - - + V - - (p A - P .A' - P A )g -
e e ar e ax ar e ax e e I I ax
- {T(L) -1£ [(peAe - PiAi - pA)g -lc jtPeDev; ] dX} :
1 ( aw aw ) 1 aw a 2 w
+ "2CtnPeDeVe ar + Ve a; + "2 J.Le CD ar + pA ar 2 = O.
(8.142)
As in Section 8.5, V: is re1ated to the boundary layer growth, and hence to equation (8.141),
Similarly, the boundary condition related to shear at the free end of a cantilevered system

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW 899
is given by
[Cp + fPe)Ae + (pi - p)Adl [ 0 0 :; ] - fPe(Ae - Ai)U: [  ]
L-l L-l
* _ [ 02y ] ( * o ) [ 03y ]
+ {fPe(Ai U e + AeUe) + 2Pi A i U dl - - E + E _ o I- o 3
OX ot L-l t X L-l
- {fPe(Ae - AdUeU: + [(Pe - p)Ae + (p - pi)Adgl} [ : t-l = O, (8.143)
where Ae is an average area defined by
- l i L
Ae = - Ae(x) dx.
1 L-l
(8.144)
In the ca1cu1ations, for simplicity, Ae and f are eva1uated as if the end-piece were conica1
and, for f, assuming that separation does not take p1ace, so that
A ......,! ( A3/2_A/2 )/( Al/2_A/2 )
e-3 e I e I'
f  41 2 /[41 2 + (De - Di)2]. (8.145)
Using the same nondimensionalization scheme as in Section 8.5, and hence equations
(8.138a,b), the dimension1ess equation of motion is given by
( o ) 041} I ( 1 - a ) 2 2 2
I+Vd- -+ - u +u,-8
Oí 04 a e I
2 ] 021} { 2 } 01}
- [y(2 - Ye + Yi) + eCtUe](1 -) o2 + y(2 - Ye + Yi) + eCnU e o
I [ 1/ + Yi ] 1/2 [ Ye - 1 ] 1/2 [ ( l-a ) 2 ] } 021}
+ 2 Uj + 1 + U e -
Ye+Yi Ye+Yi a OOí
I [ Ye-I ] I/2 ] 01} 021}
+ e(cnu e + cv) - + ---z = O,
Ye + Yi Oí Oí
(8.146)
where all the dimension1ess parameters, Y, a, , etc. are defined by equations (8. 138a,b ).
It is reca11ed that, in view of equation (8.141), the approximate re1ationship u;/u e
[(1 + a)/af has been used, where
a = 1 +0.65eOCt,
a = 0,65 ct!a,
£0 = xo/ De(O),
Ct = (2/n)C ft.
(8.147)

900
SLENDER STRUCTURES AND AXIAL FLOW
The boundary conditions for a free end ( = 1) are a2f}/a2 = O and
( a ) a3f) I [ _1 ] 1/2 [ 2 ( I-a ) 2 ]
I+Vd- 3- Xe fU e e a e +8-
aT a Ye + Yi a
[ -2 ] 112 } I 2
8 + Yi a 2 f} 1 - a -2
+ 2U i - + fu; ( - ) (1 - 8 ) - Xey[(2 - Ye)a e
Ye + Yi a aT a
J af} I ( l_a ) 2 [ ye_I ] 1/2 -2 J af)
+yd a + fU e -;;- Ye + Yi (1 - 8) aT
_ { [1 + f(Ye - l)]a e + Yi } a2f} _ O
Xe - ,
Ye + Yi aT 2
(8.148)
where a e = 1 (1 + 8 + 8 2 ) and 8 = Di! De; here Xe is used as in (8,30), rather than X: as in
(8.138c). If there is no separation over the free end, then equation (8.140) holds, namely
f = 4£2 X: /[4£2 X: + (1 - 8)2].
(8.149)
This may be an opportune place to say a few words about the hybrid Ga1erkin method
used to obtain solutions - also used in Section 8.5.
We may separate variables by seeking complex solutions of the form
f}(, T) = \II()eiú)"
(8.150)
where ev is a dimensionless complex frequency. Clearly, stability or in stability may be
assessed accordingly as the imaginary component of ev is positive or negative, Substituting
into equation (8,146) and into the four pertinent boundary conditions, we obtain a set of
ordinary differential equations of the form
4 dr \II
L fr(, ev) dr = O,
r=O
(8.151a)
and
3 [ dr\ll ]
Lg!(ev) dr =0,
r=O =j
j = 1,2,3, 4, l = 2 = O, 3 = 4 = 1.
(8.151b)
The solution of these equations is accomplished by a Ga1erkin-like method, utilizing
Fourier-series comparison functions (Hannoyer 1972). The periodicity of the series must
be greater than unity, in order to satisfy the boundary conditions at  = O and  = 1 and to
allow for continuity of \II and its derivatives; for convenience, a periodicity of 2 is chosen,
such that
00 in1r 00
\II() = Lan e M = L{!32n-l sinnJr + !32n cosmT},
-00 "II 2 O
(8.152)

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
901
where the tJ are comp1ex. Hence, equations (8.151a,b) may now be written as follows:
4 dr
D(, w) == L fr(, w) dr [tJ2n-1 sinnJr + tJ2n cosnJr] = O,
r=O
3 , ( dr 00 ]
g;(W) dr ?;[tJ2n-1sinnJr+tJ2nCosnJr] =j =0,
(8.153)
(8,154)
and we impose
1 1 D(, w) sin pJT d = O, 1 1 D(, w) cos pJT d = o, p = 1,2, ...
(8,155)
Each of equations (8.155) yie1ds a homogeneous re1ation between the tJ; the four boundary
conditions (8.154) are a1so 1inear and homogeneous. Hence, all the equations may be
written in the fonn
[A(iw)]{tJ} = {O};
(8.156)
the requirement for nontrivia1ity yie1ds a vanishing detenninant, fcom which the va1ues of
w may be obtained.
8.6.2 Some theoretical results
ln genera1, the dynamics of the system subjected to both interna1 and externa1 flows
concurrently is simi1ar to that with just one of the two flows. It wou1d be of practica1
importance if it were possib1e to detennine an equiva1ent flow which wou1d ha ve the same
dynamica1 effect on a given beam as the two flows simu1taneous1y.
If one neg1ects frictional, dissipative, base-drag and gravity forces, and assuming that
the internal and external fluid s are the same [so that Yi = _8 2 (2 - Ye)], the equation of
motion (8.146), simplifies to
a411 2 2 a211 ( Ye - 1 ) 1/2 - a 2 11 a211
a4 +(Ue+Ui) a2 +2 Ye+n (Ue+8ui) aa. + a. 2 =0;
(8,157)
boundary condition (8.148) is similarly simplified, Subject to the assumptions just made,
equation (8.157) is that of a tubular beam subjected to a single external or internal flow
with velocity U = (u; + u;)1/2, if u; + u; remains equal to (u e + 8Ui)2 or (u i + u e /8)2,
-2 - --2
i,e. for Ue/Ui = CI - 8 )/28 and Ue/Ui = -28/(1 - 8 ), respectively. [lncidentally,
provided that U e / u i remains constant in (8.157), it is always possible to find an equivalent
single flow, but corresponding to a beam of different density (i.e, a change in Ye and Yi
would be involved),] However, unless the beam is supported at both ends or is blunt at the
free end, the boundary conditions for the two simultaneous flows cannot be matched by
those of a single flow; nevertheless, for a cantilevered beam with a tapered end, the effect
of the two flows can stilI be accommodated by an equivalent extemal flow provided that
a shorter end-piece is assumed (thus reducing the value of f in the boundary condition).

902
SLENDER STRUCTURES AND AXIAL FLOW
With the assumptions made in the foregoing, the conditions for divergence may be
determined easily from the simple form of equation (8,157)
a 4 \11 a 2 \11
- + ( u2 + u 2 ) - - O
a4 e i a2 -
(8.158)
and, for example for cantilevered beams, the boundary conditions
\11(0) = \11'(0) = \11"(1) = O,
\11111(1) + f(1 - 8 2 )u;\II'(1) = o.
(8.159)
The results calculated in this manner are shown in Figure 8,57, In fact, these dia-
grams are remarkably c10se to the exact limits of stability calculated by equation (8,146)
6n
Clam,ed- free "
5n -----
..- .'
4n ...
...
U e 3n "
" "
2:tr
0.5 }
0,625 J(I - ;p)
0,75
I
o n 2:tr 3n 4n 5n 6n
(a) u j
M
5n
4n
u e 3n
2n
Stable
n \
O n 2:tr 3n 4.7r 5n M
(b) u i
Figure 8.57 Theoretical critical flow velocities for divergence of (a) cantilevered and (b) clamped-
clamped tubular beams subjected to both internal and external flows, obtained assuming C n = CI =
y* = e = Vd = O (Hannoyer & Pa'idoussis 1978).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
903
and, where applicable, equation (8.148). Particularly, the curve corresponding to the first
instability (lowest divergence zone) constitutes a 'universal', albeit approximate, bound-
ary of in stability for all cases involving both internal and external flow. The higher zones
of instability are of mainly academic interest since (i) there are regions of flutter inter-
spersed therein which cannot be determined in this manner, (ii) in the case of cantilevers,
these boundaries really represent solutions where ú) = O, and not necessarily thresholds
of in stability (i.e. they possibly correspond to points where the system regains rather than
loses stability), (iii) most importantly, only the first loss of stability may reasonably be
predicted by linear theory.
8.6.3 Experiments and comparison with theory
The tubular beams were cast in special moulds from various type s of silicone rubbers in
the manner described in Appendix D. The tubular beams, normally 25.4 mm (1 in.) in
diameter and about 500 mm long, were fitted onto a special metal tubular support at the
upstream end, which was mounted in the middle ofthe test-section via four aerofoil struts.
ln the case of beams supported at both ends, a similar lower (downstream) support was
provided, but with a sliding fit, to allow axial movement. Cantilevered tubular beams were
cast cy lindrical, and then the external surface of the free end was ground to a streamlined,
ogival shape.
The internal flow (also water) was supplied independently of the external one, through
the upper tubular support via a modification to the water tunnel- see Figure 8.58; at the
downstream end of the beam, the internal flow discharged into the external fluid.
To deaerator .- .- -,
Diffi.1ser
--"flow
.
i
TW11ing vanes 
Grids
//
(
Test-section i
I
]
oj
Po
'O
E
o
u

Figure 8.58 The modified arrangement of the water tunnel, so that a separately controlled internal
flow may also be supplied to the tubular beams, in addition to the external flow.

904
SLENDER STRUCTURES AND AXIAL FLOW
The general procedure was to fix the external flow, and then vary the internal flow,
normally starting from zero and going up.
8,6.3(a) Genera! observations
If the bearn was supported at both ends, divergence was the first instability to develop
with increasing flow, either internal or external. However, if the bearn was cantilevered,
divergence did not occur by the sole action of the internal flow (cf. Section 3.5), nor was
it observed with external flow when the free end was blunt.
The critical flow velocities for divergence were difficult to pinpoint because, contrary
to linear theory, the on set of divergence did not occur all of a sudden at a sharply distin-
guishable flow velocity, but rather more gradually - cf. Section 8.3.2; moreover, changes
in the plane oť divergence and random small-amplitude vibrations compounded the diffi-
culty. The maximum amplitude for divergence normally did not exceed one or two bearn
diameters.
Flutter, with a modal shape clo se to that of the second bearn mode, generally superseded
divergence at higher flow velocities. In the case of cantilevered bearns with a blunt end, it
was the only in stability observed. For cantilevered beams with a tapered end, establishing
the upper limit of buckling was sometimes difficult, as flutter often overlapped the diver-
gence zone; in such cases, the modal shapes of the two modes involved are quite similar
and transition can be distinguished only by the frequency, one in stability being oscillatory
and the other not. In the case of clarnped-damped bearns, the transition from divergence
to flutter was a continuous process. Flutter started in the form of a travelling wave, with
wavelength roughly equal to the length of the beam, propagating downstrearn. Increasing
the flow further caused the motion to become brisker and the frequency of oscillations to
increase; the travelling wave became less noticeable, and the oscillation more nearly of a
standing-wave formo
Irrespective of support conditions, the modal shape associated with flutter was not sta-
tionary, but involved the superposition of a standing wave and smaller amplitude travelling
waves, apparently propagating at different speeds in opposite directions.
Flutter due to external flow, irrespective of support conditions, usually involved much
smaller arnplitudes than that due to internal flow, with limit cycles typically of one to two
beam diameters. In flutter induced by internal flow, oscillations sometimes developed to
the point of hitting the wall of the test-section; moreover, the onset of stability was more
clearly defined. Consequently, it was preferable to investigate the inftuence of internal
flow on an in stability caused by the external flow, rather than vice versa.
As the flow velocity was increased further, a succession ofhigher-mode flutter instabil-
ities were encountered, with the beam never regaining stability completely between
instabilities. The transition between two type s of flutter occurred abruptly.
8,6.3(b) Comparison of experiment and theory
Two different tubular beams were tested with both ends clamped, but with the lower
end free to slide axially. For the stiffer one (Silastic type B), the maximum avail-
able flow velocities allowed only divergence to occur. The theoretical and experimental
critical flow velocities are compared in Figure 8.59. The theoretical curves have
been calculated with the full theory, although the stability boundaries are seen to be

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
905
12
8
o
10
U e 6 ....-......-.q Max,
2

o
00<1
o

o
4 6 8 10
u j
4
o 2
Figure 8.59 The effect of interna! and external flow velocities on stability of c1amped-clamped
tubular beams, Comparison of experiments with theory (c n = CI = 0.008, f = 18, fO = 10,8"=
0.5, Ye = 1.9, Yi = -0.03, y* = 0.0024, X = C v = Vd = O). Experiments with Silastic
type E: o, divergence; _, flutter, Experiments with Silastic type B: \7, divergence (Hannoyer &
PaYdoussis 1978),
not too far from those obtainable by the approximate analysis of Section 8,6,2 -
cf, Figure 8.57.
It is seen that theory and experiment s are about 25% apart; moreover, the predicted
narrow region of stability between divergence and ftutter could hardly be noticed in the
experiments. The most likely reason for the discrepancy in the on set of divergence is
related to imperfections of all kinds which contributed to the reduction of the divergence
ftow velocities in the experiments. Also, to allow axial sliding, the c1amped condition at the
lower support could not be realized fully; note that the critical ftow velocity for c1amped-
pinned beams is approximately 30% lower than for c1amped-clamped ones. Nevertheless,
the onset of ftutter is underestimated by theory, contrary to some of the reasons just given
for divergence. Since the system does not regain stability in-between, nonlinear effects
(notably associated with the external ftow) are likely responsible for delaying transition
to ftutter.
Similar experiment s were conducted with cantilevered tubular beams of two different
types of rubber, all of the same length (8 c::: 20). The results for beams with a blunt end are
shown in Figure 8.60. In this case instabilities due to external ftow alone are not possible,
and stability is controlled by the internal ftow, It is noted that the effect of the two ftows for
cantilevered beams is in general quite asymmetric (see also Figure 8.61), in contrast to the
case of c1amped-clamped beams. Thus, if the system is stable, increasing Ui will always

906
SLENDER STRUCTURES AND AXIAL FLOW
U e 6
Stability
8
10
12
ID
8
4
2
o
2
4
Ui
Figure 8.60 The effect of internal and external ftow velocities on stability of cantilevered tubular
beams with a blunt free end. Comparison of experiments with theory (c n = CI = 0.008, £ = 20,
£0 = 10, "8 = 0.5, Xe = C v = Vd = O) for beams of two different materials: ., silastic type E
(Ye = 1.9, Yi = -0.03, y* = 0.0024);, silastic type B (Ye = 1.7, Yi = -0,07, y* = 0.001).
Numbers in the figure are selected theoretical or experimental critical values of úJ. Lines represent
theoretical boundaries; 0, some individual theoretical points (Hannoyer & PaYdoussis 1978).
precipitate flutter. However, the effect of then increasing U e is to stabilize the system,
especially at low flow velocities (for U e < 3),
Agreement between theory and experiment is reasonably good. It is noted that the flutter
frequencies, marked on some ofthe experimental points and on the theoretical boundaries,
also agree reasonably well, The theoretical results were obtained with the dissipative forces
in the material of the beam taken equal to zero, for simplicity and also since it was shown
theoretically that their effect on stability is rather smal!.
ln Figure 8,61 are shown the boundaries of stability for cantilevered tubular beams
with tapered ends. In this case, there are also instability zones mainly due to externa!
flow. Thus, for small internal flow velocities, as U e is increased the beam first buckles,
then generally regains stability, and then flutters; subsequently, at higher U e , divergence is
superimposed on flutter at regular intervals, whi1e flutter switches to higher modes, It is af
interest that the two divergence regions shown here are similar to those of Figure 8.57 for
f(1 - 8 2 ) -::::.0,7 (here 8 = 0,5, f = 15/16), For high internal flow velocities, on the other
hand, the behaviour of the system becomes dominated by internal flow and is similar to
that of Figure 8,60, unless U e is sufficiently large,
Finally, it is important to note that stability may result by a combination of interna!
and external flows, which individually at the same flow velocities would have caused

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
907
8
12
lO
U e 6
4
2
o
U j
Figure 8.61 The effect of internal and external flow velocities on stability of cantilevered tubular
beams with a tapered free end. Comparison of experiments with theory (c n = Cf =0.008, 8 = 19,
80 = 10, '8 = O,5,8Xe = I, C V = Vd = O). Silastic type E: 0, divergence; _, flutter. Silastic
type B: V, divergence; T, flutter. Numbers in the figure are selected theoretical or experimental
critical values of ú), Lines represent theoretical boundaries; O, some individual theoretical points
(Hannoyer & PaYdoussis 1978).
instabilities, (The opposite is also trne, but not unexpected.) For example, for U e = O,
Ui = 5, the system would be unstable (by flutter); yet, with U e = 3 flutter may be postponed
tilI Ui '::::: 8, Similarly, for Ui = O divergence occurs at U e '::::: 2,5; however, with Uj = 3,
the onset of in stability may be delayed tíH U e '::::: 6.
Finally, a set of experiments was conducted with tubular conical cantilevers subjected to
internal and external axial flow simultaneously. These experiments have not been reported,
as the dynamical behaviour was found to relate to those for sole internal or external flow
in quite a similar manner to the case of cylindrical beams subjected either singularly ar
simultaneously to internal and external flow.
8.6.4 Concluding comments
The stability characteristics af tubular beams supported at both ends is relatívely straight-
farward. Increasing either the internal or external flow velocity, ar bath, the system
eventually loses stability by divergence; íts subsequent behaviaur wíth íncreasing flow
ínvolves a successíon of flutter and divergence instabilítíes.

908
SLENDER STRUCTURES AND AXIAL FLOW
ln the case of cantilevered tubular beams, however, the behaviour of the system is
much more complex, depending on the values of internal and external dimensionless flow
velocities and on the shape of the downstream end. Cantilevered beams with a blunt free
end can only lose stability by flutter in a process dominated by the internal flow; if the free
end is streamlined, however, a complex sequence of divergence and flutter instabilities
may occur. The theoretically predicted behaviour has been supported by experiments, and
agreement between theory and experiment is reasonably good.
It has been shown that the state of a system subjected to both internal and external flows
cannot generally be inferred from knowledge of its state when subjected separately to the
internal and the external flows.
Of course, in practical terms, a designer is not interested in the intricacies of the suc-
cession of instabilities to which the system may be subjected, but rather to the critical
conditions associated with the first instabi1ity encountered with increasing internal or
external flow velocities. With this in mind, the simplified analysis of Section 8.6.2, relat-
ing exclusively to divergence, provides approximate 'universal' stability criteria for the
first divergence. These criteria (the lowest curves of Figure 8.57) are applicable to tubular
beams with clamped ends, and to cantilevered beams where the internal flow velocity is not
excessively high (Ui < 3). For high internal flow velocities, the first instability encountered
by cantilevered beams is flutter; a very rough conservative estimate for the onset of flutter
(provided u e < 8 approximately) may be obtained by neglecting the effect of the external
flow a1together, which yields a critical flow velocity Ui ::::: 5.
8.7 PARAMETRIC RESONANCES
Here, we consider the dynamics of cylinders in harmonically perturbed axial flow. Such
perturbations may arise in industrial systems (hence in confined flows), in the pumping
system or due to self-excited flow oscillations; or they may arise in two-phase flows, e.g.
in the slug flow regime. In unconfined flows such perturbations may be related to wave
action of systems not too-deeply submerged,
The main interest here is to explore the existence of parametric resonances, much as was
done in Section 4.5 for internal flow. The dynamics in single-phase flows is considered
first, following the work of Pai"doussis (1975b) and Pai"doussis et al. (1980a,b); then the
dynamics in two-phase flows, following the work by Hara (1975a,b, 1979, 1980) and Hara
& Yamashita (1978), as well as Shin's (1978) review.
8.7.1 Theoretical model
The basic equation (8.9) still holds. However, to obtain the final equation of motion,
account must be taken of the fact that U is no longer constant.
Specifically, equations (8.2) and (8.18) now become
( a 2y a2y 2 a 2 y dU ay )
FA=XpA -+2U-+U -+-- ,
at 2 axat ax 2 dt ax
(8.160)
ap ( Stot ) ( dU )
-Ach ax = FL S - pAchg + pAch dl '

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
909
where all the symbol s are as in Section 8.2. Thus, following the same procedure, the
following dimensionless equation of motion is obtained:
ai/III + r/'" + {xu 2 - 8 [f + (1 - 2v)n] - [!ecfu 2 (1 + h) + y + fJl/2 ů J
x [(1 - !8) -  ] - !(1 - 8)CbU2} rJ" + 2XfJl/2u' + [!eCju 2 (1 + h)
+y + (1 + X)fJl/2 ů J r/ + !ecffJl/2u + [1 + (X - 1)fJ] ij = O, (8.161)
where the dimensionless quantities are as in (8.28a); ()' = a()ja and n = a()ja-c.
This replaces equation (8.27 a) for this case, when u = u (-r); specifically, the flow velocity
is assumed to be hannonica1ly perturbed, so that
u = uo(l + f.t cos úJt),
(8.162)
in which f.t is relatively small. The dimensionless frequency úJ is related to the dimensional
radian frequency Q by equation (8.28b),
The system under consideration is defined in Figure 8.62 for a clamped upstream end
and a pinnedlrotationally-constrained downstream one, to correspond to the experimental
system of Figure 8.66(a). The boundary conditions in this case are
ry(O, -r) = ry'(O, -r) = ry(1, -r) = ryl/(1, -r) + Kry'(l, -r) = O,
(8.163)
with K = qLj EI, CL being the rotational spring stiffness (Figure 8.62). However, other
boundary conditions are a1so considered.
Containing channel
x
y
Clamped support
Flow velocity
u= U o (1 + fJ. cos.Q t)
g
Flexible cylinder
Pinned support
with rotation-
resisting spring
(stiffness: CL)
Figure 8.62 The system under consideration, showing a cylinder c1amped upstream and pinned
and rotationally constrained at the downstream end (Pai'doussis et a/. 1980a).

910
SLENDER STRUCTURES AND AXIAL FLOW
On1y simple parametric resonances are considered, rather than combination resonances
(Section 4.5.1). lndeed, since the ana1ysis follows that of Section 4.5 very c1osely, no
detai1s are given here, except to say that, in the Galerkin discretization, the 4>r() in this
case satisfy a11 the boundary conditions, for examp1e, (8.163). The existence of parametric
resonance regions is exp10red by Bo10tin's (1964) method, as detai1ed in Section 4.5.
It is reca11ed that primary resonances occur in the vicinity of w/w n = 2, , , ...,
of which the first is the principal primary one; W n is one of the eigenfrequencies of the
system - the value of which of course varies with Uo. Simi1ar1y, secondary resonances
occur in the neighbourhood of W / W n = 1, 1, 1, etc.
Some typical resu1ts for cylinders pinned at both ends are shown in Figure 8.63( a,b). Sev-
eral points should be remarked upon. (i) Even with zero dissipation in the cylinder (a = O),
the 'nose' of the parametric regions does not reach f.L = O, as a resu1t of flow-induced
damping in this case of extema1 flow. (ii) The nose of the first-mode regions for secondary
instabi1ity occurs for w/ WOl < 1 in Figure 8.63(a), since Wl (uo # O) < úJ()1 == Wl (uo = O).
As Uo --+ O, one has W/WOl = 1 and 4 for the first- and second-mode instabi1ity regions,
since WOl = :rr 2 , W02 = 4:rr 2 = 4WOl. Therefore, since WOl < W2 < úJ()2, the nose of the
second-mode secondary regions occurs at 1 < W/WOl < 4. Simi1ar1y, in Figure 8.63(b),
the first-mode principal primary regions occur at W/WOl < 2, and at W/WOl < 8 for the
second-mode ones, (iii) As usua1, the principal primary regions (w/w n ::: 2), c1ustered
here about W/WOl ::: 1.5 for the first mode and W/WOl ::: 7.5 for the second, are the most
extensive and can be excited at the 10west values of f.L; the regions immediate1y below
them in Figure 8.63(b) correspond to W / W n = , (iv) The 1arger extent of the second-mode
resonances as compared to the first-mode ones in Figure 8.63(b) is because of division
of the ordinate by úJ()1 rather than úJ()2. Thus, the peculiar 10cation and extent of these
parametric resonance regions is an artefact of the nonnalization by úJ()1 throughout; if
a floating nonnalization had been used, and W / Wl and W / W2 had been p10tted, then the
curves in a11 cases wou1d have been centered OOout 1 for the secondary regions, and OOout
2 and  for the primary ones - but the confluence of so many curves wou1d have made
them difficult to read.
It is seen that, with increasing flow, parametric resonances are general1y possib1e with
progressively smal1er values of f.L: for examp1e, for f.L = 0,2, a secondary resonance cannot
occur when Uo = 2 and 2,5, but is possib1e when Uo = 3. Moreover, increasing flow makes
the resonance regions wider. It is also noteworthy that for sufficiently low flow velocities
no parametric resonances are possib1e at al1 (at least for O < f.L S 0.5). These observations
may be explained as follows: on the one hand, the amplitude of the osci1lating component
of the flow is f.LUo, that is, it is proportional to the flow ve10city; on the other hand, flow-
induced damping, which arises from the viscous fluid forces, also increases with flow
ve10city, but not quite linear1y; c1ear1y, it is only when the amplitude of the oscil1ating
parametric force becomes sufficiently 1arge to overcome the influence of flow-induced
damping that parametric resonances become possib1e.
Typica1 parametric resonance regions for a cantilevered system with a b1unt end t are
shown in Figure 8.63(c,d). The corresponding Argand diagram for the eigenfrequencies
is shown in Figure 8.64. Consistently with the deliberations of Section 8,3,3(b,e), there
is no divergence for f = O, but flutter is possib1e, occurring at ucf = 10.52, in the
third mode,
tFor analytical convenience; the boundary conditions for a blunt end (Xe = O. f = O) are very simple,

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
911
5
(a) ........
.'
4 Uo  25 uo=2
uo=3 7
3 Second mode regions
;:; 6

s
2 2 2 First mode regions
Uo
First mode regions
2.5 uo=2
Uo
uo=3 uo= 3 Uo = 2.5
O 0,1 0,2 0.3 0.4 0.5 O 0.1 0,2 0,3 0.4 0,5
ft ft
(c) (d)
5 5
uo= 8
4 4

 3 3
S
2 2
O
0,1
0,2
0.3
0.4
ft
0,5 O
O,)
0,5
0,2
0.3
0.4
ft
Figure 8.63 (a,b) Parametric resonance regions for a pinned-pinned cylinder ({3 =0.1, BCJ = I,
o = y = n = r = h = O, X = 1; WOI = 1T 2 ); - , ex = O; - - - , ex = 0.003: (a) secondary
resonances; (b) primary resonances. (c,d) Parametric resonance regions for a cantilevered cylinder
with a blunt end ({3 = 0.5, BCf = 0.25, Cb = 0.1, o = h = y = O, X = 1; W02 = 22.034);
////1 , Ua = 6; lli.il, Ua = 8; \ \ \ \ \, Ua = 10; -, ex = O, - - -, ex = 0,003: (c) secondary
resonances; (d) primary resonances (Paldoussis 197 5b).
Flow-induced damping in cantilevered cylinders is higher than in cylinders supported
at both ends, because in addition to viscous damping there is also damping due to the
mean inviscid fl.ow, this being a nonconservative system (cf. Figures 4,26 and 4.24 in
the internal fl.ow case). As a result, parametric resonances in this case occur only in two
ranges of flow velocity: (i) for very 10w ua (e.g. Ua = 0.5), and then for large J.l (typically
J.l > 0.45); moreover, the resulting resonance regions are narrow, and diminished further
if ex =1= O; hence, these resonances are practically not very important; (ii) for relatively high
Ua, as shown in Figure 8.63(c,d), for example.

912
SLENDER STRUCTURES AND AXIAL FLOW
Considering first the secondary regions in Figure 8.63(c) it is noted that, for Uo = 6,
there are two distinct resonance regions: the region around WIW02 :::: 2 is associated with
the third mode, while that at WlW02 :::: 4.5 with the fourth mode; there are no resonance
regions associated with lower modes. These apparently peculiar results reftect the fact that
š = Im(w)IRe(w) is lower for the higher modes - cf. Figure 8.64. For uo = 8 and 10,
the two region s found for Uo = 6 coalesce into one.
Considering the primary regions in Figure 8,63( d) next, it is noted that for Uo = 6 there is
a third-mode resonance centred about WIW02:::: 4.7. For uo = 8, there are two regions, the
upper one being the principal primary region associated with the third mode, wl W3 = 2,
and the lower one corresponding to w I W3 = , For Uo = 10, there are two interconnected
regions, the lower part being associated with the third mode, while the upper one probably
with the fifth mode,
Also shown in Figure 8,63(c,d) is the effect of dissipation (ex = 0,(03). For Uo = 6,
the secondary resonance regions are eliminated entirely, while the primary one is much
diminished. For uo = 8 and 10, it is the primary regions which suffer most. These
peculiarities reftect the intricate dependence of the W n on uo, as seen in Figure 8,64.
Figure 8.65(a) shows the principal resonance regions for a clamped-pinned system for
three values of the rotational stiffness K, and for values of the other parameters inftuenced
by the experimental system to be discussed in Section 8.7.2. Apart from the downward
shift of the regions as K is decreased, reftecting the corresponding reductions in Re(w n ),
it is seen that the parametric resonance regions are qualitatively quite similar. The pro-
gressively smaller va1ues of J.Lrnin for resonance to occur as K is reduced simply reftect the
corresponding diminution of the damping factor Š,
Figure 8.65(b) shows the effect of SCj and of confinement, which affects X and h
together. [It is recalled that X = [1 + (DI DCh)2J / [1 - (DI DCh)2J and h = DI Dh =
(D I Dch) I (1 - DI Dch), Dch being the channel diameter.] The effect of increased con-
finement is to reduce the extent of the parametric resonance regions. In the case of the
second-mode principal primary regions shown in this figure, the effect of confinement is
not very strong, mainly shifting the location of the resonance in the (J.L, w)-plane, In other
cases, however, for example, for the first-mode regions, the effect can be stronger. The
effect of increasing SCt and hence increasing damping is to diminish the extent of the
resonance regions, as expected.
Finally, Figure 8.65( c) shows the effect of increased dissipation on the princi pal primary
resonance region: the extent of the region is diminished and the corresponding J.Lmin
increased,
8.7.2 Experiments
The basic apparatus is shown in Figure 8.66(a). The experiments were conducted with
silicone rubber cylinders (see Appendix D), 10-14 mm in diameter and 300-500 mm long,
mounted centrally in a transparent cylindrical pipe either 320 or 400 mm in diameter,
in which a harmonically perturbed axial ftow of water was established. The pulsating
component was generated by a plunger pump, driven by a reciprocating mechanism,
adapted from the experiments described in Section 4.5.3, and providing pulsations in the
usable frequency range of I-16Hz.
All experiments were conducted with nominally clamped cylinder ends. The upper
end was positively clamped: the cy1inder was gIued onto a brass 'nose cone' (of a

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
913
10'
. '-
10 1 9 ---.? _---
First mode -L_
J' L ?-  ----_? --'-1.
10 6 ?
 5] 709-- _, j
 'V S d 8 d 4 2' ,10 10 Thlrd mode ""-.2
O I econ mo e ., \
o o ' o
""":Z 'I
Values of u
20
-10
O
10
20
30 40
(aJ)
50
60
Figure 8.64 Argand diagram for the complex eigenfrequencies of the cantilever with a blunt end
ofFigure 8.63(c,d) with varying uo: -, ex = O; - - -, ex = 0,003 (PaJ:doussis 1975b),
140
80
120
-K=O .....-.....-'
-K5 .....-
.....-.
- - K = 00 .....-.
/
/
/
(
".
(b)
120 .
80
100
aJ
100
60
aJ
120
(c)
60
40
1st-mode regiOn]
C_. -'-'-'-'-'
-,
100
20
aJ
80
(a)
O
0,0
0,2
11
04
0,6
60
0.0
0.2 11 0.4
0,6
Figure 8.65 (a) The principal resonance regions associated with the first and second modes of
a c1amped-pinned cylinder with different rotationa1 spring constraints at  = lCP = 0.467, Y =
3.98, ex = 0.0028, X = 1.26, h = 0.519, SCj = 0,5, Cb = 0.2, uo = 3.61): -, K = O;
-e-e-, K = 5; -'-, K = 00. (b) The effect of confinement (affecting X and h) and of SCj
on the second-mode principa1 primary resonances for K = 1.9 and other parameters as in (a):
-, SCj = 0.5; - - -, sCj = 1.0. (c) The effect of dissipation in the cylinder for thc sys-
tem as in (a) and K = 1.9: -'-, ex = 0.0010; -, ex = 0.0028; - - --, ex = 0.0060
(Pai'doussis et al. 1980a).

914
SLENDER STRUCTURES AND AXIAL FLOW
Controlled water flow
with constant velocity U o !
I
PIungerpum  ,
I
To ..... ---i
reciprocating i
mechanism ,
Hot-fi1m anemometer
Test-section
Test cylínder
Flow velocíty-
U= U o (1 + I'- cosQ t)
To collecting tank 
on weighíng scales
(a)
120
1
Stroke = o 63 cm
I
I
I
I
I
I
I
I
I
I
,
I
I
I
I
I
I
I
I
I
20 J
100
80
(J)
60
,
I
I
l
l ::: 3 20 cm,
,/ I'"
l "
I "
/' 1111
I "
J" ..."",,1
40
Detail ofupper
cylinder support
1st-mode
principal resonance region
o
0,0
(b)
0.2 0.4
I'-
0,6
Figure 8.66 (a) The experimental system via which the harmonie perturbations are added to the
steady mean ftow. (b) The dashed lines show examples of the manner in which J1. and lJ) vary
together for fixed values of the plunger pump stroke in one experiment, ilIustrating the diffieulties
of 'eovering' fulIy the resonanee regions with reasonably small strokes (PaYdoussis et a[, 1980b).
hemi-ellipsoidal shape); this, via four stream1ined spider legs, was connected to a brass
ring of the same internal diameter as the test-section pipe, which was clamped securely
to the rest of the test-section when the latter was screwed to the piping above. The lower
support was a sleeve, into which the end of the test cylinder was inserted, norrnally to
a depth of 7.5 mm, with a sliding fit - thus al10wing sliding in the axial direction. In
practice, so as not to impede sliding, smalllateral clearances were permitted in the sleeve,
which resulted in a less-than-perfect clamped condition at the lower end.
A1though in reality the support condition involved is complex and nonlinear, it was
found adequate to model it as a pinned support with a rotational spring resisting rotation
at the end, The spring stiffness, CL, could then be detennined such that the measured
first-mode natural frequency be equal to the theoretical value. The theoretical circular first-
mode natural frequency, QI, was detennined by Rayleigh's method (Bishop & Johnson
1960; Meirovitch 2001), using y I L = [2 - (1 + b)3 + b4],  = xl L, which yields
the following dimensionless frequency equation:
(5wI + 1512 - 18y)b 2 - (15wl + 2520 + 48y)b + (12wI - 42y) = O,
where Wl = (mL 4 1 EI)I/2QI' y = mgL 3 I EI, b = (K + 4)/(K + 6), and K = qLI EI;
m is the mass of the cylinder per unit length, and L is the free length of the cylinder.
Hence, uti1izing the experimental QI in conjunction with this equation, one may find b,
and hence K, After conducting several experiments in zero flow (Pai'doussis et al. 1980b),

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
915
an average value of K = 1,9 was detennined, and this is the value used when comparing
with theory.
Prior to conducting the experiments, some tests were done with a hot-film probe to
investigate the shape of the velocity profile in the ftow annulus, with the flexible cylinder
replaced by a rigid one. Tests were made with pulsating flow, in order to determine if the
flow profile was deformed and if there were any phase differences introduced at different
locations within the annular gap; the signal from a wall-mounted pressure transducer was
used as reference for the phase angle. Over the pulsation frequency range of which the
apparatus was capable and the range of pulsation amplitudes used in the experiments, no
phase difference could be found; the velocity profile in the presence of pulsating ftow
retained the same shape as in steady flow.
The mean flow could be obtained from the calibrated reading of the hot- film anenometer.
When the flow had a pulsating component, its amplitude f.LUo could be determined from the
flow-velocity trace on a storage oscilloscope or a chart recorder; then, from the calibration
charts, the maximum voltage of the sinusoidal curve corresponds to Uo + f.LUo, while the
minimum to Uo - f.LUo; hence, both Uo and f.L may be detennined,
For each experiment, the stroke of the plunger pump and the mean ftow velocity
were fixed, and the frequency varied gradually. Then the stroke was changed, and then
Uo, and the same procedure followed. For a constant stroke, as Q was varied, so was
f.L, as seen in Figure 8.66(b), which makes experimention for testing the theory not
straightforward.
8,7.2(a) General observations
The behaviour of the system in pulsating axial flow is first described qualitatively, before
considering the results quantitatively. The general behaviour of the system was qualitat-
ively the same in the two test-sections of different diameter, as well as for cylinders of
different length and diameter.
At low frequencies, the system was stable, Very small oscillations could be observed,
which were partly due to the response of the cylinder to the turbulent pressure field and
partly to transmitted mechanical excitation. Increasing the pulsation frequency beyond a
threshold value caused large-amplitude harmonie oscillations to occur spontaneously; their
amplitude increased with pulsation frequency, reached a maximum of up to 1  cylinder
diameters peak-to-peak, and then decreased, until it faded completely. The frequency of
pulsation was noted to be twice the frequency of oscillation, and the modal form was
similar to that of the second mode of the beam; so, these were likely principal primary
resonance oscillations associated with the second mode, as is shown definitely in the next
subsection,
The ease of pin-pointing the onset of instability decreased as f.L was diminished; this
was because the amplitude of the resonance became smaller, so that there was no longer
a step-change in amp1itude of oscillation at the threshold pulsation frequency, but merely
a change in character of the vibration: from a random one to one having an appreciable,
but difficult to separate, harmonie component. Within the limits of the apparatus and the
system parameters tested, this second-mode principal resonance was the only one that
could be pin-pointed with accuracy.
Several differences in the pattern of observations made in the foregoing should be
noted. At high strokes of the plunger pump (high f.LUo), there were bursts of organized

916
SLENDER STRUCTURES AND AXIAL FLOW
periodic oscil1ation at pulsation frequencies inferior to the threshold frequency where reg-
ular parametric resonance oscil1ations established themselves, (Similar observations were
made in experiments on parametric resonances of pipes conveying fluid in Section 4.5.3.)
AIso, at very high values of f.1, after the onset of parametric resonance, the osci11ations
persisted to the maximum frequency attainable by the apparatus; beyond a certain fre-
quency, however, there sometimes was a change in character of the oscil1ation into
what appeared to be a combination resonance. At small strokes of the plunger pump,
it was found that a certain time was necessary for resonance osci11ations to develop;
unless one allowed for this, the trne threshold of resonance could be missed. Finally, and
importantly, if the value of f.1 was sufficiently small, no parametric resonances could be
detected at all.
Let us now tum our attention to first-mode principal resonances, which, although pre-
dicted to exist by theory, were not observed in the experiments. Reasons for this may be
discussed with the aid ofFigure 8.66(b), showing theoretical fixst- and second-mode prin-
cipal resonance regions for one of the cylinders used in the experiment s (for Uo = 4.50).
Superimposed on these are traces of nondimensional pulsation frequency ev versus amp-
litude f.1, as generated by the pulsation apparatus; the three strokes of the reciprocating
mechanism shown, in the range investigated may be characterized as 'smalľ, 'medium',
and 'very large'. It is seen that a stroke larger than 3.20 cm is necessary to reach the
first-mode resonance region - although, for other cylinders used in the experiments,
such a stroke was sufficient. However, strokes greater than 2.5 cm were normally avoided
because, when combined with high frequencies, they produced unacceptably high levels
of mechanical vibration and nonsinusoidal velocity perturbations, This gives a partial
answer for the non-observation of first-mode resonances. Another possible reason is that
the resonance amplitude was too small to emerge above that due to turbulence excitation -
see Section 8.7.3.
Experiments were then conducted with a cylinder shorter than usual, to bring first-
mode resonances within the (f.1, ev )-range which the apparatus was comfortably capable of
generating, Even so, only with undesirably high strokes would these resonance oscil1ations
begin to emerge, but with much inferior amplitude and regularity of motion than those
associated with second-mode resonance. In fact, although in the neighbourhood of the
theoretical region there were definite indications of greater regularity of small quasi-
harmonie mot ion in the generally stochastic response ofthe cylinder, no c1earcut resonance
oscillation could be identified,
The conc1usion that must be reached from these experiments is that there may indeed
have been a small first-mode resonance region there, but either it was impeded from
developing by interference from the relatively high turbulent buffeting,t or that it was an
extremely small-amplitude resonance which could not be detected visually.
8. 7.2(b) Comparison between theory and experiment
Typical results forthe second-mode principal resonance region are shown in Figure 8.67 -
in (a) and (b) for a system with D = 13.6mm, L = 416mm, Dch = 400 mm, while
tl n early experiments with a much more turbulent incident flow, no parametric resonance osci11ations
whatsoever could be detected. even in regions where they were later found, after the apparatus was modified
and the incident flow smoothed.

140
140
SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
917
Uo = 2,30
(a)
120
120
100
(J)
100
(J)
80
60
- Theory
40 . Experiment
o
0.2 0.4 0,6 0,8
f.l
140
140
Uo = 2.80
(b)
120
120
100
(J)
(J)
100
80
60
40
o
0,2 0.4 0,6 0,8
f.l
Uo = 3.93
(c)
80
60
40
o
0,2 0.4 0,6 0,8
f.l
Uo = 4,37
(d)
80
60
40
o
0,2 0.4 0,6 0,8
f.l
Figure 8.67 Experirnental boundaries of second-rnode princi pal pararnetric resonance cornpared
with theory; system pararneters for (a) and (b): tJ = 0.465, e = 32.2, lY = 0.0030, K = 1.9,
X = 1.265, h = 0.519, y = 3.98; for (c) and (d): tJ = 0.446, e = 43,6, lY = 0,0036, K = 1.9,
X = 1.236, h = 0.481, y = 11.09 (Pai'doussis etal. 1980b).
in (c) and (d) D = lO.4mm, L = 439mm, Dch = 320 mm, Additional results may be
found in Pa'idoussis ef al. (1980b,c).
In these experiments, an attempt was made to vary the parameters as much as possible,
in order to assess the effect of some of them on the degree of agreement between theory
and experiment. From the results obtained, it was conc1uded that there is no significant
deterioration of this accord, as any one of these system parameters was changed, Unfortu-
nately, 1imitations ofthe apparatus did not permit a very wide variation of most parameters.
For example, if DCh - D were very small, it was found that in the course of parametric
resonance the cylinder would impact on the test-sec ti on, which rendered the theoretical
assumptions invalido
The main purpose of these experiments, however, was not to test extensively the effect
of parameters, but (i) to ob serve the behaviour of the system, and (ii) for some arbitrary sets
of parameters, to test the validity of the theoretical model. These aims may be considered
to have been satisfied.

918
SLENDER STRUCTURES AND AXIAL FLOW
8.7,2(c) Concluding remarks
ln the presence of harmonie perturbatíons to the mean flow velocíty about a cylindrica1
body, parametric resonance osci1lations may develop. Of these, the most important are
assocíated with princípal primary resonances. For the experiments just described, the
princípal primary resonances assocíated with the second mode were predominant, a1though
it cannot be conc1uded that this is a genera1 result.
It was found that the theoretical model is capable of predictíng the pulsation amplitudes
and frequencíes necessary to induce the observed parametric resonances. Accordingly,
one may use this theoretícal model with some confidence.
One important finding is that if the turbulence level in the flowing fluid is sufficíently
large, parametric resonances may be suppressed or obscured. Indeed, this may be the
reason why first-mode resonances did not materialize in these experiments. This may also
explain the paucíty of practica1 observatíons of occurrence of parametric resonances in
the field.
8.7.3 Parametric resonances in two-phase flows
Parametric resonances ofaxially disposed structures in slug flow have been studied extens-
ively by Hara (1975a,b, 1979, 1980) and Hara & Yamashita (1978), and a compendium
of this work is given here; see a1so Dimentberg (1991). Periodic variations in momentum
occur naturally in the slug-flow regime of two-phase flows, and they are the root cause Df
these resonances (see also Section 4.5.6). However, there are other situations, as well, in
whieh two-phase flows display periodic unsteadiness, for example, due to density-wave
and parallel-channel instabilities.
ln slug flow, slugs of water, separated by gas pockets, travel along the channel, as shown
diagrammatically in Figure 8.68, with the same velocity as the mean flow, U.
Hara (1979, 1980) considers a rod of rectangular cross-section, mounted vertica1ly in
a flow channel, in up-flow. Considering that in this case the added mass is M = M(x, t),
with x being measured from the bottom upwards, the following simplified equation of
t=ť
Uf3T
t= T
o
x
L
Figure 8.68 Illustration of water slugs travelling along the flow-channel in which a cylinder is
mounted axially, in an idealized slug-flow (Hara 1980),

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
919
motion is used:
a 4 y ( a 2y a2y 2 a 2y ) ( aM aM ) ( ay ay )
EI-+M -+2U-+U - + -+U- -+U-
ax 4 at 2 axat ax 2 at ax at ax
a 2 y ay a 2 y
- mgx- - mg- + m- = B(P2 - PI),
ax 2 ax at 2
(8.164)
where B is the rod width, and the right-hand side represents the loadiog due to unsteady
pressure differeoces on the two surfaces of the rod, normal to the direction of motion.
Presuming that the rod is simply supported, the equation is discretized via Galerkio's
method, using y(x, t) = Li Yi(t) sin(i7rxjL).
Further aoalysis is cooducted with a one-mode approximation, in which a viscous
damping term has been included and gravity terms neglected, that is,
s" [ 2 j C (')s (/)S ] .
(m + M l1 )YI + Cl1 + ( U7r L)M l1 + MlI + UM l1 YI
+ [EI(7rjL)4 - (U7rjL)2M1 + (U7rjL)(Mi)C + UMfíC)] Y I = PI, (8.165)
where ClI is the viscous damping coefficient, and
2 [L , ( 7rX ) . ( 7rX )
MI = L 10 sm T M(x,t)sm T dx,
(.)S 2 i L. ( 7rX ) [ aM(X, t) ] . ( 7rX )
MIl = - sm - sm - dx,
L o L at L
and similarly for Mi1 s , with aM(x, t)jax io the square brackets; superscript c deootes
an integral with the last term in the integrand replaced by cos(7r x j L); and PI == PI (t) =
(2B j L) Jo L (P2 - Pd sin(7r x j L) dx.
Denoting the added mass of a water slug by Mo aod the period of arrival to any given
location aloog the rod by Ts, we can write
N-I
M(x, t) = Mo L 8{x - U(k - I)T s - Ut}
k=1
+ M08{x - U(N - I)T s - Ut} h(tjf3T s ),
(8.166)
where N = mod(Lj UT s ) + 1, and 8 is the Dirac delta functioo - see Figure 8.68; f3 is
defined in Figure 8.68, and O ::::: f3 ::::: 1; the function h is defined by h Cr) = O for. < O

920
SLENDER STRUCTURES AND AXIAL FLOW
and "C > 1, and h ("C) = 1 for O :::: "C  1. The damping, mainly caused by the water slugs, is
represented by c(x, t)(aylat), with c(x, t) having a si1IŮlarfunctional fonn as in (8.166),
but with co instead of Mo, Co being the damping coefficient of a single water slug. An
analogous fonnis taken for p(x, t), butwith Tp instead of Ts, where Tp is acharacteristic
period of the pressure fluctuations. Substituting these expressions into those for Mil etc.
gives Ml = (Mol L)[N - 1 + Q(t)], Mfl = (Mol L)R(t), Cli = (co I L)[N - 1 +
( (.)c (/)C (')s (/)S O -l' k k (21 )S( )
Q t)], Mli +UM II = Mli +UM II = ,1or( -1)T s  t:::: Ts; PI = L t
for (k - l)T p :::: t :::: kT p ; k = 1,2,3, ...,; where
N-I
Q(t) = - L cos«2nU I L) [(k - I)T s - t]) + [1 - cos«2nU I L) [(N - I)T s - t])Jh,
k=1
N-I
R(t) = L sin«2nU I L) [(k - l)T s - t]) + sin«2nU I L) [(N - l)Ts - t])h,
k=l
N-I
S(t) = L Pk sin( (nU I L) [(k - l)T p - t]) + PN sin«nU I L)[(N - l)T p - t])h,
k=l
Hence, equation (8,165) becomes
{ Mo } .. { Mo Co } .
m + T [N - 1 + Q(t)] YI + TR(t) + L [N - 1 + Q(t)] YI
+(J2 {EI(Ž)2 - :O U 2 [N_l+ Q (t)]}YI = S(t).
(8.167)
Considering now Q(t), R(t) and S(t) to be periodic, expressing them in Fourier-series
fonn and us ing only the leading tenns, we may write
Q(t) = qo + ql cos(27ftIT s ), R(t) = n sin(2ntIT s ), S(t) = Sl sin( 2nt IT p );
1 i Ts
qo = - Q(t)dt = {J,
Ts O
2 i Ts ( 2nt )
ql = - Q(t) cos - dt,
Tso Ts
etc.
(8.168)
When these are substituted into (8.167), one obtains
(1 + EI cos wst)i\ + [(2nU I L)E2 sinwst + E3 cos wst + K] YI
+ w5 {I - [Unl(woL)]2 EI coswst} YI = PI sinwpt,
(8.169)

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
921
Mo ql Mo rl Co ql co(N - 1 + {3)
EI = """""--, E2 = """""--, E3 = -, K =  ,
M L M L M L ML
PI =  , w6 = GJ 2 [  (i)2 -  U 2 (N - 1 + {3)J.
 Mo 211' 2n
M = m + -(N - 1 + {3), Ws = -, w p = -.
L Ts Tp
We proceed, dividing by (1 + EI cos wst) and introducing a new transformation,
YI = exp[- t (nU / L)E2 sinwsT dT]XI, whereby equation (8.169) takes the form of
a Mathieu-type equation (McLachlan 1964; Nayfeh 1981):
- . 2
Xl + 2wo Xl + Wo (1 - EO COSwst) Xl = PI sinwpt,
(8.170)
where  = K/2wo andEo = EI [1 + (Un /wOL)2]+E2 [2nU ws!(w6L)]. This may further
be transformed into canonica1 form,
Xl + 2-/Ci Xl + (a - 2q cos 2T)XI = PI sin r2t,
(8.171)
inwhicha =4 (WO/ws)2 , q = 2 (WO/ws)2 EO, r2 = 4(w p /w s )2, 2T = wst. Equation (8.170)
being a nonhomogeneous Mathieu equation, we distinguish the fundamental solutions and
a particular one. The former are as in Section 8.7.1, i.e. the primary ones for ws!wo ::::
2, j, . ,., and the secondary ones for Ws / Wo :::: 1, !, , ,.. The particular solution is a1so
investigated, and it is found that (i) if w p = Ws and  « 1, the solution becomes
unstable for ws/wo -:= 1, ! and 1, and (ii) if EO« 1, when wp/wo = 1 _ 2.
Extensive experiments were conducted (Hara 1975b; Hara & Yamashita 1978), in air-
water flow; the flow itself and the response of a 10 x 2 mm 2 rectangular-section stainless
steel bar immersed in it were studied.
The experimenta1 response for different void fractions in the bubbly and slug-flow
regimes is plotted in Figure 8,69. It is seen that, genera1ly, the responsein slug flow is higher
than in bubbly flow. However, the vibration is caused main1y by pressure fluctuations on
the rod surface; this is clear from another plot of the response, S;, versus a measure of the
mean-square of the differentia1-pressure fluctuations (not shown here). Yet another plot,
of S; versus frequency, shows a broad peak at the first-mode rod frequency only. Hence,
the conclusion is that the vibration is mainly forced by these pressure fluctuations, whi1e
parametric resonances, if any, must be 'buried' in it. Here it is recal1ed that two-phase
(forced) flow-induced vibration is much larger than for single-phase flow.
Hara & Yamashita (1978) have subsequently reca1culated the response as S; =
S; /[ SJ p /(ll - (Wdp/WO)21)], in which SJp is the variance of differentia1 pressure fluctu-
ations and Wdp their dominant frequency, thereby excising the effect of forced vibration
from the response. Gratifyingly, the parametric resonance response now emerges, as seen
in Figure 8,70, at WO / Ws = !, I and  (i.e, Ws / WO = 2, I, j), as predicted by theory.
It is interesting to note that the secondary resonance ws! Wo = 1 is the weakest; hence
the supposition in Section 8.7.2(a) is probably sound that this resonance region may have
existed in the single-phase experiments, but was buried in the turbulent buffeting response.

922
SLENDER STRUCTURES AND AXIAL FLOW
10 3
M
G ,,-
B F C D "....
i>'r1 i'- .N .R
line}" Q
A ......"" I L
,"'- P/i
I
I
/line II O, bubble
,
(): bubble + slug
I .: slug
I
oQ
I
'"
:;-
2, 10 2
"!,,
10
2
3 4
U (mis)
5
6
Figure 8.69 Variance of the vibrational strain, S, versus two-phase (air-water) flow velocity
(Hara & Yamashita 1978).
8
p

:, o : bubble
1\
"
I () : bubble + slug
I
I
I .: slug
I I
: ,
, ,
I \
,
: I
, ,
I ,
I I
I I
I ,
I ,
I ,
, .' I'
I Q', C/B
I
K I 'A
.R
: I .D
I ,
I \
IN I
 I F
I .,/ "G
L M /.1 '."
. ./E
0.5
1.0
wo/ws
1.5
'" 6
O'
:I:'"
S

'g 4
2,
<'"
2
o
Figure 8.70 Modified variance of the vibrational strain, S;, versus the dominant frequency Ws
(expressed as wo/ ws), showing the existence of a parametric resonance response in the rod vibration
(Hara & Yamashita 1978).
8.8 COMPLEX SUPPORT CONDITIONS
In all the foregoing, 'simple' support conditions (e,g. c1amped, spring-constrained, ar
free) have been presumed to exist at the two ends of the cylinder, As mentioned in
Section 8.4.3(c), however, in many cases there exist intermediate supports as wel1 (e,g. for

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
923
33.1 36,0 37.4 38,0 41.4 44.6 47.6 52.4 55,7 59.9 Hz
Figure 8.71 Mode shapes of a typical multispan tube (Au- Yang 1999,2001),
heat exchanger tubes and nuclear fuel rods) - cf. Section 3.6; refer also to Pettigrew
et al. (1978), Gorman (1975b), Au-Yang (2001) and Pettigrew (2001), An example ofthe
modal shapes and corresponding frequencies of a multispan heat exchanger tube is shown
in Figure 8,71. The natural frequencies are closely spaced; for example, despite the very
different modal shapes, the third- and fourth-mode frequencies are only 0,6 Hz apart, This
clearly poses problems in the 'normal' forced vibration response calculations in which it
is assumed that these frequencies are not closely spaced.
The dynarnics of multispan cylinders in axial flow has received little attention, in contrast
to the same system in cross-flow, as discussed in Section 8.8,1.
Furthermore, in some cases the intermediate supports are loose, by design, and hence
only partially active; that is, theyare active only when static or dynarnic deflections of the
tubes result in positive, though perhaps temporary, contact with the supporting structure. In
such cases, impact with the supports and strong nonlinearities become part of the problem,
as discussed in Section 8.8.2.
Finally, in some cases the cylinders are clustered and flexibly interconnected, Hence,
the dynarnics of the whole cluster should be treated together, as discussed in Chapter 9.
8.8.1 Multispan cylinders and tubes
Unlike for pipes conveying fluid where intermediate supports do not affect the flow, the
situation is more complex for extemal axial flow: the intermediate supports both disturb
the flow downstream and provide additional sources of excitation. The premier examples
of this type of system are heat exchanger tubes and nuclear fuel rods; in these cases,
additionally, the flow properties vary with axiallocation because of heat transfer, As all
these effects are dependent on the precise nature of the supports and the design oť the
system, a generic model is difficult to conceive,
So far as stability is concemed, there is no question that analytica1/numerical model1ing
oť multispan cylinders in ax:ial flow, incorporating variable flow conditions, singularities

924
SLENDER STRUCTURES AND AXIAL FLOW
and other excitation sources along the length, is feasible; nevertheless, to the author's best
knowledge, no work on this has been published. Perhaps the main reasons are that (i) the
fundamentals of the dynamic behaviour are broadly understood by analogy to single-
span systems, and (ii) the margin between actual flow velocities in most engineering
applications and the predicted critical flow velocities is large enough for the approximate
single-span critical velocities (obtained by analyzing stability of a single span in isolation)
to be equally meaningful as those of the multispan system. Hence, there has been limited
academic and no practical imperative for pursuing this type of study.
Nevertheless, there are aspects of the dynamics well worth investigating, for example
the existence of energy trapping modes, as has been done for periodically supported pipes
with internal flow (Section 3.7.3).
With regard to subcritical, turbulence-induced vibration prediction in multispan sys-
tems, for example heat exchanger tubes, there is lively engineering interest. Referring
to Chen & Wambsganss' analytical model of Section 8.4.2, it is clear that it is equal1y
applicable to multispan systems, provided the eigenfrequencies of the multispan system
are known and the general form of the analysis is used, that is, equations (8.115)-(8.117);
thus, the simplifications arising from neglect of coupling between modes, leading to
equation (8.118) et seq. are not admissible. The reader is referred to Au- Yang (1986b,
2001) for a discussion of cross-flow-induced turbulent buffeting of single-span and
multispan tubes. Since the fluid properties vary with axial location, perhaps the most
convenient way to compute the response would be via a finite-element discretization of
the system,
However, as discussed in Section 8.4, there exist substantial uncertainties in specifying
the excitation field, to which those associated with the intermediate supports must be
added. Consequently, in practice the response of a single span is computed instead, being
representative of that of the multispan system - the loss of accuracy being presumed to
be of the same order as the uncertainty in the excitation. The same applies to the empirical
tools of Section 8.4.1(d).
8.8.2 Loose supports
As already mentioned in Section 8.4.3(c), additional mechanisms of damping normally
arise in multispan systems, because of the fact that intermediate supports are often loose,
thus giving rise to (i) vibro-impact and rubbing, and (ii) squeeze-film effects; refer to the
work cited therein. These effects should clearly be taken into account when computing
the subcritical vibration of such systems.
However, there are other very interesting, purely dynamical effects associated with loose
supports, For example, when contact at one or more of the tube support plates (TSPs) is
inactive, then 'mode localization' and 'energy trapping modes' arise (see Section 3.7.3).
An example from Yeh & Chen's (1990) work is discussed here, involving a 15-span tube
(Figure 8,72) under no-flow conditions. t When all the TSPs are active, all the modes are
clustered within the 'passing frequency bands' of travelling waves along the tube. In this
case, the first two passing bands and corresponding natural frequency ranges are (23.9-
54.2Hz) and (100.4-152.0 Hz). The intervening frequency ranges (e.g. 54.2-100.4Hz)
tThe mode shapes and eigenfrequencies of multispan cylinders are nonnally calculated by commercially
available computer codes: e.g. NASTRAN, PAL (used in the calculations of Figure 8.72), ANSYS, ABACUS,
and so on - or more particular codes, such as PIPEAU and WECAN.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
925
15 14 13 12 11 10 9
8
7
6
5
4
3
2

t Outlet
Tube sheet
Iulet t
Tube sheet
Tube support plate
(a)
(b)
(c)
 ---
(d)
\J'
Figure 8.72 (a) Schematic of the 15-span heat exchanger tube. (b) Mode shape of the first mode
(23.9 Hz) when all TSPs are active. (c,d) Mode shapes of the first mode when one of the TSPs is
inactive: (c) TSP2 inactive (11.3 Hz); (d) TSP5 inactive (9.52 Hz); (Yeh & Chen 1990).
are 'stopping bands', in which waves do not propagate freely but are attenuated. However,
if one or more of the TSPs are inactive, waves in the hitherto stopping band may propagate
within the 'defectively' supported part of the tube, but not beyond; they are thus trapped. An
example of one ofthe modes (the first) when all TSPs are active is shownin Figure 8.72(b),
while two examples when one of the TSPs is inactive in Figure 8.72(c,d), both of lower
frequency and both showing intense mode localization. Refer also to Chen (1973) for
mode localization in curved tubes.
For a system with intermediate supports, some of which are loose, the critical velo-
city would be substantially lower than for the same system with all supports active.
If the flow velocity were higher than this lowered critical value, the cylinder would
diverge or flutter as the case may be, temporarily at least, until the amplitude became
large enough for the cylinder to touch, even if obliquely, one or more of these inact-
ive supports - thereby rendering it or them active, whereupon the system would be
restabilized. This is so if the contact is established inelastically; otherwise, the contact
would involve impact and rebound, involving strong nonlinear effects, ln this regard,
an inactive/active support may be modelled as a nonlinear spring: of stiffness zero
when inactive and very large after contact, Large nonlinearities of this type have been
found to give rise to interesting nonlinear dynamics and chaos in internal flow systems
(Section 5,8,1). Indeed, the impact oscillator is one of the c1assical problems giving rise to
chaos (Moon 1992, 1999; Thompson & Bishop 1994); see also Peterka & Vacík: (1992) and
Peterka (1997),

926
SLENDER STRUCTURES AND AXIAL FLOW
With external tlow, most of the work has been done for clustered loosely suppor-
ted cylinders in cross-fiow, intermittently subjected to so-called tluidelastic instability
- an amplified oscil1ation - see, for example, Chen (1987), Blevins (1990), Price
(1995), Examples of this type of work may be found in Chen et aZ. (1985), Axisa
et aZ. (1988), Yeh & Chen (1990), Paidoussis & Li (1992), Pa'idoussis et aZ. (1993a),
Mureithi et aZ. (1995), and Chen & Chen (1996). Some work has also been done with
articulated cylinders in axial flow, discussed in Section 8.12.
8.9 TOWED AND SELF-PROPELLED CVLlNDERS
8.9.1 General introduction
Interest in the dynamics of towed ships goes back to the days when transport by horse-
drawn barges in rivers and canals was common practice, Nevertheless, the first substantive
paper on the subject was by Strandhagen, Schoenherr & Kobayashi (1950), despite the
fact that the physics of the system was well understood long before, in conjunction with
stability of kite balloons and airships moored to a mast [e,g, Bairstow et aZ, (1915), Munk
(1924), Glauert (1930), Bryant et aZ. (1942)],
Strandhagen et aZ. (1950) established the following important criteria for stability of a
towed ship: (i) the point of attachment of towrope should be ahead ofboth the centre of mass
and the centre of pressure of the (static) lateral hydrodynamic forces acting on the ship; (ii)
the ship should be stable when moving untowed; (iii) in cases where (ii) is not satisfied,
then the system could be rendered stable by either short or long enough towropes. It is
noteworthy that the criteria for stability, at least for the linearized theory of small departures
from course, apply to all towing speeds, so that for a given configuration a (rigid) towed
ship is either stable or unstable irrespective of how fast it is being towed. Instabilities are
of two distinct types: (i) yawing, that is, zero-frequency amplified motion, which in aero-
elasticity would be referred to as 'divergence', and (ii) oscillatory instability, where the
system, when disturbed, oscil1ates about its position of rest with increasing amplitude -
what we would call 'flutter' if flexural forces were involved. For infonnation of a more
general nature on ship and marine hydrodynamics the reader is referred to Bishop & Price
(1979) and Newman (1977), for example,
More recently, interest in the stability of submerged towed bodies has revived, mainly in
connection with sonar applications. Here the body housing the sonar device is towed deeply
submerged by surface craft, for the purpose of hydrographic survey, submarine detection,
or location of schools of fish. Examples of this type of work are by Strandhagen & Thomas
(1963), Richardson (1965), Patton & Schram (1966), Laitinen (1967), Jeffrey (1968),
Schram & Reyle (1968), Abkowitz (1969), Calkins (1979), Sanders (1982), Ablow &
Schechter (1983), Kato et a1..(1986) and Poddubny et a1..(1995).The stability problem for
the sonar-type towed bodies is of course quite similar to that of a towed glider, balloon,
or other pendant vehicle; see, for example, DeLaurier (1972), Etkin (1998), The work
just cited deals with the dynamics of the towed body system as a whole; the geometry of
the towed body for these applications tends to be fairly complex, and the analysis quite
elaborate.
A considerable amount of work also exists on the equilibrium configuration and the
dynamics of towing cables - where the tlow is not necessarily axial - starting with

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
927
Relf & Powell (1917) and McLeod (1918); see reviews by Casarella & Parsons (1970) and
Choo & Casarella (1973). See also studies on cables and cable-body configurations arising
from specific manoeuvres, for example, Skop & Choo (1971), Choo & Casarella (1972),
Ablow & Schechter (1983), Zhu & Rahn (1998). Also refer to work on the dynamics of
moored floating hoses, for example, Brown & E1liott (1988), Bree et al. (1989).
However, the main interest here is on towed flexible bodies. Work on this was initi-
ated in 1956 with Hawthorne's invention of the Dracone (Hawthorne 1961),t a flexible
sausage-like container towed behind a small craft, and used for the transportation of oil
and other lighter-than-water cargos, including the sea transport of fresh water to arid lands
(e.g. to some of the Aegean islands from the mainland) - see Figures 8.73 and 8.140(a)
.
(a)
(b)
(c)
Figure 8.73 (a) Stem view of a 52 m 3 Dracone under tow at 7 knots; note stabilizer at stem and
dead-straight tow, (b) The 52m 3 Dracone filled with kerosene, executing a U-tum, (c) A 500m 3
Dracone, air-inflated after discharging fresh-water cargo, leaving the island of Santorini, Greece
(Dunlop Dracones 1965).
tThis was the time ofthe 'Suez crisis', when shipping from the Persian Gulfhad to circumnavigate the Cape
of Good Hope to reach Europe, and large ocean-going tankers were in short supply.

928
SLENDER STRUCTURES AND AXIAL FLOW
and (b). The Dracone is only partially (80% or 90%) submerged [Figure 8.73(a) and (b)].
After delivery of the cargo, it is either rolled up and loaded on the towing vessel, or filled
with air and towed back, most of it above surface [Figure 8.73(c)].t
More recently, another application has motivated work in this area: towed 'acoustic
streamers', otherwise known as 'seismic arrays' or 'streamer cables' (Telford et a/. 1976;
Poddubny et al. 1995). These are extremely slender, neutrally buoyant plastic tubes
(1-3 km long and only 60-70 mm in diameter), towed behind a surface vessel, usually
slightly submerged to avoid white-water seas; typically, several are towed from the same
ship, arranged in parallel, ""' 60 m apart (Davies 1998). They house arrays of hydro-
phones, and they are used for undersea geological survey, especially for locating oil and
gas deposits. How such systems work is illustrated diagrammatically in Figure 8.74. Mod-
em acoustic arrays used for military purposes (submarine detection) are similar, though
typically less slender (""'250-650 m long, 30 mm in diameter) and are towed deeper
(Grosenbaugh 1998),
Yet another application involves towing metal pipes underwater to the offshore location
where they are needed, either as articulated strings or pre-welded pipelines several kilo-
metres long; the pipes are made neutrally buoyant while in motion by fi1ling them with air
and/or fitting them with buoys and draping chains over them; when not in motion, they
settle on the sea tloor. * For an application related to large-scale liquid C02 storage in deep
oceans, it is proposed to tow the complete pipeline in the manner shown in Figure 8.75.
This can be a more economical means of transport for such marine applications than by
surface vessels, and more convenient for eventual installation,
Finally, it was suggested (Hamy 1971) that strings of cylindrical or quasi-cylindrical
containers, towed underwater by a small submarine or self-propelled by the first module
of the string, could be used to transport liquid cargos to and from arctic ports, avoiding
the hazards of surface transportation in ice-infested seas. The containers in the so-called
'sea-chain' (Figure 8.90) could be either tlexurally tlexible or, more likely, rigid.
ln all these applications, stability is essential: (i) in the Dracone system, to avoid
excessive stresses in the container; (ii) in the case ofthe acoustic arrays, to be able to know
precisely the instantanwus geographic location ofthe part ofthe sea-bed being probed (in
this case not only stability but absence of low-frequency vibration is important) - see,
for example, Dosso & Riedel (2001); (iii) in the case of the pipe-strings, to avoid such
undesirable events as part of the string suddenly breaking surface in front of unsuspecting,
unrelated shipping; (iv) in the sea-chain case, for obvious reasons.
The first analysis of stability of the Dracone was by Hawthome; it was later exten-
ded and refined by Pa'idoussis (1968, 1970) who also conducted experiments, Pao
(1970) and Pa'idoussis & Yu (1976). More recently, further work on the subject was
done by Lattore & Zachariadis (1983), Theodoracatos & Calkins (1986), Sudarsan
et al. (1997, 1999) and Bhattacharyya et al. (2000). Some ofthis work is outlined brietly in
Sections 8.9.2-8.9.5.
tA charming tale connected to such a retum-trip after delivering water to one of the Greek islands is this, As
seen in Figure g,73(c), the Dracone (/:;,PÓlKWV in Greek, a dragon) is much larger than the fishing ves sel towing
it. The constabulary of another island, being passed along the way at dusk, received frantic telephone calls
from concemed islanders, asking them to intervene urgently and rescue a poor fishing cai:que 'being chased by
a horrendous monster' ,
*The oil industry is notoriously secretive; hence, information on these last two applications is hard to come
by, despite intensive aerial photographing of one another's systems by the rival concems.

SOLllARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
929
Figure 8.74 Diagrammatic view (not to scale) of a towed acoustic array (Poddubny et al. 1995).
Waterline
Pulling
cable
4 Buoys
Large diameter pipeline
Seabed
Figure 8.75 Large-diameter (64 in,) towed pipeline - for use in deep ocean storage of liquid
C02 (Sarv & John 2000).
Related work on the stability of very s1ender towed cylinders, represented as pinned-
free strings and modelling towed acoustic arrays - inc1uding Ortloff & Ives' (1969),
Lee's (1981) and Triantafyllou & Chryssostomidis's (1985) work - has already been
discussed in Section 8.3.l(e), 8.3.2(g) and 8.3.7; it is not substantially discussed further in
what follows. However, two important papers by Dowling (1988a,b) are, in Section 8.9.7;
they provide the link between analyses neg1ecting flexural rigidity and those taking it
into account. Some work on self-propelled cylinders is discussed in Section 8.9.8.

930
SLENDER STRUCTURES AND AXIAL FLOW
8.9.2 The Hawthorne-Pai.doussis model
ln the literature, one commonly finds reference s to the 'Pa'idoussis model' or 'the
Pa'idoussis equation ofmotion', usually referring to the model in Pa'idoussis (1968). How-
ever, the essential model was first proposed by Hawthome (1961). Hence, the heading
here is the only proper one.
The model presented in this section describes the model equations, as corrected by
Pai'doussis (1970) - corrected for the same error in the viscous forces discussed in
Section 8.2.2. It is recalled that several researchers (Ortloff & Ives 1969; Pao 1970;
Pao & Tran 1973; Lee 1981; Lee & Kennedy 1985) have unfortunately based their work
on the earlier, erroneous equations, The repercussions of using the erroneous equations
are discussed in Sections 8.9.3 and 8.9.7.
The system under consideration is shown in Figure 8.76. It is immediately obvious that
it is identical to that of Figure 8.1, except that the cylinder is now fitted with an ogival
nose as well, the upstream end of which is connected by a towrope to a point presumed
fixed vis-a-vis the flow; in this idealized model the towrope is assumed to be massless.
AU the assumptions made in deriving the equation of motion in Section 8.2,1 still hold,
Hence, without further ado, the equation of smalllateral motions for a quasi-cylindrical
flexible body of revolution of cross-sectional area A (x) is
0 2 ( 02y ) ( O o ) 2
- EI- +pA -+U- y
ox 2 ox 2 ot OX
( Oy ay ) dA 1 ( PA ) ( Oy Oy )
+pU -+U- -+-CNU - -+U-
ot ox dx 2 D at ox
1 2 [ 1 L A(x) ] 02y 02y
- -pU C2A(L)+cT -dx -+m- =0,
2 x D(x) ox 2 ot 2
where C2, which had hitherto been denoted by Cb, is the base drag. For a uniform cylinder,
or over the cylindrical part of the body in Figure 8.76, this equation becomes
04y ( O o ) 2 1 ( MU ) ( Oy Oy )
EI-+M -+U- y+-CN - -+U-
ox 4 ot OX 2 D ot OX
1 2 [ L-X ] o2y o2y
--MU C2+Q- -+m-=O,
2 D ox2 at 2
(8.172)
(8.173)
where the diameter, D, and the added or virtual mass M = pA are now constant; the
forces on the nose and tail segments of the body are taken into account in the boundary
--------_.-1f-
-----L----- ------
x
Figure 8.76 Idealized system of a towed cylinder with noncylindrical 'nose' and 'tail' segments
(Pai'doussis 1968).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
931
conditions. Of course, this equation for a cylinder in a tlowing stream also holds for a
towed flexible cylinder if we identify U as the towing speed, provided the towrope forces
are also taken into account in the boundary conditions, and provided the towing craft
moves horizontally in a straight course.
The boundary conditions are derived in a similar way as in Section 8.2.4. We could, of
course, use equation (8.172), together with boundary conditions stating that (i) at the down-
stream end, x = L, the bending moment and shear force are zero, and (ii) at the upstream
end, x = 0, the bending moment is zero, but the shear force is equal to the normal com-
ponent ofthe towrope pul!. However, as we are only looking for the general characteristics
of the dynarnical problem, we instead proceed, more conveniently, as follows: (i) we use
equation (8.173) which satisfactorily applies over the uniform, cy1indrical part of the body
as the equation of motion; (ii) the forces acting on the noncylindrical ends are lumped and
incorporated in the boundary conditions. For this proces s to be meaningful, the lengths of
the nose and the taH, lt and 12 respectively, must be small compared to L, yet large enough
to permit the use of slender-body approximations. Taking lt « L, 12 « L, we may fur-
ther simplify the problem by considering y and the lateral velocity v to be approximately
constant over °  x  II and L -12  x .:S L, and by neglecting the skin frictional forces
over the same intervals. Hence, integrating equation (8.4) with Fpy = O, utilizing (8,2)
with v = [(ay/at) + U(ay/ax)], and incorporating the forces arising from the tow-rope
pull, P, we obtain
l il aQ l il ( a a ) y l il a2y
-dx - /J - + u- (pAv)dx - P [ - J - m-dx = 0,
o ax o at ax s x=o o at 2
1 L aQ l L ( a a ) l L a2y
-dx - h - + u- (pAv)dx - m---rdx = 0,
L- 1 2 ax L- 1 2 at ax L- 1 2 at
s being the length of the towrope; the parameters JI and h, equal to unity according
to slender-body inviscid-tlow theory, are introduced to account for the theoreticallateral
force at the nose and the tail, respectively, not being fuHy rea1ized because of (i) the lateral
flow not being truly two-dimensional, (ii) boundary-layer and separation effects, and (iii)
the possible use of fins. The towrope pull P is equal to the tension in the cylinder at x = O,
plus the form drag at the nose, that is,
P = !MU 2 (ct +C2+CTL/D),
where ct is the form drag coefficient of the nose segment. Substituting P into the above
equations and assuming y and v to be constant over the intervals of integration, we obtain
[ a3y ( a y ay )
EI+/JMU -+U-
ax at ax
1 2 ( L ) Y a 2y ]
+ -MU CT- +ct +C2 - + (m + /JM)XI2 = O,
2 D s at x=o
[ a3y ( a y ay ) a2y ]
-EI-hMU -+U- +(m+hM)x2---r =0,
ax at ax at x=L
(8.174)
(8.175)

932
SLENDER STRUCTURES AND AXIAL FLOW
where
Xl =.!. ti A(x) dx and X2 =.!. [L A(x) dx.
A 10 A 1L-lz
Here M = pA, and A and D are quantities pertaining to the cylindrical part ofthe body.t
The other two boundary conditions are obtained by making the reasonable assumption
that the bending moments at X = O and X = L are negligible, or
[ a2 ] = [ a2 ] = o.
OX x=o ax x=L
The advantages in this method of analysis, in which the shape characteristics of nose
and tail are absorbed in the two parameters f1 and h, are obvious. The disadvantages are
equaHy obvious: although we can estimate f1 and h, we cannot easi1y calculate them.
As in Section 8,2.4, the range of f1 and h is taken to be between zero and unity, the
latter limit here representing well-streamlined, gradually tapering nose or tail, and no flow
separation; it is obviously much more likely for f1 to appr:oach unity than for h. In the
case of a blunt taH, on the other hand, h -+ O.
The equations of motion and boundary conditions are tendered dimensionless by the
use of equations (8.28a) and the additional ones pertaining to the nose section,
(8.176)
Xl = x1fL,
A = siL,
(8.177)
yielding
a 4 11 [ 1 1 ] 0211 0 2 11
- + u 2 1 - -ecT(l-) - -C2 - + 2f31/2u-
04 2 2 a2 aar
1 2 al1 1 1/2 011 a211
+ -ecNu - + -ecNf3 u- + - = O,
2 a 2 ar ar 2
which is a simplified foun of equation (8.27b), and
a 2 11 a 3 11 2 al1 1 2 1/2 al1
02 = 03 + f1u o + ZU [(UT + Cl + C2) I A] 11 + f1f3 u aí
a 2 11
+ [1 + (f1 - 1) f3] Xl ar 2 = O at  = O, (8.179)
0 2 11 0311 011 011 0 2 11
- = - + hu 2 - + hf31/2u- - [1 + (h -I)f3]X2- = O at  = 1,
02 a3 a ar Oí 2
(8.178)
where X2 = x21 L is the same as Xe in Section 8.2.5. In principle, f3 may vary from O to 1;
however, in most ca1culations, nuH buoyancy is assumed, so that f3 = !.
Let us now consider motions of the cylinder of the foun
11(, í) = Y(Oe iw <,
(8.180)
tThe forces arising from form drag have been taken to be tangentially a10ng the cylinder, lfthey are taken to be
in the x-direction (along the flow), then terms 1cI MUz(ay fax) and 1czMUz(ay fax) shou1d be incorporated
in equations (8.174) and (8,175), respectively.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
933
w being a dimensionless frequency defined by w = [(M + m)/EI]I/2QL 2 , where Q
is the circular frequency of motion which in general is complex. The method of solu-
tion is to express Y() as a power series in  (cf, Section 8.2.7), so that 17(, 't') =
[Lo Arr] exp(iw't'), where the Ar are generally complex. Substituting into (8.178)
and (8.179) we eventually obtain a 4 x 4 determinant, which must vanish for non-trivial
solution; this provides an implicit relation between w, on the one hand, and the di men-
sionless towing speed u and the system parameters, on the other. A sufficient number of
terms in the power series must be used to approximate adequately the shape ofthe body in
the course of its motions, and hence its eigenfrequencies. It was found that up to 40 terms
are necessary in order to determine w to three significant figures. Altematively, of course,
the Rayleigh-Ritz or an extended Galerkin-type solution could be used instead.
Certain modifications and refinements to this basic model are proposed in Pa'idoussis &
Yu (1976), all related to the boundary conditions, as follows. (i) The nose is considered to
be a half-ellipsoid, and the inviscid forces on it are formulated by means of potential flow,
rather than slender body theory, following Jones (1927). (ii) The tail is considered to be a
truncated ellipsoid, and the inviscid forces on it are formulated in the same way as for the
nose, integrating over its truncated length, and presuming that separation occurs exactly
at the surface of truncation. Thus, it is possible to predict approximately the equivalent of
the empirical factors ft and h and the base drag on the tail. The analysis for (i) and (ii)
is too involved to give here in detail, but may be found in the paper, (iii) The moments
on the nose and tail and the frictional forces thereon are tak:en into account, small though
they may be. (iv) The nose centreline is not considered to be necessarily aligned with the
towrope; hence, the term u2[(t:CT + CI + C2)/ A]17 in (8.179) is replaced by
U2[(t:CT + CI + c2)][(17/ A) - (a17/a)];
(8.181)
this is a significant modification, and would apply in cases where the towrope has vanishing
flexural stiffness at the point of attachment to the cylinder [refer also to Yoder (1973)].
However, most of the results presented in the following have been obtained with the
simpler theory (Pa'idoussis 1970).
Finally, a finite element model has been formulated (Bhattacharyya et al. 2000) via an
appropriate statement of Hamilton's principle and Pa'idoussis' (1970, 1973a) expressions
for the hydrodynamic forces.
8.9.3 Dynamics of towed flexible cylinders: theory
Typical Argand diagrams for the dynamics of a towed cylinder with ideally well-
streamlined nose and tail and A = 1 (i.e. a towrope as long as the cylinder) are shown in
Figure 8.77. Both modes in Figure 8.77(a) emanate from w = O, and for u = O correspond
to rigid-body modes: one lateral (the 'zeroth'), the other 'criss-crossing' (the 'first mode').
Increasing u, however, results in w =1= O; thus, restoring forces become operative, due to
flow-induced tension, for instance, as well as some flexure, and these modes therefore
become quasi-rigid. As seen in the figure, the first mode is associated with Re(w) > O
for all u > O up to u  3.6. The zeroth mode, however, remains on the Im(w)-axis, as
shown in the two diagrams on the left ofFigure 8,77(a), up to u  3.7, whereupon one of
its branches merges with one ofthe first-mode branches.

934 SLENDER STRUCTURES AND AXIAL FLOW
1.5 4
1.5
Zeroth
mode
2 36
.£
.o
:!
 365 ti:)
O O O o t
 5 6 I
t2:::f u "
37 

-2 :J
3.6
3.5 3.5
2 2
3 3 -4
O 2 4 6 8
(a) £-(w) = O £-(w)
0,4
0,2
 O

-0,2
-5
-10
-15
O
10
20
(b)
40
£-(w)
50
.£
.o
'"
ti)
.£

'"
=
60
70
Figure8.77 Thedimensionlesscomplexeigenfrequenciesofaflexiblecylinder(fl = 1 - CI = 1.
h = 1 - C2 = 1, SCN = BCT = 1, A = 1, Xl = X2 = 0.01) as a function of the dimensionless
towing speed u: (a) the zeroth and first mode, where the two diagrams on the left show theevolution
of the zeroth mode on the Im(w)-axis; (b) the second and third modes (Pa'idoussis 1970),
ln terms of stability, the zeroth mode is unstable for u > O, since Im(w) < O for one of
its branches; this is ayawing (zero-frequency, divergence-type) instability, The firstmode
also is unstable for O < u < 0.74, where Im(w) < O, though this is not discemible in
the scale of the figure; this is an oscillatory instability. Thus, the dynamical behaviour at
low towing speeds is as follows: (i) for O < u < 3,7 t the system is subject to a yawing
instability in the zeroth mode; (ii) for O < u < 0.74 it is also subject to an osci11atory
tFor cylinder I in the experiments (Section 8.9.4), u = 3.7 corresponds to 2.1 mJs (:::: 7.6 km/h); that is,
U = O,57u mJs, For the 100 ft long D,I Dracone in Hawthome (1961), U:::: O,60u mJs.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
935
in stability in its first mode; (iii) the yawing ceases at u  3,7 and is replaced by a combined
zeroth/first-mode oscillatory instability, operative up to u  6.3.
However, for considerably lower towing speeds than u = 3.7, the system is subject to
'flexural' or 'snak:ing' instabilities, as shown in Figure 8.77(b); for u = O the frequencies
of these modes correspond to the second- and third-mode eigenfrequencies of a free-free
beam (if the zero-frequency rigid-body mode is called the first) - hence the reason for
the mode-numbering nomenc1ature adopted. It is seen in the figure that the system loses
stability at u  2.7 and u  4.7 in its second and third mode, respectively.
Of course, all this (and what follows) has been obtained by linear theory, and it is not
known at this point if any of the dynamics, other than the initial yawing instability, is
physically meaningfuI. However, as shown in Section 8.9.4, there is a fair correspondence
to experimental observation.
Further calculations have been conducted for systems with a less well-streamlined tail,
taking h < 1 and C2 = 1 - h. It is found that the system is not unifonnly stabilized
in its first mode by the blunter taiI. Thus, the ranges of instabi1ity of the first mode for
various values of h are as follows: O < u < 0.74 for h = 1; O < u < 1.66 for h =
0.8; O < u < 1.65 for h = 0.6; and O < u < 0.70 for h = 0.4. Hence, the system is
least stable in its first mode, not for a perfectly streamlined tail, but for a somewhat less
perfectly stream1ined one. The second and third (flexural) modes, on the other hand, are
both unconditionally stabilized as the tail is made blunter; thus, the threshold of instability
ofthe second mode is at u = 2.83 for h = 1, at u = 3.85 for h = 0.6, and at u = 4.38
for h = 0.4,
Figure 8.78 shows the dynamical behaviour of a system with all parameters the same
as before, but ft = 0.7, CI = O and h = 1 - C2 = 0.7. We note that the effect of a less
than perfectly streamlined nose is to destabilize the system in all its oscillatory modes.
Thus, the first mode is unstable for O < u < 2.80, while the second and third modes lose
stability at, respectively, u = 3.21 and u = 5.40.
Also shown in these figures (dashed 1ines) is the behaviour of a system with SCN =
SCT = 0.5 and all other parameters the same. This system is unstable in its first mode for
O < u < 3.43 and loses stability in its second and third modes at u = 3.04 and u = 5.14,
respectively. Thus, decreasing SCN and SCT destabilizes the system, since reducing CN
and CT diminishes damping and tension on the cylinder. t
At this point it is useful to discuss briefly the differences in the predicted dynamical
behaviour if one uses the erroneoUs equation of motion (Pai'doussis 1968).+ Con-
sider the dynamics of the system of Figure 8,77, for instance. The behaviour of the
zeroth mode is virtually identical. The osci11atory instabi1ity associated with the first
mode, however, involves much larger negative Im(w); also, the range of oscillat-
ory instabi1ity is larger (O < u < 3.1 approximately), For a more extensive comparison,
refer to Pai'doussis (1970). It may be conc1uded that the erroneous and corrected the-
ories predict qualitatively similar dynamics, at 1east Jar a range oj parameters not
tWe may of course regard the smaller values of eCN and eCT as representing a smaller e; but it would be
wrong to conclude that reducing the slendemess of the system renders it less stable, since the definition of
u involves L: the critical values of U are in fact higher, Instead, this should be viewed as a reduction in the
viscous tenns (CN and CT), implying a reduction in flow-induced damping and tension; in this perspective, it
is physically reasonable to conclude that the system is destabilized.
*The main reason for considering the 1968 theoretical results at all is that they are much more extens-
ive than others generated thereafter, with a more systematic study of the effect of several of the system
parameters.

936 SLENDER STRUCTURES AND AXIAL FLOW
20
10
3 3 

4
2 2
4
2
...
4 4 
7.5 CI)
O O
...
1 
7 '"
3 2 2 6 :5
-2 0 2 4 6 8 10
(a) (w)=O (w)
0.4


3
Second mode
I
Third mode
...

CI)
O
..!:!

'"
=
;::>
55 65
2
0.2
O
3
O
I 3,3
I
3,3?
,
,
-0.2
15
25 45
(w)
(b)
Figure 8.78 The dimension1ess comp1ex eigenfrequencies of (a) the zeroth and first
modes and (b) the second and third modes of a flexib1e cylinder (/1 = 0.7, CI = O, h = 1 - C2 =
0.7, A = 1, XI = X2 = 0.0l) with varying u: -, SCN = SCT = 1; - - -, SCN = SCT = 0.5
(Pa'idoussis 1970).
too different from those in Figures 8.77 and 8.78, but with non-negligible quantitative
differences.
One may, therefore, draw the following general conc1usions - qualitatively valid for
both versions of the theory: (i) for optimal stability, the tail should be blunt (12 small, C2
large), the nose should be well-streamlined (fl -+ I, CI -+ O), and the towrope length

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
937
should be short (A small); (ii) a system that is unstable by yawing, within a range of
towing speeds, can be stabilized by blunting the tail, but not by manipulating the length
of the towrope; (iii) in some cases it is possible to stabilize a system which is unstable
at low towing speeds, by towing it faster, within a specified range of towing speeds.
Conc1usions (i) are not contrary to the reported experience with rigid bodies, On the other
hand, conc1usion (ii) may sound surprising; the fact is, however, that the onset of yawing
is not a function of A, nor is its cessation; see also Section 8.9.5. This is also trne for
ft. Finally, conc1usion (iii) is characteristic of the dynamical behaviour of towed tlexible
cylinders, The reader is referred to the stability diagrams in Pa'idoussis (1968; figures
8-11) which are qualitatively sound but quantitatively tlawed; an example is reproduced
here in Figure 8.79.
These general conclusions hold trne also for the resu1ts obtained by the Pa'idoussis &
Yu (1976) model, though they differ appreciably quantitatively, for reasons that have not
yet been fully elucidated. A sample Argand diagram is shown in Figure 8.80, It is seen
that the first mode is stable at low values of u, unlike what is predicted in Figures 8.77(a)
and 8.78(a), The zeroth-mode behaviour, however, is similar; so is the dynamics of the
system for the second and higher (tlexural) modes. Also shown is a comparison with
Pa'idoussis' (1970) theory, modified to have the same form of the towrope force as in
equation (8.181), and taking ft and h such that agreement be close.
A sample stability diagram is shown in Figure 8.81. Here Dl is the 'trnncation ratio'; for
example, Dl = 0.6 signifies that 40% of the tail e11ipsoid has been chopped off, while the
nose is a full e11ipsoid, AI L long. In general, the most important differences in the results
obtained by this modified theory are that (i) the first-mode oscil1atory instability occurs
for u significantly larger than zero, and (ii) its extent, in terms ofthe parameter space over
which the first-mode instability occurs, is much smaller.
Another example from this modified theory is shown in Figure 8.82 on the effect of A.
Here it is noted that Dl = 0.8 corresponds to quite a blunt tail U2 :::: 0,5 or 0.6). Again,
the limited extent of the first-mode oscil1atory in stability is quite evident.
Some calculations have also been conducted via the Bhattacharyya et aZ. (2000) finite-
element form of essentially the Pa'idoussis (1970) theory, for a ship-towed cable-array
system (L = 150 m, £ = 1705) fitted with a tail rope-drogue. First, the steady-state
configuration [shape of towrope (or 'scope') and the tension thereon, and the depth at
which the cylindrical array is towed] is determined. Then, stability is considered. It is
found that for a scope length of 100 m (A = 0.67), the system is stable at 12knots
(1 lrnot = 1.852 km/h) but develops yawing at 16 knots; for a scope of 500 m (A = 3.33),
however, the system remains stable to at least 20 knots.
8.9.4 Experiments
A number of experiments were conducted (Pa'idoussis 1968) with tlexible cylinders held
in tlow by a length of string attached to their upstream end. The main purpose was (i) to
discover whether the various instabilities predicted by theory do in fact occur, and (ii) to
provide some experimental data, particularly regarding the thresholds of instabilities, for
quantitative comparison with theory.
The tlexible cylinders were made by casting liquid silicone rubber ('Silastic') into
specially made moulds (Appendix D). The cylinders were made hollow, so that they

938
10
9
8
7
6
u 5
4
3
2
SLENDER STRUCTURES AND AXIAL FLOW
I
Third-mode 
osciJIatory instability "
I
,-
,,/
,,..."
/:,iJ.
, OJ IJ} First mode
..............!!d e "'--- T ---
.... /'
....'........ / First-mode
...., ......?sciJIatory t instability
\ .................
" ........
'...... ..........
.... ....
.... ........
--
zero!h!!). -"":.-,:",
---s:
-: - - r:.£.Dl1d
Stable region Yawmg _ :'-=-de--
_--- -stmode '....,
I
I
I
First-mode !
osciJIatory instabiJity I
Yawing and first-mode
osciJIatory instabiJity
o
O
0.1
0,2
0.3
0.4
0.5
12
0.6
0.7
0,8
09
Figure 8.79 Stability map showing the effect of the tail shape for a t1exible cylinder
(JI = 1 - CI = 1, C2 = 1 - 12, SCN = SCT = 1, A = 0.5, XI = X2 = 0.0l), obtained with
the t1awed theory of Pai:doussis (1968).
t O
P O O
8
3 3
7. -I
O
Zeroth mode
10
8
,
5
10 10 3
8,75
8 2
5 a
I
2.75 
1.5
10
8
,,121--------....
/' 7,25 ---,-
I
I
/ First mode
/
,,"
--'
Values of u
2
4
(w)
6
Figure 8.80 The eigenfrequencies of the zeroth and first modes of a system with sc N = SCT = 1,
A = 5, an ellipsoidal nose (AI = 0.015) and tail (a = 1), as functions of towing speed, u: -,
Pai'doussis & Yu's (1976)theory; - - -, Pai:doussis (l970)theorywith JI = 0.6035, 12 = 0.6046.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
939
12
8
Stable region
:,,-- r '1,
 \' ...
'- ; '\?Q
\ ,Oq; '
, 
Seco'nd-mode
osci11atory
instability
J r:::
First-mode
osci11atory
instability
"
10
u
6
4
2
o
0,0
0.2 0.4 0,6 0,8
Blunt tai1-a- Streamlined tai1
1.0
Figure 8.81 Stability map showing the effect of the tail truncation ratio ex on stability, for a system
with BCN = BCT = 1, AI = 0.015 and A = 1, according to Pai'doussis & Yu's (1976) model.
,
12 ---\-.;:-- ::..."'...:'e- -- ------___________
\ p;\\ Q 1 ------------
" Third-mode
\ " II " b " l " Third mode
" OSCI atorymsta lity ___ ______________________
10" _------
'...-----------... - Second-mode
osci11atory instability
8
Second mode
u 6
First-mode
osci1latory instability
Stable region
4
Zeroth mode
2
Yawing
o
0.1
0,5
5
10
50
A
Figure 8.82 Stability map showing the effect of A for a system with BCN = BCT = 1, AI = 0,015
and ex = 0.8(f2 ::::: 0.5); (Pai:doussis & Yu 1976).

940
SLENDER STRUCTURES AND AXIAL FLOW
"
..,
\ :'\
(a)
<JCJCJ D
. "
::::  \ "'" - - -
" .-
,.'
9
I I I
O 10 20 30
(mm)
(b)
A
B
c
D
Figure 8.83 (a) The construction of the flexible cylinders used in the experiments and the shape
of the end-pieces (for cy1inder I): 1, nylon thread (towrope); 2, perspex nose end-piece; 3, perspex
plug glued to produce water-tight seal; 4, perspex adaptor for receiving end-pieces; 5, silastic
rubber cylínder; 6, 8.35 mm (1/4 in,) diameter hole for buoyancy; 7, perspex spacers to prevent
rotation of adaptors; 8, glued threaded plug; 9, perspex taíl end-piece. (b) The four end-pieces used
interchangeably (PaYdoussis 1968),
would be neutrally buoyant when submerged in water. In fact, when submerged, the
manufactured cylínders would very slowly sink; however, the measured specitic gravity
was never greater than 1.01. At its two extremitíes the cylínder was titted with threaded
protrusions (see Figure 8.83) over which could be screwed tapered perspex (plexiglas)
end-pieces, Several shapes of tapered ends were made in order to assess their effect on
stability, Two cylínders were used: cy1inder I (353 mm long without the end-pieces,
D = 17.3 mm, E/ = 9.59 x 10- 3 N m 2 ) and cylínder II (249 mm long, D = 13,7 mm,
E/ = 3.64 x 10- 3 N m 2 ).
The experiments were conducted with the cylínder hanging vertically in down-flow, in
the centre of the glass test-section (76.2 mm in diameter) of a water circulating rig. Some
experiments were also conducted in a horizontal test-section of larger diameter. The usual
precautions were tak.en to ensure reasonably uniform flow conditions in the test-section for
the fully-developed turbulent flow conditions extant over most ofthe flow range of interest.
The maximum attainable flow was "'-'4 mis. The length of the nylon-string 'towrope' could
be varied,
Experiments were also conducted more recently by Sudarsan et al.. (1997) with more
slender cy1inders (e = 50 and 150) in a towing tank. The cylinders were made hollow,
of either nylon (specimen 1: D = 14 mm, L = 0.7 m, E/ = 3.54 x 10- 2 N m 2 )
or rubber tub ing (specímens 2 and 3: D = 8 mm, L = 0.4 and 1.2 m, respectively,
E/ = 2.37 x 10- 2 N m 2 ), and they were slíghtly negatively buoyant L8 = 0.44 (nylon)
and 0.38 (rubber)]; they were towed attached to a streamlined strut, 0.25 m underwater,
using a nylon towrope.
8.9.4(a) Generalobservations
ln the following, some observations from Pa'idoussis' (1968) experiments with cylínder I
titted with an elongated nose and tail are given in detail. Then, the effect of other tail and
nose shapes and the effect of towrope length are discussed in briefer formo The Sudarsan
et al. (1997) experimenta1 observations are discussed last.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
941
(i) Cylinder with elongated nose A and tail A. The dynamics of the system for A = 1/2 is
described first. At very low flow ve10cities (u < 0.8; U < 0.46 mis) the system was stab1e
and the cy1inder remained in its centra1 position of rest. At higher flow, 'criss-crossing'
oscillation deve1oped, that is, osci11ation during which the tai1 moved in the opposite
direction to the nose, about the position of rest, such that the two 1imiting positions
of the cy1inder formed an X [refer to Figures 8.84(a) and 8.85(a)]; the instantaneous
inc1ination of the string re1ative to the position of rest was in the opposite direction to that
of the cy1inder. The frequency and amp1itude of osci11ation increased with flow; thus, at
u = 0.8, w = 2.1; at u = 2, w = 3,2; at u = 2.9, w = 4.4. These were low-frequency
oscillations corresponding to 0,3 to 1 Hz, The amp1itude increased to the point where
the cy1inder a1most touched the flow-tube. At the lower flows, motion of the cy1inder
invo1ved no perceptib1e flexure; at the higher flows there was appreciab1e flexure, with the
tail 'dragging' during osci11ation. These oscillations were interpreted as corresponding to
first-mode oscillatory instabi1ity.
For u > 2.3 the cy1inder assumed a bowed ('banana') t shape in the course of oscillation.
At u > 3 the amp1itude of oscilIation decreased, and at u = 3.6 the osci11ation ceased
whi1e the cy1inder retained a banana shape.+ This is interpreted as the point where first-
mode oscilIatory instabi1ity ceases, but the system is stilI unstab1e in its zeroth mode, in a
form of yawing invo1ving flexing ('buckling').
This buckled shape was retained up to u = 4 (U  2.3 mis), where amp1ified second-
mode osci11ations began [Figures 8.84(b) and 8.85(b )]. The motion deve10ped in one plane,
butafter a few cyc1es of oscillation changed to a three-dimensiona1 motion invo1ving whir1-
ing of the cy1inder, ln shape the oscillations looked 1ike amp1ified free vibrations of a free-
free beam in its first flexura1 mode. The frequency of oscil1ation was about 2.4 Hz (w = 17).
The amplitude of oscillation increased with flow up to u  5, where the osci11ation
became discontinuous, It appeared as if the cy1inder osci11ated in its first-mode (criss-
crossing) again, with a second-mode oscillation superposed; however, the amp1itude of
osci11ation was quite 1arge and the cy1inder frequently hit the flow-tube. This continued
up to u = 7.2 (U  4.1 mis) where third-mode amp1ified oscillation deve10ped [Fig-
ures 8.84(c) and 8.85(c)]. The frequency was about 7 Hz (w = 49). Whir1ing took p1ace
in this case a1so.
The behaviour of the system for A = 1 and A = 1/4 was essentia11y as above. For
A = 1, the amp1itude of oscillation was generally much 1arger, and hitting the flow-tube
rendered observation 1ess re1iab1e. This was exaggerated for A = 3, and the observations
were very difficu1t to interpret.
For A = 1/8 the observations were considerab1y different. In this case criss-crossing
oscillation was not observed. Instead, the cy1inder took a skewed configuration at u > 1.2.
This presumab1y corresponds to stationary yawing at low u, where the fluid forces are
too small to cause flexing of the cy1inder. Increasing the flow gradually transformed the
skewed cy1inder to a banana-shaped one, as before. Evidently this corresponds to a case
where, in the absence of first-mode oscillatory instabi1ity, yawing asserts itself even at
low flow. Moreover, in this case second-mode oscillation, which started at u = 5,1, was
not disrupted, continuing in a regu1ar manner up to u = 7.5, where third-mode oscillation
tHere the U,S, Department ofTrade definition of a banana is more apt than the European Union version,
*This stationary shape is not shown in Figures 8.84 and 8,85, but the buck1ed cylinder looks as in
Figure 8,84(b), panel 8; see also Figure 8,86(b),

942
SLENDER STRUCTURES AND AXIAL FLOW
-------------.....
o .... N M 'i' L.{i tO
a: .... .- .- .- .... .... ....
----.--.--.--....-
.... N C"? "d' 11) ID ,... co
_---.----.-------------
o .... N M  L..'l 1.0
0'1 .... .... .... .... .... .... ....
--------------
.... N M V m w  
______---...____--...o'""___----....
- N (') "ct' lr \D f'- co
--------
.. '\; ......  ln t.O ro
----. ------ .---.. ---- --. - -
 N ("') '<t ID tO !'- CO
].g
"O
.,:,
<:.2
<1)'"
"'
:3 O <.)
s='-'
..9]
. 
<I) ,
=-g8
tes;
"0.,:,-
;És='"
O'...
e: e-
"" '" 5
 :o
e, '03
 
<.) . '-'
s= 0lJ =
<I) = O
::;::S'ooooI._
0""''''
<I) '" 
I:ne=
E <.)"[;
I '"
-"'o
'7 'e
' <.)
03
0lJ
'E
I)Č S
3: i
<
é:
I ... d"'-"
 :E .2 
= . Cd  .
'> S a 6'
'_  (,) ...t"'-
g2
'5 8.   .
eo-ge
:-= .c. o
'"i' o I  "O
'<I) ... :9 'Ť :03
= <I) OlJN"
'(:) -s ';::: '-'C
 -s »8;'::
e ' :a . 
o '.o  tJ.)
<t=  =:-::1 t::
'" <I) <.) a
<1):-::1   ...
!3<1)<I)
 "O . "O ,
<.::iaOlJO
 .....s e =
'e -..  -O .g
:S8S=
g8S
..9
o.....;::: o
u  <.).c <I)
I'-"( ...-.. '-' "'O

I)Č.= gp  -8
i :.=.... t:: '...
»...a.c::
<.) S ... ...
<1)<1)4-<
:1"":0><00
1Ji-I._ t3) "-'
3;

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
943
started. lust before the establishment of the regular second-mode oscillation (w = 23),
an unusual, lower frequency oscillation (w = 7.7) developed at 4.6 < u < 5,1; it was
not unlike second-mode oscillation in shape and it was quite regular. A summary of these
observations is given in Table 8.11.
(ii) Cylinder with elongated nose A and other tail shapes. The effect of tail shape on
the dynamics is summarized in Table 8.12; more detailed descriptions may be found in
Pa'jdoussis (1968). The effective complete stabilization of the system via a very blunt tail
is remarkable.
Table 8.11 The effect of towrope length on stability for cylinder I with nose
A and tail A - see Figure 8.83(b); (Pai'doussis 1968).
Description A = 1 A = 1/2 A = 1/4 A = 1/8
Criss-crossing osci1lation u 0,8-3.3 0.8-3,6 0,8-3,6 - a
(J) 2-4 2-7 2-5
Stationary yawing u 3,3-3.8 3,6-4,0 3,6-4,2 1.2-3,7
Second-mode osci1lation u 3.8 4,0 4.2 5,1
threshold (J) 17 17 18 23
Second-mode with superposed
first-mode oscillation u 4..8 5.0-7,2 5.0-7.2 4.6-5.1 b
Third-mode oscillation u 7.0 7.2 7.2 7.5
threshold (J) 49 49 49-56 49-56
a -: signifies that the phenomenon described did not oeeur,
b Regular low-frequeney oseil1ation, in shape like seeond-mode, preceding higher frequency
second-mode oscil1ation, ú) = 7,7,
Table 8.12 The effect of tail shape on stability for cylinder I with nose A and
A = 1/2 - see Figure 8,83(b) for shapes A, B, C and O (Pai:doussis 1968).
Description Tail A Tai1 B Tai1 C Tai1 D
Criss-crossing oscillation u 0.8-3.6 0.8-1.6 1.1-1.6 _a
(J) 2-7 1.9-2.3 3.2 _a
Stationary yawing u 3,6-4,0 _a _a _a
Stable region u _a 1,6-4,8 1.6-5,5 0-7.4 e
Second-mode osci1lation u 4,0 4,8 5,5 - a
threshold (J) 17 16 15 - a
Second-mode with first-mode
oscillation superposed u 5.0-7,2 _a - a - a
Third-mode osci11ation u 7.2 7.1 7.4 b - c
threshold (J) 49 48
a -: signifies that the phenomenon did not oceur.
b Third-mode oscil\ation not fully developed,
C Maximum flow available corresponds to u = 7.4.

944
SLENDER STRUCTURES AND AXIAL FLOW
Table 8.13 The effect of towrope length on stability for cylinder I with nQse A and
tail C; see Figure 8.83(b); (Páidoussis 1968).
Description A = 1 A = 1/2 A = 1/4 A = 1/8
Criss-crossing oscillation u 1.1-2,5 1.1-1.6 - a - a
W 2-3 3.2 - a - a
Stationary yawing instabi1ity u - a - a - a _a
Stable region u 2.5-5,2 1.6-5.5 0--5,9 0--7.4 c
Second-mode osci1lation u 5.2 5,5 5,9 - a
threshold w 13 15 17 _a
Third-mode oscillation u 7.4 b 7.4 b _c _a
threshold w _a _a
a -: signifies that the phenomenon described did not occur,
b Third-mode oscillation not fully developed.
C Maximum fiow available corresponds to u = 7.4,
The effect of shortening the towrope on stability of the system with medium-blunt tail is
summarized in Table 8.13. Clearly, shortening the towrope has a defini te stabilizing effect.
(iii) Cylinder with elongated nose and elongated .finned tai/. The cylinder in this case
was the same as in (i), except that the tail had a rectangular fin in one diametral plane. The
fin was of the same width as the cylinder diameter and extended over the full Iength of
the tail. The dynamical behaviour of the cylinder in this case was essentially as if the fin
were not there, at least up to the point of inception of second-mode oscillation, Oscillation
tended to be in the plane of the fin. In the second- and third-mode instabi1ities, oscillation
would develop in the plane of the fin; then a slight rotation of the plane of motion would
disrupt oscil1ation. Accordingly, second- and third-mode oscillation was discontinuous,
ln a subsequent experiment, a second fin was placed on the tail at 90° to the first.
Once again, this proved to be an unsatisfactory way of stabilizing the system, Essentially,
the only difference to the case without fins was that second-mode amplified oscillation
occurred at slightly higher flow velocities, that is, at u = 4.5 versus 3.8. The oscillation
was continuous and no whir1ing took place.
(iv) Cylinder with blunt nose and elongated toU, ln one experiment the medium-blunt
end-piece C was used as the nose. The tail was the same elongated end-piece as in (i).
The observations displayed no essential qualitative or quantitative difference from those
of (i). This would appear to contradict the theoretical predictions that blunting the nose
destabi1izes the system, On reflection, however, perhaps theory and experiment do not
disagree so much. In this connection, two points must be made, First, theory actual1y
predicts that reducing JI destabilizes the system. Since reduction of JI or h from unity
comes about mainly by flow separation, we can see that, whereas an end-piece used as the
tail may have a low value of h, the same end-piece used as the nose does not necessarily
have a low value of JI, Secondly, increasing the nose form-drag coefficient CI actually
stabilizes the system (in the theory we usual1y take CI = 1 - JI).
(v) Cylinder with ring stabilizer. In this case a rubberring, 19.9 mm diameter, was fitted
concentrically over the cylinder just ahead of the tail end-piece [cf. Figure 8,73(a)]. The

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
945
nose and tail were as in (i). The effect on stability was notable. There was neither criss-
crossing oscillation nor yawing, the system remaining stable up to u = 3.7. For u > 3.7
there were small, damped, second-mode oscillations which increased in amplitude and
continuity with flow. At u > 4.4 the oscillations were essentially continuous, but the
amplitude did not exceed  D. It was difficult to decide whether the system was unstable
or not, because of the relatively small amplitude of motion. Sirnilarly, with third-mode
instability; third-mode, small amplitude, continuous oscillations developed at u = 6.4 and
persisted to maximum flow, Thus, the ring stabilizer was fully effective (cf. Figure 8.73(a»,
so long as the towing speed was not very high; however, it was not nearly as effective as
the very blunt tail D, as described in (ii),
Finally, Pai'doussis' (1965, unpublished) experiments with a horizontal system in
a horizontal flow displayed qualitatively similar behaviour. Figure 8.86 shows the
towed cylinder in (i) criss-crossing oscillation, (ii) static yawing (buckling), and
(iii) second-mode oscillation. These are typical for 4.5 < A < 17. The criss-
crossing oscillation did not occur for A < 4.5. Figure 8,87 shows (i) divergence,
(ii) second-mode and (iii) third-mode oscillation of a pinned-free cylinder (A = O);
fourth-mode oscillation was also observed, but no quasi-rigid-body criss-crossing in
trus case,
A major weakness of the Pa'idoussis experiment s is that they were conducted in relat-
ively confined flow (Dch/ D = 4.5 and 5.6); for large amplitude motions, for example,
those associated with the criss-crossing oscillation, the nose and tail reached the wall
boundary layer, if not the wall itself, thus deleteriously affecting the observations. Hence,
the Sudarsan et a1. (1997) experiments to be discussed next, conducted in a wide towing
tank, are particularly useful.
Some typical observations are summarized in Table 8.14. Once more, it is observed
that a combination of a blunt tail and, to a lesser extent, a relatively short towrope can
effectively stabilize the system. There are considerable differences, however, vis-o-vis the
earlier experiments. Thus, comparing Tables 8.11-8.13 and Table 8.14, it is very clear that
criss-crossing (first-mode) oscillation does not occur, while static yawing does in its stead
as the prevalent form of in stability at low flows. One possible reason could be the much
larger values of € in these experiments, Another could be the nonexistence of interfering
side-walls anywhere near the oscillating cylinder,t
The yawed state at low tow-speeds involved a straight towrope, along the equilibrium
configuration, and the cylinder rotated slightly about its nose in the horizontal plane; at
higher tow-speeds, however, the entice system was displaced off the line of equilibrium
(with the towrope and the cylinder at two different angles).
Another noteworthy result is that for € = 150 the system is unconditionally stable at
low u - no doubt because of the increased drag forces - cf. Table 8.3 (Section 8,3.2)
and the discussion related thereto.
AIso of interest are the results of experiment s with different types of drogues. A rope-
drogue was found to be ineffective, a disc-shaped one effective only for the € = 50 system,
while a funnel-shaped one was found to stabilize the system in all cases.
t On the other side of the coin, it should be remarked that there is a limited time available for steady towing,
The towing tank is long (82 m); yet, excluding acceleration and deceleration phases, the useful available time
is only 17 s at U = 1,5 mis, and shorter for higher U, This corresponds to only 1-3 cyc1es of oscillation of
first-mode instability; moreover, the establishment of oscillatory instability does take time, Nevertheless, it
should have been observed if operative,

946
SLENDER STRUCTURES AND AXIAL FLOW
(a)
(b)
(c)
Figure 8.86 Dynamics of a horizontal towed flexible cylinder showing (a) criss-crossing oscil-
lation (multiflash photograph), (b) yawing (buckling), and (c) second-mode oscillation (multiflash
photograph),
(a)
(b)
(c)
Figure 8.87 Dynamics of a horizontal towed flexible cylinder for vanishing towrope length
(pinned-free cylinder): (a) divergence (yawinglbuckling); (b) second-mode oscillation; (c) third-
mode oscillation.

SOLITARY CYLINDRICAL STRUcrURES IN AXIAL FLOW
Description
Tapered taH
Blunt taH
Table 8.14 TypicaI observations from the experiments by Sudarsan et al. (1997).
947
A = 1
A=I
A = 0,1
Specimen 1 (nylon cylinder, I': = 50)
Stationary yawing
Second-mode flutter
u 0,8-2.1 0,8-1.5 1.5-2,6
u 2.5 3,0 3,5
w 12-16 11-17 14-16
Specimen 2 (rubber cylinder, I': = 50)
u 0.5-1.3 0.5-1.1 _a
U _a _a _a
Stationary yawing
Second-mode flutter
Stationary yawing
Oscillatory instabilities
Specimen 3 (rubber cylinder, I': = 150)
_a _a __b
4,1 _a __b
u
u
A = 0.1
1,5-2,3
_a
_a
_a
_a
__b
__b
a_: phenomenon not observed up to maximum flow,
b_ _: experiment not conducted,
8.9.4(b) Summary
For a flexible towed cylinder, the typical dynamical behaviour with increasing dimen-
sionless towing speed u, if the tail is not blunt and the tOWTope not too short, may be
summarized as follows: (i) at low u, a 'criss-crossing' oscillation develops, with the cyl-
inder inclination opposite to that of the towrope, or altematively stationary yawing; (ii)
at slightly higher u, a narrow region of stability occurs in some cases, or a1tematively a
region of yawinglbuckling; (iii) at higher u, second-mode and then third-mode (flexural)
oscillation develops.
If, however, the tail is blunt or the towrope sufficiently short, the system remains stable
for apparently all towing speeds.
8,9.4(c) Comparison with theory
It is first noted that in terms of qualitative agreement, the theoretical results in Figures 8.78
and 8.79, for instance, agree with Pai'doussis' (1968) experimental observations in sub-
sections (a) and (b) for a not too blunt tail and not too short a towrope. Thus, at very low
towing speeds the system is subject to first-mode oscillatory instability ('criss-crossing'
oscillations) and yawing, the former ceasing at slightly higher towing speeds, while
yawing persists (presumably corresponding to the observed banana-shaped buckling).
At yet higher towing speeds, second-mode flexural oscillations take place, followed by
third-mode ones at even higher towing speeds.
We next consider quantitative agreement for one specific case: cylinder I, fitted with
quite well streamlined nose (A) and tail (A) [Figure 8.83(b)], € = 20.4 and A = 1.
Based on the experience gained with cantilevered cylinders in axial flow (Section 8.3,2),
the theoretical calculations are for h = 1 - C2 = 0,7 (and 0.8), JI = 0.8, CI = o,t
t Although the nose ís of tbe same shape as the taH, there is no flow separation over tbe nose, hence a higher
value is warranted for JI than for h.

948
SLENDER STRUCTURES AND AXIAL FLOW
Table 8.15 Dynamics of a towed flexible cylinder (s = 20.4, A = 1, nose A and tail A;
see Figure 8,83); comparison of theory (SCN = SCT = 1, JI = 0.8, CI = O, h = 1 - C2 =
0.7, XI = X2 = 0.01; results marked by :j: are for h = 1 - C2 = 0,8) with experiments
(Pai'doussis 1968, 1970).
Description Theory Experiment
Criss-crossing osci1lation (first-mode osci11atory in stability) r 2S 0,8- 3,3
u (0-3.1)*
0-3.3 2-4
ú) (0-3,OY*
Stationary yawing (zeroth-mode
in stability alone) u 2.5-3.5 3.3-3.8
Second-mode osci1lation threshold u 3.5 3,8
ú) 15,5 17
Second-mode with first mode
osci11ation superposed u 4.7-7.2 4.4-6,8
Third-mode osci1lation threshold u 6,} 7.0
ú) 47 49
£CN = £CT = I, A = I, Xl = X2 = 0.01. The comparison is made in Table 8.15,
Agreement between theory and experiment is genera11y fair. However, a number of points
should be raised, as follows,
(i) The experimenta1 criss-crossing oscillation has been identified with the theoretica11y
predicted first-mode oscillatory instability, despite the fact that in most cases theory pre-
dicts that the system is a1so subject to a static yawing in stability over the same range of
towing speeds, This is supported by the observed frequency characteristics of the oscilla-
tion and the observed effect of varying A, for instance, being essentia11y as theoretica11y
indicated for the behaviour of the first mode, It has thus been presumed that oscillatory
instabi1ity is the preva1ent form of in stability. There is, however, an a1temative inter-
pretation of the observed behaviour, name1y that criss-crossing oscil1ation is a nonlinear
manifestation of yawing, that is, a dynamic divergence. This may be postulated, but cannot
be proved by linear theory.
(ii) The experimenta1 onset of the first-mode oscillation does not occur at u = O
as predicted by theory, presumably because a term of the type £C {JI/2(a'l/ar:) as in
equation (8.27b) has not been incorporated in the theoretical model.
(iii) Agreement between the new theory and experiment is least satisfactory in predicting
the point of cessation of criss-crossing oscil1ation - though this is reatly stretching the
jurisdiction of linear theory, Here it might be argued that h = 0.7, C2 = 0.3 may be
too severe for low towing speeds; comparison with theoretica1 values ca1culated with
h = I - C2 = 0.8 (shown in parentheses in Table 8.15) yie1ds better agreement, as
anticipated.
ln fact, severa1 other questions remain. More careful and extensive experiments, inc1ud-
ing experiments with rigid cylinders in a much wider channe1, and a more sophisticated
theoretica1 model would al1 be desirable.
We next consider comparison between theory and experiments by Sudarsan cf al. (1997).
As seen in Table 8.16 agreement is quite reasonable. The degree of agreement is similar
to that displayed in Table 8.15. Again, it should be remarked that stationary yawing is the

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
949
Table 8.16 Dynamícs of a towed flexíble cylínder (8 = 500r 150); comparison wíth
the Pai:doussís (1970) theory; (Sudarsan et al. 1997).
Description
Theory
A = 1
Experiment
Theory
Specimen 1, tapered tail
0-0,5
Stationary yawíng
First-mode oscillation
Second-mode flutter
u
u
u
ú)
0--3.4
0--1.1
2.8
15
Stationary yawing
Second-mode flutter
u
u
ú)
0--2.9
3.8
13
0.8-2.1
_a
2.5
12-16
_a
3.4
17
Specimen 1, blunt tail
1.5-2.6
3.5
14-16
0,4-3,0
_a
_a
Stationary yawing
Second-mode flutter
u
u
0--1.8
2.9
Specimen 2, tapered tail
0--2.3
_a
0.5-1.3
_a
Stationary yawing
Second-mode flutter
_a
Specimen 3, tapered tail
u
4.4
_a
4.1
__b
__b
A =0,1
Experiment
0,8-1.5
_a
3,0
11-17
1.5-2.3
_a
_a
0.5-1.1
_a
__b
__b
Specimens 2 and 3, blunt tail
TotaIly stable (both theory and experiment)
a _: not observed (to maximum available u),
b _ _: not investigated.
dominant 10w-u instability here, rather than eriss-erossing - this time both in theory and
in experiments.
8.9.5 Rigid towed cylinders
We eonsider exaetly the same eonfiguration as in Figure 8.76, but impose the restriction
that the body be rigid, Thus, the system is redueed to one of two degrees of freedom. The
generalized eoordinates may be taken to be the lateral displaeement of the eentre of mass,
Yc, and the angle that the body makes with the x-axis, <p. Aeeordingly, the displaeement
at any point is given by
Y = Yc +x<p.
(8.182)
For simplicity, we assume that the eentre of mass eoincides with the geometrie eentre
of the body, the tapered-end lengths Xl and X2 being smal1. For eonvenienee, we now
measure X from the eentre of mass, so that the body extends from X = -  L to X = 1 L.
lnstead of deriving the equations of force and moment balanee independently of the
previous work, we proceed as follows. We integrate equation (8.173) formal1y, to obtain an
equation of foree balanee; similarly we integrate the produet of the forees in (8.173) times
x, to obtain the equation of moment balanee. The boundary eonditions are ineorporated
through the integral of the first term of these equations. Thus, using (8.173)-(8.176),

950
SLENDER STRUCTURES AND AXIAL FLOW
two coupled equations are obtained, which may be written in dimensionless form as
follows:
{[2 + Xl(1 + Ji) + X2(1 + h)] (-úi) + [SCN + Ji - h] (wi)
+ [CTPjA]} H + {-! [Xl (1 + Ji) - X2(1 + h)] (-ú})
+ [2 - (fl + h)] (wi) + [€CN + JI - h - (1j A)cTP ]) <P = O, (8,183)
{- [Xl(1 + Ji) - X2(1 + h)] (_w 2 ) - U(Ji + h)] (wi)
- [CTPjA]} H + ([i + Xl(1 + Ji) + X2(1 + h)] (_w 2 )
+ [(fl - h) + 214 SCN] (wi) + [(1jA)CTP - (fl + h)]} <P = o, (8.184)
where 1J = yc!L = Hexp(iwT),ifJ = <pexp(iwT),A = sjL,s = LjD,Xl = xljL,
X2 = x2jL, T = UtjL, W = QLjU, CTP = SCT + Ci + C2, and M = m has been used.
For nontrivia1 solution, the determinant of the coefficients of H and <P in these equations
must vanish, yielding a quartic in w,
w 4 + Aw 3 + Bw 2 + Cw + E = O, (8.185)
where E = (cTPj2A)(SCN - 2h).
Modified equations are obtained for different assumptions as to the directions of nose and
tail form drag, linearization of the towrope forces, etc. Indeed, in this case of a rigid body,
the nonHnear equations would be tractable and could have been used. However, the main
aim here is to compare the dynamical behaviour of the rigid body to that ofthe flexible body,
so as to gain further insight into the dynamics of the latter. Recalling that the dimensionless
flow ve10city in the case of a flexible body is defined as u = (Mj E I) 1/2U L, the dynamica1
behaviour of the rigid body should approach that of the flexible one as u --+ O.
ln fact, it is found that the eigenfrequencies of the rigid-cylinder system are very c10se
to those of the flexible system for the apparently arbitrary value of u = 0.7. On reflection,
however, the rigid-cylinder dimensionless frequencies are Wrb = QLj U, while those
oftheflexiblebody arew fb = [(M+m)jE/]1/2QL 2 = [(M+m)jM]1/2 u QLjU =
v'2u QL j U for m = M. Hence, for equa1 dimensional frequency Q in the two cases, we
have Wfb = v'2uw r b, and Wfb = Wrb for u = ljv'2 :::::: 0.707.
The va1ues of Wrb are independent of U, because of the nondimensionalization scheme;
typically one obtains (i) two real va1ues (one positive and the other negative) associated
with the zeroth mode and yawing instability, and (ii) two complex ones associated with the
first-mode and oscillatory instabilities. These observations reinforce the interpretations of
the nature of these modes made in Section 8.9.3.
Yawing (w = O) in the case of the rigid cylinder simply means E = O in equa-
tion (8.185), which shows that it is independent of CT P, A and JI. This is not trne,
however, for the osci11atory instability.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
951
0,8
0,6
h
0.4
0,2
o
0,1
0,5
5
A
10
50 100 1000
Figure 8.88 The effect of A on stability of a towed rigid cylinder (C2 = 1 - 12, £CN = £CT =
0.5, XI = X2 = 0,01): -, JI = 1 - CI = 1; - - -, JI = 0.8, CI = ° (Paldoussis 1970).
A typical stability diagram is shown in Figure 8.88. It is noted that yawing is independent
of A and that it occurs when E = O, that is, for 212 = !ecN. The oscillatory in stability
zone, however, is larger for JI < 1; it is also larger as eCT becomes smaller (PaIdoussis
1970; figure 16). The results for JI = 0.7, not shown in Figure 8.88, are of interest,
because in that case the oscillatory instability practically covers the whole plane; for
example, for aU 12 > 0.013, if A = 0.1. t Finally, it should be remarked that these
stability diagrams have been obtained without the benefit of equation (8.181), the effect
of which (if applicable) could be important, particularly for large A.
8.9.6 Mechanisms of instability
We first consider static instabilities and specifically yawing. The first thing to recognize
is that yawing must involve angular motion as opposed to pure translation. This is evident
upon considering the cylinder momentarily displaced parallel to the x-axis; in this case the
forces acting on the cylinder are exactly as in the equilibrium configuration, except that the
tow-rope exerts a restoring force on the body. We next imagine the cylinder momentarily
displaced such that the y-displacement of the nose is positive and that of the tail negative
as in Figure 8.89(a). Then, considering boundary conditions (8.174) and (8.175), we note
that the inviscid hydrodynamic force on the nose is JI M u 2 (ay fax), while at the tail it is
- 12M U 2 (oyfox), producing a moment tending to exaggerate the original inclination.*
Two more conclusions may be drawn: (i) yawing may occur for all O ::: JI ::: 1, provided
12 -::j:. O, although the yawed configuration would vary; (ii) if 12 = O (and hence for
h  O), no yawing is possible since the destabilizing moment would no longer exist.
If the cylinder is flexible, there would be an additional force due to flexure,
M U 2 (o2y fax 2 ), as shown in Figure 8.89(b) - cf. Section 8.3.3(a), The bowed, buckled,
'banana' shape seen in the experiment s arises from an interaction of the nose and tail
forces together with this flexural force.
tIt is of interest to remark that, for rigid-body motions of a free-free body (not towed) of revolution in
potential flow, Wu et a/. (1989) find that (i) for a blunt nose the body is statically unstable and dynallŮcally
stable, while (ň) for a sharp nose (JI 1= O) the body is statical1y and dynamically unstable.
* Referto Durand (1936) and Hawthome (1961), and to the work cited in the first paragraphofSection 8,9,1.

952
SLENDER STRUCTURES AND AXIAL FLOW
 t
+
u

Figure 8.89 Major lateral forces on yawing cylinders under tow: (a) rigid body; (b) flexible body
(Hawthorne 1961).
Considering the quasi-rigid criss-crossing oscillatory instability next, the same forces
as in Figure 8.89(a) are operative, but in this case Corio1is forces proportiona1 to
MU (a 2 y laxat) wou1d a1so p1ay a role, a1ways opposing rotation.
Concerning the flexura1, second- and higher-mode oscillatory instabilities, the same type
of ana1ysis as in Section 8,3,3(c) may be emp1oyed, invo1ving the caIcu1ation of the rate
of work done by all the forces, d W I dt, and hence the work done per cycIe of oscillation.
Making use of equations (8.173)-(8,175) and adopting the form of the towrope force in
(8.181), we obtain t
 W = (1 - f1)M U 1 T [y2 + U Y y']x=odt - (1 - 12) 1 T [y2 + U yy']x=Ldt
+ i M U 2 C} 1 T [y y't=o dt + i M U2C21T [y y'Jx=L dt
- iCN(M U ID) 1 T 1 L (ý2 + U Ý y')dx dt
+ iCT(M U 2 I D) 1 T 1 L (ý y') dx dto (8.186)
Oscil1ations are amp1ified, that is, instability arises, for  W > O, both for flexura1 and
quasi-rigid modes. C1ear1y, therefore for very small U and neg1ecting frictiona1 effects,
stability is govemed by whether (1- f1) Jt y2 dt - (1- 12) Jt y2 L dt is positive or neg-
ative; hence, a well streamlined nose (JI -+ 1) and a b1unt tail (/2 -+ O) wou1d promote
stability. For higher U, however, the situation becomes more comp1ex, as U I Ý y' l > y2
may now obtain, and y y' may be either positive or negative; the overbar signifies the
tu the form in equation (8.174) is used for the towrope force, then additional C2 and (LjD)CT terms
involving [ý y'1x=o appear in  W,

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
953
mean va1ue over one period of oscil1atio n. (It i s noted that, from Figures 8.8 4-8.8 6, for
oscil1atory instabilities we generalIy have [ýy']o being strong1y negative, and [ýy']L a1so
negative but with smalIer abso1ute va1ue.) ln genera1, stability wil1 depend on the mag-
nitudes of f1, 12, Ci, C2, Ý5, ýI, etc., and no simp1e genera1 ru1es can be formu1ated beyond
the statement of equation (8.186). N everthe1ess, considering on ly the inviscid force terms
and presuming that f1 < 1, 12 < 1 , fYY'jo < O and [ý y'h < O, it is of interest that for
Ó W > O it is nec essary to have (i) [ý2 + U Ý y']L < O as for cantilevered cylinders, but
(ii) [ý2 + U ý y']o > O; in both cases, this destabi1izing energy transfer is counteracted
to some extent by the effect ofthe Ci and C2 terms - cf, Section 8.3,3(e).
8.9.7 Acoustic streamers: Dowling's work
If the towed cy1indrica1 body is an acoustic streamer (seismic array), it is very slender;
hence, for smalI motions, flexura1 forces may be ignored and the cylinder modelIed as a
string; refer to Section 8.3.7. In general, there is a critical point a10ng the string, where the
restoring force represented by the tension term -!M U 2 [C2 + cT(L - x)j D](a 2 yjax 2 )
in equation (8.173) is cancelIed by the centrifuga1 term M U 2 (a 2 yjax 2 ), This, as will
be seen, creates mathematica1 difficu1ties, as the solution has to be extended across this
critica1 point, in order to app1y the boundary conditions at the downstream end; it a1so
raises physica1 concems as to what might happen at that point, and downstream of it,
to disturbances travelling a10ng the cylinder. This prob1em has been dea1t with fulIy by
Dowling (1988a). This masterly, virtuoso analysis is unfortunate1y too long and difficult
to reproduce here in fulI, but the outline in what follows gives a flavour of it, as welI as
some of the main results.
8,9.7(a) Outline of the analysis
The equation of motion (8.173) may be re-written as
a 2 y a 2 y ( a 2y a2y )
m-= [ T(x)-MU 2 ] --M -+2U-
at 2 ax 2 at 2 axat
1 ( a y ay ) a4y
--pDUCN -+U- -EI-,
2 at ax ax 4
(8.187)
where
T(x) = T(Ls) + !pDU 2 CT(L s -x), (8,188)
Ls being the length of the acoustic streamer. t C1ear1y, this effective tension vanishes at
x = Xc. a point presumed to 1ie on the cy1inder, that is, for
T(Ls) 2 1
1 > - > 1 - -é:C T = 1 - -é:C T
- MU2 - 7í 2'
in which é: = Lsj D and CT = (4j7í)CT.
(8,189)
tDowling's notation is slightly different, for example, using Jr C and Jr C (the primes being inserted here
to avoid confusion) instead of CN and CT - see also Appendix Q,

954
SLENDER STRUCTURES AND AXIAL FLOW
Equation (8.187) is east in dimensionless form by defining
3 EI
E = 1pD U2 CT L3 '
Y 'n
1] = L = Y()eh.t,
x
 = L'
r2L
w--
- U'
(8.190)
where L is a sealing length, here taken to be equal to Ls for simplicity, resulting in
d 4 y d 2 y dY
E3__( -)-+b-+iwbY=O
d4 c d2 d '
(8.191)
in whieh
b =4iw/(ecT)+cN/CT, (8.192)
where the notation in (8.28a) has been adopted and m = M = p(n/4)D 2 taken. (The
reader is alerted to the eompletely different meaning of the two epsilons, é and E,)
Away from regions with intense deform.ation gradients, the flexural eontribution is
neg1igible, E being smal1, and equation (8.191) reduees to
d 2 Y dY
(c - ) d2 - b d - iw bY = O,
(8.193)
which is a seeond-order equation with a singular point at  = c'
The general solution of (8.193) is found to be
Y() = p  [iwb(c - )]n + Q(c _ )I-b  [iwb(c - )]n ,
 n!(n + b - 1)!  n!(n + 1 - b)!
n n
(8.194)
where P and Q are arbitrary constants - see, for example, Inee (1956). The first part of
the solution is analytie, but the seeond has a braneh point at c [in the eomplex variables
sense; see, for example, Wunseh (1994)]; both series eonverge for al1  and are finite at c'
Consider now a streamer with a free end, such that T(L s ) = O. For a typical problem,
c lies between O and 1, and both inequalities (8.189) are satisfied for sufficiently large é.
The diffieulty arises with the braneh point in the seeond part of (8.194), which requires
that the relevant braneh eut be found in order to extend the solution aeross  = c and
to be able to apply the boundary eonditions at  = 1. Moreover, the second part of the
solution displays large gradients in the vicinity of c, and henee flexural forees beeome
important, thus necessitating the use of the fuHer form of equation (8.191) to detennine
the defleetions. To aehieve this, the method of matehed asymptotic expansions has been
used (cf. Appendix N), in which the 'inner' bending (flexural) solution is matched to the
'outer' tension-dominated solution, equation (8,194),t
Both (8.191) and (8.193) may be re-written in terms of r = c - , and the outer
solution is then given by an equation in terms of r, denoted by <po(r), For the inner
solution, the spatia1 eoordinate is stretched via R = r / E, and solutions are sought for
<I>(R, E) = <l>o(R)+E <1>1 (R)+..., in which <1>0 satisfies (d 4 <l>0/dR 4 )-R(d 2 <l>o/dR2)-
b(d<l>o/dR) = O. EventuaHy, after laborious work, a solution is found involving four
tHere it is remarked that, if the erroneous fonn of the Pai'doussis equation is used, these difficulties are
fortuitously circumvented [Dowling (l988a. section 2)1, Thus, this correction not only has non-negligible
quantitative effects on the dynamics, as discussed in Section 8.9.3, but it can also affect the fonn of the solution.

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
955
constants, which may be related to P and Q by matching this to the outer solution.
Retuming to the spatial variable , the total solution may be expressed as follows: (i) for
e -  » E, that is, far upstream of the critical point,
Y() = P  [iwb(e - )]n +(Q2+Q3)(e-)1-b  [iwb(e - )]n ; (8.195a)
 n!(n + b - 1)!  n!(n + I - b)!
n=O n=O
and (ii) for points downstream, I »  - e » E,
Y()  P  [iwb(e - )]n _ ( - c)l-b (Q2 e -i:1rb + Q3 ei1Tb )
 n!(n +b -1)! (1- b)!
n=O
lb 5 9 3b
1. ( - e)'2 -4Er2 -'e 'e
+7T 2 (l-b)!(b-2)! (Q2 e 1 + Q3 el ),
e = 7T (b - ) + j [( - e)/E]3/2,
(8.195b)
in which the constants P, Q2 and Q3 may be deterrnined from the boundary conditions.
The solution in (8.195a) shows that upstream of the critical point the displacement
varies over a nondimensionallength scale of order unity, The first series in the solution
is always slowly varying and so is valid for all . The second has a branch point at e
and large gradients near it. It is this second series solution that has been extended across
 = e' leading to equation (8.l95b), which also deterrnines the appropriate branch cut;
it shows that, generally, the solution downstream of e contains wavelike disturbances
of short wavelengths, of order E. Hence bending stiffness has an important effect in this
region.
Leaving the boundary conditions at  = O unspecified for the moment, those at  = I
are applied to (8.195b); that is, d 2 y Id2 = O and E3(d 3 y Id3) + [hl(SCT )][dY Id +
iwY] = O, a simplified forrn of (8.175) for x21 Ls --+ O. One obtains a solution which,
surprisingly, has quite different forrn according to whether Re(b) = C NI CT is greater
or smaller than 1/2. In the limit of small bending stiffness and now confining ourselves
to points upstream of the critical, these two solutions may be combined into a single
statement with the aid of the Heaviside unit function, H ( - Re(b)), t name1y
I  [ iwb ( " - ,, )] n [ I ] sin e
Y() = P  S e 'i + H _ _ Re(b) e
n=O n!(n + b - 1)! 2 sin(ee - 7Tb)
x b(-b)! w2l1+b( _ )l-b  [iwb(e - )]n I
(b + 1)! e  n!(n + I - b)! '
n=O
(8.196)
t H(x) = I for x  O, and H(x) = O for x < o.

956
SLENDER STRUCTURES AND AXIAL FLOW
for c -  » E and wZ small, where Z = 1 - c = (ECT ) -I and (Je = :;r (b - n +
(l/E)3/2; P is detennined from the upstream boundary condition, It is of interest that
this final expression has been found not to depend on h.
Well upstream of c, the solution reduces to
Y() = P  [iwb(c - )]n .
L n!(n + b - I)!
n=O
(8.197)
For Re(b) >  this solution holds throughout c -  » E, but for Re <  only within
c -  » Z; once c -  becomes comparable to Z, the distance from the critical point
to the free end, then the second series in (8.196) becomes important. Hence, whenever
C NI CT < , the cylinder displacement has large gradients upstream ofthe critical position.
The solution downstream of c has a more complicated fonn, involving the full expres-
sion (8.195b) and the boundary conditions, but, according to Dowling, is of1ittle practical
interest.
8.9.7(b) Stability of the towed flexible cylinder
ln tenns of stability, Dowling considers only the case of a fixed upstream end, which from
(8,197) is equivalent to
00 (iwbc)n
L n!(n + b _ 1)! = O,
n=O
The complex dimensionless eigenfrequencies may be determined from this expression,
and stability of the system thereby assessed. Here, it is of interest to remark that these
eigenfrequencies are functions of b andc, andhence of CN ICT and 8CT, but independent
of U - consistently with the approximations made, This therefore implies that ifthe towed
cylinder is unstable at any tow speed, it is unstable for all tow speeds!
Introducing
(8,198)
f(w) = (b - 1)!  (iwbc)n , (8,199)
L n!(n + b - 1)!
n=O
f(w) is analytic in the lower half of the complex w-plane, and the number of zeros, N,
within a dosed contour r in this region is given by Cauchy's theorem:
N =  J f'(w) dw,
2:;ri fr f(w)
f'(w) denoting the derivative of f(w), evaluated tenn by tenn.
Example calculations have been conducted for 8CT = 60 - a long streamer! - and
cNlcT = 0.25 and 0.75. In both cases, N = O is obtained, signifying that f(w) has
no zeros in Im(w) < O, for IwlZ < 1 at least. This proves that such typical acoustic
streamers are stable for all towing speeds. These results are not inconsistent with those in
Figure 8.13 for instance, where it is seen that for sufficiently large 8C N and 8CT flutter does
not occur for cantilevered cylinders; but divergence is possible in that case (Figure 8,12),
depending on h,
(8,200)

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
957
8.9,7(c) Forced vibration and other work
Dowling (1988a) also considers the propagation of disturbances generated at the upstream
end (e.g. due to ship manoeuvres). Only the results are commented upon here, namely:
(i) at low frequencies, disturbances propagate along the streamer virtually unchanged
in amplitude, with a phase velocity equal to the towing speed, in what is known as
'worm-in-a-hole' motion; (ii) at higher frequencies, disturbances are effectively atten-
uated, increasingly for higher ú). The effect of a rope drogue is a1so investigated (Dow ling
1988a).
Finally, an extensive study of a negatively buoyant towed cylinder is undertaken
(Dowling 1988b), so that the cylinder is towed at a certain depth, and the cable from
the surface ship to the cylinder has a catenary-like shape. lt is again found that the towed
cylinder is stable, and so the effect of disturbances due to meandering motions of the
ship are investigated; it is found that the towrope acts as a low-pass filter for transverse
oscillations. The effect of ship-motion unsteadiness is also investigated, and it is found
that a significant proportion of such disturbances is genera11y transmitted to the cylinder
at all values of the frequency.
8.9.8 Concluding remarks for towed cylinders
The dynamics of towed flexible cylinders has been discussed, perhaps at greater length
than is warranted, so as to show that, a1though the fundamentals are well understood,
(i) the available models could benefit from re-examination and refinement, and (ii) more
extensive experiments in large water tunnels or towing tanks would be useful.
Other than very long towed cylinders have been found to be subject to a multitude of
rigid-body and flexural instabilities. Nevertheless, Dow ling (1988a,b) has shown that very
long and slender cylinders, such as those used in acoustic streamers, are generally stable
when the upstream end is considered to be fixed, and it is perhaps because of this type of
result that enthusiasm has flagged for further intensive research in this area.
In all the discussion on towed cylinders, the possibility of periodic vortex shedding
from a less than perfectly streamlined nose or tail has been ignored. Nevertheless, it can
arise; see, for instance, N audascher & Rockwell (1994), Hourigan et al. (2001), Pa'idoussis
et al. (2002) and the references cited therein.
8.9.9 Self-propelled/articulated cylinders
Pipes, sometimes bundled, are often towed underwater to offshore locations, to be used in
the installation of pipelines and other offshore systems; see, for example, Patel & Sayed
(1987), Sarv & John (2000). The individual pipes could be welded together (Figure 8.75) or
not permanently interconnected. In the latter case the system is articulated, and it resembles
the self-propelled Sea-Chain system proposed by Hamy (1971), shown in Figure 8.90. This
system consists of a number of flexibly interconnected cylindrical modules, propelled by
a power module in front.
The stability of towed and self-propelled articulated systems has been examined by
Pa'idoussis (1986). Articulated systems are properly the subject of Section 8.12. However,
since the results are compared to those of the continuously flexible system, it is convenient
to inc1ude this discussion here.

958
SLENDER STRUCTURES AND AXIAL FLOW
Figure 8.90 The sea-chain system (Hamy 1971),
I (o)
I
I
, (Jm

t>
X
(b)
l
p.
x
I(':I
Figure 8.91 Definition diagrams for (a) a system with a gripped (c1amped) nose; (b) a towed
system; (c) a self-propelled system (Pai'doussis 1986),
8.9.9(a) Genera! considerations and hydrodynamic forces
Three versions of the system are shown in Figure 8.91; the 'gripped-nose' (cantilevered)
version is used in some preliminary work. The system consists of N - 1 flexible joints
interconnecting N rigid cylinders, each of length I and cross-sectional area A, except
for the first (comprising a streamlined nose) and the last (comprising a rounded tail) of
length ell and eNI, respectively, where el, eN ::::; 1 - cf. Section 3,8.2, For notational
convenience, the jth cylinder is assigned a length
li = cil, Cj = 1 - (1 - el)Dlj - (1 - eN)DNj,
(8.201)
where Dri is Kronecker's delta. The equations of motion are derived by the Lagrangian
method, and the generalized coordinates used are (i) el to eN for the cantilevered one,
with el = O, (ii) el to eN and e tr for the towed one, and (iii) el to eN and y for the
self-propelled system. For the self-propelled system the propulsive force Pr, sufficient for
steady forward speed U, is in the direction shown. For simplicity, motions are confined to
a horizontal plane; also, no stabilizers of any kind are considered.
The inviscid forces acting on the cylinder may be expres sed in terms of the kinetic
energy of the fluid. The lateral velocity of the fluid on the ith inclined cylinder moving
with lateral velocity Vi (),  being the distance along that cylinder from its upstream end,

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
959
is given by
Vfi(O = Vi (O + uei,
(8.202)
where
i-I
Vi(O=y+IL>kěk+ěi' O,::::,::::cil.
k=1
Hence the kinetic energy of the fluid associated with lateral motions may be written as
(8.203)
N rc il
Tf = E 10 !pAv}ŽC)dr
(8.204)
ln addition, there are nonconservative inviscid forces acting at tbe nose and tail of the
system (Section 8.2.4). Consider a control volume encompassing the tail, of sufficient
widtb tbat tbe flow is in its original undisturbed state after leaving it. The rate at which
lateral momentum flux, Mt, enters the control volume is
Mt = (1 - ft)pAUvfN(cNl) = (1 - ft)pAU [VN (CNl) + UeN],
where VN(CNl) is tbe lateral velocity of the tail, and ft is a parameter corresponding
to tbe shape of tbe tail - what was heretofore denoted by h;t for a streamlined end
ft -+ 1, whilst for a blunt end ft -+ O. This momentum flux gives rise to a lateral force,
Ft = -Mt, or
Ft = -(1- ft)pAU [VN (CNl) + UeN],
(8.205)
which accounts for the loss of lateral momentum flux at the tail. Similarly, the
corresponding force at the nose is
Fn = -(1 - fn)pAU [VI (O) + Ue1J.
(8.206)
Considering next tbe normal (N) and longitudinal (L) frictional forces as in
Section 8.2.2, we have
FNi() = !CN (pAl D) U [Vi() + UeiJ,
Fu () = !CT (pAl D) U 2 , (8.207)
and a base drag at the taH,
Fb = !CbPAU 2 .
(8.208)
For the self-propelled system,
N
Pr:::::= !pAU 2 [Cb + cTs(11 D)], s = I>j,
j=1
(8.209)
for small motions; similar1y for the towed system.
tl n this subsection, to avoid confusion with the numerical subscripts pertaining to individual cylinders,
JI, h and C2 are replaced by Jn, Jt and Cb, where the subscripts stand for 'nose', 'tail' and 'base'. Moreover,
the fonu drag at the nose (CI) is assumed to be zero, for simplicity.

960
SLENDER STRUCTURES AND AXIAL FLOW
8.9.9(bJ The equations of motion
The cylinders are interconnected by smooth pin-joints, with rotational springs of stiffness k
constraining intercylindermotion (see, e.g., Figure 8.115). Hence, the kinetic and potential
energies of the chain of cylinders may be written as
N r il
Te = L 10 mv?()d,
i=1 o
N-l
Ve = k L (Bi+l - Bj)2 .
i=l
(8.210)
To obtain the equations of motion for the self-propelled system, the generalized forces,
Qm associated with Bm, m = 1, . . . , N, and Q y associated with y, arising from the forces
in equations (8,205)-(8.209) are obtained; similarly for the towed system, Qm and QOtr
are obtained. Then using the Lagrange equations, the equations of motion are derived;
they are given in Appendix S.
8.9.9(cJ Nondimensionalization and solution
The equations of motion are rendered dimensionless by using
( ) -1/2
!'= 2ms 4 Z 3 jk t,
( ) 1/2
u = ms 2 Zjk U,
r]=yjL,
(8.211)
with s = 'Lf= 1 C j; for details of this and other aspects of the derivations, see
Herman (1975) and Pa'idoussis (1986).
Solutions of the linear discrete system of equations, of order N (or N - 1 for the
cantilevered system), are then obtained by standard means - see Section 2.1.2.
8.9.9(dJ Transition from discrete to continuous
Some ca1culations were done for the cantilevered system (gripped nose) to validate the
model, by comparing to the equivalent continuous system (taking k = E I j Z). It was
found that the system with six degrees of freedom (N = 6; el = eN = ) adequately
approximates the continuous one insofar as first- and second-mode behaviour is concemed
- see Section 8.12.1.
The same may be supposed to be approximately trne for the self-propelled and towed
systems to be considered next.
8.9.9(eJ Dynamics of the self-prope/led system
A typical Argand diagram is shown in Figure 8.92, At u = O this system is essentially
a free-free articulated beam. However, unlike the towed system, one of its rigid-body
modes (the zeroth) remains at w = O, because in accordance with this formulation of
the frictional forces there is no resistance to latera1 motion, unless it is accompanied by
incidence to the free stream. The first (yawing) mode remains on the Im(w)-axis. It is
seen that the system is subject to yawing for O < u < 3.7. t Second-mode flutter occurs at
tThis is an artifice of the model, as fonnulated here. The whole system can travel at an angle, and there is
no restoring force to bring it back to straight course.

SOLL'ARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
961
II
i +2.4
i i
i
h6
i
4,0 i
'" 1.0
!
!
,
3,7, ,0,0
 ;0,5
3,5, 
rl'1.5
I ,J2,5
1st mode
7.45
4
7.3 2
7,225
-2
7,1
-4
7,°_6
°
8
32
16
(w)
24
Figure 8.92 Argand diagram of the lowest two dimensionless eigenfrequencies w for varying u,
forase1f-propelledsystem(N = 6,cCN = CCT = 1, fn = 1, ft = 0.8,q = O,2,el = eN = 1/2).
The first-mode locus, on the 1eft of the figure lies entire1y on the Im(w)-axis, as does part of the
second-mode locus for high u (Pai'doussis 1986).

\I
"
II
\I
\I
\I
"
"
II
"
"
"
"
"
"
\
\,
"
"
"

"
"
"
"
"
\

(a)
(b)
,
"
"

"
"
"
"
"
'
II
1\
"
"
"

"
\
"
'
\\
\\
\
"
"

/t

"
"
"
"
j'

\,
I "
r=-T \
2 'II
II
rlT .JI
4

(c)
(d)
Figure 8.93 Moda1 shapes of the self-propelled system of Figure 8.92: (a) first-mode, quasi-
rigid-body yawing (divergence) at u = 0,5; (b) second-mode stable oscillation at u = 0,5; (c) first-
mode divergence at u = 3.0, showing c1ear1y some flexura1 content; (d) second-mode oscillatory
instability at u = 4.0. In (b) and (d) the shape is drawn at four different parts ofthe cyc1e ofmotion;
T is the period (Pai'doussis 1986).
u :::: 3.5; this instability is eventua1ly transformed to divergence (the locus is shown only
for u ::: 7.0), and the system eventually regains stability at u :::: 7.23. However, it has
already lost stability in its third-mode flutter at u :::: 5.6 (not shown),
The moda1 shapes are shown in Figure 8.93. It is seen that the first mode at u = 0,5 is
sensibly of rigid-body type; at u = 3, however, an appreciable degree of flexing is seen
to take place. The second-mode shape is similarly seen to be c10se to the c1assical one for

962
SLENDER STRUCTURES AND AXIAL FLOW
Table 8.17 The effect of the tail parameter ft on stability
ofa self-propelled system (SCN = SCT = SCt, fn = 1, Cb =
1 - ft (Pai'doussis 1986).
Parameters
Values of u
Range for yawing Plutter threshold
SCt = 1, ft = 0.8
SCt = 0.2, ft = 0,8
SCt = 1, ft = 0.4
0-3.7
0-3,3
0-3.8
3.5
3.4
5,0
10 6,9
8 7
6
72
4
7.3
3 825,u. ..
3,8  o
37 1.0
3H.. 2.0 -2 7.5
-4
-6 80
-8
6,0
6 (w) 8
"'..4,8
"".
Zeroth mode ....-/""'"
"'.'O ....... ......
'./7,0
Figure 8.94 Argand diagram of a towed articulated system for varying u (SCN = ser = ],
A = I, fn = ], ft = 0,8, N = 6, el = e2 = ]/2); (Hennan 1975),
a free-free beam in Figure 8.93(b) at u = 0.5, but becomes degenerate in Figure 8.93(d)
at higher u.
The effect of the tail shape on stability is summarized in Table 8,17. It is seen that
blunting the taH generates a significant window of stability (3.8 < u < 5.0). It does
not e1iminate yawing, however; it is clear that stabilizers are essential for that, as seems
reasonable on physical grounds: without them the system is inherent1y unstable.
8.9.9(f) Dynamícs of the towed system
A typical Argand diagram is given in Figure 8.94, The system is seen to be unstable at low u
in its zeroth mode (yawing), whi1e being stable in its first mode. At u = 3.8, the system
is restabilized. At u :::: 4, however, an osci11atory instabi1ity develops, emanating from
the restabi1ized zeroth-mode locus, Therefore, the dynamics is simi1ar to that depicted
in Figure 8.80, albeit for sma1ler Ji (fn here) and h(ft here) in the latter, if some mode
veering between the two modes is taken into account. In this respect, it is remarked that
for the discrete system a clear distinction is made here between ()tr and ()I, which is similar

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
963
Table 8.18 Comparison of stability characteristics of
self-propelled and towed articulated systems (N = 6) and
towed continuous system (N = (0) for (eCJ = I, fn = I,
ft = 0.8, Cb = 0.2; e] = eN = i (Paldoussis 1986),
System
Values of u
Yawing range Flutter threshold
Self-propelled (N = 6)
Towed (N = 6, A = 1)
Towed (N = 00, A = 1)
Towed (N = 6, A = 10)
0-3,7
0-3,8
0-3,5
0-3.8
3,5
4.0
4.1
4.0
to the treatment in equation (8.181). Higher modes are subject to flexura1 instabilities as
in the continuous system (not shown).
Comparing to the self-propelled system, in this case there is no stationary mode-10cus
at w = O; in fact, neither the zeroth nor the fi.rst mode remains on the Im(w)-axis. These
differences are re1ated to the fact that for the self-propelled system the propu1sive force
a1ways points in the direction of the nose, el, whi1e in the towed system the tow-rope pull
points in the direction e tr . Hence, the two systems shou1d become simi1ar if
(i) Pr were artificially restricted to be a10ng the x-axis for the se1f-propelled
system, and
(ii) A -+ 00 for the towed system, as approxímate1y verified numerically.
The self-propelled and towed system stability ís compared in Tab1e 8.18. It is seen that if
one does not differentiate between which mode does what, but simp1y consíders the broad
aspects (stability, yawing or osci11atory ínstability), then the dynamics is remarkab1y
simi1ar. So ís the dynamics of the discrete (N = 6) and continuous (N = (0) systems,
a1so shown,
One important conc1usion from this section is that for a towed articu1ated system, if the
number of articu1ations is not small, say N  4, the dynamics is quite similar to that of
the continuous1y flexib1e one.
8.9.10 Locomotion of slender fish
Tbe premier examp1e of a se1f-propelled slender body is that of swimmíng slender fish,
far more important and interesting than what is presented in Section 8.9.9. This is a
huge subject area. No justice can be do ne to it in a mere section of a book, not even if a
who1e chapter were devoted to it. Indeed books on the subject do already exist (Gray 1968;
B1ake 1983; Vide1er 1993). Here, a very bríef account is given, on the aspects most dosely
re1ated to other topics in this chapter, concentrating on the mechanism of swimming of
slender fish, and providing some of the key references.
An excellent discussion on 'how do fishes swím' is given by Lighthi11 (1970a), outlining
in simp1e terms the various mechanisms of propu1sion by undu1atory motion which have
evo1ved in different zoologica1 orders of fish, including that of anguilliform motion (named
after Anguilla vulgaris, the common eel) which is of particu1ar interest here, and its
modified fonn of carangiform swimming.

964
11
10
SLENDER STRUCTURES AND AXIAL FLOW
9
10
0.1
8
7
6
5
4
3
2
(a)
1.0
9
0,9
8
0.8
7
0,7
6
0.63
O)
a
0.5 
5
4
0.4
3
0,3
2
0,2
(b)
Figure 8.95 Anguilliform swimming (a) in the common eel (Anguilla vulgaris), L = 70 mm,
with /)"t = 0.09 s between frames; (b) in the dogfish, L = 350 mm and /)"t = 0.1 s (Gray 1968;
Blake 1983),
The anguilliform mode of propulsion is characteristic of the orders of Apodes (eels)
and Allotriognathi (e.g. ribbon-fishes) which have continuous dorsal and ventral fins of
more or less constant depth; any tapering at the tail is rather abrupt and hence we may
approximate it as having a vertica1 trailing edge, Some Selachii (sharks, dogfishes) and
Chondrostei (sturgeons) a1so swim in the angui1liform mode,t As seen in Figures 4.38(a)
and 8,95 and as discussed by Lighthi1l (1969, 1970a,b), this mode ofpropulsion involves
a wave progressing towards the tail with increasing amplitude, and with a phase velocity
a1ways larger than the forward speed of the fish.
8.9. 10(a) Dynamics of anguilliform mode of propulsion
The ana1ysis here follows Lighthi1l (1960a, 1970b). Consider a slender fish of cross-
sectiona1 area Sex), swimming with steady forward speed U. The flow velocity due to
displacements y(x, t) of the fish, that is the perturbation velocity, is v(x, t) = (ay jat) +
u(ayjax). According to slender-body ('elongated body') theory, the local flow past the
cross-section S (x) is identica1 to the 2-D potentia1 flow that would result from motion of an
tThe swimming of cod (Gadus morhua of the order of Anacanthini) can also be analysed in the same
way if one imagines that the gaps between its three dorsal fins are interconnected by vortex sheets, thus
hydromechanically fonning a single long fin; similarly for the ventral fins (Lighthil11970b, section 3).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
965
infinitely long cylinder of diameter D(x) equal to the depth (vertical dimension) of Sex),
with velocity v(x, t) through fluid at rest. This flow has a momentum M(x)v(x, t), where
M (x) = pA (x) = p [D2 (x)] is the virtual or added mas s of the fish per unit length for
lateral motions (Section 2.2). Hence, as in equations (2.84) and (8.2),t the lift per unit
length of fish, L(x, t), is equal and opposite to the rate of change of lateral momentum
over Sex), that is,
L(x, t) = - (  + U  ) [M(x)v(x, t)].
at ax
The rate, dW jdt, at which the fish does work to generate y(x, t) is
(8.212)
dW a {i L ay l i L 2 } [ ay ] L
- = - - Mvdx - - Mv dx + U -Mv ,
dt at o at 2 o at o
(8.213)
where M =: M(x) and v=: v(x, t). Thecontributionofthefirst and secondtime-derivative
tenns to work over a period of oscillation, Ll W, or equivalently the mean value over a
long time, W, is zero. Realizing that M(O) = Obut M(L) t= O, the last term gives
W = M(L)U {Ly(Y + Uy1)]X=L} ,
(8,214)
where overdots and primes denote aOjat and aOjax, respectively. This is identical to
(8.47) for f = O if all viscous terms are ignored;:t: the difference in sign reflects that W
here is the mean work done by the fish on the fluid, This expression may be interpreted
as the mean of the product of the lateral velocity yd=: y(L)] and the rate of shedding of
lateral momentum (M v) L U at the trailing edge.
As already suggested, W here mus t be positive for propulsion - unlike the arguments
made in Section 8.3.3(c). Hence, yI > IU.YLYI, though YLY may be negative - see
Figure 8.95, where it is seen that over most ofthe cycle the tail executes a lagging-dragging
type of motion, similar to that discussed in Chapter 3,
It is next realized that not al1 of the work in (8.214) goes towards propulsion. In fact,
the rate of shedding of kinetic energy of lateral fluid motions is U[!MV 2 ]L' If P is the
mean forward thrust, then the rate of doing work available for producing the trust is P U;
hence
P U = W - u[ !Mv2t,
and, using (8.214), the mean thrust is
(8.215)
P = !M(L){ (yi) - U2 (y)2 }.
(8,216)
tThe difference in sign between FA in (8.2) and L(x. t) in (2,84) and here is that in the former FA is taken
as a ď Alembert force, whereas here a positive lift is taken to be in the +y-direction.
*It is important to stress that f = O here, since the tail of eel-like fishes may be regarded as being terminated
by a vertical trailing edge, Moreover, although the tail is thin, the virtual mass M(L) is controlled by the depth
D(L), being equa! to p [*1r D 2 (L)] rather than pS(L),
The question is whether a fish can swim in a perfect (zero-viscosity) fluid was first posed by Saffman (1967)
and is further discussed in Wu's (I 971d) excellent review, As we can see in this analysis, this is perfectly feasible,

966
SLENDER STRUCTURES AND AXIAL FLOW
The same result is obtained via a more rigorous analysis by Lighthill (1960a;
appendix) - see also Section 8.10, as well as the ingenious and very simple analysis
in Lighthill (1969, 1 970a), accessible even to the layman.
The propulsive (Froude) efficiency may be defined as I] F = P Ul W , which from (8.214)
and (8.215) gives
1 (y + Uy')i
I]F = 1 - - (8.217)
2 [Y(y + UY/)]L
ln all the foregoing, the motions y (x, t) are not unrestricted. They must satisfy total
force and moment balances, taking into account the inertia of the fish, name1y
l L a2y l L ( a a ) { ( ay ay ) }
m(x)-----zdx = - - + u - M(x) - + u - dx,
o at o at ax at ax
(8.2l8a)
l L a2y l L ( a a ) { ( ay ay ) }
xm(x)-----zdx = - x - + U- M(x) - + U- dx,
o at o at ax at ax
(8.2l8b)
where m(x) = pS(x) is the mass of the fish per unit length, the fish being presumed to
be neutrally buoyant. Clearly, any movements violating (8.218a) and/or (8.218b) would
automatically produce reactions, or 'recoils', in the form of rigid-body movements.
Considering now (8.217), it is clear that if, on the average, Cy + U y/) L is small compared
to h, one obtains a high value of efficiency. Obviously, this means h and y must
be negatively correlated, leading to the characteristic backward sloping tail vis-a-vis its
velocity over most of the cycle.
On the other hand, unlike the instability of a cantilever conveying or immersed in axial
flo w (Sections 3. 2.2 and 8.3.3), h and (y + Uy')L must be positively corre1ated; that
is, (y2 + U yy/) L must be positive for positive thrust. Also, from (8.217) it is clear that,
although it is desirable for (y + UY/)L to be small for high efficiency, it should not be too
small, lest the thrust be reduced too much; thus, a compromise is needed.
Lighthill then considers motions in the form of a standing wave and shows that I] F in
that case cannot exceed . Then, motions of a trave1ling-wave form
y(x, t) = f(x)g (t - xle)
(8.219)
are considered, where g(t) is an oscillatory function, for example, cos(t - x le), and it is
shown that
1 (1 - U le)2 f2(L) g'2 + U 2 f'2(L) g2
I]F = 1 - - -
2 (1 - U le)f2(L)g'2
From this, it is clear that a positive rate of doing work is possible only for e > U.
Furthermore, to make I]F approach unity it is desirable to have f'(L) practically at zero.
Indeed, it is found to be desirable to let f (x) increase gradually from zero at x = O to
a maximum, with f'(L) = O at the trailing edge; however, recoil considerations modify
these statements. Lighthill eventually arrives at the following set oť rules for high efficiency
with minimal recoil: (i) confine motions to be rear part of the fish where fish mass is low,
(8.220)

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
967
since typically posterior cross-sections are laterally compressed ('flatteneď); (ii) keep
v(= y + Uy') low for given ayjax by having c near U, thus reducing water momentum,
in spite of the high virtual mass near the tail; (iii) let the waveform have a positive and
a negative phase in the region of substantial amplitude, which would tend to cancel out
the angular recoils produced by each. As shown in Figure 8.95, the type of anguilIiform
swimming favoured by fish satisfies these rules.
8.9.10(b) Carangiform mode of propulsion
ln this type of swimming, the basic anguillifonn method of propulsion has been modified:
the anterior part of the fish has lost its flexibility and the amplitude of undulation becomes
significant only in the posterior half, or even one-third, as seen in Figure 8.96. A wave
stilI passes backwards, but with steeper amplification with x. As before, motion of the
trailing edge lags behind that in front. This mode of propulsion is used by several orders
of fish, including the Isospondyli (herring, trout, salmon) and some of the Percomorphi
(perches, snappers, red mullets, barracuda), When further modified by a large lunate tail,
some of the Percomorphi (tunnyfishes, mackerel, swordfishes), as well as dolphins and
some sharks have developed into the fastest swimmers in existence. t
The confinement of undulations in the neighbourhood of the caudal fin produces a gain
in efficiency. To understand this, it is first pointed out that when the fish pushes laterally on
a water slice with velocity v, the lateral momentum Mv is generated immediately (at the
speed of sound). However, continued pushing produces a certain additional momentum,
more gradually, through the action of viscous forces, by generating streamwise vorticity
shed in the cross-flow. At the same time, the water kínetic energy increases considerably
above the ! M v 2 of the irrotational flow. The net effect of these could be a reduction in
propulsive efficiency. To determine this increase in lateral momentum it is necessary to
calculate the vorticity-related force at different locations along the fish and to integrate
it with respect to time - a difficult task, nevertheless successfully albeit approximately
tackled by LighthilI (1970b); similarly, for the increase in the wasted energy. The analysis
is beyond the scope of this treatment. Qualitatively, however, it is shown that in carangifonn
motion the rafid development of the wave tends to keep the waste kínetic energy down
to near ! Mv, whereas slower acceleration would have allowed the generation of too
much vorticity from the top and bottom of the fish - specially if there is a continuous
fin there,
5
7
9
10
Figure 8.96 Carangiform swimming mode [after Gray (1968)]; (Lighthill1970a).
tFor example. tunnyfish can achieve speeds of 20 mis (72 km/h). Dolphins can emise at about 20 knots.

968
SLENDER STRUCTURES AND AXIAL FLOW
c=

Figure 8.97 The jet-like streaming generated by the caudal-fin trailing edge casting off a regular
pattem of vortices (Lighthill 1969),
In fact, the disappearance of the continuous fin is but one of the morphological changes
mak:ing carangiform swimming efficient. Since the transfer of momentum occurs only
in the posterior part of the fish, unbalanced forces would develop causing recoil and
loss of efficiency. This has led to two further modifications in fish-body morphology:
(i) a reduction of the cross-sectional depth just before the caudal fin, where the amplitude
is increasing very rapidly and the side-force caused by the rate of change in momentum
transfer is most severe; this reduces the virtual mass and hence the local rate of momentum
change; (ii) the dorsal (and ventral) fins have become larger, increasing the local virtual
mass to better resist recoil due to tail motions.
In this discussion we have so far ignored the vortex street that forms in the wake of fast-
moving fishes with large caudal fins of high aspect ratio, that is, a lunate tail; for example,
in the order of Percomorphi, the Scombridae (e,g, tunnyfishes, mackerel), Istiophoridae
(sailfishes, marlins) and Xiphidae (swordfishes) - also in sharks and cetacean mammals,
A two-dimensional view of this vortex street is shown in Figure 8.97. However, unlike the
usual Bénard-von Kármán vortex street generating a drag, the vortices here are reversed,
generating a tlow something like a propulsive jet; thus, in effect, the caudal fin propels
the fish on 'the jet propulsion principle', Lighthi11 (1970b) provides a 2-D analysis ofthis
also, as does Wu (1961) earlier in conjunction with undulating foils. This is the key to
why lunate tails are so effective (Lighthi11l970b; Chopra 1976; Triantafyllou et a/. 1993).
Finally, it should be mentioned that the wak:e is really three-dimensional and the vortices
are really vortex rings [see, e.g. Drucker & Lauder (1999)], mak:ing the analysis more
difficult but the swimming even more efficient.
8.9.10(c) Other forms of aquatic locomotion
Even leaving aside completely different methods of propulsion, for example, by the pure
jet mode employed by squid, there are other methods than those already discussed: for
example, ostraciiform locomotion where only a small tail osci11ates and the large body
remains almost inert, methods relying on the effective manipulation of drag forces, as in
ciliary propulsion, and other 'non-body modes' of swimming, for example by the use of
the fins alone, as in labriform and balistiform swimming (Blake 1983). In some of these,
the hydromechanics is very interesting and it has been attacked with the aid of a variety of
tluid-mechanical tools, for example, slender-body and actuator-disc theory (Blak:e 1983;
Blak:e & Lighthill 1990; Lighthi1l1990a---c),
None of these fascinating topics are discussed here, but some reference s are given
in subsection (d); nor is the intriguing Gray's Paradox [see, e,g. Blak:e (1983)] which has
done so much to motivate research on delaying transition in boundary layers.

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
969
8.9.1O(d) A selective, brief literature review
The brilliant and inftuential work by Lighthill (1960a,b, 1969, 1970a,b, 1971) ought to
be mentioned first; much of it, on this topic, may be found also in Lighthill (1975).
Semina1 workhas been done by Wu (1961, 1971a-d), Newman & Wu (1973) and Chopra
(1974, 1976, 1977); see also the work by Siekmann (1962, 1963) and Pao & Siekmann
(1964).
Work in this area has continued apace ever since; here we give an antho1ogy of some
of the very recent pub1ications, in which further reference s may be found: Gillis (1997,
1998), Carling et al.. (1998), Cheng et al. (1998), Anderson et al. (1998), Williams
(1999), Drucker & Lauder (1999), Wolfgang et al.. (1999), Triantafyllou et al. (2000),
Zhu et al. (2002), in some of which the 1atest too1s of the trade are used, for examp1e
CFD solutions of the Navier-Stokes equations coup1ed to the swimming-fish dynamics
and Digita1 Particle Image Velocimetry (DPIV) in the experiments.
The more physical discussions with minimum mathematics by Lighthill (1970a),
Bschorr (1988) and Triantafyllou & Triantafyllou (1995) and others pro vide excellent
reading to motivate the uninitiated. The books by Gray (1968), Blake (1983) and Vide1er
(1993) are also required reading.
Yet, despite all the research effort and the great progress achieved, there still remain
many questions unanswered, even on anguilliform swimming (Gillis 1996).
Intimately related to this work is that on propu1sion by ftapping or undu1ating foi1s
(p1ates). This work has been motivated part1y to provide understanding of fish locomotion
via the physically simp1er system of an undu1ating ftat p1ate and part1y to use such a
system for the propu1sion of watercraft. The interested reader is referred to Wu (1961),
Reece & Siekmann (1964), Uldrick & Siekmann (1964), Kelly et al. (1964), Botman
(1965), PaIdoussis (1976a), Triantafyllou et al. (1993), Streitlien & Triantafyllou (1998),
Anderson et al. (1998), and Wolfgang et al. (1999). A few words are given in subsection (e)
on the re1ation between ftow-induced instabilities and self-propu1sion.
Concerning swimming at smal1 Reyno1ds numbers (e.g. by worms and microscopic
organisms), the reader is referred to Tay10r (1951, 1952a,b), Reyno1ds (1965), Jahn &
Votta (1972), Chwang & Wu (1974), Chwang etal. (1974), Brennen (1974), Brennen &
Winet (1977), Ped1ey & Kess1er (1992), Si1vestre (2002), and many others.
8.9. 10(e) Undulatory propulsion and self-excited oscillations
The hydroe1astic ichthyoid (fish-1ike) propu1sion system has already been discussed briefty
in Section 4.7.2. A p1ate is forced to execute undu1atory motions by two pipes attached
to it on either side and driven to limit-cycle oscillation by internal ftow. The motion is
remarkab1y like that in anguilliform locomotion of fish, as may been seen in Figure 8.98.
The brass p1ate in the figure is 51 x 343 mm, 0,20 mm thick, and the tygon pipes of 9,5 mm
externa1 diameter, 1.6 mm thick (Saroudis 1974); the p1ate is attached to a massive brass
ho1der, itse1f mounted on a watercraft (Figure 4.38).
Dne reason for revisiting this probIem is to clarify how the motion of the pipe and
p1ate can satisfy the confticting requirements for energy transfer: (i) th at (ji + U jLY D
be negative for pipe instability, that is, limit-cyc1e motion, and (ii) that (ji + U YLyD be

970
SLENDER STRUCTURES AND AXIAL FLOW
a
,
I
I
4---
Xl
Y l
6---
--3
4
(a)
(b)
Figure 8.98 (a) A series of images of a swimming eel, shown with its snout aligned to the
same longitudinal position, and the figure-of-eight path followed by the tip of the tail- with the
points 1-7 sampled at equal time interval s (Gray 1968; Gillis 1996), (b) Enhanced drawing from
a photograph of a stroboscopically illuminated flexible plate, 343 mm long, driven to limit-cycle
motion by internal flow (in attached pipes on either side) and mounted in external flow in a water
tunnel (thanks to Vassilios Vainas and John Saroudis).
positive, a1beit small, for efficient propu1sion. The answer is of course trivially simp1e: in
the first U = Ui, the internal tlow velocity, whi1e in the second U = Ua, the externa1 tlow
ve10city (forward speed). In both cases hy is negative, but Ua < Ui by a 1arge margin.
Some further interesting comments on the energy transfer mechanisms in se1f-excited and
self-propelled systems are given in Iwatsubo (1981).
Based on pre1iminary experiments (Saroudis 1974; Pai'doussis 1976a), it was concluded
that, although the propu1sion system works well, it is 'un1ike1y to riva1 the propeller
in efficiency'. t Other experiments with mechanically undu1ating tai1s, for examp1e by
Botman (1965), also yie1ded disappointing efficiencies. This was so, unti1 Triantafyllou &
Triantafyllou (1995) [see also Wo1fgang et aZ, (1999)] achieved propu1sion efficiencies of
86% with their tlapping-foil 'RoboTuna'.
The 1.24 m 10ng robotic tuna that the Triantafyllou duo buiIt was internal1y powered so
that its tlexib1e tail wou1d execute carangiform undu1ations. Initial1y, the resu1ts obtained
were disappointing in this case a1so, until it was rea1ized that the wake was missing the
most attractive attribute obtainable with a lunate tail: the jet-like tlow generated by vortices
shed by the tai1, as already discussed in subsection (b). It was found that for most fish the
tai1 tlapping frequency satisfied the Strouhal-number relationship St = f D / U = 0,25 to
0.35, D being the tai1 depth. Once the frequency of tlapping in the ToboTuna was adjusted
tHowever, as pointed out by Triantafyllou & Triantafyllou (1995), small propellers are not all that efficient,
typically achieving propulsion efficiencies of 40%,

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
971
to St  0.3, the vortex street and jet flow were established, and the efficiency soared to
unprecedented values - a truly remarkable achievement.
8.10 NONLlNEAR EQUATIONS OF MOTION
As has been made abundantly clear in Chapter 5, several important aspects of the dynamical
behaviour of a system can only be predicted by nonlinear theory; for example: (i) the
existence of the post-divergence bifurcations; e.g., the observed flutter may in fact not arise
in the manner predicted by linear theory, if at all; (ii) the transition from one dynamical
state to the next; (iii) the amplitude of divergence, that is, location of the fixed points,
and limit-cycle amplitude and frequency for flutter; (iv) the exploration of non-standard
dynamics, such as the quasiperiodic or aperiodic regime observed between second- and
third-mode flutter.
In this section, a weak1y nonlinear equation of motion is derived in a Hami1tonian
framework for a cantilevered cylinder, assuming the centreline to be inextensible; in the
linear limit it is identical to that o btained in Section 8.2. The kinetic and potential energies of
the cylinder itself are discussed in Section 8.10,1. The fluid dynamic forces are introduced
in tenns of virtual work expressions, separately for the inviscid forces and the hydrostatic
and frictional forces in Sections 8.10.2 and 8.10.3, respectively. The nonlinear equation
of motion is given in Section 8.10.4, and the boundary conditions in Section 8,10.5.
The equations of motion if the centreline is extensible are given in Section 8.10.8 and
Appendix T.3. Finally, an important correction to the nonlinear equations of motion of a
fluid conveying pipe with supported ends given in Section 5.2 is presented in Appendix T.4.
8.10.1 Definitions and preliminaries
The system consists of a cantilevered flexible slender cylinder of radius R, length L,
flexural rigidity E I and mas s per unit length m, centrally located in a channel of radius
Ro and subjected to an axial flow ve10city U, as shown in Figure 8.99. The undefonned
cylinder axis coincides with the X -axis, and is in the direction of gravity if the system is
vertical. The cylinder is cantilevered, generally fitted with an ogival end-piece at the free
end, considered to be short relative to the overall length of the cylinder. The equations of
motion are derived ignoring this noncylindrical segment of the cylinder, which is taken
into account with the boundary conditions.
The basic assumptions made for the cylinder and for the fluid are that (i) the fluid is
incompressible, (ii) the mean flow velocity is constant, (iii) the cylinder is slender, so that
Euler-Bemoulli beam theory is applicable, (iv) although the deflections of the cylinder
may be large, strains are small, and (v) the cylinder centreline is inextensible.
The derivation of the equations of motion is given here as in Lopes et al. (2002) with
sufficient detail to be followed, but omitting some of the steps in the interests of brevity,
The assiduous reader is referred to Lopes et al. (1999a, 2002) and to Appendix T for more
details.
Two coordinate systems are used: the Lagrangian (X, Y, Z, T), associated with mater-
ial points on the undefonned cylinder, and the Eulerian (x, y, z, t), associated with the
defonned state of the cylinder. The displacement of a material point is thus, u = x - X, v =
y - Y and w = z - Z - see Figure 8.100. For a slender cylinder and motions with the

972
SLENDER STRUCTURES AND AXIAL FLOW
gl Ul 1
Flexible cantilevered
cylinder
 R
i
i
i
 Ro
i
i
i
Figure 8.99 Diagrammatic view of a cantilevered cylinder in axial flow, in the test-section of a
circulating system (Lopes et al. 2002),
o
x
y
() ós
óy
G
Figure 8.100 Diagram defining the coordinate systems. A material point G on the neutral axis
of tbe cylinder at curvilinear coordinate s is located at G(X, Y) before deformation and G(x, y)
afterwards; so tbat its displacement is {u, v} = {x - X, Y - y}, For a point Pat a distance y from
tbe centreline, at the same cross-section, tbe displacement is {u - y sin 81, v + u(cos 81 - I)};
(Lopes et al, 2002),

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
973
cylinder centreline in the (X, Y)-plane, we have Y = O and z = Z = O. Thus, fonnally,
the Eulerian and Lagrangian systems are related by
x(X, T) = X + u(X, T),
y(X, T) = Y + v(X, T),
z= Z,
As the cylinder centreline is assumed to be inextensible,
(oxjoX)2 + (oyjoX)2 = 1.
t = T.
(8.221)
(8.222)
Hence, in this case, one may use the curvilinear coordinate along the cylinder, s,
instead of X; s = X. Dne may thus obtain the curvature K along the defonned cylinder
(Section 5,2,1),
o2yjos2
V I - (oyjos)2
The equation of motion is derived via Hamilton's principle,
l t2 1 t2
8 £ dt + 8 W dt = O,
tj tj
where £ = Tc - Ve is the Lagrangian, Tc and Ve being the kinetic and potential energies
of the cylinder, and 8 W the virtual work by the fluid-related forces acting on the cylinder,
The derivation of the forces due to the fluid proceeds in a similar manner as in Section
8.2. Thus, the inviscid, viscous and hydrostatic forces are detennined separately - rather
than together, say by direct application of the Navier-Stokes equations, The separate
derivation of inviscid and viscous forces may be justified further by the fact that the
fonner are dominant, considering a large enough Reynolds number. The inviscid forces
are derived via slender-body potential flow theory, while the viscous forces are fonnulated
semi-empirically. This approach simplifies the analysis considerably and has been shown
to give acceptable results (Pai'doussis 1966a,b, 1973a, 1979; Pai'doussis et a/, 2002; Semler
et a/. 2(02).
The equation of motion is derived correct to third order O(E 3 ), for y = v '" O(E) and,
via equation (8.222), u '" O(E 2 ). Hence the expressions for the components ofthe virtual
work 8W must be correct to O(E 3 ), while the energy expressions to O(E 4 ).
ln the derivations, the following relationships have been found useful:
8x = - (y' + yt3) 8y + i S (li + yl2y") 8yds + O(E 5 ),
i L g(s) [1 S j(S)8 Y dS] ds = i L [[L g(S)dS] j(s)8yds,
[ L I " 1 12 [ L 1 12
(L - s)y y ds = --(L - s)y + -y ds.
s 2 s 2
Nonlinear expressions for the kinetic and potentia1 energies of the cy1inder itself have
been obtained when deriving the nonlinear equations of motion of a pipe conveying fluid;
see Chapter 5 and Semler et al. (1994) for detai1s. The kinetic and potential energies are
1 1 L 1 1 L 1 L
Tc = -m V e 2 dX, Ve = -EI K 2 dX -mg xdX,
2 O 2 O O
K=
(8.223)
(8.224)
(8.225a)
(8.225b)
(8.225c)
(8.226)

974
SLENDER STRUCTURES AND AXIAL FLOW
where Ve = xi + ýj is the velocity of a cylinder element, and K its curvature, The limit
L is really L - l, where l « L is the length of the ogival end, here ignored. After
considerable manipulation and use of (8.222) and (8.225a), while keeping in mind the
orders of magnitude of the various quantities, we obtain
1 t2 1 t2 1 L I 1 s
o I::dt = -m ji + y' (/2 + y' y')ds
tI tI o o
-y" 1 L 1 s (/2 + ylyl)dSdS} oydsdt + ťJ(E 5 ),
(8.227a)
1 t2 1 t2 1 L
o Vedt = E I [y'1II + 4y' y" ylll + yl/3 + yl/" y'2] oydsdt
ti tI o
1 t2 1 L
- mg [- (l + b /3 ) + (L - s) (y'1 + Y"y'2)] oydsdt + ťJ(E 5 ),
tI O
(8.227b)
where 0 ' = 80j8s and (') = 80j8t.
In accordance with the approach adopted, an element ofthe deformed or oscillating cyl-
inder is subjected to the following set of forces, as shown in Figure 8,10 1: the inviscid fluid
dynamic force FAOS, the norma! and longitudinal viscous (frictional) forces, FNOS and
FLOS, respectively, and the hydrostatic forces Fpxos and Fpyos in the x- and y-direction,
respectively. t
8.10.2 The inviscid fluid dynamic forces
These forces are obtained via slender-body potential flow theory, following dosely
Lighthill' s (1960a) formulation. The motion of the body is not represented by the deflection
F py ÓS
F px ós
F L ÓS
pAgÓs
jI
Figure 8.101 An element os ofthe cylinder, showing the forces acting on it (Lopes et al. 2002).
tThe 8s here is an elementallength, not to be confused with the variational 8x and 8y in the foregoing,

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
975
y
v y
---- ------ - --- -- --- -- - -- -. . - -... -. -. -- - -- --- - -- --- -- - ,7- ----- - --- - - - - = "1
,,'i ../ )
,'/ ,/';
,/' ../:
/,' ,/' :
,I' i 1 .,/ :
,// ,// ;
/," .,/' ,
X Uf
x
Figure 8.102 An element of the cylinder used for the determination of the relative fluid-cylinder
velocity V and of the angles 81 and 82 (Lopes et al. 2002),
of its centreline; rather, the motion of every point of the cylinder is taken in to account.
Hence, the coordinate y is no longer equal to the displacement v, but is of the form
y = y + v, where Y describes the po si ti on of a point in its original state.
We consider a displacement y(X, t) of the cylinder in the y-direction, away from its
stationary, straightconfiguration. In Figure 8.102, we introduce the unit vectorpair (i), jI),
respectively tangential and normal to the centreline, at angle e) to (i, j).
We consider next an element of the cylinder as in Figure 8.102, and define the rel-
ative fluid-body velocity, V = jr + x - Uf, in which Uf is the mean axial flow
velocity relative to the deforming cylinder. Then, projecting this on ji, leads to V =
ycose) + (Uf - x) sine I. From (8.221) we have axjax = 1 + aujax, and hence
e) = tan- I [(ayjaX)(l - aujax)] + O(E 5 ). From this, keeping in mind the orders of
magnitude of y and u, we obtain
el = y' - u'y' - !y'3 + O(E 5 )
(8.228a)
and cos el = 1 - !ef + ocet), sin el = el - ief + Ocel); therefore,
cosel = 1 - !y'2 + O(E 4 ),
sine) = y' - u'y' - !y'3 + O(E 5 ),t
(8.228b)
where it is understood that X has been replaced by s in the derivatives. Hence, returning
to V, we can write
V(X ) ' ,).,2 U I' I U '3 ., /" ( 5 )
,t =y+UjY - 2: YY - fUY - 2: fY -xy +V E .
(8.229)
t This fonn can also be used for extensible cylinders. For an inextensible centreline, this and other expressions
are eventually simplified considerably; thus, since u' = -! y'2, sin lil = y' + 0(10 5 ).

976
SLENDER STRUCTURES AND AXIAL FLOW
Next, the velocity Uf needs to be related to U, the flow velocity relative to the unde-
formed cylinder. Considering the three-dimensional velocity potential 4>, we may write
Uf = a4>/ax and U = a4>/ax. Then, since a4>/ax = (a4>/ax)/(ax/ax), using the
expression for ax/ax obtained previously we have
( au ) 4
Uf = U 1 - ax + O(E ).
(8.230)
Hence, equation (8.229) becomes
V (X ) . u , 1, 12 2U 1 1 1 f.3 'I /'1 5
,t = Y + Y - 2:YY - u Y - 2:Uy - xy + v(E ).
(8.231 )
The velocity potential may be expressed as
4>(X, Y, Z, T) = ux +4>o(X, Y, Z) + 4>1 (X, Y, Z, T),
(8.232)
where U X is due to the mean flow, 4>0 to variations in the body cross-section (here nane),
and 4>1 to motion of the body. This potential must satisfy a number of conditions: (i) the
fluid velocity normal to the outer channel is zero; (ii) the fluid does not penetrate the
cylinder; (iii) the solution must be 2Jf-periodic around the cylinder, and be even with
respect to Z.
The pressure is determined via the Bernoulli equation,
a4> 1 2 1 2
P - Poo = -p- - -p(V4» + -pU .
at 2 2
(8.233)
After considerable manipulation, outlined in Appendix T.l, one eventually obtains the
pressure distribution on a cross-section of the cylinder,
P - Poo = -p f {  + [ U ( 1-  ) - (  + U )]  } 4>1
aT ax aT ax ax
 ( a4>1 ) 2 _  a4>l a4>l } O ( E5.
+ 2 ax ax ay ax + )
(8.234)
Hence, integrating over the circumference, we obtain the lateral force (lift) per unit length
in the Y -direction:
FA(X T)=-pJ {[ + ( U ( I- ) - ( +U ))  ] 4>l
, fs x aT ax aT ax ax
( ) 2 }
1 a4>1 a v a4>1 a4>1 5
+- - ---- (-dZ)+O(E).
2 ax ax ay ax
(8.235)

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
977
Then, after determining </JI subject to the boundary conditions given below (8.232), the
nonlinear expression of the lift is found in Appendix T.l to be
FA (X, T)
 -pA {[ ( a + (U (1 - :; ) - ( :; + U :; ) ) a: )
x (xv-x G; :; +2U :; :; )-xv G; )')] -XV :; : } +0(,')
(8.236)
where
R 2 + R 2
o
X = R2 _ R2
o
are virtual mas s coefficients.
We notice that, as Ro becomes large, X -+ 1 and X -+ 1, and in that case
1 (R5 + R2)(R5 - 2R2)
and X =
2 (R5 - R2)2
(8.237)
FA (X T)=- {  + [ U ( 1-  ) - (  + U )]  }
, oT OX oT OX OX
X [ V- ( +2U ) - V (  ) 2 ] M+ MV oV +O(E 5 )
oT OX oX OX 2 oX 2 OX OX
(8.238)
where V = (ovjoT) + U(ovjoX), and M = pA. It is noted that the linearized version
of this expression is identical with equation (8.2), derived here more rigorously than
heretofore. For the purposes of this analysis, equation (8.238) is considered to be valid
even for cases where Ro is not so large, simply by replacing M = pA by M = XpA in
the equation,
8.10.3 Hydrostatic and frictional forces
The hydrostatic forces F px and Fpy, the resultants of the steady-state pressure p acting
on the cylinder, are derived by the procedure in Section 8.2.
Consider an element 8s of the cylinder as in Figure 8.101, momentarily frozen and
immersed in fluid on all sides. Hence, in addition to F px 8s and F py 8s, the resultants on
the normally wet surfaces, there are additional forces, pA and pA + [o(pA)jos]8s on
the two flat, normally dry, surfaces of the element. The net resultant of all these forces is
known: it is the buoyancy force. The pressure is assumed to be of the form p (x) = a +bx -
which covers both purely hydrostatic and pressure-drop-modified pressure distributions.
Consequently, one may write
[ -F px - :s (pA cose d ] 8si + [F py - :s (pA sined] 8sj
= - j}pndA = - fffVpd(vol) = (- : i) A8s,
(8.239)

978
SLENDER STRUCTURES AND AXIAL FLOW
where n is the outwards pointing norma!. Then, using the relations linking 8j8x, 8j8X
and 8j8s, the fact that 8Aj8X = O, and referring to the inset diagram in Figure 8,100,
one obtains t
8p 2 8 8p 4
-F px = -A cos 01 + pA-(COS01) - -A + O(E ),
8x 8X 8x
8p . 8. 5
F py = ;-ACOSOl smOl + pA-(smOl) + O(E ).
uX 8X
Next, using expressions (8.228b), we obtain
-F px = _yf2 A(8pj8x) - y'y" Ap + O(E 4 ),
F py = (y' - u'y' - y/3) A(8pj8x)
+ (y'1 - u"y' - u'y" - y'2y") Ap + O(E 5 );
(8,240)
it is recalled that 0 ' = 80j8s == 80j8X ¥= 80j8x.
Furthermore, by assu11Ůng the lateral movement of the cylinder to have a negligible effect
on the axial pressure distribution in the fluid at large, one can write (Pai'doussis 1973a)
8p 1 2 D
A- = --pDU CT- + pgA,
8x 2 Dh
(8.241 )
where Dh is the hydraulic diameter, and CT is a friction coefficient - cf. equations
(8.244). Re-writing the derivative with respect to X via one of the relationships of the last
footnote and integrating from X = s to X = L, we obtain
Ap(s) = Ap(L) + (PDU2CT  - pgA) [(L - s) -l L y'2dS] + O(E 4 ),
(8.242)
an expression used in the last steps of the formulation of the equation of motion.
Finally, introducing these expressions into equations (8.240), we obtain
-F px = yf2 ( PDU2CT - pgA ) - y'y"Ap + O(E 4 ),
2 Dh
I II 13 ( 12 D )
F py = (y - u y - y ) -'2 pDU CT Dh + pgA
+ (y'1 - U"y' - u'y" - yI2yll) Ap + O(E 5 ).
(8.243)
tThe following re1ationships are recalled: (i) x = X + u, thus ax/ax = 1 + au/a X; (ii) ax/as =
1/(1 + e), e being the centre1ine extension; here (l + e) = [(l + au/aX)2 + (av/aX)2f/2; (iii) ax/as =
(ax/aX)(ax/as) = (1 + u')/(l + e) = cOSOl' For an inextensib1e centreline, e = 0, ax/as = 1, ax/ax :::
a a(pA) ax a ap
1 - 1y12. Hence, for examp1e, -(pA cos O) = -cosO) + pA--(cosOI) = -Acos 2 0) +
as as as ax ax
a
pA-(cOSOl)'
ax

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
979
Next, we proceed with the formulation of the viscous, frictional forces, on the basis
of the semi-empirical expressions proposed by Taylor (1952a). These are considered to
be adequate, unless confinement of the ftow by the channel is very severe - in which
case, they could be obtained from the unsteady pressures derived analyticaIly by Mateescu
et al. (1994a,b), for instance, but unfortunately not in do sed formo An altemative is to use
the semi-empirical data, but which again are not available in easily usable dosed form,
compiled in Pai"doussis (1998); however, these can be fitted in the framework of the Taylor
formulation, and hence we proceed with thal.
The expressions proposed by Taylor are
FN = !pDU 2 (CN sin i + CDp sin 2 i),
FL = !pDU 2 CT cosi,
(8.244)
where CN and CT are fnction coefficients and CDp a form-drag coefficient; i is the
angle of attack. In some of the analysis, distinct CN and CT are used; frequently,
however, the simpIified form CN = CT = Cf is tak:en instead, FN and FL act in
the -jI and il direction, respectively. Expressing i = &1 + &2, we note that we
have already expressions for &1 in equations (8.228a,b). Proceeding similarly, we have
(h = tan-I{(oy/ot)/[U f - (ox/ot)]} - see Figure 8.102 - and hence
y xy 1 y3 5
&2 = U f + U2 - 3" U 3 + O(E ).
f f
(8.245)
We can, therefore, find i, cos i and sin i, as follows:
, 1 Y 1 1 xy 1 ( 13 y3 ) O 5 )
I =y +--uy +--- y +- + (E ,
U f U2 3 U 3
f f
1 ( y' Y y2 )
cos i = I - - y,2 + 2- + 2 + O(é),
2 Uf Uf
. ,. 1 ( '3 ,2' "2 )
., 1 Y "xy 13 Y Y Y Y Y 5
SIn I = Y + - - u y + - - - y + - + - + - + O(E ).
U f U2 2 U 3 U f U2
fff
(8.246)
Substituting these in (8.244), and relating Uf to U through (8.230), we obtain
[ ( , " . . 1 ( '3 12' 1 '2 ))
1 2 , Y yu 1 1 xy 13 Y Y Y Y Y
FN = 2 PDU CN Y + U + Ti - u y + U2 - 2" y + U3 + U + U2
+ CDp (y'2 + 2 Y + t: )] + O(E 5 ), (8,247)
1 2 [ 1 ( '2 y'y i )] O 4
h = 2 PDU CT 1 - 2" y + 2ij + U2 + (E),
The quadratic terms in the expression for F N need to be modified in order to obtain forces
which are odd with respect to y' and y, thus forces aIways opposing motion. TriantafyIlou
& Chryssostomidis (1989) have shown that terms of the form [y' + (y / U)]2, which are

980
SLENDER STRUCTURES AND AXIAL FLOW
found in the normal viscous force, could be written as [y' + (y j U)] I (y' + (y j U» I directly.
ln the same spirit, expressing y'2 as y' I y' I. 5'2 as y I 5' I, and so on in F N, we obtain
[ ( . . I ' , 1 ( '3 12' "2 ))
FN = lpDU 2 CN y' +!.... + yu _ u'y' + xy _ _ y,3 +  + y y + y y
2 U U U2 2 U3 U U2
+ CDp (y'Iy'1 + y'lyl  1y'ly + yYzI )] + 0«('5). (8.248)
On the other hand, the longitudinal force is even with respect to y' and y, and hence no
such modification is necessary.
8.10.4 The equation of motion for inextensible centreline
The virtual work associated with the fluid-dynamic forces may be expres sed as
1 12 1 12 1 L
8W dt = {[-F px + FL COSel + (FA + FN) sined 8x
  o
+ [F py + h sine l - (FA + FN) cosed8y} dX dto
(8.249)
Substituting in the above equation the expressions for FA, F px etc. derived in Sections
8.10.2 and 8,10.3, and utilizing equations (8.225a-c) and (8,228a,b), one obtains with the
aid of equation (8.224) and expressions (8.227a,b) the nonlinear equation of motion:
(m + M)y + 2MU,Y' (1 + y'2) + MU 2 y" (1 + y'2) - Myy'(Y' + Uy")
+ tpDU 2 CN (y' + b'3) - tpDU 2 CT(L - s) (y" + h'2yll)
- Ap(L)(y" + y'2y") + [tPDU2CT(DjDh) +mg - pgAJ
x [y' + ty,3 - (L - s) (y" + y'2yll) ]
+ EI(y'''' + 4y'y"y'" + y"3 + y""y'2) - tpDCNY l s y',Y' ds
1 2 ( Y 1 y' ý2 1 y,2 Y 1 y3 )
+ zPDU CN U - 2: U2 - 2:ij - 2: U3
+ 1 DU 2 C ( ' 1 ' 1 + y'lyl + yly'1 + 5'IYI )
2 P Dp y Y U U2
- myli 1 L l s (,Y'2 + y'Y')ds ds + 2M(,Y' + Uy") l s y'Y' ds
- Myli 1 L (yy' + 2U,Y'y' + U 2 y"y')ds + (m + M)y' l s (y2 + y'y')ds

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
981
+ y" 1 L {AP(L)y'y" + PDCTy2} ds
+ pDU2y"(CT - CN) 1 L (y'2 + Yj ) ds + O(E 5 ) = O.
(8.250)
ln this equation, it is realized that, unless there is a drogue at the free end, teL) would
norma1ly arise from a base drag, in which case Ap(L) may be expressed as '2PD2U2Cb,
where Cb is the base drag coefficient. Defining next the dimensionless quantities
s
 =-,
L
Y '[' = ( E 1 ) 1/2 ,
1) = I,' m + pA L2
(m - pA)gL 3
y = E1
L D 4
8 = D ' h = Dh ' Cb = ;Cb.
( PA ) 1 / 2
U = E1 UL,
4 4
CN = -CNy CT = -CT,
Jí Jí
(8.251)
pA
fJ = m + pA '
4
Cd = -CDp,
Jí
one obtains the dimensionless equation of motion:
[1 + (X -1)fJ] ij + 2UJI3x' (I + 1)'2) +U 2 X1)" (I + 1)'2)
X 1)' (fJ' +uJI31)") + U28CN [1)' + h,3] -  U 2 8cT(1-) (1)" + 1)'21)")
Cb U 2 (1)" + 1)121)") + ( U28GT h + y) [1)' +  1)'3 - (1 -l;) (1)" + 1)'21)1I)]
+ 1)111' + 41)'1)"1)111 + 1)"3 + 1)111'1)'2 - 8CN fJ i 1)'' d
+ 1 U 2 ( "ffJ , 1 fJ '2' 1 "ffJ,,2 1 fJ3/2 '3 )
'2 8CN U1) - '2 U2 1) 1) - '2U1)1) - '2 U3 1)
+ U28Cd (1)' 11)'1 + '{: ( 11)'1 + 1)'II) + t2  II)
-1)"(1 - /3) i1i ('2 + 1)'W) dl;d + 2X (/3' + U J131) ,,) i 1)''dl;
- X 1)" i 1 (fJij1)' + 2U J13'1)' + U 2 1)"1)') d + 1)' [I + (X -I) /3] i ('2 + 1)' W) d
+ 1)" i 1 { Cb U 2 1)'1)" + 8CT fJ2 } d
+ U21)" (8GT - 8CN) i l (1)'2 + '{: 1)' ŤJ) dl; + O(E 5 ) = O, (8.252)
where ()' = a()/a. (') = a()/a..

982
SLENDER STRUCTURES AND AXIAL FLOW
The linearized version of this equation is identical to equation (8.27b) for unconfined
flow (X = 1, h = O), for zero dissipation, o = O for an unrestrained free end, and c = o;t
and to (8.27a) for confined flow, if additionally CN = CT = cf. It should be mentioned
that damping due to dissipation is quite small relative to the flow-induced damping in
cantilevered cylinders. For other boundary conditions, however, it may be important to
consider it by replacing 1 in the bracketed factor of ff by [1 + a(a/ar)] as in (8.27b).
The equations of motion of the system if inextensibility of the centreline is not invoked
are given in Section 8.10.8 and Appendix T,3.
8.10.5 Boundary conditions
It is supposed that at its free end the cylinder is terminated by a short, ogival end, the
cross-sectional area of which varies smoothly from A to zero in a distance I « L. Further,
this ogival end is assumed to be rigid, so that its motion is determined solely by the values
of displacement and velocity at s = L -1; y, y', V and oy are all constant with s. The
boundary conditions are derived in Appendix T,2 to first order, that is, correct to O(E).
It is found that the boundary conditions at s = L - 1 are
- Ely'" + [fM(y + Uj/) + my]se - fMU(y + Uy') + (m - pA)gy'se
+ pDU2CThy'se + pDUCN(Y + Uy')se = O, (8.253)
y" = O,
where
l i L
Se = - A(s)ds,
A L-I
l i L
Se = - D(s)ds,
DL-I
(8.254)
and where it is understood that A = AIL-I, D = DIL-I, y' = y'IL-I, Y = yIL-I,
h = D / Dh, M = xpA. Typical values for Se and Se, by way of illustration, are as
follows, For a conical end, D(s) = (D/1)(L - s), Se = 11, Se = 1; for a para-
boloidal end, D(s) = (D/0) .y'L -s , Se = l,se = 1; for an ellipsoidal end,
D(s) = D {1 - [(s - L + 1)2/ 12J) 1/2, se = 1 and se = 1. The boundary conditions
at s = O are, of course, y(O) = O, y' (O) = O.
Other than the approximate empirical correlation between end-shape and f provided
by Figures 8.23 and 8,24, there have been two attempts to determine f quantitatively. For
conical and conoidal ends, Hannoyer & Paidoussis (1978) have proposed f = 4s; /[4s;+
(Do - Di)2] = 4€2X; /[4€2 X ; + (1 - ;5)2], where"8 = Di! Do in cases where the
cylinder is hollow and conveys fluid intemally also, Di and Do being the inner and outer
diameters of the cylinder; here "8 = O. A more elaborate method has been developed by
Paidoussis & Yu (1976) for truncated ellipsoidal ends, which however does not provide
an exp1icit expression for f. In the first case it is presumed that separation does not occur
at all, whi1e in the second that it occurs exactly at the location where the ellipsoid is
truncated,
tUnearization causes the terms involving Cd in (8,252) to vanish, contrary to the heuristic introduction of
an equivalent C introduced via (8,13) in the linear equations.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
983
Finally, equations (8.253) may be written in dimensionless form as follows:
-1]'" + Xe [(1 + (xl - I)) ij + xIUnlj'] + ( !xeecN - xl) (Unlj + U 2 1]')
+ ( !U 2 ecTh + Y) Xe1]' = 1]" = O at  = I, (8.255)
where
Xe = sel L ,
Xe = sel L.
(8.256)
8.10.6 Methods of ana.ysis
The nonlinear equation of motion (8.252) of the cantilevered cylinder is of third-order
magnitude; furthermore, the boundary conditions (8,255) are time- and flow-velocity-
dependent. This renders the problem non-standard and the solution procedure more
complicated. As a first simplification, the partial differential equations (8.252) and (8.255)
are transformed into a set of second-order ordinary differential equations using Galerkin's
method. However, since the boundary conditions are time-dependent, different solution
approaches are possible, as discussed in Lopes et a/. (1999b).t
Let us write for simplicity the equation of motion in the form F (1], U) = O. Then, with
the boundary conditions added, the boundary value problem may be formulated as
F(1](, r), U) = O,
1](0, r) = 1]' (0, r) = 0,
1]"(1, r) = -1]"'(1, r) + B(1](1, r),U) = 0,
(8.257)
(8.258)
where B (1], U) represents a complementary term in the end-shear boundary condition due
to the tapering end. An altemative way of formulating the problem is the following:
F(1](, r), U) + 8( - l)B(1](, r), U) = O,
(8.259)
1](0, r) = 1]'(0, r) = 0,
1]"(1, r) = 1]"'(1, r) = 0,
(8.260)
where 8( - 1) is the Dirac delta function. With these two formulations in mind, three
methods may be proposed to discretize the system, as follows.
Method (a) consists of utilizing the eigenfunctions <I> j () of the problem 1]"" + ij = O,
that is, the dry cantilevered-cylinder equation of motion, subject to boundary conditions
(8.258) to discretize the system and apply them to the problem (8.257). In Method (b),
the same eigenfunctions <I> j () are used, but they are applied to an 'expanded domain'
of the problem, which effectively means that the time-dependent boundary condition, the
last of (8.258), is added to the equation of motion, that is, the expression [-1]'" (1, r) +
B(1](1, r), U)] is added to the left-hand side of (8.257) via a Dirac delta function. Finally,
in Method (c), the cantilever beam eigenfunctions 4> j () satisfying (8,260) are used directly
to discretize equation (8.259).
tOf course, an altemative would be to use the Rayleigh-Ritz method,

984
SLENDER STRUCTURES AND AXIAL FLOW
Jt has been shown by Lopes et al.. (1999b) that Method (b), although requiring only
a small number of modes to yield extremely accurate results, is difficult to implement
and is numerically very time-consuming, especially for a nonlinear problem. On the other
hand, Method (c) is easy to implement and leads to more accurate results than Method (a),
provided enough tenns are used - cf. PaIdoussis (1998, Section 4.6.2). Hence, this is the
method that is presented here.
The infinite-dimensional model is discretized by Galerkin's technique with the canti-
lever beam eigenfunctions, <P j (), used as a suitable set of base functions and with q j Cr)
as the corresponding generalized coordinates; thus,
N
TJ(, T) = L <Pj()qj(T),
j=1
(8.261)
where N represents the number of modes. Substituting expression (8.261) into (8.252),
multiplying by <Pi () and integrating from O to 1, leads to the following matrix fonn:
Mi} iiJ + Ci} qj + Ki} qj + Tijk qjlqkl + Sijklqjl tlk + Si}k qj Iqkl + ti}k qj Itlkl
+ CXijkl qj qk ql + f3ijkl qj qktll + Yijkl qj tlkql + TJijkl tlj qktll + JLijkl qj qkql = O;
(8.262)
Mi}, Ci} and Ki} correspond to the mass, damping and stiffness matrices,
respectively, while CXijkl, f3ijkl, Yijkl, TJijkl, JLijkl, Tijk, Sijt. Si}k, tijk are related to the
nonlinear tenns.
The mass, damping and stiffness matrices are defined by
Mi} = [1 + (xl - 1)f3]Xe<Pi (1)<pj (1) + [1 + (X - 1)f3]8i),
Ci} = ( !xeecN - xl) U/'fi<Pi(1)<pj (1) + x IU/'fixe <Pi (1)<pj(1) + 2x U /'fibi}
+ !UecN/'fi8i} ,
Ki} = (YXe + !U 2 (ecNX e + eCThXe) - x1U 2 ) <Pi(1)<pj (1) + XU 2 Ci}
+ ( !U 2 ecT(1 + h) + Y) (di) - ci})
(8.263)
+ ( !U 2 e(cN + cTh) + Y) bij + A.J8i} - ! U2C b C i},
where the constants, bij, Ci}, di}, defined in Section 3.3.6(b), are
bij = 11 <Pi<Pj d,
Cij = 11 <Pi<P} d,
di} = 11 <Pi<P} d.
(8.264)

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW 985
Furthermore the nonlinear coefficients in (8,262) are defined by
5 2 fl "" [ 1 2 ] 1 fl ",
CXijkl = "2 xU 10 <Pi<Pj<Pk<P1 d + "2 U e(CN + cTh) + y "210 <Pi<Pj<Pk<P1 d
- [ U2ecT(1 + h) + y]  li (1 - )<Pi<P}<Pk<P;' d + 41 1 <Pi<P}<P<P;" d
+ fl <Pi<P"<P<P;' d + fl <Pi<P""<Pk<P; d - xu 2 fl <Pi<P' ( fl <Pk<Pí' d ) d
10 ] 10 ] 10 J 11;
+ U2Cb J.' ",,,j (1.' ",,,f'  )  - U2Cb J.' Mj",,,;, 
+ U2e (CT - CN) J.' Mj (1.' ",,,; d) d,
ft'j"  XUJP { H I Mj",,,; d - Hl Mj","I  - 2 J.' ",,,j (1.' ",,,;  ) d
+ 2 J.' Mj (fo' ",,,f d) } - UJPecN J.' Mj",,,/ d,
+ U JP*T - CN) J.' ",,,j (1.' ","I d, ) d" (8.265)
YIj"  -  xft l' ",,,j"k"; d - (1 - ft) J.' ",,,j (tfo' ",,,;  d ) d
+ [1 + (X - 1),8] li <Pi<P) (11; <Pk<P; d ) d + 2x,8 li <Pi<Pk (11; <p}<p; d ) d
-  ,8ecN li <Pi<Pk (11; <p}<pí d ) d - l f3UN li <Pi<P}<Pk<P1 d
+ ftecr l' ",,,j (1.' Ml d ) d,
1 f33/2ecN f I
'f/ijkl = - :4 U 10 <Pi<Pj<Pk<P1 d,
I"j kI  - (1 - ft) l' "I"j (1.' J.' ",,,;   ) d, - X ft J.' Mj (1.' ","I d ) d
+ [1 + (X - 1),8] li <Pi<P) (11; <Pk<P; d ) d,

986
SLENDER STRUCTURES AND AXIAL FLOW
1 U 2 t I I - 1 I7i t I
rijk ="2 £Cd 10 4>i4>jl4>kld, Sijk = "2 Uvf3 £Cd 10 4>il4>jl4>kd,
_ 1 u I7i {I I
Sijk ="2 vf3£Cd 10 4>i4>jl4>kld,
1 i I
tijk = -f3£Cd 4>i4>jl4>kl d.
2 o
8.10.7 Discussion: the equations used in Section 8.11.1
It is noted that, in an ear1ier derivation (Lopes et a/. 1999a), the expressions for the
hydrostatic forces F px and F py are slightly different, name1y
1 12 ( 1 2 D ) '" 4
-F px = -y -pDU Cr- - pgA - y y Ap + O(E ),
2 2 Dh
( I I I 1 13 ) ( 1 2 D )
F py = y - 2u y -"2 Y -"2 PDU Cr Dh + pgA
+ (y'I - u"y' - 2u ' y" - yl2y") Ap + 0(E 5 )
instead of (8.243), having been obtained with the simp1ifying assumption that 8pj8x =
8 p j 8 X, which is correct to first order, but strictly incorrect to third order. The final equation
of motion is also slightly different from (8.252), as a resu1t. The differences re1ate to
terms invo1ving the gravity parameter y (which had to be split into two components, one
associated with mg and the other with pgA) and the confinement parameter h (Lopes
et al. 2002, appendix),
The calcu1ations presented in Section 8.11.1 have been done with this ear1ier ver-
si on of the equation of motion. However, in these calcu1ations, h = O throughout,
and y = O or 1.9. When y = h = O, the two versions of the theory are identica1.
The differences in the dynamics when y is as smal1 as 1.9 are minima!. t Hence,
a1though some smal1 quantitative differences must exist between resu1ts presented and
those obtainable with equation (8,252), the results are expected to remain qualitatively
unchanged,
8.10.8 Equations for extensible centreline
If the inextensibi1ity condition is not invoked - either to obtain a more accurate formu-
1ation for cantilevered cy1inders, or because it does not app1y, as in the case of cy1inders
with supported ends - the equation of motions is different (cf. Chapter 5). In this
case, the deformation cannot be defined in terms of one variable (y); two are neces-
sary, namely u and v, defined in Figure 8.100. The derivation is outlined in Appendix T.3.
tOifferences begin to be appreciable for y  0(10),

SOLlTARY CYLlNDRICAL STRUCTURES IN AXIAL FLOW 987
The dimensionless form of these equations is
(1 - f3)i1- X (f3iiv' + 2Uj{jiJ'v' +U 2 v"v') - A (u" + v'v") - v"v'" - v'v""
1 ( )U 2 ( ..fP., '2 ) 1 f3 .2
28 CN -CT UVV +v + 48CT v
+ 8CTU2(1 + h) [u' - !v,2 + (1 -  - !o) v'v"]
8CdU2V' [v'lv'l + Yf (v'liJl + iJlv'l) + t2 iJliJl] + [! U2c b(l - o)
+ ['o + (1 - 2v)ITo]v'v" + y[u' - !v,2 + (1 -  - !o)v'v"] + O(E 5 ) = O,
(8.266)
{I + (X - l)f3}ii + 2Uj{jXiJ' (1 - iv,2) +U 2 xv" (1 - v'2)
- xv'(f3i1 + 3uj{jů' + 2U 2 u") - X (4Uu' + 2j{jů + j{jiJv')(j{jiJ' + Uv")
- xf3iiv r2 + [ !U2cb(1 - o) + ['o + (l - 2v)ITo] (-v" + v'u" + v"u' + v'2v")
+ 8CdU2 [ v'lv'l + ..fP (v'liJl + iJlv'l) + L iJliJl ] - A ( v'u" + v"u' + J. v ,2 v " )
2 U U2 2
+ v"" - (8v' v" v", + v' u"" + 2 v,2 v"" + 2 v,,3 + 2u' v"" + 4u" v'" + 3u'" v'')
+ !8CTU 2 (l + h) {- !v,3 + [u - u(1)(l - o)]v" + (1 -  - !o) (-v" + v'u"
+ v"u' + vr2v")} - 8(cN - CT)U 2 [v'3 + v'u' +  iJvr2 +  t2 iJ 2 v']
1 ( h)U 2' 1 U 2 [ "fP, "fP,. f3. . 1 f33/2 '3 ]
+ -8 CN + CT v + -8CN -v + -u v + -uv - --v
2 2 U U U2 2 U3
+ y {v' - !v,3 + [u - u(l)(l - o)]v" + (1-  - !o) (-v" + v'u" + v"u'
+ v'2v") } + O(E 5 ) = O,
(8.267)
where primes denote derivatives with respect to  = X / L and dots with respect to ..
To obtain these equations, in addition to the dimensionless parameters in (8.252); the
following are also necessary: the dimensionless extemally imposed uniform tension [',

988
SLENDER STRUCTURES AND AXIAL FLOW
pressurization n, and axial flexibility A, defined by
- 2
r=TL/EI,
n =pAL 2 /EI,
A = EAL 2 /EI;
(8.268a)
moreover, for simplicity, the lateral and longitudinal displacements have simply been
replaced by
u -+ u/L, v -+ viL.
(8.268b)
For the cylinder fixed at both ends, it is obvious that the following boundary conditions
mu st be satisfied:
u(O) = v(O) = u(l) = v(l) = O.
ln addition, if the cylinder is simply supported and r = O, one obtains
(8.269a)
v"(O) = v"(l) = O,
(8.269b)
while for the clamped-clamped cylinder,
v'(O) = v'(l) = O.
(8.269c)
8.10.9 Concluding remarks
ln this section, a weakly nonlinear equation of motion, correctto 0«('3), has been derived
for the dynamics of a cantilevered cylinder in axial flow via variational methods; in a
consistent manner, linear boundary conditions have also been obtained for the case of the
cylinder being terminated by a rigid, ogival end. The equations of motion for a cylinder
with both ends supported have also been derived.
The altemative of treating the cylinder together with the tapering end(s) as a single
slender body of variable cross-section, which would offer an altemative means for dealing
with the variable-area end-effects, has not been done; it is nevertheless feasible and perhaps
ought to be done.
These equations are probably not the definitive nonlinear equations of motion for this
system since they have not been obtained by a unified nonlinear treatment of the fluid
mechanics. Nevertheless, they provide a reasonable and useful tool for the exploration
of the nonlinear dynamics of the system, which has hitherto been impossible. How suc-
cessfully do these equations capture the trne dynamics of the system has to be judged by
comparing theoretical predictions with experimental observations - see Section 8.11.
8.11 NONLlNEAR DYNAMICS
Interest in the nonlinear dynamics of cylinders in axial flow is fairly recent, and as a
result not too much work has been done as yet. Nevertheless, as observed in the experi-
ments [Section 8.3,2( c,d)], there are some interesting nonlinear features deserving further
study. In addition, of course, there are the usual questions of predicting the amplitude ať
divergence and flutter and the limit-cycle frequency, feasible only via nonlinear theory.
Work on the nonlinear dynamics ofaxia1ly moving, or travelling, beams and strings
is older, and a considerable number of publications exist, some already referenced in
Section 8,3,8. Some recent papers are by Behdinan & Tabarrok (1998), Behdinan et al.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
989
(1998), Pe1lieano & Ziri1li (1998), Fung et aZ. (1998), Chakraborty et aZ. (1999),
Pe1lieano & Vestroni (2000, 2002), Oz et aZ. (2001), Evangelinos et aZ. (2002) and Lueor
et aZ. (2002). Despite its fair1y e10se re1ationship to the subjeet at hand, none of this work
is diseussed here for brevity. In what follows some of the work on the nonlinear dynamies
of cylinders in axiaZ flow is presented.
8.11.1 Cantilevered cylinders in axial flow
8. 11. 1 (a) Theoretical results
The dynamies of this system has been studied by Paldoussis et aZ. (1999) and Semler
et aZ. (2002); see a1so Lopes et aZ. (1999b) and Augu (1999). The ea1eu1ations have been
done with an earlier versi on of the equation of motion than that presented in Seetion 8.10.4,
as diseussed in Seetion 8.10.7.
The method of ana1ysis has been outlined in Seetion 8.10.6. Solutions ofthe diseretized
equations are obtained either by a Finite Differenee Method (FDM) based on Houbo1ťs
method, diseussed in detail by Semler et aZ. (1996), or by AUTO (Doede1 & Keméves
1986). The first is an initia1-va1ue prob1em-so1ver: the equations of motion are integrated
numerieally for one initia1 eondition at a time, and the state of the system is obtained at
any time thereafter; henee, the fina1 steady state must be a physieally attainab1e one, that
is, a stab1e attraetor. AUTO, on the other hand, is a eontinuation prob1em-so1ver based
on a eolloeation method. It determines the stability of statie and/or periodie solutions, by
eomputing eigenva1ues and/or F10quet mu1tipliers of a known solution (Appendix F), and
then eontinues to do the same as a parameter is varied. This method, however, eannot
'follow' quasiperiodie or ehaotie solutions.
A typiea1 bifureation diagram is given in Figure 8.103, ln this and what follows, the
following parameters are used, so that the results may be direetly eompared to experiments:
f3 = 0.47, y = 1.9, e = 20.5, CN = CT = cf = 0.024, Xe = 6.67 x 10- 3 , Xe =
7.85 x 10- 3 (quasi-e1lipsoida1 free end), f = 1 - Cb = 0.7; U = 1.57U, w = 1.19Q;
the number of modes used in the Ga1erkin expansion is N = 6 for U < 9 and N = 8 for
higher U.
This diagram shows the variation of the amplitude of the first generalized eoordin-
ate, ql, with inereasing U; the behaviour of the tota1 system, Li: 1 CPi (l;)qi Cr), is
qualitative1y simi1ar. In this figure, a solution on the U-axis (ql = O) represents the
origina1 eonfiguration, that is, the inert eylinder hanging downwards in water flow. A
nonzero solution eou1d represent either a nontrivia1 statie equilibrium position (say, a
post-buekling solution) or the amplitude of a limit eyc1e (flutter). The solution eou1d be
stab1e (full, thiek lines) or unstab1e, and henee physieal1y unattainab1e (dotted or thin
lines).
Bifureations are determined mathematiea1ly by the eigenva1ues in the ease of a fixed
point (denoted by A., with A. = O or A. = :1:iw), and by the F10quet mu1tipliers in the ease of
a periodie solution (denoted by A, with A = + 1, -1 or eomp1ex eonjugate with I A I = 1).
A eomp1ete deseription may be found Appendix F. In order to simp1ify the diseussion, the
resultshavebeensubdividedintodifferentsubseetions: (i)U < 9, (ii) U > 9, (iii) U rv 8.6.
(i) The dynamics for U < 9. The dynamies of the system for U < 9 is diseussed first,
with the aid ofFigure 8.103.

990
""
<!)
 0.05

o
8 o
'"
<!)
.
 -0.05
<!)
;::
<!)
CI
-0.1
-0,15
-0,2
-0,25
O
0,25
0,2
0,15
0,1
SLENDER STRUCTURES AND AXIAL FLOW
lniti<il equilibrium
\ _ RestabilisatiÓn \
\u:klin (secnd mode forrn) , //
Buckling (first mode forrn) ..j
/i
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Dimensionless f10w velocity, ólt
Figure 8.103 Typical bifurcation diagram for tJ = 0.47, f = 0.7, y = 1.9, eCN = eCT =
0.5, Cb = 0,3, Xe = 6.67 x 10-3, Xe = 7.85 x 10- 3 , cd = O, X = 1 and h = O, showing the first
generalized coordinate, ql, as a function of U: -, stable solutions; . . . , -, unstable solutions
(Semler et al. 2002).
(i) The zero solution, corresponding to the original equi1ibrium state, is stable up to
U = 2,1. At this flow velocity, a supercritical pitchfork bifurcation corresponding
to one eigenvalue equal to zero (A = O) occurs, leading to divergence in the first
mode; ql subsequently increases with U.
(ii) For U > 3 approximately, ql begins to decrease, reaches zero, and then increases
again over a small range of U. This is because the first-mode divergence slowly
changes into one of second-mode shape with increasing flow. Subsequently, for
U > 5.25, the system is restabi1ized over a short range of flow velocities through
a subcritical pitchfork bifurcation (A = O at U = 5.25).
(iii) Almost immediately after, at U = 5.5, the trivial 'zero' solution loses stability
again through a supercritical Hopfbífurcation (A = ::i::iw) which leads to flutter in
the second mode, corresponding to periodic osci1lations around the origin.
(iv) Thís periodic solution becomes unstable through a torus bifurcation (two complex-
conjugate Floquet multipliers cross the unit circle, that is, A = a ::i:: ib with
IAI = 1) at U = 8.52 prior reachíng the x-axis at U = 8.97. The 'IAI = ľ
bifurcation usually corresponds to the appearance of a second frequency in the
periodic response, that is, to quasiperiodic solutions which cannot be 'followed'
by AUTO, Moreover, this unstable solution can also be seen as emerging from a
subcritical Hopf bifurcation of the origin at U = 8.97, i.e. with A = ::i::iw,

SOLlTARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
991
(v) Just before the aforementioned bifurcation, an unstable third-mode ftutter solution
arises from a subcritical Hopf bifurcation of the trivial solution at U = 8.94
[see also Figure 8.106(a)]. This unstable solution becomes stable at U = 8.44
through a saddle-node bifurcation (A = +1) which corresponds to a tuming or
limit point (Moon 1992), So, again, stable periodic solutions may be seen.
A few additional comments on the dynamica1 behaviour of the system may be useful.
(a) The dynamics, as described in the foregoing, is further il1ustrated in Figure 8.104,
where the shape of the cylinder is shown as a function of time for some
o (a)
O I
0.2
0.3
0.4
0,5
 O (b)
O) 0.1
'OJ
c
] 02
o
C> 03
J 04
J 05
O (c)
0,1
02
0,3
0.4
05
(d)
(e)
(f)
-0.3 -o 2 -O I O 0,1 0.2 0,3 -0,3 -0,2 -0,1 O 0.1 0.2 O 3
Dimensionless transverse displacement, IJ
Figure 8.104 Shape of the cylinder for various values of the dimensionless flow velocity, U,
and for the same parameters as in Figure 8,103, (a) First-mode divergence (buckling) at U = 3,
(b) second-mode (shape) buckling at U = 5, (c) second-mode flutter at U = 6, (d) third-mode
flutter at U = 9, (e) third-mode flutter around a buckled position (combination of buckling and
flutter leading to an asymmetric response) at U = 11,8, (f) fourth-mode flutter at U = 13.7, but
with f = 0.6 instead of f = 0.7. The last two figures have been computed with 12 modes (Semler
et al. 2002).

992
SLENDER STRUCTURES AND AXIAL FLOW
representative cases. For U = 3, the cylinder is not moving and its shape is
similar to that ofthe first eigenmode of a cantilevered beam [Figure 8. 104(a)] , For
U = 5, the amplitude is much smal1er and the shape, displaying an inflexion point,
is similar to the second eigenmode of a cantilevered beam [Figure 8.104(b)]. For
U = 6, the cylinder is oscillating, and the shape again corresponds to a second
beam eigenmode [Figure 8.104(c)]. Final1y, for U = 9, the cylinder is oscillating
with a high contribution of the third beam-mode [Figure 8.104(d)].
(f3) The transition between second- and third-mode flutter is discussed in detail, farther
in this section.
(y) From Figure 8.103, it is obvious that linear and nonlinear analyses yield different
results; this is the case because some bifurcations are subcritical, which means
that 'nonlinear solutions' exist for flow velocities below the bifurcation points
predicted by linear theory. Nevertheless, because the system is restabilized prior
the dynamic in stability, predictions by linear theory are in fact quite adequate for
flow velocities below U = 8, which explains why agreement between linear theory
and experiments is indeed 'good' [Section 8.3.2(d)].
(8) The sequence found, that is, {stable} -+ {divergence} -+ {stable} -+ {osci11ation
in the second mode} -+ {osci11ation in the third mode}, is essentially the same
as described in the experiments [Section 8.3,2(d,t)], which is very satisfactory; a
more complete discussion is presented in Section 8,l1.1(b),
(e) Of course, the results shown here are valid quantitatively only for the paramet-
ers chosen. Nevertheless, these parameters are representative of what has been
obtained over a much larger range (Lopes et al. 1 999b; Augu 1999).
(ii) The dynamics for U > 9. In this velocity range, the bifurcation diagram and hence the
dynamical behaviour become more complicated. Indeed, from Figure 8.103, it is possible
to see many different solution branches, most of them unstable. The third-mode ftutter
discussed previously becomes unstable at U = 9.57 through a pitchfork bifurcation (/\ =
+ 1) which breaks the symmetry of the response with respect to the X -axis. This gives rise
to two new solutions, vne being unstable and the other stable. Physically, corresponding
to the stable solution after U = 9,57, the cylinder oscillates around a new equilibrium,
which means that divergence and ftutter coexist in the response. This is similar to what has
been observed by Paidoussis & Semler (1993a) - see Section 5.7.3 - for the dynamics
of a cantilevered pipe with an intennediate spring support, conveying fluid, The new stable
solution also becomes unstable at U = 11,16, so that, apparently, no periodic solution
exists thereafter. However, for U > 9, the bifurcation diagram of Figure 8.103 has been
obtained with N = 8 (eight mode approximation); from a computational point of view,
this has been the maximum that could be used in conjunction with AUTO. Hence, because
'fourth-mode' solutions have been observed by Pai'doussis et al. (2002), and because a
thorough study ofFigure 8.10 suggests that it might be possible to obtain them numerical1y,
FDM was used to integrate directly the equations of motion with N = 12 modes. Some
sample phase plane plots are given in Figure 8.105 and some additional cylinder shapes
in Figure 8.104(e) and (f),
At U = 11,8, the response in Figure 8.1 05(b) is periodic but asymmetric, in contrast to
the case of U = 6 shown in Figure 8.1 05(a). Later, a period-doubling bifurcation occurs
(A = -1) and a double loop can be observed [Figure 8.105(c)]. This new stable solution
again loses stability through another period-doubling bifurcation, leading to 'period-four'

993
SOLI'J.ARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
10
8 (a)
6
4
2

, O
-2
-4
-6
-8
-IO 4 -O 3 -0.2 -0.1 O 0.1 0.2 0,3 0.4
1](1)
, O
-I
-2
-3
-0,1
3
(e)
2
4
3
2
(b)
3
(d)
2
, O , O
-I
-I
-2
-3 -2
-4 -3
-0.1 -O 05 O 0.05 0,1 -0,1 -0,06 -0,02 O 002 0.06 0.1
ql ql
4 3
3 (c) (f)
2
2
, O , O
-I -I
-2
-3 -2
-4 -3
-0.1 -0,05 O 0,05 0,1 0,15 -0.1 -0.06 -0.02 O 0.02 0,06 0,1
ql ql
Figure 8.105 Phase plane plots for various values of U demonstrating the transition between third-
and fourth-mode flutter, for the same parameters as in Figure 8.103. (a) 'Symmetric' response
for U = 6 [corresponds to Figure 8.104(c)]; (b) 'asymmetric' response for U = 11.8 [see
Figure 8.104(e)]; (c) period-two oscillations for U = 12.2; (d) period-fouroscillations for U = 12.3;
(e) period-two oscillations with a higher frequency for U = 12.4; (f) fourth-mode flutter ('period
one') for U = 12,8; (b-f) show clearly the 'period-doubling' phenomenon (Semler et al. 2002).
oscillations, as illustrated in Figure 8.105(d) for U = 12.3. However, rather than devel-
oping the well-known period-doubling cascade (Pai'doussis & Semler 1993a; Pa'idoussis
1998, Section 5.8.1), here a 'period-bubbling' occurs instead. This leads back to 'period-
two' [Figure 8.105(e)] and to 'period-one' oscillations [Figure 8.105(£)]. Hence, through
this sequence of bifurcations, no quasiperiodic/chaotic motion could be found.

994
SLENDER STRUCTURES AND AXIAL FLOW
At higher flows, fourth-mode oscillations could be found (e.g. for f = 0.5 and U =
14.5); the transition from third to fourth mode is supposed to fol1ow the same pattem as
described for the second-to-third mode transition in what follows.
(žii) The dynamics for U '" 8.6. Finally, we retum to the discussion of the dynamics
in the vicinity of U = 8.6, where 'atypicaľ behaviour can be observed, because of the
presence of different stable attractors. For clarity, a zoomed version of the bifurcation
diagram of Figure 8.103 is given in Figure 8.106(a): not only are the different periodic
solutions clearer, but it is also possible to see that there is a small region for which both
the second-mode and the third-mode ftutter solutions are stable. Figure 8.106(a) has been
obtained by AUTO, which is not able to take the initial conditions into account. Hence,
FDM has been used to see if these two solutions could be found. This, indeed, is 80:
the two possible attractors for U = 8.50 are displayed in Figure 8.106(b), obtained with
different initial conditions, which confirms the accuracy of both numerical solutions.
As already mentioned, at U = 8.52, the second-mode ftutter becomes unstable through
a torus bifurcation (A = a :::I:: ib with IAI = 1). Thus, in theory, quasiperiodic solutions
are possible. This is illustrated in Figure 8,106(c), where again two different attractors
are displayed (for two sets of initial conditions), for U = 8.53, that is, just after the
bifurcation, The aperiodic solution is sensitive to the initial conditions and is mainly
composed of two unstable solutions. It is definitely proved through power spectral density
calculations, but mainly through Poincaré sections such as shown in Figure 8.106(d),
that the response converges either to the third-mode limit cycle or to the 'nonperiodic'
attractor.
This transition has been observed near one of regions B ofFigure 8.10, but similar beha-
viour has been observed for different parameters in the C regions (a B region corresponds
to the occurrence ofthe third-mode solution before the disappearance ofthe second-mode
solution, while in regions C the two Hopfbifurcations of the original equilibrium, i.e. from
a linear point of view, disappear completely). For example, the C region exists for f3 = 0.6
and f = 0,8 for which a bifurcation diagram is shown in Figure 8.107: even though no
solution emerges from the original equilibrium, the dynamics is still very similar. From
the results obtained using AUTO (shown in the figure) and FDM (not shown), the follow-
ing sequence has been observed: periodic solution at U = 8.50, quasiperiodic solution at
U = 8.67 (corresponding to the discontinuity in the curve), followed by periodic motions
atU = 8.70, and then third-mode ftutter atU :::: 9.0. Moreover, the quasiperiodic solution
is composed of the two flutter modes, even though no limit cycle exists concurrently.
Consequently, regions B and C are qualitatively similar since the different transitions
are the same.
(iv) The linear stability map (Figure 8.10) revisited. In the foregoing, the nonlinear
dynamical behaviour has been described and it is now fully understood. With this under-
standing, most of the dynamical behaviour can 'easily' be extracted from Figure 8.10:
the regions where the cylinder is stable, where it 'buckles', where it flutters in the second
or the third mode, and where exist regions of quasiperiodicity. Consequently, the same
logic may be used to predict the behaviour of the system for U > 13 (approximatel y, and
for the values of f of interest) by presuming the existence of some fourth-mode solu-
tions following the third-mode oscillations. Depending on the value of f, it is possible to
assume that two types of transition leading to a fourth-mode periodic solution are possible:
from a stable equilibrium (for f = 0.5), or from quasiperiodic solutions (for f = 0.7

0,1
0,08
0,06
.;
0,04
0.02
O (a) . . mu ..{\ 
6.5 7 7,5 8 85 9 95 10
11M
4
3
2
,.; O
-I
-2
-3
(c)
-4
-O 1 -0.06 O,02 O 0,02 0,06
qj
4
2
..; O
-2
(b)
-4 -0.08 -0.04
O
qj
0,04 0,08
2
,.; O
r-'''''",
F '-' \ .)
r; -.J.., ,...
'I  "-': \
" .
('" 'i - .__.::=:':::. ..)
"'-. -..,.-1
-I
(d)
-2
0.1 -0.1 O,06 O,02 O 0,02 0,06 O 1
qj
Figure 8.106 Dynamics in the transition region, U  8.6. (a) Zoomed version of Figure 8.103 in
the region of interest, obtained with AUTO; (b) phase-plane plot for two initial conditions showing
the coexistence of the two periodic solutions, at U = 8.50; (c) phase-plane plot showing the
periodic and the non-periodic attractors, at U = 8.53; (d) Poincaré section representing the trace of
the quasiperiodic attractor (Semler et al. 2002),
0.12
('
r

7,0
8.0
---........0
9,5
8,5 9,0
11M
10,0 10.5
0,10
0.08
..; 0.06
0,04
0.02
O
7,5
Figure 8.107 Bifurcation diagram in region C ofFigure 8.10 for which the two Hopfbifurcations
do not exist, in the neighbourhood of the 'transition zone', showing the first generalized coordinate,
ql, as a function of U: - , stable solutions; . . . , unstable solutions; f3 = 0.60, f = 0.8, y = O,
eCN = eCT = 0.5, Cb = 1 - f = 0.2, Xe = 0.00667, Xe = 0.00785, Cd = O, X = I and
h = O, N = 6 (Semler et al. 2002).

996
SLENDER STRUCTURES AND AXIAL FLOW
through region B where third- and fourth-mode solutions coexist, or for f = 0.75 through
region C where the two Hopf bifurcations do not exist). To verify numerically that these
assumptions are valid, a high enough number of modes needs to be used in the compu-
tation: 13 modes are necessary to find the 'approťriate' solution in regions B and C for
12 < U < 14. Nevertheless, using 12 modesonly, forU = 13.7, f = 0.6and,B = 0.47,
it was possible to find a 'fourth-mode' solution, as predicted.
This leads to the conc1usion that the bifurcation diagram in Figure 8.103 is not correct
for U > 11. The correct behaviour is the following: third-mode fiutter followed by
period-bubbling, quasiperiodic behaviour, and finally fourth-mode fiutter.
8. 11. 1 (b) Comparison with experiments
Some typical cy linder shapes, both theoretical and experimental, are presented side by side
in Figure 8.108, for different values of U and for the same parameters as in Figure 8.103.
The fiow velocity in the theory and in the experiment may have been slightly different;
also, y = O in theory, while y = 1.9 in the experiments (but this is too small to have a
significant effect).
Figure 8.108(a) and (b) represents the case of a first-mode-shape buckling. The theor-
etical point (at U = 2.8) has been chosen such that the amplitude is the largest, which is
not necessarily the case in the experiment. In general, the theoretical first-mode buckling
amplitudes are larger than the experimental ones; this may be partly due to the third-order
approximation made when deriving the equation of motion. Figure 8.108(c) and (d) rep-
resents the second-mode-shape buckling (U = 5.2) and, in this case, good agreement
between theory and experiment is observed. This is also the case in Figure 8.108(e) and
(f) where second-mode flutter is clearly demonstrated (U = 6.1). Finally, third-mode
flutter (U = 9.1) is shown in Figure 8.108(g) and (h), again very similar in theory and
experiment.
The equations of motion developed by Lopes et al. (2002) being two-dimensional, it is
of course not possible to capture the three-dimensional motions observed by Pa1doussis
et al. (2002) - see Section 8.3.2(d). Nevertheless, it is quite possible that quasiperiodic
solutions found in theory may develop and may be attracted to a three-dimensional solution
with a proper theoretical model that could exhibit that.
We next proceed with quantitative comparisons, Thresholds of divergence and second-
mode fiutter obtainable by linear theory (identical to those determined from bifurcation
diagrams) have been compared in Section 8.3.2, showing fair agreement with the experi-
ment. Some of these comparisons are summarized in Table 8.19 here, where some
nonlinear features of the behaviour are also compared. For the purposes of Table 8.19,
the free-end shapes #115, #15 and #2 [see Figure 8.23(b)] are considered to be 'well
streamlined', while #91, #35 and #92 to be 'medium well streamlined' ,
It is seen that the thresholds of divergence and second-mode flutter are in reasonable
agreement, although the theoretically predicted restabilization has not been observed;*
in practice, in these experiments, the system never really becomes totally straight before
tNeedJess to say that getting one such solution is very (computer) time-consuming.
t ln this respect, it should be said that for sJightly different parameters than those chosen, one can obtain
theoretical results in which there is no restabilization, For example, in Figure g,ll(a), this can be done by
reducing fJ, which would be nonphysical, since the value of fJ is certain; however, the same applies to other
system parameters, va1ues for which are less surely known.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
997
o
0,1
0,2
X 0.3
0.4
0,5
(a) 0,6
O
0,1
0,2
X 0.3
0.4
0,5
(c) 0,6
O
0,1
0.2
X 0,3
0.4
0,5
(e) 0,6
O
0.1
0,2
X 0.3
0.4
0.5
(g) 0,6
-0.3 -0,2 -0.1 O 0.1 0,2 0.3
rJ
(b)
(d)
(t)
(h)
Figure 8.108 Shape of the cylinder: qualitative comparison between theory (a,c,e,g) and experi-
ment (b,d,f,h) showing successively: buckling of first-mode shape, buckling of second-mode shape,
second-mode flutter and third-mode flutter (Semler et al. 2002).

998
SLENDER STRUCTURES AND AXIAL FLOW
Table 8.19 Flow velocity for the different bifurcation points: comparison between theory
and experiment.
Phenomenon
Theory
Experiment
f = 0.7 f = 0.6 Well-streaml. Medium streaml.
se f = 0.5 se f = 0.25 tail tail
Divergence threshold, Ucd
Divergence, second-mode shape
Restabilization
Second-mode flutter threshold, Ucf
Quasiperiodic, U
Third-mode flutter threshold, U
2,10
4,5
4.79
4.82
8,50
8.60
2.50
?a
5,02
6,03
8.91
a Signifies uncertainty as to its existence,
bOr sometimes none.
1.8-2.3
4,1-4.4
2,0-2,7
3,9-4.4
5,1-5.4
6.9-7.1
7.0
5,1-5.7
7.2-7.4 b
7.0-7.5 b
Table 8.20 Comparison between theory and experiment of
the maximum tip amplitudes and the nondimensional frequency
(denoted as Hz*) ofthe limit cyc1e oscillation,
Quantity
Max. amplitude at divergence, 17(1)max
Max. second-mode flutter amplitude, 17(I)max
Max. third-mode flutter amplitude, q(l)max
Second-mode limit cycle frequency (Hz*)
Third-mode limit cycle frequency (Hz*)
Theory Experiment
0.25 0.10-0.12
0.15 0.08-0.10
0.11 0.08-0,10
2.5 2,7
6.7 6,7
the onset of seeond-mode flutter, although the amplitude beeomes of tbe seeond-mode
shape. For further diseussion the reader is referred to Semler et al. (2002).
It is also seen tbat the third-mode flutter in tbe experiments oceurs at considerably
lower U tban predicted, and so does tbe quasiperiodic behaviour. In tbe experiments, qua-
siperiodicity manifests itself as a 'hesitation' between seeond- and third-mode flutter-
in appearanee sirnilar to ehaotie behaviour; in the PSD (power speetral density), there
are two or more frequeney peaks of sirnilar amplitudes. In some eases, quasiperiodic
behaviour is not clear or does not oceur, and this is marked as 'sometimes none' in the
table. Here it should be remarked that the theoretieal ealeulations with N = 6 are at the
limit of being reliable for high values of U (e.g. U > 7). It has been found, by conduct-
ing ealculations witb fewer degrees of freedom, that inereasing N would result in lower
values for the bifureation values of U for quasiperiodie behaviour and for third-mode
flutter.
Another diserepaney between theory and experiment is that in the latter a hysteresis
has been observed in the onset of seeond-mode flutter; tbis has not been reprodueed in the
theory, the bifureation being always supercritica1.
Table 8.20 shows a eomparison between the amplitudes and tbe dimensionless limit-
eycle flutter frequencies. It can be seen tbat the theoretieal amplitudes are larger than tbe
experimental ones, but tbe frequencies, botb for seeond- and third-mode flutter, are in
good agreement.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
999
8, 11, 1 (c) Concluding comments
The nonlinear dynamics of the system has been discussed and some linear aspects recon-
sidered. The general dynamical behaviour is confirmed to be as follows. The cylinder first
lose s stability by divergence in its first mode, the amplitude of which increases with ftow;
this is gradually transformed to divergence of predominantly second-mode shape, before
the system is restabilized - in most but not all cases. At slightly higher ftow velocities the
system loses stability by second-mode ftutter, which at stil1 higher ftows is succeeded by
third-mode ftutter. The transition between these two forms of ftutter encompasses a zone
of aperiodic or quasiperiodic motions. It is of special interest that multiple solutions may
coexist in some parameter ranges, and they depend on initial conditions.
Both linear aspects (the critica1 ftow velocities forthe princi pal bifurcations) and nonlin-
ear aspects (amplitudes, limit -cyc1e frequencies, transition between the various dynamical
states, quasiperiodicity) have been compared with experimental observations. Broadly,
theory and experiment are in good qualitative and reasonably good quantitative agree-
ment. The key bifurcations are predicted within 7-25%, while the amplitudes to "V 1 00%
only, and the limit-cyc1e frequencies to 7%. In this regard, it should be said that the
theoretical behaviour, especially at higher-ftows, was found to require a high number of
Ga1erkin modes, in fact more than could comfortably be accommodated in the numerical
calculations.
Some aspects of observed behaviour (e.g. hysteresis) are not reproduced by theory. The
same is trne for the observed 3-D orbiting oscil1ations, since the theory is strictly 2-D.
Hence, there is room for improvement. Moreover, the results are sufficiently interesting
to suggest that further work on this problem would be profitable. In particular, other
interesting non1inear dynamics may be found in regions of the parameter space not yet
explored.
8.11.2 Cylinders with simply-supported ends
The dynamics of this system is equally interesting, particularly the question of the existence
of the post-divergence coupled- mode ft utter predicted by linear theory. It is recalled that the
existence ofpost-divergence coupled mode ftutterfor pipes conveying .fluid has exercised
the Applied Mechanics community considerably (Section 3.4.1 and 5.5.2); in the end,
Holmes (1978) proved by nonlinear theory that this is not possible.
Figure 8.109 gives a bifurcation diagram for a pinned-pinned cylinder in axial ftow
with f3 = 0.47, £CN = £CT = 0.5, y = O, h = O and X = 1 and A = 10 4 ,obtained
with AUTO for equations (8.266) and (8.267), using two beam modes for the v-direction
Galerkin expansion and two string modes for the u-direction. The system loses stability
by a pitchfork bifurcation at U = 3.1416(::::: Jr), which leads to divergence; the new fixed
points are stable. There is a second bifurcation, also emanating from the origin, atU ::::: 2Jr,
but the solution resulting therefrom is unstable.
These are static solutions, and the results have been obtained while ignoring the non-
linear inertia terms, which makes solution by means of AUTO feasible. In considering the
question of post-divergence ftutter, however, such terms cannot be ignored. Preliminary
solutions have been obtained via FDM, and it has been found that post-divergence ftutter is
possible. The spatio-temporal content of these oscil1ations is il1ustrated in Figure 8.110 for
the same parameters as in Figure 8.109, displaying (a) quite harmonic time-waveform and

1000
SLENDER STRUCTURES AND AXIAL FLOW
0,04
0.03
,;; 0,02
0,01
°0
_ _ _ _ L_ __ _ _ _ _ _ _ _ _ _ _.1=::::::: = _ _ L _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
4 8
Dimensionless flow velocity, I1lt
12
Figure 8.109 Bifurcation diagram for a pinned-pinned cylinder in axial flow, in terms of the
first component of the lateral Galerkin expansion, ql, versus U for parameters given in the text:
-, stable solutions obtained by AUTO; - - -, unstable solutions by AUTO; _, FDM solutions
(Coulomb 2000),
0.4
0.2
,;; °
-0,2
-0.4
° 0.5 1.0
(a) r
0.5
-0.5
°
0,5
S
1.0
(b)
Figure 8.110 (a) Time trace of ql for the same system as in Figure 8.109 and U 7,5;
(b) dimensionless v(, T) versus  (Coulomb 2000),
(b) the familiar nonc1assica1, nonstationary character ofthe flutter mode (cf. Figure 3.13).
A finite and substantial initial perturbation is necessary to precipitate these oscillations:
for U = 6.5, ql (O) > 0.2; for U = 7,5, ql (O) > 0.25. Beyond U ::::: 11.2, no convergent
oscillatory solutions are found.
The flutter oscillations appear to circumnavigate the two potentia1 wells associated
with divergence, which coexists with flutter, as shown in the bifurcation diagram of

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
1001
05
. osci1latory solution _ _ _: _ _ _ _ "-----
D static solution
o
--  .
-0,5
O
2
4
6
....
10 12 14 16
8
(a)
0.4
,;-
" 0.3
"
S
"


..a 0,2

....
"
;-
] 0,1
00
(b)
0,6
.,t
. .
.
.....: .
.....,.
..
..
..
- ..._,
..
..
..
- -.- -
.
.
- - .. --
>c
.
....,
2
4
6
8
0--
------ - -- ---- .---- --, - --- -'- " -'-- 1 '
. : , _ I
- "__ __._+ . ! _. .  ! I .: " .
. . . . . . .  . . . . . . .
,.,
"
mm
.-
-0_6
O
2
4 6
Flow velocity, dlI
8
10
(c)
Figure 8.111 Bifurcation diagrams of the dimensionless transverse amplitude li versus U, for
cCI = 0.5, rr = r = r = y = O and (a) f3 = 0.47, A = 104; (b) f3 = 0.47, A = 200;
(c) f3 = 0,1, A = 200, The solutions in (a,b) were obtained with two modes each in the axial and
transverse Ga1erkin expansions, and three each in (c); (Picot 2001).

1002
SLENDER STRUCTURES AND AXIAL FLOW
4
o
.... (a)
""
j;;; -4
""
 0.10 0.35
'5 8
o

o
(b)
-8
-0.50
o
Displacement, v
0.50
Figure 8.112 Phase-plane plots of ii versus v for the system of Figure 8.111(c) showing
(a) quasiperiodic flutterfor U = 7.7, and (b) chaotic flutter of U = 8.4 (Picot 2001).
Figure 8.11I(a), where only one of the two possib1e static solutions is shown, It is seen
that the amplitude, v = v / L, is too large to be realistic. This, together with the fact that the
frequeney (Figure 8.110) is also quite large, shows that this solution is physiea1ly atypical. t
Hence, solutions for smaller values of A, namely A = 200, were tried; it is reca11ed
that A = (4L/ D)2. The results are much more physica1ly reasonab1e for this less slender
system. As seen in Figure 8.111 (b), the amplitude is about ten times sma1ler, and the fre-
quency a1so, from 28 to 3,5 dimensionless Hz approximately (not shown); the oscillation,
a1though periodic, is no longer harmonie. Moreover, in this case the oscillatory solution
bifurcates from the static solution via a Hopf bifurcation.
Ifthe value of {3 is also decreased, to {3 = 0.1, the bifurcation diagram in Figure 8.lll(c)
shows that, for high enough U, the system appears to become chaotic. By constructing
phase-plane plots and PSDs, it is seen that initial1y the ftutter becomes quasiperiodic and
then chaotie. Only two phase-plane plots are shown here in Figure 8.112. It is seen that in
the chaotie regime, the cy1inder oscillates around one of the now unstab1e potentia1 wells
and then the other, repeating this switching aperiodica11y.
It is conc1uded that, just as seen in the experiments, post-divergence oseillations are
indeed predicted to occur by non1inear theory - but they do not emanate from the trivial
equilibrium, eontrary to what is predicted by 1inear theory. Indeed, this is consistent with
tThis large-amplitude oscillation has now been shown to disappear when the number of modes in the
Galerkin solution is increased sufficiently.

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW 1003
the energy considerations of Section 8.3.3( d) and equation (8.48): for flutter to arise from
the trivial equilibrium position, one needs C N f. CT, which is not the case in these resu1ts
(computed with CN = CT = cf). What is shown here is that post-divergence flutter is
generally a non1inear phenomenon. Nevertheless, the va1ues of Ucf are not too different
f£Om those predicted by linear theory (Tab1e 8.1), namely Ucf :::::: 6.3-6.5, or f£Om the
experimenta1 va1ues (Figure 8.20). Refer a1so to Modarres-Sadeghi et al. (2003),
8.12 ARTICULATED AND PENDULAR SYSTEMS
Some work on towed and se1f-p£Opelled articulated systems has already been presented
in Section 8.9.9. However, the topic is considerab1y b£Oader. For instance, contro1 £ods in
some nuc1ear reactors, and sometimes fue1 e1ements, are mounted 1ike the counterweights
used for fire-doors and garage doors. They invo1ve one or more interconnected, essential1y
rigid neutron-absorbing bund1es, hung in spherica1-pendu1um fashion in a tube, wherein
there is coo1ant flow, as shown in Figure 8,131 (a) for instance. By lowering them into the
reactor core, precipitous1y in emergencies, the reactivity is controlled. Quite sophisticated
mode1s of the system have been p£Oposed by Hennig & Grunwa1d (1984), Grunwa1d &
Miiller (1984); they are discussed brietly in Section 8,12.3. In another set of studies,
a1so discussed in Section 8.12.3, the dynamics of a simp1ified doub1e spherica1-pendu1um
model in tlow is considered, concentrating main1y on the impact dynamics with the tlOW-
tube [e.g, Peterka (1991, 1992)].
However, before presenting this applications-inspired work, the 1inear dynamics of a
generic canti1evered articu1ated system in unconfined flow is briefly discussed first, in
Section 8.12.1. The dynamics of a generic, idealized model of an articu1ated system in
a tube is discussed next, in Section 8.12.2, based on the work of Pa'idoussis & Botez
(1993, 1995) and Botez & Pa'idoussis (1996); only p1anar motions are considered, and
emphasis is put on the non1inear and chaotic dynamics of the system.
Fina11y, articu1ated strings ('stringers') of fue1 bund1es in vertica1 tlow channels are
used in some nuc1ear reactors, often hinged at the bottom, with up-flow; see, for examp1e,
Pa'idoussis (1976b), Pettigrew (1993). Each fue1 bund1e is made up of a c1uster of fue1
e1ements (pins). The dynamics of such systems is discussed in Chapter 9.
8.12.1 Linear dynamics of cantilevered system
The equations of motion of the articu1ated canti1evered system in unconfined tlow have
been derived in Section 8.9.9; for a horizonta1 N -degree-of-freedom system they are given
in Appendix S.3; see a1so Pa'idoussis (1986).
The evo1ution of the eigenfrequencies with increasing dimension1ess tlow ve10city u
for an N = 6 system is shown in Figure 8.113. It is seen that the system is subject to
first mode divergence at u = Ucd :::::: 2.0. The system is restabi1ized at u :::::: 5.35, and then
becomes unstable by flutter at u = ucf = 5.69.
This behaviour is entirely consistent with that of an equiva1ent continuous1y tlexib1e
canti1ever in axia1 flow, of the same 1ength and with E I = kl, where k is the £Otationa1
stiffness at the joints and 1 the length of each articu1ation; the 1ast cy1inder is taken to be
!llong, which, as shown in Section 3.8, accelerates convergence of the discrete to the
continuous system. This convergence is il1ustrated in Figure 8.114 for a particular system

8
a
6

5.8
56
57
5,55
4
t t 51-.
r:1.-.- ·
ro 2
538 1 ' __o 16 1
2,0 i 20
5,3t !
52 r '! -2
. I
30 i H2
40 d40_4
16
(w)
24
32
5.59
50
-6
o
Figure 8.113 Argand diagram of the lowest two dimensionless eigenfrequencies of a cantilevered
cylinder in axial flow for {3 = 0.5, /OC! = I, ft = 0.8, Cb = 0.2: -'-, discrete system (N =
6, I j = I for j = I, ' . . , N -I, IN = il); -, equivalent continuously flexible system, The loci on
the Im(w)-axis are shown diagrammatically on the left-hand-side diagrams for c1arity (Pai'doussis
1986).
5
4
1st Mode
 3 _________...____....a....____.____-
 2
o
I :: t---:----L--------;,-:1
O 2 4 6 8 10
N
  ---------------:-m.-:"::1
I f r------.- -----.-----d'1
O 2 4 6 8 10
N
Figure 8.114 Convergence of the eigenfrequencies (real and imaginary parts) of a discrete system
to the equivalent continuous one ( - - -) as the number of articulations, N, over the fixed length ofthe
systemisincreased; for {3 = 0,5, /OC! = I, ft =0.8, Cb =0,2, IN = il and u = 1 (Pa'idoussis 1986).

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW 1005
and u = 1, It is seen that N = 4 is sufficient to model first-mode behaviour for this
u, whi1st N = 8 or 10 is required for the second mode. This is why the results for the
discrete and continuous systems are not identical in Figure 8.113. Generally, to achieve
convergence to the continuous system, N needs to be larger for higher u and, of course,
for higher modes.
Thus, provided that N is fair1y large, the discrete system behaves quite similarly to a
continuous one. For small N, however, for example, N = 2 or 3, there can be qualitative
as well as quantitative differences in the dynamics.
The linear dynamics of pendular systems in genera1ly confined flow, of the type
considered in Section 8.12.2, is discussed therein.
8.12.2 Chaotic dynamics of a pendular system
The system under consideration is shown in Figure 8.115. The annulus is considered
to be fair1y narrow, so that motions are generally small enough for linearized equa-
tions of motion to be used, yet not so narrow that the mean flow would be affected
by motions of the articulated system. For simplicity, motions are presumed to be planar.
Impacting of the articulated system with the flow channel is taken into account in an
idealized way.
The system is similar to that in Section 8,9.9, except that the fluid is confined in a
vertica1 tube and hence the pressure is not constant along the articulated system, being
a function of both e1evation and pressure drop, Nevertheless, the analysis proceeds as in
Section 8,9,9,
r-- y
x
Dch
(a)
kl
(b)
.,
CfJN
Figure 8.115 (a) Diagram of the articulated cylindrical system in a flow-channel, subjected to a
mean annular ftow. U; (b) definition diagram, showing the generalized coordinates, l{J j, the cylinder
lengths lj and the interconnecting rotationa] springs of stiffnesses kj (Paldoussis & Botez ]993).

1006
SLENDER STRUCTURES AND AXIAL FLOW
8. 12.2(a) The equations of motion
The kinetic and potential energie s of the structure are first obtained, followed by a
consideration of the fluid forces. For sufficiently small motions, such that sin q; j :::: q; j,
cos q; j :::: 1, 1 - cos q; j :::: ! q;;, the kinetic energy Ts and potential energy Vs are found to
be (Paidoussis & Botez 1993, 1995)
[ ( ' 1 ) 2 J
1 N lj J-
Ts = 2:?; 1 mj:; Zq cPq + ; cP j d; ,
1 N { ( ( j-l ) } 1 N
Vs = 2:?; 10 J mjg :;Zqq;: +;q;; d; + 2: ?;k j (q;j -q;j_d 2 .
(8.270)
(8.271)
The flow velocity over an element o; of the jth articulated cylinder is
( j-l )
Vj(;) = Ui + L)qcPq + ;cPj + Uq;j j,
q=l
and hence the kinetic energy of the inviscid component of the flow is
{ [ ( " 1 ) 2 J}
1 N lj J-
Tf = 2:?; 1 Mjn :;lqcP q + ;cPj + Uq;j d; + MjlU 2 ,
(8.272)
where Mjn = xpA j is the virtual or added mass ofthe jth cylinder subjected to transverse
motions in the confined annular space, and Mjl is the equivalent quantity for axial motions.
The added-mass coefficient X for axisymmetrically confined flow is found by potential
flow theory to be
[(1 + h)2 + h2]
X = [(1 + h)2 - h2]'
(8.273)
where h = D / Dh, D being the diameter of the articulated cylinders, and Dh = Dch - D
is the hydraulic diameter; Dch is the internal diameter ofthe flow channel (Figure 8.115)-Ť
ln addition, there is a nonconservative force related to the stream1ined free end - cf,
equation (8.205),
Fnc = (1- f)MU ( t ZjcPj + Uq;N ) ,
J=l
(8.274)
where, again, f  1 for a well-streamlined end, whi1e f  O for a blunt one,
tHere the de1inition of h confonns with the notation elsewhere in this chapter, Note that in the papers by
Pa1doussis & Botez (1993, 1995) the inverse notation is used, that is, h = Dh/ D,

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW 1007
Furthennore, for a point  along the jth cylinder, the viscous hydrodynamic forces,
adapted from equation (8.13), for CN = CT = Cf are given by
(FN)j = PDUCf ( I:ZqfjJq + fjJj + U'Pj ) + PDCd ( I:ZqfjJq +  fjJj ) ,
q=l q=l
(8.275)
1 2
(F£}j = -pDU Cf.
2
In addition, we have the hydrostatic/pressure-drop forces, (Fpx)j = O, (Fpy)j =
A(dpjdx)'Pj. By assuming that lateral movement of the cylinder has negligible effect
on the axial pressure distribution, one may write
A ( d P ) = -!pDU2Cf + pgA.
dx 2 Dh
Hence, one obtains for the jth cylinder:
(8.276)
( 1 2 D )
(Fpy)j = --pDU Cf- + pgA 'Pj.
2 Dh
(8.277)
There will be an additional pressure force at each joint, related to curvature in continuously
flexible systems and equal to pA(a 2 y jax 2 ); for the articulated system, the corresponding
tenns are taken into account in the generalized forces,
After considerable manipulation, the generalized force (moment), Qj, associated with
8cpj and the nonconservative forces (8.274)-(8.277) is found to be (Patdoussis & Botez
1993, 1995)
Qj = - i 1j (FN)j d + i l1 (Fpy)j d + t { - ili (FN)iZj d + ti (Fpy)iZj d
O O i=j+l O 10
+ i 1i (F£}iZj ('Pi - CPj) d - (dpjdx)iAiZiZj('Pi - cPj)}
1 2 2
+ '2PD U CbZj('PN - 'Pj) - FncZj, (8.278)
in which the tenn involving dp j dx is associated with the pressure forces at the joints.
The linearized equations of motion may now be obtained by application of the
Lagrange equations. A 'uniform' articulated system is henceforth considered, such that
mj = m, Aj = A, Dj = D, k j = k, etc., j = I, ..., N; and Zj = Z, j = I, .,., N - 1,
with ZN = eZ and e = 1 (see Sections 3,8.2, 8,9.9 and 8.12.1). The linearized equations
of motion may be rendered nondimensional with the aid of the dimensionless parameters:
t3=pAj(pA+m), y=(m-pA)gL 2 Njk, u=(pALNjk)1/2 U ,
£ = LjD, c = (4j7f)Cd[MLNjk]1/2 = (4j7f)Cd, cf = (4j7f)Cf'
Cb = (4j7f)Cb, h = DjDh, . = [(pA +m)L 3 Njkr 1 / 2 t, (8.279)

1008
SLENDER STRUCTURES AND AXIAL FLOW
where L = (N - !)Z and k = E I / Z, E I being the flexural rigidity oť a fictitious,
continuously flexible system to which the present system would converge as N -+ 00.
The equations oť motion obtained up to this point are valid only if impact with the
flow channel does not occur, However, in general, when the threshold of stability is
surpassed and limit-cycle motions ensue, impact with the flow channel does take place.
The nonlinear ťorces generated thereby are introduced next.
Impacting oť the articulated cylinder system with the flow channel is best modelled by
a tri1inear spring. For contact arbitrarily assumed to take place at the second articulation
(lower end ofthe first cylinder) t where the trans verse displacement is denoted by 'f} = II CPI,
the ťorce exerted by the trilinear spring may be expressed as
FT} = kt {'f} -! [I'f} + 'f}gl-I'f} - 'f}gl]},
(8.280)
where kt is the contact stiffness, and 'f}g is the maximum displacement of the system
with no contact (Figure 8.116), Reca1ling that in the equations of motion we are dealing
with moments and angles, rather than forces and displacements, the nondimensional term
associated with the trilinear spring is
..Mt = kt {CPI - ! [lcpI + CPg I - IcpI - CPg I] } ,
where CPg = 'f}g/ ft (with Zl = Z, as discussed earlier) and Kt = (ktfk)L 2 N 2 .
For convenience, particularly for analytical solutions, a cubic-spring approximation
may also be utilized to model impacting with the flow channel. The force exerted by a
cubic spring is
(8.281)
F('f}) = k c 'f}3,
(8.282)
Force
Displacement
1J g
Figure 8.116 Force-displacement curves for the realistic trilinear-spring model (- - -) for
impacting of the articulated system on the confining channel, and the cubic-spring idealization
(-); (Pa'jdoussis & Botez 1993).
t Although it has been confirmed that this corresponds to the 'natural' mode of vibration of the system, this
nevertheless represents a serious limitation in the model.

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW 1009
where ke is the cubic stiffness, ln this case, the nondimensiona1 moment tenn is
u _ 3
">'le - Ke<{JI '
(8.283)
where Ke = (ke! k)L 4.
ln what follows, calculations are confined to two- and three-degree-of-freedom systems
(N = 20r 3). The dimensionless equations ofmotion in the latter case, from which those
for N = 2 may easi1y be obtained, incorporating (8.281) or (8.283), are given by Pai'doussis
& Botez (1993, 1995):
[I + (X - 1),8] [ (1 + e) fÍ5I + O + e) fÍ52 + !e2fÍ53] + ! (3e + 4)u£CfJP<PI
+ !(2e + 1)U£CfJP<P2 + !U£CfJPe2<p3 + !(3e + 4)£CJP<PI + !(2e + 1)£CJP<P2
+ !8cJPe 2 <p3 + X (1 - f)uN JP<PI + X (2 - f)uN JPq;2 + X (2 - f)uN JPe<p3
+!(1 +h)(2e+3)u 2 N£Cf<{JI - xu 2 N 2 <{JI + x(1- f)u 2 N 2 <{J3
+ !u2N2cb(<{JI - <{J3) + !(2e + 3)NY<{JI + N 4 (2<{JI - <pz) + .Me(or .Mr) = 0,
(8.284a)
[I + (X -1),8] [(! + e) fÍ5I + (+ e) fÍ52 + !e2fÍ53] + !(2e + 1)U8CjJP<PI
+ i(3e + 1) u8C fJP<P2 + !u8cfJPe 2 q;3 + !(2e + 1)8CJP<PI + i(3e + 1)8CJP<P2
+ !8cJPe 2 <p3 - xfuNJP<PI + x(1- f)uNJP<P2 + x(2 - f)uNJPeQ;3
+ !(I + h)(2e + 1)u 2 N8Cf<PZ - xu 2 N 2 <{J2 + X (1 - f)u 2 N 2 <{J3
+ !u 2 N2Cb(<PZ - <{J3) + !(2e + 1)NY<{J2 - N 4 (<{JI - 2<{J2 + CfJ3) = 0,
(8.284b)
[I ( 1),8] [ I 2.. I 2.. I 3.. ] I 10 ,8 2, I 10 ,8 2'
+ X - Ze <{JI + ze <{J2 + Je <{J3 + 4 U8C fy pe <{JI + 4 U8 Cjy pe <{J2
+ i U8 CjJPe 3 <p3 + !£cJPe 2 <pI + !8cJPe 2 <jJz + i 8c JPe 3 <p3 - XfuN JPe<pI
- xfuNJPe<p2 + xO - f)uNJPe 2 <p3 +!O + h)u 2 N8Cfe 2 <{J3
- xfu 2 N 2 e<{J3 + !Nye 2 <{J3 + N 4 (<{J3 - <{J2) = 0,
(8.284c )
where dots now denote d( )/dí.

1010
SLENDER STRUCTURES AND AXIAL FLOW
8. 12.2(b) Dynamics of the N = 2 system
The discussion here is based on some of the work in Pa'idoussis & Botez (1993), specifical1y
on a system with s = 10, f3 = 0.4, Y = 10, sCf = 0.25, f = 0.8, Cb = 0.1, h = 2.0 and
C = 0.38, constrained by a cubic spring with Ke = 5 X 10 3 at the end of the first cy1inder.
The dynamical behaviour with a trilinear spring has been shown to be qualitatively similar.
The linear dynamics is investigated first via an eigenvalue analysis, considering solutions
of the form {cp} = {q5'} exp(). T). As seen in Figure 8,117, the system loses stability by
divergence in its first mode at u = 1.695, but is restabilized at u = 2.53. It then loses
stability by flutter (Hopfbifurcation) in its second mode at u = 2.74. Thus, qualitatively,
the dynamics is as in Figure 8.113,
A bifurcation diagram for u > 2,74 is shown in Figure 8,118, which suggests that the
system eventual1y becomes chaotic. The route to chaos is clarified via the phase-plane
portraits of Figure 8,119. For u = 2.74, there exists a stable, symmetric limit cycle
that develops after the Hopf bifurcation (not shown). The symmetry of the limit cycle
is lost by a symmetry-breaking pitchfork bifurcation at u = 2.795, the first bifurcation
6
4
3,5
2

 5 4
 o
Div,
-2
-4
3
-6
(a) -40 -30 -20 -10 O 10
(A)
30
20
10 3 4
 2,74 5
 Flutter
 O
2.74 4 5
-10 3
-20 2
-30
-5 O 5 10 15 20 25
(b) (A)
Figure 8.117 Argand diagram for the (a) first-mode and (b) second-mode eigenvalues of the
N = 2 pendular system ({3 = 0.4, y = 10, cC! = 0.25, f = 0.8, Cb = 0.1, h = 2.0, C = 0.38),
with the dimensionless ftow velocity u as parameter (Pai'doussis & Botez 1993).

SOLlTARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
1011
010
008
i:i
"
S
"
i;j 0,06
].
'6
 0.04
.

0.02
000
2.74
I I
_1iťif.;;
./',., '''J'
. "P:""',,,".: .....:;
I I I .-'.. '; ,:' -
2,76 2,78 2.80 2.82 2,84 2,86
Flow velocity, u
Figure 8.118 Typical bifurcation diagram for the N = 2 system, showing (,01 max versus u
(Pai'doussis & Botez 1993).
,e: 0,2
i3
u 0,0
o
'O
:> -0,2
-0.4
-0,6
(a)
,e:
i3 00
u
..2 -0,2
-;:
-0,4
-0,6
-0,8
-0.10
(c)
0,6
04
06
0.4
0.2
0.0
-0,2
-0.4
-0,6
(b)
06
0.4
0.2
0.0
-0.2
-0.4
-0,6
0.10 -0,10 -0,05 0.00 005
(d) Displacement, 'PI
0,10
Figure 8.119 Phase-plane plots of CPI versus (,01 at (a) u = 2,8225, (b) u = 2,8240, (c) u = 2.8295
and (d) u = 2.8310, for the system of Figure 8.ll8 (Pai'doussis & Botez 1993),
0,6
0.4
02
shown in Figure 8.118 - where the two branches are obtained with different-sign initial
conditions, At higher u, a cascade oť period-doubling bifurcations occurs, and
Figure 8,1l9(a) and (b) shows period-2 and period-4 motions. Figure 8,119(c) shows
chaotic motion for u = 2.8295, corresponding to the first den se cIoud oť points in
-0,05 0.00 0.05
Displacement, 'P I

1012
SLENDER STRUCTURES AND AXIAL FLOW
the bifurcation diagram. Figure 8.119(d) shows a periodic (akin to period-S) motion
for u = 2.831, corresponding to the sparse patch (periodic window) in the bifurcation
diagram. At higher u, the motion becomes strongly chaotic again.
The period-doubling bifurcations have been pin-pointed to occur at u = 2,8195
(period-2), u = 2,8234 (period-4), u = 2.8243 (period-8) etc., yielding a Feigenbaum
number Fei = 4.33.
The chaotic character of the motion for u = 2,8295 and other values of u has also been
iUustrated by time traces and PSDs as in Section 5.8.1. Here we show a Poincaré map
for u = 2.84 in Figure 8. 120(a), displaying a very definite structure and showing some
similarity to a M6bius strip. Figure 8,120(b) shows the variation of the largest Lyapunov
exponent, 0', with u [see Section 5.8.1 or Moon (1992)]; it is c1early seen that the system
is chaotic (O' > O) for u  2.8245 and remains so for higher u, except for a small periodic
window near u = 2.831.
Thus, it is clear that, at least for some part of the system parameter space, the nonlinear
dynamics of the system is quite complex and interesting.
AU the foregoing results have been obtained numerically via a fourth-order Runge-Kutta
integration algorithm, using T = 0,01. However, some analytical work has also been
done using centre manifold theory in the vicinity of the Hopf and pitchfork bifurcations
(see Appendix ES and H). A typical result in which the post-Hopf limit cycle amplitude
obtained numerically is compared to that obtained by centre-manifold reduction is shownin
Figure 8,121. Agreement between the two is quite reasonable, deteriorating as u distances
itself from u e , as usual.
8. 12.2(c) Three routes to chaos for the N = 3 system
A system with N = 3 but otherwise the same parameters as in subsection (b) has been
found to also become chaotic, but this time following the Ruelle- Takens-Newhouse qua-
siperiodic route (RueUe & Takens 1971; Bergé et a/, 1984). This has motivated the work
to be described next. The objective here is to see if different routes to chaos could be
obtained while making minimum parameter changes to one basic system. Without excess-
ive exploration, this has been found to be the case for one N = 3 system: by changing
the values of only two system parameters, three different routes to chaos are obtained -
which is rather rare,
The basic system parameters are: £ = 10, tJ = 0.4, y = 10, £cf = 0.25, Cb = 0,1
and a cubic-spring model for impacting, with Ke = 10 5 . The two parameters varied are
the end-shape parameter f and the narrowness of the channel, characterized by h, which
entrains a corresponding change in c. Thus, we have the fol1owing three cases:
Case 1; narrower annulus:
h = 10, C = 1.27 (Cd = 1),
h = 2, C = 0.38 (Cd = 0.3),
h = 2, c = 0.38 (Cd = 3),
f = 0.4;
f =0;
f = 0.4.
(8.285)
Case 2; wider annulus, blunt end:
Case 3; wider annulus, profiled end:
(i) Case 1: Period-doubling route to chaos. The dynamics of the linear system, i.e.
without impacting, is as follows: divergence occurs in the first mode at u = 2.04; the sys-
tem does not regain stability in this mode as u is increased; however, flutter is superimposed

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW 1013
2.5
2,0
'5
J 1.5
 1.0


0.5
0.0
0,15 282 2.83 2,84 2.85
(b) Flow velocity, u
1.5
,
,
,
1.0
I ..: . ...
.:.;;!../  .\.
.:1.....;.. ....
N 0,5
'e.
,
u 0.0
o
;;
;> -0,5
.)'
.'
.,'
...!'
-1.0
'.
,>
:!,\.. . -<
":c" ,". '. _f,".
';''7,'':'..
....oJ: ..
-1.5
-0.15
0,00
Displacement, 'P2
(a)
Figure 8.120 (a) Poincaré map for (her:) versus <'o2(-r:) when <,01 Cr:) = O for the system of
Figure 8,118 at u = 2,84; (b) the largest Lyanunov exponent, a, versus u for the same system
(Pai:doussis & Botez 1993),
1.2
o
o
o
1.0 o
o
o
'" o "
"=' 0,8 o "
, o " "
Q. o "
 o
o " "
 0,6 o ..
o "
..
o ..
. o ..
::E 0.4  ..
1/
lil
0,2 "
0,0
0,00 0,02 0.04 0,06 0,08
u-u c
Figure 8.121 The post-Hopf limit-cyde amplitude versus J.L = u - U e for the system of
Figure 8.118 with Ke = 5 X 10 3 , obtained: o, numerica11y (Runge-Kutta integration); , by
centre manifo1d reduction (Pai:doussis & Botez 1993).
at u = 2,65. When the system is analyzed by the nonlinear set of equations, it is found
that the fi.rst mode develops a secondary Hopf bifurcation at u = 2.457 and the system
displays limit-cyc1e motion thereafter; this limit cyc1e is symmetric about the fixed point
which gave it birth, but asymmetric vis-a-vis the origin, as seen in the bifurcation diagram
ofFigure 8.122(a).
Thereafter, a cascade of period-doubling bifurcations ensues, leading to chaos at
u  2.60, as seen in the Poincaré map of Figure 8.122(b). Thus, quite apart from the
mode involved and the asymmetry with respect to the origin, the route to chaos is quite
similar to that in subsection (b) and needs not be discussed further here.

1014
SLENDER STRUCTURES AND AXIAL FLOW
0,10
i::
Š 0.05
.,
g
-a.
'"
'6 0.00
8
S
. -0,05
::s
L.C. ',..,",
S FP_ ",<.;
. . <. :f!
:..&;..::;
19?
Á'.:,..: :
'<j1
-0.10
2.0
(a)
0,6
0.4
0,2

'u 0.0
o

-0,2
2,\ 2,2 2.3 2.4 2,5 2.6 2.7
F\ow velocity
//:,,:.<.::';'':'
,tl:': .
,/f':: .
/:. t .
I i.i
l ..
. 1 i .
,
,
... ",'
'.
" .
\',\
-0.4
í
'1 /
: ; .f
.' lI"
o! ,..
. . ,. .
.:;:; ::', ":')';: .::/.r'(.';' ,
-0.6
-0.08
,
(b)
-0,04 0,00 0.04
Displacement
0,08
Figure 8.122 (a) Bífurcatíon diagram showíng CPI max versus u, obtained for the N = 3 system
with the cubic-spring model for impactíng with the channel; Case 1: h = 0,1, f = 0.4, Ke = 10\
S.F.P. and L.C. stand for 'stable fixed poinť and 'limit cyc1e' , respectively. (b) Poincaré map of cP2 ('r)
versus CP2('r) when CPI ('t') = O for u = 2,6, showing chaotic behaviour (Paldoussis & Botez 1995).
(ii) Case 2: Quasiperiodic route to chaos. Another route towards chaos, found by
Ruelle & Takens (1971), is re1ated to the characteristic frequencies of a system and is
referred to as the quasiperiodic route (Bergé et az.. 1984; Moon 1992). This is the route
to chaos followed by this system.
The system loses stability in its first mode by a Hopf bifurcation, at u = u ft = 4.95.
ln its second and third modes, the real part of the eigenvalues remains a1ways negative;
so, from the linear stability point of view, only the first-mode behaviour is of interest.
We next consider the dynallÚcs when impacting takes place. Figure 8.123 is the bifurc-
ation diagram; a similar one is obtained for a trilinear rather than a cubic spring model for
the impacting (Kr = 8.2 x 103). Not the whole 'cloud-infesteď region is in fact chaotic; as
may be seen in Figure 8.124, the motions in (b,c) for u = 6.235 and 6.25 are quasiperi-
odic, as also seen in Figure 8. 125(a) and (c) for u = 6,25, The motion in Figure 8. 124(d),
however, ís chaotic, as demonstrated also in Figure 8,125(b) and (d),
ln Figure 8.125(a) and (c) the motion is quasiperiodic-2, that is, with two base frequen-
cies: tI = 0,3904 and h = 0,9273. All other peaks in the PSD may be confinned to
correspond to f = nfl ::!:.mh, withn andm integers; forexample, the third peakin thePSD
is h = 6f1 - h, and the va1ues ofthe following peaks are f4 = 7 tI, f5 = 13tI - 2h,
f6 = 4h, h = 7 h-6tI, fs = 7 h-5tI, f9 = tI+5h, flO = 7 f\+3h,etc. Theratio

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
1015
0.20
0,15
<=
'"
E 0.10
'"
u
'"
'E.
OJ 0,05
"3
,I 0,00
><
'"
::8
-0,05
-0,10
4,5 5,0 5,5 6,0 6,5
Flow velocity
Figure 8.123 Bifurcation diagram of CPI max versus u for Case 2 of the N = 3 system (h = 2,
f = O, Ke = 10 5 ); (Pai:doussis & Botez 1995).
(3) U = 6,15
6 4
4
2
c
.u
o O

-2
-4
-6
-0.2
-0,1
4
0,0
0.1
0.2
2
,£
u °
o

-2
-4
-0.3 -0,2 -0,1 0,0 0,1 0.2 0,3
Displacement
(b) u = 6,235
2
°
-2
-4
-0.2
-0,1
0,0
0.1
0.2
6
(d)u=6.3
4
2
°
-2
-4
-"?0.3 -0,2 -0.1 0.0 0.1 0,2 0.3
Displacement
Figure 8.124 Phase-plane plots of <Pi Cr) versus CPI Cr) for the system of Figure 8.123 (Case 2):
(a) u = 6.15, periodic; (b) u = 6.235, quasiperiodic; (c) u = 6.25, quasiperiodic; (d) u = 6.3,
chaotic (Pai'doussis & Botez 1995).

1016 SLENDER STRUCTURES AND AXIAL FLOW
20 O
15
O -20
2 -20 -40
.S:
'O -40 -60
O
 -60 -80
-80 -100
-100 -120
O 2 4 6 8 10 O 5 10 15 20
(a) Frequency (b) Frequency
2 . , , 3
I 2
C I
'u O O
o

-I
-I
-2 l ,-
loS
-2 , I I -3 l
-0.3 -0,2 -0.1 0,0 0.1 0,2 0.3 -0.3 -0,2 -0.1 0,0 01 0.2 0.3
(c) Displacement (d) Displacement
Figure 8.125 Case 2 system: (a,b) PSDs of ifJ\ (-r), and (c,d) Poincaré maps of ifJz(-r) versus
CP2(-r) when CP] (-r) = O. In (a,c), u = 6,25 and the motion is quasiperiodic; in (b) u = 6.40 and
(d) u = 6,35, the motion is chaotic (Pai:doussis & Botez 1995).
JI! 12, usual1yreferred to as the winding(or 'rotation') number, is W = 0.42105::::::: 8/19, a
rational number, For u = 6,2675 and 2,685, the winding numberis W = 0.53846 ::::::: 7/13
and W = 0.61538 ::::::: 8/13, respectively, thus increasing with u.
ln the quasiperiodic-2 state, motions evolve on a 'two-torus', T 2 , The transition to
a 'three-torus', T3, via a third Hopf bifurcation, signals the effective onset of chaos,
since motions on the T 3 torus are nonrobust when perturbed, the torus thus becoming
transformed into a strange attractor [Section 5.8.3(b)], This is what happens here between
the left- and right-hand panels ofFigure 8.125.
(Ui) Case 3: /ntermittency route oj chaos, ln this case [parameters in (8.285)], as the
flow velocity u is increased, both the real and imaginary components of the second-
mode eigenvalue approach zero, leading to a pitchfork bifurcation (divergence) at
u ::::::: 4.15. The system remains unstable for u > 4.15. Furthermore, a Hopf bifurca-
tion occurs at u = 4.7, in the third mode. The first mode remains stable throughout this
range of u.
Figure 8.126(a) shows the bifurcation diagram for a range of u showing the pitchfork
bifurcation and beyond; Figure 8. 126(b ) covers the range of u where chaos appar-
ently arises: 4.91 < u < 4,925. The two stable branches in Figure 8.126(a) after the

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW 1017
0.20
.
.
.
1J 0,15
S
 0.10
P..
CI>
L:: -======
-0,10 , ,
4,0 4,2 4.4 4,6 4,8 5,0
(a) Flow velocity
0.20 ,
 0.15 1:li :; I; d:' . I;: 1::tl. ;:::: !!:: : ,I.
5 ,!:II ,I 1 1 ....1,1. 1 ..".,1.,... :,,1 ; " . i . . , ! , : .,
 0.10 t!:n i I ' .:::': tI I::: :!t;!. ::: I:::::
 _lit 11.,.:,,',1" "'ill";" 'I::"
V,I 1:: i I :. . . I: '.' I . ! . : . . . .; ! '. : . . : I : I
:o 005 :;' ;:.... .,,:'..., ' ". ::.':
J _::: ":: ,:) :,,'0::/« )i,: <
-0,10
4,910
(b)
4,915 4,920
Flow ve10city
4.925
Figure 8.126 Bifurcation diagram showing fPl max versus u, obtained for the N = 3 system with
the cubic-spring model for impacting with the channe1; Case 3: h = 2, f = 0.4, Ke = 10 5 . For (a)
4 < u < 5, and (b) 4.91 < u < 4,925 (Pai'doussis & Botez 1995),
pitchfork bifurcation were determined via opposite-sign initial conditions, Thus, for
4.15 < u < 4.90941, the origin is no longer a stable fixed point, but new stable fixed
points on either side are generated, The stability of these new fixed points was investig-
ated by linearizing the system in their vicinity, As u is increased from u pf = 4.15, the
eigenvalues become pure1y imaginary, at u = 4.90942, signifying the onset of another
Hopf bifurcation and the development of limit-cycle motions for higher u. This was fur-
ther clarified by phase-plane plots (not shown), For u = 4.90941 the fixed point is still
stable and the trajectories approach that point with time. For u = 4.91, however, beyond
the Hopf bifurcation (the one corresponding to the eigenvalues of the linearized system),
an unstable limit cycle develops; the in stability is weak and, although trajectories are
attracted to the neighbourhood of the limit cycle, the limit-cycle amplitude increases very
gradually but continuously with time, as shown in Figure 8. 127(a) and (c).
As the amplitude increases further, the system becomes chaotic, in this case via inter-
mittency (Bergé et al. 1984), as shown in Figure 8.127 for initial conditions CPI = CP2 =
0, CPI = 0.1, CP2 = O, As seen in Figure 8.127(a) and (e), after the initial transient has
died out, the osci1lation is relatively stable ('laminar' phase) for 3 < i < 8, although,
as remarked in the foregoing, it actually increases very gradually with time. At a higher
tirne interval, 100 < i < 115, Figure 8. 127(t), one of the bursts of 'turbulence' is cap-
tured, which are characteristic of this type of chaos; the associated phase-plane diagram,

1018 SLENDER STRUCTURES AND AXIAL FLOW
0,6 0,6
(a) (e)
0,4 0.4
l:- 0.2 0,2
TJ
o
 0,0 0,0
-0.2 -02
-0.4 -0.4
0,00 0,02 0,04 0,06 0,08 O 2 4 6 8
4 4
2 2
l:-
'G O O
o

-2 -2
-4
-0,15-0,10-0.05 0,00 0.05 0.10 0.15 105 ilO 1I5
0.10 0.10
(c) (g)
0.05 0.05
l:-
'G 0,00 0,00
o

-0.05 -0,05
-0,10 -0.10
-O 056 -0.054 -0,052 400 401 402 403 404 405
20 20
(d)
10 10
l:-
'G O O
o

-10 -10
-20 -20
-1.0 -0.5 0,0 0,5 1.0 1362 1364 1366 1368 1370 1372
Displacement Time
Figure 8.127 Phase-plane plots of rPJ (r) versus CPJ (r) for the system of Figure 8.126 (Case 3) at
u = 4.91; (a) 0< r < 8; (b) 100 < r < 115; (c) 400 < r < 405; (d) 1362 < r < 1372; (e-h) the
corresponding time traces of rPl (r) versus r (Pai'doussis & Botez 1995).
Figure 8.127 (b), reinforces the view that, in that time period, the oscillation is chaotic.
Similar behaviour is seen in Figure 8.127(c) and (g), showing a quiescent, nearly steady
oscillation ('laminar' phase) for 400 < T < 405, and in Figure 8,127(d) and (h), showing
chaotic oscillation at higher T with several unsteady bursts.
That the oscillation is indeed chaotic for u = 4.91 was reinforced by the calculation of
the PSD and the construction of a Poincaré map, not shown here for the sake of brevity.

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW
1019
Some further calculations were conducted to investigate what type of intermittency is
involved. Figure 8,128(a) shows a long view of the displacement ({)lrnax as a function of
nondimensional time T, for O < T < 2500, where the laminar phases (represented by a
single point for each T) are easily differentiated from the turbulent ones (represented by
vertically clustered points). An expanded view is shown in Figure 8.128(b); the sequence of
dynamical behaviour may be summarized in this case as: (i) a turbulent phase of intermit-
tency (chaos) for 1160 <T < 1164; (ii) period-twomotion for 1164 < T < 1166, involving
decreasing (with T) fundamental and increasing subharmonic components; (iii) period-
one motion at the subharmonic frequency for 1166 < T < 1182; (iv) intermittency once
more, followed by period-two and then period-one motion for higher T. The succession
of period-two, period-one and chaotic regions is characteristic of type III intermittency,
as discussed by Bergé et al. (1984),
A more quantitative signature of intermittency is provided by the statistical distribution,
P(T), of interval s of laminar phases versus T, as constructed in Figure 8,129(a). It is
seen that in this case P(T) has a maximum for short times (T  40) and it decreases
exponentially at larger times. This is definitely characteristic of type III intermittency.
0,20
I I , '  I ' I :(jL I !!U:ji! j  ,; I Š
 0,10 :',J.  1ii11..1 'f. ' H
1 005 '. : «,: 4K  -
'/ ' ":: i' . .,)' '.   : t 1'1 :.:
g t.  ;;!. : ! '"  čI-! i - )
s 0,00,';' ;';q..'j .i}f.i :! ": 
 ; f ';::I UJ '  ".,,'t' i:  . .
 -o 05 : UJ V  ; , '  JlJ:!' : ',i...A;' !,f
020
Ci 0,15

 0.10
 _ ":J
:a 0,05
e
=
e 000
.
::!S -0,05
...,......
"
".
'o"
.-/
.-/
.
1200
-0.10
o
500
-0,10
1000 1500 2000 2500 1160 1170 1180 1190
Tirne (b) Tirne
(aj
1210
Figure 8.128 Bifurcation diagrams of CPI max versus< for the system of Case 3 at u = 4,91: (a) for
O < T < 2500; (b) for 1160 < T < 1210 (Pa'idoussis & Botez 1995),
20
15
:s 10
t:&..
5
O
O 100
(a) T
100
80 u
 60
II
-t.-
:c; 40
..
00
00
00
u
o.
u
Ob
'"
20
500
O
O
200
100
300
400
500
(b)
TO
Figure 8.129 (a) Distribution of interval s of laminar phase P(T) versus T; (b) number of
1aminar phases lasting longer than TO versus TO; u = 4.91, for the system of Case 3
(Pa'idoussis & Botez 1995).

1020
SLENDER STRUCTURES AND AXIAL FLOW
Attention is drawn especially to the long tail for large T; thus, for 270 < T < 450,only
one laminar phase occurs. This type of ca1culation is complemented by Figure 8.l29(b)
giving the number of1aminar phases lasting longer than TO, denoted by N (T > TO), plotted
versus TO. It is seen that trus is an exponential1y decaying function; it depends on TO and
also f.l, == u - Ucr, where Ucr is the critical flow velocity for the instability triggering
intermittency,
In the final set of computations, two return maps [Section 5.8.3(c), p. 385] were con-
structed of successive maxima of the displacement qJlmax with increasing time. Defining
the kth, (k + l)th and (k + 2)th maxima by h, h+l and h+2, Figure 8,130(a) shows
h+l versus lb which is the first return map, wrule Figure 8.130(b) shows the second
return map, involving h+2 versus h, for u = 4,91. Two notations are used in this figure:
'before LC' and 'growing LC', LC standing for limit cyc1e; they correspond, respectively,
to the period-two and period-one motion of the system preceding a turbulent burst (and
eventual reinjection into the map) - cf. Figure 8.128(b), Itis seen that the firstreturn map
0,12
I:i
" 0,10
S
"
g
]o 0,08
:a
 0,06
';;1
'"
S 0.04

+
 0,02
(a)
1J 0,075

g
]o 0.070
:a
S
S 0.065
;;1
'"
S 0,060

N
+
 0,055
,
,
,
,
,
,
,
,
,
/ " '5 + + / Growing LC
Tangent + .
',.....+
,
,
..
Before LC ' ,
,
,
,
,
,
0,00
0.00
0.04
0,06
0,08
0,02
0,10
0,12
kth maximum displacement
0,080
o lk+3 vs lk+l
A lk+4 vs lk+2
/
"
"
"
o ... ",,/
. ",,/ o
o ,,/
AO  :>/ o
0/ o
, o Before LC
0,050
0,050 0,055 0,060 0,065 0,070 0,075 0,080
(b) kth maximum displacement
Figure 8.130 Graphs of (a) the first retum map, h+l versus h, and (b) the second retum
map, h+2 versus h, at u = 4.91 for the system of Case 3; LC denotes 'limit cyc1e'
(Pardoussis & Botez 1995).

SOLITARY CYLINDRlCAL STRUCTURES IN AXIAL FLOW 1021
is tangent to a straight line with a slope of -1, The second retum map may be represented
by a curve of the type
h+2 = (1 + 2J),)h + alf + b11,
(8.286)
with a and b constant and a «b, The values of a and b depend on J)" which is difficult
to pinpoint accurately in this case, because the difference between u and U cr is so small.
Thus, at u = 4.90941, the system has a stable fixed point, while chaotic motion develops
at u  4.90942; the limit cycle occurs somewhere in-between. Taking Ucr = 4,90942,
one finds a  -5 and b  75. The important thing, however, is that both the -1 slope in
Figure 8,130(a) and the curve with an inflection point in Figure 8.130(b) are again char-
acteristic of type III intennittency, as for the Rayleigh-Bénard convection phenomenon
discussed by Bergé et al. (1984).
8, 12.2(d) Oiscussion
It has been shown that, a1though by no means universally possible for all sets of system
parameters, chaotic behaviour is commonplace for an articulated system in axia1 flow,
impacting on the flow channel. Indeed, by varying two parameters, it has been demon-
strated that chaotic motion may be attained via three different routes: by a cascade of
period-doubling bifurcations, quasiperiodicity and intennittency.
The parameters used in these computations, and the analytical model itself, have not
been selected to be specially 'realistic'; indeed, the main point has been to illustrate that a
very rich kaleidoscope of non1inear dynamics is possible for such a system, On reflection,
and by comparison to the work in Section 8,11, it is clear that chaos is mainly due to the
impacting; indeed, impacting is the only source of nonlinearity.
It has been shown, however, that a weakly nonlinear model of the dynamics, similar in
spirit to that in Section 8.10, coupled with impacting, also gives rise to very interesting
nonlinear behaviour (Botez & Pa'idoussis 1996). The results of that model are similar to
those obtained by the inherently linear model of subsection (b), but are not discussed here
in detail. Quantitatively, and qua1itatively a1so, there are differences for high values of f,
as may be appreciated from Table 8.21. The main generic difference, however, is that for
the weak1y nonlinear model (Mode1l), for high enough u the system always ends up at a
nontrivial fixed point - physically, touching the wall of the flow channel.
8.12.3 Oscillation of a spherical pendulum in a flow-channel
As mentioned in the introductory comments, spherical pendulum models for sheathed
control rods in coolant flow, impacting on the 'sheath' (flow channel) have been studied by
Grunwa1d & Hennig (1977), Hennig et a/. (1980), Hennig & Grunwald (1984), Grunwald
& Miiller (1984), Menyai10v et a/, (1986); see, forexample, Figure 8. 13l(a) and (b). These
are realistic, generally nonlinear models of the system, too complex and too compact in
their published form to be described here in detail. One of the results obtained is shown
in Figure 8.13l(c), where it is seen that the natural frequencies of the system (here the
second) increase very dramatically with flow rate. This is mainly due to the impacting
[cf., e.g. Axisa et a/. (1988), Pa'idoussis et al. (1993a)].
A simplified, idealized model of the system, in which most attention is placed on
the impact dynamics has been studied by Peterka, Figure 8.132(a). The nondimensional

1022
SLENDER STRUCTURES AND AXIAL FLOW
Table 8.21 Dynamics according to the weak1y nonlinear model (Model 1)
and linearized model (Model 2), in both cases with the nonlinear cubic spring
model (8 = 10, f3 = 0.4, Y = 1O,8cf = 0.25, Cb = 0.1, Ke = 5 X 10 5 , h =
2, C = 0.39, N = 2); (Botez & Pai:doussis 1996).
f
Model 2
Model 1
o
0,2
0.4
0,6
0.8
Periodic motions about 2 symmetric points;
chaos; period-bubbling; period-l; fixed points
Period-2, 4, 8 Period-2, 4, 8
Chaos Chaos
Fixed points
Period-2, 4. 8 Period-2, 4, 8
Fixed points
Period-l Period-2, 4,8
Quasiperiodic, chaos Chaos
Fixed points
Periodic Period-2, 4, 8
Fixed points Chaos
No chaos
Dimensions
in mm
"0 o
400
500
A-A
(a)
Po
8 X 10 3
5
4


2
o
O
2 4 6
,;, (kglh)
L1p
(b)
(c)
Figure 8.131 (a) The experimental set-up used by Hennig & Grunwald to study flow-induced
pendular oscillations. (b) Diagram of the mathematical model: a double spherica1 pendulum,
(c) Second-mode (dominant) frequency as a function of the mass flow-rate: o, experiment; -,
theory (Hennig & Grunwald 1984).
equations of motion are given by
čP + [k1 - k2U + k3 (<jI2 + q:>2 ť p ) 1/2] <ji + (1 + k4U2ťP) q:> = O,
iJ + [k1 - k2U + k3 (<jI2 + q:>2(}2) 1/2 + 2<j1/q:> ] (} = O,

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW 1023
(a)
(b)
Figure 8.132 (a) Model of the spherica1 pendu1um ana1yzed by Peterka; (b) cross-sectiona1 pat-
tems of impacting-sliding motions for different va1ues of the 'braking' coefficient, B. for tangentia1
sliding (Peterka 1991, 1992).
where dots stand for d( )jdT, T = ru, and u = U j(ZQ); kl to k4 are coefficients of the
nondimensionalized viscous, quadratic and structural-type friction coefficients. Impacting
occurswhencp(T) == CPr = jZ, where isshowninthefigure. Therelationshipbetween
the ve10cities before and after impact is given by
. .
cP+ = -rcp_,
é+ = (1 - B)é_,
where r is a coefficient of restitution and B is the 'braking' coefficient due to traction on
the wan.
The results of some sample calculations for kl = 0.019, k2 = 3.25, k3 = 0.0229,
k4 = 3.643, u = 0,3, CPr = 0.003 and r = 0.5 are shown in Figure 8.132(b) for varying
values of B, and initial conditions cp(O) = !CPr. qj(O) = 0,8(0) = 0.75, é(O) = 0.2.
A tremendous variety of cross-sectional impacting pattems is observed: planar motions
with polar rotation, more general 3-D motion, involving some rolling on the wall
or no rol1ing, and so on. Motions in an oval channe1 have also been considered
(Peterka 1991).

1024 SLENDER STRUCTURES AND AXIAL FLOW
8.13 PRACTICAL APPLlCA TIONS
Most of the practical, that is, engineering, applications of the subject matter in this chapter
have already been mentioned in the appropriate sections; indeed, it is the applications
that inspired and motivated most of this work. For example, the work on low-amplitude
vibrations (Section 8.4), two-phase flows (Sections 8.3.5, 8.4 and 8.7.3), 'unusuaľ support
conditions (Section 8.8) and pendular systems (Section 8.12) is related to systems used in
the power-generating industry. Similarly, the work on towed and self-propelled systems
(Section 8.9) is closely related to systems used for the transport of liquid cargos by sea
and river, for oil and gas exploration and submarine detection. AIso, in Section 8.9.10 on
fish locomotion, some very interesting biomimetic schemes for marine propulsion have
been presented.
Here some further discussion of the practical aspects of the subject is given and
additional references cited.
8.13.1 Nuclear reactor and other power-generating systems
Nuclear reactors feature prominently in any discussion of fluid-structure interactions
(FSI), for several reasons. First, nuclear power was for a number of decades, and stili
is, the pre-eminent source of power in several countries, notably France and Japan, and
regions, notably Ontario in Canada, and it looked like being the energy source of the
future. t Hence, a great deal of research was done in many countries, including those
where the profusion of other resources made it statistically less prominent, for example
USA, USSR, Gennany and Canada. A second reason was that many dif.ferent and tech-
nologically novel designs were proposed and some were built. These novel designs were
not only for the reactor itself, but for the peripherals as well: the steam generators, various
heat exchangers, pumps, etc., depending on the type of reactor involved. As a result, new,
foreseen and unforeseen FSI problems arose, which required solution and the develop-
ment of new design guidelines, again meaning more research. Last, but not least, was the
realization that failures in nuc1ear systems are more important than in conventional plant,
for two reasons. (i) They could endanger the health of a very large number of people,
living not necessaril y near the site of a possible accident - as in the case of Chemobyl. *
(ii) Typically, nuclear power plants are very large, 500 or 1000 MWe not being unusual;
if one is out of service for any reason, then the power-generating utility is faced with
'replacement costs' - to buy power from elsewhere - amounting to huge amounts, of
the order of one million dollars per day (one thousand dollars per MW per day)!
The interested reader is referred to Naudascher & Rockwell (1980a) - the
Proceedings of an IAHR/lUTAM Symposium on 'Practical Experiences with Flow-
Induced Vibrations' - and particularly to a paper by Pai:doussis (1980) therein, in which
a total of 52 practical flow-induced vibration (PIV) cases, their causes and implemented
solutions, are discussed; see also Au-Yang (1993).
It is rather difficult to track down and then uncover sufficient detail on 'prac-
tical cases' involving problems in industry in general, and in this field in particular
tlt may yet become so again, when the greenhouse effect begins to bite.
+The quaint but true story of an explosion in a British power plant is recalled, in which a Ione fisherman
fishing on a river miles away was struck by debris and killed; in nuclear accidents we are talking about everyone
at a radius of hundreds and thousands of miles, fishermen or not, being imperiled!

SOLflARY CYLINDRICAL STRUCTURES IN AXlAL FLOW
1025
Instrumentation pit
Control rod drive
mechanisms
Reactor vessel head
RCC guide tube
Cover plate
Core barrel
Upper core
plate
Inlet nozzle
Control rod drive
shaft
Thennal shield
Fuel assemblies
Vessel
Bafl1e plate
Lower core
plate
Diffuser plate
In-core instrumentation
guide thimble
Figure 8.133 Schematic of a PWR reactor ves sel (Axisa 1993).
(Pa'idoussis 1980). Several factors come into play, even for problems successfully over-
come: commercial secretiveness on design detai1s, corporate image, nationa1 pride, and
fear of 1itigation. Tracking down these 52 cases required not only good contacts and trust,
but considerable powers of detection. t ln this, what helped considerably was the fact that
all problems related to nucIear plant, no matter how smal1, have to be reported to the vari-
ous national regulatory agencies, for example, the US Nuc1ear Regulatory Commission
(NRC). Furthennore, a1though in some countries accessibility is severely restricted, in
others, the USA for instance, most of the infonnation is generally in the public domain.
Figure 8.133 shows a schematic of the reactor vesse1 of a PWR (Pressurized Water
Reactor), and Figure 8.134 the fuel assembly and the control rod of the same type of
reactor, Figure 8.l35(a) shows an advanced gas-cooled reactor (AGR) fuel cluster and
(b) a pressurized heavy-water-moderated and -cooled reactor (PHWR) CANDU-type fuel
bundle. Figure 8,136 shows two types of steam generators. What is quite obvious in a11
t As a result, the writing of the paper (Pa1doussis 1980) was one of this author's most arduous but also most
enjoyable experiences, aIlowing bim (a my stery fan) to indulge in true detective work!

.i
gp
í5.
'"C
>=i
\1>-
:o
Q.
:3'
'l12wmm:nat":1
 1 '-""l";" 
o  
'"C  \I> >=i
'E :g
::cQ. '"C \I> 
'" 8 &; u
'"8 03
I-.. &;
"O
tJ
g
U
'"C
]

'S 
\I> 't::
'" Q.
'" '"
 >=i
\I> 
&;.8

o
...
]
g 
u 8
:o
Q.
:3'
á
,$
1a0..0\
.£3 <.s 2:
BZ5
",'-''''
.a -ť:
u <") Q)
"iJQ
,a-rId
<.s8.c::
, ;'
 j..j t:
,  [JJ
. '-' '-'
..1-6
QJgp
d gp o
ť'fe
. -I/')
......QJó
<E
......  I
 '-'
,=
<I:  -g
dd]
<.9_
 S 15
'-'<").....
II)
I"'i'
 '-'
oc Q
t;
s
.... QJ U
1)

.a
p.
""
>=i
\I>
e
g
o

.g -g 
(I:j ..-4

,:::..0--
us>'
CI1":O
 s '
.".<.sQJ'"
-",g(
OI
oc......"1j-
 8
ClI -
"" o.. o
<.sJ:j
.. r..... d
 08

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW 1027
these figures is that there is a profusion of cylindrical elements (solid, tubular, and shell-
like) subjected to flow, some axial, some cross-flow, some annular, or more complex,
It is not surprising, therefore, that understanding fluid-structure interactions (FSI) in
such systems is important. Also not surprising is that, in practice, such systems are not
strangers to flow-induced vibration (PIV) problems, some catastrophic (short-term) and
some long-term, for example, fretting and fatigue.
Some statistics may be useful to show the relative importance of FSI/FIV-related prob-
lems. In 1973, of the 13 cases involving shut-downs and power reductions reported to
the US NRC, two were due to flow-induced vibration, Also, of 62 reported PWR steam-
generator problems, five were caused by PIV and ten by water-hammer. Bearing in mind
the expense in shutting down nuclear plant, this rather low incidence of FSI/FIV-related
problems does not render them unimportant! ln the end, there is no shortage ofreportable
cases.
Figure 8.137 shows one of the most spectacular cases of the damages caused by flow-
induced vibration, as well as some 'run of the milľ type - but nevertheless obviously
serious cases. t
Of the 52 cases in PaIdoussis (1980), 33 involved cross-flow about bluff-bodies -
most1y cy1inders in heat exchangers and nuclear reactors. Ofthose, though not the topic of
this book, 5 (+7) involved vortex shedding, with or without acoustic resonance, 10 (+6)
fluidelastic instabi1ity, 7 (+4) buffeting and flow impingement, where the numbers out-
side the parentheses are cases where the mechanism has been identified with certainty,
while those within parentheses are additional cases where the mechanism is merely judged
as the most likely.
Of more relevance to this book are the nine cases involving internal and external axial
flow, caused by excessive turbulence, noise, swir1, acoustic coup1ing, or more exotic
mechanisms. One sample case is presented in Section 8,13.2. Also relevant are the 10 cases
involving annular and leakage flow, some of monumental destructiveness, the subject of
Chapter 11.
The interested reader is referred to Wambsganss (1967a), Shin & Wambsganss (1977),
Pettigrew (1977, 1981), PaIdoussis (1980, 1981, 1983, 1987), Chen (1987), Blevins (1979,
1990), Axisa (1993, 2001), Pettigrew ct aZ, (1998), Weaver ct al. (2000) and Au-Yang
(2001) for useful reviews of problems encountered and state-of-the-art reviews,
8.13.2 A sample practical case
This is Case 40 in PaIdoussis (1980, pp. 67-68), reproduced here almost verbatim, in (a),
(b) and (c).
8. 13,2(8) The problem
In BWRs, fuel assemblies are fitted in 2 mm thick rectangular 'channel boxes'
[Figure 8.138(a)], placed in honeycomb fashion in the reactor core [Figure 8,138(b)].
Most of the flow passes axially through the channel boxes, In some locations, between
tS ome fascinating non-nuclear incidents remain intriguingly under wraps; for example, the case of a heat
exchanger in a petroleum refinery, in which a triple resonance of the tubes, vortex shedding and acoustics
completely destroyed it within 24 h of operation, Other non-nuclear incidents remain totally unknown to the
wider engineering community, simply because they do not have to be reported!

1028
SLENDER STRUCTURES AND AXIAL FLOW
Steam
nozzle
Upper
sheU
Swirt vane
moisture
separators
F eed water
ring
Feed water
nozzle
(a)
Secondary
manway
Steam outlet
(one oftwo) 
F eedwater in/et
(one oftwo)
(b)
Figure 8.136 Sketches of (a) a U-tube type and (b) a 'once-through' type steam generator
(Pai'doussis 1980),
(a)
(b)
Figure 8.137 (a) A spectacu1ar case of flow-induced vibration damage, and (b) some 1ess spec-
tacu1ar cases showing flow-induced fretting wear (Shin & Wambsganss 1977; Pai'doussis 1980),

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW
1029
Bail handle
Upper tieplate
Pnel cladding
Pnel bnndle
Pnel rod
interim spacer
Pnel channel
F inger spring
(typical of 4)
Lower tieplate
Nose piece
(a)
Control rod
Fnel rods
Core snpport
plate
(b)
Figure 8.138 (a) A BWR fuel bundle in a fuel channel; (b) part of the core of the BWR, showing
four fuel bundles in place (Pai'doussis 1980).
Control rod
In-core
blade flux monit
00000000 00000000
00000000 00000000
00000000 00000000
00000000 00000000
00000000 00000000
00000000 00000000
00000000 00000000
00000000 00000000
00000000 00000000
00000000 00000000
00000000 00000000
00000000 00000000
00000000 00000000
00000000 00000000
gggggggg 00000000
00000000
t 't Channel
Pnel assembly
(a)
or

Tope guide
t
In-bnndle
flow
Core plate
P(t)
Core plate
(b)
Figure 8.139 (a) A cross-sectional view of the fuel assembly of Figure 8.138, showing also a
control rod blade and an in-core flux monitor. (b) The redesigned fuel bundle, showing the by-pass
leakage flow (Pa'idoussis 1980),
the ehannel boxes, there are thin instrument tubes of various types, for example, LPRMs
(Ioeal power range monitors), SRM (souree range monitors), as shown in Figures 8.138(b)
and 8.139(a). Each LPRM eontains miniature fission chambers for measurement of1oeal
neutron flux, and a ealibration tube for a traversing in-core probe (TIP), They are plaeed in
a guide tube, D '" 19 mm, with evenly spaeed holes along its length for water eireulation
to eool the deteetors. The LPRMs are installed in 'thimbles' permanently mounted at the

1030
SLENDER STRUCTURES AND AXIAL FLOW
bottom of the core lattice, penetrating the reactor vesse1 bottom, and are also supported at
the top, but are unconstrained through the rest of the core region, with an overall length
of L  4 m (Behringer et al. 1977),
In BWR/4-type reactors, as opposed to the earlier BWR/3s, 25.4 mm 'by-pass' holes
were drilled in the lower core support plate, to improve coo1ing ofLPRMs, The TIP signa1s
tipped operators that something was wrong, when an unusua! peak in the 1-4 Hz range
appeared in the neutron flux PSD. This occurred in one reactor first; later, at least twelve
BWR/4s were involved,
It was discovered that the signal was due to vibration and impact of the instrument tubes
on the comers of the channe1 boxes, in some cases fracturing them and producing holes as
large as 89 x 127 mm - a serious matter because coolant flow is thereby diverted from
the fuel in the channel, also causing cross-flow around the fue1 rods; missing pieces of
channel box, carried away by the coolant, created added worry. In most cases power was
reduced to 40% until the problem was solved.
In one reactor, of the 192 channels inspected, 65% were considered rejects; four had
been perforated (US AEC 1975; Docket 50298-465). In another, which also had 'poison
curtains' (for neutron absorption) in the channe1 box interstices, the curtains were found
to vibrate and impact on the channels; upon removing them, the problem was not solved,
because then the in-core instrument tubes were found to impact on the channels, a problem
that 'was hidden "behind the curtains" for the first two years' (Vema 1976).
8. 13.2(b) The mechanism
The problem was connected to the 25.4 mm by-pas s holes in BWR/4s, which allowed
highly turbulent, high velocity axial water flow, emerging in jet form from the lower support
plate, to excite the slender and flexible instrument tubes to sufficiently high amplitude to hit
the channe1 boxes. For the LPRMs, the fundamental natural frequency was in = 2.5 Hz;
in full-sca1e cold hydraulic tests, impact levels of 4-8 g were recorded. Unfortunately
no geometrie or flow data have been re1eased. A fundamental program to investigate
the effect of turbulence (Mu1cahy et aJ. 1980) in annular flows has shown that, a1though the
vibration of a cy 1inder in very narrow annular flows is little affected by upstream turbulence
(Dh/ D = 2,85, Dh being the hydraulic diameter), in wider annuli (Dh/ D = 4,71)
the effect is very strong.
8. 13.2(c) Cure
The solution, or 'cure' , ofthe problem consisted of (i) plugging al1 by-pas s holes in the core
bottom support plate; (ii) drilling new holes in the tie-plate of the channel boxes, which
restores the requisite by-pass flow [Figure 8.139(b)]; by directing this flow downwards,
towards the core support plate, prior to mixing with the total by-pass flow, which is
upwards, a more uniform flow is produced (Anonymous 1976). No further problems have
been reported.
8,13.2(d) Concluding general remarks
In this, as in many of the cases discussed in Paldoussis (1980), it is seen that (i) the cause
of the problem, although simple, was not easy to discover; (ii) the cure was simple and

SOLITARY CYLINDRICAL STRUCTURES IN AXIAL FLOW 1031
practical, as opposed to something overly ingenious; and (iii) the cure did not result in
reduced efficiency of operation; in other words, it could well have been part of the original
design, had problems been anticipated. This last point is, of course, easy to say with the
benefit ofhindsight. In many cases, the poor state ofknowledge at the time, or incomplete
awareness of pertinent work in other spheres of engineering activity, made this impossible.
8.13.3 Towed and propulsion systems
All of the work on towed systems is either directly related to or inspired by engineering
applications.
The Dracone system has already been presented in Section 8.9.1 - see Figures 8.73
and 8. 140(a) and (b). In addition to transporting petroleum products and fresh water, its
use has been broadened to storing and transporting spillage in environmental clean-up
AECOVERY TOWING MOORING
HND HOSE RING
III

NYLON  NOSE CONE
TOW ROPE ""
lOADlNGJDlSCHARGe CONNEC1lON
SLACK DIAMOND MARKERf NAVIGATIQN LlGHT
-

/ \ \
Mool RlNG 1W;N HULL
STAISIUS£R LlGHTFLOAT
(a)
(b)
(c)
Figure 8.140 (a) Diagrammatic layout of the Dracone barge; (b) a nose-end view of a beached
500 tonne unit undergoing hydraulic test; (c) an Aquarius bag at port (Anonymous 1998a,b,c).

1032
SLENDER STRUCTURES AND AXIAL FLOW
operations (Anonymous 1998a,b). It is currently being used in more than 20 countries
around the world in varying sizes, ranging from 4.5 m 3 capacity (7.5 m long) to 1100 m 3
(91.5 m long). A feasibility/economics study may be found in Lattore & Zachariadis
(1983). Another system for short-haul sea-transport offresh water has recently appeared
[Figures 8,140(c)], involving a flatter-topped bag. t
ln the case of towed seismic arrays (acoustic streamers), information beyond what
is already given in Section 8.9.1 is almost impossible to obtain from either the oil/gas
exploration concems or from the military using them for submarine detection. However,
it is well known that these systems are widely used [refer, e.g. to Anonymous (1998d)].
Work on the related topics of deploying/extruding cylinders and high-speed travelling
band s and trains is clearly also pertinent to engineering applications,
Finally, the work on biomimetic devices for marine propulsion has already been men-
tioned in Section 8.9.10 - see, for example, Triantafyllou & Triantafyllou (1995). An
off-shoot of this type of work, involving the use of foils for electricity production, is
discussed in Chapter 10.
t Alas, despite the author's repeated attempts, the manufacturers failed to provide additional information,
other than a publicity brochure.

9
Clustered Cylinders in Axial Flow


9.1 INTRODUCTION


The dynamics of c1ustered slender cylinders in axial flow is fundamentally similar to
that of a solitary cylinder, except in one important aspect: motions of the cylinders are
coupled. As one cylinder oscillates, the pressure field around all the others is modified
and unbalanced, and hence the other cylinders are induced to oscillate also.
In Chapter 8, we have dea1t to some extent with applications involving c1ustered
cylinders in a simple manner, taking into account only the confinement effect due to
the presence of adjacent cylinders or other structures, but treating each cylinder as if it
were alone. In this chapter, this is remedied.
Fluid coupling is discussed briefly in Section 9.2, followed by a discussion of the dynam-
ics of clustered cylinders in axial flow and flow-induced vibration of c1ustered cylinders
in Sections 9.3 and 9.4. Finally, the dynamics of an articulated system of cylinder bundles
(c1usters), but with the c1ustered cylinders moving in unison, is discussed in Section 9.5.
There is a hea1thy body of literature on the fluid mechanics per se ofaxial flow past
an array of parallel cylinders, conceming velocity and vorticity distributions in the inter-
cylinder passages, This c1early is of great interest for heat transfer considerations, for
example, in nuc1ear reactor cores and heat exchangers. It wil1 not be discussed here. The
reader is referred to the papers by Hooper & Rehme (1984) and Eifler & Nijsing (1967),
for instance, and the references cited therein.


9.2 HVDRODVNAMIC COUPLlNG


9.2.1 General discussion


Hydrodynamic coupling arises whenever more than one cy1inder 1ies in proxllmty
of another or others, and the group is immersed in a quiescent or flowing 'dense
fluid', that is, a fluid of non-negligible density. If lateral motions of the cylinder,
ar of the fluid, resu1t in flow separation, things can become analytically intractable.
Otherwise, potential flow may be used to study not only hydrodynamic coupling-
particularly inertial (or added mass) coupling - but also the cross-flow through
groups of cylinders; see, for example, Da1ton & Helfinstine (1971), Balsa (1977),
Pa'idoussis ef aZ. (1984b) and Beale (1993),t in the first of which it is said that


tS ee also Bea1e & Spa1ding (1999) for viscous cross-flow,


1033




1034
SLENDER STRUCTURES AND AXIAL FLOW
this type of study goes way back, to Hicks (1879), who solved the problem of two
cylinders. t
Here, however, we are interested mainly in axial flow and hence zero mean cross-flow,
though the aforementioned studies are still pertinent and useful. Hence, we are primarily
concemed with inertial coupling coefficients, that is, virtual or added mass coefficients
(see Section 2,2.2), but also the damping coefficients (Section 2.2.3).
The dynamics of groups of paral1el cy linders in inviscid quiescent fluid has been studied
theoretical1y by, among others, Chen (1975a-c), Yamamoto (1976), Pa'idoussis & Suss
(1977a), Chung & Chen (1977), Pa'idoussis et al. (1977), Weppelink (1979), Dalton (1980),
Pa'idoussis & Besanon (1981), Lin & Chen (1981) and Lin & Raptis (1986), in the last two
considering an 'acoustic fluid', thus taking compressibility effects into account. Chen's
work in 1975, in particular, has been seminal, in the sense that it is responsible for
sensitizing the research community to coupling per se and to the interesting features of
coupled motions of large or smal1 groups of cylinders.
Early experiments on groups of cylinders (smal1 clusters or rows) have been done by
Moretti & Lowery (1976) and Chen & Jendrzejczyk (1978).
The nature of (inertial) coupling is discussed next, with the aid of some typical results.
ln contrast to most parts of the book, these are presented before first giving the methods
by which they could be obtained, for two reasons: (i) to motivate the analysis and (ii) to
make the rest of the literature review more easily understandable.
An obvious consequence of inertial coupling is that, instead of one added (or vir-
tual) mass, the same for each cylinder, we now have an added mass matrix -
cf. Section 2.2.2(iv) - the elements of which play the same role as influence coeffi-
cients in structural dynamics. Thus, whereas a system of K identical cylinders in vacuo
has but one first-mode natural frequency, one second-mode natural frequency, et seq.,
the system in a dense fluid generally possesses groups of natural frequencies, each group
comprising 2K frequencies, although some may be repeated. Hence, we have a group of
2K modes with first flexural-mode shape, another of 2K modes of second flexural-mode
shape, etc., where the modes in each group differ from one another in terms of differing
cross-sectional pattems of motion.
The effect ofinertial coupling is illustrated in Figure 9.1. For this very simple system and
because of symmetry, the eigenmodes are segregated into two modes involving y-motion
only, and two with z-motion only, as seen in Figure 9.l(b); specifical1y,
Mode 1 :
Mode 2 :
Mode 3 :
Mode 4 :
VI = V2,
VI = -V2,
WI = W2,
WI = -W2,
WI = W2 = O,
WI = W2 = O,
VI = V2 = O,
VI = V2 = O,
0)1 = 6.83;
0)2 = 9.71;
0)3 = 8.33;
0)4 = 8,92;
where the dimensionless eigenfrequencies given above, defined as in (8.28b), are al1 of
the first ftexural-mode shape. General1y, as wil1 be seen more convincingly later, modes
implicating motion of relatively higher fluid kinetic energy are associated with relatively
lower eigenfrequencies, and vice versa. The group of modes with second and higher
flexural-mode shapes have identical cross-sectional modal pattems to those of the first.
tHicks' (1879) is a most interesting paper, treating all kinds of situations involving two cylinders in a fluid,
It is also a period piece: a 54-page paper with spectacular mathematics, for example elliptic functions, but only
two measly diagrams. Present-day referees would have had a fit!

CLUSTERED CYLINDERS IN AXIAL FLOW
1035
@,
(b)B088
(a)
VI
V< 1  A A  A  A  A A  I\  A.  A A  M  A
O V lV  \ft  6 V V 8W  o V  2  'V I  Vlv6 V t 8  V 20
(c)
w,
W2
Dimensionless time, T
(d)
Figure 9.1 (a) Definition diagrams for a system of two pinned-pinned cylinders in quiescent
dense fluid contained by a circular channel with G c = G w = 1/4. (b) The cross-sectional patterns
of the four modes of the system (I corresponds to W\, et seq.). (c) Theoretical response of the
system when cylinder I is given an initial displacement in the y-direction, and (d) in the z-direction
(Pai:doussis & Bensanon 1981).
If cylinder 1 is given an arbitrary initial displacement, VI (O) = 1, with first-mode shape
in the y-direction, then cylinder 2 is induced to osci11ate also, as shown in Figure 9.1(b).
Since the initial conditions do not correspond to an eigenmode, a11 modes are genera11y
involved in the ensuing motion; in this particular example, however, since there is no
coupling between y- and z-direction modes, only modes 1 and 2 are involved. Similarly,
if cylinder 1 is given an initial displacement WI (O) = 1, again both cylinders osci11ate,
Figure 9.1(c), their motion now involving a synthesis of modes 3 and 4, Moreover, in
this case, because W3 and W4 are c1ose, the characteristic beating phenomenon is c1early
visible.

1036
SLENDER STRUCTURES AND AXIAL FLOW
Table 9.1 Eigenfrequencies of a cluster of four pinned-pinned cylinders: (a) in a
circular channel as in Figure 9.1(a), and (b) in a square channel; G c = G w = 1/4
(PaYdoussis et al. 1977).
Confinement
Wl
W2
W3,4
W5,6
W7
Ws
(a)
(b)
8,56
8.51
8,66
8,57
8.69
8.64
9,62
9.39
9.75
9.40
10.13
10.08
Another examp1e, for a four cy1inder duster in a rigid tube (channe1), is shown in
Figure 9,2. For this and subsequent discussion, we define the geometry in tenns of
smal1est intercylinder gap gc smal1est cylinder-to-wall gap
G c = - G w = (9.1)
cylinder radius li' cy1inder radius
For the G c = G w = 1/4 system and pinned-pinned cy1inders, we have the set of first-
mode (meaning first flexura1-mode) eigenfrequencies given in Tab1e 9,1. In this case,
because of symmetry, we have two pairs of repeated eigenfrequencies, The cross-sectional
pattems of motion associated with these modes are shown in Figure 9.2(b), where it is of
interest that the amp1itudes of the four cylinders are equa1 for al1 the distinct eigenfrequen-
cies (though the directions are different), but for the repeated eigenfrequencies we have
one pair of vectors 10nger by 1.88: 1 than the other pair. These cross-sectiona1 pattems are
identica1 for the second, and higher, flexura1-mode groups.
Furthennore, these 'nice' cross-sectiona1 pattems of the eigenmodes are obtained only
after the modes have been orthogonalized (obvious1y this affecting only the repeated-
eigenfrequency modes). Otherwise, one obtains 'odd-100k:ing' cross-sectiona1 modal
shapes, such as shown in Figure 9.2(c) for these modes.
It is a1so of interest that the cross-sectiona1 pattems of Figure 9,2(b) as well as the
ordering of the eigenfrequencies depend on the confining geometry - even though
the eigenfrequencies may not be significantly different (cf. Tab1e 9.1). Thus, for the
square-section confining channe1 and still G c = G w = 1/4, the 10west mode (mode 1)
has the modal pattem ofmode 2 in Figure 9.2(b), and vice versa;t moreover, the ratio of
the vectors for the repeated eigenmodes is no 10nger 1.88: 1 but 1.25: 1. Incidental1y, the
same switching of mode pattems 1 and 2 vis-a- vis Figure 9 .2(b) occurs for circu1ar con-
tainment but higher G w ; for examp1e, for G w = 1.75, as shown via response ca1culations
by Curling (2001), and for higher G w in Figure 9.15.
Final1y, Figure 9.2( d) shows the effect of proximity (here for G c = G w) on the extent of
the first-mode frequency band. It is seen that increased inertia1 coupling via tighter spacing
makes the frequency band spread, whereas at G c = G w  10 the eight eigenfrequencies
are do se to one another and not too different from the value for an iso1ated cylinder in
unbounded fluid. If individual curves are p10tted for each of the w, then one can observe that
they cross for sufficiently small G c = G w . as discussed above; see figure 4 in Pa1doussis
et a/. (1977a).
Another way of illustrating coupling is by 100k:ing at the PSDs of the motion, as shown
in Figure 9.3 for a four-cy1inder duster (Chen 1975c). Here too, vibration of cy1inder 1
tThe moda1 pattems in figure 6 of Pai'doussis et a/. (1977) actua1ly correspond to the system in a square-
section channel, and not as stated therein. This error has found its way into some subsequent publications, for
example in Pai'doussis (1979) and Naudascher & RockweII (1994, figure 7.62).


(a)
-,!..45°'
r
/

5°
r-
(b)
/5

CLUSTERED CYLINDERS IN AXIAL FLOW
1037
'::___::.-::::: ::::;;i 3 ':
....._------"......
,
./
/

'\..
.JI
2
3
4
/
'\..

/

JI
'\..'
6
,
./
,
/

7,
/
8
./

'\..
10.4
1730 73 ..y-29.3°
-,- "\ ......;. -ť
3 4 9,6
\ J  Solitary cylinder,

unbounded fluid
w
8,8
65 t \ '1 8 1.5 0 - 29,3° Band of w for
T T cross-sectional
5 6 modes 1 to 8
! 8,0
\ / -
(c) O 2 4 6 8 10
(d) G c = G w
Figure 9.2 (a) A duster of four pinned-pinned cylinders in quiescent dense fluid contained in a
circular channel with G c = G w = 1/4, and pro forma the first two axial mode shapes. (b) The
cross-sectional modal pattems of the eight modes of the system, as obtained after orthogonalization;
(c) the modal pattems of the repeated eigenfrequencies (cf. Table 9.1) without orthogonalization.
(d) The frequency band for the first flexural-mode group of eigenfrequencies as a function of
interstitial and cylinder-channel spacing for G c = G w (Pai'doussis et al. 1977).

1038
SLENDER STRUCTURES AND AXIAL FLOW
10
9
8
7
6
5
4
 
eS
8 1
' O
 2
' 1
'"
 O
3
2
1
O
2
1
Oť:. x
Q)G)
r<D
a sin wt
Tube 1
y-direction
Tube 4
x-direction
5
4
3
2
1
O
4
3
2
1
O
4
3
2
1
O
3
2
1
60 O 10 20 30 40 50 60
Frequency (Hz)
(b)
Figure 9.3 Resonance and coupling effects for a c1uster of four cylinders in unconfined, quiescent
dense fluid, when cylinder 1 is excited hannonicaUy in the direction shown: (a) for a loosely spaced
geometry, G cx = 2, G cy = 3; (b) for a more tightly spaced one, Gcx = 0.10, G cy = 0.15
(Chen 1975c).
Tube 4
y-direction
induces motion of the other cylinders also, If the cylinders are relatively far apart, then
a single peak is seen in the PSD, as in I:"igure 9.3(a) - but a broader peak than for
a solitary cylinder because of the presence of eight dosely spaced first flexural-mode
eigenfrequencies (though some are repeated), For the more tightly arranged array of
Figure 9,3(b), the eigenfrequencies are sufficiently separated to give rise to multiple peaks
in the PSD.
ln Figure 9.4, cylinder 1 is constrained to move in one direction with a constant amp-
litude, at a frequency corresponding to 'constrained resonance', whi1e the other cylinders
are free. Nonnal resonance, obtained if the system is excited by a harmonie force, with
all cylinders free to osci11ate, would occur at one of the eigenfrequencies of the system.
Constrained resonance, on the other hand, occurs at other frequencies. t
ln Figure 9.4(a), cylinder 1 is constrained to osci11ate in the y-direction at ev = 8.0. It
is seen that the other two cylinders are excited in both directions (because such are the
modes), with increasing amplitude. Amplification in the y-direction is particularly rapid.
The same is shown in Figure 9.4(b) for a row ofthree flexible cylinders, when cylinder 1 is
excited at ev = 9.90 in the z-direction. Cylinder 3 amplification is specially rapid, despite
O 10 20 30 40 50
Frequency (Hz)
(a)
tFor example, for the simple two-cylinder system of Figure 9.1, the ordinary resonances for y-direction
motionoccuratwl = [n 4 /(m22+m2I)]I/2 andW2 = [n 4 /(m22-m2I)]I/2, whileconstrainedresonanceoccurs
at W cr = (n 4 /m44)1/2, such that WI < W u < W2; similarly for the z-direction (pai:doussis & Besan4ton 1981).
Here mij are elements ofthe total mass matrix; see Section 9,2.2.

v< ]
o
CLUSTERED CYLINDERS IN AXIAL FLOW
1039
'fY
@N
, I , I
2 3 4 S 6 7 8 9 10
v< ] ?  A f\ (\ f\ f\ f\ !\ A f\ A
o 1 '-' Y V 3 V 4 V s V 6 V V 7 V 8 V 9V  O
V< ] _  0  f\ 1\ /\ A f\ 1\ f\ A r
o 1 "'2 V 3 V N VS V 6 V 7V W V 9 Vľo
1 1
Wl .
o V'v '\...J0 '-' .J"V' ..../'v '-/ '"\....,".J"'v
o 1 2 3 4 S 6 7 8 9 10
W< ]
O
W< ]
O
(a)
"'1'......,....,....;:;:.r>.AAp,r
r_ - -     v V V'
2 3 4 S 6 7 8 9 10
r ......v......v0
2 3 4 S 6 7 8 9 10
Dirnensionless tirne, T
 CD CD CD 
Wj  1 f\ f\ f\ f\ f\ f\ f\ ti f\ f\ (\ f\ f\ f\ f\
o V VI VN V3V4V V S V 6 V VN V 8 V9V \'i o
W< ]
O
W3 ]
O
(b)
(',1\1\f..,f\f\(\f\f\f\f\f..,f\f\[\,
 1 VF VN 1J4VSV V6VN V 8 V V9V lO
_"01\f\f\AAAAAAA
í- '2 v 3 v '4  s V V 6 V  V 8 V V 9 V lO
DlrnenslOnless hrne, T
Figure 9.4 (a) 'Constrained resonance' of a c1uster of three pinned-pinned cylinders in dense
fluid (G c = G w = 1/4) when cylinder 1 is excited to osci1late in the z-direction at a constant
amp1itude and frequency w = 8.0, while cy1inders 2 and 3 are free. (b) Constrained resonance for
a row of cylinders in quiescent fluid (G c = I, G w = 00), when cylinder 1 is excited to oscillate
in the z-direction at a prescribed amplitude and w = 9.90, while cylinders 2 and 3 are free
(Pai'doussis & Besanon 1981).

1040
SLENDER STRUCTURES AND AXIAL FLOW
this being farther away from the cylinder generating the oscillation than is cylinder 2. t
The interest in these resu1ts from the practical point of view is self-evident: resonance-
like conditions may materialize at frequencies other than those of trne resonance and at
some distance from the excitation source. The discussion so far has ignored the effect
of viscosity. As shown for simpler systems (Section 2.2.2), this is pennissible so long
as the Stokes, or oscil1atory Reynolds, number is not too small - physical1y, provided
the fluid-fil1ed gaps and the frequency are not too small, nor the fluid too viscous - see
also Section 9.2.3. In general, for reasonable systems, viscous effects modify rather than
overturn the dynamical behaviour discussed in the foregoing.
Two typical sets of experimental results from Chen & Jendrzejczyk (1978) are compared
to theory in Figure 9.5, showing the steady-state acceleration response of (a) a row offive
cylinders when cylinder 5 is excited in the y-direction, and (b) a three-cylinder cluster,
again when one of the cylinders is excited in the y-direction, in both cases as a function of
the forcing frequency. It is seen that agreement with theory is remarkably good, especial1y
in the first case. In the theoretical calculations, damping is taken into account, considering
it to be equal to the uncoupled modal damping of one of the cylinders in water. In the
comparison of Figure 9.5, the magnitude of the excitation force used in the analysis is
obtained from the experimental values, such that theoretical and experimental accelera-
tions be equal for the cylinder excited at 50 Hz. Even with this 'calibration', however, the
degree of agreement is remarkable; for those with no first-hand experience, it should be
stres sed that such experiments are anything but simple.
The foregoing discussion demonstrates that there are several interesting facets in the
dynamics of c1ustered cy1inders, coupled via the interstitial fluid. These are explored
further in other sections of this chapter. N ext, some of the analytical methods used to obtain
hydrodynamic coupling characteristics, such as those in Figures 9.1-9.5, are presented.
9.2.2 Analytical methods for inviscid coupling
Inviscid, or inertial, coupling is discussed first. The 'c1assicaľ method developed by
Paldoussis & Suss (1977a) and Paldoussis et al. (197.7) is presented first, to be followed
by a discussion of other methods.
Consider a c1uster of K identical cylinders immersed in a 1iquid-filled channel (normal1y
a rigid container). Strictly speaking, the hydrodynamic forces at each cross-section depend
not only on local displacements, but also on the slopes of the cylinders. However, upon
assuming that displacements and slopes are smal1, it is reasonable to consider that motions
at any cross-section are as if the cylinders remained paral1el to one another. Thus, by
taking the hydrodynamic field to be two-dimensional, the problem reduces to detennining
the added mass matrix of a c1uster of paral1el, infinitely long, flexibly supported, rigid
cylinders.
The system under consideration is shown in Figure 9.6. Any fluid velocity vector is
equal to V <jJ, where <jJ is the velocity potential satisfying
V 2 <jJ = O
(9.2)
tl n this simple case, analyticaI solution ofthe systemis feasible, forexample by the use ofLaplace transforms
(Besanon 1978, appendix F), It is shown that, whereas the growth rate for cylinder 2 is linear in 1", that of
cylinder 3 contains a quadratic term in 1",

Ir)
...
o o
"O
00 .5,....:..
,.....,
'" N -00
r-- e; ;;"1-
"'0'\
o » =-
;;j o
r-- ..c::
 u  
o '" ::s
u '" O" ,....., '"
..c:  'f""""I'G)'
E-<
o I>/) II t!
'" ,5 ,,"O
'" e í.J
o
'" "'" '-" .....,
o 
'" ... =
""1" N 0""1" N o '" ""1" N o ""1" N o ""1" N o 00 \O ""1" N o ... o
 o..c::
"Ou
8-: ='-"
8 :.=;::
.::i o 
e 'G 00 I o
 0..c::
 '" g ;> '"
<!': '"
"" r-- «:1 «:1
11 o » 'O-g
u
r-- = o....
a u ",.
't:: '" g. = '"
CI> '"  o 
Po .g = = = = '"
 g .Si g g o I>/)
 '" O'
 N  c<"I t> . N " c<"I ,5 "'<'1
CI> u CI> e u CI> CI> u e = ...
.1:: .g '" o o
 :;:;  .o :;:; .o .o .o '" o
'Ó   'Ó   "'" 'p "O
   "" "" «:1 =
o ...'
o-
N o '" o N o ""1" N 00 '" '" -;;.,
""1" N 0""1" ""1" N ""1" o ""1" N o o '"
ÍI 'UOIUl:!!:!:J:JU :!qnHo :!pm!!dw1! :!1!s-Ápu:!s  '" =
1;j o
o..c::
C;J 
.... ----
'fN
-6'11
«:1
o "
....í.J
"',-"
o ...
-=B
... '"
g .a
Q'"
» o t
5 0"0
 .g .s
o ::s "O-
u O" aY'
..c: 
E-< .... o
N '" ""1" '"  ]
u u u u CI> 'a s....
.g .o .g .o .o V>  'e: «:1
E-<  E-<   '" o ...
o <£
""1" N o ""1" N o ""1" N o ""1" N o \O ""1" N o'" o o
=  s
e-lJ o «:1
o  
00 o..c::
@  .oE-<
 '" g =----
6"" o r-- 0.0
'" '-"
= 6> o  .!a d
CI> r--  S'
a e '" ::s o..c::
't:: O"
u '"  u:
Po
><  ____ «:1
 N c<"I ""1" '" .5f "O
u u .g  u  B
.o .o .o '" II) 'u
  E-<  '" "'" . 
c:l\o
o .
""1" N o ""1" N o ""1" N o ""1" N o \O ""1" N o'" 
ÍI 'UO!1!l:!p:J:J1! :!qm]O :!pm!!dw1! :!1!S-Áp1!:!S  

1042
SLENDER STRUCTURES AND AXIAL FLOW
z
Ro
Figure 9.6 Definition diagram, coordinate systems and symbols used in the determination of the
velocity potential of a cluster of cylinders in dense quiescent fluid (Pai:doussis & Suss 1977a),
in the yz-plane, with boundary canditions requiring that (i) the velocity ať the fluid normal
to the flow channel be zero on its surface, and (ii) the velocity oť the fluid normal ta the
surface oť each cylinder be equal to the velocity oť the cylinder in that direction,
The potential li> is taken as the sum oť the potentials li> j, j = 1,..., K, due to the
presence oť the cylinders in the channel, t such that
K
li> = L li> j (r j, () j ) .
j=l
(9.3)
By virtue oť the linearity oť (9.2), we have
V 2 cpj(rj, ()j) = O,
(9.4)
whose solution is oť the form
00
li> j (rj, (Jj) = L {AnjrJ cos n(Jj + BnjrJ sin n(Jj
n=l .
+Cnjrjn cos n()j + Dnjrjn sin n()j} .
Each oť the cp j has been written in terms oť (r j , () j ), the coordinate system centred on
the jth cylinder; but, to apply the boundary conditions, it is necessary to expres s CPj in
terms oť coordinates centred on each oť the other cylinders. This can be done by (a) the
use oť the coordinate transťarmation
(9.5)
rjej(}j = ri ei8i - Rije i1frij ,
(9.6)
tlt is more correct to consider each cp j to be due to the presence of cylinder j rather than due to its motion,
ln that sense, each cp j may be viewed as the velocity potential due to a source-sink doublet representing the
cross-section Dr a cylinder in an otherwise unperturbed medium,

CLUSTERED CYLINDERS IN AXIAL FLOW
1043
where Rij is the centre-to-centre distance between cylinders i and j, with i = O reserved
for the coordinate system at the centre of the channel, and (b) using series expansions for
[ . (j , ] ::I::n
ri e ' i - Rije'1/fij .
After considerable manipulation the potentia1 due to cylinder j, written in terms of
coordinates centred on cylinder i, is
00 { n ( _l ) n-m R,:.-mr'!1n'
i IJ 1 .
CP J ' (ri , ej) =" " { Anj cos [mei + (n - m) o/ij ]
L., L...J m!(n - m)!
n=l m=O
. 00 ( _1)n (n + m - 1) !rj
+Bnj SIn [mei + (n - m)1frij]) + L n+m
m=O m!(n - 1)!Rij
x {Cnj cos [mei - (n + m)o/ij] - Dnj sin [mei - (n + m)O/ij]}}, (9.7)
which converges for ri < Rij, or
cP (ri , ei) = f { [same as first sum in equation (9,7)]
n=l
00 (m _ l)!R-n
+ L ,'J ,m {Cnj COS[mei - (m - n)1frij]
m=n (n - l).(m - n).r i
+Dnj sin[mei - (m - n)1frijJ) },
(9.8)
which converges for ri > Rij (Suss et a/, 1976). Hence, the total potential in terms of
coordinates centred on cylinder i, denoted by cpi, may be written as
K
cpi = L * CP) + CPi,
j=1
(9.9)
where the starred summation excludes j = i,
Boundary condition (i), stated below (9.2), may be applied to each of the partial
potentials separate1y, and is expres sed as
acpJ/aro Iro=Ro = O,
j = 1,2,3"."K,
(9.10)
where cpJ is obtained from equation (9.8) with i = O, unless a cylinder, say cylinder q, is
located at the centre of the outer channel, in which case cp2 is obtained from equation (9.5)
directly, Boundary condition (ii) may be written as
; Ir;=R; = O,
i = 1, 2, . . . , K,
i i= /;
(Jcpl I . e .. e
- = Wl cos f+Vl SIn f,
(Jrz n=Ri
(9.11)

1044
SLENDER STRUCTURES AND AXIAL FLOW
for l = 1, 2, 3, . . . , K, where Wl and VI are the ve10cities of cylinder l (Figure 9.6).
Equations (9.11) imply that the constants Anj to Dnj must be of the form
K
Anj = L (anjlWl + anjlVl) / R't l ,
1=1
K
Cnj = L (YnjlWl + CnjlVl) Rj+l,
1=1
K
Bnj = L (13njlWl + bnjlVl) / R't l ,
1=1
K
Dnj = L (8 n jlWl + dnjlVI) Rj+l.
1=1
(9.12)
Combining expressions (9.12), (9.7), (9.9) and (9.11), and also expressions (9.12), (9.8)
and (9.10), one obtains the following relations for the anjl to dnjl:
[{ 13 'I } { a 'I } ]
K 00 (_1)n-m 'R,!.-m R,!,-I n) n}
L* L n. I} 1 bnjl sin(n -m)1/fi' + anjl cos(n -m)1/fi'
( m - 1 ) ' ( n _ m ) 'R-1 -anjl ) 13njl }
j=1 n=m . . }
-anjl bnjl
K * 00 (_1)n(n + m -1)!R;n-1 R+I [{ n<l } .
+ '" '" } n}l sm(n + m)1/f"
  (n - 1)!(m - 1)!R n + m Ynjl I)
}=I n=1 }
Cnjl
{ ynjl } ] { amu } { Ymu } { 8Im8U }
+ njl, cos(n + m)1/fij + m ami,l _ m Cml = O ,
8n}1 13m,l 8mtl O
-dnjl bmu dmU 81m 8 U
(9.13)
where m = 1,2,3, ..., l = 1,2,3, ..., K, i = 1,2, ..., K;
00 (_1)n-m n !R3jm RO-I
L ( _ 1)'( _ )'R n - I
n=m m . n m. j
[{ anjl } { 13njl } ]
l cos(n - m)1/fOj + :: sin(n - m)1/fOj
bn}l an}l
[ {  'I }
m ' R m-n R n+1 
m. O" C '
_ }} n}l cos(m - n)1/fo'
 (n -l)!(m - n)!RO'+i =:;: ,J
{ 81 } ]
- dnjl sin(m - n)1/fOj = {O},
Yn}l
Cnjl
(9.14)
where m = 1,2,3,..., and l, j = 1,2,3,..., K. If there is a cylinder at the centre of
the outer channel, then in equation (9.14) we must have j =1= q and utilize the additional

CLUSTERED CYLINDERS IN AXIAL FLOW
1045
equation
1 { anql } +1 { ynql }
( Ro ) n- anql _ ( Rq ) n Cnql ={O},
 ql  l
, bnql dnql
where n = 1,2,3, ..., 1= 1,2,3,..., K.
N ow, the z and y components of the inviscid hydrodynamic force on cylinder i, denoted
by Fz and Fy, respective1y, may be expressed as
(9,15)
{27f
Fz = Jo pi Iri=Ri Ri cos (}id(}i,
{27f
Fy = Jo pi Iri=Ri Ri sin (}id(}i,
(9,16)
where pi is the perturbation pressure and may be found from Bemoulii's equation,
pi = -pa</Ji lat.
(9.17)
PerfoffiÚng the operations required by expressions (9.16) and (9.17) on </Ji, we obtain
K
F i R 2  { .. .. }
Az = p7r i L SilWl + eilvl ,
1=1
K
F i R 2  { .. k .. }
Ay = p7r i L KilWl + ilVl ,
1=1
(9.18)
i = 1,2,3, "" K, where from equation (9.13) with m = 1, it is found that
Sil = -Oil - 2Yli/, eil = -2cli/, Kil = -2oli/, kil = -Oil - 2dli/,
(9.19)
Oil being the Kronecker delta. The Sil, eil, Kil and kil are the e1ements of the dimension1ess
added mass matrix. For a disp1acement vector
{U}={W1,W2,'" ,WK; Vl,V2,..' ,VK}T,
(9.20)
the dimension1ess added mass matrix is t
[ [8]
[Ma] = [K]
[e] ]
[k] ,
(9.21)
For a c1uster of K flexib1e or flexib1y supported cylinders in quiescent fluid, the equation
of motion is
a4Ui 2K a 2 Uj
EI ---;r + L (pAMij + mOij) ----z = O,
ax j=l at
i = 1,2,...,2K,
(9.22)
for smali 1atera1 motions, where E I, A and m are, respective1y, the flexural rigidity, cross-
sectiona1 area and mass per unit length of each cylinder, p is the fluid den si ty, Mij the
t An unexpected application of basically the same analytical approach has been utilized by Hara & Kohgo
(1986) in a model for tbe determination of tbe added mass in two-phase llow.

1046
SLENDER STRUCTURES AND AXIAL FLOW
e1ements of [Ma], and Ui is an element of {u} of equation (9.20). Equations (9.22) may
be written in dimension1ess form as follows:
a 4 a 2
a4 {ry} + [M] aT 2 {ry} = {O},
(9.23)
where
 = xl L, T = [EII(pA + m)]1/2 (ti L 2 ), (3 = pAI(pA + m),
{ry} = {u}IL, [M] = {3[M a ] + (1 - (3)[I], (9.24a)
[/] being the unit matrix and L is the 1ength of the cylinders. The 2K dimensionless eigen-
frequencies obtained from this equation are re1ated to the dimensiona1 circu1ar frequencies,
Qj, by
úJj = [Cm + pA)1 E/]1/2 QjL 2. (9.24b)
The free-vibration results presented in Section 9.2.1 have been obtained by solving (9.23).
This ana1ysis has been extended to the case where the cylindrical channel is also free
to move (PaIdoussis et al. 1977; appendix 1).
A similar method is used by Chen (1975c) for the determination of the mass matrix
of the system, but in unbounded fluid. This has later been extended by Chung & Chen
(1977) to deal with fluid bounded by a cylindrical channe1, that is, for the same system as
in the foregoing, However, it is of interest that, instead of expressions (9.3) and (9.5) for
the ve10city potential, Chung & Chen use
K
<P = <Po + L<Pj,
j=1
(9,25)
where
00 n
L ro
<Po = _ 1 (aon cos n()o + bOn sin n()o) ,
R n -
n=1 O
00 R+1
<Pj = L + (ajn cosn()j + b jn sinn()j).
r.
n=1 }
(9.26)
ln particular, comparing (9.5) to (9,26), the expression for <Pj in the latter would appear
to be incomplete and more appropriate for an unbounded fluid (in which Anj = Bnj = O
in (9.5), since <Pj -+ O for r -+ (0); but there is an additional partial potential <Po in
(9.26), associated with the presence of the outer channel. Yet, despite these differences,
the numerica1 results obtained by the two formulations have been found to be the same.
This sufficiently exercised the authors, not to mention various referees, for a fonnal
demonstration of equivalence to become useful, which is done in PaIdoussis et al. (1977,
appendix 2). The same basic method has been used by Lin & Chen (1981) to obtain the
added mass in compressible fluids.
The degree of validity of these 2-D ana1yses has been tested by a 3-D potential flow
ana1ysis (Weppe1ink 1979; Van der Hoogt et al. 1979). Ca1cu1ations are done for clusters
of 2-25 cylinders, and G c = 0.5 to 2.0. For LI R = 50 and 100, the difference between

CLUSTERED CYLINDERS IN AXIAL FLOW
1047
2-D and 3-D in the e1ements ofthe added mass matrix is 1ess than 1 % - but it is expected
to rise for sma11er L / R.
Other approaches to the added mass coefficients of c1ustered cy1inders have been
developed in conjunction with work on cross-flow around such c1usters, whi1e making
the severely simp1ifying assumption that there is no flow separation and that the flow
field is potentia1. For our purposes, however, where there is no mean cross-flow velocity,
such methods are, in princip1e, as good as the one just described. Da1ton & Helfinstine
(1971) consider each cy1inder as a doub1et in unifonn flow, plus the appropriate image
doub1ets due to mutual interaction of the cy1inders, thus obtaining an infinite-series com-
p1ex velocity-potentia1 function; however, ana1ysis becomes overlaborious for systems
of more than a few cy1inders, and singularities arise for c10sely spaced cy1inders. The
same approach in followed by Yamamoto (1976). Balsa (1977) solves the potentia1 flow
prob1em for a genera1 array of cy1inders in cross-flow by utilizing matched-asymptotic-
expansion techniques, and so does Chen (1978) by solving the Lap1ace equation in tenns
ofinfinite Fourier series. This is a1so done by Pa'idoussis et al. (1984b) in a more comp1ete
fonnulation. By cross-comparison, it has been found that results obtained by Dalton &
Helfinstine's method agree dosely with those from PaYdoussis et al., but the same can-
not be said for Ba1sa's, as discussed in Pa'idoussis et al. (1984b). Yet another ana1ytical
solution, again for a potentia1 cross-flow, has been obtained by Bea1e (1993); extens-
ive computations have been conducted for c1usters of different geometrie pattems and
pitch-to-diameterratios of 1.05-100 (G c = 0.05-99).
The added mas s matrix may a1so be determined via the finite-element method. One
such computer code has been developed by Suss et al, (1976) and Pai'doussis et al. (1977),
using triangu1ar finite elements - see Appendix U. Not surprising1y, the same results
are obtained, but computationa11y 1ess efficiently. However, the finite-element method is
easi1y adapted to dea1 with noncircu1ar cy1inders in arbitrari1y shaped channels, whi1e the
c1assica1 method just presented cannot.
Of course, the object in determining the hydrodynamic coup1ing is to obtain the vibra-
tion characteristics of the cluster in dense fluid. However, if the number of cylinders is
1arge, determining a11 the individua1 eigenfrequencies and eigenmode pattems can be quite
tedious. t ln such cases, one is more interested in the ranges of frequencies within which
1ie al1 the modes of first flexural-mode shape, second flexura1-mode shape, et seq. One
way of doing this is by means of homogenization schemes, in which the discontinuous
medium (invo1ving rigid cylinders interspersed in the fluid) is replaced by an equivalent
continuum. Such schemes have been developed by Schumann (1981) and P1anchard and
co-workers (Conca et al. 1990, 1994, 1995); for geometrica11y periodic cy1inder c1usters,
the flow potential is decomposed into Bloch waves - see Bensoussan et al. (1978).
These powerfu1 methods are too involved to present in detail here; the interested reader
is referred to the specia1ist books on the subject (Conca et al. 1994, 1995). The same
applies to the 'flow-path network' methods of Tomita (1984) and Tomita & Sasaki (1984),
deve10ped especially for seismic applications, where all cy1inders move in unison as if
rigidly connected.
ln all of the foregoing, cylinder motions are presumed to be sufficiently sma11 compared
to the intercylinder gaps, so that the geometry may be considered to remain sensibly
tO ne heroic such effort was made by Ibnou-Zahir (1984) for a system of 400 cylinders in a square-section
channel!

1048
SLENDER STRUCTURES AND AXIAL FLOW
una1tered. If this is not so, because of 1arge amp1itude motions, the added masses become
dependent on disp1acement and hence on time, as treated by Gorman & P1anchard (1988).
Finally, it shou1d be mentioned that a who1e chapter on the topic of hydrodynamic
coup1ing may be found in Chen (1987, chapter 3) containing a wealth of usefu1 information;
see a1so Rousseau (1994). For examp1e, Figure 9,7 gives the upper and 10wer bounds of
/L, which is an 'effective' added mass coefficient, such that the 10wer (1) and upper (u)
bounds of the group of natural frequencies for a cluster of K cylinders, Wnf,K, may be
determined from those of a sing1e one, Wnf, 1, via
I m + pA
W fK = Wn f l,
n , m + /LmaxpA '
u m + pA
Wnf = Wnf,l ,
m + /LminP A
(9.27)
where mis the mass ofthe cy1inder and pA the fluid disp1aced mass. af course, /Lmax and
/Lmin may be determined empirical1y for each array by calcu1ating all the eigenfrequencies
via equation (9.23). However, Chen (1 977 a, 1987) deve10ped an ingenious alternative, of
fundamenta1 interest, as follows. As the added mass matrix [Ma] is symmetric, it may
easi1y be diagonalized; the e1ements of the diagonalized matrix are its eigenva1ues, /L, of
which /Lmax and /Lmin are the 1argest and smal1est. It is shown that this scheme a1ways
gives results which correspond to those found by solving (9.23).
4,0
1.0
---K=l
--- K= 7
---- K= 19
-K=37
00
000
00
3.0
c
e
 2,0
Š
:::t
000
000
000
0,75
@@@@
0000@
@0000@
000000@
80000@
000@@
@0@@
o
0,25
0,50
1.00
G c
Figure 9.7 The upper and lower bounds of the effective added-mass coefficient, ftmax and ftmin,
as functions of G c , defined in (9.1) for various generic clusters of identical cylinders, such as those
shown (Chen 1977a, 1987).

CLUSTERED CYLINDERS IN AXIAL FLOW
1049
9.2.3 Analytical methods for viscous coupling
If the cylinders are very clo se to one another or to other structures, or if the fluid is very
viscous, it is nasonable to expect that the hydrodynamic coupling would be affected
by viscosity, More precisely, it may be stated that such effects are not negligible if the
osci11atory Reynolds number (or Stokes number), defined in what follows, is small.
A finite-element method for the added-mass and viscous-damping coefficients of two
cylinders osci11ating in a confined viscous fluid has been developed by Yang & Moran
(1979). A classical, rigorous method for a system of K cylinders has been developed by
Lin (1987b), and a more approximate approach by Pa'idoussis & Suss (1977a); these last
two are discussed in what follows.
9.2.3(a) Un' s method
The physics of this method is identical to that in Section 2.2.3 which could profitably
be consulted. The linearized Navier-Stokes and continuity equations for motion of the
perturbed fluid for the system ofFigure 9.6, but here taken to be unbounded,
oV 2
p- = -VP+t-tV V,
at
v.V=O,
(9,28)
are used, in which the flow-field is presumed to be two-dimensional, V
Introducing the stream function 1/f as in Section 2.2.3, we obtain (Lin 1987b)
{v, w}.
2 ( 2 10 )
V V - -- 1/f = 0,
vat
(9,29)
where v is the kinematic viscosity; this equation governs the diffusion of vorticity in the
fluid. It is next assumed that motions, including those of fluid particles near the cylinders,
are harmonie , and so take 1/f = <(J exp( -i2t); then, defining the viscous wavenumber by
k = (1 + i) .J r2f(2v), equation (9,29) becomes
V 2 (V 2 + k 2 ) <{J = O. (9.30)
Its solution consists of the solution of
V 2 <{JI = °
(9.31a)
and
(V 2 + k 2 ) <{J2 = W,
V 2 W = O.
(9,31b)
(9,31c)
The general solutions of the two Laplacian equations have the same form,
00 _ R,!,+l
<{JI, W = L [liim cosmei + bim sinmei],
r,
m=O I
(9.32)
where Ri is the radius of cylinder i, and {jim and bim are arbitrary constants. (Since the
fluid is unbounded and <{JI must vanish at infinity, the part of the more general solution

1050
SLENDER STRUCTURES AND AXIAL FLOW
invo1ving r;n in the numerator must be discarded.) Then the solution of the inhomogeneous
equation (9.31b) is easily found to be
00
<P2 = L [lžjm cosm{}j + bjm sinm{}j] Qm(rj),
m=O
Rm+l
Qm(r;) = --T-m + AimH)(krj),
k r i
where H) (krj) is the Hankel function ofthe first kind - the second-kind once again being
discarded to ensure that the viscous waves trave1 outward from the surface of the cylinder;
the first term in (9.33b) is the particu1ar solution and the second term the homogeneous
solution of (9.31b). After substituting (9.33b) into (9.33a) and comparing with (9.32), it
is seen that the particu1ar solution of equation (9.31b) and the genera1 solution of (9.31a)
have the same form, and hence may be combined into
(9.33a)
(9.33b)
. 00 R'!'I+1
<pi = L [aim cosm{}i + bim sinm{}d 7;
m I
(9.34)
the homogeneous solution of (9.31 b) may simi1arly be expressed as
00
<P = L [cjm cos m{}j + dim sin m{}d H) (kri),
m=O
(9.35)
where the constants aim to dim are to be determined from boundary conditions. As pointed
out by Lin, equation (9.35) represents a spatia1 form of the viscous waves generated near
cy1inder i, which decay rapid1y, their amp1itudes diminishing by exp(2:rr) in a distance
2:rr (2v / Q) 1/2. The viscous waves are therefore most1y confined within a boundary layer
of thickness (2v / Q) 1/2 on cylinder i. This boundary 1ayer is common1y referred to as the
penetration depth (Batchelor 1967; Schlichting 1968). In this ana1ysis, the possibi1ity of
streaming (a mean ftow) due to viscous waves is exduded, as convective terms are not
induded in the equations of motion.
The tota1 solution to (9.30) is now expressed as
K
<P = L [<pl + <p] .
i=1
(9.36)
The rest of the ana1ysis consists of determining <P, and hence the constants in (9.34) and
(9.35), such that the boundary conditions on each of the cy1inders are satisfied. This part
of the work, which follows dosely that in Section 9.2.2 and that of Chen (1975b,c), is
outlined in Appendix V.
The viscous hydrodynamic forces on cylinder i are
i 2 :n:
F1 = [ <jr cos {}i - <jo sin {}j ] Rj d{}j,
O =
i 2 :n:
F = [ <:r sin {}j + <:0 cos {}i ] Ri d{}i,
O =

CLUSTERED CYLINDERS IN AXIAL FLOW
1051
where T: r and T: e are the components of the hydrodynamic stress tensor, see equa-
tion (2.64), such that
i i avj
Trr = -p +2f-J,- a '
ri
T:e = -f-J, [ aVj _ vj +  aVj ] ,
ari ri ri aei
where {Vr, Ve} are the flow velocity components in the {r, e} frame. The hydrodynamic
force components are then written in terms of <p and <p as follows:
1 27r [ a i ]
i ' -iQt <Pl i .
Fz = -IQe pRi ri- + <P2 Smei dei,
o ari
1 27r [ a i ]
i . -iQt <Pl i
Fy = lQ e pRi o ri ari + <P2 cos ei de i .
(9.37)
Substituting (9.34) and (9.35) in these expressions, we eventually find
{ F } = -iQpJT R e -iQt  [ Clil
F' I L Til
Y 1=1
O'il ]{ WI } ,
{3il VI
(9.38)
where WI and VI are the components of velocity amplitude of cylinder l in the z- and
y-directions, respectively.t The Clil, O'il, Til and (3il are given in Appendix V; they are
complex, and so equation (9.38) implies the existence of both velocity- and acceleration-
dependent forces.
A very useful analysis for a solitary cylinder is given by Lin (1987b), yielding
{ FZ } = _iQpJTR2e-iQt [ 1 + 2 HI)(kR) ] { W } ,
Fy kH61) (kR) V
(9.39)
which is exactly the same result as that given by Stokes (1851).
For vanishing v or high frequencies, that is, as the viscous wavenumber k -+ 00, the
functional of the Hankel functions tends to -2, and hence (9.39) becomes identical to
(9.18) for K = 1 and harmonie motions, since -iQW = w and -iQV = v.
For 10w frequencies or high viscosity, such that k becomes small, the Hankel functional
asymptotically tends to -4/(8 2 1n 8), where 8 = kRo The hydrodynamic forces then may
be written as
{;:} = iQpJT R 2 e- iQt [A - iB] {} ,
JT / 4
A = 1 + ,
s2(lns)2
1
B--
- s21ns'
R
s = v0.f/
8 = (  r/2
(9.40)
tTo bring out the similarity to the analysis of Section 9,2.2, -iQWI exp( -iQt) may be replaced by WI, and
-iQVI exp( -iQt) by VI, where WI = WI exp( -iQt) and VI = tlI exp( -iQt) are the components of cylinder
displacement - cf, equation (9.18) and Appendix V,

1052
SLENDER STRUCTURES AND AXIAL FLOW
These are exactly the same as the results obtained by Stokes (1851) and Rayleigh (1911).
For vanishing frequency (Q  O), the hydrodynamic force on the cylinder becomes
infinitely large.
Lin (1987b) presents several sets of computations for c1usters of 2-6 cylinders, giving
the coefficients in equation (9.38) in graphical form, first as functions of G c [defined in
(9.1)] and then of Gc!(8/ R) = gc!8, in which gc is the minimum intercylinder gap and 8
is the penetration depth. Reca1ling that 8 = (2v/Q)lj2, it is obvious that
s = Qg = Q(G c R)2
2v 2v
is a form of the Stokes or oscillatory Reynolds number. Accordingly, we would expect
that viscous effects should become important for small S, and negligible for large S -
cf. Section 2.2.3.
This is seen to be so in Figure 9.8. For.fS < 0.05, the effect ofviscosity is important,
and the hydrodynamic coefficients diverge dramatically away from those obtained by
potential ftow [from Chen (1977c)]. For.fS > 0.05, however, the coefficients are c10se
to the inviscid ones, and essentially the same for .fS '" 0(1), or even 0(0.1).
To see what this means physically, let us take R = 1.27 cm, Q/2n = 20 Hz and water
at 20°C as the liquid (so that v :::::: 10- 6 m 2 js), For G c = 0.1, one obtains.fS :::::: 10, and so
very c1early in such a case one may safely ignore the effect of viscosity in hydrodynamic
coupling. For the same basic system, for viscous effects to become important, one needs
to have G c '" 0(10- 3 ) or a frequency of '" 10- 3 Hz, or some equivalent combination of
unrealistic values.
However, it must be stres sed that this discussion pertains only to the real parts of the
coupling matrix and hence to the added mass coefficients. The imaginary parts - not
(9.41 )
30,0
7.0
20.0
R
f('1)G)
gcT00
gclR = 0,1
2,0
"'::::
, 10,0
"':::

"'::::
o, -3,0
"':::
o
0,0

-8.0
10- 1 10°
S 112
10 1
-13,0
10- 2
-10,0
10- 2
10- 1 10°
S 112
10 1
Figure 9.8 The real part (superscript R) of some of the hydrodynamic coefficients defined in
equation (9.38) as functions of S l /2 = [g;Q/(2v)]1/2 for a square cluster of four cylinders and
G c = gc! R = 0,1; similar plots for -r l and -r , and also for f3 and f3 , are available (Lin 1987b).

CLUSTERED CYLINDERS IN AXIAL FLOW
1053
given in Figure 9.8, but see Appendix V - are related to the damping matrix, As just
discussed, these latter may be negligible insofar as frequency ca1culations are concemed;
however, they are important for response ca1culations, as they affect the amplitude.
9.2.3(b) Pai'doussis & Suss' method
This approach is much more simple-minded, but it is useful because it can easily be
extended to the case of a cluster in an axially flowing fluid, which is needed in Sec-
tion 9,3. Consider a cluster of cylinders (Figure 9.6) oscillating in viscous fluid. Dissipative
effects maybe takeninto accountby adding the term iPDCD L71 CijUJu jjat) toequa-
tion (9.22), where CD is the viscous force coefficient for a solitary cylinder in quiescent
fluid and Cij are the elements of the dimensionless viscous coupling matrix, [C], Hence,
equation (9.23) now becomes
a 4 1 a a 2
4{ry} + -£c,81/2[C]_{ry} + [M]---z{ry} = {O},
a 2 ar ar
(9.42)
where most quantities are defined in (9.24a), £ = Lj D and c = (4jn)CDL(pAj EI )1/2,
The actual relative velocity between the jth cylinder and the fluid is not simply av j jat
and aWjjat, but aVjjat - Vyj and aWjjat - VZj, where Vyj and Vzj are the mean flow
velocities resulting from motions of all other cylinders. Now, the fluid velocity at any
point in the yz-plane, as if cylinder j were absent, may be expres sed as
aj 1 aj .
Vz(Tj, Oj) = - cos Oj - -- smOj,
arj rj aOj
aj 1 aj
Vy(rj, Oj) = - sin Oj + -- cos Oj,
arj rj aOj
(9.43a)
where
K
j (r j , Oj) = L' I (r j , Oj),
;=1
the starred summation excluding i = j, Clearly the velocities V z and Vy vary around
the circumference of the cylinder, It is reasonable to take a mean value to calculate the
frictional forces; hence defining
(9.43b)
1 1 211:
Vzj = - Vz(Rj,Oj)dOj,
2n O
1 {211:
Vyj = 2n 10 Vy(Rj,Oj)dOj,
and proceeding as before for determining the added mass matrix, but with the jth cylinder
absent, one eventually obtains the following expressions for viscous forces in the z- and
y-directions:
K
1 L { awI aVI }
-pDCD 1;'1- +g'l- ,
2 J at J at
1=1
K
1 L { aWI aVI }
-pDCD (J'I- +S'I-
2 J at J at '
1=1
(9.44a)

1054
SLENDER STRUCTURES AND AXIAL FLOW
and for the added damping (viscous coupling) matrix:
[ []
[Ca] = [0']
[g] ]
[s] ,
(9.44b)
where jl, gjl, O'jl and Sjl are given by jl = Sjl = 1 and gjl = O'jl = O for j = l, and
are equal to jl = -Vjl, gjl = -njl, O'jl = -Vjl andSjl = -njl, respectively, for j =f=.l,
given by
{ Vjl } K. 00 [ R' n-[ [{ cx nSI } { f3nsl }
'!.l = L L (_1)n-ln ( --.!!.. ) ansl cos [(n -1)1frjs] + .!.nsl
v JI Rs f3nsl CXnsl
- s=l n=l
n J '1 bnsl . -ansl '
J J
] n+l [{ Ynsl }
x sin [(n -l)1frjs] + (-1) n n ( :;s ) 8:1 cos [(n + l)1frjs]
dnsl .
J
{ Onsl } ]
+ dnsl sin [(n + 1)1frjs] ,
Ynsl
Cnsl j
(9.45)
where l, j = 1, 2, . . . , K, l =1= j, and where the constants CXnsl to dnsl are the same as in
Section 9.2.2, but evaluated with the jth cylinder missing. From the above, one has the
elements of [C a ]. Therein lies the attractiveness of this approach: once the added mass
matrix is determined, it is an easy matter to compute the viscous coupling matrix, [Ca].
The elements [Ca] have been calculated for several different geometries, and then the
eigenfrequencies by solving equation (9.42). For the four-cylinder system used as an
example in Section 9.2.2 and Oe = Ow = 1/4, taking C = 0.03 gives (VI = 8.5502 +
0.0488i, versus (VI = 8.5503 for C = 0- see equation (9.24b). However, once more,
it is the imaginary part of the frequency that is important - for example, for response
calculations. Moreover, the modal shapes ofthe damped system differfrom those discussed
before, in that orbital motions ofthe cylinders are predicted; this is so, even if only interna1
dissipation in the cylinders is tak:en into account.
ln practice, the elements of [Ca] would probably be determined experirnentally, but
lumping therein structural and other types of damping. This is perhaps the reason why this
topic, by itself and for quiescent fluids, has not received more vigorous attention.
9.3 DYNAMICS OF CLUSTERED CYLlNDERS IN FLOW
The first to examine the dynamics of clustered cylinders, as opposed to a single one,
in axial flow was Chen (1975c), who obtained the equations of motion for cylinders in
unbounded flow and presented a sample Argand diagram. This was followed by a study
of clusters in bounded axial flow (PaIdoussis & Suss 1977a) which was later extended and
complemented by experiments (paIdoussis 1979), clusters with both interna1 and externa1
flow (PaIdoussis & Besanon 1981), furtber experiment s (paIdoussis et al. 1982), and

CLUSTERED CYLINDERS IN AXIAL FLOW
1055
c1usters of flexibly interconnected cylinders (Pa'idoussis et al. 1983a,b). In what follows,
the ana1ytical model of Pa'idoussis & Suss (1977a) is presented, followed by a discussion
and typica1 results from this and subsequent studies. Work on vibration due to turbulence
is presented in Section 9.4.
9.3.1 The equations of motion
Consider the system of Figure 9.9, the same as in Figure 9.6 but in this case with the
fluid flowing axially rather than quiescent. The system consists of a c1uster of K para1lel,
vertical, flexible slender cylinders in axial flow, contained by a rigid cylindrical channel.
The cylinders are identica1 Euler-Bemoulli beams of diameter D, cross-sectional area
A, length L, flexural rigidity E [, and mass per unit length m, supported at one or both
extremities, The fluid is incompressible and the unperturbed flow unifonn and steady, in
the x-direction. The lateral deflection of the jth cylinder is Vj(x, t)j + Wj(x, t)k, and
motions and deflection angles are supposed to be small.
Following the procedures of Section 8.2, the equation of motions of the jth cylinder in
the y-direction is given by
a 5 v' a 4 v' a [ av. ]
E*[----L + E[-----.!.... + FY + FY, - - (T + pA) ,---:!..
atax 4 ax 4 AJ NJ ax J ax
av. a 2 v,
- F L .---:!.. + m-----.!.... = o , J I 2 K
J ax at 2 . = , ,..., ,
(9.46)
J(,
'>=.z
(T+ pA)}

(FL)/h
J(T+pA)
(T+ pA)} + Jx j óx
(a)
x
(b)
Figure 9.9 (a) Overall view ofthe system and detinition ofthe displacements v j (x, t) and W j (x, t)
of the jth cylinder; (b) forces acting on an element 8x of the jth cylinder, with the flexural forces
not shown (Pa"idoussis 1979).

1056
SLENDER STRUCTURES AND AXIAL FLOW
with a similar equation for motions in the z-direction; E* is the viscoelastic constant for
dissipation in the cylinder, Fl j and F"k j are, respectively, the lateral inviscid and viscous
hydrodynamic forces per unit length in the y-direction, p is the pressure in the fluid, T the
tension in the cylinder, and FLj is the viscous axial hydrodynamic force per unit length.
The term (T + pA) j is detennined from the axial force balance equation,
o ( OP )
-(T+pA)j+ mg--A +FLj
OX ox
( y y ) OVj ( Z z ) OWj
+ FAj+FNj ox + FAj+FNj OX =0.
(9.47)
It is found that FNj and FAj are both of 0(8), whereas FLj '" 0(1), Hence the last
two terms in (9.47) may be neglected, Proceeding as in Section 8.2.3 and recalling that
F L == FLj = !pDU2Cf,integrationfromxtoLgives
(T + pA)j = (T + pA)jlx=L + (mg -  A) (L - x) + PDU2Cf(L - x), (9.48)
where it has been assumed that the pressure drop is linear or piecewise linear, Furthermore,
assuming that the frictional force coefficient is uniformly valid for all structural elements
within 8x, one obtains the equiva1ent of (8,18),
op 1 2 ( D )
-A- = -pDU Cf - - pAg,
OX 2 Dh
(9.49)
where Dh is the hydraulic diameter. Moreover, as in equations (8.14) and 8,20), Dne
obtains
(T + pA)jlx=L = 8 {T + (1- 2v)pA}
{ I 2 op } ( 1 ) 1 2 2
-o -pDU Cf +mg - -A -L + -(1-8)pD U Cb
2 ox 2 2 '
(9.50)
where 8 = O applies when the downstream end of the cylinder is either free Dr free to slide
axial1y, and o = 1 when the two ends are c1amped, so that the length between supports
is fixed; T is the mean extemally applied tension, p the pressure at x = ! L, v Poisson's
ratio, and Cb the base drag coefficient.
Then, substituting equations (9.47)-(9.50) into (9.46) yields
( o ) o4v' { _ [ 1 ( D )
E* ot + E I ox 1 + Fl j + F"k j - 8 [T + (1- 2v)pA] + '2PDU2Cf 1 + Dh
] [( 1 ) ] 1 2 2 } 02Vj
+(m - pA)g 1 - '28 L - x + '2 PD U (1 - 8)Cb ox 2
[ 1 2 ( D )] OVj 02Vj
+ (m-pA)g+-pDU Cf - -+m--Z=O,
2 Dh ox ot
(9.51)

CLUSTERED CYLINDERS IN AXIAL FLOW
1057
with a similar equation in W j for motion in the z-direction. Hence, by comparing to
equation (8.21), it is clear that the equation of motion has the same form as for a sol-
itary cylinder, except for the lateral fluid-dynamic forces - the only ones affected by
coupling - which are discussed next.
Proceeding in exactly the same way as in Section 9.2.2 and making the same
assumptions, the inviscid fluid-dynamic forces in this case are
Fj = - i p ( :t + U o ) 4>j.ds
(9,52)
and a similarone forthe z-direction-ratherthan (9,16) and (9.17) which are for U = O;
bere S is the cylinder perimeter, The velocity potential 4> bas to satisfy V 2 4> = Oas before
and the same boundary conditions (9,10) and (9.11), except for the last one on the surface
of a moving cylinder, in which WI and VI are replaced by DWI / Dt and DVI / Dt, where
D/ Dt == (%t) + U(%x), that is,
04>1 I DWI DVI
- = -cosel + -Sinel,
orl r[=R[ Dt Dt
I = 1,2, ... K,
(9.53)
Hence, the final result is as in equation (9.18) but with %t replaced by D/Dt,
namely
z 2 K { D2W[ D 2 vI }
FA } ' = pnR/f:J €jl-Z + ejl '
1=1 Dt Dt
(9.54)
K { 2 2 }
Y 2 D WI D VI
FA } " = pn Rj L Kjl-Z + kjl '
1=1 Dt Dt
where the K ji, k ji, € JI and e JI are given by equation (9.19).
Tbe viscous lateral forces Fj and Fj are considered next. We continue using the
form proposed by Taylor, equations (8,11)-(8.13), such that FN = !pDU2Cti +
!pDCD(oh/ot) for a displacement h, where the angle of incidence of the incoming
flow is i = tan- I (oh/ox) + tan- I «oh/ot)/ U), As in Section 9.2.3(b), we recognize that
the fluid velocity approaching the cylinder is not purely axial, but it contains smalllateral
components Vy and V z due to motions of adjacent cylinders. Hence assuming that Vy and
V z are ofthe same order of magnitude as U-I (oh/ot), the angles ofincidence for the jth
cylinder are
1 ( OVj ) I [( OVj ) 1 ]
iyj = tan- ox + tan- ar - Vyj U '
(9.55)
1 ( OWj ) I [( OWj ) 1 ]
izj = tan-  + tan- ar - Vzj U '

1058
SLENDER STRUCTURES AND AXIAL FLOW
Hence the viscous lateral forces may be written as follows:
y ( aVj aVj ) ( aVj O )
FNj = pRjUCf at + u ax - Vyj + pRjCD at - Vyj ,
z ( aWj aWj ) ( aWj O )
F N J , = pRjUCf - + u- - Vzj + pRjCD - - V zj ,
at ax at
(9.56)
where V, Vz are the values of Vyj and Vzj when U = O.
The velocities Vyj, Vzj may be obtained rrom the velocity potential, keeping in mind
that the oncoming flow, vis-a-vis which the cylinder has a certain angle of incidence, is
undisturbed by the presence of that particular cylinder. The fluid velocity at any point in
the yz-plane, as if cylinder j were absent, may be expres sed as
a</Jj 1 a</Jj
V ( r' O,) = - sinO' + -- cosO'
Y J' J ar' Jr' 00, J'
J J J
(9.57)
and similarly for V z , with </Jj exactly the same as (9.43b). The velocities found in this way
vary from one point to another on the circumference of the cylinder. It is reasonable to
take a mean value for calculating the frictional forces, and accordingly define
1 1 2"
Vzj = - Vz(Rj,Oj)dOj,
2n O
1 1 2"
Vyj = - Vy(Rj,Oj)dOj.
2n O
(9.58)
Proceeding in a similar manner as in Section 9.2.3(b), these are found to be
K
L . { DWI DVI }
V'= v'I-+n'I-,
ZJ J Dt J Dt
1=1
K
"'. { DWI DVI }
Vyj =  Vjl Dt +njl Dt '
1=1
(9.59)
where the coefficients v j I to n ji are given by equation (9.45), and the starred summation
exc1udes i = j.
Hence, the normal viscous forces are given by
K K
(F Z ), = !pDUC f '" {  'I DWI + g 'I DVI } + !pDCD '" {  'I aWI + g 'I aVI } ,
N J 2  J Dt J Dt 2  J at J at
1=1 1=1
1  { DWI DV1 } 1  { awI aVI }
(Fk)j = '2pDUCf  (jjl Dt + Sjl Dt + '2 PDCD  (jjla( + Sjlar '
1=1 1=1
(9,60)
where jl, gjl, (jjl, Sjl are equal, respectively, to -Vjl, -n JI, -Vjl, -njl for j =1= 1, and
jl = Sjl = I, gjl = (jjl = O for j = [,

CLUSTERED CYLINDERS IN AXIAL FLOW
1059
By substituting equations (9,54) and (9,60) into (9.51) and into the equivalent one for
the z-direction, one obtains the fina1 fonn of the equations of motion. Defining the
dimension1ess quantities
x _ ( EI ) 1/2!...- pA
g = L ' r - m + pA L2' {3 = m + pA '
n= pAL 2 Y _ (m- p A) g L 3 L h-
EI' - EI ' s= D ' -Dh'
C - C L( PA ) 1/2 a - [ I ] 1/2 E * 4
- Jr D EI ' - E(pA+m) L2' Cb = ;Cb,
T L 2
r=-
EI'
4
cf = -Cf,
Jr
(9.61)
( PA ) 1/2
u = - UL
EI '
Vj-K
TJj = -,
L
TJj = wj/L,
j=1,2,...,K,
j = K + 1, K + 2,..., 2K,
the dimension1ess equations of motion may be written in matrix fonn as follows:
a 5 q a 4 q a 2 q a 2 q
A-+I-+C-+E-
ara4 ag 4 agar ag 2
a 2 a a a 2
+ F ---.! + G-.! + H-.! + M---.! = o
ag 2 ag ar ar 2 '
(9.62)
where A = aI, I = unit matrix,
C = 2{31/2 U M a = 2{31/2u U %n,
F = [!SCfu 2 (1 + h) + Y] I,
E = u2Ma - {8 [r + (1- 2v)n] + (1- 8)!CbU2
+ (1- !8) [!SCfu 2 (1 +h) + y] }I,
(9.63)
G = !scfu 2 C a + [y + !SCju 2 h] I
= lscfu 2 [ [jiJ [gjiJ ] + [ y + l SCju 2 h ] I,
2 [UjiJ [SjiJ 2
H = (!SCj{31/2 u + !SC{31/2) C a ,
M = {3Ma + (1 - {3)I, q = {TJ1, TJ2, .,., TJ2K}T .

1060
SLENDER STRUCTURES AND AXIAL FLOW
9.3.2 Solution of the equations of motion
Solutions are obtained via Galerkin's method, by letting
00
r/j(, 'l') = L l/Ji(g)pf ('l'),
i=1
(9,64)
where the l/Ji () are the beam eigenfunctions satisfying the same boundary conditions as
those applying to the problem at hand, and pf ('l') the corresponding generalized coordin-
ates, Here we confine ourselves to cylinders with supported ends, for otherwise the shape
of the free end and the specia1 boundary conditions in Section 8,2.4 would complicate
matters.
Substituting (9.51) into (9.50) and applying the Ga1erkin procedure, we obtain
AA;8riPr + 1A;8riPr + C L hriPi + E L CriPi + F L driPi
i i
+ G L hriPi + B8riPr + M8riPr = O,
r = 1,2, . , , , 00,
(9,65)
where Pr = {pl), p2) , . . . , p;K} T, Ar are the dimensionless eigenvalues of the beam
equation, and
hri = 11l/Jr()l/J;()d,
Cri = 11l/Jr()l/J;'()d,
dri = 11 l/Jr(g)l/J;'()dg,
which are tabulated in Table 3.1.
Now truncating the infinite series at i = N and defining the globa1 vector
P = {Pl, P2, ..., PN }T,
(9.66)
equations (9.65) may be written in the fonu
Mp + Cp + Kp = O,
(9.67a)
where M, C, K are 2K N x 2K N matrices made up of N submatrices, each of order 2K,
Mi} = M8i}, Ci} = AAi8i} + Chi} + B8ij,
/Ci} = lAi 8i} + ECij + Fdi} + Ghij,
(9.67b)
where 8i} is Kronecker's delta, and i, j = I, ..., N. Mter transfonnation of (9.67a) into
first-order fonu, these equations are solved as a standard eigenva1ue problem, yielding the
eigenfrequencies and eigenvectors and hence the eigenfunctions of the system.
9.3.3 The dynamical behaviour
The loci of the eigenfrequencies of the lowest members of the first and second flexura1-
mode groups for a typical system of three pinned-pinned cylinders are plotted in

CLUSTERED CYLINDERS IN AXIAL FLOW
1061
2.5 9
2.95
1.75 5
I

1.5
2,9916
0,5
1.496
C89
rvaluesofu
2.5 1.75
BA.
Unstable
-0,5
1.5
-I 2,9917
2,9916 C
1.75_ 5
295
-9
2,5 °
8
16
(w)
24
Figure 9.10 Argand diagram of the complex eigenfrequencies, ú>, of the lowest members of the
first- and second-mode groups for a three-cylinder pinned-pinned system (f3 = 0.1, eCf = 0,25,
<5 = 1, r = TI = y = a = c = O, G c = G w = 1/4) as function of u (Pa'idoussis & Suss 1977a).
Figure 9.10 as an Argand diagram, with the dimensionless flow velocity, u, as para-
meter. Atlow u, itis seen thatIm(w) > O, and so flow damps the system, With increasing
u, the frequencies of osci1lation, Re(w), diminish; eventually that of the first mode van-
ishes at u :::::: 1.495, and the locus bifurcates, one branch moving up and the other down
the Im(w)-axis, as shown in the diagram on the left of the figure for clarity, At slightly
higher ftow (point A), Im(w) vanishes as well for one branch ofthe locus, indicating the
onset oť divergence (buckling). At yet higher ftow (point B), the second branch follows
suit, indicating the onset of another divergence, At slightly higher u, the two branches
coalesce and leave the Im(w)-axis at point C, indicating the onset of coupled-mode flut-
ter. t The behaviour of the system resembles that of a single cylinder in unbounded flow
(cf. Figure 8.5); however, the critical flow velocities for the various instabilities are very
much lower. Thus the critical flow velocity for divergence, Ucd, for a solitary cylinder in
unbounded ftow is 3.143, whereas here it is 1.496; the corresponding flow velocity for
flutter, ucf, is 6.41, while here it is 2.992.
t As before, we refer to this as coupled-mode flutter, despite the fact that it is branches of the same mode
that coalesce here.

1062
SLENDER STRUCTURES AND AXIAL FLOW
Table 9.2 Stability bounds for each member ofthe six-
mode group for the three-cylinder system ofFigure 9.10.
Mode number UA == Ucdl uB = u c d2 Uc = ucf
1,2 1,496 Uc 2,992
3 1.74  Uc 3.48
4,5 2.52 5.04 5,10
6 3.84 7,72 8,07
@
22, 8° av"" .
50,8: 1 .
- -
\ 
, . o.........
W 1 = 1.78 + O,028i W2 = 1.78 + O,028i .
W3 = 4,76 + O,03li
-. 80,8° '-('\2,8 0 30°? .

. - . I I
. "-
I I 
w4 = 7.79 + O,02li Ws = 7,79 + O,02li w6 = 9.28 + O.Olli
1,2
W 1.78 + O,028i
A 2 /A j 0,535,3,98
Mode
3
4.76 + O,031i
I
4,5
7,79 + O,021i
0.535,3,98
6
9,28 + O,Olli
1
Figure 9.11 The six cross-sectional modal pattems corresponding to the system of Figure 9,10 for
u = 1.46. The associated table gives the eigenfrequencies and the ratio of the second to first-mode
amplitude, A2/ AI; in al1 cases A3/ A2 = 1 (Paldoussis & Suss 1 977a),
The other members of the first and second flexural-mode groups behave similarly. The
critical flow velocities associated with the second eigenmode are very slighůy higher than
those of the first (the difference being in the fifth or sixth significant figures), but the
critical flow velocities for the other members of the first and second flexural-mode groups
are higher, as may be seen in Table 9.2. It is noted that in some cases the gap between
the second divergence and flutter is wider than shown in Figure 9.10. It is also noted that
'third-mode' (third flexural-mode group) divergence occurs between u = 4.4 and 4.5.
Finally, it should be mentioned that the frequency loci within each group remain distinct
at all flow velocities,
The cross-sectional modal pattems and eigenfrequencies of the first-mode group at
u = 1.46, just before divergence, are shown in Figure 9.11, with the corresponding rel-
ative amplitudes being given in the inset table, The amplitudes of cy1inders 2 and 3, A2
and A3, are always the same; moreover, for modes 3 and 6, aU amplitudes are the same
and motions are symmetric. It should be noted, however, that the cross-sectional pattem

CLUSTERED CYLINDERS IN AXIAL FLOW
1063
Table 9.3 Effect of some system parameters on stability
of a three-cylinder system, in comparison to the 'standard
case' of Figure 9.10.
System parameters Ucd ucf
Standard case 1.496 2.992
y = 10 1.48 3.00
TI = 5, v = 0.3 1.64 3,06
[' = -5 1.05 2,80
Clamped ends 2,99 4,33
8
ucf(single cylinder in
unbounded flow)
6
ucf

Ucd (single cylinder in
unbounded flow)
u 4
2 
""-
ucd
 G w =2G c
. Gw=!G c
o Gw=Gc
00
2
3
G c
4
5
6
Figure 9.12 The effect of contiguity between cylinders (G c ) and proximity to the channel wall
(G w ) on stability of the three-cylinder system (f3 = 0.1, E;cf = 0,25,8 = 1, ex = c = [' = TI =
Y = O). Unless otherwise denoted, G c = G w (Pai'doussis & Suss 1977a).
associated with the lowest critical ucd and ucf' which is also that with the lowest fre-
quency at u = O, is a1tered with changes in geometry. Thus, for G c = G w = 5, the
lowest critical Ucd and ucf are associated with the symmetric pattem of the third panel of
Figure 9.11. The cross-sectional modal pattems have been found to be insensitive to flow
velocity.
The effect ofvarious parameters on stability is investigated next. The effect of increasing,
in tum, cCf from 0.25 to 1.0, f3 from 0.1 to 0.5, and a from O to 0.003 alters Ucd and ucf
less than 0.3%. The effect of gravity (y), pressurization (O), compression (-r) and
of clamping the ends is shown in Table 9.3. The effect is qualitatively the same as for
a single cylinder in unbounded flow. However, the destabilizing effect of compression
is particularly important in this case: Ucd = 1.05 could correspond to dimensional flow
velocities in the range of practical engineering applications.
Finally, the effect on stability of contiguity between cylinders is i11ustrated in
Figure 9.12. It is seen that even at G c = G w = 5, that is, at gaps equal to five cyl-
inder radii, the critical flow velocities are still somewhat lower than those for a solitary
cylinder. As G c = G w diminish, so do the critical flow velocities, quite dramatically.

1064
SLENDER STRUCTURES AND AXIAL FLOW
A1so shown in the figure is the effect of altering G w while keeping G c the same; the
effect is quite substantial, indicating that proximity of the cylinders to the channe1 plays
a significant role.
It is of interest to compare this theory to the simpler one of Section 8.2.3. For the three-
cylinder case considered here, the average of the absolute values of the diagonal terms of
Ma is 2.64. Using this value for X as defined in Section 8.2.3, one obtains Ucd = 1.94,
versus 1.496 in this theory, and ucf = 3.88 versus 2.992. Clear1y, the simpler theory, which
does not take coupling oj motions into account but just bulk confinement, underestimates
the destabi1izing effect of proximity to other, moving cylinders.
Further calculations are presented for a looser c1uster of Jour cylinders with c1amped
ends (G c = 1, G w = 5.02), these values correspond to those of an experimental system
(Section 9.3.4). The dynamics of the first and seventh members of the first flexural-mode
group are shown in Figure 9.13. It is seen that the system loses stability by divergence
at Ucd :::::= 4.85. The subsequent behaviour in Figure 9.13(a) is somewhat different from
that in Figure 9.10: the system is restabi1ized in that mode at u :::::= 6.9, and then single-
mode fiutter ensues at ucf :::::= 7.0. The behaviour of other members of the first-mode
group is quite similar - cf. Figure 9 .13(b), but here we have coupled-mode flutter and no
post -divergence restabi1ization.
The stabi1ity-instabi1ity boundaries for the whole group are shown in Figure 9.14. It is
seen that, after the system becomes unstable in the lowest of these modes, with increasing
fiow it becomes unstable in other modes in rapid succession, without regaining stability
overal1. Thus, at u = 8, the system is unstable in no less than seven of its modes.
The loci of the second fiexural-mode group are rather simi1ar. Thus, the lowest second
mode, that is, the mode with the lowest frequency at u = O among those with second
fiexural-mode shape, is subject to divergence at u :::::= 9.55.
As mentioned earlier, for u f= O the system is mathematical1y non-self-adjoint and
the eigenfunctions are no longer orthogonal - that is, normal modes no longer exist.
Moreover, the fiexural mode shapes vary with fiow, and indeed the nomenc1ature of
'first-mode', 'second-mode', etc., becomes more one of convenience than one of fact.
The evolution of first fiexural-mode-group modal shapes with fiow is illustrated in
Figure 9.15(a)-(c) for the same system as that of Figures 9.13 and 9.14; both axial and
cross-sectional modal pattems are shown. Several important features are noted. Before
discussing these, however, two explanatory notes should be made. First, it is noted that
in Figure 9.15 the oscillatory behaviour of the modes is animated by showing the flex-
ural modal shapes at different times in a cycle of oscillation; however, in so doing, the
contribution of Im(w) has been suppressed, for ease of graphical presentation. In reality,
for a mode with Im(w) > O, the amplitude of oscillation decreases progressively within
a cyc1e offree oscillations, the opposite being the case when Im(w) < O. Second,once
a mode bifurcates, there are two pure1y imaginary frequencies, whereas before there had
been only one (on the right-hand side of the complex frequency plane); hence, although at
u = O the first-mode group would comprise eight eigenfrequencies and, therefore, eight
modal shapes, at sufficiently high fiow this number may be larger. Thus, at u = 4.85 in
Figure 9,15(a) there are nine such eigenmodes (note that there are two modal shapes in the
first panel), the first mode having bifurcated; the two branches of this mode are identified
accordingly as to whether they lie on the positive or negative Im (w )-axis by WI + and WI_,
and the corresponding modal shapes by full or dashed lines; at u = 6.95, Figure 9.15(b),
there are 13 independent eigenmodes, five loei having bifurcated, and so on.

CLUSTERED CYLINDERS IN AXIAL FLOW 1065
15 15
10 10
6,5 5.5 10.8 8
5 10,9 5
6,8 4,85 7.8
O,5 n Values of u 05 Stable

  7,5 6,5 4,5
4.5 3.5 2.5 1.5 +---- 2,5
-----. O
06,97 7.1 O 9.4 O
10.98 Unstable
-0,5 -0,5
6,8 4.85 7,8 11.2
-5 10,9 8 -5
6,5 5,5 10,8
-10 -10
-15 -15
O 10 20 30 40 O 10 20 30
(a) (w) (b) (w)
Figure 9.13 Argand diagrams of the complex eigenfrequencies of (a) the first and (b) the seventh
mode, of the first flexural-mode group, of a cluster of four clamped-clamped cylinders in a circular
channel, as functions of u. For this system, a = 0, f3 = 0.467, y = 2.65, /) = 0, sCJ = 0.17, C =
Cb = 0, h = 2,54, G c = I, G w = 5,02 (Pai:doussis 1979),
10
8
u 6
4
O
c:::J Stable
_ Buckling
O F1utter
2
Figure 9.14 Regions of stability and instability for the system of Figure 9.13 when consid-
ering the whole of the first flexura1-mode group, (Note that other instabilities arising from the
second flexural-mode group also exist, starting from u = 9.55, but are not shown for clarity.)
(Pai:doussis 1979),
Several features noted for so1itary cy1inders in axial flow hold trne for clusters also,
as seen in Figure 9.15. Thus, substantial higher beam-mode components are evident in
the flexural mode shapes for higher u, This is already discernible in Figure 9.l5(a), but
in 9.l5(b) and (c) several modes appear to have more of a second than a fi.rst flexural-mode
shape. At u = 10.75 [not shown here; see figure 9 in Paidoussis (1979)] the third flexural-
mode contribution is quite strongly apparent. Also, because of their non-self-adjointness,

1066
SLENDER STRUCTURES AND AXIAL FLOW
Im I) (I r =-I< )1
W!+  o + I 84i, w!_  o - 1.64i
W z  7 99 + O 103i
I@I()II@I()I
W3  10,96 + O 105i
W 4 =W 3
Im I;II@ <=> II
W 5  14.74 + O 106i
W6=W5
I <> !)<II@ <> 1<)1
W7  18.08 + O 108i
(aj
Ws  20.58 + O,l1li
Figure 9.15 (a) The axia1 and cross-sectiona1 moda1 shapes for the first flexura1-mode group of a
cluster of four cy1inders in a circu1ar channe1, showing a1so the associated eigenfrequencies, In the
case of a bifurcated mode (pure1y imaginary eigenfrequencies) the positive branch is drawn with
full1ines, and the negative branch with dashed 1ines. Parameters as in Figure 9,13 and dimension1ess
flow velocity u = 4,85; calcu1ations conducted with five comparison functions (Pai'doussis 1979),
the flexural modal shapes do not generally have stationary nodes, indicating the presence
of travel1ing wave components.
It is of interest that whereas at u = O the flexural mode shapes of each group of modes
are identical [cf. Figure 9.2(a)], this is no longer trne with flow; indeed with increasing
flow this becomes more pronounced, which is evident from Figure 9.15(a)-(c).
Considering next the cross-sectional pattems of the modes, it is first noted that
the rnirror-image symmetry associated with repeated eigenfrequencies at u = O [Fig-
ure 9.2(b)] no longer holds for u -1= O, as may readily be seen by comparing the fifth and
sixth modes of Figure 9.15(a), Nevertheless, something akin to it survives in some cases;
for example, if the third and fourth modes in Figure 9.15(a) are compared, it is noted that
the orbital pattems are quite sirnilar, but clockwise in the former and counterclockwise in
the latter.
Orbital modal pattems are always associated with repeated eigenfrequencies, whereas
modes with distinct eigenfrequencies are associated with planar motions (straight-line
pattems in the diagrams). By exarnining all sets of third/fourth and fifth/sixth modes
in Figure 9.15 and some more in Pa'idoussis (1979), the basic pattem of the motions
emerges. The main characteristics of the third- and fourth-mode pattems are (i) that there
is diametral symmetry in the motions (i.e. cylinders 1 and 3 t perform identical motions, as
t As in Figure 9,2, the cylinders are numbered counterclockwise in the plane oť the paper, and cylinder I is
at the upper right-hand comer,

CLUSTERED CYLINDERS IN AXIAL FLOW 1067
lro <@@> I) <I I@  j( )1
Wl = 5 49 + 0.019i Wl+ = O + 9.45i, Wl-= 0-9 290i
I ;II@ I( >1
W3+  O + 9.48i, W3- = O - 9292i w4+ = W)+, w4-  w3_
I II@];<I
ws+  0+5 13i, ws- 0-4,92i w6+ =wS+,W6- =ws_
I <:>- I) <I !@ <:>- 1<)1
W? = 949 + 0.1I8i
(b)
Wg = 14 53 + O 129i
IW I)(II@ 1<)1
Wl = 1693-12.60i wl= 12.31-0.299i
II-:';II@I<)I
w3+ = O + 8,52i, w)_  0- 8.46i w4+ = w3+, w4- = w)_
I I;II@ I
ws+ = O + 10 59i, ws-= 0- 1044i w6+ =WS+,w6-= ws-
I ])(II@ <:>-1<)1
W7+ = O + 5.48i, w7- = O - 5,28i
(c)
Wg=956+0,122i
Figure 9.15 (b) u = 6,95; (c) u = 8.00 (Pa'idoussis 1979).

1068
SLENDER STRUCTURES AND AXIAL FLOW
do cy1inders 2 and 4), and (ii) all cy linders move basical1y towards one side of the channel,
rather than in opposite directions; apart from that, they appear to be combinations of two
fundamental pattems typified in impure forms, one by the fourth mode in Figure 9.15(b)
and (c) and the other by the negative branch ofthe third mode ofFigure 9.15(c). Sil1Ú1ar1y,
the basic characteristics ofthe fifth- and sixth-mode pattems are (i) diametral symmetry in
the motions, and (ii) some kind of opposition in the motions of differently moving pairs of
cy1inders. The two essentia1 e1ements from which these modal pattems are synthesized are
shown in impure forms by the criss-crossing pattem of the fifth mode of Figure 9.15(c),
or the sixth of Figure 9.15(b), and the paralle1 opposed-pair pattem of the fifth mode of
Figure 9.15(b).
ln contrast, the distinctive modal pattems assoeiated with the nonrepeated eigen-
frequencies are remarkably stab1e and persist for al1 flow ve1oeities. These are the
'explosion-imp1osion' pattem ofthe firstmode, the 'rhombic pinch' pattem ofthe second,
the 'diagona1 attraction-repu1sion' pattem assoeiated with the seventh mode, and the
'rotary' pattem associated with the eighth. These same pattems recur for the higher groups
of modes; cf. Section 9.2.1 and Figure 9.2.
A few remarks might usefully be made on the form of the moda1 pattems of modes
associated with pure1y imaginary eigenfrequeneies. In the case of distinct eigenfrequen-
eies, for examp1e, the second mode of Figure 9.15(b) and the seventh of Figure 9.15(c),
the cross-sectional modal pattems associated with the two branches of the bifurcated locus
are identical. This is not true for modes with repeated eigenfrequencies; in some cases the
pattems associated with the positive and negative branches are essential1y simi1ar but not
identical, for examp1e, those ofFigure 9. 15(b), but sometimes they can be quite different,
as those assoeiated with úJ3+ and úJ3- in Figure 9.15(c).t The flexural modal forms are
different for the two branches of bifurcated mode loei: they are very near1y mirror images,
in the sense that those of the negative branches may be obtained by flipping the positive
branches over, so that x = L lies over x = O and více versa.
The remarks made in this section, although illustrated by a specific set of resu1ts, are
quite general and app1y in suitab1y modified form to other systems, for examp1e, three-
or six-cy1inder systems, possessing some inherent geometrie symmetry; see, for instance
PaYdoussis & Suss (1977b) and Figure 9.16.
Divergence in this case occurs at Ucd ::::: 2.992, essentially the same as for a three-
cylinder c1usterwith the same G c and Gw(Ucd ::::: 2.991). Furtherdiscussion on the modal
coupling characteristics may be found in Section 9.4. 1 (b).
9.3.4 Experiments and comparison with theory
The experiments were conducted in a special water tunne1 (Figure 9.17) invo1ving a trans-
parent vertical test -section, approximate1y 20 cm in diameter and 75 cm long. The ve10city
profi1e was flat and uniform over a central portion of the test-section, covering at 1east
15 cm in diameter; the highest flow ve10eity attainab1e was 6 mis.
The flexib1e cylinders were made of si1icone rubber by casting in special mou1ds
(Appendix D). Each flexib1e cylinder was fitted with metal discs which cou1d be screwed
tThe extent to which the cross-sectional pattems are identical for the positive and negative branches depends
appreciably on the specific techniques utilized by the computer subroutine which detennines the eigenvectors
ofthe system.

CLUSTERED CYLINDERS IN AXIAL FLOW
1069
45° , . ."
.J ' /.
, ....
'" " 45°-L 
'.
"... / /J
,: ...
W3 = 5.446 + O,025i W4 = 7.282 + O,026i
" 45°-:L ,( 
..'" 45°"
.,'J /J
, ...
,( . " /
/. '"
... ,
W7 = 8.097 + O,022i
W8 = 9.187 + O.013i
46° L Ir 59°
....J --.:::\
,: 31°L
....J
. /
\
,
Wt = 3.166 + O,031i

41°
--.::: ....-
31°
"
",
......
.'.....
45°-L-
,"J .
" 41°
,'J
/ ../
#'
Wz = 3,166 + O,03li
'""
A
Ws = 7.452 + O,025i
71°...,-'
IJ ď- "
1 ...' /
: I
'",50°
--.:::
\.
, "
t
'
W6 = 7.452 + O,025i
r
\
-710
--.:::.
..... . . o...-.
W9 = 9.354 + O,Ol4i WtO = 9,354 + 0.014i

Figure 9.16 The cross-sectional modal shapes for a system of a cluster of five pinned-pinned
cy1inders in axial flow; u = 1.35, G c = G w = 1/4 (Pa.'idoussis & Suss 1977b).
onto different end assemblies to reproduce clamped or pinned (simply supported) bound-
ary conditions; these, in turn, fitted into support assemblies at the upper (upstream) end
and the lower end, the fonner well streamlined, The supports and an exploded view of
the cylinder assembly are shown in Figure 9.18, The 10wer support for each cy1inder was
such that axia1 sliding was possible, although, if desired, sliding could be eliminated by
tightening of a set-screw, thus permitting the establishment of both conditions character-
ized by 8 = O and 8 = 1 in the theory. In the latter case, it was possible to pre-tension the
cylinders (to introduce a nonzero value for í). t
Most experiments were conducted with c1usters of four identica1 cylinders in radially
symmetric arrangements such that G c = 1,2 or 3, with the cylinders either c1amped
or pinned at both ends and allowing axial sliding. The cylinder and channel radii being
fixed at 1.26 and 10.24 cm, respectively, G w varied with G c in a prescribed manner;
G c = 1 corresponded to G w = 5.02, and G c = 3 to G w = 3.61. Experiments were
tBy attaching known masses to the lowest points oť the cylinders before tightening the set-screws.

1070
SLENDER STRUCTURES AND AXIAL FLOW
Vent To drain
+ +
------------------------------
-+
Heat exchanger
t Manometers
Pump
DitTuser
To de-aerator
+
-+
.-
Figure 9.17 Schematic of the water tunnel used in the experiments with c1usters of flexible
cylinders in axial flow.
1. Upstream support ogive
2, Airfoil support bars
3, Supports resting in niches
in test-section
4 Downstream support
5. Lower support friction shoes
6. Setscrew to prevent sliding
(a)
'D-z
g
t
x
+
6 4
':liiO
fjil
(b)
Figure 9.18 (a) The upper and lower support arrangements for a system of four cylinders, and
(b) an exploded view of a cylinder assembly with a c1amped upper end and a pinned lower end
(Pai:doussis et al. 1982).

CLUSTERED CYLINDERS IN AXIAL FLOW
1071
carried out with sets of cylinders of different lengths: 52, 49.5, 47 and 41.9 cm. With
the 47 cm cylinders further sets of experiments were also conducted: namely, with axial
sliding prevented, with the cylinders c1amped upstream and pinned downstream, and with
c1usters of three and two cy1inders.
ln the first set of experiments (Pa'idoussis 1979), the observations and assessment of
stability were made visua1ly/optica1ly, as well as photo- and cinematographically. In later
experiments, however, strain-gauge measurements were a1so made (Pa'idoussis ef al.
1982).
9.3.4(a) General observatíons
At low flow velocities, the system was quite stable. Very small amplitude oscillation was
sometimes visible about the positions of rest of the individua1 cylinders.
At sufficiently high flow velocity, the system was observed to lose stability by divergence
(buckling) - the doser the cylinder spacing, the sma1ler was the critica1 flow velocity.
The cross-sectional pattem involved was not always the same. For c10sely spaced systems
(G c = 1,1/2) the 'implosion' pattem was observed most often (vide the first mode
of Figure 9.15, and a1so the discussion in Section 9.3.2), and sometimes the equivalent
'explosion' pattem; in some cases, however, it was that of the 'rhombic pinch' [Le. the
pattem of mode 2 of Figure 9.15, or mode 1 in Figure 9.2(b)]. For less dosely spaced
systems (G c = 2,3) the pattem most frequently observed was that of rhombic pinch;
another was one where pairs of adjacent cy1inders moved directly towards each other, a1l
motions being para1lel (which may be considered as the vector sum of the explosion and
rhombic pinch pattems); in addition, other modal pattems were a1so observed, induding
the implosion-explosion one. Maximum displacement occurred at, or just downstream of,
the cylinder mid-points in a1l cases.
According to theory, for the particular set of system parameters involved, this system
should a1ways lose stability by divergence in the explosion-implosion pattem - for 1/2 ::::
G c :::: 3, at least. In the experiments, on the other hand, as mentioned above, this was
the case mostly for small inter-cylinder spacings, but commonly not for G c = 2 and
3. The only reason that may be advanced for this lack of complete agreement between
theory and experiment is the following. For G c = 1, theory predicts that divergence of
the first and second modes of the first flexural-mode group occurs at appreciably different
flow velocities, typically 10-15% different, This difference decreases with increasing
G c ; for example, at G c = 2 it can be less than 5%. Accordingly, if in the experiments
the trne critica1 flow velocity for the first buckling instabi1ity is exceeded by even a few
percent before the pattem becomes visibly identifiable, the moda1 shape one would see
may actua1ly be a composite of that of the first two modes, or even entirely that of the
second cross-sectiona1 mode.
Divergence usually developed gradual1y, its amplitude increasing with flow; typica1ly,
the maximum amplitude attained was three-quarters of a cylinder diameter, as the flow
was increased to 10% above the threshold flow velocity. Occasional1y, however, especia1ly
in cases of small intercylinder gaps (usua1ly for G c = 1), a phenomenon of accelerated
buckling was observed where, once divergent motion began in the implosion pattem, it
continued until the cylinders touched one another dose to their mid-points. Clearly in
such dosely spaced arrangements, as the cylinders get doser, the hydrodynamic forces
become appreciably stronger, pushing the cylinders even doser together, and so on.

1072
SLENDER STRUCTURES AND AXIAL FLOW
The buckled form of the cylinders would not always remain stable. In some cases,
for instance, cylinders in an implosion pattem would move outwards spontaneously to
an explosion pattem. Another source of unsteadiness occurred when cylinders carne
close together, as when in a rhombic pinch pattem, or close to the wall; a high fre-
quency oscillation would then ensue, described as 'shuddering' by most observers - and
would persist until the cylinders moved further apart, This phenomenon is likely related
to local flow separation just downstream of the point of smallest distance between the two
curved cylinders.
With increasing flow, the buckling pattem would change, Thus, if the system was in
the implosion or explosi on pattem, at higher flow it often changed to the rhombic pinch
pattem, Once this occurred, it was not uncommon for the pattem to altemate spontaneously
between these two, while the flow velocity remained fixed. As the flow was increased
further, pattems sirnilar to those of the third to sixth modes of Figure 9.15 could be
observed, but did not all occur for all of the arrangements studied. Some transitions from
one pattem to another, and back again, gave the impression of oscillatory behaviour, albeit
without a regular frequency.
At higher flow, something akin to flutter would develop, where the cylinders oscillated
erratically in terms offrequency, in the sarne pattem in which they were buckled. When the
flow was increased beyond this point, the oscillation would become progressively more
regular and its frequency higher. This allowed one to bracket a range of flow velocities
within which the onset of flutter occurred, without being able to pinpoint it accurately.
The oscillation was never quite continuous or regular, being disrupted by quasi-quiescent
interludes during which the system appeared to be buckled in a specific modal pattem.
It was apparent, even to the casual observer, that more than one in stability was operative
concurrently: one or more divergence instabilities, corresponding to the quiescent pattems
observed, and an oscillatory instability. Flutter almost invariably was in predorninantly
the second flexural mode, and in the first flexural-mode pattem depicted in Figure 9.15.
Clearly, these observations are quite close to theoretical predictions.
With further increase in the flow, the oscillation frequency continued to increase -
again in accord with theoretical predictions (Figure 9.13); third beam-mode components
started becoming visible, and the cross-sectional pattem of oscillation became less clear as
amp1itudes increased, to the point where cylinders invaded one another's normal position
of rest. Of course, if pre-tensioning was applied, or if the cylinders were clamped rather
than pinned, the amp1itudes were considerably smaller,
One interesting aspect of the observations is that the diagonal attraction-repulsion pat-
tem and the rotary one of the seventh and eighth modes ofFigure 9.15 were neverobserved.
Presumably, this is because by the time divergence in these modes materializes, other
modes have become unstable by flutter - according to theory at least.
Finally, the behaviour of the two- and three-cylinder clusters was quite sirnilar. The
commonest cross-sectional modal pattems observed were of the explosion-implosion
form; for two-cylinder systems, the cylinders moved diagonally apart, without preference
as to the plane of the motion.
Figure 9,19 shows some photographs of four- and three-cylinder systems at various
post-divergence flow velocities. Photographs (a)-(e) are for pinned-pinned systems (with
the downstream end free to slide axially); (a) shows clearly the implosion pattem, while
(c), (d) and (e) show the equivalent explosion pattem - the last one for a system of
three cylinders; (b) shows a case of the rhombic pinch pattem. Photographs (f)-(j) show

CLUSTERED CYLINDERS IN AXIAL FLOW
1073
(a)
(b)
(c)
(I)
(g)
(h)
(d)
(i)
(e)
(j)
Figure 9.19 Photographs of four- and three-cylinder systems either (a)-(e) pinned, or (f)-(j)
clamped at both ends - but free to slide axially at the downstream end, (a) K = 4, G c = 1,
u = 2,86 (V = 1.74 mis); (b) K = 4, G c = 1, u = 3.40 (V = 1.94 mis); (c) K = 4, G c = 1,
u = 4.77 (V = 2,55 mis); (d) K = 4, G c = 3, u = 4.16 (V = 2.37 mis); (e) K = 3, G c = 1,
u = 4,55 (V = 2.77 mis); (f) K = 3, G c = 2, u = 5.46; (g) K = 4, G c = 1, u = 4.87;
(h) K = 4, G c = 3, u = 8,24; (i) K = 4, G c = 3, u = 7.20; (j) K = 4, G c = 2, u = 9.42
(Paldoussis 1979).
systems clamped at both ends (and free to slide axia11y): (f) and (g) show the implosion
pattem, in the former case for a system of three cylinders. Photographs (h), (i) and (j) are
atmuch higher flows, relative to the onset ofthe first divergence, than the others; (h) shows
a pattem where the two cylinders closer to the ob server seem to be in a pinching pattem,
while the two others in an explosion pattem; (i) shows fair1y well-behaved osci11atory
motion (flutter) in an essentially second-mode implosion-explosion pattem; (j) shows
typica1, less clear, osci11atory motions at yet higher flow.
9.3.4(b) Measurements and comparison with theory
Unlike the behaviour observed in experiment s with solitary cylinders in essentia11y
unconfined flow where the system is usually subject to a sing1e in stability at a time, which
would subside prior to the onset of another, here both theory and experiments indicate

1074
SLENDER STRUCTURES AND AXIAL FLOW
that, once the system becomes unstable, it never regains stability; moreover, the system is
subject to several instabilities at once. Therefore, the only point which may unreservedly
be used for comparison is that of onset of the first instability - which, strictly speaking, is
the limit of applicability of linear theory in any case. However, a secondary point of com-
parison wi1l be the onset of flutter, to the extent that it may be detennined experimentally,
and subject to the reservations discussed in the foregoing.
Divergence was detennined by monitoring continuously the deflection of the cylinders
and recording it as a function of flow, in the neighbourhood of the presumed critical flow
velocity. The onset of divergence could then be detennined in the normal way as the
intercept with the flow axis, of the tangent to the flow-amplitude curve. The accuracy of
thus detennining the critical flow velocity Ucd was not better than within 5%.
Flutter, as already discussed in the previous section, was difficult to pinpoint because of
concurrence with other instabilities; moreover, as shown by theory, its onset is associated
with smal1 or vanishingly small frequencies of osci11ation [cf. Figure 9.13(a) and (b),
respectively], compounding the difficulty of detennining its on set. Had the osci11ations
in the range of flutter been more regular, then the critical flutter flow velocity, ucf, could
have been detennined as the zero-frequency intercept of the tangent to a frequency-flow
curve - at least if its onset was as in Figure 9.13(b) rather than 9.13(a). However, this
was not feasible and, at best, only a range of flow ve10cities could be specified within
which flutter established itself as an identifiable phenomenon. The beginning of this range
is characterized by flip-flopping motion of irregular frequency, in which not al1 cylinders
moved in phase, while at the end of the range an almost continuous and essentially regular
oscil1ation could be observed.
Three sets of boundary conditions were tested: nominally c1amped or pinned at both
ends, or c1amped at the upstream end and pinned at the other.
Some typical results for c1usters of four cylinders, 49.5 cm long and pinned-pinned ar
clamped-clamped, with the downstream end axially sliding, are shown in Figure 9.20.
Where more than one experimental point is shown, this indicates that the experiment was
repeated after the system was disassembled and reassembled, thus providing a measure af
repeatability.
If one considers the theoretical and experimental values of Ucd and ucf for the pinned-
pinned cylinders as shown in Figure 9.20(a), it is noted that agreement for the onset af
divergence is very good, in al1 cases being better than lO%,t whi1e agreement for flutter
is to within 8 or 15%, depending on whether one compares theory with the lower or
upper extreme of the experimental range. The dashed horizontallines correspond to the
theoretical values of Ucd and ucf for a solitary cylinder in unconfined fluid - to which the
system tends asymptotical1y, as G c and G w -+ 00, Clearly, agreement between theory
and experiment is quite satisfactory in this case.
Agreement in the case of the c1amped-clamped system of Figure 9.20(b) is much less
c1ose. This was pin-pointed to be, at least partly, due to imperfect c1amping conditions at the
lower, axial1y sliding support. Clearly, to allow sliding, some 'play' was necessary, which,
in tum, al10wed some, albeit smal1, rotation. To account for this properly, a nonlinear
correction is necessary. However, an approximate correction is possible by modelling the
lower support as a pinned one, with a rotational spring constraint of dimensionless spring
t AU percentage discrepancies between theoretical (x) and experimenta1 values (y) are calculated as
(x - y) I[ (x+ y) 12], since it is not obvious whether the theory is being checked againstexperiment, or více versa,

CLUSTERED CYLINDERS IN AXIAL FLOW
1075
constant Kt. With the new boundary conditions, new beam eigenfunctions and eigenvalues
are necessary in the Galerkin scheme; the characteristic equation is easily found to be
Kt (1 - cos A cosh A) - A (cos A sinh A - sin A cosh A) = O as shown in Patdoussis (1979;
appendix). The value of Kt was determined from experiments in air, with the lower end
lubricated by water. t For the different conditions described in the footnote, a range of Ki
was determined: 7.6 ::: Ki ::: 14. Theoretica1 and experimental values of Ucd for pinned-
pinned cylinders (unaffected by this modification) and clamped-clamped cylinders (to
be henceforth called 'clamped-sprung') for four different values of L:j: are compared in
Figure 9.21. Agreement with the thus modified theory for the clamped-sprung cylinders
is much better than in Figure 9.20.
Figure 9.22 shows theoretica1 and experimental va1ues of Ucd and ucf for cylinders
clamped upstream and pinned downstream. Agreement is remarkably good, being better
than 11 % - surprisingly, but not too significantly, even for ftutter.
ln some cases where ftutter developed reasonably clearly, some measurements of the
frequency of osci1lation were made, as the ftow velocity was increased. In view of the
general behaviour of the system, as discussed in the foregoing, the measured values could
not be more than ::I: 1 0% accurate. These frequencies for a pinned-pinned system of four
cylinders (with G c = 2, and other parameters as in Figure 9.20) are compared with the
theoretical ftutter frequencies, in dimensionless form, in Figure 9.23. The dimensional
frequencies were of the order of 1 Hz. It is noted that, according to theory, all modes of
the first-mode group lose stability, in succession, within the interval of u involved; the
theoretical Re(w) shown in the figure are associated with unstable loci only. Clearly,
because of the nature of the theoretica1 behaviour, one can compare the observed fre-
quencies to only the ensemble of unstable modes, rather than to any one in particular.
Equally clearly, experimental and theoretical points are in the same range, which may be
considered to lend further support to the validity of the theoretical model.
Additional experiments and comparison with theory for two- and three-cylinder clusters
and for cylinders in which sliding at the downstream end is prevented are described in
Pa'idoussis (1979) and Pa'idoussis et al. (1982). An example is shown in Figure 9.24 for
three-cylinder clusters with pinned-pinned and clamped-sprung end conditions, for two
different values of DI/ R and varying Dz/ R. The error bars indicate a fair1y large degree
ofuncertainty, specially for the clamped-sprung cases [in panels (c) and (d) ofthe figure].
Nevertheless, agreement between theory and experiment in these more carefully conduc-
ted experiments is sufficiently good to conclude that the theoretical model is effectively
validated.
9,3.4(c) Transítíon to buckled state
The onset of divergence, being the first to be encountered with increasing ftow, has received
special attention. Figure 9.25 shows the manner in which cylinder deftection increases with
u in four different cases.
tIn these experiments, the first-mode natural frequency was measured for different tightness of fit, penetra-
tions of the lower pin into the support, amplitude of motion, etc, These were then compared with theoretically
predicted first-mode frequencies obtained by a simple Rayleigh method, where the dimensionless first-mode fre-
quencyw) = [m/EI]I/2Q[ L 2 isfoundfrom(5wJ+1512-18y)b 2 -(l5wI+2520-48y)b+(l2wJ-42y) =
O, in which b = (Kj +4)/(Kj + 6); in tenns ofthe dimensiona1 spring constant, CL, here Kj = CLL/ El.
*For the theoretical calculations shown, r:: and y were calculated for L = 52.1 cm. However, the theoretica1
results using the actual L are essentially coincident with those shown.

1076
10
9
8
7
6
u
5
4
3
2
10
(a)
SLENDER STRUCTURES AND AXIAL FLOW
--
Pinned-pinned
2
G c
3
4
9
8
7
6
u
5
4
3
2
10
(b)
10
;:
Á
o 8
o
8
Clamped-clamped
2
G c
3
4
Figure 9.20 Theoretical and experimental values of the dimensionless critical flow velocities for
divergence, Ucd, and flutter, ucf' of a four-cylinder system with pinned or c1amped supports, for
three inter-cylinder gaps, G c ' (a) Pinned-pinned; (b) clamped--clamped cylinders. The theoretical
calculations were done with parameters f3 = 0.467, y = 2.65, 8 = 0, 8 = 20.71, cf = 0.016, a =
C = Cb = O (Páidoussis 1979),
4
3
ucd
2
10
(a)
6
5 
4
4 3 0 2 3 4
(b) G c

2 3
G c
Figure 9.21 Theoretical and experimental values of the dimensionless critical flow velocity for
divergence, Ucd, of several systems of different lengths, with either pinned or nominally c1amped
end supports. The theoretical parameters are the same as in Figure 9,20 for the pinned system;
similarly for the clamped system, but with the lower support modelled as pinned and constrained by
a rotational spring of nondimensional stiffness K;. (a) Pinned-pinned cylinders: O, L = 52,1 cm;
0, L = 49,5 cm; !:::,., L = 47.0 cm; _, L = 41.9 cm; -, theory; (b) clamped--clamped cylinders:
same notation for experimental data; - top, theory (Kj = 14); - bottom, theory (K; = 7.6);
(Pa'idoussis 1979).

1077
CLUSTERED CYLINDERS IN AXIAL FLOW
u 5
9
8
7
6
4
3
2
10
2
4
5
3
G c
Figure 9.22 Theoretical and experimental values of the dimension1ess critica1 flow ve10city for
divergence, Ucd, and flutter, ucf, of a system of four cy1inders clamped at the upstream end and
pinned downstream, for three intercylinder gaps, G c. Parameters as in Figure 9.20, except £ = 18.69
and y = 1.77 (Pa'idoussis 1979).
14
12
.
10
8
a

6
4
2
6 7 8 9
u
Figure 9.23 The dimension1ess observed frequencies of oscillation for a cluster of four pinned-
pinned cylinders compared to the theoretical Re (w) values of the unstab1e loci of the system;
G c = 2. other parameters as in Figure 9.20 (Pa'idoussis 1979).

1078 SLENDER STRUCTURES AND AXIAL FLOW
7 7 -.
1----------------- -------------------------
6  6 
5 5
Solitary cylinder
4 in unconfined fluid 4
u
3 - ----------------------- 3 ----- -- ---------------- --
! I '
ucd ucd
2 2 Theory
Expenment
D]/R = 1.5 D,/R = 2,0
O 00
O 2 3 4 5 2 3 4 5
(a) (b)
9 9
8  8 
7 7 ucf
6  6
5 5 
u
4 4
3
2
D/R 15
O
O 2 3 2 3 4 5
(c) DzlR (d) D 2 /R
Figure 9.24 Comparison between experimental and theoretical dimensionless critical flow velo-
cities for divergence, Ucd, and flutter, ucf' for a three-cylinder system with different values of
Dli R and D21 R. (a), (b): Pinned-pinned ends; (c), (d): clamped-sprung boundary conditions with
K* = 7.6, Othersystemparameters: f3 = 0.467, t: = 16.67, y = 1.38, h = 0.176, cf = 0.016, Ci =
Cb = O (Pai'doussis et a/. 1982).
It is seen that for the pinned-pinned system with axial sliding permitted - left-hand
diagram in Figure 9.25(a) - divergence developed most abruptly and cleanly. Divergence
for the clamped-clamped case with axia1 sliding prevented - right -hand diagram in (b) -
is the most gradual, since deflection in this case involves axia1 extension of the cylinder;
clearly, deciding what Ucd would be in this case is most difficult. The clamped-clamped
cases with axial sliding - right-hand diagram in (a) and left-hand diagram in (b) - are
intermediate. t
The three regions, marked 1, 2 and 3 in Figure 9.25(a), correspond to regimes of dif-
ferent dynamical behaviour: region 1 is the subcritical regime, where system behaviour is
dominated by turbulence-induced smal1-amplitude, first flexural-mode vibration; region 2
trn the 'clamped-sprung' cases ofFigure 9,25(a), the clearance for sliding was relatively larger than for the
'axially sliding, clamped-clampeď case ofFigure 9.25(b), for which Kj  00,

CLUSTERED CYLINDERS IN AXIAL FLOW 1079
1.6
 1.2 Pinned-pinned
;:: R
"  -I-]D
E
"
 J 1
]- 0,8 Dz
'" (DI@
 
I
I
. 0.4 I
I
::E I
I
I
I
I
0,00 2 4 6 8
(a) u
0.5
o Clamped-clamped
 0.4
;:: Axially sliding
"
E lower sup port
"
u 0.3
os
'i5..
on
-6 I R
... DZ
 0,2
,5 +-
>- Dz . Dz
u
..:,
 0,1
0.0
O 2 4 6 2 4 6 8
(b) u
Figure 9.25 (a) Typical well-defined thresholds of divergence for pinned-pinned and clamped-
sprung three-cylinder systems with axially sliding bottom supports, obtained by optical meas-
urements, showing also three of the regimes of dynamical behaviour, 1, 2 and 3, encountered
with increasing dimensionless flow velocity, u; dimensionless parameters: f3 = 0.467, € =
16,67, y = 1.38, h = 0,176, Cf = 0.016, Cb = 0.0; for the pinned-pinned system: DI! R = 1.5,
D21 R = 1,25; for the clamped-sprung system: Dl IR = 1.5, D21 R = 2,0, (b) Typical transition
to buckling of an instrumented clamped--clamped four-cylinder system; dimensionless parameters:
f3 = 0.466, € = 20.6, y = 1.93, h = 0.197, cf = 0.027, Cb = 0.20, D21 R = 0.75; for the case of
an axially fixed 10wer support: r = O, TI = 141, v = 0.47 (Pai:doussis et a/. 1982).
is a transition region where deflection increases rapidly; and region 3 corresponds to the
buckled state. In this latter state, the cylinders are severely bent, often in a cross-sectional
pattem where they all have moved toward the centre of the channel; especially when the
cylinders are free to slide at the lower support, they may even bend enough to touch one
another. Vibration about this bent state is considerably abated. However, in some cases,
the cylinders snap back and forth from an inward collapsed form to an outward one.
ln experiment s with strain-gauge instrumented cylinders it was possible to obtain PSDs
of the vibration signal, as shown in Figure 9.26. The experimental band of dominant
vibration frequencies was found to always be in the neighbourhood of the theoretical
eigenfrequencies of the first flexural-mode group. The second- and third-mode groups of

1080 SLENDER STRUCTURES AND AXIAL FLOW
w
O 5 10 15 20 25 30 35
0,003
----- u = 3
1 I -u=4
-,-.- u=5
,.--, !Iiji,
 0,002 li i I
N !ii 
8 I '\
:o-
.....
o 4 
Q
 0.001  I .
",
, IV 
\\!
0.000 O 2 5 6
Figure 9.26 Observed power-spectral-densities for the mid-cylinder lateral displacement, 8, of
a cylinder in a cluster of four clamped-clamped cylinders, for three different dimensionless flow
velocities u, Frequency resolution: l1f = 0.024 Hz; 500 averages, System parameters: f3 =
0.466, s = 18.6, y = 1.42, h = 0.197, r = O, I1 = HS, v = 0.47, D2/R = I; D2 is definedin
Figure 9.25 (PaYdoussis et al, 1982).
40
Theoretical
? c;> frequencies
30 I o ? ofmode
c;>
w 20
6
. .
6 Ó
10 -... Average experimenta!
frequency band
0-----0 Extreme va1ues of
exp. frequency band
O
O 2 3 4 5 6 7
u
Figure 9.27 Comparison of the theoretical eigenfrequencies of the first flexural-mode group with
the observed vibration frequency band, at different values of the dimensionless flow velocity, u, for
a system of four clamped-clamped cylinders. Dimensionless system parameters: f3 = 0.466, s =
20.6, sCf = 0.556, y = 1.93, h = 0.197, r = O, I1 = 141, v = 0.47, D2/ R = 0.75 (PaYdoussis
et al. 1982).

CLUSTERED CYLINDERS IN AXIAL FLOW
1081
frequencies could also be discemed, but with very small energy content; see, for example,
Figure 9.43. Figure 9.26 shows clearly that with increasing flow (i) the amplitude of
vibration is increased and (ii) the vibration frequency band becomes broader and shifts
toward lower frequencies, as predicted by theory.
Within each PSD plot, it was found difficult to pin-point with certainty individual
frequencies, as (i) the dominant frequency peaks within the band shifted from one exper-
iment to the next, and (ii) the number of the dominant peaks was sometimes smaller, but
usually larger than the theoretical number of 6-8. However, the frequency band itself
remained sensibly unchanged and well defined, and it was decided to compare theoretical
and experimental bands of frequencies, rather than individua! frequencies. A typical such
comparison is made in Figure 9.27, where agreement with theory is seen to be reasonably
good. It should be notOO that the flow velocity at which the minimum frequency within the
band is essential1y zero corresponds to the onset of divergence, which is in good agreement
with theory.
9.3.5 Clustered, flexibly interconnected cylinders
ln all engineering applications involving clustered cy1inders subjected to flow, for example
in heat exchangers, nuclear reactors, transmission 1ines, and bundled offshore risers,
the cy1inders are neither totally unconnected to one another, nor are they supported by
utterly rigid frames; rather, they are normally interconnected by flexib1e, sometimes flimsy,
structura1 elements. Thus, in the case of nuclear fuel rods, the thin end-plates used in
some types of nuclear systems (Figure 9,28) cannot be considered rigid. Spacers used
in heat exchangers, transmission 1ines and offshore risers are similarly flexible, often by
design.
ln view of the interesting dynamics obtained for clustered cy1inders, coupled via the
interstitia1 fluid, it is of interest to explore the dynamics of structurally interconnected
cylinders, first in vacuum and then in axial flow, as originally done by Pa1doussis et al.
(l983a,b ).
9.3.5(a) Flexibly interconnected cylínders in vacuum
The end-plate supports of Figure 9,28(a) and (b) could well be modelled as in Fig-
ure 9.28(c), Each of the small struts connecting cylinders i and j is flexible in terms
of in-plane stretching and bending, and may be modelled by springs of stiffnesses K ij and
K[}, as shown in Figure 9.29(a) and (b), Simi1arly, and quite generaI1y (Le. irrespective
of the kind of connectors used), we have out-of-plane bending and torsion, modelled by
two rotationa1 springs of stiffnesses Cij and C[}, acting in two perpendicular directions,
as shown in Figure 9.29(c)-(f).
The appropriate boundary conditions are obtained via straightforward force and moment
balances. However, the kinematics is quite involved, and hence is not given here; only the
final results. Nevertheless, to give a flavour of the method, we give some of the pertinent
relationships for the in-plane components. Thus, the deflection-induced forces are
Fjj = Kij(Rjj - R),
F!, = K, 8.R.
IJ IJ IJ'
(9.68)

1082
SLENDER STRUCTURES AND AXIAL FLOW
(b)
Figure 9.28 (a) Typical fuel element 'bundle' used in CANDU-type nuclear reactors; (b) end
view of the plate to which the ends of the cylinders are attached; (c) idealized model of end-plate,
showing intercylinder and cylinder-to-channel connectors (Pai:doussis et al. 1983a),
where R, Rij and J)"R;j are shown in Figure 9.29(a). With the help of the diagrarns in
Figure 9.30, one finds
Rij - R = cos ei(-Wi + Wj) + sin eg(-Vi + Vj),
J)"Rj = sin eg(-Wi + W j) - cos ei( -Vi + Vj),
(9.69)
with similar ex p ressions for Rii - R. and J)" R' " Considerin g then a force balance at the
II II
upstream end of the cylinders, x = O, one obtains, to first-order approximation,
3 N
a Vi" ' 2 o I i }
EI ---:f +  Kij SIn eij{Vi - ( - Oj)Vj
ax . 1
J=
N
+ L(Kij - K: j ) sineg COSei {Wi - (1 - O)Wj}
j=l
N
+ L K[j cos 2 eg {Vi - (1 - O)Vj} = O,
j=l
(9.70)
where oj is Kronecker's delta, and N is the number of cylinders in the cluster.
Proceeding similarly and taking moment balances, the dimensionless boundary con-
ditions which should be used in conjunction with equation (9.65) or (9.22) are the

CLUSTERED CYLINDERS IN AXIAL FLOW
1083
Cylinder i ; :
(a)
(c)
(e)
Massless
bar
(b)
(f)
Figure 9.29 (a) In-plane deformation of a connector, involving in-plane stretching and bending;
(b) detinition of the springs of stiffnesses Kij and K[j' (c) Out-of-plane bending and (d) torsion
ofthe connector; (e, f) detinition ofthe rotational springs Cij and C[} resisting out-of-plane bending
and torsion (Paidoussis et al. 1983a).
y
y
 R'
IJ
I......)
-"
R
Rij
C>
z
z
Figure 9.30 Diagrams for the analysis of in-plane k:inematics of the connector (Paidoussis et al.
1983a).

1084
SLENDER STRUCTURES AND AXIAL FLOW
following:
N
E(a311i!a3) -l)gij + J(){l1i - (1 - 8)l1j}
j=l
N
'"', '
- (hij - hij){I1N+i - (1 - 8j)I1N+j} = O,
j=l
N
E(a311N+i!a3) - L(fij + g;){I1N+i - (1 - 8)I1N+j}
j=1
N
- L(hij - h;j){l1i - (1 - 8)11)} = O,
j=l
and
N
E(a211i!a2) + Laij(a/a){l1i - (1- 8)l1j}
j=l
N
+ Ldij(a/a){I1N+i - (1 - 8)I1N+)} = O,
}=1
N
2 2 '"' '
E(a I1N+i!a ) + bij(a/a){I1N+i - (1- 8j)I1N+)}
}=1
N
+ Ldij(a/a){l1i - (1- 8)11)} = O,
j=l
(9.71a)
(9.71b)
at = O(whereE = -1)and = 1 (whereE = 1), for i = 1,2,...,N; in these
equations, Jij, gij, etc., stand for
Jij = kij sin 2 eg, gi} = kij cos 2 eg, hij = ki} sin eg cos eg,
I'!. = e. sin 2 e9. g . = e. cos 2 e9. h(. = e. sin e9. cos e9,
JiJ IJ IJ' IJ IJ IJ' IJ IJ IJ IJ'
2 e o ,. 2 e o b . 2 e o , 2 e o
aij = Cij cos ij' +Cij SIn ij' ij = Ci} SIn i}' +C ij COS ij'
di J ' = (c i J ' - C ,) sin eO, cos e9.
IJ IJ IJ'
(9.72)
and are constants for a given system. The dimensionless quantities are related to the
dimensional ones by
Wi Vi
l1i = L' I1N+i = L '
, L 3 ,
k.. = -K..
IJ EI 'J'
L 3
kij = -Kij,
EI
x
 =-,
L
L
Ci} = -Cij,
EI
, L,
c" = -c..
'J EI 'J'
with Kij to C;j as in Figure 9.29,
(9.73)

CLUSTERED CYLINDERS IN AXIAL FLOW
1085
12
10.99
----------------- --  --
10 m ode gro

8
7,85
-------- - mode group---
---2:rr
A
6
4
1st-mode group
:rr
2

Within each band t,
there are six eigenvalues
o
10- 3
(a)
10 1
cll
10 3
10- 1
12
@
8 ---- -
10
--- 3:rr
6
2:rr
4,73
4
2
--:rr
o
10- 2
(b)
10 2
10- 1
10° 10 1
cll
10 3
Figure 9.31 Evolution of tbe three lowest groups of eigenvalues of the system shown inset, for
constant kij and k;j and varying Cij and c;j' all of which are of tbe order of magnitude of ClI (see
Tables W.l and W.2): (a) for large in-plane stiffnesses [kij  0(10 4 ), k;j '" 0(10 2 )]; (b) for small
in-plane stiffnesses [kij, k;j '" 0(10- 4 )]; (Pa'idoussis et a/, 1983a).
New beam eigenfunctions are then derived, satisfying (9.71a,b) which are used as
comparison functions for the system t in vacuo, and in quiescent or flowing fluid,
The eigenva1ues, Ak == ú)/2, have been computed for a geometricaUy symmetric system
of three interconnected cylinders, each of which is a1so attached to the channel, shown
diagrammatically, inset in Figure 9.31. The results are shown in Figures 9,31 and 9.32,
Mainly to minimize computational difficulties associated with the occurrence of repeated
eigenvalues, but also to reflect more dosely a 'real' case (Figure 9.28), not all spring
constants of a kind were taken to be equal, but merely of the same order of magnitude;
thus, C21/Cl1 = 8/5, C3t!CIl = 1, and so on, in the case ofFigure 9.31; these ratios were
chosen arbitrarily and are given in fuU in Appendix W,
For the calculations of Figure 9.31(a), kij and k;} are kept constant and large [kij '""
0(10 4 ), k;} '"" 0(10 3 )], while the out-of-plane stiffnesses are varied. (It is realized, of
course, that this independent variation of some of the spring stiffnesses, while others
are kept constant, is for analytical convenience and that it is not physically realistic.)
The abscissa is Cll, but, as may be seen in Appendix W, all the Ci} and C;} are varied
proportionately to Cll' Moreover, in the figure only the band of the lowest three groups of
eigenvalues is shown within each mode group.
tUnlike cases or independently supported cylinders, here the comparison functions involve ull or the
structura1\y coupled cylinders, rather than just one.

1086
SLENDER STRUCTURES AND AXIAL FLOW
It is seen that for very smalI Ci}, <i' the six first-flexural-mode eigenvalues are essen-
tialIy coincident and equal to the classical value of 7r for idealIy pinned-pinned cy1inders.
Similarly, the second- and third-mode groups are c1ustered about 27r and 37r, respectively,
but the eigenvalues are increasingly spread out for these higher mode groups; this is phys-
ically reasonable, as higher flexural modal shapes imply larger end-moments, so that even
very weak out-of-plane springs do not have a neg1igible effect on the coupled dynamical
behaviour of the system.
With increasing Ci}, <}' the band of each group of eigenvalues spreads out, which char-
acterizes stronger intercylinder coup1ing. For large enough Ci}, <}' the system eigenvalues
approach those of an ideally c1amped-clamped beam. Yet even there, fairly strong coup-
1ing sti11 exists for the higher modes; although the spring constants are large, they are not
quite infinite!
Similar results are shown in Figure 9.31(b), where in-plane stiffnesses are now very
low [kij and k;} ,-.., 0(10- 4 )]. It is seen here that ifthe Ci}, C;} are large, the eigenvalues
approach those of a beam with idealIy transversely 'sliding' end conditions, that is, zero
slope and no constraint to lateral movement (Bishop & Johnson 1960), Hence, in addition
to those shown, there is an additional group of six nearly null eigenvalues, lying essentially
on the CI 1 -axis, corresponding to rigid-body motions ofthe cylinders, paralIel to the x-axis.
They wi11 further be discussed later.
At the left-hand end of Figure 9.3l(b), alI springs are very weak, and the three groups
of eigenvalues approach those of a free-free beam, namely O, 4,73, and 7.85. The group of
eigenvalues approaching zero is associated with the other six possible rigid-body modes
of free-free beams, involving rigid-body rotation about the cylinder midpoint; these
c1early evolve from corresponding mode shapes of a s1iding-sliding beam, which are
half-sinusoids with a midspan node. (The formerly discussed paralIel motion rigid-body
modes are of course sti11 there, with nearly nulI eigenvalues.) It is remarked that the first-
mode eigenvalues shown are sti1l somewhat away from zero at Ci} '" 0(10- 2 ), but the
progression to zero cannot but be very gradual, as c;}' cij, kij and k;}, although small, are
stilI finite. Once more, intermediate values of Ci}, C;} give the broadest band of system
eigenvalues for each mode group, that is, they are associated with maximum coup1ing in
cylinder motions.
A more interesting set of results is shown in Figure 9.32, where Ci}, C;} are fixed at
fairly high values, whi1e the kij are varied from '" 0(10- 1 ) to '" 0(10 5 ), and the k;}
proportionately but two orders of magnitude smaller. Thus, this figure may be thought of
as the continuation of the previous one, from the state represented at the right-hand side
ofFigure 9.31(b) to high kij and k;j'
For small kij, k;}, it is noted that, in addition to the eigenvalues c1ustered about 7r and
27r (those about 37r are not shown), there now appears a new set lying between A = 0.4
and 1,2 which were absent in Figure 9.31(b). Noting that k;} and k i } at the extreme left of
Figure 9.32 are, respectively, one and three orders of magnitude higher than those at the
extreme right ofFigure 9.31 (b), these 'new' eigenvalues may be identified as the evolution
of the set of nulI eigenvalues formerly lying on the abscissa.
At the extreme right of Figure 9.32, once more alI the spring constants are high and,
predictably, the eigenvalues tend to cluster around those of a clamped-clamped beam.
The most interesting part of this figure, however, lies in the middle range of in-plane

CLUSTERED CYLINDERS IN AXIAL FLOW
1087
10
..
.
------------;:::--:------- ! r
, 6 -______--__________lo ------------ 2/::"
--------  ---------- -: .------11------ r
. .
. .
; .. I
---------r o ------- I .------8.:.---n: ..
SiX / 'S . ·
.. e
e
. e'
SiXA'S --..: o.
8
\O-I
4
2
$
o
\0-2
10°
\01
\02
\03
10 4
10 5
k ll
Figure 9.32 Evolution of the three lowest groups of eigenvalues of the system shown, for fairly
large and constant cij, C;j [cij, C;j  0(10 2 )] and variable kij, all of which are of the order of
magnitude of kll, and with the k;j two orders of magnitude smaller, o: Stiffnesses as given in
Table W.l; +: stiffnesses as given in Table W.3 (Pai'doussis et al. 1983a).
stiffnesses. Thus, whereas for lower kij the three groups of eigenvalues are quite distinct,
for kij > 0(50) approximately, the first- and second-mode groups become intertwined
into a single group of 12 eigenvalues. Sirnilarly, for kij > 0(5 x 10 2 ), all three groups of
eigenvalues become interspersed. This means that the eigenvalue bands become so large
as to intersect. On the other hand, as the kij are further increased, the eigenvalues once
more begin to separate into distinct mode packets: for kij rv 0(5 x 103), there is a clearly
distinguishable first-mode group; for higher kij, the second and then the higher-mode
groups become once more distinct.
Thus, it is seen that very intricate and unexpected coupling effects arise when the
different stiffnesses involved in structural coupling are varied. However, for the purposes
of this book, this is but a preamble to the dynarnics when quiescent or flowing fluid is
present.
9.3.5(b) Flexibly interconnected cylinders in flow
The equations of motion are the same as in Section 9.3.1, but a simplified version is used
in the calculations by considering zero gravity (y = O) and no axial sliding (8 = 1).
A set of eigenfrequencies (the first and second flexural-mode bands) are shown in
Figure 9.33, in vacuum and quiescent fluid, The in vacuo eigenfrequencies correspond
exactly to the lowest set of eigenvalues shown in Figure 9.31 (a) and (b). The main effects of
the additional, fluid coupling are: (i) to render the frequency bands wider; (ii) to prevent
the severe narrowing of the bands for low and high Cij, <j' because fluid coupling is
operative throughout.

1088
SLENDER STRUCTURES AND AXIAL FLOW
25
f3=0
-"--4- f3 = 0.1
10
20
15
#,k---'"
/
/
I
,
,
,
I
,
/
/
l
/'2
---;>" 1C
__.._--,.IIT
w
10
5
--.
/,,,,
,
,
,
" -4
, kij-({)(1O )
c::(>
5
$
10- 3
10- 1
10 1
Cl1
10 3
o
Figure 9.33 The band of eigenfrequencies of the lowest mode group in vacuum (f3 = O) and in
stilI fluid (f3 = 0.1) as Cij and <j are varied, proportionately to Cli. Upper figure: the in-plane
stiffnesses are large [kij  0(10 4 ), k;j  0(10 2 ), vide Table W.l]; lower figure: the in-plane
stiffnesses are small [kij, k;j  0(10- 4 ), vide Table W,2]. Other system parameters: r = TI =
u = O, G c = G w = 1/4 (Pai:doussis et al. 1983b).
The effect of flow for a typical three-cylinder system with vanishingly large in-plane
stiffnesses and small \Jut-of-plane ones is illustrated in Figure 9.34(a). It is seen that
the evolution of the first flexural-mode eigenfrequencies with increasing u is similar
to that of a system with only fluid coupling. For higher u than shown, divergence in
each of these modes takes place, at u  1.6, 1.63, 2.0, 2.85 and 2.85 for the first five
eigenmodes.
On the other hand, for 10w in-plane and intermediate out-of-plane stiffnesses (such
as in the middle range of Figure 9,32), the results become far less regular, as shown
in Figure 9.34(b). In this case the first and second flexural-mode groups become inter-
twined, and one may distinguish one grouping of three modes which, for low u,
lie in the range 3 < Re(w) < 6, another grouping of six modes in the range af
10.5 < Re(w) < 14, all ofwhich are shown in Figure 9.34(b), and another three modes
at higher Re(w). The third-mode group remains distinct. The nature of these modes is
rather perplexing, and they are discussed further when the eigenfunctions of the system are
examined.
The cross-sectional modal shape characteristics of the lowest 12 modes of the system
are shown in Figure 9.35(a,b). The flexural mode shapes are shown in Figure 9.35(c)-(f).
Several points should be made, as follows, (i) The flexural mode shape ofFigure 9.35(c)
is associated with modes 1-3, while that of Figure 9.35(d) with modes 4-6, Similarly,

CLUSTERED CYLINDERS IN AXIAL FLOW 1089
0,05
1.9
0,04
 0,03

0,02
0,01
0,°°0 2 4 6 8 10 12 14
(a) (w)
0,05
2,01
,-., 0,03


2.0 0,8
0,01
0,6 0,2 0,2
0,2
0.00
° 2 4 6 8 10 12 14
(b) (w)
Figure 9.34 (a) The evolution of the six lowest eigenfrequencies with increasing u, shown as an
Argand diagram, for a three-cylinder system with low out-of-plane stiffnesses and high in-plane
stiffnesses [cij, c;j  0(10- 1 ); kij  0(10 4 ), k;j  0(10 2 ), as given in Table W.l, with Cll =
1.73 x 10- 1 ]. (b) The evolution ofthe nine lowest eigenfrequencies for a system with quite low in-
plane stiffnesses and intermediate out-of-plane stiffnesses [kij  0(10), k;j  0(10- 2 ), Cij, c;j 
0(10 2 ), as given in Table W.3, with kll = 6.50]. Other system parameters: f3 = 0.1, r = TI = O,
eCf = 0.25, h = 1.26, G c = G w = 1/4 (Pai'doussis etal. 1983b).
the mode shape of Figure 9.35(e) and (f) are associated with mode s 7-9, and
modes 10-12, respective1y. (i i) The cross-sectiona1 moda1 patterns of modes 1-3 are
essential1y the same as those for mode s 4-6; the same app1ies for mode s 7-9 and 10-12.
(iii) The flexura1 moda1 shapes in (c) and (e) resemb1e the first, or 'nu1I' rigid-body mode of
a sliding-s1iding beam, while those in (d) and (f) those ofthe second (first flexural) mode.
From points (i) and (ii), it is clear that the 12 modes concerned involve two intertwined
flexura1-mode groups: the first, consisting of modes 1, 2, 3, 7, 8 and 9, and the second,
comprising modes 4--6 and 10-12. However, although these mode groups are mixed, it is
still possible to identify them.

- .....
g<£
o 00
'.z: «1)
l2Ji l2;]  

00 _
O 
.....-0
../" ; U O
../" ';;' <U 8
£-
II  
'" oD 13
s s '-'><
.. <U
'r:<e
o
<U
ul  II '-'
;:,:'-Ó
.....-0
,...., <£a

Z'V)
........   .,f
II II 
S 0\-0<'"1
S <U 000
1380\
101) -
..8
;r 4-< ...
5r o -o 
e g  c 8 :;-,
/ -"   -" L <U I=i 
  ::1
>. .0
00 <'"1 -o
<U :03
£-o
a'-'
,...., 'SN
s6  ťS"":.s
-go
8'8::
iJ a 8.g
<U ..... o
f  f   <£ 8
-00.....
6\ /  (} I t , g,<£
oo
<U ..c: 4-<
£00'-'
--o
'Sa
00 o .
e 8 0\
«1)  .8
13!'
J a Q.,><OO
..-o..g
- 8 o
-  ---- '
"      3 08
83-
'čd 00
I=i <U
0-0
@ II 6> II '.0 o
N N on '" U 8
s s .
00 00
00...
o ><
..... <U
 u I=i
LJ <U <U
c E5£
c
-- '
C 4-<
+  o
\ I  '-'00
Ir) e
tr)8
e s @ I,...., ,...., o\
Q.,
s   f:-
= 
.-g
r..8

CLUSTERED CYLlNDERS IN AXIAL FLOW
1091
A final point of interest in Figure 9.35 concems the existence of orbita! pattems of
motion, for example, for modes 5 and 6, ln this case, unlike what has been found when
fluid coupling alone is present, this is believed to be related to the fact that the flexural
mode shapes in the y and z directions are not identical; for example, in one direction they
may be as in Figure 9.35(d), whi1e in the other the quasi-node at the cylinder midpoint is
much more diffuse.
9.3.5(cJ Concluding remarks
It has been shown that, provided that either the in-plane or the out-of-plane spring con-
stants are either small or large, the system behaviour is similar, but not identical, to that of
a system of independently supported (structuralIy uncoupled) cylinders. Thus, the charac-
teristics of the system in vacuum tend toward those of solitary beams with the appropriate
boundary conditions, as in such cases there is effectively little intercylinder coupling of
any kind; also, in axial flow, system behaviour is similar to that of systems discussed in
Section 9,3.3,
When one set of spring stiffnesses is of intermediate magnitude, however, then struc-
tural coupling effects become pronounced, and the frequency bands of the so-called mode
groups become considerably broader; but these mode groups remain distinct. The broaden-
ing of the frequency bands becomes more pronounced in the presence of fluid (Figure 9.33),
and more so in the presence of flow (Figure 9.34), The relative importance ofhydrodynamic
and structural coupling depends on the fluid density and 'tightness' of the cylinder geo-
metry, as well as on the magnitude of the spring stiffnesses, a1l of which have been
characterized by appropriate dimensionless numbers.
The most interesting case arises when all kinds of springs are of intermediate strength.
Then the curious phenomenon occurs that the first- and second-mode groups, or even
alI the lowest three-mode groups, merge, As shown here, this may be thought to be a
manifestation of extreme widening of the frequency bands of the different mode groups,
leading to intertwining of the eigenfrequencies. One may stilI identify sets of 2N modes
as belonging to distinct mode groups, but the ordering of the modes becomes irregular: as
shown in conjunction with Figure 9.35, the first mode group there consists of modes 1, 2,
3, 7, 8 and 9 of the system, where these modes have been nurnbered in ascending order of
the eigenfrequencies.
For stability considerations, for systems with firmly supported ends (i,e, fair1y large
kij and k;), the transition from one limiting case of spring stiffnesses to the next is
smooth, and one may consider the pinned-pinned system as giving the most conservative
estimate of the critical flow velocity. However, for low kij and k;j' considerably smalIer
critical velocities may be obtained,
9.3.6 Clustered cylinders simultaneously subjected to internal and
external flow
The dynamics of a solitary cylinder concurrently subjected to intemal and extemal flow has
been treated in Section 8.6, where most attention is devoted to cantilevered cylinders,
Here, the dynamics of clustered cylinders with both ends supported is discussed; heat-
exchanger tubes in axial tlow is the principal engineering application.

1092
SLENDER STRUCTURES AND AXIAL FLOW
.'--...
z
Figure 9.36 A cluster of tubular cylinders simultaneously subjected to internal and bounded
external axial flow (PaIdoussis & Besanon 1981).
The equations of motion in the absence of interna1 flow are presented in Section 9.3.1.
A1though the externa1 fluid introduces intercy1inder coup1ing, this is not so for the interna!
flow; hence the effect of interna1 flow on each of the cy1inders is as if it were a10ne, that
is, as in Section 8.6.2. Therefore, adding the effect of interna1 flow to the equations of
Section 9.3,1 is a re1ative1y easy task, and the derivation is not given here; the interested
reader can find it in Paldoussis & Besanon (1981).
Consider a system of K c1ustered cylinders (Figure 9.36) subjected to an externa1 axia!
flow of ve10city Ue, with each cy1inder containing an interna1 flow of ve10city Ui. The
corresponding fluid densities are Pe and pj, and the cylinder cross-sectiona1 areas Ae
andAj.
The equations of motion may then be written in the following matrix form:
0 5 1/ 0 4 1/ 0 2 1/ 0 2 1/ 0 2 1/ 01/ 01/ 0 2 1/
A o40T +1 04 +C aOT +E o2 +F 02 +G o +H OT +M oT2 =0, (9.74a)
where
A = al, I == 2K x 2K unit matrix, C = 2 (fJ}/2 Uj I + fJ;/2 U e Ma) ,
E = u; Ma - {o[r + (1- 2v)(n e - nj)] + (1 - !o)[y - Ye + Yi
+ !e Cj(1 + h)u;] + (1 - 8)8 - unl,
(9.74b)

CLUSTERED CYLINDERS IN AXIAL FLOW
1093
G = 1£ Cf u; C a + [1£ Cf u; h + Y - Ye + Yi ]1,
F = [18Cj(1 + h)u; + Y - Ye + YdI,
H = [18 Cj /3;/2 U e + 18 C /3;/2]C a ,
{ I 2 2K ] T
M = /3e Ma + (1 - /3e)I, ." = 1], 1] "." 1]
in which Ma and C a are given by equations (9.21) and (9.44b), respective1y. The various
dimension1ess quantities are
 = xlL, 1]i = w i IL, 1]HK = vi IL,
for j = 1, 2, ' , , , K,
/3 - PeAe PiAi Pe A e L2 Pi A i L2
e - , /3i = , ne = EI ' ni = EI
m + PeAe + PiAi m + PeAe + PiAi
Ye = Pe A eg L3 I EI, Yi = pi A ig L3 I EI, Y = mgL 3 I EI, r = TL 2 I EI,
U e = {PeAeIEI}l/2 VeL, Ui = {PiA;jEI}l/2 ViL, 8 = LID,
h = DI Dh, ex = {EII(m + PeAe + piAi)}1/2 11,'; L 2, Cf = (4/7r)Cf'
C= (4/7r)(PeAel EI)1/2 CD L, e =DbL 2 I EI,
1: ={EI I(m + PeAe + piAi)}1/2 ti L 2, (9.75)
D being the externa1 diameter of the cylinder; j and j + K are superscripted to avoid
confusion with the subscript i here denoting 'internaľ.
As before, 8 = O corresponds to an axia11y unconstrained downstream end and 8 = 1 to
an axially fixed one. In the fonner case, there are distributed tension and pressure forces
a10ng the 1ength of the cy1inder - due to buoyancy and frictiona1 effects - as well as
a fonn (base) drag Db at the free end, which is fonnu1ated as in equation (8.133) and
depends on both Ui and U e ; Dh stands for the hydrau1ic diameter in a cross-sectiona1 plane
of the system. In the case of an abso1utely fixed end (8 = 1), there may exist an externa1
tension, T, as well as 'pressurization' effects due to the mean externa1 pressure, Pe' and
interna1 pressure, Pi' eva1uated at X = 1 L; V denotes the Poisson ratio.
The boundary conditions, for c1amped ends, say, are
." = a1Jla = O
at  = O and  = 1,
(9.76)
with equiva1ent expressions for other types of end support.
The methods of solution are the same as those with on1y external flow, presented in
Section 9,3.2,
The dynamica1 behaviour of a c1uster of three c1amped-c1amped cy1inders with increas-
ing flow is i1lustrated in Figure 9.37 for: (a) externa1 flow only (u e t= O, Ui = O); (b) both
flows (u e t= O, Ui = 6); (c) interna1 flow only (ue = O, Ui t= O). The parameters for these

t-
o-:
t-
"''''"
"1
'"
N
'-O
O) O)
"8"8
E E

 
0<= <=
t':]
<"I
"O


on
'-O
'"
O
o-:
00
]
t':
,.-
4-;
O
on
O)
"

L-"O">
<"I
'"
'-Ď
t-
'"
'"
'"
O
o-:
'"
M
O
o-:
-
M
'-Ď
","'"
..
.g
<= E
O">
00
O">
O
"1
<"I
'"
'-Ď
o
..

 O)
"0"0
<=I O
<"I E

:o
4-;
O
'"
O)
.E

O"> 00
00 00
0">0">
00
'"
....;
'"
N
"!
'"
"<
'"
N
"'"
N
'"
N
<"I
'"
'"
"'"
-i
'-O
t-8
t- '
- o'"
....;....;<
t-
t-
o
>!I"";
"'"
-i
u
c;.,
o
(ro)-«<p
'-O
"'.,-;
š O)
><"0
 O) o
 = E
-
'"
"<
'"
"'"
-i
N
N
I
N
'"
I
t-
o-:
N
'"
N
'"
N
"'"
o
N
'"
N
'"
N
"'"
N
M ,-.,


N
N
N
-
o
N
'"
N
'"
N
"'"
N

N 
N
o
';-
I
á

.. Q,)
'" (.) g
0..0
-
O Q) '"
... ..c ,-
bO.... Q)

0-5 '"
S Q) Q)
"'!'o-5
 Z ,.
 . g;
 13'
q:: '!a .E
"2>
O - Q)
(.) :: ...
Q) ,-
"'o.,g
"2 II .;:: ,....:.,
 ooo
.... :: bOO\
.S
<.1::(.)"0=
Q)ao
..<:!"O....<»
<::iQ)"'=
.....'é ... «I
O čij Q) '"
..c>"2
(.)  
«I::.....
Q) . O
 \O O).
O II '" 
'" «I 
... "- Q) O
]:::o
S --- "'" ';J
Q)6<ú
S -i:i:g '"
.... Q) "'.
""é ... óiI
 .,g
S! . 
Q)::=-
-5oo
'B 1I.<;::f.,g
... _ o.....
a :: .... =
___ Q) o
v.)'C\S::::= (1)
Q) "-' cd '1"""1
'<:)"0 -
= = o..
Q)«I -
g."2 Ě
Q) r- «I (.)
r-:"'«I
á3  ' ť
, Q):a
Q)B-5
 Š 1t: '<:)
- «I O O
S'  
00......-5
(.) "O -ťb
-5.s
..... a = 'Zl
o....'-"'
CI'} ti} cd c:
S Q) -€3 ,-
«I -5 =
5h-5 
«1.- 
:.a
"OS
aB..c
bO"''''
< .
... I
Q) ---
r-"Oa
!"'j,S1
='%N
f I Q)
=-5
,!2P 13 =
E-<o

,-.,
e,
,-..
$

CLUSTERED CYLINDERS IN AXIAL FLOW
1095
calculations constitute a so-called 'standard set' ,
1
G c = G w = -,
4
f3e = 0.32,
f3i = 0.26,
£ cf = 0.25,
v = 0.3,
8 = O,
ex = Ye = Yi = Y = r = ne = ni = e = C = O,
(9.77)
some of which willlater be varied.
There is nothing unusual about Figure 9.37(a). It is similar to Figure 9. 13(a), except
that the cylinders here are in a tighter cluster. The system loses stability by divergence at
point B where w = O, for Ue,cd  3.077. The system is restabilized (point C), and then
loses stability by single-mode flutter at point D.
The dynamics of the system with relatively high internal flow velocity superposed
(Ui = 6) is illustrated in Figure 9.37(b). Qualitatively, the dynamical behaviour is similar
to the case of Ui = O. The instabilities, however, occur at much lower values of U e . Thus,
Ue,cd  0.988 in this case, a three-fold reduction vis-a-vis Figure 9.37(a), The reason
for this reduction is that there is a synergistic effect between interna1 and external flows
(cf. Figure 8.57(b) and the discussion of Section 8,6.2). From Figure 9.37(c) it is seen that
Ui,cd  6.315. Hence, it is not surprising that, with Ui = 6, the necessary U e to precipitate
in stability is much reduced vis-a-vis Ui = O.
Figure 9.37(c) for U e = O and Ui variable is similar to that for a solitary cylinder
with internal flow at low f3 and no external fluid loading (cf. Figure 3.9). Indeed, the
critical flow velocity, Ui,cd  6.315 is only 0.5% different from 2n, t the critical flow
velocity for a clamped-clamped cylinder conveying fluid (Section 3.4.1). At first sight,
it may appear surprising that the confined external fluid has no effect on Ui,cd, though
it generally does on the eigenfrequencies: for example, the first-mode eigenfrequency at
Ui = O is Wl  16 here vis-a-vis Wl  22 in Figure 3.11. However, it is realized that
the hydrodynamic coupling is proportional to frequency and frequency-squared. Hence,
as divergence is approached, this effect becomes progressively weaker, and it vanishes at
the point of divergence - more precisely when Re(w) = O.
Df course, had the tubular cylinders been cantilevered, they would have lost stability
by flutter under the action of internal flow; in that case, confinement in dense fluid would
have resulted in severe destabilization of the system.
The effect on geometry on the critica1 flow velocities for divergence under the combined
action of internal and external flow is considered next. Figure 9.38 is a stability map
for clamped-clamped system with the standard parameters given in (9.77) but for three
different G c and G w , and for different numbers of cylinders (K) in the cluster.
It is seen that, irrespective of G c, G w and K, Ui,cd = 2n when U e = O, for the reason
discussed in the foregoing. For U e ::/= O, however, it is seen that the critical U e is diminished
considerably through the effect of internal flow. Not surprisingly, the effect of the number
of cylinders, K is rather weak; more important are the interstitial distances G c and G w .
Indeed, with increasing K but for given G c and G w, the critical stability curve tends
asymptotically to a limit, as the ratio of the area covered by the cylinders to that occupied
by the surrounding fluid becomes more uniformIy distributed in the cross-section.
tlndeed, this small difference is due to the numerics being more involved here than for a solitary cylinder
with internal flow.

1096
7
SLENDER STRUCTURES AND AXIAL FLOW
6
5
4
U e
3
2
1, 2, 3 cylinders
between two curves
4 cylinders
5 cy linders
Stable
o
2
3
4
U.
I
Unstable
1 cylinder
5 cylinders
5
6
7
Figure 9.38 Stability map for the threshold of divergence of c1usters of c1amped-clamped
cylinders simultaneously subjected to internal and bounded externa1 flow, showing the effect of
(i) G c and G w and (ii) the number of cylinders in the c1uster; system parameters as in (9,77);
(Pai'doussis & Besanon 1981).
3
2
U e
Pinned-pinned cylinders
Stable
o
2
U j
Unstable
3
4
6
5
7
Figure 9.39 Stability thresholds for a three-cylinder c1uster with G c = G w = 1/4, simul-
taneously subjected to internal and external flow, showing the effect of (i) boundary conditions
and (ii) departures (as noted in the figure) from the standard parameters (9.77); (Pai'doussis &
Besanon 1981).

CLUSTERED CYLINDERS IN AXIAL FLOW
1097
The effect of varying some ofthe other parameters on stability is shown in Figure 9.39.
It is seen that the effect is qualitative1y quite similar to what has already been observed
for a solitary cylinder - cf. Section 8.3.l(c).
9.4 FLOW-INDUCED VIBRA TIONS
Here we consider the sma1l-amplitude flow-induced vibrations occurring at flow ve1ocit-
ies smaller than the lowest critical value for instability, otherwise known as 'subcritical
vibrations' . The readeris referred to Section 8.4, where the subject is discussed extensive1y,
mainly for a single flexible cylinder in flow. Neverthe1ess, in the experiment s discussed
therein, c1ustered cylinders were often involved, though they were treated as if each cyl-
inder were alone. Broadly speaking, the treatment in Section 8.4 may take into account
confinement by adjacent cylinders or other structures, but not the fact that the motions are
coupled. Intercylinder coupling is fully taken into account here.
The excitation field, here as in Section 8.4, is assumed to be mainly due to flow turbul-
ence. Moreover, the amplitudes of vibration are sufficiently smal1, Yrrns/ D '" 0(10- 3 ),
for the vibration to be presumed not to affect the excitation fie1d significantly. Thus, the
vibration is treated as due to extraneously induced excitation in Naudascher & Rockwell's
c1assification (Section 1.2).
Work in this area, taking coupling into account, has been done by Paldoussis & Gagnon
(1984), Paldoussis & Curling (1985), Curling & Paldoussis (1992a,b, 2003), Gagnon &
Paldoussis (1994a,b). The discussion that follows is based on that work.
9.4.1 General vibration characteristics
9.4.1 (a) The experimenta/ set-up
Experiments were conducted as in Section 9.3.4, with essentially the same apparatus
but with the cylinders now always instrumented. The vibration was so small as to be
invisible to the unaided eye. The instrumented cylinders contained, embedded in them,
a cruciform silicone rubber 'implanť, possessing two pairs of diametrically opposite flat
surfaces, on which were glued strain gauges (Gagnon & Paldoussis 1 994b, figure 4);
trus implant was carefully positioned in the mould, and the rest of the cylinder was cast
around it (Figure 9.40) - refer to Appendix D, ln the finished cylinder, the strain gauges
were close to the surface for maximum sensitivity, and they lay in a longitudinal position
yie1ding nearly equal outputs for flexural deflections in the first, second and third modes
of a clamped-clamped beam. The strain gauges were calibrated statical1y by imposing
lateral deflections on the cylinders, of known beam-mode shape.
Most experiment s were conducted with clusters of four flexible cylinders, in a square
pattern, centrally located in the test-section [Figure 9.4l(a)]. Different intercylinder
spacings were tested, characterized by the parameter
G c = (minimum inter-cylinder gap)/ R,
as in equation (9,1), where R is the cylinder radius. Axial sliding was prevented. Some
experiments were also conducted with mixed clusters of flexible and rigid cylinders,

1098
SLENDER STRUCTURES AND AXIAL FLOW
Figure 9.40 Photograph of the silicone rubber insert for mounting the strain gauges in the flexible
cylinder of the same material which is cast around it; the cylindrical mould is shown open, for
viewing, by removing the other half (Gagnon & Pai:doussis 1994b),
(a)
Figure 9.41 Diagrams of the upper support structure, showing the geometry of the system in the
containing channel (G c = 0,75): (a) the cruciform supportforthefour-cylinder cluster; (b) the upper
support plate for the 28-cylinder cluster. The cross-section of the cruciform fins and of the intercyl-
inder ligaments in the upper support plate were suitably streamlined (Gagnon & Pa'idoussis 1994b),
a 19-cylinder cluster of 'polar' geometry, and a 28-cylinder cluster of square geometry
[Figure 9.41(b)].
9.4. 1 (b) Vibration characteristics
Figure 9.42 shows typical power spectral densities (PSDs) of vibration of one cylinder
in the four-cylinder cluster, in the direction of the solid-line vector in (a). PSDs in the
dashed-line direction and for other cylinders are similar. It is recalled that two of the

CLUSTERED CYLINDERS IN AXIAL FLOW 1099
0,2
u = 1.0

08- 0,1 i th,b,:
00 
(a)
O
(b) O 2 4 6
1.5 16
u=2.0 u=3,O
8

'O
-
x
N O
:r: 2 4 6 O 2 4 6
N (d)
 180
Q u=4.0 u=5,O
'"
 120
th,b,
40
60
O O
O 2 4 6 (f) O 2 4 6
(e) Frequency (Hz) Frequency (Hz)
Figure 9.42 Power-spectral-densities of vibration of one of the clamped-clamped cylinders in
the four-cylinder cluster of Figure 9.41: (a) diagram showing direction in which vibration was
measured, indicated by the solid-line vector; (b)-(f) experimental PSDs and theoretical frequency
bands (th.b.) ofthe first flexural-mode group of eigenfrequencies, for various flow velocities. System
parameters: G c = 0,75, L = 460 mm, D = 25,3 mm, E = 2.65 X 10 6 N/m 2 , m = 0.58 kg/m,
E* = 2,3 x 10 3 kg/m s (Pai'doussis & Gagnon 1984),
X 10- 2
6
X 10- 4
X 10- 6
9

N
:r:
"4
ť
Q
g; 2
5
00
Figure 9.43 Measured PSDs and theoretical eigenfrequency bands (th,b.) for the lowest three
flexural-mode groups of the four-cylinder system of Figure 9.41(a), for u = 4.5 (Pai:doussis &
Gagnon 1984).

1100
SLENDER STRUCTURES AND AXIAL FLOW
system eigenfrequencies are repeated once, so it might be expected that each PSD should
display six discrete, identifiable peaks - which is c1early not the case, Partial reasons
for this are that the eigenfrequencies are c10sely spaced and the system is damped; also,
perhaps, they are unequally excited. However, it is possible to see individual peaks if the
vibration is measured in the characteristic or modal directions of the system, as discussed
later t - see Figure 9.45.
It is noted that the spectra of the observed vibration in the PSDs in Figure 9.42 agree
remarkably well with the predicted theoretical band (th.b.) of the first tlexural-mode group
of eigenfrequencies, Moreover, equa11y remarkably, the theoretically predicted shift of
this band towards lower frequencies and increase in the vibration bandwidth, as the tlow
velocity is increased, are also c1early demonstrated in the experimental results.
Furthermore, although the dominant tlow-induced vibration was evidently in the fust
flexural-mode group of eigenfrequencies, it was nonetheless possible to detect vibration,
of much smaller amplitudes, in the second and third flexural-mode groups of eigenfre-
quencies, as shown in Figure 9.43 - where attention is drawn to the different ordinate
scales for the three mode groups. Once more, it is remarkable that the observOO envelopes
of vibration 1ie within the corresponding theoretical bands.
At this point, it is instructive to recall the theoretical cross-sectional moda1 pattems
for this system and geometry, as shown in Figure 9.44. The pattems for the repeated
eigenfrequencies are generally orbital as in Figure 9.44(a); but, if a sma11 asymmetry is
artificially introduced, then they become planar as in Figure 9.44(b).* It is important to
notice that these cross-sectional modal pattems define two characteristic modal planes
or directions: one radial, the other tangentia1. Thus, modes 1, 5 and 6 (predominantly)
and 7 involve radial motions in the sense defined in Figure 9 .44( c), in the R 1, R2 etc.
directions; similarly, modes 2, 3 and 4 (predominantly) and 8 involve tangential motions,
that is, motions along TI, T2 etc.
Figure 9.45(a) and (b) shows, for two different flow velocities, the experimental vibra-
tion PSDs in the radial and tangential planes, defined, respectively, by the so1id- and
broken-1ine vectors in the inset diagram - together with the theoretical characteristic
frequencies (labelled 1-8) of the first-mode group. Once more, it may be notOO that (i) the
tlow-induced vibration bandwidth and the theoretical band of eigenfrequencies a1most
coincide and (ii) for the higher flow velocity, the vibration bandwidth has become wider
and has shifted downwards.
Furthermore, by comparing the experimental vibration peaks in the PSDs of
Figure 9.45(a) and (b) with the theoretical cross-sectional modal pattems of Figure 9.44,
it may be seen that (i) most of the power in the radial-plane PSDs is in the vicinity of
the eigenfrequencies Ji == 'h/21l' = [EI/(m + pA)L 4 ]I/2(WI/21l') and 15 = 16 (and a
small peak may be discemed at h), II all of which are associated with mode s involving,
tFor further discussion, refer to Curling (1982, pp. 294-303).
+Later calculations (witb slightly different parameters or asymmetry) yield /s and /6 pattems witb tbe
smaller vectors reversed (Gagnon & PaYdoussis 1994a).
The qualification of 'predominantly' indicates tbat, although the diagonal pair with tbe larger vectors
involves radial-direction motions in modes 5 and 6 [Figure 9.44(b)], the other, smaller amplitude pair involves
tangential direction motions. Moreover, a given cylinder may be associated witb tbe larger vector in Ws and tbe
smaller one in W(;. Presumably, tbe modal pattem would switch between tbose of Ws and W6 in Figure 9.44(b)
continually, and hence tbe orbital pattems probably are the most realistic representation of actual motion,
111ft is measured in seconds and , is the dimensionless time, tben, from w, = Qt (botb in radians), w is
tbe dimensionless radian frequency (in radians), Q is in rad/s and / = Q/21f in Hz,

CLUSTERED CYLINDERS IN AXIAL FLOW
1101
,
,
\
\
\
\
\
"- 0 '" \ :"/',
" "i.: " I /.' :"
i< j N0
........- .../
-- -----
----
Wl
7 .. ..,
, ".
. :.
..../;?"
" /',.:
(a)
W3
/ ..,
. :,',
" /
(b)
W3
(c)
R''tl'
T2:y\0
R3 T3
W2
,. x",
:,.,'" :""
, .. ."
: '; -:'0"';
'. . . . . . . .
W4 =W3
:" '.
. . ", '"
 ,"
W4""W3
:s. ..' 0 '"
. '. .
. " .
. ,
:/'('"
 .' :, . 
W7
.... ,"
;,':
P:'
Ws
", ,"
':..\ ','.,
..  ,.
, ,
:. ". " .:
Ws
0 "'  '.,
. ., .
. .. .
. " .
 "' 0 '"
. '. .
, , '
. .. .
. ,
Ws
 "" i:? '"
. .. .
, ,
, ,
/7',
7.",
W6 = Ws
..' / '
:"':' '
/ , ''/:
. , .
W6 "" Ws
Figure 9.44 Theoretical cross-sectional modal pattems (a) for a symmetric system and (b) for an
artificially slightly asymmetric system (G c = 0,75, G w = 5.3). (c) Definition of the characteristic
planes/directions: 'radia!' (Rl, R2, etc.), 'tangential' (Tj, T2, etc,); (Pai'doussis & Gagnon 1984;
Gagnon & Pai'doussis 1994b).
purely or predominantly, radial cross-sectional modal pattems; (ii) most ofthe power in the
tangential-plane PSDs is in the neighbourhood ofthe eigenfrequencies 12, h = J4 and fs,
which are associated with tangential cross-sectional modal pattems; (iii) some individual
peaks in the tlow-induced vibration PSDs correspond very dosely to the theoretical free-
vibration eigenfrequencies: for example, at JI, J5 = J6 and fs; indeed, in Figure 9.45(b)
one could identify each of the theoretical eigenfrequencies of the system with a prominent
peak in the corresponding PSDs.
Hence, it is seen that not only is there doser spectral correspondence between experi-
ment and theory if vibration is measured along the characteristic planes of the system, but
that there is also modal correspondence in the following sense: tlow-induced vibration
occurs randomly in al1 directions, but predominantly in the characteristic directions of the
cross-sectional modal pattems predicted for free vibration of the system at the frequencies
associated with these pattems.

1102 SLENDER STRUCTURES AND AXIAL FLOW
2 20
aL u=20 CD QXID u=3.0
,..., CD24678
M
::: 0'0  f!
X " ," 
f I i.11
N ,'\, I y'
;; 00 I t'l I :
, " 10
' " l 
'"'   >{ ,
s , ! \
, .' \
,
 1
CI)

0 0 o
2 3 4 5 6 O 2 3 4 5 6
(a) Frequency (Hz) (b) Frequency (Hz)
10 x 10- 3 "
I I
N ' , ))0
' ,
;; I ,
/ \
"' 5 / \ '00
C- I ,
o ,
 OV ,
o ,
CI)  \
U
5 6
(c)
270
 180

Q)
 90


.d O

-90
O 2 3 4 5 6
(d) Frequency (Hz)
Figure 9.45 Measured PSDs for the four-cylinder system of Figure 9.41(a): (a) for u = 2,0 and
(b) for u = 3,0; -, in the radia1 direction; - - -, in the tangentia1 direction (see Figure 9.44(c».
Comparison of the PSD peaks with the theoretica1 eigenfrequencies, shown as numbered cursors;
for examp1e, circled 2 corresponds to W2 in Figure 9.44. (c,d) The corresponding cross-spectral-
density magnitude and phase measurements, for the two cylinders indicated [in inset to (c»), at
u = 3.0 (Paldoussis & Gagnon 1984).
This last observation is further supported by cross-spectral density (CSD) measure-
ments of the vibration of two adjacent cylinders in the cluster, as shown in Figure 9.45(c)
and (d), for CSD magnitude and phase, respectively. It is seen that the magnitude of
the CSD between the two radia 1 planes (solid line) has prominent peaks clo se to fl and
h, both of which correspond to radia1-pattem modes; similarly, the CSD magnitude
between the two tangentia1 planes has a merged peak in the vicinity of the h and !3 = f4
frequencies and another coincident with Is, corresponding to tangential pat tem mode s
[cf. Figure 9.44(a) and (b)].
The phase infonnation in Figure 9.45(d) provides new insights. Considering the radial-
planes CSD phase (solid line) first, it is seen that at f1 and h the phase is sensibly

CLUSTERED CYLINDERS IN AXIAL FLOW
1103
0 0 and 180 0 , respectively; by comparing with Figure 9.44(a) and (b), it is seen that this is
exactly the theoretical prediction. Similarly, the tangential-planes CSD phase (broken line)
is 180 0 at h and 0 0 at /8, again in agreement with the theoretical pattems of Figure 9.44.
The correspondence in phase in the case of the repeated characteristic frequencies is les s
straightforward, For example, looking at the modal pattems of /5 and /6 in Figure 9 .44(b)
together and considering the larger (dominant, radial-direction) vectors only, it is seen
that the 'top' pair of cylinders vibrate mainly radially and 180 0 out of phase, as in Fig-
ure 9.45( d), but the left or right pair vibrate radially in phase. This appears to contradict the
symmetry of the system. The same ambivalence is obtained for the 13, /4 pair, for tangen-
tial vibration, In fact, upon examining Figure 9.44 carefully, one might at first conclude
that the phase difference for adjacent cylinders at a repeated frequency is indeterminate.
However, the conundrum is resolved if one considers that, as is likely, the cylinders do not
oscillate in both pattems of the repeated frequency together, that is, simultaneously, but
now in one and then in the other (Curling 1982,2001), Thus, as indicated by the long-time
average CSD measurements in Figure 9.45(d), the 'top' pair of cylinders, and indeed by
symmetry any other adjacent pair of cylinders, in the long-time average vibrate roughly
180 0 out of phase, while instantaneously they may vibrate in either the /5 ar /6 pattem, or
permutations thereof, and similarly in either the 13 ar /4 pattem or their permutations; t
in a purely random, unbiased situation, each permutation at a given frequency is equally
likely to occur, Similarly, intercy1inder CSD measurements of a diagonal pair of cylinders
(see computed results in Figure 9.46) indicate that a diagonal pair of cylinders also vibrate
180 0 out of phase in the long-time average, while they must also conform to the modal
pattems already described and possible permutations thereof.
The paradox, more fully but succinctly, is this. If one computes or measures CSDs
between adjacent and also diagonal pairs in radial or tangential directions at a repeated
frequency [Figures 9.45(c) and (d) and 9.46], one finds that on the average: (i) cylinders
1 and 2 [see Figure 9.44(c)] vibrate outof phase; (ii) cylinders 2 and 3 vibrate outof phase;
(iii) but the diagonal pair of cylinders 1 and 3 do not, as could logically be deduced from
symmetry, vibrate in phase! This then, if one presumes concurrent /5 and /6 vibrations (or
concurrent!3 and /4 vibrations), is a contradiction, in view ofFigure 9.44(b). If, however,
one considers vibration in either the /5 ar /6 pattem (and similarly either the 13 ar /4
pattem) [see Figure 9.44(a)], that is, not in both simultaneously (but over different spans
of time), then (a) the radial U5 = /6) or tangential (13 = /4) vibration of adjacent pairs
is out of phase on the average, (b) that of diagonal pairs is also out of phase on the average,
full stop; exactly as obtained in Gagnon & Pai'doussis and as shown in Figure 9.46. This
a posteriori shows that the supposition that the two intercylinder pattems do not, at any
instant, occur simultaneously must be sound.
Final1y, the measured/calculated intercylinder CSDs give us pair-by-pair performances
'on the average', but they do not describe the overall performance of the cylinders as a
group. By examining all pairs of cylinders, we can then deduce the action of the entire
group in the long-time average. The complete picture requires also an analysis of mixed
radial and tangential directions at the repeated frequencies [Gagnon & Pai'doussis (1994a,
figure lla,b); Curling (1982, pp. 119, 194,324-330,615-621)]. In the latter reference,
the reader can find a very pertinent analysis of modal vibration pattems related to repeated
frequencies in a three-cylinder system.
tMeaning the actual pattems ofFigure 9,44(a). not those ofthe slightly asymmetric system in Figure 9.44(b).

1104
SLENDER STRUCTURES AND AXIAL FLOW
1.0
0,8
1:
... 0.6
g

... 0.4
-=
8
0,2
0,0
(a)
270
180 ..........
'""' 90
OD
! O
...
'"
'"
-= -90

-180
-270
(b)
1.0
0,8
'""'
J.,
... 0,6
j
0.4
8
0,2
.........
0,0
(c)
270
180
'bD 90
! O
...
.a -90

-180
-270
(d) O 20
...........
40
Frequency (-)
60
80
Figure 9.46 (a) Computed coherence and (b) CSD phase ofthe radial (-) and tangential (.., ,)
vibration components of a pair oj diametrically op po sed cylinders in the four-cylinder c1uster, for
u = 3, G c = 0.75. (c,d) The same for apair oj adjacent cylinders (Gagnon & PaYdoussis 1 994a) ,

CLUSTERED CYLlNDERS IN AXIAL FLOW
1105
A more quantitative comparison between experiment and theory is undertaken after this
latter is presented.
9.4.2 The basic analytical model
The basic model is developed in Pa'idoussis & Curling (1985), and the presentation here
is based thereon. Consider a general c1uster of cylinders as in Figure 9.9. The equations
of small motions may be written as
.c [1J(, T)] = f(, T),
(9.78)
where.c is a system of K linear space-time differential operators, directly obtainable from
equation (9,62), involving the system parameters and accounting for all flow-induced
forces associated with the mean flow field; 1J(, T) is the 2K -vector of dimensionless
cylinder displacements, in terms of the dimensionless axial coordinate, , and time, T,
defined in equation (9.61); f(, T) represents the 2K -force-per-unit-length-vector associ-
ated with the perturbation flow field - that is, it represents the instantaneous resultants of
surface pressure fluctuations in the lateral Zj- and Yj-directions (Figure 9.47), Tbe system
is discretized via Galerkin's method, through the expansion
N
1J(, T) = L <PI()ql(T),
1=1
(9.79)
where <PI () are appropriate comparison functions, q/ (T) are the corresponding generalized
coordinate vectors, and N is the number of comparison functions, sufficient to adequately
approximate the response 1J(, T). Equation (9.78) may then be written in the form
Mij + Cci + Kq = F,
(9.80)
Ys
Pr (, 8, ,)
Ps (', 8', ,)
Zs
Zr
Figure 9.47 The random pressure fields on two neighbouring cylinders and the Cartesian coordin-
ates along which the resultant forces on the cylinders are resolved (Pa'idoussis & Curling 1985).

1106
SLENDER STRUCTURES AND AXIAL FLOW
where M, C and K are, respectively, the 2K N x 2K N mass, damping and stiffness
matrices, the elements of which are constants for a given system, and
q = {ql, Q2, ..., QN}T, F = 1 1 {<Plf, <P2f, .,., <PNf}T d, (9.81)
The forces (per unit length) on the right-hand side of equation (9.78) are assumed to
be random and ergodic, and hence so is the response. By definition then, the spatia1
cross-correlation of the response is
{ I 1 T / 2 }
RT/iT/j(,f,'f)= lim - l}i(,!')11j(/,!'+'f)d!',
T-+oo T -T/2
i, j = 1, 2, . , . , 2K,
(9.82)
where 'f is an arbitrary delay in!'. With the aid of equation (9.79), this may be written as
N N
R TJiTJj (, /, 'f) = L L <Pz() <Pn(f) Rq/,iqn,j ('f),
Z=1 n=1
(9.83)
whereqZ,i is the ithelementofsubvectorqz(!') inequation (9.79) or(9.81); cf. Appendix R.
Now, in tenns of the cross-spectra1 density of q, Sql,iqn.j (w), one may write
I 1 00 iú.>r
R q / ' q .('f)=- S q/ ' q .(w)e dw,
./ n.J 2n -00 ,/ n,J
(9,84)
where
Sq[,iqn,/w) = }imoo {[QZ,i(-W, T)Qn,j(W, T)]},
in which Q is the finite Fourier transfonn of q, that is,
(9.85)
1 T / 2
QZ,i(W, T) = qZ,i(!') e-iú.>T d..
-T/2
The matrix of the Ql,i may be determined by the finite Fourier transfonn of equation (9.80),
retaining only the steady-state-producing tenns, that is,
(9.86)
Q(w, T) = H(w)(w, T),
(9.87)
where the matrix of the frequency response functions is
H(w) = [-w 2 M+iwC+Krl
and  is the finite Fourier transfonn of F. More specifica1ly, the (I, i)th component of Q
is
(9.88)
2KN
QZ,i(W, T) = L Hp(I,i),p(W)';Jp(w, T),
p=1
(9,89)

CLUSTERED CYLINDERS IN AXIAL FLOW
1107
where, in view ofthe structure of equations (9.81),
{3(1, i) = 2K(1 - 1) + i, p(w, T) = 1 1 <Pa(p)()Fb(p)(, w, T )d,
(9,90)
in which F is the finite Fourier transform of f, that is, the components of f(, i) in
equation (9.78) or (9.81), and
a(p) = {largest integer .:S (2K + p - 1)/2K}, b(p) = p - 2K[a(p) - 1]. (9.91)
The subscript ranges are: {3 = 1, 2, . . . , 2K N, a = 1, 2, . . . , N, b = 1, 2, ' , . , 2K.
The complicated indicial form of these equations and the index ranges arise because of
the fact that the vectors 1/ and f [equation (9,78)] are associated with K different cylinders
and two different sets of directions, Zj and Yj; moreover, q and F [equations (9.79)
and (9.81)] involve components associated with the N different terms of the Galerkin
expansion.
Upon combining equations (9.83)-(9.85), (9.89) and (9.90), the final form of the
response cross-correlation may be found, as follows:
RT/ir/j(' ',"f) =  f oo ST/iT//' ', w)e iWT dw,
2n -00
(9,92)
where the spatial-response cross-spectral density is given byt
N N 2KN2KN
ST/iT//' ', w) = L L <PI() <Pn(') L L {H;(l'í)'P(W) Hf3(n,j),q(w)
1=1 n=1 p=1 q=1
X 1 1 d111 d2<Pa(p)(r)<pa(q)(2)Sfb(p)/h(q)(1'2'W)},
i, j = 1,2, ..., 2K, (9.93)
Sfh(pJfb(q/1, 2, w) = Tlmoo {[Fb(P)(1' -w, T )Fb(q)(2, w, T)]},
(9.94)
The excitation in the above equation, Sfh(p)/b(q) (1, 2, w), is the set of cross-spectra1
densities of the forces per unit length fb(p) (1, i) and fb(q) (2, i) acting on the cylinders,
where b(p), b(q) = 1,2, ..., 2K; when they are known, the response, in terms of ST/iT/j
and RT/iT/j ' maybe calculated. Fina11y, by taking  = ', i = j and"f = O, the mean square
response 1Jf(, i) may be determined, and hence (1Jí)nns' Multiplication by 8 2 [or 8 for
(1Jí)nns] gives the response with respect to the cylinder diameter, D.+
A more genera1 analytica1 model may be found in Gagnon & Pai'doussis (1994a)
where, for instance, mixed bundles of flexible/rigid cylinders are treated, as are some
of the particularities of the experimental system to which the theoretical results are later
compared.
tNote that H*(w) = H( -w); thus these are used interchangeably.
+The response here is computed in Cartesian directions, For the equations needed to transform the results
to arbitrary polar directions, see Curling (1982, pp, 29292),

1108
SLENDER STRUCTURES AND AXIAL FLOW
The excitation is assumed to comprise the random wall-pressure ftuctuations in the
generally turbulent ftow. Hence, by simple resolution of the pressure field into Cartesian
force components (see Figure 9.47), one may easily obtain
Sfi/j(, f, w) = { L3 } 2 R2 {2n dO (27r dO'Spr(i)Ps()(' f, O, O', w)Trig(i, O) Trig(j, 8'),
E I 10 10 J
(9.95)
where L 3 / E I is the factor rendering dimensionless the forces per unit length in
equation (9.78), R is the radius of the cylinders,
SprPs(' f, O, O', w) = lim .!.. [Pr(, O, -w, T)Ps(', O', w, T)],
T-+oo T
(9.96)
Pr being the finite Fourier transform of the pressure Pr, and
[ r C i ) S ( i ) Tri ( i O )] = ! [i, i, cos O]
, , g, [ . K ' K . O]
I - , I - , SIn
when i = 1,2,..., K,
when i = K + 1, K + 2, ' . . , 2K.
(9,97)
To date, no fully analytical characterization for the wall-pressure cross-spectral dens-
ities, Spr p\' is available for arbitrary clusters of cylinders in axial ftow. Moreover,
measurements of the same tend to be difficult. Most of the available data in the open
literature pertain to solitary cylinders centrally located in a channel, though some do
exist for clustered cylinders also [e.g. Ohlmer (1973), Mulcahy et a/. (1982)] - see
Sections 8.4.1-8.4.3; however, these measurements pertain to wal1-pressure correlations
on a single cylinder in the cluster and are therefore insufficient for the purposes of the
present analytical model, where intercylinder correlations are also required. A more ambi-
tious set of measurements for clusters, but for only one geometry and a limited range of
Re, was made by Curling & Pai'doussis (1992a,b) - see Section 9.4.3.
9.4.3 The excitation field and the response
9.4.3(a) An approximate, yet successful model
This model (Curling 1982; Pai'doussis & Curling 1985) predates the last set of measure-
ments made in the 1990s; yet, though requiring some empirical input which is not easily
available for yet unbuilt systems, it remains as successful as some of the later, more refined
and elaborate models. Anyway, something needs to be said about this model, because it
is the one used for obtaining the extensive comparisons with experiment undertaken by
Gagnon & Pa'idoussis (1994a,b), presented in Section 9.4.4,
The turbulence-induced ftuctuating pressure field is assumed to be homogeneous.
Following Bakewell et a!, (1962) and Bakewell (1964, 1968), one may write for the
real component of (9.96)
Re {Sprpr (Xl, X2, 01, (h, Q)}
= [e- bIS " I cos(27l' S;)] { (1 + C SJ) [2 - exp( -dSJ)] r l Spp(Q),
(9.98)

CLUSTERED CYLINDERS IN AXIAL FLOW
1109
where Spp(Q) is a homogeneous wall-pressure PSD, and
1.fP !:!.() w
2 € (27T)U e '
(9.99)
where Ve is the average velocity of convection of wa1l-pressure disturbances from
Xl to X2,t U e = (ul V)V e being its dimensionless counterpart [see equations (9.61)],
and (b, c, d) = (0.7, 10,80), for two points on the same cylinder, X2 - Xl, longitudinally
apart and R!:!.B (i,e. at the shortest lateral distance) circumferentially+ apart. It is further
assumed that the same fonnulation holds for points on different cylinders in the cluster
via the assumption of the shortest lateral fluid distance, Le. if only R!:!.() is replaced by
Lmin «()rl, ()s2), the shortest lateral distance through the fluid between point 1 on cylinder r
and point 2 on cylinder s. Hence, furtherdefining!:!. = X2 -Xl and R!:!.() = R«()2 -()1) or
the appropriate shortest lateral fluid distance, and determining the imaginary component
of (9.96) (Curling 1982; Curling & Pai:doussis 2003), we can write
(X2 - xd(Q/27T) .fP(2 - l)W R!:!'()(Q/27T)
S = = , So =
Ve (27T)U e Ve
[exp( -bIS I)][exp( -i27T S)]
S (!:!.é!:!.() Q) = S (Q)
PrPs 5" [1+cS][2-exp(-dSo)2] pp ,
(9.100)
in which Spp(Q) == SprP,(O, O, Q), the homogeneous wall-pressure PSD.
Reverting now to a dimension1ess fonnulation, equation (9,95) may be written as
( 3 ) 2
/ RL -blS I -i21TS
Sf;Jj('  , w) = EI e  e Spp (w)Kij,o (w),
(9.101)
in which
Spp(w) = [ (m+ E ;A)L4 f/2 Spp(Q)
and a new function has been defined,
(9.102)
K.. o = {21T dB1 {21T dB2 Trig(i, ()d Trig(j, ()2)
IJ, 10 10 (1 + c S)[2 - exp( -dS)]'
i, j = 1, 2, ' . , , 2K.
(9.103)
The integrations in (9.103) have been carried out numerically and curve-fitted (Curling
1982), and expressions for them are given in Appendix X.1 for points on the same cylinder
or a cylinder pair.
tFor frequencies greater than 1000Hz, Bakewell et al. (1962) found that U e decreases from approximately
0,7 Ut, to approximately 0,64 UL where Ut,is the pipe centreline velocity, which is equivalent to "-' 1.20U,
and U is the mean flow velocity in the pipe (Schlichting 1968), At a lower frequency (,,-, 300 Hz), they found
U e  O,96Ut, 1.16U,
*The shortest lateral distance R 110 on the same cylinder equals RIOz - 011 if O  102 - 011 < 1T and
R[21T - !02 - Ol!] if 1T  !Oz - Ol! < 2:rr,
 In an alternative nondimensionalization, the factor (R L 3 / E I) is replaced by (2£ /:rr), but then the factor
multiplying Spp(rl.) becomes (32J1f £2)/[E 3 (m+ pA)]1/2 (Gagnon & Pa'idoussis 1994a); this follows through
to equation (9.104).

1110
SLENDER STRUCTURES AND AXIAL FLOW
Fina11y, substituting equation (9.101) into (9.93), we obtain the response cross-spectra1
density:
( RL3 ) 2 N N
ST/iT/j(' (, w) = EI Spp(w) L L (I>t() Q>n()
/=1 n=l
2K N 2K N
X L L H;(l,i)p(w)HfJ(n,j)q(w)Ka(p)a(q),(w)Kb(p)b(q),()(w),
p=l q=l
(9.104)
where
K/n,(w) = 11 dl 11 d2 Q>/(1) Q>n(2)e-blS1 e-i21TS,
/, n = 1, 2, , , , , N,
(9.105)
expressions for which are given in Appendix X.2. Now, everything is known, except for the
power-spectra1-density ofthe fluctuating pressure, S pp (w) = []S pp (Q), equation (9.102).
Most of the avai1ab1e data of the one-sided differentia1 t wall-pressure PSDs, G pp (Q),
nondimensiona1ized as G pp (Q)/(p2 U 3 Dh), is p10tted versus f Dh/ U in Figure 9.48(a),
where Dh is the hydrau1ic diameter and f = Q/(21(). The important thing to note here is
that the data cover a band of more than three orders of magnitude. Since the response is
linear1y proportiona1 to the excitation, the band in possible response amplitudes could be
different by a factor of 1000: 1 in PSD, or'" 30: 1 in r,m.s. which is c1ear1y unsatisfactory.*
The reasons for the ordering of the data (in terms of the fiat-region magnitudes at low
f Dh/ U) remain obscure. Note that (i) the pipe-fiow data should have been the quietest
(lowest), for examp1e those of Bakewell et a/" since enormous care was taken in every
way to ensure remova1 of acoustic components (sand-boxes, gravity feed, etc.), yet they
1ie somewhere in the middle of the who1e set; but remember that these were originally
point measurements, so some acoustic component may have remained; (ii) the Mulcahy
et a/, and Lin et a/. data a1though they are, respectively, without and with spoi1ers are
sensib1y in a band, while the Curling and Ohbner et a/. data (definitely without spoi1ers!)
are substantial1y above them, This suggests that there are systemic, non-turbulence-related
differences between the data sets.
Severa1 attempts have been made to reduce the spread in G pp (Q), main1y in tenns of
the Reynolds number, Re = U Dh / v, and D / Dh. Perhaps the most successful result is
shown in Figure 9.48(b), in which
G pp (Q) [Re/(p2 U 3 Dh c 2 ) ] == G pp (Q)/(f,L P U 2 c 2 ) (9.106)
tOne-sided PSDs are those defined for O  Q  00 ooly and are denoted by Gpp(Q), so that
G pp(Q) = 2 Spp(Q). The overbars on p, here G pp (Q), denote dijferential pressure measurements, obtained
from diametrically opposite points on a cylinder cross-section, These are preferred to single-point measure-
ments, to eliminate plane-wave acoustic components. Where only single point measurements are available, they
have been converted via G pp (Q) ::::: 2 G pp(Q); (Curling & Paldoussis 1992a).
*This is similar to wa1l-pressure turbulence data in cross-flow over cylinder clusters (Blevins 1990,
figures 7-11),
*The range of Re in the data shown is quite large: 14 x 10 3 -21 x 10 3 for Curling's (1990) data,
18 x 10 3 -92 x 10 3 for Gagnon's (1994), 63 x 10 3 -285 x 10 3 for Ohlmer et al. (1972) and Ohlmer (1973),
170 x 10 3 for Bakewell et al, (1962), 116 x 10 3 --483 x 10 3 for Mu!cahy et a/, (1981) and Lin et al. (1981),
and 1120 x 10 3 for Clinch's (1969) data,

dl 10- 6
(a)
I "'
'<>''''
'<>. ::::,
'" q"
::t
(b)
CLUSTERED CYLINDERS IN AXIAL FLOW
1111
10- 3
10- 4
10- 5
10- 7
10- 8
10- 9
1000
100
10
0.1
0,01
0,001
0,001
0,01
0.1
10
100
fDh/U
Figure 9.48 (a) The mean dimensionless differential wall-pressure PSDs from various sources
(Paldoussis & Curling 1993); (b) renormalized version of PSDs (Curling & Paldoussis 2003).
Source of data: +, Ohlmer et al, (1972) and Ohlmer (1973); ., Clinch (1969); L, Curling (1990)
and Curling & Pai:doussis (l992a,b); x, Lin et al. (1981) and Mu1cahy et al. (1981); *, Bakewell
et al. (1962); ., Gagnon (1994).

1112
SLENDER STRUCTURES AND AXIAL FLOW
is plotted versus f Dhl U, where c 2 = [1 + (DI Dh) l.2Sr1.8 (Curling & Paidoussis
1992a,b, 2003). Some improvement vis-a-vis Figure 9.48(a) is noted, but not too much,
However, the main virtue of this new scheme is that it succeeds in collapsing data
from the same experimental facility (rig) into distinct bands, for example, the Gagnon
data, The failure to collapse data from different rigs into a single thin band il1ustrates
further that undetermined features of each rig (and accompanying measurement system)
affect the results. What the nature of these systemic differences is remains unknown.
Likely, they are the effect of severa1 parameters: detailed geometry, intercylinder gaps
(array 'density' and density distribution, as distinct from the globa1 measure of Dh),
turbulence scale, roughness of the cylinders, extraboundary-layer content due to entrance
conditions, etc,
9.4.3(b) More elaborate models and practical considerations
An extensive set of measurements of the pressure field in a square array of four cylinders,
surrounded by eight half-cylinders and thus 'fi1ling' the cross-section of the flow channel
(G c = 0,17), has been made by Curling (1990) - see also Curling & PaYdoussis (1992a,b).
PSDs and CSDs of the differential pressure fluctuations, as well as new 10ngitudina1 and
circumferentia1 pressure correlations were measured, on the same cylinder and on adjacent
cylinders, and their dependence and characterization in terms of Reynolds and Strouhal
numbers, and a1so on the effect of orientation (i.e. for locations facing the narrows of
intercylinder gaps or facing less confined space, near or far from the wa1l of the flow
channel), These have subsequently been used to construct two variants of a new ana1ytica1
model (Curling & Paidoussis 2003), one of which requires only one empirical input:
. a severity factor' , indicating the degree of quietness or otherwise of the flow rig, t
These models are not given here for the following reasons: (i) they have been insuffi-
ciently tested, since measurements of a11 the data required by the models are very scarce;
(ii) the version ofthe model requiring only the use ofthe 'severity factor' has not proven to
be very successful for quantitative prediction of the vibration; (iii) the version of the model
in which empirica1 input of a differentia1 pressure PSD, G pp (Q), needs to be supplied
has been much more successful, but not better than that provided by the simpler model in
Sections 9.4.2 and 9.4.3(a), Thus, any advantage gained by using these models is washed
out by the uncertainty as to what G pp (Q) should be. The interested reader is referred to
Curling & PaYdoussis (2003); it is certain that if and when more information becomes
available, the procedures and thinking developed therein wi1l be useful in arriving at a
more successful ana1ytica1 model.
For the present, there are three options open to the designer, The first is to use the model
of Sections 9.4,2 and 9.4.3(a), coupled with in situ measurement of the point-pressure
PSD, Spp(Q) or Gpp(Q), on one of the cylinders - if the system or one sufficiently
similar exists. The second, though one that could be overconservative, is to use the model
of Sections 9.4.2 and 9.4.3(a) with the top curve or envelope ofFigure 9.48(b). The third
is to revert to the empirical expression of Section 8.4.l(d), equation (8.101) which, as
shown in Table 9.4 (in Section 9.4.5) is reasonably successful even for clusters.
tThe general form of the equations for the response in the 'new' model is essentially the same as that of
equation (9.104), except for the use of differential rather than point pressures throughout. The model is 'new'
in the sense that the interpretations and characterizations of the PSD, Spp(Q), and the factors Kij,e and Kln,
are more elaborate and hence more complex,

CLUSTERED CYLINDERS IN AXIAL FLOW
1113
9.4.4 Comparison between theory and experiment
In this section, experiments on the 4- and 28-cylinder dusters described in Section 9.4.1
and Figure 9.41 are compared with a slightly modified form of the theoretical model of
Sections 9.4.2 and 9.4.3(a) - see also Section 9.3.1 - following the work of Gagnon
& Pa'idoussis (1994a,b). The modifications to the model were introduced for it to rep-
resent more dosely the experimenta1 system (e.g. the effect of the rigid end-pieces at
the cylinder ends). Also, a modified material damping representation was introduced,
corresponding better to the experimental system, namely FD(X, t) = IJ.1(a 5 yj3x 4 3t) +
U(3 5 j3x 4 3t) LdYrjRe(Qr)] and sinúlarIy for z-displacements, in which IL is the
viscoelastic and A the hysteretic damping coefficient - where in the latter term the dis-
placement is expres sed in terms of its modal components, Yr, of corresponding frequency
Qr. The dimensionless counterpart of IL is a as in (9.61) and that of A is Y = Aj E. The
caIculations have been conducted for
€ = 18.57, h = 0.197, a = 0.865 x 10- 3 , Y = 0.022,
f3 = 0.466, Ye = 10.52, YF = 9.16, r = O, n = 109.6, (9.107)
v = 0.47, cf = 0.016, C = 0.039, G c = 0.75;
in the simpler form of the theory utilizing equations (9.62) and (9.63) unmodified, Y = Ye-
VF. G pp (W), andhence Gpp(W) = 2S pp (w), has beendeterminedby in situ measurements
(Gagnon & Pa'idoussis 1994a).t
9.4.4(a) The 'basic' four-cylinder system
Typica1 theoretica1 (dotted Iines) and experimentaI (solid Iines) PSDs for u = 3 are
compared in Figure 9.49, for vibration in the radial and tangential directions as defined
in Figure 9.44(c), nondimensionalized with respect to the cy1inder diameter D.+ The
scale for the first-mode-group theoretical results is on the right-hand ordinate, while for
the experimentaI ones it is on the left; the sca1es for the second and third flexuraI-mode
groups, which are greatly amplified in the figure, is given in the caption. It is remarkable
that not only the first, but also the second and third flexural-mode groups have been picked
up by the sensing system, and that they are in broad qua1itative and quantitative agreement
with theory - best for the first flexural-mode group and worst for the third.
These and subsequent measurements have been obtained using a two-stage 'turbulator'
upstream of the test-section, the first stage consisting of a perforated plate to promote
tThe measured differential PSDs, G r>r> (£o). were essentially fiat over the frequency range of 1-25 Hz for
u = 1-5, corresponding to a maximum f Dh/U = 4.54 for the four-cylinder system and f Dh/U = 0,987
for the 28-cylinder one,
fll is noted that the PSD data in Figures 9.49 and 9,50, and a!so Figures 9,53-9,55, are equal to
(s2 /2Jr)Gij = (s2 /Jr)Sij; thus an additiona!factorof 1/2Jr has beenemployed in frontof equation (9.104)
(Gagnon & PaYdoussis 1994a), Moreover, the response PSDs in the radial and tangential directions have been
obtained from those in the Cartesian directions using the rectangular-to-polar transfonnations mentioned earlier
[vide footnote to the paragraph following (9.94), This was necessary because, in the cross-sectionally 'finite'
square-pattem systems discussed here, significant differences exist between the radia! and tangential direc-
tions in respect to cylinder response. In cross-sectionally 'infinite' such systems, no transfonnations would
be necessary, as there would be no directional differences in the response PSDs (Curling & Paidoussis 2003,
appendix B,I), The radial and tangential directions were selected here because they correspond to the modal
directions of the system, and they coincide with the 'principal' directions of the 'field' of response PSDs
(Curling 1982, pp. 294-296; 2001),

1114
SLENDER STRUCTURES AND AXIAL FLOW
X 10- 6
X 10- 6
5
8
4
6
:t 3
O
 2
o
(a) x 10- 6 x 10- 6
5 8
4
6
:I: 3
O 4
Cf.) 2

2
o
O
30
120
(b)
60 90
Frequency (-)
Figure 9.49 Vibration PSDs in the (a) radial and (b) tangential directions of motion for the three
lowest flexural-mode groups for the four-cylinder cluster (K = 4, G c = 0.75), at u = 3, showing
the ability of the instrumentation to pick up extremely small vibration levels in the third mode-
group (i.e. in the third flexural mode of the cylinders): -, experimenta1; .,., theoretica1. The
left-hand ordinate is for the first-mode-group experimental PSDs; the right-hand ordinate for the
corresponding theoretica1 PSDs. The second-mode-group scales are 8.0 x 10- 8 forthe experimenta1
and 2.5 x 10- 8 for the theoretical PSDs; the third-mode-group sca1es are 2,5 x 10- 9 for the
experimental and 4 x 10- 10 for the theoretical ones. System parameters as in equation (9.107);
(Gagnon & Pa'idoussis 1 994b).
mixing, and the second consisting of a wire mesh to reduce turbulence scale and intensity.
This turbulator achieved uniformization of the turbulence field without creating too much
additional turbulence intensity - that is, from 0,1 to 1,5% only.
The radial- and tangential-direction PSDs for the same system and u = I, 3 and 5
are shown in Figure 9.50, where from here on the third flexural-mode group has been
suppressed.
Overall, considering the complexity of the system, there is excellent qualitative agree-
ment between theoretical and experimental PSDs for the first-mode group and u = 1
and 3. This agreement deteriorates considerably at low frequencies for u = 5, but only
in the radial-direction, where the system according to theory is very c10se to the onset
of divergence. Nevertheless, the broadening of the peaks, the spreading of the frequency
band and shifting of the band to lower frequencies with increasing u observed in the
experiments is generally well tracked by the theoretical resu1ts.

o o
o '"
N
o
o
o
.,.,

I
o
x
II
"
. ......
....

I

x
o
.,.,
o
...,
o
N
o
...
(-)aSd
8
N
o


o
.,.,


x
II
"
.......

I
o
x
o o
.,., ...
HaSd


o
'"'
...
N
x
'C

'"
II
"
'C

x.... ....
o
...
N
(-)aSd
o
'"'
...
N
'C
I

'"
II
"
x
'C
I

x.... ...,
o
...
..., N
HaSd
o
o .,.,
'" N

o
N

'C
I

.,.,
II
"
x
..o...o...o.....o. ..
'C
I

x..............,
o

'"
'"'

(-) aSd
o
o
...,
'"
.,.,
N
o
N


'C

.,.,
II
"
x
........
..... ..
'C
b
x .1'.... -:::::::.0"
o

'"
'"'

(-)aSd
00
00
o
,....;- oS íi)
II S II
::! o ;::: :o
Cd
"sb9:
íi) <;! ...... ,
t- U X '"
0'£  
11..........-8
"g . 'a
oS r;:;-p..
..,fII
II " ;::: S
 c: 
___  I '"
",8°0
 b.O  '-'
ox
 '8 c: V) II
u6 ao
. t) --=:. 3
..c 'f
].... II °
....... t.8 .....-4
s...---x
""''t '<t
5 b o
"Bos.....-4 
o = X
.= o
'O oV:
]
t:b
- .....-4
I:: S  X
sv:
06s N
oS '5  ,-,
=0..6.....
Z.E II
 I , 3
= 00
'" 0..1"
S5
..... =' 
'ge x
=oo
:.o<!)'8.,.;
'86
]6'0'3
Jo; S r.rJ
<!) =  
oS"'<;!
.s  ,2 .g
-g '" 
= o  ;::
o  u -;:1
'.;:1 '" (/J ....
 'O --" "s
:9ao..'"
;>-...  o
..... o 8<;!

",c.s::: Ó 
@.9'8
p..6s
=' E o..
10 o c.s::: 
=-i.c: .... ....
ClI "',,,s o
!;V)...u
b/:) 'O .c:
 a 'ťf
o
'"'
:r
J
o
'"
o
o
e
0 0
00
o
'"'
:r
I
o
N
o
o


1116
SLENDER STRUCTURES AND AXIAL FLOW
ln quantitative terms, the experimental peaks of vibration correspond very well to the
theoretical ones in the case of the first ftexural -mode group, but 1ess well for the second one.
Indeed, for the second ftexural-mode group, the experimental frequencies are consistently
higher than the theoretical va1ues, by a progressively larger amount as u is increased. This
could be explained by the fact that the vibration in all modes occurred simultaneously.
The vibration in the modes of the first ftexural-mode group, which has the highest 1eve1,
generates a nonlinear tension effect, proportional to fo L (a W I/ax)2dx, where W I is the
r.m,s. lateral deftection in the first-mode group and L is the 1ength of the cy1inders. The
system therefore becomes effectively stiffer for coexisting modes with higher frequencies
(the modes of the second group), since this additiona1 tension is present over the who1e
cyc1e of higher frequency modes and is always positive, albeit its value changing slowly
over the cycle. Also, judging from the broadness of the peaks, theory tends to underestimate
damping, by an increasing amount as u is increased. This may reftect an inadequacy in
modelling viscous damping.
Overall, the quantitative agreement with theory is reasonably good, ln this respect the
height of individual peaks may be rnis1eading at times, and the overall power around a
particular frequency peak is a better measure. However, broadly speaking, one can compare
sca1es, whereby it is seen that theory consistently overestimates the observed PSDs by a
factor of 1.5-4 approximately - corresponding to r.m,s, discrepancies of 1.2-2 in round
figures,
The coherence t in the motion of two adjacent cy linders in the c1uster and the corres-
ponding CSD phase are shown in Figure 9.51 (a,b) for the radial-direction components of
motion and in Figure 9,52(a,b) for the tangential components, For the pair of adjacent
cylinders, only peaks corresponding to distinct eigenfrequencies in each mode-group and
in each direction are expected to be observed (Figure 9.44): two in each case. This indeed
is the case, Thus, for the radial-direction CSD at u = 1, in the first ftexural-mode group
we see in Figure 9.5l(a) dimension1ess radian-frequency peaks at 20 and 29 (rad) approx-
imately, and second mode-group peaks at 55 and 75 (rad) approximately; simi1arly, we
see two peaks for each mode-group in Figure 9.52(a) for the tangential direction motions,
The correspondence between analytical and experimental results in Figures 9.51
and 9.52 is even more striking than in Figure 9,50. Thus in Figures 9.51(a) and 9.52(a):
(i) the coherence frequency peaks agree quite well; (ii) the evolution with u of these
peaks, in terms of shifting to lower values of w and of becorning broader, is quite similar;
and (iii) the relative amplitude in the peaks and their shapes are sirnilar in theory and
experiment. Furthermore, the measured CSD phase is very c10sely predicted by theory,
the only areas of discrepancy being associated with regions of low coherence, which is to
be expected.
The importance of the c10se correspondence in the results of Figures 9.51 and 9.52 is
that (i) the spectral content of the vibration and (ii) the intercylinder modal pattems and
other modal characteristics of the vibration predicted by theory were actua]]y observed in
the experiments with excellent agreement overall, thus validating an important component
of the theoretical model.
It is worth mentioning in this context that, if the incident excitation field across the
cylinders is nonuniform, as was the case before the turbulator was installed, then the
tIt is recalled that coherence = ICSDli2/[ (PSD) I (PSD hl, where the subscripts denote the two measurement
points.

II
"
o o
r- 00
N ....
o O O
CI'> CI'>
I
(í!ap) aS\Jqd
o
00
....

ID -.r
O O
N
O
00
O
(-) a:>uaJaqo;)
ff)
II
"
o
r-
N
I
o o O
CI'> CI'>
I
(í!ap) aS\Jqd
o
00
....
I
o O
r- 00
N ....
ff)
II
"
o
ID -.r
O O
N
O

00
O
(-) a:>uaJaqo;)
o
00
....
O
...
B
CJ>
..a
u
...
<1)
"O
.S
;;.
u

;::I
.£
<1)
iJ
c2:2
"Ť
O\
uO\

0\ 
 5
5b;S
,_ o:!
i:l..
o(!
s:::
Ng
"ObO
c3
..... '-"
CJ>i::'
!J O
"O <1)
s:::..s:::
,- ...
;;.
U
ó;.:
s::: s:::
.9 Š
<;; 'I::
&
;> K
s::: <1)
. 1-
... ..
;.rjV)
, "O
]
....,
E! -
.........
 II
 ::::
"&.2
O,...::.
<ZJo:!
U;::;
---- "Ť
"sa\
"O 
 5b
8 ti::
s:::
<1)
...
<1)
..s:::
O
U
V')
II
"
o
ID
1:
;>,
u
O o:
-.r "
g.
"
...

O
N
O
O
r-
N
I

O
r-
N
I
O
00
....
I
O O
r- 00
N ....
O O O
CI'> CI'>
I
(í!ap) aS\Jqd
O
00
O
....
11'1
0\

=
OJ)

V')
II
"
O
ID
1:
;>,
u
 []
g.
"
...

O
N
----

ID -.r
o o
N
O

00
O
(-) a:>uaJaqo;)
,-.,


II
'"
o o
r-- 00
N
o O O
a-. a-.
I
(i!:Jp) :JS1!qd
o
00
I
II
'"

 "'"
O O
(-) :Jou:JJ:Jqo:)
N
O
00
O
o
r--
N
I
o O O
a-. a-.
I
(i!:Jp) :JS1!qd
o O
00 r--
- N
I I
o O
r-- 00
N -
cO')
II
'"
o
 "'"
O O
N
O

00
O
(-) :JOU:JJ:JqO:)
o O
r-- 00
N
o O O
a-. a-.
I
(i!:Jp) :JS1!qd
..
o
00
O
 "'"
O O

(-) :Jou:JJ:Jqo:)
o
00
...

'"
;:::I
U
...
I!)
"O
,5
>.
'-'
.!.
;:::I
..8
<1)
£
...
c.S
.D
,-,"'"
3g:;
-
0\ .
 
i5

Il..

NI::
]8
tU
-o
'-'
<1)
]0
, I!)
£
'-' ,
....
o
s......
, I::
 Š
:9 '5
;> o.
I:: 
.S (I)
. I'
"O
I "
]
.... I::
 tU
O().....
 ,
-
'C5 II
I!) ::!
'" ...
,.g..8
0.,-,
Q
CI)-
u-.:t:
,-, 0\
.DI!)
;o
I:: 101)

go

I!)
..c
o
U
,-,

'"
II
'"
$
:I
>,
u
 5
6-
Q)
...

o
N
o
00
-
I
o
o
r--
N
I

o
00
:I
>,
u
5
6-
Q)
...

N
o
N
\I)







CLUSTERED CYLINDERS IN AXIAL FLOW
1119
degree of agreement with the theoretica1 model (which presumes unifonnity) deteriorates
considerably, and both the excitation and response PSDs differ from one cylinder to
the next. A1so, in clusters, the differentia1 wa1l-pressure PSDs tend to vary around the
circumferences of the cylinders. They can vary significantly from directions of maxima
to those of minima, and hence influence the fina1 ca1culation of the response significantly
(Ohlmer 1973; Curling & Paldoussis 1 992a) ,
9.4.4(b) Variations of the four-cylinder system
Many other experiments have been conducted with variations of the same basic system,
to study the effect of (i) closeness ofthe cylinders (G c ), (ii) the number of cylinders in the
c1uster (K), (iii) some of the cy1inders being rigid while the rest flexible, etc. In aU of
these, the experimenta1 results were in good agreement with theory and they provide new
insights into the dynamics and vibration of the system; see Gagnon (1989), Gagnon &
Pa'idoussis (1994b). Here only a sample of these results is presented,
Theoretica1 and experimenta1 PSDs for different G c are shown in Figure 9.53, for only
the first flexura1-mode group. Here, unlike the previous case, the results presented are for
a diagonal pair of cylinders. The foUowing features are noted.
(a) The width of the mode-group frequency band is considerably broader for the closer
spacings (smaUer G c ).
(b) The amp1itude of motion is not dramatica1ly different for the three values of G c .
In this respect, it should be reca1led that, for this particular cluster geometry, as
the cylinders come closer together, the mean flow distribution is changed: more
fluid bypasses the centra1 core in-between the cylinders, as flow resistance in the
core increases; this could imply a higher turbulence intensity being generated in the
central-core flow, but not necessarily a higher turbulence excitation, the mean flow
velocity in the core being lower.
(c) AU theoretica1 ca1culations were conducted with the excitation PSD measured for
the G c = 0.75 system. Accordingly, the theoretical PSDs for G c = 1.00 and
1.50 are not claimed to be quantitatively correct; they are included to reinforce the
trends displayed by the experimenta1 results, and to show these trends more clearly.
Nevertheless, theoretical and experimental PSDs are reasonably close.
(d) Coherence and phase, the latter not shown here, are not dependent on the excitation
PSD, and the agreement between theory and experiment in Figure 9.53(e) and (f)
is quite good. Three peaks are predicted by theory in each of the two orthogona1
directions, since in this case the vibration of a diagona1 pair of cylinders is being
correlated (as opposed to adjacent cylinders in Figures 9.51 and 9.52), In the exper-
imenta1 data the situation is less clear-cut, for example with regard to the equiva1ent
of the third theoretica1 peaks. t
(e) As expected, the coherence in the motions of the diagona1 pair of cylinders is
diminished as their distance apart increases, and this is found both theoretica1ly
and experimentaUy, with the experimenta1 coherences genera1ly lower than the
theoretica1 ones.
tThe results shown in Figure 9,53(e) and (f) are for motions of the two cylinders in the radial direction,
Similar results for the tangential direction are not shown, for brevity.

1120 SLENDER STRUCTURES AND AXIAL FLOW
x 10- 6 Theoretical X 10- 6 Experimental
10 5
8 4

J..,
Q 6 3
CI)

Oj 4 2
'i3
os
P:::
2
(a) O (b) O
X 10- 6 X 10- 6
10 5
:I 8 4
Q
CI) 6 3

3
i:: 4
"
I:>JJ
 2
(c) O
1.0
0,8

J..,
"
(j
d

"
..d
o
U
(e)
15 25
Frequency (-)
30
30
(f)
15 20 25
Frequency (-)
Figure 9.53 First mode-group vibration PSDs for the basic four-cylinder cluster for u = 3,
showing the effect of G c : (a, b) theoretical and experimental PSDs in the radial direction;
(c, d) theoretical and experimental PSDs in the tangential direction; (e, t) theoretical and experi-
mental coherence in the radial-direction vibration for the diagonal pair of cylinders 1 and 3: -,
G c = 0,75; "" G c = 1.00; - - -, G c = 1.50 (Gagnon & PaYdoussis 1994b),
(f) The results for u = 3 are typical ofthe who1e flow-ve1ocity range tested, Those for
10wer u, for examp1e, u = 1, tend to show better agreement between theory and
experiment, whi1e those for u = 5 worse,
In addition to the foregoing, some usefu1 physical insights may be gained from these
results, as follows.
Two important features are noted in the PSDs and coherence of Figure 9.53 in the
radial-direction vibration [pane1s (a, b) and (e, f)]: there is one frequency peak (w :::: 23)

CLUSTERED CYLINDERS IN AXIAL FLOW
1121
that remains sensibly constant, irrespective of G e ; similarly, in the tangential-direction
PSD [panel s (c,d)], the W :::::::: 21 peak is relatively little affected by G e . At the same time,
there are other frequency peaks that are greatly affected by G e, notably the lowest in the
radial direction and the highest in the tangential one. The question is: why?
The reason becomes clear by cross-reference to Figure 9.44, in which the dominant
pattems of motion are shown for each of the modes (peaks). The W :::::::: 23 frequency peak
corresponds to the repeated root Ws = W6. For these modes, diagonal pairs of cylinders
move in unison, while there is little coherence in the motion of adjacent cylinders. Thus,
there is very little 'squeezing' of the interstitial fluid, and hence little sensitivity to G e;
most of the effect comes from the surrounding fluid, which is sensibly the same for
all G e . In this respect, it is of interest to note that the added mass associated with the
pair of diagonal cylinders vibrating in this mode of motion is about twice that of an
unconfined single cylinder; it is then not surprising to remark that w :::::::: 23 :::::::: w se , where
W se is the unique eigenfrequency of an unconfined single cylinder! For tangential-direction
vibration, cross-referring to Figure 9.44, it may be verified that W = W3 = W4 :::::::: 21.
Once again, the inter-cylinder pattem of motion is such that, by similar reasoning to the
foregoing, one can explain the insensitivity of this type of motion to varying G e.
ln contrast, consider the case of the lowest radial-direction dominant frequency [Fig-
ure 9053], corresponding to WI in Figure 9.44. In that 'implosion-explosion' pattem of
inter-cylinder motion, in which al1 cylinders participate equal1y, the amount of 'squeezing'
of the interstitial fluid is indeed maximal, and so is the effect of G e o
ln the case of the highest frequency peak in the tangential direction [Figure 9.53(c)
and (d)], the pattem of motion involves a rotary (twisting) motion of all cylinders in
phaseo The closer the cylinders are to one another (smaller Gc), the more 'common' (ioe.
'shared') is the entrained fluid, leading to lower added mas s per cylinder, and hence to
higher frequencieso
We next consider the effect of the number of cylinders, K, on vibration. For the one-
cylinder test, the cy linder was placed clo se to the centre of the channel. For the two-cy linder
test, two diagonal cylinders in the four-cylinder cluster were removed, leaving the other
diagonal pair.
The theoretical results for K = 1,2 and 4 in Figure 9.54(a) and (c) show the radial- and
tangential-direction PSDs, respectively. It is seen that the single cylinder has but one peak,
at the same frequency in both directions, as expected. The two-cylinder cluster displays
two peaks in each direction, while the four-cylinder c1uster displays three peaks in each
direction, as discussed before. The experimental results [Figure 9.54(b) and (d)] represent
an imperfect rendition oftheoretical predictions: for example, in Figure 9054(d), there are
spurious peaks for both the single cylinder and the two-cylinder c1ustero
As to the effect of K on the dominant vibration frequencies, similarly to the effect
of G e already discussed, it is noted that some frequency peaks in Figure 9.54 - in
fact, the same as those in Figure 9.53 - are insensitive to changes in K. Thus, the
W :::::::: 23 frequency is essentially identical for the K = 2 and K = 4 systems, the fluid
motion in the inter-cylinder modes involved is but slightly impeded by the additional two
cylinders [Figure 9.54(a) and (b)]; similarly for the W :::::::: 21 frequency in Figure 9.54(c)
and (d). The opposite situation is exemplified by the lowest-frequency peak in the radial
direction, which for K = 2, involves a pattem of motion with the two cylinders moving
in opposition [w :::::::: 21.2, Figure 9.54(a) and (b)], whereas for K = 4 all four cylinders
move in opposition thus generating a dramatic decrease in the frequency to W :::::::: 15.8.

1122 SLENDER STRUCTURES AND AXIAL FLOW
x \0-6 Theoretical X \0-6 Experimental
12 5
4
:I: 9
(::) 3
r/) ,
p.. 6
';;J II 2
'i3 I I
'" I I
 3
I I
/ \
,I \ _w.
(a) O (b) O
x \0-6 X \0-6
12 5
:I: 4
9
(::) ,
r/) I 3
p..
'$ 6 ,:1'
i:i " li'. 2
" I \/:
..
o:: I ,
 3
I .
/ .
." .' .-,
O O
10 15 20 25 30 10 15 20 25 30
(c) Frequency (-) (d) Frequency (-)
Figure 9.54 Comparison of vibration PSDs at U = 3: -, solitary cylinder; ..., two-cylinder
cluster; - - -, four-cylinder cluster; (a, b) radial direction vibration and (c, d) tangential-direction
vibration (Gagnon & Pai:doussis 1 994b),
Similar insights are gained from the analysis of mixed flexible/rigid cylinder c1usters.
For example, the results for a three-flexible/one-rigid cy1inder c1uster (IF/3R) were found
to be quite similar to those for a solitary cylinder (IF); hence, the co upi ing characteristics
in, say, 4F c1usters are associated with motion of the cylinders, rather than with their mere
presence.
9.4.4(c} The 28-cylínder cluster
The main advantage of conducting experiments with c1usters of only a few cylinders
(two or four) is that the modal pattems are few and simple, and hence the interpretation of
results is easy. At the same time, there are two drawbacks to these particular four-cylinder
tests: (a) the c1uster is too 'open', in the sen se of being located in the middle of a large
conduit, and (b) in engineering applications, it is much more common to have c1usters
with many cylinders,
The 28-cylinder c1uster is shown diagrammatically in Figure 9.41 (b). Only four of the
cylinders were flexible and instrumented, and they were centrally located; the rest were
rigid. As for a four-cylinder c1uster, there are still eight eigenmodes for the four flexible
cylinders, but here they are all distinct: four radial- and four-tangential-direction modes.
The PSDs for u = 1 and 3 are shown in Figure 9.55. The four distinct theoretical peaks
are more-or-less discemible in the experimental data for the first flexural-mode group at
u = 1, for both the radial and tangential PSDs. The degree of distinctness of the peaks

CLUSTERED CYLINDERS IN AXIAL FLOW 1123
XI9 XI9
50 250
40 200
1: 30 150
O
jC 20 100
10 50
o O
X 10- 6 X 10- 6
3 12
2.5
2,0
--
-.!-
O 1.5
<J:)

1.0
0,5
0,0
O
(a)
50
40
1: 30
O
jC 20
10
3,0
2.5
1:
O 1.5
<J:)

1.0
0,5
0.0
O
(b)
u3
,
.
.
.'
"
"
, '
9
6
3
O
80
20
40
Frequency (-)
60
X 10- 9
X 10- 9
.....,
O
X 10- 6
X \0-6
12
u=3
9
6
3
O
80
20
40
Frequency (-)
60
Figure 9.55 (a) Radial-direction PSDs and (b) tangential-direction PSDs for the 28-cylinder
duster (4F/24R), for u = 1 and 3: -, experimental PSDs, scale for first mode-group on the
left; ..., theoretical PSDs, scale on the right. The second mode-group expanded scales for the
experimentalresults (and the theoretical ones in parentheses) are: for u = 1,0.4 x 10- 9 (0.4 X 10- 9 );
for u = 3,90 x 10- 9 (40 x 10- 9 ); (Gagnon & Pai'doussis 1994b).

1124
SLENDER STRUCTURES AND AXIAL FLOW
deteriorates with increasing u; t however, the genera1 theoretical trends are reproduced
fairly well by the experimenta1 data, at 1east in the tangentia1-direction PSDs,
The theoretica1 second flexural-mode group displays less than four peaks; indeed in
the tangentia1 direction there is essentia1ly only one broad-banded peak. This, most prob-
ably, is due to the effect of increased damping for the higher frequencies associated
with the second-mode group; the effect is stronger in this case than for the four-cylinder
cluster, because the hydraulic diameter is much sma1ler. The experimenta1 frequencies
are genera1ly higher than the theoretica1 ones, for the same reason as that advanced in
subsection (a).
The coherence between the tangentia1 components of vibration of a diagona1 pair of
the four flexible cylinders and the corresponding phase are shown in Figure 9.56, The
coherences and frequencies are in good agreement; so is the phase, provided that the
corresponding coherence is not too low. In the case of the radia1 direction (not shown),
a1though agreement in phase is good, it is not as good for coherence.
9.4,4(d) The amplitude of vibration
The theoretica1 and experimenta1 radia1- and tangentia1-direction r,m.s, dimensionless
amplitudes of vibration are plotted versus the dimensionless flow velocity in Figure 9,57 (a)
and (b) for the basic four-flexible-cylinder cluster and in Figure 9,57(c) and (d) for the
28-cylinder cluster.
It is seen in Figure 9.57(a) and (b) that agreement between theory and experiment is
remarkably good for the four-cylinder cluster, for u =s; 4 - considering the complexity
of the system; discrepancies at the high flow range are a maximum of 30% for a11 u and
0(20%) for u =s; 4. Alas, the discrepancy for the set of data for u =s; 1 is considerably
larger - see Table 9.4. Agreement for the 28-cylinder cluster [Figure 9.57(c) and (d) is
less impressive (a maximum discrepancy of 44%), but is still quite good.
Theory consistently overestimates the amplitude of vibration.:j: Two reasons may be:
(i) the underestimation of flow-induced damping; (ii) the inherent assumption in the
Table 9.4 Comparison ofr.m.s. amplitudes, Yrms/ D, from (i) Pai:doussis empirical
expression, equation (8.101), (ii) the CUrling, Oagnon & Pai:doussis (COP) theory
(Sections 9.4.2 and 9.4.3), and (iii) the Oagnon (1989)/Oagnon & PaYdoussis (1994b)
experiments for the 4- and 28-cylinder c1usters of Figure 9.41,
u 4-cylinder cluster 28-cylinder cluster
Empirical Experiment CGP Empirical Experiment CGP
expression theory expression theory
0.5 0.30 0,09 0.46 0.11 0,14 0,80
1.0 0.67 0.49 0.78 0.24 0.54 1.13
1.5 0.87 1.49 1.76 0.31 1.29 1.68
2.0 0.97 3,04 3.54 0.34 2,28 3.51
tResults for u = 4 are given in Gagnon & PaYdoussis (l994b); divergence is reached at u = 5 for this
dus ter.
* A proIIŮsing reduction of this overestimation is achieved by Curling & Pai'doussis (2003), by imposing a
zero-frequency/zero-PSD condition to the excitation PSD, instead of a fiat extension to f = O.

CLUSTERED CYLINDERS IN AXIAL FLOW
1125
1.0
0,8
1:
'" 0,6
u
<=
'"
....
'" 0.4
..s:::
o
U
0,2
O
u=1

...!.-
'"
g
'"
....
'"
..s:::
o
U
180
.
.
.
.
.... .
biJ 90
'"

'" O
'"
tO
 -90
-180
-270
u=1
270
180
biJ
'"

u=3
-270
O
20
40
Frequency (-)
60
80
(b)
Figure 9.56 (a) Coherence and (b) CSD phase of tangential-direction vibration of a diagonal pair
of the four-flexible-cylinder kemel in the 28-cylinder cluster, for u = 1 and 3; -, experimental;
..., theoretical (Gagnon & PaYdoussis 1994b).

1126 SLENDER STRUCTURES AND AXIAL FLOW
x 10- 3 Four-cylinder cluster X 10- 3 28-cylinder cluster
12 12
1:
 9 9
,€ l'
.
Q. .
ij . x
6 6 x'
C;;
'.jj
e
IOl) 3 3
c

O O
(a) x 10- 3 (c) X 10- 3
12 12
jC
1: .
.
 9 . 9
 '"
.
Q. x
S 6 6 .
'" )c.
3 .
'5 )(.
CI) .
IOl) 3 3 .
 .)(
. .
.)(..
O O
O 2 3 4 5 O 2 3 4 5
(b) Velocity (-) (d) Velocity (-)
Figure 9,57 Theoretical and experimental dimensionless r,m.s. vibration amplitudes versus u for
(a, b) the basic four-cylinder cluster, and (c, d) for the 28-cylinder cluster; (a, c) for radial-direction,
and (b, d) tangential-direction vibration: -, experimental; ,.., theoretical (Gagnon & Pa'idoussis
1 994b).
theoretical model that the boundary layer excitation is axially and circumferentially
uniform, whereas in the experiments fully developed conditions were likely not achieved
much upstream of cylinder mid-Iength; even then, the excitation field was found to vary
circumferentially (Curling & Pai'doussis 1992a,b). Nevertheless, prediction of vibration
amplitude is generally very good, especially when compared to previous attempts utiliz-
ing empirica1, semi-empirica1 correlations and other semi-analytical means of predicting
vibration amplitudes, which achieve only order-of-magnitude agreement (Section 8.4).
9.4.5 Concluding remarks
Some experiments and an elaborate analytical model for turbulence-induced vibration of
clusters of cylinders in axial flow have been presented. The theoretical and experimental
results, and their comparison, have provided valuable additional insights into the modal
and spectra1 characteristics of the vibration, vis-o-vis Sections 9.3.3 and 9.3.4, as well as
providing a predictive too1.
Further work, on simplified versions of the theory, for example, by neglecting excita-
tion coupling, or considering the vibration to be entirely in the first flexural mode, are

CLUSTERED CYLINDERS IN AXIAL FLOW
1127
discussed in Curling & Paldoussis (2003), as are the savings in computer time by successive
simp1ifications in the model, which can be very substantial. A1so presented therein are
more sophisticated versions of the analytical model. Nevertheless, the major weakness
of these models remains that, unless the in situ level of the wall-pressure PSD is used in
the model, the discrepancy between predicted and actual vibration amplitudes could be
large, as discussed in conjunction with Figure 9.48: the excitation PSDs can vary from
one apparatus to the next by a factor of at least 100, and hence the r.m.s. amplitudes by a
factor of 10.
ln the absence of any prior knowledge of the excitation field, the designer has two
options: (i) to calculate a range of vibration amplitudes, using the upper and lower bounds
of G pp/ (f,L p U 2 c 2 ) from Figure 9.48(b), or (i i) to use the empirica1 expression, equa-
tion (8.101). Sample comparisons are given in Table 9.4,t showing that the empirical
expression is not too bad - recalling that it is valid strictly for u < 1.
Retreat to the empirica1 expression is regrettable, though in many cases necessary. To
improve matters, a sustained effort and an extensive set of measurements of the excitation
PSDs are necessary, in systems of varying geometry and other parameters, inc1uding
idea1ly 'quieť and realistica1ly 'noisy' systems - thereby making the use of the more
elaborate theory of Curling & Pa'idoussis (2003) practicable.
9.5 STRINGS OF CLUSTERS
9.5.1 Motivation
ln boiling light-water and light-water-cooled heavy-water-moderated nuc1ear reactors
(BLWRs and LWCHWRs), c1usters (bundles) of fuel elements or rods are stacked ver-
tica1ly one on top of the other and held together by a tensioned central support tube
running through the centre of the whole fuel-bundle 'string' (or 'stringer') as shown in
Figure 9.58(a) and (b) for a CANDU-BLWR. Each of the fue1 bundles is similar to that
shown in Figure 8.l35(b), except that the centre is free of fuel rods, to let the centra1 support
tube through. Each fue1 string is located inside a pressure tube which contains the axia1ly
and upwardly tlowing coolant. The string is tlexibly supported at the bottom and is either
free or simply supported at the top, where it is terminated by a neutron-tlux suppressor.
The totallength is of the order of 7 m and hence the string is quite tlexible, norma1ly
touching the pressure tube near the top - the c1earance being of the order of 2 mm.
Inevitably, therefore, some tlow-induced fretting is expected. Each fue1 element in the
bundle consists of cylindrica1 U02 pellets encapsulated in a zirconium alloy (zirca1oy)
tubular sheath. On the latter are welded small 'wear pads', which are sacrificially worn to
protect the tubular sheath itse1f. The design is such that the rate of wear of the wear pads
and the pressure tube be consistent with the design life of these components.
ln one case, however, involving a CANDU-BLWR with fuel strings free at the top,
in Genti11y, QC, Canada, it was found that accelerated fretting had taken place at the
uppermost fuel bundles of 40% of the high-tlow fuel channe1s Figure 9.58(c); the wear
was preferentially to one side. It was determined that the wear was caused by excessive
tThe experimental and theoretical values are the vectorial sum of radial- and tangential-direction amplitudes.
indicating the maximum possible r,m.s, vibration amplitude in all directions around the cylinder,

1128
SLENDER STRUCTURES AND AXIAL FLOW
a
o
or\
II
'"
"
'i3
S

o
><

Outlet
Fuel bundle
(=< 27 kg)
10.4 cm
Central
support
tube
o
o
'"
Pressure
tube
a
""
o
Spring
assembly
Shield plug (b)
a
r-:
(a)
I Centralizing pads
2 Spacers
3 Bearing pads @
4 U0 2 fuel pelIets (ž)
5 Zircaloy 4 sheath
6 De1ineating disc
7 End cap
8 End plate
(c)
Figure 9.58 (a) Schematic of a fuel string of the CANDU-BLWR; (b) a fuel bundle, containing
18 fuel elements; (c) photograph of the pathological wear that occurred in some of the fuel channels
at Gentilly (Pai'doussis 1980).
axial-tlow-induced vibration and repeated impacting on the pressure tube. It was theorized
that the relatively highly tlexible string had become unstable by tlutter or divergence, in
the latter case the uppermost part of the string pressing on the pressure tube and chatter-
ing against it. This was one of the practical 'cases' of tlow-induced vibration (Case 42)
discussed in Paldoussis (1980).
Extensive analytical model1ing was undertaken of the dynamics of the fuel string in
tlow (Pai'doussis 1975c, 1976b; Mason et a/. 1978), With the aid of the resultant analytical
model, the stability of the system could be assessed. Contrary to expectations, however,
it was found that under design conditions the system was free of instabilities. The real
problem, it was discovered, was much more mundane: due to manufacturing error, some
of the tlux suppressors at the top of the string were assembled slightly eccentric; the
steady tlow then forced the string to bend to one side and chatter against the pressure tube
(Pai'doussis 1980).
Despite this disappointing dénouement, it is considered well worth presenting this mod-
el1ing work in summary, as the system is quite general and the model may well be used for
other nuc1ear fuel systems also, for example the British SGHWR and the lta1ian Cirene
reactors - see, for example, Figure 8.135(a). Indeed, it may be used for any system

CLUSTERED CYLINDERS IN AXIAL FLOW
1129
involving c1ustered cylinders vibrating in stili or axially flowing fluid, where the cy1inders
move together rather than individua1ly as in the foregoing; that is, in this case intercylinder
motions are neglected. t
9.5.2 Modelling of cluster-string dynamics
The elementary model to be outlined here has been developed by Pa'idoussis (1975c,
1976b). The basis of the mathematical model is that the fuel string behaves like an
articulated structure, each bundle being an elemental unit of this structure, The overa11
defonnation is assumed to arise by (i) a so-called parallelograrnming mode of defonna-
tion, involving a net shear ofthe bundle by flexing and twisting ofthe end-plate 1igaments
and by bending of the fuel elements, and (ii) by bending concentrated at the articulation
points, involving the flexing of the central support tube (CST) and of the end-plates. All
fuel elements in a bundle are assumed to move parallel to one another. The derivation of
the equations of motion is via application of the Lagrange equations.
9,5.2(a) The structural model
The string defonns by shear (para11elogramming) which involves movement of one end-
plate of a bundle relative to the other whi1e remaining paralle1 to it. This involves the
angles of deformation Xi (Figure 9.59). Additional defonnation is possible by bending
the end-plates and the CST, involving the angles 1/1'i. Thus, a system of N bundles c1early
has 2N degrees of freedom, For convenience, we choose as generalized coordinates the
angles 1/1'i and </>i == 1/1'i + Xi.
Modelling the resistance to parallelogramming by rotational springs of stiffness Ci and
that due to bending of the CST and of the end-plates by k7 and ki, respectively, the total
potential energy of the structure is given by
1 N { {I [ P-l ] } 1 N
VS=-"2E ]oPm p g lq</>+</> d +"2Ekp-1(1/1'p-1/1'P-l)2
N N
1,,* 21" 2
+"2 Lkp (</>p+l-</>p) +"2 LCP (</>p -1/1'p)
p=l p=l
(9,108)
for small motions, where mp is the mass of the pth bundle per unit length and lq is the
length of qth bundle, q = O, , . , , N; for notational convenience, 1/1'0 = </>0 = O.
The kinetic energy of the structure is
{ [ ] 2 }
1 N Ip p-l, ,
Ts ="2:; 1 mp lq</>q +</>p d ,
(9.109)
cf. Section 8.12,2.
tThe frequencies of intercylinder motions and those of a string of clusters are usually well separated - by
a factor of 10: I at least.

1130
SLENDER STRUCTURES AND AXIAL FLOW
Figure 9.59 Simplified model of the structural deformation of the string (much exaggerated),
involving shear (paralle1ogramming) and bending (PaYdoussis 1976b),
9.5.2(b) Fluid forces
The fluid forces are derived in a similar manner as in Section 8,12.2. The conservative
hydrodynamíc forces are taken into account in terms of a kinetic energy of the fluid, using
slender body theory, name1y
{ [ ] 2 }
1 N I p-l
Tf = '2:; 1 p Mp  lq q +  p + Up</>p d ,
(9.110)
where Mp is the added mass of the pth bundle for lateral motions. The rest of the forces
are expressed in a similar fashion as in equations (8.274)-(8,277). They are as follows:
(i) the nonconservative fluid dynamic force at the free end,
F"<  -(1 - f)MLUL [lP4>p + UL'/'N] ,
(9.111)
where the subscript L here denotes quantities associated with the free end;

CLUSTERED CYLINDERS IN AXIAL FLOW
1131
(ii) the 10ngitudina1 and norma1 frictiona1 forces (subscripts L and N),
1 2
(Fd p = "2nppDUpCf,
(FN)p = n Pp D Up Cf [ I: lq 4>q +  4>p + Up 4Jp ]
q=O
+ in p D CD [f.i,,p, H ,pp] ,
(9.112)
where n is the number of fue1 e1ements in each bundle, D the fue1-e1ement diameter,
and so on (the notation being the same as e1sewhere in this chapter);
(iii) the pressure forces, (F pJ p = O and
[ 1 2 D ( au ) ]
(Fpy)p=- "2nppDUpCf Dh +ppgAp+ppApU p  p 4Jp.
(9.113)
Eva1uating the virtua1 work, the genera1ized forces (of all forces not in the kinetic and
potential energies) are obtained,
t- t-
Qj=- loJ(FN)jd+ loJ(FP\)jd
N { t l i rz
+ L - 10 I (FN)i lj d + I (Fp,)i lj d + 10 I (Fdi Ij(4Ji - 4Jj)d
i=Hl O O O
( ap ) } 1 2 2
- - Ai li Ij(4Ji - 4Jj) + -PL Deq UL Cb Ij(4JN - 4Jj) - Fnclj; (9.114)
ax i 2
Fnc, (FN)j, (Fdj and (Fp,)j are given by equations (9.111)-(9.113), and
(ap fax) j Aj = (FPy) j j4Jj; Deq is anequivalentdiameteratthefree endforbase-drag ca1-
culations. For detai1s, see Pa1doussis (1976b), Furthermore, since genera11y there is boi1ing
somewhere a10ng the string and two-phase ftow thereafter, Pp, Up and Mp are supposed
to be variable a10ng the string, but piecewise linear; for example, Pp = Pp + p( - !li).
The subscript in PL and UL denotes va1ues at the free end.
Thus, everything is now determined and, in princip1e, one may derive 2N linear ordinary
differentia1 equations which may be put into matrix form,
[M] {t} + [C] {:} + [K] {} = {O}.
(9.115)
However, in practice, the task is not simple, particularly when N is large, For this reason,
a symbolic manipulation package was used, named FORMAC (Trufyn 1970), to obtain
the equations of motion by performing the differentiations and integrations required; the

1132
SLENDER STRUCTURES AND AXIAL FLOW
printout ofthe coefficient of tP2 alone is one page long (Pa'idoussis 1975c).t This is one of
the first recorded instance s of deriving equations of motion in the Lagrangian framework
by means of a symbolic manipulation package.:j: An altemative, matrix operator approach
is presented in Mason et aJ. (1978).
9.5.3 Dynamical behaviour
Extensive calculations have been conducted for an N = 4 system, mainly for checking,
and for an N = 12 system, which is capable of mode1ling the real system.
A sample calculation for a system with no boiling and with plausible but hypothet-
ical parameters is shown in Figure 9,60; the bottom two joints are modelled as pure
pin-joints [see Figure 9.58(a)]. The dimensionless complex eigenfrequencies Wi are plotted
50 1 ' 20" ,
408 
30 ']
16. -16
1667 20 8/\
o i------- - ------------------_:-==:::.-=.o----
@ -10'8
   12
'20
-20
.18
-3D ! (a) ,
o ID
20
, t ,
- I st and 2nd mode loci
- 3rd mode locus
I LI' . ,
30 40 50 60 70
1. '24. 24'
24..20  \
ID .22 h 162_. \ '.16 \ 16 .1624
16. rOV 16" \ . 16
11,.....-.....8 7 1'8 -8 .8 .8 .8 \ 8 20
O --18t:----4\----" -- \ I ' ;---7)---8-----ij--1-J-i-ii QiI;
-ID
(b) 20'
- 20 O 100 200 200 400 600 800 1000 1200 1400
.'We-(w)
,
20
90
IDO
Figure 9.60 Typical Argand diagrams showing the dimensionless complex frequencies, W, as
functions of U for an N = 12 system: (a) First, seeond, and third mode loei; (b) 4th-12th mode loei
(Pa'idoussis 1976b).
tThe probability of error for derivations by hand for the N = 4 system was found to be reasonably small;
for the N = 12 system, however, it was close to unity,
*Reca11ing that this work was performed back in the mists of time, relatively speaking, this was then quite
avant garde.

CLUSTERED CYLINDERS IN AXIAL FLOW
1133
as Argand diagrams, with the dimensionless flow velocity, u, varied; here
u = (M3LN / k'3) 1/2 Ul,
[ 3 ] 1/2
ev = (M3 + m 3) L N / k'3 Q,
where Q is the dimensional radian frequency and L the tota11ength, For detail s of the
other parameters, see Pai'doussis (1976b); Argand diagrams for other systems are given in
PaYdoussis (1975c) and Mason et a/, (1978).
It is noted that when u = O, the string is buckled by gravity in its first and second
modes - because of the lack of restraining stiffness at the bottom two pin-joints. With
increasing flow the positive and negative branches of these two modes coa1esce and leave
the Lm (ev)-axis at u ::::: 4, indicating the onset of coupled-mode flutter, However, the
system buckles once more at u ::::: II and remains buckled tilI u ::::: 20. It is also noted that
at u ::::: 12, the system loses stability by flutter in its third mode. This instability subsides,
and is followed by flutter in the fourth mode, then in the fifth and sixth modes (recall that
these are linear theory predictions). Clearly, had this system been the first iteration in a
design process, it would have been rejected out of hand.
Typica1 mode shapes of the second and sixth modes of the system under flow conditions
are shown in Figure 9.61, displaying the characteristic travelling wave content,
8
9
7
10
6
11
5
2
4
3
Figure 9.61 'Animated' mode shapes of the second and sixth modes of the N = 12 system under
flow conditions (Pa'idoussis 1976b).

1134
SLENDER STRUCTURES AND AXIAL FLOW
Other systems and the effect oť varying selected parameters may be found in Pai'doussis
(1976b) and Mason et al. (1978),
9.5.4 Concluding comments
The main finding of interest is that, at flow velocities far removed from those at which
individual fuel elements may be subjected to hydroelastic instabilities, the whole fuel
string may be unstable by virtue of its lower effective rigidity.
The elementary model presented proved to be an effective design tool [see Reimer et al.
(1975), Lelli (1975), Mason et al. (1978)].
As mentioned in Section 9.5.1, however, when realistic parameters are inserted in the
model, it is found that the CANDU-BLW design is safe. It is interesting, nonetheless,
that (i) this possibility of trouble was not foreseen at the design stage, (ii) judging by the
resources and time devoted to the task, this possibi1ity was quite real, and (iii) real troubIe
was averted, most probably by the instinctive good sense and in-bui1t conservatism of
good designers.
9.6 SUMMARY AND CONCLUSION
Clustered cylinders in den se axial flow develop fluidelastic instabilities similarly to solitary
cy1inders, but at lower flow velocities; the tighter the array, the lower is the critical flow
velocity. The transition from turbulence-induced vibrations (the buffeting régime; see
Figure 8,38) to in stability occurs through progressive broadening of the hydrodynamically
coupled band of vibration frequencies as the flow velocity is increased; los s of stability by
divergence (for cy1inders with supported ends) then occurs when the lower bound oť this
band reaches zero frequency. Agreement between theory and experiment is generally very
good. However, even in cylinder arrays, these instabilities materia1ize only for u > 1,5
usually, which 1ies beyond the operating range of most conventional engineering systems. t
Perhaps the most interesting aspect of the work presented in this chapter relates to
hydrodynamic coupling (and, where appropriate, structural coup1ing also). In particular,
the variety of coupled intercy1inder pattems of motion that arise therefrom and their
relationship to the spectral characteristics of the vibration are of special interest,+ both
in still and in flowing fluid, in the latter case induced by turbulence or as a feature of
instabilities, These are amply elucidated in'Sections 9.1-9.4.
The physical systems investigated are dosely related to engineering systems, in nudear
reactors and heat exchangers mainly. Hence, some comments of direct interest to designers
follow. The state of understanding of the various mechanisms underlying vibrations and
instabilities of c1ustered cylinders in axial flow may be said to be good, at Ieast for single-
phase flows. Hence, a design procedure may be proposed, as follows.
(i) It is ascertained that u < 1.5, so that fluidelastic instabi1ities are unlikely to occur.
tUnconventionally flexible engineering systems ofthis type do exist: forexample, the plastic-pipe condensers
proposed by Ecopol (Pai:doussis 1993, section 6,6) and 'Flexa' heat exchangers (Anonymous 1993; Paidoussis
1993).
*Of course, fmm the utilitarian point of view, a designer is interested mainly in amplitude and frequency
information, irrespective of pattems, symmetries, etc, Then, of course, some look at a cut diamond (a brilliant)
and see but a lump of coa1. Designers not interested in the inherent beauty in the objects they design are not
good designers, generally speaking!

CLUSTERED CYLINDERS IN AXIAL FLOW
1135
(ii) If the system is known to contain sizable harmonic flow-velocity perturbations,
the design should be such that their frequency I be:::l:: 25% away from 21n, where
ln is the range of the first group of flexural-mode eigenfrequencies; altematively,
upstream turbulators may be uti1ized to suppress parametric resonances,
(iii) The turbulence-induced ('subcritical' or buffeting) vibration amplitude is com-
puted, to ensure that no intercylinder impact occurs, or that this impact does not
exceed fretting criteria - or, for that matter, that the vibration does not exceed
fatigue criteria. The turbulence-induced vibration amplitude may be estimated
either semi-empirically or analytically - remembering that the re1iability of pre-
diction may be as low as within one order of magnitude, as discussed in Section 9.4.
(Clearly, more work needs to be done here to improve reliability of prediction. t )
Finally, in Section 9.5, the dynamics of so-called strings of c1ustered cy1inder bundles
has been examined, The dynamics is qualitatively similar but more complex. For vertical
systems with a free end, gravity-induced buckling may arise, resulting in accrued 1iability
to fretting,
tlt is recalled that an unscheduled shutdown of a 400 MW nuclear power plant (say, to replace overfretted
fuel or adjacent components) results in lost revenues of the order of US$ I million per day.

10
Plates in Axial Flow


10.1 INTRODUCTION


The dynamics of plates and panels (generally curved plates) in ftow is a huge topic,
covering an important part of the field of aeroeZasticity. However, virtually nothing is
said specifically on the aeroeZasticity oj pZates in the classical books on aeroelasticity,
for example Bisplinghoff et aZ. (1955) and Fung (1955). For that, one has to tum to more
specialized texts, for example, Dowell (1975, 1995), Df particular interest here is Dowelľ s
(1975) excel1ent monograph, AeroeZasticity oj PZates and Shells, which, apart from the
difference in pub1ication date, should in principle cover exactly what one would expect to
find here. This is not exactly trne, however, because in Dowelľs work strong emphasis is
placed on aeronautical applications and more particularly on strongly compressible and
supersonic ftows, which, after all, is what gave the subject its main impetus: applications to
high-perfonnance aircraft and missile skins. t The same is trne ofLibrescu's (1975) book
which, additionally, stresses anistropic plate and shell problems, as well as in Bolotin's
(1963) and Buivoľs (1975) books, Here, the emphasis is on incompressible and low-
Mach-number subsonic ftows.
In addition, of course, there has always been interest in the hydroelasticity of plates [see
e.g. Naudascher & Rockwell (1994)], motivated specially by applications to submarine
skins, and more particularly to boundary-layer transition to turbulence and noise gener-
ation, as already remarked in Chapter 7. An excellent, compact review may be found in
Lucey et aZ. (1997). Furthennore, new app1ication areas, such as paper making and wind-
ing, high-speed tape winding and spinning disks in disk -drives, have added new dimensions
and particularities to the topic. Finally, there has been a steadily increasing interest in the
dynamics of plates of composite materials and, more recently stilI, smart materials, which
is beginning to spill over to such plates in axial ftow, thus to the technological1y important
application areas.
The net result of all the above is that the literature on the subject is quite enonnous.:j:
To do justice to it all, just as for the topics of Sections 7.9 and 8,9,10, requires a book on
its own. Faced with this problem, the author has opted for this solution: (i) to present in
Section 10.2 a general, simplified analysis, making the necessary links to the methods and
physical behaviour presented in Volume 1 and in Chapters 7 and 8 here, (ii) to present some
representative studies, in Section 10.3, stressing the kind of results obtained, and (iii) to


tpanel flutter occured on the skin of early Satum rocket flights that eventually put a man on the Moon in the
early 1970s,
+The reference list in Dowell (1975) has 102 publications, while Komecki (1978, 1979) has 131, in each
case not covering everything; obviously, more has appeared since then! A conservative estimate would be a
total of 300-400 papers. In a recent review by Mei et aJ. (1999), exclusively on nonlinear panel flutter in
supersonic and hypersonic flows, 83 references are cited.


1137




1138
SLENDER STRUCTURES AND AXIAL FLOW
give a reasonable list of references in each subsection of Section 10,3, which, together
with the reference s just cited, would be required reading for anyone wishing to do further
work on this topic.
Concerning the plate itself, the simplest possible equation for l-D motion is utilized
widely, namely
8 4 w a 2 w
V 8x 4 + pph 8t 2 = O (10.1)
for a plate vibrating freely in vacuum, in which V = Eh 3 /[12(1 - v 2 )] is the flexura1
rigidity, E being Young's modulus, h the plate thickness and v the Poisson ratio; Pp is
the plate density, so that pph is the plate mass per unit area. The reader is referred to
Timoshenko & Woinowsky-Krieger (1959), Leissa (1969), Soedel (1993) or Ventsel &
Krauthammer (2001), for instance, for more accurate and more particular equations of
motion, including pertinent equations for nonlinear motions. In this regard, similarly to
shells, nonlinearities begin to play a role for amplitudes of deflection of the order of the
plate thickness.
For free osci1lations in quiescent jluid, this equation becomes
8 4 w 8w 8 2 w
V 8x 4 + Cdfit + p p h(1 + I-q) 8t 2 = O,
(10.2)
where JL = pa/(pph) [cf. equation 7.49], P being the fluid den si ty, and a a characteristic
plate dimension (e,g, the radius of a circular plate or the width of a rectangular one); y is
the so-ca1led nondimensiona1 added virtua1 mass incrementa1 factor (NAVMI factor)-
see, for example, Kwak & Kim (1991), Amabili & Kwak (1996) and Section 10,3.1;
cf. equation (7.25)],t ln genera1, y depends on the shape of the plate and the mode of
vibration. Cd is a damping coefficient, representing in a simplified manner both dissipation
in the plate and viscous fluid damping.
An a1ternative form of (10.2) is
a 4 w 8w 8 2 w
V 8x 4 + Cdfit + pph 8t 2 = !:!.p,
(10.3)
where !:!.p is the differential pressure on the two surfaces of the plate due to its motion
w(x, t). For a quiescent fluid, !:!.p would effectively yield -p p hJLy(8 2 w/8t 2 ), where
pphJL y is the added mass per unit area; but this equation also applies to jlowing jluid, in
which case !:!.p represents the total, coupled fluid-dynamic forces (including added mass
effects).
These equations are appropriate for infinitely wide plates. For plates of finite width
("3-D plates"), a 2-D model should be used, and the equivalent of equation (10.1) is
( 8 4 W 84w 8 4 W ) 82w
V 8x 4 + 2 8x 2 8y2 + 8 y 4 + pph 8t 2 = O,
(10.4)
and the other equations are similarly modified.
The nonlinear equations of motion are discussed in Section 1O.3.7(a).
tThis treatment of the added mass is approximate, but it is a good approximation; see Section 10,3,1.

PLATES IN AXIAL FLOW
1139
10.2 GENERAL ANAL VSIS
10.2.1 Infinite plate in incompressible fluid flow
The presentation here follows that of Komecki (1978) dosely, which itself is a synthesis
based on earlier work by Miles (1956), Dugundji et aJ, (1963) and others.
10.2. 1 (a) General formulation
The plate is considered to be infinite, both in width and length, supported by an elastic
foundation, and subjected to a flowing fluid on both sides, as shown in Figure 10,1.
Its equation of motion is a modified form of (10.3), namely
a 4 w a 2 w aw a 2 w
V- 4 - T- 2 + Kfw + Cd- +m p - 2 = p - p,
ax ax at at
(10.5)
where T is the tension in the plate, Kf the spring stiffness of the elastic foundation,
and p and p are the perturbation pressures as shown in the figure, It is reca11ed that
V = Eh 3 /[12(1 - lJ2)] is the flexural rigidity, mp = pph the mas s per unit area, and Cd
a damping coefficient.
We consider perturbations w (x, t) of the plate from the stretched-straight equilibrium
state along the x-axis, in the form of a travel1ing wave
w(x, t) = Re {w o e-ikx+iQt} == ne {w o e-ik(X-Ct)} ,
(10,6)
where the amplitude wo, the phase velocity c and frequency Q are generally complex, and
the wavenumber k real, positive or negative (related to the wavelength by Ikl = 2il' IA).
We next substitute (10.6) into (10,5) and obtain
mpk2[c - c 2 + i(Cdlmpk)c]w = p - p, (10,7)
where
c = m;1(Vk 2 + T + Kfk- 2 ),
( 10,8)
Ul
'I
I
z = W (x, t)
.......
--
/
Figure 10.1 A plate in axial flow; geometry and some of the parameters used (Komecki 1978).

1140
SLENDER STRUCTURES AND AXIAL FLOW
Co being the phase ve10city in the p1ate in vacuo (in the absence of damping a1so), obtainab1e
by substituting (10.6) into (10.5) with p; = P2 = Cd(aW/at) = O.
The perturbation pressures are determined by potentia1 flow theory in this simplified
ana1ysis, The potentials are expres sed as
Wj(X, z, t) = UjX + <I> j(x, Z, t),
j = 1,2,
(10.9)
where j = 1 app1ies far z > O and j = 2 for z < O, and where <1>1 and <1>2 must satisfy
a 2 <1> . a 2 <1> .
J + J O
ax2  = ,
and (i) the impermeability condition,
a <I> j I = aw + Uj aw ,
az Z=Zw at ax
j = 1,2;
(10,10)
(10,11)
with Zw = :I:!h + w(x, t) :::::: O for a very thin p1ate executing very small motions (cf.
Section 7,2.3), and (ii)
a<l>1
- = O at z  00;
az
a <1>2
- = O at z = - H.
az
(10.12)
Using Bernoulli's equation and proceeding as in Section 7.2.3, one eventually finds
* ( a<l>j a<l>j ) I
p, =-p -+U,- ,
J at J ax
z=o
(10.13)
App1ying boundary conditions (10.11) and (10.12), one obtains
pi = - [Plkl(Ul - c)2] W,
P2 = [pk(U2 - c)2 Coth(kH)] w.
(10,14)
Hence, equation (10.7) becomes
c 2 - c6 - i(Cd/mpk)c + J-L[(U1 - c)2 + &(U2 - c)2] = O,
(10.15)
where
J-L = p/(Ikl mp),
a = coth(2]'(H/)"'),
(10,16)
in which k = 2]'(/)", is treated as an independent system variab1e.
Before proceeding further, it is noted that, by means of (10.6), the first of equa-
tions (10.14) may a1so be written as
P [ a 2 w a2w a 2 w ]
P *-- -+2U 1 -+U2-
1 - Ikl at 2 ax at 1 ax 2 '
(10,17)
whereby the similarity to the perturbation pressure expressions for internal flow in a pipe
and externa1 axial flow over a cy1inder (Chapters 3 and 8) and in ar over a shel1 (Chapter 7)
becomes crysta1 clear.

PLATES IN AXIAL PLOW
1141
10.2,1(b) Stability, critica! wavenumber
We may now proceed to examine stability of the plate. If Im(Q) == Im(kc) < O, the
system is unstable. !f, additionally Re(Q) = ° when Im (Q) < O, the in stability is of the
divergence type; otherwise, if Re(Q) =1= 0, it is of the flutter type. t
Taking Cd = ° and solving (10.15), we obtain
f.-t(U1 + &U2) =r= [c5(1 + f.-t + f.-t&) - f.-t(Uf + aUl) - f.-t2&(Uf - Ui)]1/2
Ci,2 = 1 + f.-t + f.-t&
(10,18)
Below the threshold of divergence, Ci :::; ° and C2 > O. Divergence occurs when Ci vanishes,
at U = Ued, For U > Ued, no signals can propagate upstream (unattenuated), Flutter occurs
by coalescence of the two phase velocities, such that Ci = C2 = cef' at U = Uef; thereafter
the phase velocities become complex.
For Ul = U2 = U, we have
U d _ co
e - [JL(1 + &)]1/2'
Co[f.-t(1 + &)]1/2
cef = [1 + f.-t(1 + &)]1/2'
As & increases with decreasing H, it should be realized that the system is destabilized by
proximity to the lower, fixed boundary.
For Ul = U, U2 = O, we have
co[1 + f.-t(1 + &)]1/2
U f -
e - [JL(1 + &)]1/2
(10. 19a)
co co[1 + f.-t(1 + &)]1/2
Ued = "fii ' Uef = [f.-t(1 + f.-t&)]1/2 '
"fii
cef = {(1 + JL&)[1 + f.-t(1 + &)]}1/2'
For Ul = U, U2 = ° and pz = 0, we have
(lO,19b)
co
Ued = "fii ,
co(1 + f.-t)1/2
Uef = "fii ,
"fii
cef = (1 + f.-t)1/2'
(10.19c)
Similar expressions may easily be obtained for the case Ul = 0, PÍ = O, U2 = U.
In all cases, it is evident by inspection that Ued < Uef' so that divergence occurs
prior to flutter. (For Cd = O, the system is neutrally stable beyong the critical divergence
speed, until the flutter speed is reached. Paradoxia1ly, however, for Cd =1= O, the system is
dynamically unstable for U > Ued; see Section 10.3.3.)
A typical plot of C versus U for the system leading to equation (lO,19c) is shown in
Figure 10,2. Divergence occurs at point D, and flutter at F. For U > Vef' c becomes purely
imaginary: one branch ofthe locus with Im(c) < ° and the other with Im(c) > O.
A very compact way of presenting Ued and Uef versus JL is il1ustrated in Figure 10.3.
It is stressed that the ordinate is CO/ Ve, and hence Ucd is indeed below Uef. AIso, the
abscissa changes at JL = 1, to 1/ JL, to cover the range to f.-t -+ 00.
tS ee Figures 7,70(a), 7,108 and 8,34, for instance, and the discussion pertaining thereto,

1142
SLENDER STRUCTURES AND AXIAL FLOW
1.0
0.5
'"
 0,0
-0.5
-1.0
0.0
1.0
2.0
3.0
U/co
Figure 10.2 The dependence of c on U, both nondimensionalized by co (Komecki 1978).
15
10
I
I
I
/
/
/
/
".
,,/
,,/
/"
,,/
,,/
/"
//
,,/
/
,/
//
,,/
".
/
".
".
/
/
/
/.
I.
I.
Flutter
Divergence - - - --
Ul = U, U 2 = O
t::t
'"
"
05
Unstable
O
O
f/-
1If/-
O
Figure 10.3 The critical flow velocity U c versus f.1, == p/(km), where U c = Ucd for divergence
and U c = Ucf for flutter, for the case Ul = U, U2 = O (Figure 10.1). Note that the ordinate is
(Uc!cO)-I; (Komecki 1978),
The dimensionless f.L and ff are clearly functions of k (and >..), and in Figure 10.3 both
co and f.L are thus functions of the wavenumber; hence, so are Ued and U e !, The question
naturally arises as to which wavenumber (or wavelength) is critical. This is addressed
according to two different viewpoints, considering that (i) the critical wavenumber, ke,

PLATES IN AXIAL FLOW
1143
corresponds to the 10west critica1 flow ve1ocity; or (ii) it corresponds to the most severe
in stability, namely that with the largest negative Im(Q).t
We consider the second criterion first. For U > Uef' we norma11y have a complex phase
velocity, c = CR + iC[, For the case of Ul = U2 = U and ér = I(H -+ (0), we have
2f.L(U 2 - c 2 ) - c 2
[Im(Q)]2 == (kc )2 = k 2 o o
I (1 + 2f.L)2
(10.20)
Plotting Im(Q) versus k, one can find the required maximum, and hence ke, For
example, for a trave11ing sheet of paper with vanishing Vand K f, such that c5 = T / m =
158 m 2 /s 2 and f.L = 16.4/ k with k in m- l units (Komecki 1978), it is found that ke =
16,7 m- l , and hence Ac = 0.37 m.
For the first criterion to be satisfied, we must have aUed/ak = O or a U cf lak = O, with
Ued and Uef as in equations (10. 19a,b,c). In this way, one obtains Aed and Aef. Considering
the case Ul = U, U2 = O and (10.19b) with ér = 1 and T = O, one obtains
Aef  Aed = 2n(3V/Kf)1/4,
(10,21)
The two criteria genera11y give different answers. Of more interest, however, is to note
that if Kf = O, then Aef = Aed -+ 00. Moreover, since f.L = p/(km p ), Ued = Uef = O!
This at first sight paradoxical finding is in fact quite reasonable, if one considers that the
plate is entirely unsupported and infinite1y 10ng! Here refer to the discussion on wave
propagation in pipes (Section 3.7.2),
Komecki (1978, pp. 407-412) a1so considers the effect of damping by obtaining
solutions to the full form of equation (10.5). The roots in this case are complex; for
U2 = O, pz = O, they are
1 [ . k ]
q,2 = 1 + f.L (f.LU =F Co Lq) + likTco(d =F 2) ,
(10,22)
where
1 { 1/2 } 1/2
1,2= ..ti (u2+4d2f.L2(U/cO)2] ::I:: u ,
u = 1 + I-L - f.L(U /cO)2 - d 2 , d = Cd/(2mplklco).
The critica1 flow ve10cities for d = O, Ued and Uef' are given by (1O.19c). The critical
velocity for divergence remains unaltered. However, there is a profound difference in the
post-divergence dynamics, which may be summarized as follows:
(i) for U :S Ued,
Re(q) :S O, Re(c2) > O;
Im(kq)  O, Im(kc2) > O;
tThe question of a critical wavenumber should strictly be addressed by means of more sophisticated,
nonlinear theory.

1144
SLENDER STRUCTURES AND AXIAL FLOW
(ii) for Ved < V ::::: Ve!,
Re(c2) 2: Re(q) > O;
Im(kq) < O, Im(kc2) > O.
(10.23)
It is therefore c1ear that there is something like a 'single-mode' flutter that arises as soon
as Ved is exceeded, prior to the 'coupled-mode' flutter at the va1ue of Ve! quoted above.
This is related to the destabilizing effect of damping, as discussed in Chapter 3 for pipes
conveying fluid; see a1so Appendix C and Grotberg & Reiss' work on para1lel-plate model s
of pulmonary passages discussed in Section 7.9.5(b), subsection b.3 - the latter being
very dosely connected to this work, Refer a1so to Section 10.3.3. A more sophisticated
analysis is given by Carpenter & Garrad (1986).
The important thing, from the physica1 point of view, is that for a1l V > Ved the
system is unstable by flutter, and this instability is independent of the va1ue of d (and
hence Cd).
In addition to Carpenter & Garrad (1986), the reader is also referred to other papers on
compliant wa1ls and coatings (Section 7.4.1); for example, the Gad-el-Hak (1986,2002)
and Carpenter ef a/, (2000, 2001) reviews.
10.2.1(c) Other configurations
Other configurations may be ana1ysed in the same simple manner. Figure 10.4 gives some
oť them, but no detail s are given here in the interests of brevity.
_ z t:!:1
U -----  I x
= H
7.T.777T7 ///////// 
(a)
U
:-- zl _--__ ,
h- -___- -z> LS --___- n X
f-- I --I
(b)
- ---- --
lx
H
L
Zl
- '\J ____ '\J------.!Z..
1-----
-----
...Z),--------___
! :x
H
l
U

(c)
(d)

l--j
U _-- -==:----:
u--':' - ----
-h---__-Á----- c,,'
r '
H x
j
/:}:i//////l .
SJ= J-___- k x
j
TT'/ TTT/777 / / / / / / / / / / / / .
(f)
(e)
Figure 10.4 Other systems that may be analysed by the same methods as used in Section 10.2.1
(Komecki 1978),

PLATES IN AXIAL FLOW
1145
10.2.2 Plate of finite length in incompressible flow
10.2.2(a) Unconfined f10w, simply supported plate
The presentation here follows dosely Dugundji et al. (1963), Kornecki (1971a, chapter 9)
and Kornecki et al. (1976). The system under consideration is shown in Figure 10.5,
involving a baffled t plate of finite length and infinite width in axial flow. Such plates are
often referred to as two-dimensional in the sense that the third dimension, y, is either
infinite, or, in any case, there are no variations in displacement in that direction, The
fluid forces are deterrnined via potential flow theory; hence, drawing on the work in
Section 10.2.1, the governing equation is
04W 02w * ( 0<1> 0<1» I
1J-+K f w+p h-=-p =p -+U-
ax 4 p ot 2 at ax
z=o
(10.24 )
and, assurning simply supported edges, the boundary conditions are
a 2 w
ox 2 = O
w=O,
at x = O and x = L.
(10.25)
Expressing the associated unsteady flow field by means of a time-varying source distri-
bution along the x-axis for O < x < L, Bisplinghoff et al. (1955) obtain the perturbation
velocity potential in the fonu
1 {L ( ow aw ) [( X - U ) 2 ( Z ) 2 ]
cp(x, Z, t) = 2n 10 ar + U a; ln -C- + L du,
(10,26)
z
u
Figure 10.5 A plate of finite length and infinite width in axial flow (Weaver & Dnny 1970).
tA baftled plate is one embedded in a rigid wall; that is. in this case, a flexible plate with semi-infinite rigid
extensions, for x < O and x > L,

1146
SLENDER STRUCTURES AND AXIAL FLOW
where u is a dummy variable. This eventua11y leads to
* P 1 L ( a 2 w a2w ) ( X-U )
-p =- -+U- In - du
7f o at 2 auat L
pU l L ( aw aw ) ( 1 )
+- -+U- - du,
7f O at au X - U
(10,27)
The general solution ofthe integro-differential equation resulting from (10,24) and (10.27)
is difficult, and so a Galerkin-type solution is introduced by Dugundji et al. (1963),
assummg
N N
w(x, t) = L wm(x, t) = L qm(t) sin(m7fxj L).
m=1 m=1
(10.28)
The aerodynarnic pressure is considered to be that on an infinitely 10ng wavy
(sinusoidal) wan,
N
<I>(x, z, t) = L [Clm(t) sin(m7fxjL) + C2m(t) cos(m7fxjL)] !m(Z);
m=1
(10.29)
this is reasonable, except in the immediate vicinity of x = O and x = L, particularly for
high m. The velocity potential must satisfy the Laplace equation and be finite at z ----+ 00,
so that lm (z) = exp( -m7fzj L), Then, applying the impenneabi1ity condition (10.11),
-p* = p t ([ - ( :7f ) d:; + U 2 ( : ) qm]
m=1
( m7fx ) [ d q m ] ( m7fx ) }
xsin L - 2Udř- cos L '
(10.30)
Substituting this expression together with (10.28) into (10,24), we obtain
N f [ 1 ( p,* ) d 2q ] ( m7fx )
E l Bmqm + ú) 1 + m7f dt 2m sin L
( AI.l* ) 1/2 dQm ( m7fx ) J O
+2 - -cos - =,
ú) dt L
(10.31)
where
pU 2 L 3
A=
V
ú) = ( V ) 1/2
O hL4
Pp
* pL
I.l =-,
pph
4 Kf L 4
Bm = (m7f) + ----v- - m7f A.
(10.32)

PLATES IN AXIAL FLOW
1147
Applying Galerkin's procedure and introducing -r = wot, we obtain
( * ) d2 N ( 8m d )
1 +   + (AfL*)1/2" qr + Bm qm = O,
Jrm d-r 2  m 2 - r 2 d-r
r=1
(10.33)
where r + mis an odd integer. Hereafter, the ana1ysis is restricted to two modes only, m
and m + 1. Seeking soIutions of the fonn
qm(-r) = w m exp(h),
we obtain
[(1 fL* ) ,2 B ] A ' A * ) 1/2 8m A - O
+- A + m Wm+A( fL 2 2 Wm+1- ,
mJr m - (m + 1)
[ * 1/2 8(m + 1) ] A [( fL* ) 2 ] A
A (Afl ) 2 2 W m +1 + 1 + A + Bm+1 W m = O,
(m + 1) - m Jr(m + 1)
(10,34)
which yie1ds the characteristic equation in A. Divergence is found to occur when Bm = O,
thatis, for A = [(mJr)4 + (Kj L 4 jD)]jmJr, or
[ ] 1P
Ued = (  ) (mJr)4 + (KjL 4 jD)
pL3 mJr
(10.35)
This implies that the critica1 aerodynamic pressure is minimum at m = med, the nearest
integer satisfying
med  ; ( K; 4 r (10.36)
Dugundji et aZ. give an example, the results of which are in Figure 10.6, showing the
frequency versus flow ve10city (in mph, miles/hr). It is seen that Ued = 101.55 mph
(45.39 mis), corresponding to m = med = 8. The system is restabilized at U =
lO1.77mph. Soon after, coupled-mode flutterdeve1ops, at U = I03.13mph (46.10 mis).
The mode shape at the onset of flutter is shown in Figure 10.7, demonstrating c1early
the travelling wave nature of the flutter. This mode shape, a1though for a high va1ue of m,
is strikingly similar to those shown for a string-pipe conveying fluid (Figure 3.13) and for
a she11 conveying fluid (Figures 7.9 and 7.113).
The complete soIution for incompressible flow, using the full fonn of equation (10.26)
and (10.27), has been obtained by Kornecki et aZ. (1976). First, the perturbation pressure
is refonnulated into
_p*= 1 t [ L2 a2w +2UL a 2 w +U2 a2W ] Inlx_ld_U2R(X) I ,
Jr L 10 at 2 a at a2
(10.37)
R(x) = U 2 {[w'(1) + (LjU)ŮJ(1)]1n(1-X) - [w'(O) + (LjU)ŮJ(O)]1nx}; (10.38)

1148
SLENDER STRUCTURES AND AXIAL FLOW
20
5
15
N-
e;

'"
&10

10
5
10 20 30 40 50 60 70 80 90 100
V (mph)
Enlargement
Ve!
99 100
Figure 10.6 Prequencies of free oscillation, R.e(rl.) /2n, versus flow velocity for the two-mode
analysis of Dugundji et al., showing the eighth and ninth modes, and the critical flow velocities for
divergence, Vcd, and flutter, Ve!, in mph (miles/hr); 1 mph = 1.609 km/h = 0.447 mis, (Dugundji
et a/. ] 963).
I I
t=O
1 \
\ 1
t=I/8
vii
\ 1-
V\../í\./1'\/V t = 1/4
\ \
 t=3/8
1 1
\ \
 t=1/2
1 1
 t=(5/8)T
Figure 10.7 Mode shape ofthe plate at the threshold offlutterfordifferentparts ofthe cyc1e; Tis
the period (Dugundji et al. 1963).

PLATES IN AXIAL FLOW
1149
in which the dimensionless X = x j L,  = u j L have been introduced, but tis dimensional;
()' = a( )jax, (') = a( )jat. The integrand on the right-hand side of (10.37) resembles
the well known expression for the perturbation-pressure-related forces on a pipe conveying
fluid [see, e,g" equations (3.1) and (3.2)]; the function lnlx - I may be viewed as the
effect of spatial memory. By analogy then, it is quite reasonable to expect the plate with
supported ends to lose stability by divergence, and the cantilevered one (Section 10.3.6)
by flutter.
Hence, for the determination of divergence of plates with simply supported ends, using
equation (10.37) and taking Kf = O, equation (10.24) may be written as
D d 4 w pu 2 11 1 d2w _ , -, _ )
- - - - -lnlx - Id + w (O) ln x - w (l)ln(1 -x) = o,
L 4 d:x4 Jr L o d2
(10.39)
subject to boundary conditions (10.25). Invoking again the analogy with the pipe problem
suggests that p* should be expressed in terms of local curvature, d 2 w j d2. Komecki et al.
thus introduce a new variable 1J =  - x and expand the integral in equation (10.39) in a
Taylor series about 1J = O,
l 1 nw 00 w
- d l: 2 ln lx - Id = L A n (x)----=t=2,
Jr O " n=O d:f'1
(10.40)
where
1 I I-x 1 {
An(x) = - 1Jn ln 11Jld1J = (1 - X)n+I
Jrn! -x Jr(n + 1)!
x [ ln(1 - X) -  ] - (_X)n+I [ ln X -  ] } .
n+l n+l
Substitution of (10.40) into (10.39) transforms the latter into an ODE with an infinite sum
of derivatives, easier to handle than the original integro-differential one:
d 4 w -2 1 00 d n + 2 w 1 )
=4 - U LAnW _ +2 + - [w'(O)lnx - w'(1)ln(1-X)] = O,
dx n=O dx n Jr
(10.41)
where the dimensionless flow velocity U is defined by
U 2 =pU 2 L 3 jD;
(10.42)
U 2 is often denoted as A 2 or A 2 and is called the dimensionless dynamic pressure.
Proceeding then to obtain a solution of (10.41) by Galerkin's method with a single
normal mode, w(x) = W sinJrx, one obtains
Jr3 -2 1 2 (I 00 )
2 - U -; 10 (ln X) sinJrx dx + Lan = o,
n=O
(10.43)

1150
SLENDER STRUCTURES AND AXIAL FLOW
Table 10.1 The critical flow velocity for divergence, Ued, expressed as pU;:dL3/(Vn3) ==
U; d/n3: Komecki's (1971a) comparison between Ishii's and Weaver & Unny's theories with that
of Dugundji et al" with more recent results for m = 1 by Ellen and Komecki et al. added,
Reference
Number of antinodes m = med
2 3
1.00 8,00 27,0
1.35 9.41 31.6
1.30 8.4
1.29
1.32
Dugundji et al. (1963)
Ishii (1965)
Weaver & Unny (1970)
Ellen (1973)
Komecki et a!, (1976)
where
a n = nn [I [ xn+1 (  -ln X ) f(x) ] dx,
(n + I)! 10 n + 1
in which f(X) = (_l)n/2(1 - cos2nX) for n = O, 2, 4,..., and f(x) = (_l)n+I/2x
sin 2nx for n = 1, 3, 5, . . , , Evaluation of the integrals while retaining five terms in the
summation in (10.43), yields U; d = 40,8, a little higher than that in equation (10.44b) in
what follows for m = 1 and KJ = O; see Table 10.1.
Weaver & Unny (1970) have studied the same system but with water flow t obtaining
solutions by a Fourier integral approach (cf. Section 7.2.5). The results are qua1itative1y
similar, but the gap between the flutter and divergence boundary is wider. Weaver & Unny
give the following simple expression for first-mode divergence:
Ued = I.14n3/2 ( p3 ) 1/2,
which is reminiscent of expressions for the critical velocity of a simply supported pipe
if V j p L 3 is replaced by E I j M L 2 . It is pointed out that this differs but a 1ittle from
Ued = n 3 / 2 (V j p L3)1/2 obtained in 1960 by Flax and by Dugundji et a/. (1963).
Komecki presents a comparison between results obtained by Dugundji et a/.
[equation (10.35) with KJ = O] and Weaver & Unny, on the one hand, and those of
Ishii (1965) who uses the complete form of the velocity potential, equations (10.26)
and (10.27), on the other, reproduced here in Table 10.1, but with some newer results
added. It is seen that the relative discrepancy between Ishii's 'exacť solution and that of
Dugundji et a/. decreases with increasing med. The Weaver & Unny solution is doser to
Ishii's. The most accurate solutions are considered to be the more recent ones.
Both Weaver & Unny's and Ishii's solutions have been criticized in a remarkable paper
by Ellen (1973), and later by Matsuzaki & Ueda (1980).
Employing asymptotic expansions of the generalized pressure integrals, Ellen (1973)
obtains solutions for subsonic compressible flow and low reduced frequency, D..Lja
(a being the speed of sound), app1icable not only to two-dimensional (infinite aspect
(l0.44a)
tS ee also Kantor & Kvitnitskii (1988).

PLATES IN AXIAL FLOW
1151
ratio, ARt) panels, but also capable of yielding information on the effect of AR on the
critical ftow velocity. The ana1ysis is very laconically presented and, a1as, too complex to
give here in detail.
E11en proves that there is an error in the lower limit of one of the integrals in the Weaver &
Unny analysis. Likewise, he finds errors in Ishii's analysis, However, in both cases, the
predicted threshold for divergence is unaffected! Remarkably, E11en is able to obtain a
dosed form solution for Ucd, namely
pU 2 L 3 n 5
(1 - ;)1/2V = 2[nSi(n) _ 2] :::: 40, (1O.44b)
where Si(z) = f(sinf)d, which for Mach number M == U fa = O is very dose to
(1O.44a); cf. Table 10.1.
The post-divergence dynamics, however, is affected. Thus, unlike Weaver & Unny's pre-
dictions, E11en finds that ftutter occurs as soon as the divergence threshold is exceeded -
which is dose to what Dugundji et al, find (see Figure 10.5). t
Matsuzaki & Ueda (1980) re-exarnine the problem and extend E11en's analysis, with no
strong limitation imposed on the reduced frequency, obtaining a very rigorous solution.
They give a very valuable discussion ofthe problem encountered in a purely incompressible
ftow ana1ysis for odd modes, for example, m = 1 in equation (10.28), because the so-ca11ed
radiation condition,
1im<l>(x,z,t)=O as J x 2 +Z 2 -+oo,
(10.45)
is violated. This problem is circumvented by Weaver & Unny, for example, by considering
a nearly incompressible solution, thus avoiding the boundlessness of the pressure integral
for M = O. However, Matsuzaki & Ueda confirm the zero-order error in one ofthe integrals,
as found by E1len and as a1ready discussed, They then exarnine Ishii's ana1ysis and compare
numerically the generalized forces therein with their own, reaching the conc1usion that (i)
Ishii's analysis is correct for compressible ftows, but (ii) it is incorrect for incompressible
ftow and odd wavenumbers m, because it violates the radiation condition.
A number of additional comments are made here, in point form for the sake of brevity.
(i) In compressible but subsonic ftow, the dynamical behaviour is similar to that just
described, (ii) ln supersonic ftow, divergence is ruled out and the system loses stability
by ftutter (Kornecki 1974). (iii) ln incompressible or subsonic ftow, the post-divergence
dynamical behaviour of the system must rea11y be assessed by a nonlinear model, as often
stated in connection with other systems,
A comparison of finite- and infinite-length analytical results for the case of finite-width
plates is presented in Section 10.3.2,
Experiments have also been conducted (Dugundji et al. 1963; Ishii 1965). These are
difficult experiments, because the plates are very thin, and ensuring flatness and freedom
from imperfections and locked-in stresses is problematic.
ln the Dugundji et al. (1963) experiments, aluminum plates (L = 2540mm, width
b = 610 mm, h = 0.46 mm) were used on an elastic foundation in air-ftow, generally at
a nonzero angle of attack. Considerable and troublesome static deftections of the panel
tIt is recalled at AR = span/chord = (plate width)/(plate length).
+This is likely because Weaver & Dnny consider water flow, whereas the others air flow - cf. shells
conveying fluid (Chapter 7).

1152
SLENDER STRUCTURES AND AXIAL FLOW
z
u
-
H
U///t////,//J,U/ 1
x
Figure 10.8 A plate of infinite width and finite length in confined flow (Guo & Pai'doussis 2000a).
developed, much below the flutter threshold; whether they were a developing divergence
is not c1ear,t The flutter amplitudes were of the order of A = 12 mm, so that Aj h was
large. Some aspects of the experimenta1 results, notably the flutter speed, were in good
agreement with theoretical predictions,
Ishii (1965) used alullŮnum plates (L = 300 mm, b = 68 mm, h = 0,04 - 0.17mm)
under tension in air-flow. Good agreement with theory was obtained for the divergence
threshold. In some cases, first-mode-shape divergence was succeeded by S-shaped diver-
gence, and then by flutter [cf. Section 8.3.2(c»). Two kinds of flutter were observed:
(i) a high-frequency 10w-amplitude flutter for the thinner plates, and (ii) a 10w-frequency,
higher-amplitude flutter for the thicker plates that is rellŮniscent of osci11atory divergence,
The reader is also referred to Lucey & Carpenter (1992,1993), Lucey et a/, (1997) and
other papers dealing with the stability of comp1iant coatings.
10.2.2(b) Confined flow; effect of boundary conditions
The same problem as in subsection (a) is considered by Guo & Pai'doussis (2000a), that
is, a plate of infinite width and finite length, but in confined flow, as shown in Figure 10.8.
Some interesting observations on the effect of boundary conditions are made.
Theequation ofmotionis equation (10,3) with Cd = Oand w(x, t) = w(x) exp(H'2t). A
potential flow analysis is considered, so that the perturbation flow must satisfy V 2 <1> = O,
such that Ap = -2p[(a<l>jih) + U(a<l>jox)]z=o is the perturbation pressure difference
across the plate, The equations are expres sed in dimensionless fonn using the following:
* pL
 =xjL,  =zjH, H = HjL, J-t =-,
pph
Q = ( P;h Y/2 QL2, U * = ( P;h Y/ 2 UL,
U = U * J-t*1/2, T = ( Q jQ)t, w ()eiTIT = w(x, t)jL,
- ( p h ) 1/2 - L3
q;(, Oe iQT = ; <I> (x, z, t), p()eiQT = V Ap.
(10.46)
tHow exactly this squares with Ellen's finding that ftutter follows immediately the onset of divergence is
not c1ear. However, the physical model differs from the ideal one analysed theoretically.

PLATES IN AXIAL FLOW
1153
Hence, the dimension1ess governing equations in the frequency domain are:
d 4 w -2_
d4 -Q w =-p,
( - -*(Jq; ) I
p = -2f.L* iQq; + U - ,
(J =o
(10.47)
(J2q; 1 (J2q;
-+---0
(J2 H 2 (J2 - .
(10.48)
The boundary conditions for q; are
q; 1 =0,
-+:J:oo
(Jq; ! -O
(J =1 - ,
(10.49)
1 (Jq; 1 .-- -*d w
=- =lQW+U-,
H (J =o d
[ - -*(Jq; ]
p = -2f.L* iQq; + U - = O,
(J =o
0::::::1,
(10.50a)
<O and >l.
(10.50b)
Solutions are obtained via Galerkin's method; thus,
M
w () = L Am wm (),
m=l
(10.51)
in which W m () are the beam eigenfunctions satisfying the appropriate boundary
conditions at the 1eading and trai1ing edges of the p1ate.
The perturbation pressure is determined next. Taking the Fourier transform of q;(, n,
the second of equations (10.48) and (10.49) are (1/ H2 )((J2ip/(J2) - a 2 ip = O,
(Jip/(JI=l = O, where ip(a, n is the Fourier transform of q;(, n and a is the transform
variab1e. Solving these and taking the inverse transform, we obtain
q;(, n = i: B(a) [e-Hlal + eHlal(-2)] eia da,
(10.52)
where B(a) is a function to be determined. Inserting this into the first of equations (10.48)
we obtain
p() = i: B(a) [ -2f.L* i( Q + a U *) (1 + e- 2Hl al)] eia da.
(10.53)
Substituting q; and p into (10.50a,b) 1eads to a pair of dua1 integra1 equations in B(a). By
using dp/d == p'() as the basic unknown, however, the two may be reduced to one;
taking its inverse Fourier transform and incorporating (10.50b), we obtain
B(a) = [4JTf.L*a( Q +a U *) (1 +e- 2H lal)rl11 p'()e-ia d.
(10.54)

1154
SLENDER STRUCTURES AND AXIAL FLOW
Now, from (10.52) and (10.54), we have
1 acp l = / 00 B(a)lal ( -I+e-2H1al ) eiada= tN(,1])p'(1])d1], (10.55)
H a =O -00 10
where
N( I: ) _ _ / 00 lal tanh( H Jal) ia(-IJ) d
s,1] - _ _* e a,
4JtJL* -00 a(Q+aU )
Substituting (10.55) into (10.50a) yields an integral equation for p' ():
(10.56)
1 1 N(, 1])p'(1])d1] = i Qw () + u * w' (),
0.:::::::1.
(10,57)
Multiplying this by i Q + U* (d/d) leads to
11 G(, 1])p'(1])d1] = _ Q 2 w () + i2 Q u * w'() + U *2 w" (),
(10.58)
in which
G(, 1]) = _ / 00 lal tanh(Hlal) eia(-IJ) da
4Jt JL * -00 a
1 { I 1 00 - }
= - - + [tanh(H a) - 1] sin [a( - 1])] da .
2JtJL*  - 1] O
(10.59)
Equation (10.59) is a Cauchy-type singular integral equation, from which p' () may be
obtained numerically by the Gauss-Chebyshev method (Erdogan & Gupta 1972), and
hence p().
The perturbation pressure may be expressed as
p() = _ Q2 p(M) () + i 2Q U * p(G) () + U *2 p(K) (),
(10.60)
where the three components of p() are obtained by retaining on the right-hand side of
(10,58) tenns associated with w (), w '() and w "(), respectively; the superscripts M, G
and K stand for mass, gyroscopicity and stiffness, respectively.
Substituting next equation (10,51) and (10.60) into (10.47), we obtain
M
L [_ Q 2 (M:!,} + M!?) + i2 Q U * G) + K:!,} + U *2 K)J Am = O,
m=1
r=I,2,...,M,
(10.61)

PLATES IN AXIAL FLOW
1155
in which
M;:) = 11 Wm () W r() d,
G) = 11 pG)() W r() d,
K;,r;) = 11 pK)(O w r(O d,
where the superscripts pand f stand for plate and fluid.
Calculations with M = 10 in (10.61) have been conducted to obtain the eigenfrequenc-
ies, even though quite good accuracy is obtained even with M = 1, Typical plots of the real
andimaginarypartsoftheeigenfrequencies, Q R and Q ], ofthe fust twomodes for(a)c1am-
ped-clamped, (b) c1amped-pinned and (c) free-free plates are shown in Figure 10.9, The
notation for the boundary conditions follows that in Chapters 3 and 8, for instance; thus a
c1amped-pinned plate is one c1amped at  = O and simply supported at  = 1.
It is seen in Figure 1O.9(a) that the c1amped-clamped plate loses stability by divergence
at U ;d ::::: 8; after a brief restabilization it loses stability at U ;f ::::: 17 by coup1ed-mode
flutter. The behaviour of pinned-pinned plates is similar but the critical flow velocities
are smaller; for large H , U cd = U ;d IL *1/2 ::::: 3.3. This, when sca1ed by a factor of 2 for
flow on one side only, corresponds to U d/7r3 ::::: 1.40, a little higher than the values in
Tab1e 10.1.
The free-free system [Figure 10.9(c)] loses stability by c1assical coupled-mode flutter.
It should be remarked here that the rigid-body modes in this case have been ignored,
The dynamical behaviour of the c1amped-pinned plate [Figure 1O.9(b)] is broadly
similar, but for the fact that there is a small amount of flow-induced damping for all
U * > O, most easily visible in the Q ] curve associated with the second mode. If the
boundary conditions are reversed so that the plate is pinned-clamped, then the corres-
ponding Q [ changes sign (not shown); thus, the system is predicted to be unstable by
flutter for infinitesima11y small U *.
This behaviour is reminiscent of that shown in Figure 7.38 for shells conveying fluid,
which has subsequently been shown to be incorrect (Section 7.3.4). It is tempting,
therefore, to suppose that this finding also is a numerical artefact, for example due to
insufficiently accurate comparison functions, W m (). There is, however, a difference. In
the simple theory used to compute the pressure loading on shells, the added mass is con-
stant for any given (n, A) combination. For the case of a pipe conveying fluid, this is
even c1earer: the mass involved is the mass per unit 1ength of the fluid conveyed. Thus,
one obtains for the work done on the system,  W = 0, equations (3,6) and (7.123),
respectively, for pipes and shells with supported ends.
For these aforementioned systems, more fundamenta11y, the local fluid forces depend
only on the local displacement. This is the device a110wing us to use a constant added
mass. t For the plate problem, however, the equivalent of the added mass depends on the
whole tlow field (in the domain), cf. equation (10.37), and also on the particular component
ofthe perturbation pressure considered, as in equations (10.60) and (10.61). Refer also to
Section 4.4.4, for instance, and Blevins (1990),
M;,r;) = 11 pM)() Wr () d,
K;:) = 11 w ;;:'() W r() d,
tFor the shell problem the added mass is not constant but, still, for given (n, .1..) pair it is.

1156
50
40
30
10:'" 20
Icf 10
O
-10
O
(a)
40
30
laJ 20
10:'" 10
O
-10
-20
O
(b)
60
40
laJ
10:'" 20
O
-20
O
(c)
SLENDER STRUCTURES AND AXIAL FLOW
1,2
..,.....",..,',!.:.......................................
...,...."..,.,!...................................."".
".
5
10
15
2 .'
/...."..........I.:..-- ""'. ..{
""....."""-"l""_,,_.._,,_.,,",,..,,..,,:...!.
2
4
6
8
10
12
14
2
..............
..."...'('1 +2t
(1 + 2)
1,2
'....., (1 + 2f
........,.........
5
10
U*
15
20
Figure 10.9 The real (-) and imaginary (......) parts ofthe dimensionless eigenfrequencies,
Q R and Q I respectively, of the lowest two modes, denoted by the numerals 1 and 2, as functions of
the dimensionless flow velocity U *: (a) clamped-damped, (b) clamped-pinned, and (c) free-free
plate; f-L* = I, H = 1 (Guo & Paldoussis 2000a).

PLATES IN AXIAL FLOW
1157
ln view of equation (10.58), we can express the perturbation pressure as
p() = _ Q 2 m l () w() + i2Q m2(g) V* w' (O + m 3(OV*2 w"(),
which may be re-written in dimensional form as follows:
p(x, t) = ml (x) W + 2m2 (x) U Ů!' + m3(x)U 2 w", (10.62)
where (') = oOjot and ()' = JOjJx. The rate ofwork done on the plate per unit width
by the fluid is
dW i L
- = - p(x, t)Ů! dx.
dt O
Considering periodic osci11ations, we can write
w(x, t) = We(x) cos Qt + Ws (x) sin Qt,
(10.63)
(10.64)
and thus each of the components in (10.62) has two parts; for example, for ml we have
m le and m 1s. Hence, the work done on the plate over a period of osci11ation T = 2Jr j Q is
 W = -Jr [ _ Q2 1 L (mel - msl)W e Ws dx
+ 2U Q l L (m e 2 w Ws + ms2 w wc)dx
+ U2iL (m e 3 w Ws - ms3 w; We)dX] ,
(10.65)
Ifthe upstream and downstream supports ofthe plate are the same, then one ofthe W e and
Ws is symmetric and the other antisymmetric with respectto ! L. From equations (10.58),
(10,59) and (10.62), it is concluded mel (x), "', m s 3(x) are all symmetric with respect to
 L. Hence,  W = O, and single-mode flutter cannot occur, though clearly divergence
can and does; so can coupled-mode flutter - at least according to this linear theory
(cf. Section 3.4.1).
If the supports at the two ends are different, however, then  W i O generally. Also, if
for one set of conditions, say for a clamped-pinned plate,  W < O, then osci11ations are
damped. Inverting the boundary conditions is equivalent to reversing the sign of U, and
hence from (10.65) it is clear that  W > O for the pinned-clamped plate,
It should finally be noted that the flow-induced positive or negative damping for plates
with both ends supported, if any, is very small [see e.g. Figure 1O.9(b)] and is easily
overwhe1med by any dissipative damping present, Hence, divergence remains the principal
in stability of practical interest.
Figure 10,10 shows the critical flow velocity U c d == U; d 1..1.,*1/2, expressed as a universal
result for all J,I, *, as a function of H for different boundary conditions.
10.2.3 Absolute and convective instabilities
ln all of the foregoing, and much of what is presented in Section 10.3 and elsewhere
in this book, in stability is viewed in a temporal sense, as the growth of amp1itude with
time. Thus, in equation (10.6) a rea1 wavenumber is considered, and instability (stability)

1158
SLENDER STRUCTURES AND AXIAL FLOW
Clamped-clamped
8
Clamped-pinned
6

*"
I::::'u
I 4
I::::'c
Pinned-pinned
2
06
H
Figure 10.10 The dimensionless critical flow velocities for divergence U c d == U;df.L*1/2 as func-
tions of the dimensionless height of the channel H for plates with different boundary conditions
(Ouo & Pai:doussis 2000a),
o
O
0,2
0.4
0,8
12
is assessed from the complex frequency accordingly as Im(Q) is negative (positive).
However, another possibility, especial1y meaningful if the system is subjected to periodic
forcing, is to consider a real frequency and a complex wavenumber, thereby obtaining a
spatial assessment of stability. A third possibility is to consider the response to an impulsive
force F(x, t) = 8(x)8(t), which, by Fourier transfonnation, inc1udes tbe previous two
cases, and which allows us to distinguish between absolute and convective instability.
Thus, an absolute instability is one in which the response to the impulse grows both in
time and space (eventual1y covering the whole domain); a convective instability is one that
grows as a wave packet is convected away from its source, while simultaneously being
attenuated witb time at tbe source. Figure 10.11 gives a diagrammatic view of an impulse
response due to a pulse at t = O and x = O, bounded by two rays of zero amplification in tbe
(x, t)-plane: outside the wedge, disturbances decay, while witbin they grow exponentially.
Thus, the system in (a) is absolutely unstable, while in (b) it is convectively unstable. t Itis
understood tbat these are asymptotic notions of stability, related to long-time behaviour.
This topic is important because it deals with how imposed disturbances grow in flows,
for example, when considering stability of shear layers, mixing layers and otber advecting
flows; see, for ex ample , Gaster (1968), Drazin & Reid (1981), Huerre & Monkewitz
(1985, 1990), Monkewitz & Nguyen (1987), Morkovin (1988), Chomaz et al. (1988),
Huerre (2000), The same approach. however, can also be used when considering stability
of compliant surfaces and plates subjected to axial flow (Brazier-Smith & Scott 1984;
Carpenter & Garrad 1986; Gad-el-Hak 1986; Crighton & Oswell199l; de Langre 2000,
2002; Carpenter et al, 2000, 2001). For tbe matbematical foundation (in the plasma physics
context), refer to Bers (1983). Because of the dispersive nature of the media concerned
(i.e. the phase velocity is a function offrequency), it is natural to characterize tbe growth.
tThese terms have been adopted from plasma physics; refer for example, to Bers (1983), Huerre &
Monkewitz (1985), Carpenter & Garrad (1986), Huerre (2000),

PLATES IN AXIAL FLOW
1159
....... .
....... .
........ .
....... .
........ .
........ .
......... .
......... .
......... .
......... .
.......... .
.......... .
.......... .
.......... .
........... .
........... .
............ .
.... ...... 
!i!i!i!i! : i!:!::::::::::: :::::::1..1.//
x
x
(a)
(b)
Figure 10.11 Sketches of typical impulse response for (a) an absolutely unstable system and
(b) a convectively unstable system (Monkewitz & Nguyen 1987); see also Figure 10.13.
or otherwise, of disturbances in terms ofthe group velocity, detined by c g = M:?lak. We
must also generally consider both Q and k to be complex.
ln what follows, it is very important to note that, consistently with what is presented in
Sections 10.2.1, 10.3 and 7.4, it is assumed that the wave amplitudes are proportional to
exp [-ik(x - ct)] == exp [-ikx + iQt];
(10.66a)
so, instabi1ity (exponential growth) arises when Im(Q) < O. In the papers considering
absolute and convective instabilities, however,
exp [ik(x - ct)] == exp [i(kx - Qt)]
(10.66b)
is commonly taken, so that instabi1ity is associated with Im(Q) > O.
Consider the development in space and time of intinitesimal perturbations. Thus, con-
sider the impulse response (the Green's function) G(x, t) due to an impulsive loading
f(x, t) = 8(x) 8(t) - see Bers (1983), de Langre & Ouvrard (1999). The system is such
that it is stable for V < Ve and unstable for V > Ve. The dispersion relation - as given
by equation (10.15), for instance, in which the dependence on k is implicit - may be
written as
D(k, Q) = O.
For V < Ve, G(x, t) is an evanescent wave packet - that is, vanishing both spatially and
temporally. For V > Ve, the instabi1ity is characterized by the long-time response at the
impulse point, G(O, t). If limHooG(O, t) = O, the instability is convective; the amplitied
wave packet is convected downstream. If, on the other hand, limt-+ooG(O, t) = 00, the
in stability is absolute.
Considering rays x I t = const. as in Figure 10.11, this is equivalent to considering the
long-time behaviour of the wavenumber ko along the ray x I t = O. This complex ko has
by detinition a zero group velocity,
aQ
ak (ko) = O,
and the corresponding frequency, commonly called the absolute frequency, is Qo == Q (ko).
Thus, the following criterion detines the boundary between the two types of instabi1ity:
(10.67)
Im(Qo) > convective instability;
Im(Qo) < absolute instability;
(10.68)

1160
SLENDER STRUCTURES AND AXIAL FLOW
2
2
I O O
I...,"'
I
-1 -1
-2 -2
-2 -1 O 2 -2 -1 O 2
(a) kR/U (b) k R / U
Figure 10.12 Pinching behaviour in the complex k-plane, used in the detennination of Im(Qo)
and on deciding via equation (10.68) whether the instability is connective or absolute: (a) for
úJ! = QJ/fJ2 = -0.20; (b) úJ! = -0.15; pinching occurs at úJ! = -0,1939. Here f, fJ and Q are
dimensionless fonns of k, U and Q (de Langre & Ouvrard 1999).
200 400
180 350
160
300
140
120 250
'-J 100 200
80 150
60
100
40
20 50
O O
-50 O 50 100 -50 O 50 100 150 200
(a) x (b) x
Figure 10.13 The impulse response of a fiexural pipe - the long-wavelength approximation of
two fiat plates with fiow in-between - showing (a) an absolute instability and (b) a convective one
(de Langre & Ouvrard 1999).
refer to the discussion related to (10.66a,b). This criterion is, in fact, incomplete. At the
transition boundary, the saddle point (ko, Qo), satisfying D(k, Qo) = (oD(k,Q)jok)b o =
O, must also be associated with a pinching in the complex k-plane between two branches on
either side of the excitation pulse (Figure 10.12). t The pinching itself is not the deciding
factor in the transition between absolute and convective instabilities. Rather, it is the
discriminant: it al10ws us to obtain the value of Im(Qo) and thereby decide the question
via (10.68). To this end, Im(Q) is varied tilI pinching is observed.
Figure 1 0.13( a) and (b) gives graphicalIy G (x, t) from an example on wave propagation
in a long pipe conveying flow, which is shown to be equivalent to the system of two
tOespite appearances, pinching is not the same as the mode exchange/veering in the complex Q-plane
encountered in Chapter 3; refer, for example, to Huerre (2000),

PLATES IN AXIAL FLOW
1161
infinite plates in sinuous motion with flow in-between, in the limit of long wavelengths
(de Langre & Ouvrard 1999).
This is a very simplified treatment of a complex and mathematical1y involved topic,
even for simple physical systems, such as that of an undamped plate or membrane in
potential axial flow. Once dissipation is included, then further complexity is added; this,
then, becomes par exeellenee the system demonstrating the princip les first studied by
Landahl and Benjamin; see, for example, Carpenter & Garrad (1986) and Appendix C.
A sample problem is discussed in greater detail in Section 10.3.4.
In addition, there is considerable interest in loeal and global instabilities. What has
been presented so far is related to local instabilities. lf, however, conditions evolve slowly
along the x-direction, or if the system is subject to boundary conditions upstream and
downstream, then it is of interest to consider how these concepts apply global1y (in
domains covering several wavelengths); see, for example, Chomaz ef al. (1988), Huerre &
Monkewitz (1990) and Doaré & de Langre (2002a).
A good example in the context of this book is concerned with systems subjected
to gravity-induced or friction-induced tension. For example, as found by Doaré &
de Langre (2002b,c), a long hanging pipe can, from the dynamical point of view, be
divided into two parts: (i) the upstream part which, because of the very high tension, does
not affect the behaviour of the downstream part (being essential1y inert and not transmit-
ting any information to the downstream part); (ii) the downstream part which is where
instabilities manifest themselves and which does respond to excitation (which propagates
downstream and upstream up to where the 'upstream part' begins). Another example is
that of the very long string cylinders in axial flow discussed in Section 8.3.7(b), where
again most of the upstream part is inert. In both cases, the instability is global, but it is
confined to the downstream part, the extent of which may nevertheless be determined by
simple loeal analysis, The upstream part is 10cal1y stable, If a pipe is horizontal, however,
the gravity-induced tension is absent and the unstable system would be global1y unstable
everywhere, no matter how long.
10.3 REPRESENTATIVE STUDIES
In this section, a number of arbitrarily chosen, representative studies are presented, rather
laconical1y, showing some typical results. These studies convey the flavour and give some
of the physical understanding of each topic discussed. Additiona1 reference s are given in
each case, but not an exhaustive literature review.
10.3.1 Plates in contact with quiescent den se fluids
This topic is quite important, since thin flat plates are often used in practice as end-
plates in cylindrical or rectangular containers and reservoirs. If the fluid contained is
dense, generally a liquid, the added mass per unit area can be of simi1ar magnitude as the
corresponding plate mass; thus, its effect on stresses and vibration frequencies may be
large. Refer also to Section 7.2.2.
The added or virtual mass may of course be determined by finite element or boundary
element methods (FEM or BEM) - refer to the introduction to Section 7.3.3. How-
ever, approximate analytical methods have been available for a long time, going back to

1162
SLENDER STRUCTURES AND AXIAL FLOW
Lamb (1921) who ca1culated the added mass of a c1amped circu1ar 'baffied' p1ate, that is,
a p1ate c1amped a11 around and placed in a matching circu1ar aperture in an infinite rigid
plane wa11; a considerab1e amount of work on this has been done since then. Considerably
less has been avai1ab1e for rectangular p1ates, unti1 much more recendy.
Most of the approximate ana1ytica1 methods are based on the assumption, attributed to
Lamb, that the 'wet modes' (the modes of vibration of the p1ate in contact with fluid) are the
sameas the 'drymodes' (thoseinvacuo)t -referred tofromnow as 'Lamb'sassumption'.
Ca1cu1ations based on this assumption result in the so-called 'nondimensiona1 added virtual
mass incremental' (NAVMI) factors (Kwak & Kim 1991) introduced in Section 10,1. The
use of NAVMI factors has been further deve10ped recently and their accuracy vis-a-vis
more comp1ete theory assessed, notab1y by Kwak (1996) and Amabi1i & Kwak (1996).
Consider a circu1ar p1ate in vacuo, the equation of motion of which is
/'
a 2 W
DV 4 w + pph ---z = O,
at
(10.69)
I
where V 4 = V 2 V 2 , and V 2 is the Lap1ace operator in po1ar coordinates. Assum-
ing axisymmetric boundary conditions, the solution may be expres sed (Leissa 1969) in
the form
00 00
w(r, (), t) = L L wnm(r) cos n() f(t),
n=O m=O
(10.70)
where
wnm(r) = Anm ln ()"nmrja) + C nm In ()..nmrja),
f(t) = cos fU,
(10.71a)
(10.71b)
in which n and m represent the number of nodal diameters and nodal circ1es, respective1y,
a is the radius of the plate, and ln and In are Besse1 and modified Bessel functions of the
first kind. The 'dry' circu1ar frequencies may be found by application of the boundary
conditions; they are re1ated to the frequency parameter Anm by
Qnm = ( A:m )2 ( P:h ) 1/2
(10.72)
Consider next the p1ate in contact with fluid on one side. We first present the ana-
1ysis based on Lamb's assumption, considering a specific mode shape (n, m), following
Amabili & Kwak (1996).
In inviscid fluid, the fluid-dynamic 10ading on the p1ate is obtained via potentia1 flow
theory, in terms of the velocity potentia1 <I>(r, (), z), satisfying V 2 <1> = O in cylindrical
coordinates, and (8<1>jaz)lz=0 = -w(r, () on the p1ate and (a<l>j8z)lz=0 = O on the
rigid wa11 (baffie), where w(r, () = w(r, (), O) - cf. equation (10.70) - for the mode
concerned, Further, assuming
<I> (r, (), z) = rjJ (r, z) cos n()
(10.73)
tThis can be justified by a perturbation analysis [with respect to j.i, defined after (10,78)], showing that,
to first order, the wet eigenvalues but not the eigenfunctions are changed vis-a-vis the dry ones, Thus, this is
accurate for air, but progressively deteriorates for denser fluids (Dowell 2002),

PLATES IN AXIAL FLOW
1163
and using the Hankel transfonn, after considerable work Amabi1i & K wak (1996) obtain
4> (rja, O) = a 2 oo Hnm(a) Jn(r) d,
(10.74)
where Hnm(a) is a complicated functiona1 ofthe Bessel functions, with mas in (10.70),
The reference kinetic energy of the plate, that is, exc1uding f(t) in (10.70), is
1  27r l a
T; = -pph w 2 rdrde,
2 O O
(10.75)
and the reference k:inetic energy of the fluid is
Tj=p{27r {a [ 4> 84> ] rdrde.
2 10 10 8z z=o
ForllŮng the Rayleigh quotient (Bishop & Johnson 1960; Meirovitch 2001) for coupled
vibration (Zhu 1994), one eventually obtains the remarkably simple expression
(10.76)
Qv
Qf = ,
.J I + f3nm
(10.77)
re1ating the natural frequency in fluid, Q f, to that in vacuum, Qv, where
T*
f
f3nm = - = Ynm f.L
T*
p
and f.L = paj(pph); f3nm is computable from the foregoing, and Ynm iS  he AVMlfactor.
Considering now the more general problem in which the wet and modes are not
assumed to be identical, it is noted that a1though the n wavenumbers, e orthogonal and
hence may be considered separately, this is not so for the m wavenumbers, Utilizing the
Rayleigh-Ritz method, in this case we take (Amabili & Kwak 1996)
(10.78)
N
iň(r, O) = cos nO L qm wnm(r),
m=O
N 00
4> (rja, O) = a 2 E/m i Hnm(a)Jn(r) d,
(1O.79a)
(10.79b)
where the comparison functions W nm and the Hnm are as in (10.71a) and (10.74); the qm
are unknown parameters to be deterllŮned, On this basis, the reference k:inetic energies
and the potential (strain) energy are
T; =!O" pphqTIq,
1 T M
Tf = '20" paq fq,
V p = !(0"'Dja 4 )qTpq,
(10.80)
where O" = a 2 o/ n , with o/n = 2n for n = O and o/n = n for n > O, I is the N x N unit
matrix, Mf is the added mass matrix, and P is an N x N diagona1 matrix with elements

1164
SLENDER STRUCTURES AND AXIAL FLOW
Pij = (jij Ai and the Ani as in (1 0,71 a). Therefore, constructing the Rayleigh quotient
and minimizing it with respect to the qm leads to
-2
Pq - Q (I + JL Mf)q = O,
(10.81)
where Q 2 = Q2(a 4 jD)pph is the dimensionless radian frequency, Q being the
dimensional one,
It is important to note that Lamb's assumption corresponds to considering no coupling
between the m wavenumbers, thus leading to diagonal matrice s in (10.81). This then
corresponds exactly to the simplified analysis and leads to equation (10.77),
Some typical NAVMI factors, Ynm, for different (n, m) are given in Figure 10.14. As
expected on physical grounds, the smaHer the n and m, the larger are the Ynm, since the
quantity of fluid displaced is relative1y larger.
The added mass matrix has also been calculated by the fuH fonu of (10.81) and the error
in the approximate (NAVMI factor) analysis is computed, as shown in Figure 10.15 for
n = O. It is seen that the error in the fundamenta1 mode (m = O) is nil, but it is measurable
for m > O, increasing with fluid density as characterized by JL. Sti11, the percentage error
is re1atively small; besides, n = O represents the most severe test. Hence, equation (10.77)
gives a very good engineering approximation and has the benefit of simplicity.
The more difficult case of a rectangular plate has been treated by Kwak (1996). Com-
pared to the circular plate, in this case the boundary value problem does not admit a
0,8
0,5
__m=O;s
----6- m = 2; s
-D-m = o; c
----6-m = 2; c
-+-m=\;s
-o-m=3;s
-o-m=\;c
-o-m = 3; c

o
t>

Si 0.4

Z
0,2
0,\
o
O
2 3
Number ofnoda\ diameters, n
4
Figure 10.14 NAVMI factors, Ynm, for simply supported (s) and clamped (c) circular plates for
various n and m for v = 0,3 (Amabili & Kwak 1996),

PLATES IN AXIAL FLOW
1165
m=O
o
-I
,-.,

.... -2
o
t::

-3
-4
-5
O 5 10 15 20
J.l
Figure 10.15 The error in the approximate (NAVMI factor) vis-a-vis the Rayleigh-Ritz evaluation
of the added mas s for n = O, v = 0,33, as functions of JL = pa/(pph) (Amabili & Kwak 1996).
closed-form solution. The Green's function approais used to solve the fluid boundary
value problem, in which the fluid domain is discr9lzed into small rectangular panels; thus,
the method is similar to BEM.
Assuming a potential function associated with sources of unknown strength distributed
on the plate, the source strength is determined by matching it to the normal velocity at the
interface. If the plate has width a in the x-direction and length b in the y-direction, for
a point (x p, y p), the velocity potential is given by
4J ( ) - - / ( v(x p , Yp, t) dS
xp, Yp, t - 2n 1s (x p _ x q )2 + (yP _ yq)2 '
(10.82)
where v is the velocity at the interface and S the plate area. The beam eigenfunctions are
used as admissible functions in a Rayleigh-Ritz scheme, and the integration in (10.82) is
carried out numerically, by dividing the surface area into m sma1l panels, thus, permitting
the evaluation of the kinetic energy by summing over these panel s,
1 m /1 [ JcI>' ]
Tf = 2 P L ,cI>iT dSi.
i=1 S, Z z=O
(10.83)
From here on, the analysis follows the same path as for the circular plate, leading
to an equation of the form of (10.81). Considering this to be diagonal, which again
amounts to considering that there is no intermodal coupling and hence that Lamb's
assumption/approximation is exact, one obtains the corresponding NA VMI factors.
Typical results for Ynm where n and m are the number of nodal lines in the x and
y-direction, respectively, are shown in Figure 1O.l6(a). These are compared to some
analytical results by Kim & Kim (1978) for semi-infinite plates adapted for comparison

1166
SLENDER STRUCTURES AND AXIAL FLOW
0.4
 0.3
t>


Z 0,2
0.1
1.0
15
2.0
bla
2.5
3.0
(a)
0.25
0.20
 0.15
g
 0,10
0,05
0.00
O
20
40
60
80
100
(b)
IA
Figure 10.16 (a) The NAVMI factors, Ynm, for simply supported rectangular plates as functions
of length/width ratio for various (n, m): -, Kwak's NAVMI factors; ---, Kim & Kim (1978)
results for semi-infinite plates, appropriately adapted; symbols, experimental results by Kim et a!,
(1979). (b) Error in added mass, plotted versus f-[ = pal Psh, when using NAVMI factor vis-o-vis
fuller theory (K wak 1996).
with finite plates - therefore, exact agreement should not be expected - and some
experiments by Kim ef al. (1979). Agreement)Š' quite good.
Again, these results are compared with t obtained by the fuU Rayleigh-Ritz ana1ysis
in Figure 1O.16(b) for various (n, m) cotnbinations. In this case, the error in using the
NA VMI factors is even sma11er than those in Figure 10.15.
There is a large body of literature on this topic, Sample papers, further to those a1ready
cited, are by Espinosa & GaUego-Juárez (1984), Amabili (1996, 2001), Kwak (1997) and
Cheung & Zhou (2002). Sample papers on floating platforms and interactions with the
marine environment may be found in Eatock Taylor & Ohkusu (2000). Further references
may be found in aU of these papers.
10.3.2 Finite-width and finite-width-and-Iength plates in flow
The system considered first is a ba:ffled plate of finite width but infinite length, as
shown in Figure 10.17. This topic was studied by, among others, Dowell (1966a),

PLATES IN AXIAL FLOW
1167
Dowell & Ventres (1970), E11en (1972, 1973) and Epstein et al. (1995) for generally
compressib1e flow, and Weaver & Unny (1972) for incompressib1e flow.
The basic equation is (10.4), with p* on the right-hand side and a viscous damping tenn
sometimes inc1uded as in (10.3). The most genera1 solution may be written as
w(x, y, t)  {.J,... w. cos(n7ty/b) + .J,.,. w. sin (n7ty/b) } e-;(2H/Á)(>-''',
(10.84)
where b is the p1ate width, A the wave1ength and C the wavespeed, satisfying w = O,
a 2 w/ay2 = O at y = :l:: b. The velocity potential, which must satisfy V 2 <1> -
(1/a 2 )[a 2 /at 2 + 2U a 2 /ax at + U 2 a 2 /ax 2 ]<I> = O, a being the speed of sound, takes
the form
<I> (x, y, z, t) = <tJ(y, z)e- i (2JT/A)(X-ct) ,
(10,85)
and
aA. 1 { ow + U OW for Iyl  b
'I' _ ot ax 2 (10.86)
 - 1
z=O O for Iyl > lb,
as well as being bounded at x -+ :l::oo and satisfying the radiation conditions at z -+ 00.
Dowell (1966a) considers the simpli1ied prob1em in which on1y the 1irst tenn in (10.84)
is considered, that is, n = 1.
Expressing <tJ (y, z) in the fonn of a Fourier integra1, the following characteristic equation
for C is found (Dowell1966a):
c 2 - c6(X + 1/X)2 + f.L(U - c)2 F(1J) = O, (10.87)
where
Co = .,.. (_ V ) 1/2
- J< X = A/(2b),
pph b 2
:rr (CO cos2 d
F(1J) = '210 (1J2 + 2)[(:rr/2)2 _ 2]2'
f.L = pb/(pph),
:rrb [ 2 ( C ) 2 ] 1/2
1J=T 1-M I- U
and  is a dummy variab1e. In the above, a notation simi1ar to Komecki's (1979) has been
adopted; note differences in de1initions of Co and 1J vis-a-vis Dowell (1966a).
Solving (10.87), for incompressible flow (M = O) one obtains an explicit soIution,
since then 1J is not a function of U, name1y
CI,2 = {f.L U FO(1J):l:: [c6(X + 1/X)2(1 + f.L FO(1J)) - f.LU 2 FO(1J)]1/2}
x [1 + f.L FO(1J)]-I, (10.88)
in which FO(1J) is F(1J) for M =0. As discussed in Section 10.2.1, the flutter boundary
occurs when CI = C2 = ccf, that is, when the radicand vanishes, yie1ding
[ 1 ] 1/2
Ucf = Co (X + I/X) 1 + ,
f.L F o(1J)
Ccf = f.L Ucf FO(1J).
(10.89)

1168
SLENDER STRUCTURES AND AXIAL FLOW
z
Fluid flow
x
Figure 10.17 A finite-width, infinite length, baffled plate in axial flow (Weaver & Unny 1972).
40
30
20
16
 12
;;:,
8
6
4 2,0
3 3,0
2
\0-2 2
4 6 8 \0- I
2
4 6 8 1.0
!1
2
4 6 810
Figure 10.18 The effect of the mas s ratio I.L and Mach number M on the critical flow velocity U e ,
nondimensionalized by co: -, flutter; ---, divergence (Dowell1966a),
However, there is a divergence boundary a1so, arising when CI = 0, which occurs at
[ 1 ] 1/2
Ved = co (I + 1jI) f.-t FO(17)
(10.90)
ln these expressions, FO(17) has to be computed numerically, ln the case of compressible
.flow (M > O), F(17) is a function of Vand hence the critica1 V, that is, Ve!, is determined
implicitly from solutions of (10.87), _ /
Typica1 resu1ts are shown in Figur.18. For subsonic flow it is seen that Ved < Ve!;
for small f.-t (air-flow), Ve!  Ved  cf. equations (10.89) and (10,90). For supersonic
flow divergence does not arise (Komecki 1974), and hence on1y Ve! is shown.

PLATES IN AXIAL FLOW
1169
The effect of damping on stability, so far ignored, is profound - see Section 10.3.3.
Suffice it to say here that, for subsonic flow, any small amount of damping causes the
flutter speed, U e !, to decrease to Ue' where the superscript O denotes the value without
damping. This is similar to what has been found in infinite panels in incompressible flow
(Section 10.2,1). In fact, the dynamical behaviour in subsonic flow is quite similar to that
in incompressible flow, but Ue is proportional to (1 - M 2 )1/4. It is clear, therefore, that
for M = I, U e ! == Ue  O, indicating thatfor supersonic flow all plates are unstable
by flutter (at any flow velocity satisfying M > 1), provided there is a small amount of
damping. However, flutterfor  < U < U! is weak, that is, IIm(c)1 is very small, and
only becomes important as U! is approached - which may therefore be regarded as the
effective flutter threshold after all. These intriguing questions have been discussed for the
first time in Dowell's (1966a) seminal paper,
Weaver & Vnny (1972) consider the problem of an infinite-Iength plate in den se fluid
with a free surface, at a generally finite distance H above the plate. Mainly flutter is
discussed, As H  00, the critica1 conditions are identica1 to Dowell's, Typica1 res-
ults are shown in Figure 10.19. It is seen that proximity of a free surface to the plate
(small HlA) great1y stabilizes the system, acting in the opposite way to a solid boundary
(Section 10,2,1), This is physica1ly reasonable, as the perturbation pressure radiates away
from the plate at the free surface,
Epstein et al. (1995) consider the case of an infinite-Iength plate in a duct; that is, with
rigid walls d: 1 H above and below the plate, The results obtained are similar to Dowell's,
but now as H is decreased the system is destabilized, Some typical results for M = O are
shown in Figure 10.20.
One important aspect of divergence or flutter in both sub- and supersonic flows is that,
unlike the case of infinite-width plates (Section 10.2.1), the critical wavelength does not
go to infinity (in the absence of an elastic foundation): it is finite and it is of the order
of twice the plate width for a plate in unconfined flow, as shown in Figure 10.21; with
confinement, Ac can be appreciably higher, depending on f.L and M.
A question of great interest is this: how well does the foregoing analysis apply to finite-
length, that is, finite-length-and-width plates, also known as '3-D' plates? The critical
aerodynamic pressure, that is, the dimensionless flow velocity squared, for low aspect ratio
(AR = biL) plates, that is, for Lib  00 discussed in the foregoing is compared to that
obtained from finite-Iength analyses ofbaffled 3-D plates in Dowell & Ventres (1970), As
seen in Figure 10.22, the infinite-Iength (AR  O) analysis agrees reasonably well with
the finite-Iength (finite AR) analyses for Lib = 10 and even Lib = 2 for subsonic and
low Mach number supersonic flows - although the two results for M < 1 correspond to
divergence rather than flutter; this is not so, however, high Mach numbers.
Plates of finite width and length have been treated in a 'standing-wave' analysis by
Ellen (1973), considering plate deflections of the form
Wnm(x,y) = Wo sin(n;rrylb)sin(m;rrxIL)
(10.91)
for plates covering O ::: x ::: L, O::: y ::: b, with simply supported edges.
The stability of low aspect ratio (AR = biL) plates is discussed by considering asymp-
totic expressions for the perturbation pressure, ln the limit of AR  O, Ellen finds that the
most unstable mode is with m  00, such that mAR is finite. Specifically, an asymptotic
expression for p U 2 b 3 IV = f(mAR, M) for divergence and n = I is found; from this,

1170
SLENDER STRUCTURES AND AXIAL FLOW
14,0
12.0
Unstable (flutter)
4,0
20
Stable
0.1
1.0
10,0
100,0
11
Figure 10.19 The effeet of mass ratio 11 and depth H (in terms of H / A) on the eritieal velocity of
a flat plate in flowing dense fluid with a free surfaee; A is the wavelength (Weaver & Unny 1972).
14
12
II =0.01
10
o 8
-.S:!
 6
_ =j),!. - - - -
4
2
=10
------------- ---------- ------ -- - - -- - --
o
2
3
4
5
Hfb
Figure 10.20 The eritieal flow velocity Ve, nondimensionalized by Co, as a funetion of dimen-
sionless duet height, H / b, for a plate of width b in a duet, for three values of 11 (Epstein
et at. 1995).
the value of mAR rendering p U 2 b 3 IVa minimum is determined. Specifical1y, for M = O,
p Uc b 3 IV = 138,
(10.92)
with mcdAR = 0.72; in fact this mcdAR is va1id for aH M :S 0.5, The value in (10.92)
agrees fuHy with that in Figure 10.22 for AR = 0.5 (Lib = 2) and M = O. The
dependence of U on AR is found to be
Ucd(AR) = Ucd(O)[1 + C(M)AR + O(AR 2 )]
(10.93)
for small AR, where C(M) is a weak function of M: C(M) = 0.151 for M = O and
0.2, C(M) = 0.149 for M = 0.4, C(M) = 0.145 for M = 0.5, and C(M) = 0.134 for

PLATES IN AXIAL FLOW
1171
3,0
M 1.3
M=O
 2.0
r<
M 3
M=oo
1.0 2
10- 2
4 6 8 10- 1 2
4 6 8 1.0 2
4 6 810
JI.
Figure 10.21 The critical wavelength, Ac, in tenns of Ac/b as a function of M for several Mach
numbers; o, resu1t for M = 1.3, obtained earlier (1963) by Cunningham (Dowell1966a).
200
150 f=2
b

"'--...
-'=>
 100
Q.
50
f=1O
b
O
O 1.0
__o-
..... ...- f = 10
/' b
/,P i = 10
/
/
./
2,0
3.0
M
Figure 10.22 Comparison of the critical aerodynamic pressure for infinite and finite plates in
roda1 flow versus Mach number, M, for a plate with pb/ pph = 0.04, b being the plate width:
-, infinitely long plate analysis; - - -, finite length plate results from various sources (Dowell &
Ventres J 970).
M = 0.8; Ucd(O) in (10.93) is obtained from (10.92).t Also, the corresponding mcdAR
agrees with what is obtained from Figure 10,21 for M = O and smal1 j-t: that is, that the
critical streamwise wavelength is approximately 2.8 times the plate width,
ln the other limit, for high aspect ratios, AR = biL --+ 00, the system looks like
that in Figure 10.5, but the far side-edges are now pinned, For AR --+ 00, the critical
pUcL3 ID == U d as given by equation (10.44b) is obtained. For smal1er AR, Ellen finds
tSince this implies knowing a specific b, it means that (10.93) in effect gives the dependence of Ucd on bl L
as L is varied, such that effectively O < biL < 1,

1172
SLENDER STRUCTURES AND AXIAL FLOW
the dependence to be
Ued(AR) = U e d(OO)[1 + A -2(0.262ln 2A - 0.251) + 0.175A -3 + O(A -4 ln A)],
(10,94)
where A = AR(1 - M 2 )1/2. This, as in Section 10,2.2, cOITesponds to n = m = 1.
Relationships (10.93) and (10.94) are graphed in Figure 10.23, where it should be
stressed thattheirrange ofva1idity does not extend to the extreme right of (a) and left of (b).
Towards the centre of the figure, that is as AR --+ 1, the two expressions do not converge,
because the criteria for computing them are different. t Clearly for the intermediate range
a fulI, as opposed to asymptotic, analysis is necessary.
ElIen (1972) has extended his asymptotic analysis to c1amped plates. The left- and right-
hand limits ofFigure 10,23 in this case are 177.6 and 364; considerably higher. The critica1
wavelength for low-AR plates in this case is approximately 1.8 times the plate width.
ElIen (1977a) has a1so considered the stability of 'isolated' rectangular plates and mem-
branes, that is, with the bafftes removed, in subsonic flow, again constructing asymptotic
expansions for the low- and high-AR cases and obtaining expressions similar to (10.92)-
(10,94). Various boundary conditions are considered, inc1uding cases where only two of
the plate edges are supported. The results are compared to those for baffted plates.
Some results for divergence of1ow-AR plates and membranes are shown in Figure 10.24.
It is noted that the relatively low va1ues of pU!db 3 IV for baffted plates are not inconsistent
with those of Figure 10,23; here these plates are subjected to flow on both sides.
The sca1loped nature of the stability curves is important. It is associated with the
changing med, the value of m giving the sma1lest Ued, as AR is varied, ln this respect,
equations (10.93) and Figure 1O.23(b) are strictly incoITect, having been constructed con-
sidering mAR to be a continuous variable, ln Figure 10.24, however, the discrete integer
nature of m is taken into account: as AR increases, med jumps to decreasing integer values.
10.3.3 The effect of dissipation on stability
As already remarked, the effect of dissipation on the dynamics of plates in axia1 flow is
profound. Part of the reason for this is related to the bifurcation we have termed 'diver-
gence' in the travel1ing wave ana1ysis, which is different from the 'normaľ divergence
that we have encountered in pipes, shelIs and cylinders in flow. Contrast, for instance,
Figure 8.34, on the one hand, and Figures 3.4(a), 3.9, 7.7 and 8.5, on the other. In the latter
case, beyond the critical point, one root has Im(w) < O and is therefore associated with
nonoscil1atory growth of disturbances, since Re(w) = O. In the former case (Figure 8.34),
however, the neutral stability condition persists beyond the divergence boundary, that is,
Im(w) = O as Re(w) changes sign; such is the case with the plate in flow a1so, only
instead of w we dea1 with the wavespeed c (for k > O). Physica1ly, therefore, this does not
necessarily represent an instability; in fact, what it means is that upstream propagation of
disturbances is no longer possible beyond Ued.
Hence, in the latter case of 'normal' divergence, for example, as in Figures 3.4(a) and
3.9, although the bifurcation itself is a1tered by the inc1usion of damping, for example,
tThat is, in (10,94), we have n = m = 1 and the minirnization is with respect to AR for M = O, while in
(10,93) n = 1 and minimization is with respect to mAR, so that m = 7 for AR  0(10- 1 ).

PLATES IN AXIAL FLOW
1173
120
100
 80
"'......
""-1 60

 40
20
O
100
(a)
Figure 10.23
250
 200
......
":co
",""
::J"
 150
10
biL
100
0,25 2
(b)
0.1 0.01
biL
0,001
The asymptotic behaviour of Ucd for finite values of AR = hl L, 'near' the limits
of (a) AR  00 and (b) AR  O according to Ellen(l973).
5
iii
4
ii
pU 2 b 3
T
2
iv
90
80
pU 2 b 3
9lJ
70
60
o
0.1 0.2 0.3
Aspect ratio, AR
0.4
0,5
Figure 10.24 Stability curves for divergence of low-AR simply supported plates and membranes
(V = O) for M = O: (i)inisolation, in flowing fluid; (ii) baffled, with flow on both sides; (iii) forming
part of a periodic array with multiple supports, with flow on both sides; (iv) membrane in isolation,
with only the leading and trai1ing edges supported, Upper curves: membranes; lower curves: plates
(Ellen 1977a).

1174
SLENDER STRUCTURES AND AXIAL FLOW
see Figures 3,14 and 8.5, the physica/ behaviour of a system as it crosses the divergence
boundary is not radically altered.
ln the fonner case of 'travelling wave divergence' (see Figures 8.34 and 7.108, for
instance), it is not obvious how the Argand diagrams would change when damping is
included. As already discussed in Sections 10.2.1 and 10,3.1, the effect can be dramatic.
The effect of damping on the dynamics of plates in axial flow has been discussed by
Dugundji et a/, (1963), Dowell (1966a), Komecki (1970, 1978, 1979), and Carpenter &
Garrad (1986), among others.
10.3.3(a) The effect of damping in incompressible flow
This question has been tackled briefly in Section 10.2.1 for an infinite plate. The following
is found: (i) the damped plate loses stability by divergence at V = Ved; (ii) as soon as Ved
is exceeded the plate begins to flutter. This has been shown to be the case for both viscous
and structural (hysteretic) damping (Komecki 1978). Thus, damping causes the effective
flutter threshold to be reduced from Vef to Ved, independently of the value of damping
(Cd or its dimensionless counterpart, d).
Komecki ties this behaviour to the mechanisms of energy transfer developed by
Benjamin and Landahl (see Appendix C), specifically to Class A waves which are
destabilized by damping and Class B waves which are stabilized by it. Thus, referring
to equation (10.23) it is clear that, for Ucd < V :::: Vef' the slower-moving waves are
destabilized by damping, Im(kq) < O, and hence are Class A waves; the faster moving
waves are stabilized by damping, Im (kC2) > O, and hence these are Class B waves.
The temporal growth rate for V > Ved is nevertheless small for V < Vf' where the
superscript O denotes 'in the absence of damping'. The amplification factor, defined by
A = exp(-21iCilco) = exp[21i(2 - d)(1 + fL)-I],
with Ci == Im(c), and fL, 2 and d defined in equations (10.16) and (10.22), gives
a measure of the growth rate or 'severity' of the instability, As seen in Figure 10.25,
provided the dimensionless damping is small, the amplification remains smal1, up to
V ::::: Vf' whereupon it increases explosively.
10.3.3(b) The effect of damping for compressible flow
Continuing the presentation for infinite plates, for the sake of simplicity, the equivalent of
equation (10.15) for compressible flow is
c 2 - c5 - idcco + fL{(V - c)2/[l- (V - c)2Ia 2 ]I/2} = O,
(10,95)
where d = Cdl(2kmpco), fL = pI(lklmp), a the speed of sound, and co as in (10.8), in
the absence of a bottom wall (ér = O) and no flow below the plate,
With no damping, a typical variation of c with V is given in Figure 10,26, showing
that at point D, corresponding to V = Ved, the backward-travelling wave ceases to exist.
Soon after, in this particular case the flow becomes supersonic, and Vef (point F) occurs
at M ::::: 1.6, This is similar to what is obtained in incompressible flow (Figure 10.2).
Defining the dimensionless quantities
C = cla,
Co = cla,
M = V la,
Me = (V - cR)la,
(10.96)

PLATES IN AXIAL FLOW
1175
30
1
1.0
20
10
A
5
4
3
2
1.1 1.2 1.3
U/d
1.4
1.5
Figure 10.25 The amplification factor, A, of travelling waves versus U / Ucd for incompressible
flow over a plate; Ji = 0.5 (Komecki 1978).
1.0
'F
0.5
o
'"
<> 0.0
-0,5
-1.0
0,0
0,5
1.0
M= U/a
1.5
2,0
Figure 10.26 The variation of wave velocity, c with flow velocity U for an undamped plate in
compressible flow; Ji = 0,16, co/a = 0.9 (Komecki 1979).

1176
SLENDER STRUCTURES AND AXIAL FLOW
in which CR is Re(c), equation (10,95) may be written approximately (Kornecki 1979) as
-2 -2 M 2 (1 M ) -1/2 2d - - O
CR - Co + f.l, e - e + CjCO = ,
ďCRCO = Cj [CR - !f.l,M e (2 - M;)(1 - M;)-3/2] ,
(10.97a)
(10.97b)
for M; < 1; and for Me > 1 or Me < -1,
d - - 1 M 2 (M 2 1) -1/2 --
CRCO  2f.l, e e - = CRCj,
C - C6 + 2dCjCO  f.l,M e Cj(2 - M;)(M; - 1)-3/2 = O,
(10.97c)
(10.97d)
The exact expressions are a1so given by Kornecki (1979).
From (10.95), in the case of d = O, it is easy now to see that a neutra1 stability condition
can 31so arise when
O  c 2 ::S c6 and O::S M; < 1,
that is, for V ::: CR and V - CR < a,
Typic31 resu1ts with damping, obtained using the full form of equations (10.97a,b), is
given in Figure 10.27; in this case 31so, the flow becomes supersonic before reaching Ve!.
It is seen that for the undamped system (d = O) the va1ues ofc remain rea1 up to the point of
coa1escence of CI and C2 at F, the flutter thresho1d, thereafter becoming pure1y imaginary.
For d =1= O (dashed 1ines), however, immediately after divergence at D, one of the roots has
Cj < O [the upper part ofthe diagram in Figure 1O.27(b)], indicating amp1ified osci11ations
immediate1yafter Ved is surpassed. A1so seen in Figure 1 0.27(b), is the fact that C j remains
small unti1 the v31ue of V! is approached; note the change of sca1e on the right-hand part
of the figure,
Itis interesting to note that, for supersonic flow, equation (1O.97c) imp1ies that, if Me> 1
and CR > O (possib1e on1y for supersonic flow), a neutra1 configuration is possib1e on1y
for d = d* == f.l,{M; /[2c6(11 - M;I)1/2]} and CR = Co. Thus, apart from that singu1ar
situation, there always exists an unstable root, and the system is unstable in 311 cases!
This rather disturbing result is indicative of the 1imitations of the simple theory and of the
idea1ized system concerned.
10.3.3(cJ Other work
The effect of a laminar boundary layer on the stability of a plate in axial flow is discussed
in Carpenter (1984), It is shown that the presence of a boundary layer, no matter how thin,
causes the effective flutter velocity to jump from Ve! down to a lower velocity, similarly
to the effect of damping. An important difference, however, is that the boundary layer
destabilizes Class B waves, whereas damping destabilizes waves of Class A.
The reader is 31so referred to Carpenter & Garrad (1986), Lucey & Carpenter (1992),
as well as the many papers on the stability of comp1iant coatings [see, e.g. Gad-el-Hak
(1986), Carpenter et a/. (2000, 2001)]; also to Higuchi & Dowell (1992) who discuss the
effect of damping on flutter of a plate subjected to a follower force.

PLATES IN AXIAL FLOW
1177
-1.0
-0,5
I
-0,1
°
0,1
0.5
Co
1.0 MF
(a)
-0,02 -0.2
-0.01
-0,1
/
/
/
I .'
I ,/.
/,///
y'
/
,//
-0,02,,/ /
....... ./ F
M D
-0,002 0.1
°
1<3" 0.002
0,5
/
D ,/
----- / /0.02
cn '
.......
,,(:/2/---------
X
0.01
I,
1- \
fCn \
0,02 \
\ c/2 \
1.1
1.2
1.3
1.4 1.5
M
1<3"
.............
"
"'<2
,
\.
_ c/2
--...-........
"
\
\
\
\
\
\
\
0.1
0.2
(b)
Figure 10.27 (a) The real and (b) imaginary components ofc = cla, CR and Cf respectively, for
Co = 0,8 and different dimensionless damping, d: -, d = O; ---, d = 0,01: -- __, d = 0,05
(Kornecki 1970, 1979),

1178
SLENDER STRUCTURES AND AXIAL FLOW
10.3.4 Plate channels conveying fluid
The system under consideration is configuration (c) of Figure 10.4. It has been discussed
in conjuction with pliable shells conveying fluid; see subsection b.3 of Section 7.9.5, as
well as Appendix P.2(d) in which the Grotberg & Reiss (1984) model is outlined. lndeed,
Grotberg and his co-workers have been in the forefront ofthe development ofparallel-plate
models for physiological flow conduits, notably for pulmonar flow passages.
Moreover, such models have been extended and generalized to deal with stability of
plate channels conveying fluid of more general interest than for strictly physiological
application - see, for example, Grotberg & Shee (1985), Prince et aZ. (1993), LaRose &
Grotberg (1997). In the last study, the linear stability is examined of a developing viscous
flow in a flexible channel, using the full Orr-Sornmeďeld equation.
Huang (1998) considers the general case of a shear flow, for example, a plane Poiseuille
flow, in a p1iab1e channel, in a paper aptly entitled 'Reversal of the Bernoulli effect and
channel flutter'. lt is shown that the well known inverse relationship between pressure and
constriction (due to wall motions), that is, the Bernoulli effect responsib1e for destabiliz-
ation, is opposed by pressure perturbations in the vorticity field, Specifically, the pressure
induced by travelling varicose waves (see Figure 10,28) opposes wall displacement, as
part of the perturbation Reynolds stres s overrides the inertia force; thus, the Bemoulli
effect is reversed, The stability of a finite-1ength thin membrane in a Poiseuil1e flow is
studied by Huang (200la). It is found that divergence and flutter occur at similar flow
velocities and may coexist. Which is dominant depends on parameters such as membrane
1ength and inertia, and on upstream and downstream fluid boundary conditions.
lmportant though these studies undoubtedly are, they are not discussed here further.
lnstead, we consider some work which ties this topic to that of pipes conveying fluid
(Chapters 3 and 4), on the one hand, and gives concrete examples of absolute and
convective instabilities in simple accessible tenns on the other, following de Langre &
Ouvrard (1999) and de Langre (2000, 2002).
10.3.4(a) Varicose motions of plate channel
The system under consideration is that of Figure 1O.4(c): two infinite plates oscillating
with opposing phase ('varicose motion'), as shown in Figure 10.28. The analysis given
here follows de Langre (2000), The equation of motion of each plate is
a 4 y a 2 y a 2 y *
1)-T+mp=p ,
ax ax at
(10.98)
y
Figure 10.28 Plate channel undergoing varicose oscillations (de Langre 2000).

PLATES IN AXIAL FLOW
1179
a fonu of equation (10.5). The perturbation pressure is govemed by
V 2 ifJ =0,
aifJ aifJ ( a a )
ay (x, h, t) = - ay (x, -h, t) = at + u ax y(x, t),
(10.99)
which gives for the upper plate
* ( a a )
Pu = -p at + u ax ifJ(x, h, t).
(10.100)
Consider motions of the fonu of a propagating wave y(x, t) = Yo exp{ -i(kx - Qt)};
consistently with all of the foregoing, we take the minus sign in the exponent (con-
trary to de Langre) - refer to discussion on equations (1 0.66a,b). The resulting dispersi on
relation is
D(k, Q; U, h) = Vk 4 + Tk 2 - m p Q2 - P (Q - Uk)2 = O.
ktanh(kh)
(10.101)
For k real, a wave is temporally unstable ifIm(Q) < O. This allows us to define a critical
flow velocity U e , beyond which there exists at 1east one unstable wave of wavenumber
k. By means of (10.101), it is thus found that U e = (T /mp)1/2 if V = O; for T = O and
V =I O, however, U e = O. Thus we retrieve the result discussed below equation (10.21),
and for the same reasons. t
The main interest here, however, is to establish the boundaries between absolute and
convective instabi1ities. Referring to Section 10.2.3, it is recalled that the instability is
convective or absolute, accordingly as Im (Qo) is positive or negative, with Qo being the
absolute frequency defined in conjunction with (10.67), which should vanish.
We define a zero group velocity in a slightly different manner,
D(ko, Qo; U, h) = O and
aD
-(ko, Qo; U, h) = O;
ak
(10.102)
it is also reca11ed that, at the transition, (ko, Qo) should be associated with a pinching in
the complex k-plane as Im(Q) is diminished - see Figure 10.12.
ln this case, because the system is conservative and symmetric in x, the eigenfrequencies
from the dispersion equation are either real or imaginary in pairs. Thus, we have an absolute
in stability when one root is imaginary, or marginally convective when all roots are rea1.
The transition between absolute and convective instability, at U = UT, therefore occurs
when Qo = QT is a double root, that is, for
D(kT, QT; UT, h) = O,
aD
af(kT, QT; UT, h) = O,
a 2 D
_ 2 (kT, QT; UT, h) = O.
ak
(10.103)
tThis, by the way, is part of the reason why, in so much of the work on the dynamics of plates, elastic
foundations are taken into account: to avoid such peculiar behaviour; altematively things are 'simplifieď by
considering membranes (D = O, T =1= O) rather than plates,

1180
SLENDER STRUCTURES AND AXIAL FLOW
For the case of (i), a plate with T = O and (ii) a membrane with V = O, and ph / m p -+ O
in both cases, de Langre finds
_p UTm3j2 21/23 2 I 0.228
UT = pVl/2 55/2 (ph/m p )I/2:= (ph/m p )I/2'
U 1/2 2 5/2
-m Tm p
UT = Tl/2 = 3 3 / 2 := 1.09,
(10,104)
respectively. The limit ph/mp -+ O referred to as the 'thin film' limit, in practice means
that h -+ O, The values of U  and U; as functions of ph/mp are given in Figure 10,29.
0,25
0,20
0,15
"-....
I:::'
0.10
0.05
O
O
(a)
t.3
1.2
1.1
e....
I:::'
0,9
0,8
0,7
O
(b)
Absolute ínstabílíty
Convectíve ínstabilíty
5
10
15
Absolute ínstabílíty
Convectíve ínstability
Stability
2
3
ph/mp
4
5
6
Figure 10.29 The dimensionless critical flow velocity fj T for transition from convective to abso-
lute in stability, equations (10.104), as function of ph/ m p (a) for a plate and (b) for a membrane:
-, general system; - - -, for the limit h --+ O; - - , - -, for the limit h --+ 00 (de Langre 2000),

PLATES IN AXIAL FLOW
1181
The instabi1ity is produced by coalescence of the forward and backward traveUing waves.
The in stability becomes abso1ute when 1arge wavenumbers which have negative group
ve10city become unstab1e.
For the membrane-channe1, the instabi1ity manifests itself first for smaU wavenumbers
when U e is surpassed. AU wavenumbers are subsequently progressive1y destabilized, up to
those causing the abso1ute instabi1ity. Forthe p1ate-channe1, the instabi1ity for U> U e = O
is again associated with smal1 wavenumbers, but the stiffness operator being ofhigher order
CDk 4 instead of T k 2 ) de1ays the destabilization of1arge wavenumbers and hence the onset
of abso1ute instability.
10.3.3fb) Sinuaus matians af plate channel
ln this case. the two p1ates oscillate with the same phase. The dispersion relationship
(Walsh 1995; de Langre & Ouvrard 1999) is
Vk 4 - m p Q2 - p[tanh(kh)1 k](Q - Uk)2 = O. (10.105)
FoUowing de Langre & Ouvrard, it is of interest that/ar shart wavelengths, that is. for
kh» 1, since tanh(kh)  klikl. this re1ation reduces to
Vk 4 - m p Q2 - (pllkl)(Q - Uk)2 = O, (10.106a)
which is identica1 to that for a sing1e flat p1ate subjected to flow [cf. Section 10.2.1;
Brazier-Smith & Scott (1984»). Far lang wavelengths, one has tanh(kh)  kh, and
(10.105) reduces to
Vk 4 - m p Q2 - ph(Q - Uk)2 = O, (1O.106b)
This 1ast relationship has the same form as the dispersion re1ationship for a pipe, once it
is realized that E/ = 2bV, where b is an arbitrary p1ate width, the mass ofthe pipe per
unit 1ength m = 2bm p, and the fluid mass per unit 1ength M == pA = phb.
The genera1 prob1em considered by de Langre & Ouvrard a1so invo1ves an e1astic found-
ation. Sometypicalresu1ts are showninFigure 10.30. Itis seen thatfor f3 = M I(M +m) <
2/3 there exists no convective instability at alt. If Kf = Kfm2/(p2E/) = O, where Kf
is the foundation constant, the instabi1ity is abso1ute for f3 < 8/9, whi1e for f3 > 8/9
it is convective. t For K f 1= O. both types of instability arise, and the transition is as in
Figure 1O.30(b,c); for examp1e, for intermediate values of f3, as in (b), the system 10ses
stability convective1y (for K f > O), and then at higher U the in stability becomes abso1ute.
For sinuous motions of a p1ate channe1, the behaviour is broad1y simi1ar to that for the
pipe: the instability is principally abso1ute. Perverse1y, however, it is main1y convective
for membranes.
The reader is referred to Brazier-Smith & Scott (1984), Huerre & Monkewitz (1990),
Triantafyllou (1992) and Peake (1997) for a more comprehensive treatment of convective
and abso1ute instabilities of p1ates subject to axia1 flow.
10.3.5 Parallel plate assemblies
Cascades of p1ates in axia1 flow have been used as idea1izations of stator vanes in flow
machines, guide vanes, etc. (Naudascher & Rockwell 1994, pp. 81-84); of particu1ar
tlt is realized !hat when fJ exceeds 8/9, the pipe should realisticaHy be analyzed as a shell.

1182
SLENDER STRUCTURES AND AXIAL FLOW
0,5
1.0
1.0
ST
0.5
0.5
Nl;J
.8-
N
i::
:><'-,
-0.5
-0.5
AI
o
CI
-0.5 AI
(c)
-10
20 O 10 20
mA U1EI
AI
O
(a)
-1.0
O
10
mAU 2 /El
(b)
-1.0
20 O
10
mAU 2 /EI
Figure 10.30 Stability diagram for a flexural pipe on an elastic foundation stiffness K f, in
terms of the flow velocity U: ST, stability; AI, absolute instability; CI, convective instability.
(a) f3 = 0.5 < ; (b)  < f3 = 0.87 < ; (c)  < f3 = 0,95 (de Langre & Ouvrard 1999).
Figure 10.31 Diagrammatic view of antisymmetric parallel-plate deformation ofthe type leading
to divergence and collapse (Miller 1960).
interest are the acoustic resonances excited by vortex shedding from the downstream end
of such plates (Parker 1966, 1967).
An equally long-standing interestexists in the dynamics ofparallel-plate fuel assemblies
used in some types of nuclear reactors - notably the engineering test reactor (ETR)
(Doan 1958) and the advanced neutron source (Swinson et al. 1992). The stability of such
systems is of most interest, particularly the static divergence and plastic collapse of the
plates at sufficiently high flows of the coolant.
Much work has been done towards theoretically modelling the dynamics of this latter
system; see Figure 10.31. Miller (1960) was the first to predict the critical flow velocity
for collapse (static divergence) by means of relatively simple theory based on wide-beam
theory - effectively equation (10.1) - and Bemoulli's theorem: thus, the critical flow

PLATES IN AXIAL FLOW
1183
z
!H
2
!H
2
y
!H
2
b
z
u
..
x
U
..
L
Figure 10.32 System considered by Guo & Pai:doussis (2000b): parallel plates with clamped
side-edges and free leading and trailing edges,
velocity, Ucd, is obtained by equating the pressure differences acrOss adjacent channels to
the elastic restoring forces on the plates, yielding (Miller 1960; Wambsganss 1967b)
( VH ) 1/2
Ucd= a pL4 '
(10,107)
where H is the interplate height (gap) through which the fluid flows, the other symbols
are as before, and a is a factor depending on the boundary conditions: a = 30 for simply
supported ends, and a = 181.2 for c1amped ones,
Johansson (1960) improved on Miller's theory by inc1uding fluid friction and flow
redistribution effects, Scavuzzo (1965) and Wambsganss (1967b) made further improve-
ments, by inc1uding nonlinear effects caused by large plate deflections. Further notable
contributions are by Rosenberg & Youngdahl (1962), Smissaert (1969), Davis & Scarton
(1985), Davis & Kim (1991) and Guo & Pa'idoussis (2000b). However, this is but a partial
list. Experiments have also been conducted; see, for example Groninger & Kane (1963),
Smissaert (1968), Swinson et al. (1992).
Here, a few words are said about the simplest fonn of these theories, considering the
flow to be inviscid and the dynamics line ar, based on the work of Guo & Pa'idoussis
(2000b), and some representative results are given.
The system under consideration is shown in Figure 10.32. It consists of a number of
paral1el plates, c1amped at their side-edges (i.e. at y = O and b) and free at their leading and
trai1ing edges (at x = O and L). It is assumed that all the plates have the same deflection
and shape at any time, but any two adjacent plates deflect in opposite directions; this fonn
of deflection is associated with the 10west critica1 flow velocity, as this involves the highest
blockage and hence largest added masses and 10west frequencies before collapse - cf.
Section 9.3.3.
The equation of motion is (10.4), with the perturbation pressure óp on the right-hand
side. The ana1ysis of the system is similar to that of Section 1 0.2.2(b) but more complex
because of the 3-D flow field and 3-D plate; it is not given here in detail. Suffice it to say
that the plate displacement is assumed to be of the fonn
M N
w(x, y) = L L Amn Wmn(x, y),
m=l n=l
(10,108)

1184
SLENDER STRUCTURES AND AXIAL FLOW
with
00
Wmn(x, y) = LankWmk(X, y),
k=O
Wmk(X, y) = Xm(x) cos(k1ry).
(10,109)
The corresponding perturbation pressure and velocity potentia1 are taken to have similar
forms; thus,
00
Pmn(x, y) = LankPmk(X, y),
k=O
CPmk(X, y, z) = <Pmk(X, z) cos(k1ry). (10.110)
Because of the antisymmetry of the flow with respect to any given plate, the perturbation
pressures on its upper andlower surfaces are equal butopposite; thus p = -2p[(acp/at) +
U(acp/ax)]z=O. The boundary conditions on cp are similar to those in Section 1O.2,2(b),
but here (acp/az) = O at Z = H, instead of H, because of the antisymmetry ofmotions
of adjacent plates.
The solution is obtained in the Fourier domain and, after inversion, a singular integra1
equation is obtained for the perturbation pressure component P mk (x), where Pmk (x, y) =
Pmk(X) cos(k:Jry); this is eva1uated numerica11y as in Section 10.2,2(b). Eventua1ly, an
equation of the form (10.61) is obtained, via which eigenfrequencies of the system may
be obtained.
Typica1 results are shown in Figure 10.33 in terms of the dimensionless frequency and
ftow velocity, defined by
( h ) 1/2
Q* = p Qb 2 ,
( h ) 1/2
U* = p Ub,
both nondimensiona1ized with respect to the plate width, b, The dynamica1 behaviour in
Figure 1O.33(a) is typical of small aspect ratio systems (here L/b = AR- 1 = 1), not
too widely spaced (H / b = 0.1). It is seen that the system loses stability at U* = 8.6 by
divergence in the first mode. It is restabilized at U* = 12.3, and then loses stability by
coupled-mode ftutter involving the second and third modes.
Figure 1O.33(b) is typica1 oflarger-AR (wider) plates. The system loses stability by
coupled-mode flutter involving the first two modes, and at higher U* by coupled-mode
Table 10.2 Dimensionless critical flow velocities, U;, for divergence (D) or coupled-mode flutter
(F), Divergence in the first mode is denoted as (D, I); in a higher mode as (D, hm), Coupled mode
flutter involving the i and j mode is denoted by (F, i + j).
Hlb Lib
0.5 1.0 2,5 10
0,1 11.1 (F, 1+2) 8,6 (D, 1) 4,7 (D, 1) 3.8 (D, 1)
0,2 13.1 (F, 1 +2) 11.6 (D, 1) 6.7 (D, 1) 6,2 (D, 1)
1.0 15.3 (F, 1 +2) 17.0 (D, 1) 10,0 (F, 3+4) 10.5 (D, hm)
5.0 15.4 (F, 1+2) 17,5 (D, 1) 10.1 (F,3+4) 11.0 (D, hm)

PLATES IN AXIAL FLOW
1185
40
30
20
Q"
*<>: 10
OJ
O
-10
O
3
2
..........
.........
".
...........
................
5
10
15
80
60
Q" 40
*<>:
OJ
20
O
4
3
2
1 ---
.-.-._.-.-.-..-.._._.::::::::::::
.-..... ...--
........ ..................:::::::::::::::::
O
5
10
u*
15
20
Figure 10.33 The real (-) and imaginary (. . , ) parts of the dimensionless eigenfrequencies,
Q and Q1 respectively, of the lowest few modes of the system as functions of the dimensionless
flow velocity U*: (a) for 11 = pbj(pph) = I, Ljb = I, Hjb = 0,1; (b) for 11 = 1, Ljb = 0.5,
H jb = 0,1; mode numbers are denoted by numerals (Guo & Pai'doussis 2000b),
flutter of the third and fourth modes also. It is recalled that, since the upstream and
downstream edges of the plates are unsupported (except at their extremities), it is not too
surprising that the system loses stability by flutter.
Other types of behaviour are also possible for different L / b and H / b combinations, as
shown in Table 10,2, It is seen that U; is lower for smaller H / b, which is similar to what is
found in Section 1O.2.2(a) for instance, and for higher L / b, which is physically reasonable.
Short and wide systems are prone to flutter, while long and narrow ones to divergence.
This can be explained physically by noting that for larger aspect ratios AR = (L / b) -1 the
support of the ends of the leading and trailing edges becomes less effective in constraining
the whole plate width, thereby rendering the system more highly nonconservative. The
dynamics at the right-hand bottom comer of the table, where stability is lost either by
higher-mode divergence or coup1ed-mode flutter, is more difficult to interpret physically.
Finally, it is worth noting that, in this case'also [cf, Section 10.2.2(b)], U cd ex: j),-1/2,
with j), = pb/(pph) here; but no such simple relation is available for flutter.

1186
SLENDER STRUCTURES AND AXIAL FLOW
The effect of dissipation has a1so been investigated, in the case where stability is lost by
divergence. Viscous damping, intemal (in the plate) or externa1 (in the fluid) has no effect
on d' but hysteretic (structura1) damping causes d to be raised drastically - however,
this dissipative model is unreliable at 10w frequencies, through which the system has to
pass as QŘ  O, AU types of damping have a destabi1izing effect on post-divergence
restabi1ization: for examp1e, for the case in Figure lO,33(a) the value of U;f is reduced
from 12.7 to 12.3 or 11.7, depending on the type of dissipation considered.
10.3.6 Cantilevered plates in flow
The possibi1ity of vortex shedding from the free downstream end of not-too-thin plates in
axia1 flow has a1ready been mentioned in Section 10,3,5, Flow-induced resonances ofthe
plate may result. In the case of multip1e para11el plates, acoustic resonances may also be
induced. The interested reader is referred to Toebes & Eagleson (1961), Eagleson et al.
(1964), Parker (1966, 1967), Blake (1986), Naudascher & RockweU (1994; pp. 120-
135, 157-158, 281-283) and Hayashi et al, (1999), for instance. The vortex shedding
per se is not treated here, although vorticity shedding at the trailing edge, associated with
osciUatory motion, is considered.
A c1assica1 paper on this topic is by Kornecki et al. (1976), involving both theoretical and
experimenta1 components. A more mathematica1 treatmentis Shayo's (1980). The prob1em
has also been studied by Huang (1995a,b), presenting a mechanism for the generation of
snoring; see also Tanida (2001), Several other studies have been conducted, some recently,
for examp1e, by Guo & Patdoussis (2000a), Yadykin et al. (2001), Tang et al, (2003), Refer
also to papers dea1ing with the nonlinear aspects of the problem in Section 10.3.7. Also,
there exists a distinct literature on the subject of 'strips', foils and flags in axia1 flow, which
is reviewed briefly in subsection (b) here.
10,3.6(a) Canti/evered p/ates
Kornecki et al. (1976) present an analysis of 2-D cantilevered plates in which the unsteady
fluid-dynamic loads are obtained according to (i) noncirculatory and (ii) circulatory theory;
in the latter, the circulation induced by the flapping free end is taken into account.
ln the noncirculatory theory, the fulI expression (10,37) is used, retaining time-
dependent terms, The plate deflection is assumed to be w(x, -r) = w(x) exp(i Q -r), which
leads to
d 4 wjdX 4 - Q 2 [w(X) - fL* h] - i Q fL*I/2 U ft - U2 Io = O,
(10.111)
where
A 1 1 1 dwjd
10 = Io(w, x) = - --=---d,
Jr o x-
A 1 1 1 { w() dw }
ft = ft (w, x) = -  + - ln lx -  I d,
Jr O X -  d

PLATES IN AXIAL FLOW
1187
x = xjL,
1 1 1
fz = fz(w, x) = - {w() ln lx - I}d,
Jr o
M* = pLjpph, U = (pU 2 L 3 jV)1/2, Q = (p p hL4jV)1/2Q
and T the corresponding dimensionless time. Then taking w(x) = L Cnwn(x) and
applying the Galerkin procedure, leads to a vanishing detenninant, from which Q i, i =
1, ..., 2N, may be detennined. When one of the Q i has Im (Q i) < O, the system is
unstable. As expected, and discussed in conjunction with equations (10.37) and (10.38),
the system 10ses stability by flutter. Taking N = 2, it is found that the dimensionless critica1
flow velocity for flutter, U cf = (p U;:f L 3 jV) 1/2, for flow on one side of the plate, is
U cf = 8.57 for M = O
- +
and U cf = 4,09 for M = O .
(10.112)
This at first sight perplexing finding is no different from the equiva1ent one for a pipe
conveying fluid, for tJ = O and tJ = 0+, discussed in conjunction with Figures 3.37
and 3.56. In the two problems, tJ or M* = O corresponds to a massless fluid, while tJ or
M * = 0+ to a fluid of vanishing but nonzero density. The explanation provided by Nemat-
Nasser et al. (1966) therefore applies equally here too. However, what is remarkable in
this case is the magnitude of the jump: by a factor of more than 2 in U cf.
The circulatory theory is based on that of thin flexible aerofoils (Bisplinghoff et al.
1955), involving the use of the Theodorsen function, C(k). Both the full version of this
theory and the quasi-steady version (C(k) = 1) applicable for 10w reduced frequencies are
employed.
Results by all versions of the theory for flow on both sides of the plate are shown in
Figure 10.34, for U cf and Q cf' and are compared to a single point from experiment s to be
discussed next. It is clear, however, that the quasi-steady theory grave1y underestimates
U cf, which means that the reduced frequency is not sufficiently 10w for this ap pro xi mat ion
to be reasonable. The dramatic reduction in U cf as M * is changed from O to 0+ is very
striking.
The experiment s are conducted in two different facilities, as shown in Figure 1O.35(a)
and (b). In both cases, the presence of the rigid stmcture downstream inhibits the full
rea1ization of circulatory wake effects. The gap c had a very strong influence on Ucf for
Configuration 1, but for Configuration 2 the effect was insignificant; hence, this is the
experiment with which theory was compared.
The comparison in Figure 10.34 shows that both circulatory and noncirculatory theories
underestimate the experimenta1 flutter velocity, but, consistently with the comments in the
previous paragraph, noncirculatory theory is doser. t The modest range of the ca1culations
with the full (unsteady) circulatory theory reflects the fact that they are not easy. On the
whole, agreement between theory and experiment is judged to be fair.
The importance ofthe gap c, at least for Configuration 1 ofFigure 1 0.35(a), i1lustrates the
importance of the fore and aft fluid boundary conditions. Another example is provided by
Horácček & Simerská's (1994a,b) theoretica1 work on the dynamics of a rigid plate hinged
at its mid-point, elastica11y supported and generally damped, and subjected to l-D or 2-D
potential flow (essentially as in Configuration 1). Immobile rigid extensions to the plate
tThe discrepancy between theory and experiment is part1y attributed to the theoretical underestimation of
the in vacuo stiffness of the p1ate,

1188
7
:r
4.-(b)
3
'-
i:::t 2
1
0.3
0,2
0,1
O ,
O 0.1
SLENDER STRUCTURES AND AXIAL FLOW
24
22
(d)
20 (c)
18
_(c) 16

IQJ
14
Ir
(b)
10
8--(a)
6
0.2 0.3 0.4 0,5 O 0,1 0.2 0.3 0.4 0.5
J..I.* /-1*
(b)
Figure 10.34 The dimensionless critical flow velocity for flutter, U cf, and corresponding fre-
quency nc f as functions of fL* for Configuration 2 ofFigure 10.35 (flow on both sides ofthe plate).
Within the figures: (a) noncirculatory theory; (b) circulatory quasi-steady theory; (c) circulatory,
full unsteady flow theory; (d) experiment (Komecki et a/, 1976).
(but not attached to it) are provided for -00 < x < O and L < x < 00, and an infinitely
long upper plate, a distance H above, similarly rigid and immobile, delimits a channel
containing the flow. It is found that for a narrow enough channel flow (Hl L = 1 I 1 O), the
damped system loses stability by flutter. t ln Horáček & Simerská (1 994a), the boundary
conditions for the perturbation pressure are applied at the ends of the plate: (a et> I ax) Ix=o =
O, [(aet> lat) + u (aet>IBx)]lx=L = o; as well as others for the velocity involving Bet>IBy at
y = O and H, Then, Horáček & Simerská (1994b) test the possibility of the perturbations
persisting beyond the plate: from x = -ft to x = L + h. The upstream effect is negligible,
If, however, the downstream boundary condition is applied at x = L + h instead of L, the
effect is important. Thus, taking /21 L = 1.5, instead of zero, almost doubles the critical
flow velocity for flutter.
Essentially the same physical system as Komecki et a/. has been studied by Huang
(1995a), with a novel application: the socio-medieal problem of snoring, which affects
about 20% ofthe adult population. Figure 1O.36(a) shows the soft palate which is thought
to be responsible; with flow on both sides when the mouth is open, the soft palate vibrates
in the pharynx during inhalation, causing the characteristic sound of snoring. Among the
several treatments practiced are (i) the cutting off of the posterior part (along white line),
(ii) the laser-assisted trimming thereof, and (iii) the stiffening of its lower surface (along
the other white line, perpendicular to the first) and attaching it to the hard palate. For the
vibro-acoustics of snoring, see Huang (1995b), Huang & Ffowcs Wi11iams (1999).
tlnterestingly. when undamped the system loses stability by divergence.

PLATES IN AXIAL FLOW
1189
u

:>
x
c
(a)
//
//,
z
u
:>
:>
NACA 64-008
airfoi\
pu! ! ! !! w (x, t)
/'
- I
- WI'I'-/ /H7'7'H-1'/ /  /; , 1
 \1.25 in - I pd 1  i t I L I
:> \Om--J c
:>
:>
x
(b)
Figure 10.35 The two experimental set-ups for a cantilevered plate in axial flow: (a) Configura-
tion 1, with the fluid in the cavity stagnant; (b) Configuration 2, with flow on both sides of the plate
(Komecki et al. 1976).
Huang uses essentially the same equations as Komecki et al. in their fully unsteady
circulatory theory; however, a purely numerical solution is adopted, whereby stability
is assessed from the evolution of the response with time. Experiments have also been
conducted, using aluminum-foil plates in a wind tunnel, in a similar arrangement as in
Figure lO,35(b), but without the downstream rigid pla!e. _
Typical results are shown in Figure 10,37, where U = U /(Q\b), Q = Q/Q\, Q\ =
(nt!I)2[V/(pphL4)]\j2withJTt!1 = 1.875, il = pb/(pph)andb = !L. These are plotted
with the amplification factor, which is defined similarly as in Section 1O.3.3(a),
G = exp(2JTQ/ /QR),
(10,113)
as parameter, The experimental results should strictly be compared to the threshold, G = 1
curve, in the absence of any nonlinear effects. As in the Komecki et al. experiment,
however, it is seen that theory underestimates the experimental flutter threshold. The
Komecki et al. data point, added to this graph, is wholly consistent with Huang's.
ln retrospect, comparing theory for G = 1 and experiment for G = 1.3-1,6 is strictly not
valid. t As pointed out by Huang (2002), though the means permitted it, no observations of
flutter with G < 1.3 could be made in the experiments, Whether this is a nonlinear
manifestation or due to the neglect of (i) acoustic radiation damping or (ii) material
inhomogeneity (e.g. due to the nonnegligible stiffness enhancement due to oxide layer
on the plate) is currently researched by Huang (2002) and perhaps others.
t Comparing theory with G = 1,45 to all experimental data. the underprediction of ftutter by theory is
around 25%.

1190
SLENDER STRUCTURES AND AXIAL FLOW
Soft palate
(a)
['.'
--
(b)
Figure 10.36 (a) The soft palate in the pharynx, in a cross-section of the human head; (b) model
for the study of palatal flutter, in which the hard palate is modelled as a rigid upstream support, and
the soft palate as a flexible cantilever attached thereto (Huang 200Ib).
4
2
00
o 2 0.4
ft
6 ... o .. 1M
,., ............It.JK)I( '" 
5 JI<: ...... ...--
4
IOJ
3
2
1
00
(b)
.............................
0,6
0,2 0.4
0.6
(a)
ji
Figure 10.37 The flutter boundary in tenns of the dimensionless flow velocity {;, frequency ratio
{2 = !;2 /!;21 and the amplification factor G, defined in the text, compared to experiment: ' , . , , , ,
theory, G = I; -, theory, G = 1.6; *, Huang's experiments; o, Komecki et a/, (1976) experiment
(Huang 1995a,c).
Huang's work has been extended by Tanida (2001), taking into account the presence of
the wind tunnel wal1s and compressibility. Figure 10,38 shows (a,b) Huang's theoretical
and experimental forms of a fluttering plate and (c) Tanida's theoretical result. It is noticed
that in the experiments the slope at the free end is much larger than in theory. It is hard
to imagine that for such large deflections the Kutta condition, implicitly or explicitly

PLATES IN AXIAL FLOW
1191
Hard palate time2
8 time6
7
0.5 1
(b)
8- 6 "
-5 O e
'" 2 
-0,5
-I
-I -0,5 O 0.5
(a) x/b
(c)
Figure 10.38 Fluttering cantilevered plates: (a) Huang's theory (ji = 0.3, fj = 3,5); (b) Huang's
experiment; (c) Tanida's theory (Huang 1995a; Tanida 2001).
assumed to hold in the theory, does in fact do so; refer also to Dowell et aZ. (1995), Dowell
(1995) and Tang et aZ. (2003). This may contribute to the discrepancy between theory and
experiment.
The analysis is extended by Shayo (1980) who considers 3-D (finite-Iength-and -width)
plates, thus using equation (10.4). Fluid-dynamic circulatory (wake) effects are neglected,
via two different approaches: (i) they are neglected in both the perturbation potential
and pressure [this then corresponds to the Komecki et aZ, (1976) noncirculatory the-
ory]; (ii) they are neglected only in the perturbation pressure. The problem is formulated
fully, but, since it is not possible to evaluate the pressure integrals exactly, an asymptotic
approximation is developed instead, for large aspect-ratio (AR) plates - similar in spirit
to those by Ellen (1972, 1973) for plates with supported ends. It is found that, for small
p,* = pLj pph, the values of Ucf according to (i) and (ii) above are close, suggesting
that perhaps circulatory effects are negligible; for p, * > 0.2, however, the results become
increasingly different, suggesting that such effects become important.
Typical results of U cf = (PU;fL3 jV)I/2 versus p,* are shown in Figure 10,39. The
discontinuity for p, * -+ O is not discussed. Clearly, as the plate width increases, the system
is stabilized for p, * < 0.47, but the opposite is trne for p, * > 47; no explanation for this is
available.
Table 10.3 gives a comparison of the results for p, * = 0.1 and 0.3 or 0.25 from the work
of Komecki et aZ., Shayo and Huang; the accuracy in some of the values given is limited by
the fact that they were read off the graphs. It is seen that the Shayo and Komecki et aZ. results
are virtually identical. Further calculations by Shayo show that the results are insensitive
to whether one or two cross-stream modes are used in the computation, showing that the
instability is dominated by streamwise deformation.
The agreement between Komecki et aZ. and Huang versions of the unsteady circulatory
theory is also close. Hence, the fact that they both underestimate the observed fl.utter
threshold lends weight to the suggestion that the circulatory effects do not materialize fully,

1192
SLENDER STRUCTURES AND AXIAL FLOW
4.8
3,0
w
.
/.'
/,
/..
/ '
/ "
//,.'
/.'
/ .'
/ "
/ "
/ .
/ .'
/ .'
/ '
/ .'
/ .
/ .'
/ "
/ .
/ .'
, .
;.......
4.0
I'tc
2,0
í
O
0.1
I
0.2 0.3 0.4 0.5
/-1*
f
0,6
Figure 10.39 The critical flow velocity for flutter, V cf versus JI * for large aspect-ratio (AR)
plates: -, AR = 00; ---, AR = 10; , , , , AR = 5 (Shayo 1980).
Table 10.3 Comparison of V cf and Qcf for flutter of a cantilevered plate, as obtained by different
researchers.
Reference
(a) Noncirculatory theory
Vef r2 e f
11*=O,} 11* = 0.3 11*=0,1 11* = 0,3
Komecki et a/. (1976) 3.2 3,9 19.4 18,2
Shayo (1980) for AR -+ 00 3,22 3,94 19,5 18,2
(b) Circulatory theory
Vef r2 c f
11* = 0.1 11* = 0.25 JI*=O.1 11* = 0.25
Huang (1995a) 1.78 2.87 20.1 19.6
Komecki et a/, (1976) 2.78 20.2
in the Kornecki et al. experiments because of the downstream rigid plate, and in the Huang
experiments perhaps because large amplitudes cause violation of the Kutta condition.t
Guo & Pa'idoussis (2000a) have also conducted ca1culations for cantilevered plates by
the same theory as in Section IO,2.2(b), neglecting circulatory effects. Typical results are
shown in Figure 10.40, where U c f := U;j J.1.*1/2 is plotted versus fJ = J.1.* /(1 + J.1.*). This
tThe c10ser agreement between experiment and noncirculatory theory than with circulatory theory should
not be interpreted as suggesting that noncirculatory theory is superior, but rather that circulatory theory needs
refinement - which is also Huang's view (2002).

PLATES IN AXIAL FLOW
50
40
30
I:::'''''
20
10
H 5,0-00
(a) O
I> :: 1 H=O.I 
O 0,2 0.4 0,6 0,8

(b) 1 +f1*
1193
Figure 10.40 (a) The critical flow velocity for flutter of a cantilevered plate in flow, in a channel,
versus f3 = f.L* /(1 + f.L*) for various values of the dimensionless channel height H ; (b) the same,
with a small amount of structural damping, y, included (Guo & Pai'doussis 2000a),
latter is similar to f3 in Chapter 3 for flow in pipes: its range is from zero for very light fluid
flows to f3 = 1 for very dense fluid flows. Since, however, this is equal to pLj(pL + Psh)
for the plate system, in this case f3  1 can a1so be achieved with L very large,
It is noticed that for f3 < 0.15 (p.,* ::: 0.18) for H :::: 5 this theory predicts
U cf ::: O, which is c1early incorrect. For lower va1ues of H , the range of p., * for which
U cf = O is predicted is larger. However, if even a very small amount ofhysteretic damping
is inc1uded [so that 'D is replaced by 'D(1 +irly )], then finite va1ues of U cf are obtained, as
illustrated in Figure 1O.40(b). However, for f3 = 0.3 and H  00, from Figure 1O.40(a),
one reads U cf < 2, which is much lower than the values in Table 1O.3(a). Clearly, there-
fore, this theory, at least for cantilevered plates, appears to be flawed. t Another, more
favourable comparison is given in Figure 1O.43(b).
It is a1so noticed that the U cf versus f3 curves display the very same 'jumps', at
f3 = 0.56, 0.85 for large H , that are observed for canti1evered pipes conveying fluid:
cf. Figure 3.30, for example.
ln most cases discussed so far, flow is on both sides of the canti1ever. As suggested by
Shayo (1980), results for flow on one side only may be retrieved from those with flow on
both sides by replacing p., * and U by 2p., * and .Ji U , respectively,
tlt is noted that, in contrast to the other analyses, the plate here is of finite length, with no extensions, The
difference in the results may be due to the boundary conditions for the fluid flow.

1194
SLENDER STRUCTURES AND AXIAL FLOW
For studies on nonlinear aspects of ftutter of cantilevered plates, refer to
Section 1O.3.7(d).
1O.3.6(bJ Canti/evered tapes, foi/s, strips and flags
The mechanism of the ftuttering of ftags has exercised researchers' curiosity for a long
time, going back to Lord Rayleigh and Thomas, as discussed by Kornecki (1971a, 1978);
nevertheless the problem is stili not funy understood.
Some remarkable experiment s have been perfonned by Taneda (1968), using muslin,
canvas and blanket materia!. Sample photographs for ftags of different lengths and AR 
0(1) are shown in Figure 1O.41(a) and (b). Once more, attention is drawn to the similarity
(a)
(b)
(c)
Figure 10.41 Multi-flash photographs of (a,b) fluttering flags, made of blanket material:
(a) L = 130mm, (b) L = 250 mm, and in both cases U =2.1 mis and f =4,7 Hz (Taneda 1968);
(c) multiflash photograph of a fluttering sheet of paper JL * = 2.70 (Watanabe et a/, 2002a),

PLATES IN AXIAL FLOW
1195
to fluttering pipes and cylinders in axial flow, Experiments have also been conducted by
Datta & Gottenberg (1975) using Mylar strips, and by Bejan (1982) using tissue-paper
ribbons. Experiments, using aluminum, cellophane, polyester and paper sheets have been
done by Watanabe et a/. (2002a); a sample photograph is shown in Figure 10.41 (c).
Watanabe et a/. also provide a useful discussion on common flutter problems in sheet- and
paper-web-handling machines; refer also to Section 10.3.9.
Sample theoretical studies are by Sparenberg (1962), Czaykowski et a/. (1966),
Yamamoto (1989, 2001), Yadykin & Levin (1998), and Watanabe et a/. (2002b).
Yamamoto uses a beam-type analysis and considers two cases (i) when the free end is
aft and (ii) when it is fore; in the latter case, the system loses stability by divergence. In
most of these cases, however, the strip is considered to be essentially a tensioned string,
with the variable tension generated by the drag (cf. Section 8.3,7) taken into account. Also
similarly to the work in Chapter 8, the fluid forces are modelled mainly by potential flow
but are modified by adding viscous forces, usually in the longitudinal direction only, and
sometimes also in the transverse direction,
A typical comparison between theory and experiment is shown in Figure 10.42, where
it is seen that theory underestimates stability, as for plates. Interestingly, the degree of
underestimation increases with decreasing strip thickness.
Watanabe et a/. (2002b) study the problem of sheet-paper flutter. t They use a beam type
model for the paper sheet and a linearized form of the Kiissner & Schwarz (1941) theory for
the inviscid component of the flow and Fung's (1955) approximation for the Theodorsen
function; however, they also include a viscous drag on the paper, and hence a variab1e
tension along the length. Typical theoretical results, not only from their own theory, are
compared to the Watanabe et a/. (2002a) and other experimental data in Figure 10.43.
Here (; = (pbL 3 / E /)1/2 U, where b is the sheet width, AR = b / L, and CT is the longi-
tudinal drag coefficient as in (8.13). First, it is seen that the theoretical results are broadly
consistent (but note the logarithmic scale). Then, a11 the theories are seen to underestimate
the experimental data. Recently, more sophisticated analyses have been undertaken, also
taking nonlinear effects into account; see Section 1O.3.7(d).
A novel application has been proposed by Allen & Smits (2001): to use piezoe1ectric
foils (called 'eels') in an ocean current, osci11ating in response to vortices shed from a bluff
body located upstream and convected along their length, thereby generating electricity.
This could possibly be modified, for sufficiently long foils, for the osci11ations to be
self-excited, even in the absence of the bluff body.
10.3.7 Nonlinear dynamics
A considerable amount of work has been done on the nonlinear behaviour of plates sub-
jected to flow; refer to the Mei et a/. (1999) review. Much of the early work, starting
in the 1950s, deals with high-speed, supersonic flow. This section comprises: (a) a brief
discussion on the geometrically nonlinear equations of motion of a plate; (b) a brief survey
of some key early references on plates in supersonic flow, with a couple of samp1e results;
(c) a somewhat more detailed discussion on plates in subsonic and incompressible flows;
(d) cantilevered plates in incompressib1e flow.
t Also applicable to 'edge-flutter' of unsupported paper web; refer also to Section 10.3,9,

1196
SLENDER STRUCTURES AND AXIAL FLOW
200
150
! 100
'c>
50
O
.
.
5
10 15
L (in)
20
25
Figure 10.42 Experimental data compared with theoretical results for the critical speed (in/s)
versus length for Mylar strips, 0.5 in wide, in air-flow: the three theoretical curves are (from bottom
to top) for h = 10-3, 1.5 X 10- 3 and 2,0 x 10- 3 in; the corresponding experimental results
(in the same order) are denoted by _, &, ? Note that I in = 25.4 mm; 200in/s = 5.08m/s;
25 in = 635 mm (Datta & Gottenberg 1975),
1000
°
100
'c>
<:::>
10
Q ooCj.._r. l!.
\Y  e:,l!. l!.
.. <. O
. .
OO$'
Ob.
.
+.--
4th mode
3rd mode
1/JI*
Figure 10.43 The dimensionless flow velocity for flutter, Úcj, versus 1/ J1 * for cantilevered sheets.
Watanabe et al. (2002a) experimental data: o, 'flag-size' paper (AR = 0,7); _, wider paper (AR =
2.3-4,0); l:::.., aluminum, cellophane and polyester sheets. Other experiments: o, Huang (1995a);
., Komecki et al. (1976). In Watanabe et al. (2002b) theory: thickest, medium, thinnest black lines,
CT = 0,0.1,0.2, respectively. Othertheories: +, Komecki etal.; thickgray line, Guo &PaYdoussis
(2002a); thin gray line, Huang; (Watanabe et al. 2002b).
1
0,01
0,1
10
10.3.7(a) Equatians af matian
Von Kármán's large-deflection plate equations are usually adequate for the analysis of
3-D plates, that is, plates of finite length and width, They are given by (Bolotin 1963;
Leissa 1969)
a 2 w a 2 F a 2 w a 2 F a 2 w a 2 F a 2 w
VV 4 w+p h----+2-----=jj.p,
P at2 ay2 ax 2 ax ay ax ay ax 2 ay2
1 ( a2w ) 2 a2w a 2 w
Eh V 4 F = ax ay - ax 2 a y 2 '
(10.114)

PLATES IN AXIAL FLOW
1197
where thenotationis as in the foregoing (see Section 10.1) and F is the Airy stress function;
x is the axia1, flow-direction. Note the simi1arity with the non1inear Donnell sha110w she11
equations (Section 7.8.2).
For 2-D p1ates, for examp1e p1ates of infinite width, the above equations become
effective1y 1inearized, and the following non1inear equation is used instead (Leissa 1969)
4 [ - Eh 1 £ ( aW ) 2 ] a2w aw a 2 w
'DV w - T + - - dx ---z + Cd- + pph = !:!"p,
2L O ax ax at at
(10.115)
where T represents an extema11y app1ied axia1 tension, and the other term in the brackets
is the non1inear deformation-induced tension, for p1ates axia11y restrained at their ends -
cf. Section 5.2.9(b).
Many variant s of these equations are often used; for examp1e, by inc1uding in-plane
forces in (10.114), or by rep1acing the first term in (10.115) by 'D(a 2 jax 2 ){(a 2 wjax 2 )j
[1 + (aw jax)2]3/2} which takes into account the non1inear curvature (Peake 2001).
10,3.7(b) Brief discussion on plates in supersonic flow
The ear1y work on the non1inear dynamics of p1ates in flow was for supersonic flow, with
obvious app1ications to high-speed aircraft and missi1e skins. Refer to the authoritative
treatments in Bo1otin's (1963)t andDowell's (1975) books, as well as Fung (1958), Dowell
(1966c, 1967a), Librescu (1965, 1967) and Ventres & Dowell (1970), for instance. How-
ever, as said in the Introduction (Section 10.1), the emphasis in the present book is on
subsonic and mostly incompressib1e, flow; hence very 1ittle will be said on the supersonic
work. The interested reader is referred to Dowell (1975), Dowell & Ilgamov (1988) and
Dowell et al. (1995).
Dowell (1966c, 1967a) uses non1inear shell equations simi1ar to (10.114) and (10.115)
for 2-D and 3-D p1ates, together with either quasi-steady or unsteady inviscid (potentia1)
piston theory. Various solution techniques yie1d supercritica11imit-cyc1e solutions with
amp1itudes of the order of the p1ate thickness; the effect of various parameters is studied.
Here, on1y one, representative result is presented in Figure 10.44, on the effect of in-plane,
axia1 tension (compression); in practice, such compression may be re1ated to therma1
expansion. It is seen that severa1 different types of dynamiea1 behaviour are possib1e:
divergence, harmonic and nonharmonic periodic flutter; see a1so Section 10.3.8.
Bo1otin (1963) distinguishes between (i) 'soft', that is, mi1d, divergence (for M < 1)
and flutter when structura1 geometrie non1inearities predominate, and (ii) 'hard', 1arge-
amp1itude instabi1ities when aerodynamic non1inearities predominate. There are many
fascinating facets of dynamica1 behaviour, for examp1e, the 'bottom1ess transonie bucket' ,
which are 1eft for the reader to exp10re in the aforementioned references.
Ho1mes (1977) recasts the equation in a form simi1ar to that pertaining to a pipe con-
veying fluid and ana1yses both systems by finite-dimensiona1 ana1ysis using the methods
of modem dynamica1 theory. His ana1ysis for the pipe prob1em is discussed in detail in
Section 5.5.2, cu1minating in the finding that pipes with supported ends cannot flutter, that
is, they do not deve10p post-divergence coup1ed-mode flutter. In the case of the p1ate, how-
ever, flutter both directly from stability or via divergence is possib1e. This Ho1mes relates
tThis is the English language edition of Bolotin's book, The Russian work on which it is based, by Bolotin
and others, dates back to the 1950s,

1198
600
SLENDER STRUCTURES AND AXIAL FLOW
500
400
IS 300
Stable
200
100
o
O
Hannonic L.C,O,
III a,b,c; IV
Divergence
-2
-4
-6
-8
-10
r/.n 2
Figure 10.44 Different stability regions for a plate in high supersonic ftow (M» I), where
-2 _ 2 3 2 1/2 _ - _ - 2 _ - _ - * _
U - pUL I[(M - I) V] and r = Rx - T L IV, for Lib - I, Ry - Rx, J-t 1M - 0,1
with J-t* = p LI pph (DowellI975); earlier versi on s in Dowell (1966c, 1967a). L.C.O, stands for
limit-cycle oscillations, and Nb for nonharmonic. The notation I, II, etc, has been superimposed to
discuss Holmes' (1977) findings.
IIIc
III
IIIb
lIla
Figure 10.45 Vector fields for the plate of Figure 10.41 (M» 1) on the centre manifold for
( U *2, r) near (107,8, -21.91) and ex = 0.005, /) = 0,1, where U *2 = pU 2 L 3 /[(M 2 - 1)1/2V],
r = TL 2 IV, ex is the hysteretic and /) the viscous damping coefficient; in this work, (107.8,
- 21.91) is c10se to where regions I, II and III meet - numerically a little different from Figure 10,44,
The 2-D stable manifold is shown schematically as a single attracting direction. The regions I-IV
correspond to those in Figure 10.44 (Holmes 1977).
to the presence of the [pU 2 L 3 j(M 2 - l)'D](ay jax) term in the equation of motions;
terms involving U 2 (ay j ax) are absent from the pipe equation. t Figure 10.45 shows the
vector fields on the centre manifold. In region I (see Figure 10.44), there is a sink at
tHowever, such a term does exist in the case of a cylinder in axia! flow, in the frictiona! forces, and flutter
does arise - see Section 8,3,3,

PLATES IN AXIAL FLOW
1199
{O}t and the p1ate is flat and stab1e; in region II, there is a sadd1e at {O}, surrounded by an
attracting limit cyc1e, hence flutter; in region III there is a sadd1e at {O} with sinks at {::I: I},
hence divergence; in regions lIIb,c which are c1ustered at the top of the divergence region
bordering flutter invo1ve a sadd1e at {O} and sinks at {::I:1}, with attracting and repelling
limit cyc1es, hence divergence ar flutter; region IIla is a flutter region, with sadd1es at {O}
and {::I:1} and an attracting limit cyc1e; region IV invo1ves sadd1es at {O} and {::I:1}, hence
divergence. This paper is often 'forgotten' in literature reviews, but repays carefu1 study.
Apparent1y unaware of Ho1mes' work, Bo10tin et al. (1998) study the same prob1em
numerically, exp10ring a1so 'remote' regions, for examp1e regions at the top of
Figure 10.44. A variety of new attractors are found, some periodic and some chaotic,
as well as several hysteretic phenomena. See a1so Section 10.3.8.
Dowell (1975) has a1so used a shear-flow aerodynamic model to consider boundary-
1ayer effects. Even though the nomina1 free-stream Mach number is transonic or
supersonic, the flow nearer the p1ate is subsonic. This has a profound effect in the nominal1y
transonic range, and it has reso1ved a major discrepancy between theory and experiment.
The literature in this area is rich. Samp1es of recent contributions are: Han & Yang
(1983), Ye & Dowell (1991), Chandiramani et al, (1995).
10,3.7(c) Plates in subsonic and incompressible flows
The asymptotic nonlinear stability of p1ates in incompressib1e flow (on one side on1y) for
1arge and small aspect ratios, AR, has been studied by Ellen (1977b), considering both
structura1 and fluid-dynamic non1inearities, obtaining (i) for 1arge aspect ratios, AR» 1,
U 2 == pU 2 L 3 IV = 40.0{1 - 0.82(hpl L)(wol h p ) + 0.75(3 - v 2 )(wol hp)2},
(10.116)
where h p == h, the plate thickness; and (ii) for smal1 aspect ratios, AR« 1,
U *2 == pU 2 b 3 IV = 138{1 - 0.52(hpIL)(wol h p ) + l.72(wol hp)2},
(10.117)
where b is the p1ate width and Wo the amplitude of divergence. Keeping in mind the
smaUness of h piL for thin p1ates, it is seen that there is a very small region of subcritica1
behaviour (due to fluid-flow nonlinearities) for w I hp of the same order as h pl L; but for
more rea1istic va1ues of wl hp, of 0(1) or more, this is swamped by an overall harden-
ing effect. This agrees with the results obtained by Dowell (1967a) for wlh p "-' 0(1)
shown in Figure 10.46. It is interesting that, because of the linear terms in w I h p in
equations (10.116) and (10.117), the two potential wells associated with divergence are
asymmetrically located vis-a-vis the origin, the positive one being slightly farther.
Lucey et al. (1997) provide an excellent, compact literature review of what they aptly
term 'this deceptively straightforward system'. They then examine the dynamics of a finite-
length plate, mainly in the post-divergence regime where coupled-mode flutter osci11ations
may arise. The equation ofmotion is essentially (10.115), but generally with a term Kfw
added for the effect of an elastic foundation. The flow is considered to be inviscid, with
velocity
U(x, t) = Ui + VcjJ(x, t),
cjJ(x, t) =  f a(xs, t) ln lx - xs(t)1 ds,
2n
(10,118)
tThis is Holmes' notation, defined also in Section 5,5,2.

1200 SLENDER STRUCTURES AND AXIAL FLOW
1200
1000
800
b 600
400
200
O
O 1.0 2,0
M
Figure 10.46 The critical velocity U cr for divergence (M < 1) and flutter (M > 1) as a function
of Mach number, for wlh p --+ O and wlh p = 1, and /1* = 0.1: -, AR == biL = 1;
- - -, AR --+ 00 (DowellI967a),
1.0
....t.v.:"''I.'''''''''''''''.''''''"''.'''''''-I _.;.........H...../.:...I>...""!...\..#..I_\. .
, :;" ....::.
.....:;.:.,....'4.:.:::#..,..:..:......... ::"):::-i"""';"""'.>:::.:"t;:'r'
.' f' '..). "
< :"'.. ...........,.......... ....-"'..............,.. \ "
 .'
.
" \.................-'" ..........,'..........".... , "
\ "- I .:/
.0::", ...
., ...;::......:::,...:.-.,...'.,.........):.:."(. ..--II':::::...:..!:.;:..:.".,...t.:::-....
,'.
.... ....;:
. :"-.\............\..,.....i...:...................,._ ...........I"'-'....................:......?..
....
o
-
X 0,0

"'"
.
-1.0
-1.2
0,0
w/h
Figure 10.47 Phase portrait for undamped plate (hl L = 1O2), showing responses at xl L = 1
at U = .J6l ::: 7.8; x denotes the location of the attractors, that is, the potential wells (Lucey et a!,
1997),
1.2
(j being the intensity of sources distributed over the plate-flow interface, XS' The solution
of the coupled problem is obtained by boundary-element and finite-difference methods.
Lucey et a/. first examine the dynamics of an undamped system without elastic found-
ation, for U == (pU 2 L 3 jV)I/2 above the critical value of .J4O = 6.3 approximately,
specifically for U = 7.8 and 15.6. The dynamics, as conjectured by Ellen (1977b), are
summarized in Figure 10.47: t small initial perturbations cause the system to oscillate
about one or the other potential well, while larger perturbations give rise to osci11ations
tlf equation (10.117) is correct, then, for hp/ L = 10- 2 , the asymmetry due tothe linearterm when wo/ hp =
10- 2 would be of the order of 1O3, hence hardly visible in Figure 10.47,

PLATES IN AXIAL FLOW
1201
about both wells. The oscillations, of essentially fundamental (first-mode) shape have
a pronounced nonlinear (nonharmonic) wavefonn; they also display a travelling wave
component (of increasing importance for higher U ), similarly to Figure 3.13 for a pipe-
string. Inclusion of damping causes the oscillation to eventually spiral to one of the two
potential wells, The reader is referred to Figure 5.9 in Volume I, where similar dynamical
behaviour is discussed for an undamped pipe conveying fluid. Here, however, periodic
phase-plane orbits are possible at U much below that for coupled-mode flutter, predicted
by Weaver & Vnny (1970) for example. Moreover, many cycles of oscillation take place
before the oscillation is finally quenched. Plots of the maximum amplitudes of oscillation
of the undamped system give the following relation:
U;  40[1 + 1.5(wo/ hp)2], (10.119)
which is c10se to Ellen's equation (10.116), However, Ellen's result is for divergence,
while here the results are for oscillations; t thus U c here is the value of U beyond which
oscillations of amplitude Wo are possible.
Even more interesting results are obtained with the plate on an elastic foundation, show-
ing that oscillations, tenned downstream-travelling divergence waves, are possible for
flowvelocities above that for divergence, (;ed = 2n: 3 / 2 , where (;2 = 3n: 3 pU 2 /(3'D K} )1/4.
Typical results are shown in Figure 10.48. In this case, the critical wavelength is seen to
be small compared to the plate length, in contrast to the predominantly first-mode shape
displayed in the absence of an elastic foundation.
The dependence of amptitude on (;2 - (;;d is shown in Figure 10.49, The actual
wavelength of the oscillation is always close to the wavelength of the divergence instability,
Aed = 2n:(3'D / K f) 1/4, hence the notation of divergence waves.
Wu & Zhu (1997) consider baffted 2-D plates in compressible fluid flow. The res-
ults show that, for post-critical flows, while the overall displacement amplitude may be
bounded, the plate vibration may be locally unstable, jumping from one equilibrium pos-
ition to another. The jumping may be random, and the vibration appears to be chaotic.
These local instabilities are controlled by structural nonlinearities,
An important contribution is by Sorokin (2000), who studies 2-D plates, yet includes the
effect of nonlinear tension as in equation (10,115) by supposing the plate to be periodically
supported - so limiting the wavelength of interest for any given support interval. Sorokin
finds two regimes for fluid loading, depending on jl, = p / (kpph), k being the wavenumber,
k = (pphQ2/'D)1/4: light fluid 10ading is defined tocorrespond to small values of jl" while
heavy fluid loading to large ones (hence for smaIl frequencies), ln the heavy fluid-Ioading
regime, nonlinearities, not only in the plate but also in the fluid equations, should be
accounted for.
The 'deceptive simplicity' of this system is further demonstrated by Peake (2001),
who studies analytically the nonlinear stability of 2-D plates, similarly to Sorokin, for
heavy fluid loading; hence, he considers both plate and fluid nonlinearities. Dashpots
are included, distributed along the plates, and special attention is paid to negative energy
waves (NEWs), that is, class A instabilities of Appendix C, which are excited by the
dashpots, It is shown that NEW s may be stabilized by the nonlinear tension, without the
tEllen' s one-mode analysis could not possibly detect these oscillatory solutions, in any case, Other instances
of one theory predicting divergence while another predicts fluner, with similar critical flow velocities, are given
in Section 1 I. 7 .2,

1202
SLENDER STRUCTURES AND AXIAL FLOW
0.3
(a)
oJ::

'()
-0.3
O
3,0
(b)
) 0.0
-3,0
O
x/L
Figure 10.48 Instantaneous profiles of the plate for an elastically supported plate with 1ight damp-
ing (š = 5%), for (; = 16.66, hp/Acd = 0.345, p/pp = 0.852, The time between curves in each
figure is !:!.t, in the sequence -, - - -, -'-, - - - . (a) Response for early times, t = O to
3!:!.t; (b) nonlinear response at later times, t = 22!:!.t to 25!:!.t (Lucey et a/. 1997).
10 1
.;n
oJ:: 10° I í

N

10- 1
10 1
10 2
1\2 1\2
U -U cd
10 3
Figure 10.49 Dependence of amplitude A of nonlinear divergence waves on flow speed, expressed
as (;2 - (;'1d: 0, nonlinear flow solution; ., linear flow solution (Lucey et al. 1997),
need of wall shear. The nonlinearities act to increase the plate restoring stiffness until it
balances the destabilizing effect of the fluid flow. The rate of working of the dashpots is
proportional to Q2 , where Q is the dimensionless frequency; hence, Q is steadily reduced
and at the saturated state of amplitude we have Q = O, that is, divergence, Thus, with
time, the oscillatory motion evolves to a static deflection of constant amplitude, as found
by Lucey et a[, (1997).

PLATES IN AXIAL FLOW
1203
Some interesting results are also obtained by Peake with regard to the effect of non-
linearities on convective instabilities. For 2-D motion, the wave-train is defocused and
decays with time. If spanwise modulation is inc1uded, however, wave focusing and hence
solitons can result (Drazin & Johnson 1989). The spatio-temporal evolution in this case
is shown to be govemed by a nonlinear Schrodinger equation. t
10.3.7(d) Cantilevered plates in incompressible flow
The work presented here relates directly to that of Section 10.3.6.
Yadykin et al. (2001) undertake a nonlinear analysis using beam-type nonlinear
equations for the structure, similar to those of Lundgren et al. (1979) discussed in
Section 5.2.8(c), thus also taking axial motions into account; the plate is considered
to be inextensible, These equations are modified by inc1uding longitudinal and transverse
frictional forces as in Chapter 8. A linear potential flow formulation for the inviscid forces
is adopted. A Galerkin-type discretization is used, and solutions are obtained numerically.
Typical results for the critical flutter velocity are shown in Figure 10.50, compared to
the experimental data of Datta & Gottenberg (1975). It is seen that the Hopf bifurcation
obtained must be subcritical, since nonlinear limit-cyc1e solutions exist below the linear
flutter threshold; this, unfortunately, further distances theory from experiments.
Figure 10.51 gives a bifurcation diagram for the free-end displacement and fre-
quency. The following regimes are distinguished: (i) limit-cyc1e motions arise for
2.1 < U < 2.38 mis, with the motion being periodic and of a standing-wave form;
(ii) a jump, mainly in frequency, occurs at U :::: 2.38 ml s, but the character of the motion
remains the same; (iii) for U > 3.08 mis the motion becomes 'irregular' and with a
strong travelling wave formo This last transition is in qualitative agreement with Datta &
Gottenberg's observations. Here the reader is referred to Figure 10.38.
Another recent study is by Tang et al. (2003). The equations of motion for the structure
are similar to those of Semler et al. (1994) for inextensible systems, while the fluid forces
are obtained by a linear 3-D vortex lattice method, as in Tang et al. (1999). The equations
of motion are obtained via Hamilton's principle, discretized, and solved numerically.
Experiments are also conducted, with a rectangular aluminum cantilevered plate, 127 mm
wide, 267 mm long, and 0.394 mm thick, in a wind tunnel.
ln the experiments, a small-amplitude limit-cyc1e oscillation (LCO) developed at U ::::
29.1 mis. Its amplitude increased with U, and at U :::: 30m/s there was an explosive
increase in amplitude. Indeed, because of the increased wind-tunnel flow blockage, this
resulted also in a sudden drop in wind speed to U = 28,7 mis. Thus, in this case also
there are two regimes: small and large amplitude LCOs. When reducing the flow, the LCO
finally disappeared at U = 27.3 mis, and again there is ajump in U, this time upwards to
U = 27.8 mis. Clear1y, the wind tunnellimitations pose problems in the interpretation of
the results and comparison with theory. In the theoretical calculations, the Hopfbifurcation
is supercritical.
A comparison between theory and experiment is shown in Figure 10.52. It is seen
that theory slightly overestimates the small-amplitude LCO threshold, the theoretical
threshold being c10ser to that of the large-LCO onset. However, theory does not track the
tThe reader is a1so referred to another series of studies on nonlinear dynamics of plates, exemplified by
Awrejcewicz et a/. (2002); the plate, subjected to longitudinal time-dependent loading, exhibits spatio-tempora1
chaos and solitons,

1204
4.5
SLENDER STRUCTURES AND AXIAL FLOW
...
4.0 ...
 ...'" ... ...
'"
g ...
::::> 3,5
6
u 3,0
o
O)
;-
tO 2,5
.
.E
u
2,0
0,0 0.1 0,2 0,3 0.4 0.5 0,6 0,7 0,8 0.9 1.0
Length of strip, L (m)
Figure 10.50 The critical flow velocity for cantilever flutter, Ve!, versus length, L: ., .,
theoretical threshold with linear theory (for N = 4 and 8 modes in the Galerkin discretization);
o, O, corresponding threshold with nonlinear theory; ., Datta & Gottenberg's experimenta1 data
(Yadykin et a/. 2001).
20 ...... ';.......:.. ....., ......:.......;
., "
, ,
. ,
! 12::::, uUjuu.[FU
4

a
8 -4
..
].
:a -12
Po.

a


i
<I)
 o
u
..
]. -I
:a
Po.
 -2
-20
. . . . .
: : : .:: .
. . ... . . .:. .. _ ...  .... . . . .. . . . .. .. . . . .. to . . ..:
: : : :.:
, , ,
. , .
. . . o.
.......... .:............. ........... .....:.............. ..........:
. . . . .
:20 :::',.:.:3ď.::.40
. ... ..
. , .
. ....
. . .. . .. .  .. . . . . . :. .......;........!........:..
. . .. . .
. . .. . .
.. o.
,. .
. . .. .
.......... ......................: - -.....:..
. ,
, ,
. ,
. ,
-3
2.0 2,1 2,2 2.3 2.4 2.5
(a) Velocity, U (mis)
. . . . .
. . . . .
15 .... '--;.......:.. .....;........ .....;
, ,
. :.
.: .
N' 12
e;
.......
;;: 9
u
<=I
<I)
& 6


. . . . .
.............--........,.................
. . . . .
. ""
. ""
; . : :.'
. . . . .
..........................................
" ,
:. .:
. : .
... _.:._ ou..;.... .. '..u ,;
, "
, "
: :. : :
3 .......: ......:.......:.............:
: : : .:. :
" ,
" .
" ,
o
2,0
3.2
3,6
4,0
, . . . .
, . . . .
6 7 . . . . .
. .......:"............... -............,_.... -...........
. . .
N'6.2
e;
"-,
;;:5.7
u
<=I
<I)
&5,2


'--.r ..----.......... ....... ........,--
, .
, .
........._--..._----........ ..-........
.......---.................. ....._--....
" ,
, ,
.........--........
4.7
4,2
2,1 2,2 2.3 2.4 2.5
(b) Velocity, U(mls)
Figure 10.51 Numerical bifurcation diagrams of (a) free-end (tip) amplitude and (b) frequency
versus flow velocity for cantilever flutter at post-Hopf bifurcation flows, showing a1so a blow-up of
the circled regions (Yadykin et a/. 2001).

PLATES IN AXIAL FLOW
1205
0.4
:::!

 0,3
,
Q.
i
] 0,2
'f
01)
]
'
 0,1
"
Q.

,,/I em --- '.
Expenment
o
27 28 29 30 31 32 33 34 35
Flow velocity, U (mis)
Figure 10.52 Comparison between theory and experiment of the dimensionless peak free-end
displacement versus U for a cantilevered plate in flow: ., experimental data with increasing flow;
o, the same with decreasing flow; -f:,,- , theoretical result (Tang et al. 2003).
explosive increase in amplitude seen in the experiments. Whether this is a weakness in
the theoretical model, which does not include aerodynamic non1inearities, or is inherent
in the plate-amplitude/tunnel-flow interaction remains to be seen.
10.3.7(e) Other work
ln most of the foregoing, inviscid flow theory is mostly used. However, increasingly
fuli Navier-Stokes codes are developed and used in the analysis; see, for example,
Gordnier & Visbal (2002), Watanabe et al. (2002b). AIso, finite-element techniques have
been used for a long time in this area; see, for example, Han & Yang (1983).
10.3.8 Buckled plates in flow
No treatment of the dynamics of plates in flow would be complete without a discussion of
this topic. For one thing, as pointed out by Moon (1992), while study ing the vibrations of a
buckled plate in supersonic flow, Kobayashi (1962) uncovered chaotic vibrations, and this
a year before Lorenz's classical paper for chaos in the atmosphere, Similar observations
of nonperiodic motions were made in another important paper by Fung (1958),
Also, in addition to Matsuzaki & Ueda's (1980) paper, already discussed in
Section 10.2.2, the analysis of buckled plates and channe1s in incompressible flow by
Matsuzaki & Fung (1977b), briefly discussed in Section 7.3.6(b), and Matsuzaki (1981)
should be mentioned, Here, however, in keeping with the aim of making this chapter
brief, ouly a discussion of Dowell's (1982, 1984) cIassical papers on chaotic motion of
the buckled plate is presented.
Dowell considers high supersonic fiow over one side of a fiat plate, subjected also to
in-plane compression, that is, the problem studied earlier by Dowell (1966c, 1967a)-
see Section 1O.3.7(b) and Figure 10.44. This is an autonomous detenninistic system.

1206
SLENDER STRUCTURES AND AXIAL FLOW
Using !inear piston theory aerodynamics, the equation of motion is equation (10,115)
withD.p = (pU/M)[(aw/at)+u(aw/ax)], where M isthe Mach number. Theequations
are discretized using the Galerk:in method, taking w = Ln a n sin(n:rrx/ L). The dynamics
become easy to grasp if a two-mode approximation is taken, leading to
A " (1 -2 - 1/2 I ( 2 2
1 + AI + r) - U A2 + UC,1 AI + AI + A 2 )Al = O,
" A -2 - 1/2 I 2 2
A 2 + 2(4 + r) + U AI + UC,2 A 2 + (AI + A 2 )A2 = O,
where numerical constants have been omitted for clarity, and An = ani h, u 2 =
pU 2 L 3 /MV, c'n = pL/(PshM), r = T L 2 /'D, T = t('D/p p hL 4 )1/2 and ()' = a()/a!'.
It is seen that divergence occurs if r is sufficiently large and negative. Plutter, of the
coupled mode variety, occurs because of the skew-syrnmetric terms involving U 2 .
ln Figure 10.44, it is seen that regions of periodic but nonharmonic osci1lations have
already been delimited. Dowell (1982) pursues this study further, numerically, and finds
that, for sufficiently high negative r, for example, -4:rr 2 to -6:rr 2 , and U 2 = 115-200,
chaotic osci1lation occurs. Figure 10,53 gives examples of representative phase-plane plots
showing (a) nonharmonic periodic motion and (b) chaotic motion. The chaotic nature of
the dynamics has been confirmed by the character of PSDs and Poincaré maps - cf.
Section 5.8,
The underlying mechanism leading to chaos is related to the proximity of the potential
wells to the ftutter attractor, Thus, it is no accident that, for a fixed r = -4:rr 2 , chaos
-2
becomes 'stronger' as U is reduced from 200 to 150 and then to 115, as done by Dowell
(10.120)
"'''' T '
'" "
'" ,
'" w ,
p,u- . ",,,,,,,'" ! "
;;m.----"'- 
(a)
8
21
17
13
9
5
24
16
w
w
w
-3
-7
-11
-15
-1.0 -0,5
(c)
o 0.5
w
1.0
o
-8
-0,7 -0.3
(b)
0.1 0,5
0.9
Figure 10.53 (a) The buekled plate system under eonsideration. (b,e) Phase-plane plots of the
2 -2 2 -2
plate at xl L =0,75, for (b) r = - 3n , U = 150, and (e) r = - 4n , U = 115 (Dowell1982).

PLATES IN AXIAL FLOW
1207
(1982, figure 5) - where by stronger it is meant that the phase-p1ane trajectories become
more comp1ex. Here the reader is referred to Sections 5.7.3 and 5.8.4 for a discussion of
chaos near doub1e degeneracies, and to Moon (1992) for a discussion of chaos in buckled
systems under extema1 excitation.
Bo10tin et al. (1998) have further studied this prob1em and discovered new attractors
and new chaotic regions e1sewhere in the parameter space of Figure 10.44, notab1y in
'remote' post-critica1 domains.
10.3.9 Travelling web, sheets and tapes
This topic has obvious app1ications to paper web (i,e. continuous rolls of paper) trav-
elling between rollers in paper-handling machines, as well as other types of thin
materia1s, for examp1e, p1astic sheet web, going through manufacturing machinery, print-
ing, and recording/p1aying machines. Web ve10cities can be re1ative1y high: 15-30mls
(50--100 km/h).
Thus, in a way, this topic is simi1ar to that of dep10ying/extruding cy1inders, discussed
in Section 8.3.8, but with a crucia1 difference. In the case of cy1inders, wire and other
string-1ike structures, the effect of added mass (and hence the tota1 dynamica1 effect of
the surrounding fluid) is minima1 if dep10yment or extrusion is in air, since the correction
due to the added mass wou1d be of the order of P I Pp::::: 10- 3 -10- 4 , where P is the air
density and Pp the density ofthe structure - c1ear1y a neg1igib1e correction. However, in
the case of a paper sheet flapping between rollers, the added mass per unit 1ength wou1d
be of the order of p(7r 14)b 2 , where b is the width of the paper; this is not neg1igib1e in
comparison to the paper mass per unit 1ength, pphb, since bl h is usually quite large t -
easi1y of 0(10 4 ), or more.
In fact, in the ear1iest theory on the stability of travelling paper, the trave11ing threadline
model (TTM) of Mujumdar & Doug1as (1976), all fluid-structure interaction is neg1ected.
Prami1a (1985, 1986) initially inc1udes air-re1ated added inertia, damping and stiffness
terms and conc1udes that TTM overestimates the thresho1d of instabi1ity by a factor of
3 or 4. Subsequently, however, this has been revised, and on1y the added mass effect is
retained (Prami1a 1987), as further discussed in what follows.
Here, the dynamics of web is discussed (a) as it trave1s between rollers in quiescent or
axia11y flowing air, taking fluid effects into account, and (b) with air-flow in the cross-trave1
direction (the so-ca11ed 'edge-flutter' phenomenon). The dynamics of web in the absence
of any fluid effects is not discussed; for this, the reader is referred to the well-deve10ped
and vo1uminous 1iterature on axia11y moving media (Mote 1968, 1972; Wickert & Mote
1989, 1990).
10.3.9(a) Dynamics of travelling web
Perhaps the first to take aerodynamic effects into account in the dynamics of trave11ing
web (Figure 10,54) is Prami1a (1985, 1986), who proposed the foUowing equation for
tensioned web:
a 2 w ( a a ) 2 ( a a ) 2
-T-+M -+V- w+m -+V- w=O,
ax 2 at ax at ax
(10,121)
tRefer to Section 8.9.10 on fish locomotion and the large added mass associated with a thin, fiat taiI.

1208
SLENDER STRUCTURES AND AXIAL FLOW
L
x
w
Figure 10.54 Travelling web in quiescent air (Pramila 1987),
where V is the speed of travel of tbe web, of mass per unit lengtb m; M =a (%pb 2 )
is the fluid added mass per unit lengtb, b being tbe web width. Generally, a < 1; see,
for example, Greenspon (1961) and Meyerhoff (1970), and refer to Section 10.3.1. In
equation (10,121), it is imp1ied tbat, so far as fluid-structure interaction is concerned,
tbe flow of air relative to tbe web is equiva1ent to a forward potentia1 flow of the same
magnitude over a stilI web, tbus justifying the second tenn in the equation, Later, Pramila
(1987) interprets this equation as meaning tbat the web entrains the forward motion of
a certain bulk of fluid of mass M per unit length, as if it were glued to tbe web, which
is c1early not true. Re, tberefore, corrects this equation by dropping the Coriolis and
centrifugal tenns, while retaining the added mass term, so tbat it becomes
a 2 w a 2 w a 2 w a 2 w
-T- +mV 2 - +2mV- + (m +M)- =0,
ax 2 ax 2 axat at 2
It is suggested further that the truth 1ies somewhere between equations (10,121) and
(10.122). Chang et a/. (1991) and Moretti (2001) re-examine the question more gen-
erally and propose that tbe second tenn in (10.121) should be Ml (a 2 wjat 2 ) + 2M2 V
x(a 2 wjax at) + M3 v 2 (a 2 wjax 2 ), where Ml = M, M2 = 2pb8*, M3 = 2pb(8* - O),
where 8* and O are tbe displacement and momentum thickness of tbe boundary layer,
respectively. Rowever, forambientair, wehaveM2, M3« m, andhenceequation (10,122)
is retrieved,
Since equation (10.122) may be difficult to accept conceptually, the following inter-
pretation may be found usefu!. The travelling band ofFigure 10.54 may be imagined to be
witbin a stationary massless and frictionless sheath, Then the system is similar to a pipe of
vanishing mass containing an ideal fluid of mass m per unit length flowing witb velocity
V, immersed in an outer stagnant fluid (so that tbe added mas s is M per unit length). t
Experiments on tensioned sheet travelling between rollers (Figure 10,54), excited to
vibrate in its fundamental mode (of frequency fr), were conducted by Pramila (1986,
1987), The dimensionless frequency is plotted versus the dimensionless web velocity in
Figure 10,55, where
-- 1/2
ft = 2L fr(mjT) ,
(10.122)
V = V(mjT)I/2,
(10.123)
tFor a fish travelling forward with velocity V, we are interested in the forces on the fish itself, and so we
have m(a 2 w/at 2 ) + M(ajar + va/ax)2 w = O (Section 8,9,10). For the extruding beam of Section 8,3.8, the
situation is similar to that of the travelling web (Mote 2002): the surrounding fluid is not flowing, and so there
shou!d be neither Coriolis nor centrifuga! terms, 2MV(a 2 w/ax ar) and Mv2(a 2 w/ax 2 ) -- unless one treats
M in these terms as the mass of the axia!-motion-entrained fluid layer or, a!temative!y, suitably reinterprets V,
so that either M  O or V  O.

PLATES IN AXIAL FLOW
1209
1.0
0,8
0,6
'''':::
0.4
+ +
0.2 +..::..--t'-._._._._.._.._._..__
"
,
0,0
(a)
'," ::  ------__________ __ I
01  , , 
° 0,2 0,4 0,6 0,8 1.0
(b)
v
Figure 10.55 The dimensionless first-mode natural frequency, II' versus the dimensionless flow
velocity, V , for paper web with (a) m = 0.017kg/m and (b) m = 0.025kg/m: -,11M theory;
- - -, equation (10.121); - - -, equation (10.122); +, experimental data (Pramila 1987).
and they are compared with analytical results from TTM theory (Mujumdar & Douglas
1976) and equations (10.121) and (10,122), It is seen that TTM greatly overestimates 71
vis-a-vis both equations (10.121) and (10.122), as it neglects the added mass. Equation
(10.122) follows the trend of the experimental data best, though one would have wished
the range of V covered in the experiments to have been larger. t It is also seen that
V c d = 1 for both TTM and equation (10.122) since, for ft = O, the added mass effect
vanishes.
Some additional work, this time on axial flow over stationary web, in practice represent-
ing air-flow for drying purposes (flowing much faster than the web), has been conducted
by Chang et al. (1991), Chang & Moretti (1991). The web is placed in a wind tunnel and
tensioned via a deadweight - thus allowing axial movement. The equation of motion in
this case is
a 2 w ( a a ) 2 a2w
-T-+M -+U- w+m-=O;
ax 2 at ax at 2
(10,124)
tNonlinear effects must be quite important for paper web, unless the supports allow some axial movement;
otherwise, divergence would manifest itself by stretching the paper, and paper is quite inextensible. Here refer
to Liu & Mote's experiments on a pipe conveying fluid: Figure 3,26 and the related discussion. The nonlinear
tension due to stretching is as in equation (10,115).

1210
SLENDER STRUCTURES AND AXIAL FLOW
0.4
Q 01
E,'
"

-,,,
Theory
Stable
0,01
0,01
0,1
(2/n) (Tlb 2 ) (psi)
0.4
 0,1
8
"S

-,,,
Theory
C
0.01
0.01
0.1
(2/n + pb 2 /2m) (T/b 2 ) (psi)
Figure 10.56 Comparison between theory and experiment for tensioned web in axial air-flow,
showing the critical flow velocity for (a) divergence (Ved) and (b) flutter (Ve!), expressed as
p V;:d' and p V;:f' for web of different dimensions (width x length): 0,3 x 10; D, 3 x 20; +,
6 x 10; x, 6 x 20 in; 1 in = 25.4 mm; 1 psi::::: 6.9 kPa (Chang et aJ. 1991).
assuming that w = wexp[ik(x - Cf)], it is easy to obtain the critical fl.ow velocities for
divergence and fl.utter:
2 T 4T
V - - .
ed - M - 7rpb2 '
2 T ( m ) T ( 4 P b 2 )
Ve! = m 1 + M = P b 2 -; + --;;;- .
(10,125)
A comparison between theory and experiment is shown in Figure 10.56. These results
should be compared to those for a cylinder with pinned ends in axial fl.ow (Figure 8,20). The
degree of agreement is similar, though here theory underestimates Ved and overestimates
Ve!, while the opposite is trne for the cylinder.
Among other pertinent work is that by Niemi & Pramila (1987), Koivurova & Pramila
(1997) and Koivurova & Salonen (1999), the last two presenting nonlinear analyses of the
problem; see also Wang (1997), Mtiftti & Cole (1999), Marynowski & Kapitaniak (2002),
and other references cited therein.
10.3.9(bJ Edge flutter
ln the manufacturing of paper or plastic-film web, there are situations where the web
is subjected to air-fl.ow in the cross-travel direction, as shown in Figure 1O.57(a). By

PLATES IN AXIAL FLOW
1211
(a)
z
(b)
Figure 10.57 (a) Web moving with velocity Vand subjected to cross-travel air-flow ofvelocity U;
(b) the assumed form of traveling-wave flutter and some ofthe pertinent parameters for the analytical
model (Chang et al. 1999),
symmetry, either ha1f of the web may be considered separate1y, simp1y supported at
the centreline (y = O) and free at the unsupported edge - but, unlike the situation in
Section 10,3,6(b), with supported side-edges, similarly to Figure 10.17. It has been found
that, due to this air-flow, the free edges of the web can vibrate violently, An analysis of
this problem. together with experiments, is presented by Chang et a/. (1999) and Chang &
Moretti (2000, 2002); a brief account is given here.
Tbe equations of motion may generally be expres sed as follows (Chang & Moretti 2000,
2002):
( a4w a 4 w a 4 w ) ( a2w a 2 w a 2 w )
'Dx ----:i; + 2V xy 2 2 + 'Dy ----:i; - Tx ----z + 2Txy - + Ty----z
ax ax ay ay ax axay ay
( a 2 w a2w a 2 w )
+m  +2V- + V 2 ----z = b.p,
at axat ax
(10.126)
where the x and y directions are shown in Figure 1O.57(b); m and V are the web mass per
unit length and travel speed, Tx. Ty and Txy are the tension components, and (Graff 1975)
V x = Ex h3 .
12(1 - vxv y )
Vy = E y h 3 ,
12(1 - vxv y )
V xy = vy'Dx + i Gh3 ,
(10.127)
where it is presumed that the web is anisotropic; G is the shear modu1us. The flow ve10city
is U and the air density p. The b.p due to air-flow is determined by !inear potentia1 flow
theory,
[ aljJ aljJ ]
b.p = 2p -:} + U -:} ,
ut uy z=o
1 ( a a ) 2
V 2 ljJ - - - + U - ljJ = O,
0 2 at ay
(10.128)

1212
SLENDER STRUCTURES AND AXIAL FLOW
in which a is the speed of sound and U the air velocity, on both sides of the web. As usual,
the impermeability condition has to be satisfied (acpjaz)lz=o = (awjat) + U(awjay)
for O < x < L, and also acpjax = O at x = O, L.
A travel1ing wave solution is assumed to apply,
w = wsin(JTxjL)eik(ct- y ),
cp = (x, z) eik(ct- y ),
(10.129)
where a first-mode defonnation in the x-direction is assumed, see Figure 1O.57(b). The
wavenumber k = 2JT j A is presumed to be real, while the frequency Q = kc may be complex,
and hence so can the phase velocity c. As in the foregoing sections, flutter occurs when
two roots of c coalesce, thereafter c becoming complex.
One eventually finds
4(U - c)w [ 1  2 ( 2nJTX )] . ( )
cp(x, y, z, t) = i - -  e- Q2nZ cos - e k ct-y ,
A qO (4n 2 - l)q2n L
n=1
O 0.130a)
J 2 [ 2 ] 1 1/2
qn =  (2 + ( 2: ) 1 _ ( U: c ) ,
O 0.130b)
where the latter is subsequently simplified by taking (U - c)ja« 1. Hence, b.p may be
detennined from 00.128). Substitution of 00.129) and b.p into 00.126), multiplying by
sin(JTx j L) and integrating from x = O to x = L, gives a quadratic equation in c, which
in dimensionless form is t
(JT 2 + 8sf.L)c 2 - 16sf.L U c + 8sf.L U 2
_ JT2 [ A 2 + 2 V xy + Vy A -2 + T A 2 + T Ty ] = O,
V x V x Trn
00.131)
where
A
A=-,
2L
U
U=-,
Co
2pL
f.L =-,
JTm
- TrnL2 (Tx -mV 2 )L2
T---
- JT2V x - JT2V x
_ c
c= -,
Co
JT 2 V x
Co = -----Z'
mL
2
s = A + 9(4 + A -2)1/2 + ...
00.132)
The condition for c to become complex in (10.131) yields the critical value
U f = ( 1 + JT2 ) [ A 2 + 2 V xy + Vy A -2 + T A 2 + T Ty ] .
&f.L   
O 0.133a)
Expressions for the corresponding wavenumber and c are also obtained. It is seen that U cf
is a function of the wavelength. However, equation O 0.133a) may be solved numerically,
tlt is noted that the assumed form for w ex sin(1f xl L) in (10,129) causes the Coriolis-type term
2V(J 2 wIJxJt) and the term J 2 wlJxJy to drop out.

PLATES IN AXIAL FLOW
1213
and hence the minimum critica1 flow ve10city for all wavenumbers may be obtained.
Chang & Moretti (2000) show that, from empirica1 information, A c::: 1'-1/4; so, a1so
taking V x = Vy = V xy , equation (10.133a) may be simp1ified to
U 2 = ( 1 + Jr2 ) [ 1'-1/2 + 2 + 2 T 1 / 2 + T Ty ] , (10. 133b)
 & 
with s c::: 1'-1/4 + 2/[9(4 + 1'1/2)1/2].
Experiments were conducted (Chang et al. 1999) in a wind tunne1 built around a portion
of an infinite-100p web machine, capab1e of web speeds of 6.6 mis. The experiments
showed that Ucf is independent of V, at these web speeds; hence, 1ater experiments
(Chang & Moretti 2000) were conducted on a stationary tensioned sheet in the wind
tunnel. Typica1 measurements of the bui1d-up in amp1itude with increasing U are shown
in Figure 10.58. lt is seen that there is a definite point for the onset of flutter. t At higher
flow ve10cities, the amplitude saturates. The flutter mode invo1ves severa1 waves in the
flow direction, but typically on1y the fundamenta1 mode in the cross-flow direction as in
the first of equations (10.129); for post-critica1 flow, however, higher mode shapes wou1d
appear.
Figure 1O.59(a) gives U cf from a collection of experiments, compared to theoretica1
prediction. The agreement is quite good. A1so, Figure 1 0.59(b) compares the experimenta1
and theoretica1 frequencies from one ofthe experiments. For a given U, there shou1d be two
va1ues of frequency, corresponding to the two theoretica1 branches given by (10.131). The
experimenta1 data cou1d not be associated definite1y with one or the other theoretica1 branch
(hence a1so making a nondimensiona1 plot such as that in (a) impossib1e). Neverthe1ess,
there is no question, from this and other simi1ar p10ts, that the agreement for flutter
frequency is a1so quite good.
10.3.10 Spinning discs
It cou1d be argued that, since there is no tru1y axia1 flow, this topic does not be10ng here.
However, as seen in Figure 10.60, as the disc rotates, there is a tangentia1 flow+ of air
on it, simi1ar to axia1 flow over a p1ate; so it does be10ng. With obvious app1ication to
high-speed computer disk drives and VCR-foi1 disks, this topic is 'hoť and research on
it is very active. Vibrations and, of course, instabi1ities are undesirab1e, as they affect the
fide1ity of read/write operations. As rotationa1 speeds increase, so do the exigencies of
accurate modelling.
Among the most significant contributions are those of Hosaka & Crandall (1992),
Yasuda & Shimizu (1992), D' Ange10 & Mote (1993a,b), Renshaw etal. (1994), Renshaw
(1998), Hansen (1999), Hansen et al. (2001) and Raman & Mote (2001). Some of this
work is theoretica1, some experimenta1, or both. In some, the disc is considered to be
rotating in unconfined fluid, confined fluid, or in vacuum; in Hosaka & Crandall, the disc
is close enough to a stationary bottom p1ate for the fluid flow in the air film in-between to
be considered 1aminar. Here a very brief summary is given, based on the work by Hansen
et al. (2001).
tUtilizing the frequency data, not given here, the r,m,s, displacement amplitudes are of the order of 3 mm,
+This induces a secondary radial flow outwards (in the boundary layer) due to centrifugal acceleration of the
fluid; to balance the mass flow, this radial flow, in tum, induces an axial flow towards the disk (Hansen 1999),

1214
10°
SLENDER STRUCTURES AND AXIAL FLOW
 10- 1
!
g
'
10- 2
10- 3
1

8
i
 o
8 o
o
U (mis)
10
Figure 10.58 Typical r.m,s. vibration-velocity amplitude, w rms , for a par ti cul ar plastic web, versus
U, showing the onset of edge-fl.utter (Chang et a/, 1999).
10 3
10 2
I't;>
10 1
(a)
1000
N
b
:>,
u
c:
g. 100
..g
...
(1)
15

10
1
(b)
10 100
[(1 + T{2/ 8sfl ) (1'-1/2 + 2 + 21'112)]112
1000
Critical
frequency .!J)
10
100
1000
U (mis)
Figure 10.59 (a) The dimensionless critical fl.ow velocity U cf for edge-fl.utter versus a functional
ofthe dimensionless effective tension T: o, paper; O, x, +, different types ofplastic; -, theory,
(b) Flutter frequency versus U for paper (T = 17.2 Nim) (Chang & Moretti 2000).

PLATES IN AXIAL FLOW
1215
t1 ! ,t
, JJ,
r*
A p*(r* ,tp ,t*)
Figure 10.60 Schematic of the spinning disc system and definition ofthe coordinate systems used
(Hansen et al. 2001).
Consider the system in Figure 10.60. The starred quantities are the dimensional counter-
parts of the same dimensionless starless symbols. Two coordinate systems are used for the
spinning disk: a ground-fixed frame (r*, cP, z*), and a co-rotating one (r*, e, z*), such that
<P = e + Q* t*. In addition to the aerodynamic loading !:!..p*, a stationary, external trans-
verse force I*(t*) is applied at (rj, <Pf) = (rj, O). The equation of motion is based on
linearized Kirchhoff plate theory (Hosaka & CrandalI1992); in dimensionless tenns, it is
02W o2w o2w [ ow ] 1
---z + 2Q- + Q2---z + D - + K[w] = !:!..p + -8(r - rf)8(<pf)f(t),
ot oto<p o<p ot r
(10.134)
where 8 is the Dirac delta function, and
w*
W=-,
h
r*
r= -,
b
a t*
K=- t=-
b' T'
_ [ 12(1- V 2 ) P p b 4 ] 1/2
T- 2
Eh
(10.135)
!:!..p*T 2
!:!..p = h 2 '
Pp
I*T2
f = pph 2 b 2 '
D is a spatial operator modelling damping, and K is a stiffness operator,
4 1 o ( aw ) 1 a 2 w
K[w] = V w - -- H'rr- - -a -,
r ar ar r 2 rprp a<p2
(10.136)
in which a rr = a;r(r, K, v)Q2 and arprp = a;rp(r, K, v)Q2 are membrane stresses due to
rotation; a;r and a;rp are obtainable from in-plane equi1ibrium and boundary conditions
(D' Angelo & Mote 1993b). The reason for inc1uding the forcing function is to accord with
experiments, where, by exciting the disc via an electromagnetic actuator, the frequency
response functions (FRF s) could be obtained. The damping is typically considered to be
hysteretic, thus proportional to av 4 w jot, but there is additional frictional damping at
the interface of clamping of the disc. The boundary conditions for the clamped-free disc

1216
SLENDER STRUCTURES AND AXIAL FLOW
are
Wlr=K = O, [ 02 + v ( 02 + ow )] = O,
or ocp or r=1
ow I = O, [ (V'2W) + (1 - v) 0 2 2 ( ow - W )] = O.
ar r=K or ocp or r=1
ln the literature on this to pic, it is common to refer to a criticaZ rota ti on speed and to
a flutter speed. The former is the lowest Q at which a transversely applied static force
can resonate a stationary wave (D' Angelo & Mote 1993b). As seen in Figure 10.61, for
the (n, m) = (3, O) mode this occurs at Q* = Q;d  2l00rpm, where the frequency of
the backward trave1ling wave vanishes - similarly to a plate for U = Ucd' Hence, the
critical rotation speed could be viewed as a divergence. For post-critical speeds, however,
this wave re-emerges with a finite frequency. It eventually becomes unstable, giving rise
to a single-mode flutter in stability. The flutter speed is that associated with the mode
developing flutter at the lowest Q.
As seen in the experimental data of Figure 10.62, at Q;d there is a jump in vibration
amplitude of '" 1 0% t - not very prominent in the scale of the figure, Thereafter the
amplitudes remain small, until the flutter threshold is reached, which is clearly the crucial
phenomenon and the one of most interest to us here. Flutter is an aeroelastic phenomenon,
and it does not occur if the disc rotates in vacuum. Hosaka & Crandall (1992) have
demonstrated that what they call rotating damping* destabilizes the backward trave1ling
waves and causes flutter.
When the aerodynamic forces are modelled as rotating damping, I:1p may be
expressed as
I:1p(r, e, t) = -C [  - Qd ; ] = -C [ ao + (Q - Qd) ; l (10.138)
it comprises a dissipative term, -C[owjot], and a circulatory one, C[Qdowjoe].
Hosaka & Crandall interpret this latter as a distributed viscous damping that rotates at
speed Qd relative to the disc; Qd may be referred to as the damping speed. Hansen (1999)
and Hansen et aZ. (1999,2001) provide discussion to give further insight into I:1p, present-
ing (i) its piston-theory form (DowellI975), in which Qd = Q, (ii) the form proposed
by Yasuda et aZ" again with Qd = Q and with C a sca1ar, and (iii) the so-cal1ed reduced
shear-laier model of Raman & Mote [see Hansen et aZ. (1999); Hansen (1999)], where
C ex (V' )-1 and Qd = aQ3/2, in which a is a function of quantities re1ated to the swirling
boundary layer,
Solution to this system of equations is effected by considering
w(r,cp, t) = Wnm(r,cp)e".t = Rnm(r)eimpeAt,
(10.137)
(10.139)
where n is the number of nodal diameters and m the number of nodal circles; the Rnm (r)
are genera1ly complex, so that the mode shapes may display curved nodallines (Hosaka &
CrandallI992). Based on observations offlutter, attention is henceforth restricted to modes
with m = O.
Details of the analysis - see Hosaka & Cranda1l (1992), Hansen et aZ. (200 1) - are
not given here, However, some of the equations obtained by Hansen et aZ. are given, so
tHere Q;d has shifted from 2100 to 2400 rpm because of the effect of surface tTactions.
*Hansen (1999) also discusses rotating mass effects.

PLATES IN AXIAL FLOW 1217
150
125
 100
g
;>.
 75
"
5-

 50
25
2000 4000 6000 8000
Rotation speed (rpm)
Figure 10.61 Measured frequencies of the (n, m) = (O, O) wave and the (3, O) forward and back-
ward traveling waves, and comparison with various versions of the theory: o, experiment; -,
zem displacement boundary; - - -, solid disk model; - - - , stress-free boundaries (D' Angelo &
Mote 1993a),
4
3 Post-flutter ..;'
.;P+
I ,/,
2 
" '"
'"O ..
B .
.
I 8
.J
p.. +0
r::.. Pre- flutter f i ..:
O
Qd
O 1000 2000 3000 4000 5000 6000
Rotation speed (rpm)
Figure 10.62 The peak -to-peak amplitude for the steady-state vibration for an unconfined spinning
disc: 0, measurements during acceleration fmm rest; +, measurements during deceleration. The
critical speed is Q;d = 2400 rpm; the flutter speed is Q;j ::::: 3300 rpm (D' Angelo & Mote 1993b).
that the results may be understood. Substitution of (10.139) into (10.134), with (10.138)
in it, results in a non-self-adjoint eigenvalue problem, and hence in nonorthogonal eigen-
functions. This, though not posing undue difficulty in determining the eigenvalues and
hence stability, it does so insofar as the nonhomogeneous form of (10.134) is concemed,
since the eigenfunctions are not orthogona1. Hence, Hansen ct al. proceed by considering
an approximate solution based on a number of reasonable assumptions, for example, that
m = O and that the curvature of the nodallines is small, namely
N
w(r, cp, t)  L Rno(r) [Zno(t)e incp + ZnO e- incp ] ,
n=1
(10.140)

1218
SLENDER STRUCTURES AND AXIAL FLOW
the overbar denoting a complex conjugate; ZnO are the so-ca11ed complex wave coordinates
associated with mode (n, O). Substituting (10.140) into the equation of motion yields N
decoupled equations of motion.
The natura1 frequencies in the co-rotating frame may be found through wO =
(Wno, K[Wno]), where ( ) denotes the inner product. They can be related to the rotation
speed through the approximate relationship
[ 2 2 J 1/2
WnO  (ú)) + SnO Q ,
(10.141)
where ú) is the natuTa1 frequency of the stationary disc and SnO is a ccntrifugal stiffening
coefficient (D' Angelo & Mote 1993a). The damping coefficient, for both structura1 and
fluid forces, is found to be CnO = (Wno, (D + C)[Wno]), and the damping speed is
Q,nO = Qd,nO(WnO, C[WnO])/(Wno, (D + C)[Wno]),
(10,142)
where Qd,nO is the actua1 speed ofthe damping force defined in equation (10.138), Because
C and D are positive definite, Q,nO < Qd,nO.
As shown by Hansen ct al., the measurement of damping, always difficult, in this case
is particularly so; the determination of the point where it vanishes is even more difficult.
Hence, considerable ingenuity and ana1ytica1 development have been necessary to enable
the reliable extraction of the critica1 parameters from the experimental data, for example
w and SnO for the backward and forward waves, the critica1 speed Qd,nO and the flutter
speed Qf'
Sample data for ambient air experiments are shown in Table 10.4 and Figure 10.63. In
the table, 2B for instance stands for the backward wave in the (2, O) mode. The correlation
in the data ranges from 0.9854 to 0.9998! It is seen that the critical speeds in air are lower
than in vacuo, This phenomenon is also manifested by lower va1ues of SnO in air, and may
be explained by added mass effects (Hansen 1999; Hansen ct al. 2001). As the values of
SnO indicate, and as shown by Hansen et al. (2001, figure 3), the natura1 frequencies ofall
modes, wO' increase with Q* as a result of the centrifugal stiffening,
Table 10.4 The natural frequencies of the stationary disc, ú), centrifugal stiffening coefficients,
and critical speeds, Q;d,nO' at ambient pressure; the latter are also given for in vacuo experiments
(Hansen et a/. 2001).
Wave At ambient pressure In vacuo
ú) (Hz) snO Q;d,nO (Hz) Q;d,nO (Hz)
lB 36.10 1.435
1F 35.83 1.362
2B 50.59 2.397 39,9 44.4
2F 50.32 2.241
3B 86,74 3.886 38.4 42,7
3F 86.36 3.608
4B 141.8 5.509 43.8 48.3
4F 140.9 5.346

PLATES IN AXIAL FLOW
1219
0.6
-.
,
u
0.5
x
FTWs n O O +
XO++
, 1!4- (!J +
O  + S +
....+ O OD9:Q
+o + C *
I:  IjJ o + o
)( x o ..
x )(x1\Q tJel!l ..n
+ + x o ;JI; '"
X o o xXx X o
1&0 o o
..+ + !;Io
 00 o o x
o x x III o o
o BTWs
GI + .+++
+ oSo
e /I f/-
x
o
"N'
e; 0.4
*0
tj"
...
B
g 0.3
..,
OIJ
c::
's,
E 0,2
'"
O
0,1 1
o
O
5
10 15 20 25 30 35 40 45 50
Rotation speed Q* (Hz)
Figure 10.63 Damping factors ao at ambient pressure for BTWs and FrWs (backward and
forward travelling waves): +, (n, m) = (1, O) mode; x, (2, O) mode; O, (3, O) mode; o, (4, O)
mode (Hansen et a/, 2001),
70
60
"N'
e; 50
o
*"
-.,;
CI 40
"O

..!" 30
*'i;
/<3
'"
"O 20
O)
O)
0-
e/).
10
O
O
... ".;,J::;;??':::>?f?;
___ ' /' " I
. - - -..:...:.:.;.>- ,;.;,",""!
- .:......
..... ... {J'*
.....,. d,nO
! Q,if
10
20 30 40 50
Rotation speed Q* (Hz)
60
70
Figure 10.64 Wave speeds w:o/n (upper curves) and damping speeds Q;;:nO (lower curves), also
showing the flutter speed Q;f predicted at their first intersection, for the (3, O) mode (Hansen et a/.
2001).
As shown in Figure 10.63, damping for the (3, O) mode vanishes at Q* :::: 47 Hz.
According to Renshaw et aZ, (1994), the fl.utter speed is defined as the rotation speed
where the resonance peak of the critica1 backward trave1ling wave is 20 dB above the
noise level of the FRF; based on this definition, Q;j :::: 55 Hz is obtained, On the other
hand, curve-fitting the data, flutter may be predicted in the Hosaka & Crandall manner:
that it occurs at the point where the wave speed, wo/n, equals the damping speed, Q;;nO'
as constructed in Figure 10.64; this gives Q;j :::: 62 Hz again in the (3, O) mode, about 13%
higher, High precision experiments are difficult to conduct, and this degree of agreement
is considered to be very good.

1220
SLENDER STRUCTURES AND AXIAL FLOW
Figure 10.65 The disc deftection at ftutter speed of an 8 in. ftoppy disk (Hosaka & CrandaIl1992).
The tlutter mode in this case, (3, O), is quite low. In Hosaka & Crandall's (1992) work,
however, for the disk close to a rigid plate, (n, O) as large as (30, O) is obtained. The disc
detlection at critical conditions (Q* :::::: 7200 rpm = 120 Hz) in the (30, O) mode for an
8-inch tloppy disk is shown in Figure 10.65,
10.4 CONCLUDING REMARKS
Even a casual perusal of the material in the chapter reveals how clo sely related this topic
is to the rest of the material covered in both volumes of this book, It is also clear, even in
this highly abbreviated treatment, that this is a varied, very rich and fascinating subject.
Most of the material in the foregoing sections deals with dynarnics and stability, rather
than tlow-induced vibrations, for example, due to turbulence. Some of the earliest work
on such vibrations has nevertheless been done for plates; see, for example, Section 7.7
and refer to Blake (1986). However, tlexible panels may also be used to passively control
duct noise - see, for example, Huang (1999, 2001 b), Huang et al. (2000). Among many
other interesting topics not discussed, the following are two samples.
One of the unexpected discoveries of linear theory discussed in Volume 1 is that of
vibration-induced flow in pipes, as opposed to tlow-induced vibration. The equivalent
phenomenon here is that of steady secondary streaming (Schlichting 1960) due to vibration
of slight1y curved or wavy plates in their own plane. This has now been extended to
random-vibration-induced tlow (Volfson & Viiials 2001).
The second topic is that of large deformation of elastic bands, which can display fas-
cinating nonlinear dynamics: solitons and spatial chaos, for example, as discussed by
E1 Naschie (1990); refer also to Awrejcewicz et al. (2002). No axia1 tlow is involved in
the studies discussed therein, the parameter varled typically being tension/compression,
However, there is little doubt that similiar phenomena wou1d arlse with tlow; a hint that
this may be so is provided by Peak's work [Section 1O,3.7(c)].

11
Annular- and Leakage-
Flow-Induced Instabilities


11.1 INTRODUCTION


11.1.1 General introduction


F1exible structures in annular flow are notoriously susceptible to fluid-elastic instabi1ities.
The same applies to plate-like or cylindrical structures subject to leakage flows, to be
defined shortly. Thus, unlike most of the systems considered in the foregoing chapters,
neither does the structure need be particularly flexible (or highly pliably supported) for
instabi1ities to arise, nor the flow ve10cities unusually high. If the geometry is right, the
system is prone to instabi1ity regardless. t
Annular flow is self-explanatory: for example, the flow over a cylinder in a generally
tightly confining channel or the flow between two concentric or eccentric shells.
Leakage flow takes its name from systems such as that in Figure 11.1 (a). The reason
for having a 'slip-joinť rather than an integral pipe to deliver the flow from B to A is for
ease of assembly, disassembly and replacement, and to accommodate thermal expansion
in the axia1 direction, whi1e the lip on the sma1ler pipe provides partiallatera1 support for
limiting possible flow-induced vibrations. However, the pressure drop in the main flow
may well induce a secondary, 'leakage' flow in the annulus past the flow constriction
(lip).:j: This flow, though generally of smaller velocity and momentum, is the one that may
cause problems (Section 11.2). By extension, the flows in the narrow passages around the
enlarged parts of the 'centre-bodies' in Figure 11.1(b,c) are also referred to as leakage
flows, though they are the main and only flows; however, in such cases, the term is usually
reserved for the flow within the narrower passages only. The bodies in all three cases in the
figure could be either axia1ly symmetric (cylindrical) or rectangular, plate-like structures.
ff they are cylindrica1, then one could equal1y label the flow as annular, and this is the
reason why the two are treated together here. However, the term 'leakage flow' is reserved
to cases where there is a narrow passage as in Figure 11.1, whereas for a narrow passage
extending from end to end the flow would be labelled annular.
Ten case studies of annular/leakage-flow-induced instabi1ities are discussed in
Pai'doussis (1980; Cases 43-52), some of monumental destructiveness; additional cases
and further discussion may be found in Axisa (1993). Very nice and useful reviews, with
emphasis on nuc1ear reactor components, are given by Mulcahy (1983a,b).


tProvided that the in-fluid natural frequency is such as to allow the appropriate unsteady fluid-dynamic
forces to develop,
*Sometimes the leakage flow is necessary for cooling purposes of one of the components involved.


1221




1222
SLENDER STRUCTURES AND AXIAL FLOW
Flow constriction 
A+:::::"'i
A ti;;;;;'}
Support rod End piece
(b)
(a)
A f=:::J:::,'::> H
(c)
Figure 11.1 Typicalleakage flow paths: (a) a tube-in-tube slip joint; (b) a geometry typical
oť some nuc1ear reactor control rods; (c) an idealized sketch oť the AGR finned 'gag bomb'
(Mu1cahy 1983a,b).
Leakage-ftow-induced instabi1ities are generically re1ated to those in hydraulic gates
and in plug and check va1ves; see, for example, Kolkman (1980), Weaver & Ziada (1980),
Weaver (1980) and Naudascher & Rockwell (1980b, 1994).
This author experienced a leakage-ftow-induced instabi1ity for the first time in 1961,
in the course of experiment s in the series described in Section 7.9.2, involving ftow in
horizonta1 collapsible tubes. To obtain a constant but adjustable upstream pressure head,
the header tank shown in Figure 11.2 was designed, The inlet ftow was slightly higher than
that needed for the collapsible tube experiment, and the overflow went over a cy1indrical
weir in the centre of the tank; the weir could be moved up and down by rotating it in the
threaded sleeve at the bottom of the tank, thus changing the static pressure at the inlet of
the collapsible tube, Pu, In these experiments, Pu was but a few centimetres of water, and
therefore the air above the free surface of the water was open to atmosphere. The original
design did not include the wooden plank shown in Figure 11.2.
In operation, it was found that there were some sma11 ripples on the free surface and,
because of the sma11ness of Pu, the author had the bright idea of attempting to eliminate
them by ftoating a wooden plank on the free surface, with a hole in its middle, nicely
c1earing the cy1indrica1 weir, and a1so nice1y creating a leakage- ftow path for the overftow. t
The result was violent latera1 inverted-pendulum oscil1ations of the weir assembly about
the threaded sleeve, which acted as a ftexible support because of the re1ative thinness
of the galvanized metal sheet from which the tank was fabricated, The oscil1ations were
violent enough to cause the wooden plank to bang on either side ofthe tank. The oscil1ations
occurred over a fair1y wide range of overftow rates and rendered the experiment inoperative,
The unexpected problem disappeared once the opening in the centre of the plank was
a1tered from circular to square, as in Figure 11.2( c).
tFor experiments on the same basic slip-joint system, see, for example, Nakao (1983) and Mulcahy (1984),

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1223
To expenment
\
(a)
Overflow
(b)
ln
(c)
Figure 11.2 (a) A constant head tank used to supply water-flow in collapsible tube experiments:
1, the square-section tank; 2, the overflow assembly; 3, threaded sleeve for adjusting height of
overflow weir; 4, perforated guard; 5, wooden plank floating on free surface. (b) Plan view of the
plank with a circular hole to clear the overflow weir; (c) with a square hole.
Of course, in terms of engineering importance the above example is trivial, but as may
be appreciated from Section 11.2, annular- and leakage-flow-induced instabilities can well
be of staggering destructiveness, for example, capable of generating forces of the order
of tens of tons, and able to move the whole core barrel (see Figure 8.133) of a nuc1ear
reactor!
A preliminary discussion on some of the mechanisms is given in the section that follows.
Then, after presenting some of the practical experiences of this type in actual engineering
systems (Section 11.2), t a brief review of the work on the vibration in narrow annuli
filled with quiescent fluid is given (Section 11.3), followed by work on the dynamics of
plate-like structures in l-D flow passages (Section 11.4), cylindrical structures in annular
flows (Sections 11.5 and 11.6), and coaxial shells (Section 11.7).
11.1.2 Preliminary discussion of mechanisms
The purpose of this discussion is to give the flavour and physical understanding of the phe-
nomena involved, by considering, at the notionallevel, a couple of engineering occurrences
of leakage-flow-induced instabilities.
The first problem (Pai'doussis 1980; Case 43) occurred in a pressurized water reactor
(PWR), in which there was 'evidence of hydraulic flutter of the control-rod blades and
stationary shim rods' - cf. Figure 8.134. There is insufficient information in the original
reports of the problem (Reed 1965; Mil1er & Kennison 1966; Mil1er 1970) as to the
real geometry. However, for our purposes, the geometry of the related model studies
t Additional practical experiences are presented in Sections 11.1.2, 11.5,3(b) and 11.7.4,

1224
SLENDER STRUCTURES AND AXIAL FLOW
Pivot
point
1 7 . 94
(a)
////////////////////////////// ///////// / /////////////////.
1w/7;)'t/7,0'/$'/;.1/7///M
(b) '//////////// //// /// // /// //"'////// // // //// /' /' ///////////
+
Lower channel
Upper channel
 -

Q.
,g
1'í +
"-'
l!.. Flow from left to right
'" Flow from right to left
Upper channel
Lower channel
(c)
Figure 11.3 (a) The experiment with a blade partly in a narrow annulus (left), used to demonstrate
leakage-flow-induced instability; all dimensions in cm. Numbered items: 1, five simulated wear
pads on each side of tbe blade (0.16 x 0.79 x 7,62); 2, stiff wire strut to take thrust load, (b) An
idealization of the system, with the wider segment on the left corresponding to the narrowest gap
due to tbe presence of the wear pads. (c) The transient pressure in tbe upper and lower subchannels
due to a transverse velocity of tbe blade, V, witb flow from left to right (upper graph), Dr reversed
(lower graph); (Miller & Kennison 1966; Miller 1970; Pa'jdoussis 1980, Case 43),
is sufficient. A rectangular blade was placed in flow in a water channel, as shown in
Figure 11.3(a), partly in a narrow channe1 ('the scabbarď, upstream end) and partly in
a wider channe1 (downstream); the basic geometry is therefore as in Figure 11.1(a), The
vertical blade was pivoted at the downstream end, and flexibly supported (pendularly) at
two other points along the blade. The simulated wear pads at the upstream end created an
added constriction there. It was found that the blade developed self-excited oscil1ation at
a flow ve10city of U = 1.22 mis, at "" 1 Hz.
The now c1assical mechanism, as e1ucidated by Miller (1970), is one of flutter associated
with negative damping - see, for example, Figure 3.4(b), The mechanism may be under-
stood by considering the idealization of the system in Figure 11.3(b). Consider first the
flow to be £rom left to right. Ifthe blade is given in instantaneous flow 'upwards',t the flow
in the upper subchannel must decrease, while that in the lower subchanne1 must increase.
Hence, as a result of these localized valving effects, fluid in the upper subchanne1 mu st
decelerate, while that in the 10wer subchanne1 must acce1erate. This results in a transient
depression of the static pressure to be10w what it originally was in the up per subchannel * -
cf. equation (2.71) - and a transient pressure rise in the lower subchannel, as shown in the
up per part of Figure 11.3( c). Hence, a net force directed upwards is generated, that is, in
tPor ease of discussion we talk of 'up' and 'down' in the plane ofthe paper, whereas in the physical system
we have left and right.
*Por flow to decelerate, there must be a pressure gradient with the pressure rising with distance from the
restrictor on the left.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1225
phase with the blade velocity. Thus, the system may extract energy from the flow - cf. the
mechanism for galloping (Den Hartog 1956) - and flutter (a negative-damping instabil-
ity) becomes possible, provided the negative damping overcomes the positive dissipative
damping in the system. Leakage of the flow from one subchannel to another, around the
blade edges, diminishes but does not eliminate the effect.
Consider next the system in Figure 11.3(b) with the flow from right to left. An instant-
aneous upwards velocity of the blade produces a decreasing outflow from the upper
subchannel and the inverse unsteady pressure effects, vis-o-vis those just described, as
shown in the lower part of Figure 11.3( c). Thus, a positive damping is generated, stabil-
izing the system. This suggests a solution of the real problem: to relocate the wear pads
from the upstream to the downstream end of the scabbard [the narrow channel on the left
of Figure I 1. 3 (a)]. It works!
There are intricacies to the problem, not discussed here; for example, for higher flows,
the vibration frequency suddenly drops and static divergence ensues; see, for example,
Mulcahy (1983a). However, in its essence, the general behaviour is as described, and
hence one arrives at one of the several 'rules of thumb' proposed by Mulcahy, namely
that a constriction at the upstream end of a narrow channel tends to destabilize the system,
while a constriction at the downstream end tends to promote stability.
The second system to be discussed is also related to a control rod in a nuc1ear reactor,
schematically as in Figure 11.1 (b) but rotated c10ckwise by 90°. If the central body is
excited into a rotational vibration, as shown in the figure (dashed lines), then the local
va1ving action and the presence of convergent/divergent flow passages may cause flutter
(Mulcahy 1984). If, however, the motion is limited to a transverse translation, and fur-
thermore a downstream constriction is added, the system is prone to divergence, with the
central body in contact with or c10se to the outer body.
The third geometry of interest is that of Figure 11.1 (c), a highly simplified sketch of a
fuel channel in an Advanced Gas-cooled Reactor (AGR). A more realistic representation is
shown in Figure 11.4. The finned 'gag-bomb' serve s to stop (gag) the flow for refueling,
while the reactor is in operation. The self-excitation mechanism in this case is very complex
(Parkin 1980; Pai"doussis 1980, Case 52; Hobson 1982; Mulcahy 1983a,b, 1984). In fact,
several mechanisms may be identified: (i) the annular flow in the narrowest gap at the
upstream end of the diffuser section [Figure 11.1 (c)] similar to that described in the first
case presented here; (ii) localized choking of the flow on the fins themselves, and the time
delay to reestablish flow which causes a component of the fluid forces to be in phase with
the gag velocity; (iii) the flow in the diffuser itse1f, of particular interest here. Essentially, if
the efficiency in the diffuser (i.e. the pressure recovery therein) is augmented as the throat
area increases, as a result of centre-body motions, it has been shown that this generates a
force out of phase with displacement, and hence in phase with velocity; again, here too,
there is a potential for self-excitation. This is further discussed in Section 11.5.
11.2 MAJOR ENGINEERING PROBLEM CASES
AU ten cases of practical experiences caused by annular- and leakage-flow-induced vibra-
tion discussed by Pai"doussis (1980) are related to nuc1ear reactor intemals t - see, for
tlt is recalled that in some countries, for example the United States, alt problems in nuclear systems have
to be reported, and the reports have to be in the public domain, This is not the case in other realms, and hence

1226
SLENDER STRUCTURES AND AXIAL FLOW
Gag drive
shaft
Bias port
inlet flow
Gag orifice
Gag fins
Gag bomb
Gag tube
Figure 11.4 A sketch ofthe finned 'gag-bomb' used to shut offthe flow in an AGR fuel channel
for refueling (Pai'doussis 1980; Case 52),
example, Figure 8.133. Those caused by leakage flow involve control rods, reaetor-eore
internals or fue1 rods (Cases 43,44,45,48 and 52). The rest, exc1usively or mainly eaused
by annular flow (Cases 46, 47, 49, 50, 51), involve the thermal shie1d and core barrel
mainly, or peeuliar components such as the feedwater spargers. Here, a compendium of
these eases is presented, mainly tomotivate the work to be presented in Seetions 11.3-11.7.
11.2.1 Nuclear reactor core barrel and thermal shield problems
Figure 11.5 and in more detail Figure 8.133 show sehematically the internal stmeture of a
Pressurized Water Reaetor (PWR), and in partieular the eore barre1 and the thermal shield,
this latter present in many but not all PWRs. Also shown are the narrow annular flow paths
between (i) the core barrel and the thermal shield and (ii) the thermal shie1d and the reaetor
pressure vessel. The core barrel eontains the uranium-oxide fuel assemblies and is therefore
very massive and generally rigid; if it moves at all, it is prone to rigid-body motions, for
example, pendular osei11ations, hinging at its upper support on the pressure vessel. The
thermal (radiation) shie1d, although not thin, has a small enough thickness/radius ratio to
such occurrences in conventional power and chemical plants, for instance, remain under wrap (and unknown
in the wider engineering community), to protect corporate image and/oe escape possible litigation,

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTlES 1227
Therrnal
shield
Figure 11.5 Schematic of a PWR, showing the flow paths around the thermal shield and in and
out of the care barrel (Pai'doussis 1980),
be considered as a shell, and to vibrate as such, typically at circumferential mode numbers
n = 1, 2, 3 or 4.
Several occurrences of severe damage have been reported, involving the thennal shield
and the core barrel. Some were trne instabilities, while others were 'jusť vibrations; in this
latter context, the reader is referred to Figure 9.4 and the associated text in Section 9.2.1,
showing that for hydrodynamically strongly coupled systems, forced vibration (from
whatever source) of one component may cause amplified oscillation of another nearby.
Figure 11.6(a) shows a PWR of a type in which there is no thennal shield. In 1973,
annular-flow-induced vibration problems were encountered, related to the core barrel of
such a reactor in the Palisades nuc1ear power plant (Pai'doussis 1980; Case 47). The core
barrel is basically a cantilevered vessel, c1amped at the upper support flange of the pressure
vessel; possible lateral vibrations are constrained by the snubbers at the bottom. Indications
of core movements (f "-' 0-6 Hz), sensed via ex-core flux monitors, led to inspection of
the reactor, revealing that (i) all expansion-compensating ring bolts were broken, (ii) the
core barrel support flange had wom a ledge in the vessel flange, thus allowing 6.5 mm
vertical movements and (iii) a groove, 1.6 mm deep and 6.5 mm wide, was wom into the
reactor vessel head [Figure Il.6(b)]. It is noted that the core barrel weighs tons!
Flow-induced rigid-body motion of the core barrel was estab1ished as the cause, due to
insufficient hold-down forces at the upper clamped support, allowing small motions; wear
began to take place, thereby allowing larger amp1itudes of motion and more wear. The
rigid-body motion was a complex combination of rocking, translation and nutation, with
the snubbers constraining the motion at the bottom into a quasi-elliptical path, No specific
excitation mechanism was advanced, other than impingement by the multi-inlet flow and
unsteadiness of the flow in the annulus, no doubt exacerbated by the narrowness of the
annular flow passage. The problem was solved by installing a new hold-down device at
the top, involving stacks of Belleville washers, thus increasing the hold-down force from
15 x 10 3 to 33 x 10 6 N.

1228
SLENDER STRUCTURES AND AXIAL FLOW
lnlet
Keys nozzle
Upper guide Core support
strneture shield
lnlet Oudet Detail D
nozzle nozzle
Reaetor vessel
i Thermal shield
I
i Core support Core barrel
-.+- barrel
i
Core i
support I
plate Detail C
Snubber Lower grid
Flow skirt Guide lugs
(a) (c)
Reaetor
ves sel
Reaetor ves sel head
1.6 mm wear
Expansion compensating nng
Upper guide
strneture flange
Detail C
Core support
barrel flange
wear
(b)

63,5 mm
(d)
Figure 11.6 (a) Schematic of a PWR in which no thermal shield is used; (b) detail of the core-
barrel support, showing the location of barrel-flange wear. (c) Schematic of another PWR, with a
thermal shield; (d) details of the lower support (Detail C) and upper end (Detail D) of the thermal
shield, where damage took place (Pa'idoussis 1980; Cases 47 and 50).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1229
Problems involving the thermal shield in another type of PWR were encountered at
the Oconee-I Power Plant in 1972 (Pai'doussis 1980; Case 50). The thermal shield in this
design is supported at the bottom on the lower grid, as shown in Figure 11.6( c) and detail C
in (d). At its upper end, the thermal shield is supported and positioned radially by spacer
blocks, at interval s around the circumference [Figure 11.6(d), detail D], the aim being to
constrain the vibration into higher circumferential shell-modes.
Inspection showed that the retaining welds on the bottom dowel pins had failed and
some deformation of the mating surfaces had taken place; the dowel holes had been
worked oversize. Clearances at the top spacer blocks were 0.2-0.5 mm, whereas they
were originally interference fits. It was proposed that unbalanced steady fluid-dynamical
forces deformed the shield, thus creating small clearances, so that vibration in the second
circumferential mode (at 31 Hz) and perhaps the third (49 Hz) became possible, again
exacerbated by the narrow annular spacing. Vibration caused fretting at the spacer blocks,
allowing larger amplitudes to develop and more fretting. The problem was solved by
improving and strengthening the lower support, and providing positive attachment to the
core barrel at the top, as well as reducing the peak downcomer flow velocities adjacent to
the thermal shield.
A more devastating but more complex set of failures occurred in another type of PWR
(Figure 11,7), in several reactors around the world, for example, SENA, SELNI and San
Onofre I, in 1967-1970 - see Daubert et al. (1971, 1974) and Pai'doussis (1980; Case 46).
Control rod drive
mechanism
Barrel
Pressure vessel
Barrel joint
Control rod absorber
Thennal shield
Core plate
Core plate joint
Shroud tube joint
Shroud tube
Tie rods
Lower casting
Figure 11.7 Schematic of another PWR in which severe damage occurred in the thennal shield
and core barrel components, causing a three-year shut-down for repair (Pa'idoussis 1980; Case 46).

1230
SLENDER STRUCTURES AND AXIAL FLOW
ln one reactor, it was found that (i) the pins connecting the three segments of the thennal
shie1d had fai1ed, al10wing them to oscil1ate and to impact on the core barrel and pressure
vesse1, (ii) one-third of the bo1ts joining the upper and 10wer part of the core barrel had
failed, débris finding its way all the way to the steam generators, and (iii) two-thirds of
the tie rods connecting the 10wer casting to the barrel had also failed. The shut-down
for clean-up and repair, using frogmen, took three years! ln another reactor, when its
supports fai1ed, the who1e thennal shield dropped to the bottom! Typical recorded core
barrel vibration amp1itudes were on1y 0.2 mm and the frequencies for pendu1ar motion
5-10 Hz; typica1 annu1ar flow ve10cities were 5-10 mis.
One mechanism proposed is that smal1 pen dul ar oscil1ation of the core barre1
induced 1arge amplitude oscil1ation of the thenna1 shield in an inverted pendu1um mode
(cf. Figure 9.4); or vice versa. The shie1d vibration generated swirling flow at the bottom
of the annulus which caused the tie rods to fai1 by cross-flow-induced vibration. It was
calcu1ated that core barrel oscillation of 1 mm amp1itude cou1d generate unsteady pres-
sures in the narrow annu1us ofO.1-O.2 atm and 10ads on the thennal shield of 20-30 tons.
The initial excitation causing pen dul ar motion could be the unsteadiness of the mu1ti-entry
flow, its division into two by the thennal shield (cf. Figure 11.3), and pump pu1sations;
gross velocity fluctuations of a few percent cou1d produce the recorded 0.2 mm core barre1
amp1itudes. In a 1 -model study (Daubert ef al. 1974), a globa1 flow in stability was found,
with 1arge eddies fonning at the top of the thennal shield, be10w the flow in1et, capab1e of
producing alternating 10ads of the order of 2 tons.
ln most cases, the cure was to e1iminate the thennal shie1d altogether, also removing
some of the outennost fuel assemb1ies to protect the pressure vesse1 from undue radiation
intensity; in one case the shield was retained, but with stronger supports. The core barre1
bolts and tie-rod arrangement were redesigned.
A clear case of 1eakage-flow-induced in stability occurred in the Big Rock Point reactor
(Pai'doussis 1980; Case 48). This is an unusual Boi1ing Water Reactor (BWR) design in
which the annulus between the thennal shie1d (2.61 m in diameter and 2.34 m 10ng) and tbe
reactor pressure vesse1 is filled with quiescent water, A Bellevil1e washer (the 'seal ring'),
on which the thennal shie1d rests flexib1y, serves as a bottom seal to prevent water from
by-passing tbe core [Figure 11.8(a)]. In 1964, several thennal shield retaining bolts were
found to be broken. Tbe support structure was stiffened and tests were conducted in 1965.
With increasing flow, the shie1d first disp1ayed 'low 1evel nervousness' at f ::::: 4 Hz, and
then sudden1y deve10ped rigid-body vibration 'of into1erab1y 1arge amp1itude', 0.6 mm,
such that its centre described an elongated ellipse. The fatigue 'usage' for the tbermal
shie1d support was 1 % per minute, clear1y untenab1e,
The mechanism proposed was leakage-flow-induced vibration, with 1eakage through the
Bellevil1e washer once a sufficient pressure differential across it is estab1ished (Corr 1970),
essential1y as described in conjunction with Figure 11.3. Taking the 1eakage to vary loc-
aBy, tbe motion becomes coup1ed to the flexibility of tbe wasber seal, Tbe cure consisted
of (i) immobi1izing the Bellevil1e washer by wedges, so that 1eakage wou1d be equal on
al1 sides, (i i) inserting ho1d-down weights in the annu1ar gap, resting on the Bellevil1e
wasber, to immobi1ize the shie1d and disrupt the annu1ar flow, and (iii) installing a top seal
[Figure 11.8(b)] to provide a positive flow-induced damping. Any one of these three meas-
ures was deemed capable of solving the problem; in any event, tbe three together did so.
Calcu1ations [Figure 11.8(c)] clear1y demonstrate tbat a small gap at the top (downstream)
seal and a large gap at the 10wer (upstream) seal promote optimal stability.

Bottom ofthennal shield
ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1231
3,81 cm
al
13lcmR
t t t t t t
(a) Flow
j"
(b) W
40


30 
I 
Unstable #
!ir 
ba 20 I
O;
:'A
$  Stable
10 

O 1
O 2 3 4 5
(c) Lower seal gap (mm)
Figure 11.8 Leakage-flow-induced instability at the location of the Bellevil1e washer (seal ring)
in a BWR: (a) schematic of the seal ring; (b) detail of the upper (downstream) seal introduced to
stabilize the system; (c) the ranges of stability and instability as functions of top and bottom seal
gaps (Pai'doussis 1980; Case 48).
For additional information on core barrellthennal shie1d annular-flow-induced vibration
problems, the reader is referred to Altstadt et a/. (1991), Pusch & Grunwald (1991), Axisa
(1993) and Section 8.12.3. A fascinating case involving annular-flow-induced instabi1ity
of a flexible weir coupled to sloshing in the Superphenix Liquid Metal cooled Fast Breeder
Reactor (LMFBR) is described briefly in Section 11.7.4.
11.2.2 Feedwater sparger problems in BWRs
The feedwater spargers used in BWRs are a spray system to distribute the reactor feed
water [Figure l1.9(a)]; they are sometimes a single pipe all around the pressure vesse1
[Figure 11. 9(b)], and sometimes individual curved segments on either side of a junction - or
tee-box, each fed individually by separate nozzles [Figure 11.9(c)]. Holes, 12.7-38.1 mm
in diameter, all along the header pipe (typically 152 mm in diameter) discharge the water
radially into the flow coming down from the steam separators toward the jet pumps.
Wedges bearing against the pressure vessel were used to preload the spargers, so as to
prevent vibrations arising from internal and external flow excitations.
Deep ('" 1.6 mm) cracks were found in the spargers of one such reactor after another,
starting in 1972, in some cases threatening the structural integrity of the system; in one
case, a crack extended 200° around the pipe (Pa1doussis 1980; Case 51), The cracks were
mostly at the junction between tee-box and header pipe, or tee-box to header sleeve. At
least 16 reactors were implicated, bot events are described here in terms of just one of them.

1232
SLENDER STRUCTURES AND AXIAL FLOW
(a)
Steam dryers
Steam separators
Driving f10w
....
....
pump
Feedwater sparger
Sparger support
bracket
restraint
(b)
Thermal sleeve
Main steam f10w
to turbine
-+
Main feed f10w
from turbine
...
....
A ([D V
Thermal sleeve
(c)
Figure 11.9 (a) Sehematie of a BWR; (b) the feedwater sparger system, where leakage-flow-
indueed in stability oeeurred in the feedwater-nozzle/thermal-shield annulus; (e) 'redesign 4' for
redueing leakage flow and alleviating vibration, Not shown in (b,e) are the small equispaeed holes
in the header that spray the water outwards, into the eore (Pai'doussis 1980; Case 51).
Initially the high-cycle fatigue cracks were thought to have been caused by inadequate
pre1oad, or loss thereof. 'Design 2' spargers were installed, with heavier pipe (schedule 80
instead of 40) and larger preload. After "'6 weeks' operation, new fatigue failures occurred.
This led to 'Design 3' spargers, involving an entirely different support bracket, but these too
indicated excessive vibration when the reactor was operated at > 85% power. The vibration
frequencies of "'47Hz compared to one of the calculated natural frequencies of 60 Hz.
Strain amplitudes of as much as 100 !-LI': were obtained, sufficient to cause fatigue failure.
Extensive full-scale test s (Torres 1980) eventually detennined that the mechanism
underlying the vibration was a se1f-excited leakage-flow-induced instability involving

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1233
0.6
0,5
Low level
vibration
I 0.4
g.
bQ
 0.3
e
"
bQ
 0.2
<
0,1
o
O
2
3 4 5 6 7
Leakage flow rate (m 3 /s)
8
9
10 X 10- 3
Figure 11.10 Results fmm full-scale sparger testing, for a nozzle of  29 cm inside diameter,
schedule 80 stainless still pipe with cold water flow, no preload, and variable gap between the
thermal sleeve and nozzle (Torres 1980),
annular leakage flow between the feedwater sparger and the thermal sleeve. A typica1 set
of experimenta1 results is shown in Figure 11.10. It is seen that the critical flow rates are
quite sma1l (ofthe order of 4 x 10- 3 m 3 /s), consideringthe large diameter and short length
of the nozzle; however, the leakage flow velocities are substantial, of the order of 15 mis.
Moreover, it is seen that the self-excited flutter is subcritical with considerable hysteresis.
To cure the problem, 'Design 4' spargers were insta1led, incorporating (i) interfer-
ence fit between the thermal sleeve and the nozzle to reduce the annular leakage flow
[Figure 11.9(c), bottom], (ii) a reduced pressure difference between sparger inlet and
discharge, and (iii) stress-concentration reduction features.
11.2.3 A final comment
ln spite of the fact that the existence of annularlleakage flow-induced instabilities and
vibrations has been known to designers of hydraulic va1ves and gates for a long time and
to nuc1ear engineers at least since 1970, the reader wil1 have noticed that this knowledge
did not exactly propagate at the speed of sound, for those concemed (i) to ensure problem-
free design ar (ii) to recognize the mechanism in a problematic design at a glance. In
this regard, complicated systems and complicated drawings tend to camouflage the salient
mechanisms that a simple sketch might reveal. Also, of course, hindsight often trivializes
the long and tedious detective work and difficult experimentation needed to corroborate
the final conclusion,
11.3 VIBRATIONS IN NARROW ANNULI
As seen in Chapters 7, 8 and 10, and in Section 11.2, there are many engineering systems
involving a cy1inder within a cylinder, separated by an annular space filled with dense

1234
SLENDER STRUCTURES AND AXIAL FLOW
fluid. This explains the interest in this topic and the many different studies undertaken to
predict the coupling characteristics and vibrations of the structures in such geometries.
ln this section, various aspects of the dynamics of such systems, involving cylinders or
cylindrical shells in quiescent fluid are reviewed briefly.
11.3.1 Inertial and viscous coupling
The added mas s matrix of coaxial cylindrical shells coupled via the annular fluid is dis-
cussed in Section 2.2,2, with the particularcase ofbeam-like cylinders or flexibly supported
ones being retrieved by taking n = I, Similarly, the coupling of coaxial cylinders in viscous
fluids and various ways for predicting the added mass and viscous damping coefficients
are discussed in Section 2.2.3. The reader is also referred to Section 9,2, where the system
of K cylinders within a fluid-filled cylindrical channel is considered; these methods can
also be used here, with K = 1.
No more about this is said here, except in Section 11.3.2 where the vibration of rods in
annuli offinite length is discussed, and in Section 11.3.3 where some numerical techniques
are described. A very useful review by Brown (1982) gives simplified expressions for the
inertia and damping coupling coefficients, for other geometries as well, and cites 246
papers. It is pointed out that the earliest recorded awareness of the existence of an 'added
mass' goes back to Du Buat (1786)t and Stokes (1843). Early work on the effect of
viscosity on solid surfaces in relative motion and immersed in fluid is by Stefan (1874)
and Reynolds (1886).*
11.3.2 Vibration of fluid-coupled coaxial cylinders
A great deal of work has been done on this topic, partly because of some of the engineering
problems discussed in Section 11.2.1, for example Bohm (1965), but also in more general
tenns, for example Fritz & Kiss (1966), Fritz (1970,1972), Yeh & Chen (1978), Mu1cahy
(1980), Pai'doussis & Ostoja-Starzewski (1981), Gamer & Chandra (1984), Svetliskii &
Grigoriev (1988); see also Sections 2.2.2 and 2.2.3, It should be mentioned that, once the
inertia and damping coefficients are determined, that is, the coupling forces, it is a simple
matter to obtain the eigenfrequencies of the coupled coaxial cy1inders - cf. equations
(9.23) and (9.42); hence, this aspect is not belaboured here.
Nevertheless, a few words wi11 be said on Mulcahy's (1980) important work treating
the forces on a cylindrical rod in a narrow fluid-filled annulus oj finite length, and its
vibration characteristics. As discussed in Section 2,2,2(e)g and elsewhere in this book
(e.g, Section 8.3.5), as the aspect ratio LI R is diminished, so is the added mass.
The system under consideration is shown in Figure 11.II(a). The outer cylinder forming
the annulus is rigid and immobile, The analysis may be considered to be an extension of
tBoth the first (1779) and second (1786) editions of this book were published in Paris by lmprimerie de
Monsieur. 'Monsieur' at the time had a much more honorific, nobiliary meaning than today (cf. Sire, Sir).
Indeed, 'Monsieur' here was a specific person: the brother of the King (Louis XIV)! The third, expanded
edition (1816), included pyrodynamics; it was written while Count Du Buat was in exile after the Prench
Resolution (1789), and it appeared posthumously after the end of the Napoleonic era (publisher: Firmin Didot,
Paris).
* Interestingly, one of the items in Reynolds' long Royal Society paper is the detennination of the viscosity
of olive oil! Clearly, interest in olive oil is not entirely a yuppie phenomenon.
This discussion could have in fact been part of Section 2.2,2,

y
ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1235
(a)
(b)
1.00 35
33
0.40
'N 31
::;
0,20  29
27
K 0,10
25
°
(d)
0,04
0.10
0,02 0,08
0.01 0.06
0,2 04 0,6 1.0 2,0 4.0 6,0 10,0 
L1R 0.04
(c)
0,02
°
(e)
x
x
I
Z
2LI
x
100,0
40,0
20.0
10,0
2
 4,0
2 2.0

1.0
0.4
0,2
0,2 0.4 1.0 2,0 4,0 10,0
rVh
.4....
......
......
0,5
1.0
2.0
1.5
0.5
1.0
L1R
2,0
1.5
Figure 11.11 (a) Plan and elevation view of a cylinder in a narrow fluid-filled annulus of finite
length, 2L, (b) The added mas s and damping multiplicative factors, Re( C) and Im (C), as functions
af the viscous penetration depth ov; (c) the force coefficient K versus LI R. (d,e) measured versus
predicted eigenfrequencies and damping ratios for the fundamental mode: 0, - , - , hl R = 0.03;
-.-, hl R = 0.04 (only theory); b., - - - ,hl R = 0.06 (Mulcahy 1980).

1236
SLENDER STRUCTURES AND AXIAL FLOW
Chen et a/, (1976) and Yeh & Chen's (1978) work to finite-Iengtb annuli. Considering
oscillations of the form
x = aRe [exp(iQt)] = aRe [exp(i-r)]
(11.1 )
and neglecting nonlinear terms in fluid velocity, the Navier-Stokes and continuity
equations may be written in dimensionless form as follows:
ap aVel> _ _ ap  ( Dv ) 2 a2Vel>
aa = o, a-r - R acp + 2 h aa 2 '
av z _ _  ap  ( Dv ) 2 a2vz aV r avcp av z
a-r - L af3 + 2 h aa 2 ' a;;; + acp + aji = o,
(11.2)
where
a=(r-R)jh, f3=zjL, Dv = (2vjQ)lj2, p=p*j(pQ2 LR ),
v r = v:jQh, Vcp = v;jQR, V z = v;jQL, -r = Qt,
(11.3)
in which v:' v; and v; are tbe fluid velocities, respectively, in tbe radial, circumferential
and axial directions, the asterisks denoting dimensional quantities, p* is the pressure, h
the dimensional annular gap, and Dv the viscous penetration depth (Landau & Lifschitz
1959). To obtain (11,2), it has been assumed that viscous damping is significant, and so
Dvj h rv 0(1), the annulus is narrow, hj R « 1, and motions are small, aj h « 1. Tbe
peculiar form of tbe nondimensionalization of {v:' v;, v;} results in dimensionless terms
of similar order of magnitude.
The boundary conditions are:
a 'n _ ( i'r ) a . A. 'n ( i.
v r = h cos cp IU::: e , vel> = - R Slll 'f' I"e e ),
v r = vcp = V z = O at a = 1, and p = Pa at
V z = O at a = O,
(11.4)
13 = ::1::1, O  a  1,
where Pa is the ambient pressure, considering fluid entrance and exit effects in tbe annulus
to be negligible, Two additional boundary conditions state that tbe shear-forces at tbe ends
of the annular region are zem.
Tbe pressure p may be eliminated fmm the second and tbird of (11.2), and the higher-
order equations resulting tbereby solved by separation of variables; for example, tbe
equation for vel> is a 2 vcpja-raa - ! (Dvj h)2 a 3 vcpjaa 3 = O and admits the solution Vcp =
(aj h)[Fl + k F4 cosh(ka) + k Fs sinh(ka)] sin cp exp(i-r), where k = (I + i)hjDv, and
Fj = Fj (13). The solutions for v r , vcp and V z involve a total of seven undetermined functions
Fj and Gj, which may be expres sed in terms of two of them by applying the first six of
(11.4). Eventually, the force per unit length is found by integration of the expression for
pand application of the remainder of the boundary conditions, giving
2 ( R )[ COSh(ZjR) ][ .. " ]
Fx = p:rr R - 1 - Re(C) X + Q Lm (-C) X ,
h cosh(Lj R)
C = k sinh kj[2 + k sinh k - 2 cosh k].
(11.5)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1237
The coefficients of X and X are effectively added-mass and damping coefficients, For
2-D flow (LI R -+ 00), equation (11.5) is the same as that in Chen et a/, (1976). As may
be seen in Figure 11.11 (b), the added mass coefficient is insensitive while the damping
coefficient is very sensitive to ov. Thus, for a constant ov, as h is reduced, the damping
increases more rapidly than the added mass - both increasing because ofthe RI h factor
in (11.5).
Fx in (11.5) is a function ofaxiallocation z. The average force per unit length, F x , may
be found to be
F x = P Jr RZ K(RI h) [Re (C) X + QIm (-C)X], K = 1 - (RI L) tanh(LI R).
(11.6)
As seen in Figure 11.11 (c), the force coefficients are dramatically diminished as LI R is
reduced; for 2-D conditions, LI R -+ 00, K -+ 1.
Mulcahy also conducted experiment s in which a cantilevered rod, a portion of which was
confined in a narrow water-filled annulus, was electromagnetically forced to vibrate. The
response was sensed by an accelerometer, and the fundamenta1 frequency and damping
could be extracted using a servo system, These difficult experiments were performed for
gaps hl R = 0,03 and 0.06, and annulus lengths 0.5 ::: LI R ::: 2, As may be seen in
Figure 11.11 (d,e), excellent agreement between experiment and theory has been obtained,
thus validating the latter,
11.3.3 Numerical techniques
Of the many numerical techniques used to determine the coupling characteristics of cylin-
ders in and around fluid-filled annuli, two are briefly discussed here: a spectral-collocation
method and a finite element method; others are discussed when treating the system with
annular flow (Section 11.5.4),
(a)
z
I
-:Tl :Tl
-
()
(b)
Figure 11.12 A cylinder eccentrical1y located in a fluid-filled cylindrical container: (a) the physical
cross-sectional plane (ar, (J); (b) the computational domain (Z, (J); (Mateescu, Pai'doussis & Sim
1994d)

1238
SLENDER STRUCTURES AND AXIAL FLOW
11.3.3(a) The spectra/-collocation method
We start with the spectral-collocation method, as formu1ated by Mateescu et a/. (1994d).
Consider a 10ng cy1inder within an equal1y 10ng cy1indrica1 tube in an eccentric config-
uration as in Figure 11.12( a), with their mean positions characterized by the eccentricity
ae, where a is the radius and e the dimension1ess eccentricity, Consider the case of either
the inner or the outer cy1inder performing smal1 transverse osci11ations,
ag(t) = aeRe[exp(iQt)],
(11.7)
where e is the dimension1ess amplitude with respect to a; the other cylinder is fixed.
The Navier-Stokes and continuity equations for 2-D flow in the annu1us are given by
aV r +  [ vr aVr + V8 aV r _ v +  ap ]
at a ar r ae r p ar
=  [ r ( r avr ) + a 2 v r _ V r _2 aV8 ] ,
a 2 r 2 ar ar ae 2 ae
a V 8 +  [ vr aV8 + V8 aVe + VrVe +  ap ]
at a ar r ae r p r ae
v 1 [ a ( aV8 ) a2V8 aVr ]
= a2 r2 r ar ra;: + ae 2 - Ve + 2ae '
a aVe
- ( rv ) + - = o
ar r ae '
(11.8)
(11.9)
(11.10)
where V r and V8 are the radia1 and circumferential ftow ve10cities induced by cy1in-
der motions, p is the cOlTesponding pressure, and p the fluid density. The boundary
conditions are:
V r = Ve = O
(11.11)
on the fixed cylinder, and
V r = (1 - 8)ag cose + 8ag sine, V8 = 8ag cose - (1 - 8)ag sine
(11.12)
on the osci11ating cy1inder, where g = dg/dt; 8 = O or 1 according1y as the osci11ations
are in the plane of symmetry (e = O) or normal to it (e = 90°).
The problem is solved in a rectangular computational domain [Figure 1l.12(b)]
defined by
r - 1
Z = 1 - 2 h(e) , e = e,
h(e) = (b 2 - e 2 sin 2 e)I/2 - e cose - 1.
(11.13)
(11.14)
Equations (11.8)-(11.12) are transformed via (11.13) and (11.14) and 1inearized for small
motions; symbo1ically, equations (11.8)-(11.10) become
.cj[Vr(Z, e), Ve (Z, e), p(Z, e); h(e)] = O,
they are given in fulI in Mateescu et a/. (1994a).
j = 1,2,3;
(11.15)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1239
Next, the spectral method (Gottlieb & Orszag 1977; Canuto et al. 1988) is employed, via
{ vr/(a Q) } ,m n { Vr,jk [(1 - 8)Ck(0) + 8 Sk (O)] }
ve/(a Q) = iE: elfU L L Tj (Z) Ve,jk [(1 - 8)Sk(0) + 8 Ck(O)] , (11.16)
piCi pa 2 Q2) j=O k=1 Pjk [(1 - 8)Ck(0) + 8 Sk(O)]
where Tj(Z) are Chebyshev polynomials of order j, Ck(O) = cos(kO) and Sk (O)
sin(kO). Substituting into (11.15), one obtains three equations of the form
tt{PJk' Plk' P]k} { ": } =o, s=1,2,3, (11.17)
j=O k=1 Pjk
where Pjk = Pjk (Tj (Z), Tj (Z), Tj' (Z), Ck(O), C(O), C' (O), S' (O); tJ), in which ()' =
d O/dZ, O' = dO/dO, and
tJ = Qa 2 /v = 2(a/8 v )2, 8v = (2v/Q)I/2, (11.18)
tJ being the osci11atory Reynolds number (Stokes number) and 8v the viscous penetration
depth. In addition, we have four relationships for the boundary conditions on the fixed
and osci11ating cy1inders:
B]k(Z, O) Vr,jk = O, BA (Z, O) Ve,jk = O,
B]k(Z, O) Vr,jk = (1 - 8) cosO + 8 sin O, B}k(Z, O) Ve,jk = 8 cos O - (1 - 8) sin O,
(11.19)
with the Bjk evaluated at Z = -1 or I, depending on which cy1inderis fixed and which
is moving.
The collocation method is then used to determine the a priori unknown coefficients
Vr,jk. Ve,jk and Pjk, by imposing equations (11.17) at (m - 1) x n points in the com-
putational domain, defined by (Z J , OK), J E {1, m - I}, K E {I, n}. These collocation
points are clustered more densely near the cy1inder walls, where larger velocity variations
occur. Similarly, the boundary conditions are imposed at n azimuthally distributed points,
OK, K E {I, n}. Altogether we have a 3m x n system of equations, which solved yield the
spectral velocity and pressure coefficients Vr,jk. Ve,jk. Pjk.
Forinviscid flow, a simplified analysis is possible using the velocity potential (Mateescu
et al. 1994d). Once the pressure is determined, the unsteady fluid dynamic force per unit
length on cylinder i due to osci11ation of cylinder K in the plane of symmetry (8 = O) or
normal to it (8 = 1), denoted by FA(t), may be determined:
FfK(t) = _ [27r P I acosOdO,
10 2=1
F8K(t) = - [27r P I ba cos E> dE>, (11.20)
10 2=-1
where i, K = 1 for the inner cylinder and i, K = O for the outer one; E> is defined in
Figure 11.12(a). Denoting the amplitude of FlK(t) by F/ K , which is real, the added mass
coefficients are
-8
a = F iK
lK p7ra2Q2 [! (rf + ri)] aE:'
(11.21)

1240
SLENDER STRUCTURES AND AXIAL FLOW
where it is reca11ed that 8, rj and r K are dimensionless, and rl = 1 for the inner cylinder
and ro = b for the outer one.
For viscous flow,
[21f
FfK(t) = 10 a Tl de ,
[21T
F8K(t) = 10 ba Tod0,
(11.22)
where
( jJ. av r ) [ . ]
TI = - p + 2- - (1 - 8) cos e + 8 srn e
a ar
jJ. ( ave 1 aV r ve )
+ - - + -- - - [ocose - (l-8)sine]
a ar r ae r
and a similar expression for To; jJ. = vp is the dynamic viscosity, ln this case, Fj is
complex. We define the dimensionless force on the inner cylinder (i = 1) as
8 -8 2 2
F 1K = F 1K /(pTC a Q a 8),
(11.23)
and hence the added mass and viscous damping coefficients (cf. Section 2.2.3) are
8 8
Cm lK '" Re(F 1K ),
8 8
C v lK '" -Im(F 1K ).
(11.24)
The added mass for concentric cylinders with inviscid fluid in the annulus are compared
with Fritz's (1972) ana1ytica1 expressions in Table 1 1.1 (a). It is recalled that Fritz finds
o o b 2 + 1
0'00 = 0'11 = Z-'
b -1
o o 2 [ b2 + 1 ]
0'10 = 0'01 = 1 + b2 1 + b2 _ 1 '
(11.25)
where b is the ratio of the outer cylinder radius to that of the inner one; the superscript
is superfluous for concentric systems, It is seen that agreement with the ana1ytica1 res-
ults is excellent, with very few tenns in the Chebyshev series. t ln fact, therein lies the
attractiveness of this method.
The results for viscous fluid and concentric cylinders are compared with the analytica1
ones by Chen et a/, (1976) in Table ll.l(b). Agreement is seen to be better than 5%.
A comparison for eccentric cylinders is made in Table 11.1 (c). Here,
e re l = e/(b - 1),
(11.26)
thus increasing with diminishing annular gap. In this case, agreement is only within 10%,
most probably because of the finite element results having not quite converged for the
mesh used,* More extensive data and comparisons are given in Mateescu et a/. (l994d)
and Sim (1991).
The effect of relative eccentricity e re l on the added mass coefficients for a cylinder
oscillating within an annulus filled with inviscid fluid is illustrated in Figure 11.13. Some
ofthe results are compared to and virtua11y overlap Chung & Chen' s (1977). As expected on
tPor concentric systems, n = 1 in equation (11.16) because of orthogonality,
* Actually, it is not clear from Yang & Moran's (1979) paper whether the results given are for FfJ or Fi I'
However, the corresponding spectral method values are doser to each other than to Yang & Moran's,

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1241
Table 11.1 Comparison of spectral-collocation to analytical and finite-
element results for added mas s and viscous damping coefficients (Mateescu
et al. 1994d).
(a) Concentric cylinders, inviscid fluid, b = 1.25; percent deviation from
Fritz's (1972) analytical results
m O % deviation O % deviation
a]] -a lO
3 4.3117 5,35 4.1457 4,39
5 4.5549 0,01 4.3355 0.01
7 4.5556 0.()00 1 4.3360 0,0001
(b) Concentric cylinders, viscous fluid; comparison with Chen et ul. (1976)
analytical results
{3
b
O
'R.e(F(] )
Spectral Chen et ul,
O
-Im(F]])
Spectral Chen et ul,
2
2
1.5
1.25
50
500
500
5000
2.375
1.914
3.112
5.016
2.38
1.85
3.10
4,90
1.016
0,282
0.668
0.546
1.04
0,28
0.68
0.54
(c) Eccentric cylinders, viscous fluid; comparison with Yang & Moran's (1979)
finite-element results; b = 2, {3 = 50
o o
Re(F]]) -Im(F]])
e re ] Spectral Yang & Moran Spectral Yang & Moran
0,1 2.382 2.21 1.093 1.12
0.3 2.453 2,30 1.153 1.23
0.5 2,563 2.45 1.460 1.51
6 2,5
0>i:, 5
4
0.00
0,25
0.50
0.75
e r e1
(a)
2,0
1.5
1.0
0.00
0,25
0.50
0.75
erel
(b)
Figure 11.13 The added mas s coefficients a?K for oscillations in the plane of symmetry, as
functions of erel = e/(b - 1) for: (a) b = 1.25 and (b) b = 2; comparison between the
spectral-collocation solution for potential flow (-0-) and Chung & Chen's (1977) solution
(Mateescu et al. 1994d).

1242
SLENDER STRUCTURES AND AXIAL FLOW
physical grounds, increasing erel, hence increasing the relative proximity ofthe cylinders,
results in increased values of the added mass coefficient; the maximum possible erel is 1.
Figure 11.14(a) and (b) shows the effect ofthe oscillatory Reynolds number, f3, and
the outer-to-inner radius ratio, b, on added mas s and viscous damping coefficients for
a concentric configuration. As expected, both increase sharply with decreasing b but
-Im(Fg) much more quickly. Also, with increasing f3 both coefficients decrease, again
the viscous one much more rapidly (cf. Figure 2.9). For eccentric configurations, as seen in
Figure 11.14( c,d), increasing f3 while keeping b constant results in decreasing coefficients,
the viscous one much more rapidly. 50 does increasing b, but for a change from b = 2 to
b = 1.25, the effect is not very strong. Increasing erel, that is, proximity of the cylinders,
increases both the added mas s and viscous coefficients, but for the maximum erel = 0.5
in the figure, the effect is not very pronounced.
10
8
10,00
6 .
4 '"""'
o::: o::: 1.00
< <
 
I
2 0,10
°
1
2
3
4
(a)
10
8
6
b
4
<

_e_e--e-e--
2
1
0,00
(c)
0,10
0,50 0.00
(d)
0,25
erel
0,01
I
2
3
4
(b)
b
<

I
I
- -
_.-
.-.-.-. -
I
10,00
1.00
0,25
0.50
erel
Figure 11.14 (a,b) The real and imaginary components of the dimensionless fluid-dynamic force
fr?l on the inner oscillating cylinder in a concentric annular configuration, versus b for varying tJ:
, tJ = 50; e, tJ = 500; 0, tJ = 5000. (c,d) FJl for an eccentric configuration, the inner cylinder
oscillating normal to the plane of symmetry to = I), versus erel: -0-, tJ = 50, b = 1.25; -e-,
f3 = 50, b = 2; --, f3 = 5000, b = 1.25 (Mateescu et a/, 1994d),

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1243
The main advantage of the spectral-collocation method versus analytical ones is the ease
of formulating the solution, especia1ly for viscous ftuids. In comparison to finite element
methods, its main advantage is that convergence is reached with a relatively small number
of equations and hence small matrices, with m of the order of 7-10 in (11.16) and n ::: 5
for eccentric configurations. t
11,3,3(b) A finite element method with moving boundaries
Next, we consider finite element analyses. 'Standard' such analyses are, for example,
those ofPai'doussis et al. (1977) and Yang & Moran (1979), Here, however, we discuss a
specific analysis, by Chilukuri (1987), which is capable of dealing with large motions by
applying the boundary conditions on the moving boundary. The governing Navier-Stokes
and continuity equations are written in Cartesian coordinates:
av x av x av x 1 ap ( a2vx a 2 v x a 2 v y )
-+vx-+v -=---+v 2-+-+- ,
at ax y ay p ax ax 2 ay2 axay
av y OVy aVy 1 op ( 02Vy a 2 v y a 2 v x )
-+vx-+v -=---+v 2-+-+- ,
at ax y ay p ay ay2 ax 2 axay
éJv x éJv y
- + - = 0,
ox ay
(11.27)
and solutions are sought for motions of the inner cylinder, say, in the x-direction
xc(t) = A sin Qt,
vc(t) = Xc = AQ cos Qt,
(11.28)
Any spatial (cross-sectional) 2-D region occupied by the fluid is transformed into a quad-
rilateral one, involving the coordinates (, 17), as shown in Figure 11.15(a), involving nine
nodes. Then, recognizing that, due to cylinder motion, these spatial regions are deformed
with time and treating time as the third dimension, space-time elements are defined in
(, 17, O as shown in Figure 11.l5(b), which are mapped into cubes, via interpolation
functions N j e ; for example,
1 9 9
V x = '2y - O L N'J v:j + (1 + O LN'Jv:j+l,
j=l j=l
N'l = H 17( - 1)(17 - I), etc.
(11.29)
where e stands for 'element' and n refers to quantities at the nth time step; similarly for
the rest: x, y, Vy and p, as given in fulI by Chilukuri.
Then, Galerkin's averaging principle is app1ied to the integrated form of the equations
of motion, and the integra1s evaluated over alI of the elements. The banded equation set
t Specifically, for tJ = 50, it is generally sufficient to use m ::=:. 6, n ::=:. 6; for tJ = 5000, m ::=:. 10, n ::=:. 4, It
is necessary to increase m with increasing b, and n with increasing erel; for example, m = 7 and n = 5 for
tJ = 500, b = 1.25, erel = 0,5.

1244 SLENDER STRUCTURES AND AXIAL FLOW
7
D n C n 4 3
'YJ
Bn+1 8 L; 6
9
A n B n A n + 1
2
(a) 5
c n + 1
\
\ Bn+l
I \
\
\ ... 
\
Lit \ t 
1 \ \ x IL,
....----4
rr Dn C n ' (-1,-1,-1) (1,-1,-1)
,,-
'"
A n
(b) B n
Figure 11.15 (a) A quadrilateral fluid area at time steps n and n + I, and its mapping into a square,
nine-node finite element. (b) Isoparametric transformation of a space-time 'volume' into a cube
(Chilukuri 1987).
for unknown new time variables is solved by Gaussian elimination at each step, while
interior nodes [e.g. node 9 in Figure 11.15(a)] are eliminated from the equation set via
nodal condensation.
Figure 11.16(a) shows the deforming mesh at two time steps, where it is noted that
a coarse mesh is used in the figure for clarity of display; in the calculations, however,
refined mesh rearrangement and refinement protocols were used for accuracy. The number
of elements necessary is not large; for example, it is of order 10 each in the radia1 and
circumferential directions for Dol D = 2 and AI D = 0.4; D is the cylinder diameter and
Do the outer channel diameter. However, the number of time steps per period of oscillation
is 200-400.
Sample results for the force coefficients, defined as in (11.23) and (11.24), for two
geometries, Dol D = 2 and 5 (corresponding to b = 2 and 5 in Mateescu et aZ.), are
given in Figure 11.16(b) and (c). In both cases, the viscous damping coefficient increases
substantially with amplitude, A. The added mass coefficient in (b) also increases with A,
while in (c), intriguingly, it decreases first and then increases with A; this last observation
likely reflects the more complex flow pattern in the wider annulus.
The values ofCm and C v arecompared with thoseofChenetal. (1976)forthelinearized
system (AI D  O), showing excellent agreement, and tothose ofYang & Moran (1979),
to within 5% for the added mass coefflcient (Chilukuri 1987).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1245
(a)
3,0
2.0
2,5
1.5
1.5
....0
O ,.",.,.",. CV
-....
_-0--
O- --0--
'"
oP"
'"
1.0 "
'"
,,'" C v
"
'"
JY'
0-....
0,5
0,0
t.l
i: 2,0
u
0,1 0,2 0,3 0.4
A/(D o - D)
I I I
0.5 1.0 1.5
A/D
0.5
1.0
0,0
0,1 0,2 0.3 0,4
A/D = A/(D o - D)
0.5
(b)
I
0,0
(c)
2,0
Figure 11.16 (a) Mesh deformation for a coarse mesh (for illustration) for Dol D = 5,
AI D = 1.5. (b,c) The effect of vibration amplitude, A, on the added mass, Cm, and damping,
C v , coefficients for f3 = rW 2 /4v = 50: (b) Dol D = 2; (c) Dol D = 5 (Chilukuri 1987).
For narrow annuli, it is not difficult to have vibration amplitudes A/(D o - D) = 0.5.
Chilukuri's work shows that precise analysis requires the application of the boundary
conditions on the moving boundary.
Another finite element method with moving boundaries has been developed by Pattani
& Olson (1987, 1988), also considering the steady streaming component of the ftow.
For symmetric bodies or other bodies oscillating perpendicular to an axis of symmetry
(e.g. bodies with circular or square cross-section), steady streaming does not create any
additional force on the bodies, but does so otherwise.
11.3.4 Squeeze-film effects
The intennediate tube supports in shell-and-tube heat exchangers (Figure 8.136) are plates
with many holes, through which the tubes pass. The support provided is generally loose, for
ease of assembly/disassembly and to allow for thennal expansion. As the tubes vibrate due
to turbulence or vortex shedding, parts of them do so within narrow ftuid-filled annuli [see
Figure 11.11 (a)] and are thus subjectto increased added mass and damping, as discussed in
Section 11.3.2. Indeed, the main source of damping in shell-and-tube heat-exchanger tube
vibration with liquid on the shell side is squeeze-film damping (Pettigrew ef a/. 1986a,b).

1246
SLENDER STRUCTURES AND AXIAL FLOW
Because of a common theoretica1 basis, treatments of squeeze fi1ms in heat exchangers
and of squeeze-fi1m dampers used in rotating machinery are dosely related. Unlike das-
sica11ubrication theory, in this case inertia1 fluid effects are not necessari1y neg1igib1e.
Moreover, in the very narrow annuli invo1ved, the type of motion (whether transverse or
orbita1), initia1 eccentricity, nonlinear effects and fluid boundary conditions at the ends
of the annular passage are a11 significant. Hence, proper treatment of this topic is quite
intricate.
Because this topic is rather specia1ized, it is not treated here, important though it
undoubted1y is. Instead, some of the pertinent reference s are cited, to help the inter-
ested reader: Mulcahy (1980), San Andres & Vance (1987a,b), Has1inger et al. (1990), E1
Shafei & Cranda11 (1991), Esmonde et al. (1992), Zhang et al. (1993, 1994), Lu & Rogers
(1995), Han & Rogers (2001). Mulcahy's (1980) ana1ysis outlined in Section 11.3.2 is a
sample of the simp1er of these studies.
11.3.5 Vibrations of fluid-coupled coaxial shells
The literature on this topic is very rich, Samp1es are: Magrab & Burroughs (1972), Levin &
Milan (1973), Krajcinovic (1974), Bowers & Horvay (1975), Chen & Rosenberg (1975),
Taymans et al. (1975), Au- Yang (1976,1978,1981, 1986a), Yeh & Chen (1977), Livo1ant
& Jeanpierre (1978), Chu et al. (1979), Chu & Brown (1981), Chung et al. (1981),
Chiba & Kobayashi (1983), Brown & Chu (1983), Tomski & Kukla (1984), Fujita (1986),
Yoshikawa et al. (1994), Horáček et al. (1995), Horáček & Kruntcheva (1997), Jeong
(1998); see a1so Sections 2.2.2 and 2.2.3. A1so, the fluid mechanics per se in such systems
has received attention; for example, for some beautifu1 free-surface wave pattems, see
Krasnopo1skaya & van Heijst (1996). Part of the reason for this extraordinary attention is
the following: the added mass of the dense fluid for a narrow annu1us can be of shni1ar
magnitude or severa1 times 1arger than the mas s of a thin shell forming the inner or outer
boundary of the annu1us. A1so, as seen in the previous section, the damping is anything
but neg1igib1e. Hence, the vibrationa1 characteristics of the structure are not just modified
by the presence of the narrow annu1us, but radica11y a1tered.
C1assica1 studies of vibration of fluid-coupled coaxia1 shells are, for examp1e, those
of Levin & Milan (1973) and Au-Yang (1976), both with theoretica1 and experimenta1
components. Finite-element ana1yses are presented by, among others, Brown & Chu
(1983), Chiba & Kobayashi (1983), Horáček & Kruntcheva (1997) and Jeong (1998).
Additiona1 experiments have been done by Chu et al, (1979), Chiba & Kobayashi (1983),
Horáček et al, (1995), and others.
The vibration of fluid-coup1ed shells coup1ed via a quiescent annu1ar fluid can be studied
with the same ana1ytical/numerica1 too1s as those deve10ped for a flowing annu1ar fluid.
Hence, for methods of ana1ysis the reader is referred to Section 11.7. Here, nevertheless,
some typica1 resu1ts for quiescent fluid are presented to illustrate the vibration of such
coup1ed systems.
The effect of inertia1 coupling for a single steel shell in a concentric rigid cy1inder, with
both the shell and the annu1us water-filled, is illustrated in Table 11.2. The shell is of radiu s
279.4 mm, 3.175 mm thick and 1.12 m long, clo sed at the bottom and open at the top; the
annu1ar gap is 25.4 mm. As in Chapter 7, n denotes the circumferentia1 and m the axia1
wavenumber (Figure 7.3). It is clear that, mostly due to the presence of the dense fluid in
the annu1us, the natura1 frequencies are diminished dramatica11y, compared to those in air.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1247
Table 11.2 Measured and computed in-water nat-
ural frequencies, 7, and ratios thereof to in-air
natural frequencies, 71 f, for a shell within a rigid
cy1inder (Au-Yang 1976).
Mode Experimenta1 Computed
n,m
7 (Hz) flf 7 (Hz) 71f
2,1 64 0,181 62.4 0.195
3,1 54 0.252 46,2 0,252
4,1 60 0.302 55,9 0.304
5,1 90 0.352 89.2 0.353
5,2 136 0.357 127.4 0.362
500
400

N 300
Č
;>.
u
s::
<1)
::I
CJ' 200
<1)
....

100
O
1 2 3 4 5 6 7
n
Figure 11.17 Natural frequencies of two coaxial steel shells coupled by a water-filled annulus:
-, D, hi = ho = 2 mm; - - -, ,hi = 3 mm, ho = 2 mm; other dimensions given in
the text. Symbols: experimental data; lines: theoretical predictions. Diagrams define in-phase and
out-of-phase modes (Lev in & Milan 1973).
The effect is more important for the lower va1ues of n - cf. equation (7.55). Agreement
between the inviscid-tlow model and experiment is very good,
Figure 11.17 shows another comparison between theoretica1 and experimenta1 natura1
frequencies for a stee1 shet1 (radius ai = 132 mm and thickness hi = 2 mm or 3 mm)
within another stee1 shell (a o = 156 mm, ho = 2 mm), both of 1ength L = 6 m, simp1y
supported, and with the annu1us water-filled. Two distinct modes are obtained for each
(n, m): for the two shells osci11ating azimutha11y in phase and out of phase, as sketched
in the figure, Agreement between inviscid tlow theory and experiment is quite good.
Finally, Figure 11.18(a) and (b) shows measured and computed mode shapes (n = 5,
m = 1) for a shell (a = 77.25 mm, h = 1.5 mm, L = 231 mm) with an interiorwater-filled

1248
SLENDER STRUCTURES AND AXIAL FLOW
(a)
z
(b)
10 0
800
.. ."'......
... .. O
. ...
, 
II!. Q
, ,
, O
g=7mm < 'j,\!#.,:').b:."-
/. , 'b
. .
,. '\.
: '.
. '\
g= 3,5 mm I g= 25,5 mm .'.
t ,. '.
I I h' a-O
i ,o'
t' I'
I i /
I i P
+' {)
". "
.!i cf
'..cP
10- 3
10- 1
600
..:
B
u
10- 2 ť
<=I
'8..
g
Q
J...J>
N
:I:

400
200
o
(c)
10-4
1.0
Figure 11.18 (a) Measured and (b) computed modes ofvibration of a cylindrical shell surrounded
by a water-filled annulus for n = 5, m = 2, Hl L = I; in (a) f = 802 Hz, and in (b) f = 772 Hz,
(c) Measured (-0-, -.-) and eomputed (-) natural frequency f, and measured damping ratio 
(-.-+-.-, -'-0-'-, -.-.-.-) of a cylindrical she11 in an oil-filled annulus as fu ne ti on s of the oH
level Hl L, for n = 4 and m = I, and for various annulus widths, g (Horáček et a/, 1995),
annulus and a rigid inner cylinder. Figure 11.18( c) shows the natural frequencies and
damping ratios for n = 4 and m = 1, with the annulus filled with oíl to various levels,
H / L, and with various annular gaps, g, Theoretical results are shown only for the natural
frequency and g = 7 mm, showing excellent agreement with experiment.
11.3.6 Forced vibrations
A considerable amount of work on forced vibration of coaxial cylinders, and especially
shells has been done: on turbulence-induced random vibration [e.g. Au- Yang & Connelly
(1977)], pump-induced excitation [e.g. Au-Yang (1979)], and for blow-down accident
conditions in nuclear reactors [e.g. Schumann (1979)].

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1249
However, we shall not dwell on this topic, devoting instead more attention to se1f-excited
motions, in Sections 11.4-11.7.
11.4 1-D LEAKAGE FLOWS
Generica11y, the system under consideration is a rectangu1ar-section channe1 conveying
viscous fluid, with one or both of the rigid side-wa11s flexib1y supported, Motions of
the side-walls are either of one or two degrees of freedom (latera1 trans1ation and/or
rotation), and the fluid dynamics is ana1yzed via 1-D flow theory. Altemative1y, the system
may consist of a flexib1y supported rigid b1ade in a channe1 with immobi1e wa11s, as
in Figure 11.3(a). This cou1d be a model of a rea1 system (see Section 11.1.2). More
generally, however, it represents a simp1ified geometry which a110ws modelling to go
further, and to do so more re1iab1y, than in cy1indrica1 configurations, 1eading to deeper
insight of the dynamics. In particular, for the system of a moving b1ade in a fixed channe1
[Figure 11.3(a)], here it is usually assumed that there is no cross-flow from one subchanne1
to the other, whi1e in cy1indrica1 arrangements the circumferentia1 flow associated with
1atera1 motions of the cy1inders cannot be ignored - so that, even if radia1 flow ve10cities
are neg1ected, a 2-D flow model is necessary.
This subject is re1ated to that of para11e1 p1ate assemb1ies (Section 10,3,5). However,
the first contributions specifically on this system are due to Mi11er & Kennison (1966) and
Mi11er (1970), discussed briefly in Section 11.1.2. The system has a1so been studied by,
among others, Bland ct a/. (1967), Hwang & Thorpe (1973), Mark (1985) and Schwirian
(1986), More fundamenta1 in their outlook are the studies by Mu1cahy (1988) and Inada
& Hayama (1988, 1990a,b, 1993), on which the materia1 presented here is based. Chro-
no10gica11y, however, work on annu1ar geometries was done before this (Section 11.5),
starting with Hobson's (1982) semina1 contribution.
11.4.1 Mulcahy's 1-d.o.f. analysis
We start with Mulcahy's (1988) one-degree-of-freedom ana1ysis, invo1ving prescribed,
symmetric 1atera1 motions of the channe1 side-wa11s, Figure 11.19( a), described by
H = H + h = H + li sin Qt;
h/H« 1.
(11.30)
The genera1 system under consideration is shown in Figure 11.19(b), where subscripts
U, D, L, R and I stand for upstream, downstream, 1eft, right and intermediate. Anywhere
in the system, we have
U = U + u,
p = p + p;
u/ U « 1, p/P« 1.
(11.31)
The pressure drop across each of the three constrictions is assumed to be proportiona1
to the dynamic pressure of the turbu1ent flow through it. Since the 10ss coefficients K are
not avai1ab1e for (i) unsteady flow and (ii) expanding or contracting constrictions, they
are assumed to remain constant at their steady-flow va1ues. However, in the case of the
intermediate constriction, the 10ss coefficient is augmented by an unsteady component k],
1inear1y dependent on h, to permit comparison with Hobson's (1982) work.

1250
SLENDER STRUCTURES AND AXIAL FLOW
:, , , " " " " " " " " " -':':, " " " , " " " " " " " ':, " " , " " " " " " " " " " " ""'i
---------------------,
L 
r: l H
L___________________
(a) """"""""""""""""""""""""""""""",',""""""""""" 'o,J
I .
L
L
L
: 1
Puf
L . R
r
UL . HL =rHI U R . H R
U
I
x
H D
1-
T PD
(b)
Figure 11.19 (a) Mode deformation of the channel; H is the mean width (also the width at rest),
and H = H (t) is the width while the channel is in motion, (b) The geometry of the channel considered
in Mulcahy's l-D analysis (Mulcahy 1988).
The shear stresses on the channe1 walls are modelled by
T = c P U 2 ,
(11.32)
where c is a known constant - determined, for example, from the Darcy-Weisbach
equation (Massey 1979), again taken to be independent of u.
Considering incompressible ftow and a control volume of width H and infinitesimal
length, conservation of mas s and momentum requires
oU dH
H- = --, (11.33)
ox dt
p ( OU + U OU ) = _ oP _ 2. (11.34)
ot ox OX H
Substituting (11.30) and (11.31) in to (11.33) and (11.34), and linearizing, leads to
equations for the steady-state,
oU
ih' = O,
and for the ftuctuating components,
-oP -2
H - = -c p U ,
ox
(11.35)
ou
ox
1 dh
=---
H dt'
-op o P - -ou -dh
-H- -h--2cpUu =pH- -pU-.
OX OX ot dt
(11.36)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1251

PA
1 1
HB
P B
. UA
---. U
. U B
Figure 11.20 Velocities and pressures at a constriction between two segments of different width
of the channel (Mulcahy 1988).
Solutions for U, and then for P, may be obtained by integration:
U=U c , u=_ dh ( X ) +Ut(t), p =_ l cp U2 x+ P c,
dt H H
p  P U 2 H :- U  U, ) +  : - 22 e> ' ) ]  + p, (.),
(11.37)
where U c and Pc are constants, and Ut and Pt are functions of t, determined from the
boundary conditions.
The boundary conditions are related to the constrictions which, as noted in
Section 11.1.2, play an important role in the stability of the system. Thus, for any of
the constrictions, with the aid of Figure 11.20 we have
1 2 1 2 1 2
PA + '2 PU A = PB + '2PUB + '2 Kp U ,
where U is the flow ve10city through the constriction. Hence, using (11.30) and (11.31)
and linearizing, we have
- 1 -2 - 1 -2 1 -2
PA+'2PUA=PB+'2PUB+'2KpU,
- - - 1 -2
PA+pUAUA=PB+pUBUB+pKUu+'2 PkU ,
(11.38)
(11.39)
in which the k component is ignored in all but the intermediate constriction, where it is
taken to be
k = 2k/(hj H /);
(11.40)
in all constrictions, no downstream pressure recovery is inc1uded in the model.
Evaluation of (11.38) and (11.39) at each constriction, provides two sets of three
equations each to solve for the four constants of integration, U L, Pe L, U R, Pe R, and
UtL, PtL, UtR, PtR, the fourth equation coming from the continuity relation UA HA
U H, with steady and unsteady components
U A = (HjHA)U, UA = (HjHA)U + (hjHA)(l- HjHA)U.
(11.41)

1252
SLENDER STRUCTURES AND AXIAL FLOW
It is noted that, in eva1uating the 10ss re1ations (11.38) and (11.39), the ve10city and
fluctuating pressure in the upstream and downstream plenums are taken to be zero:
Uu = Uu = U D = UD = O,
PU = PD = O.
(11.42)
To eva1uate the two constants and two integration functions in (11.37), U L, U R, U U
and U Dare expressed in terms of U /, and UtL, UtR, Uu and UD in terms of Ul, by means
of (11.41). Then, three steady and three fluctuating pressure 10ss re1ations are generated
by eva1uating (11.38) and (11.39) at each constriction. For detai1s, see Mulcahy (1988)
and the earlier Report cited therein.
The mean flow ve10cities in the channe1s and the constrictions may all be expressed in
the form
U A = /3A U S,
where U S is the mean ve10city for the reference channe1 width H s,
(11.43)
-2 - -
U S = 2(Pu - P D)/(p RI);
(11.44)
/3 A = H s / H A is eva1uated by substituting for the channe1 or constriction width of interest
(e,g. /3L = H s/ H L>, and
RI = /31 K/ + /3 Ku + /31 KD + 2(/3 KR + /3Z KL),
(11.45)
where friction 10ss coefficients have rep1aced (11.32) via KR = C R L R / H R, for instance.
The determination of the fluctuating ve10city components is more complicated, because
of the derivative du t / dt . Elinůnating Pt L and Pt R by using two of the three equations, the
third one is found to have the following particu1ar solution for U/:
 = T2 W S d(h/Hs) _ TI ( h ) ,
  Hs
Ws = QL /U s ,
(11.46)
where Ws is the reduced frequency, and the constants TI and T2 are given in
Appendix Y.1.
Once u / has been deternůned, the fluctuating velocities in the other constrictions and
either half of the channe1 may be determined via equations (11.41) and the first two of
(11.37). Thus,
 =  + ( /3/ - /3A )  _  dh
U A U/ /3 A H A H A U A dt '
where subscript A becomes L, R, U or D, as appropriate.
The net force on the moving channel is expressed as
(11.47)
F = -CD M Q(dh/dt) - CMIKM(d 2 h/dt 2 ),
(11.48)
where M = p (H L LL + H R LR) is the fluid mass in the channel, CD the fluid damping
coefficient, and CM I K the combined added-mass and stiffness coefficient, given in explicit
form in Appendix Y.1. This latter is partitioned into added-mass, CM, and stiffness, C K,

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1253
components via C K = CM - CM / K, where CM is the value of CM / K for V = O, which is
independent of Q, and CK is a function of Vs == l/ws.
Hence, denoting by Mw, C w and Kw the mass, damping and stiffness of one of the
sidewalls, the equation of motion is
d 2 h dh 2
(Mw + MCM)-----Z + (Cw + MCDQ)- d + (Kw + CKMQ )h = O.
dt t
(11.49)
Dynamic instability (single-mode flutter) arises when
Cw + MCDQ < O,
(11.50)
thus requiring that CD be negative and sufficiently large; it is seen from the expression
of CD, in equations (Y,3), that it is a function of Ws or Vs. No coupled-mode flutter is
possible because this is a l-d.o.f. system. Divergence, occurs when
- 2 2
Kw + (V siL) Mws CK .:::: O,
Ws -+ O,
(11.51)
requiring that C K be negative, t
Based on this simple system and because a closed-form solution is possible, a number
of important deductions may be made on its dynamics, by considering particular 'cases',
as follows.
11.4. 1 (a) Case A: the effect af entrance, exit and wa/l-frictian lasses
Here we consider the system without the intermediate constriction, so the channel width is
- 1 1 1-
H throughout. Wetake LR = LL = zL and KL = KR = zKf = ZC LIH, andconsider
the effect of K u, K D and K f on stability. The damping coefficient, a particular form of
(Y.3), is evaluated to be
1 ( L ) 2 [ 4 (KD- K U- 2 )[-2Kf + (KD-Ku)RdVi ]
CD = 4Ys H Rl - 3 Kf - 1 + R 2 V2 '
1 s
(11.52)
in which Rl = Ku + KD + 2K f and Vs = V I(QL).
Considering first the case of K f = O, this reduces to
_ 1 ( L ) 2 [l-2(Ku-KD-2KuKD)Vi]
CD--VS = (KD+Ku) 22 '
4 H 1 + (Ku + KD) Vs
(11.53)
Hence, for flutter we must have 2(Ku - KD - 2Ku KD) Vi > 1, or
1
V 2 >
s 2(Ku - KD - 2KuKD)
and
KD
Ku>
1 - 2KD
(11.54)
This means that, if the upstream loss coefficient is sufficient1y large (the maximum being
K u ::: 0.5) and the downstream loss coefficient sufficient1y small, specifically K D < 0.25,
t Another expression, in terms of the components of C K is given in Appendix Y.l,

1254
SLENDER STRUCTURES AND AXIAL FLOW
then flutter is possible for sufficiently large Vs = U IQL. Hence, the first important
conclusion is that flutter does not occur just for any entrance losses and for any value
of U , but for a specific balance of these, as given by (11.54). Furthermore, examin-
ing the form of (11.53), it is clear that CD is initially positive and increasingly large
with increasing Vs (e.g. increasing U for given Q and L), up to a maximum; it then
becomes negative as the second term in the square bracket asserts itself, as Vs is increased
further.
Considering next the case of K U = K D = O, equation (11.52) becomes
I ( L ) 2 [ 6 V 2 ]
CD = -K f Vs = 1- S2 2 .
6 H I + 4K f Vs
(11.55)
It is seen that instability is possible if K f is sufficiently small (K f ::: 1.23), as shown in
Figure 11.21 (a). Clearly, for a long enough channel, flutter is not possible. Other subcases
are considered by Mu1cahy (1988).
It is shown that static divergence is not possible. From equation (Y.5), divergence would
occur if 2 - K D + K u < O, which cannot be satisfied for realistic entrance and exit loss
coefficients in a uniform channel.
11.4. 1 (b) Case B: a constriction in a channel with ideal entrance and exit
A single sharp-edged constriction at an arbitrary location is considered, in an otherwise
uniform channeI. Thus, we have Ku = KD = O, while KI = K, AIso, fh == Hl H I = P
defines the width ratio at the constriction: a large P means a large constriction, Moreover,
we define 0= (LR - Ldl L, so that
o > O for a constriction upstream of ! L,
o < O for a constriction downstream of ! L;
thus, o = -1 means that the constriction is at the exit, and 0= 1 that it is at the
entrance.
Flutter in this case is found to occur when
8
V 2 >
s 2p2K + (P -1)(2p2K +418)'
-2(P - I)
when 8 > 3
pK
-0.3
>-
- K '
(11.56)
where it is recalled that K == KI. In realistic systems K and p are interdependent,
depending also on the geometric detaiIs of the constriction. In fact, Mu1cahy defines
Ke = p2 K as an equivalent 'contraction effect', associated with the constriction.
Typical results are shown in Figure 11.21 (b). It is seen that, if the constriction is suffi-
ciently downstream (e.g. for o = -0.66), flutter does not arise. If it is upstream (8 > 1),
flutter is possible for sufficiently large Vs or U . These results clearly fully agree with the
preliminary discussion in Section 11.1.2.
Presuming next that pand K are independent of each other and taking K = 1 or 4, the
effect of p is ilIustrated in Figure 11.21 (c), It is seen that increasing the contraction ratio
is a very effective way of increasing the stabilizing effect (8 = -I, upper curves) or the
destabilizing effect (8 = I, lower curves), Comparing the dashed curves (ft = 1. K = 4)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1255
0.15
0,10
0,05
'"
I
:::J. O

-0,05
-0.10
-0,15
O 10.0
(a)
0,2
0,1
'"
I
 O
rJ
-0.1
0,25 0.50 0,75 1.00
(b)
0.3
0,2
0.1
 /3=1
I::t:
 O
rJ ,
,
-0,1 " " /3 = 1 (K= 4)
"
"
-0.2 "
"
"
,
-0.3
O 1.0 2,0 3,0 4,0 5,0
(c) U/(fL)
Figure 11.21 (a) Case A: the effect of wall friction coefficient K f = ! c L / H on the fluid damping
coefficient CD, plotted versus U/U L) = 2JTVs for Ku = KD = O. (b) Case B: the effect oflocation
(value of 8) of a sharp-edged constriction in the channel on CD for K [ = 1.0 and contraction ratio
f3 = 4.0. (c) The effect of the contraction ratio f3 on CD, with the constriction either at entrance
(8 = I, lower curves) or at exit (8 = -I, upper curves): -, K "" K[ = I; - - -, K = 4
(Mu1cahy 1988).

1256
SLENDER STRUCTURES AND AXIAL FLOW
with the continuous ones for P = 2, K = 1, so that Ke = p2 K is the same, it is seen that
increasing p is more effective than increasing K in augmenting ICD/(LI H )21,
ln this case also, no divergence instability is possible,
11.4. 1 (c) Case O: convergent and divergent channels
ln this case, we consider a width change without a point-constriction, and without a
pre ssuross at the point of change of the width from H L to H R; thus, effectively,
Hl = H R, and K I = O. This may be thought to represent a first approximation to a
convergent ( H R I H L < 1) or divergent ( H R I H L> 1) channel, where K I = O implies
a 'smooth' transition. Calculations are conducted for a square entrance (Ku = 0.5) and
exit (KD = 1), and K f = O,
For convergent configurations, CD is positive, increasingly so for larger contraction
ratios p = H LI H R and larger LRI L. For divergent configurations, however, CD is neg-
ative for sufficiently large expansion ratios p = H R I H L, at smaller U for larger pand
smaller LLI L. Moreover, in this case static divergence is also possible for the divergent
configurations. These results are not presented in any detail, because this highly idealized
representation of convergent and divergent channels cannot be hoped to give more than
qualitatively valid conclusions,
Several other cases have been studied by Mu1cahy, not discussed here for brevity.
11.4.2 Inada & Hayama's analysis
Mu1cahy's model has been extended by Inada & Hayama (1988) to deal with tapered
passages, one of the side-walls of which undergoes translational motions. This has further
been extended (lnada & Hayama 1990a, b) to a 2-d,o,f. model, invol ving both translational
and rotational motions, as outlined in what follows.
Figure 11.22 shows the system under consideration. The upper wall of the tapered
passage can oscillate in translation with amplitude !).Z, or rotation about a hinge point at
X = L f (f for fu1crum) with amplitude !)'8. The gap H is considered to be small with
respect to the length, H l L « 1, and the amplitudes smal!, for example !). Hl H « 1.
The ftow remains attached to both walls. The ftow ve10cities in the X- and Z-direction are
---
---
--
2
8.2 exp (i.Q/)
Pin
H(X) Q
---=--.
Pex
x
o
L
Lj
Figure 11.22 Narrow tapered passage, H (X) / L « I, the coordinate system and the two kinds
of oscillatory motions considered by Inada & Hayama (l990a).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1257
u and w, respectively. The equation of motion for the X -direction t and continuity are
au au 2 a(uw) 1 ap a 2 u
at + ax + az = -p ax + v az 2 '
au aw
ax + az = o;
(11.57)
the continuity of the normal velocity components at the upper wan is written as
( aH ) aH
WIH - UIH ax - ar = o,
(11.58)
which is a generalization of (11.33). Integrating (11.57) over the channel width, we replace
u by the volumetric ftow rate per unit thickness, Q, thereby transforming the 2-D problem
into a 1-D one:
a Q a ( Q2 ) H ap 1 Q2
-+- - =-----A-
at ax H p ax 4 H2'
aQ aH
ax + ar = o,
(11.59)
where we have used
[H [H 1 ( [H ) 2 Q2
Q = 10 u dZ, 10 u 2 dZ c:::: H 10 u dZ - H'
V (  I _ I) =_!A Q2
a z a z 4 H2'
H o
(11.60)
A being the friction coefficient for channel ftow; for laminar ftow, A = 48jRe, with
Re = Qjv.
Furthermore, we define pressure loss factors at inlet and exit by in and ex, and write
P ( O ) _ P. _ ,=, P Q2(0)
- m Sm 2H2(0) ,
P e L ) _ p, '= p Q2(L)
- ex + sex 2H2(L) .
(11.61)
Introducing the dimensionless quantities
X
x= -
L'
H
h(x) = =- = l+ax,
H(O)
= [ H (L) - H (O) ]
a _ ,
H(O)
Lf
lf=T'
H(O) == Ho,
(11.62)
tWith continuity included in the first Navier-Stokes equation.

1258
SLENDER STRUCTURES AND AXIAL FLOW
the steady (mean) pressure drop, pressure at any point, and flow rate are found to be
d P P Q 2 { Id ( 1 ) 1 }
dx = Hli -h dx h - 13 h 3 '
-2
P(x) Pin ex p* = p Q
--p;:- = P* - 2 - (13 - a)l)(x), H2 '
o
ÁL
13 = 4Ho '
_ { } 1/2
Q Pin - Pex
Ho = p [(13 -a)l)(1) + !in + !exlh2(1)] ,
(11.63)
wherea isdefinedin(11.62), anďX"is thevalueoťA when Q = Q ; In (x) = ft [h(x)r n dx,
with n = 3 here. Based on these, the steady force F and moment M are detennined; see
Appendix Y.2.
The unsteady force and moment are detennined next. We first define
H(X, t) = H (X) + I::1H(X, t), I::1H(X, t) = {I::1Z + 1::18(X - L f)}eir.lt,
P(X, t) = P (X) + I::1P(X)e ir2t , Q(X, t) = Q + 1::1 Q(X)e ir2t , (11.64)
Á = I + ( : ) I Q 1::1 Q e ir2t , F = F + 1::1 F eir.lt , M = M + I::1M e ir2t ,
Then, we define the dimensionless quantities
I::1P I::1Q
P = p* ' q = Q '
y = 1 + 2 ( : ) I Q ,
1::1 Z
e=-,
Ho
1::18
i}=
(Hol L)'
w = r2to,
t
T =-,
to
HoL
to = -=-,
Q
(11.65)
and recall that Ho = H(O). Substituting into (11.60) and linearizing, we obtain
dp { iw 2f3y d ( I ) } 2 dq { 313 d ( I ) }
dx = - h + --,;3 + dx h 2 q - h 2 dx + h 4 + dx h3 e
{ 3 f3 d ( X -I f ) }
+ -(x -I f )- - i}
h 4 dx h 3 '
dq + iwe + (x -I f) iw i} = O;
dx
(11.66)
( 11.67)
also from (11,67), we have the dimensionless flow rate,
q(x) = q(O) - xiwe - (!x2 -Ip) iwi}.
(11.68)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1259
Linearizing the boundary conditions and rendering them dimension1ess, we have
p(O) = -gin {q(O) - e + I jiJ} ,
p(1) = gex h21) [q(1) - h:1) {e+ (1-1j)iJ}].
(11.69)
Next, substituting (11.67) and (11.68) into (11.66), integrating, and taking (11.69) into
account, we find p(x) as a function of a, {3, y, gin, I j, w and integra1s invo1ving h(x), as
given in Appendix Y.2. Then, integrating p (x) and (x - I j ) p (x) from in1et to exit, and
expressing q (O) in terms of e, iJ, iwe == ě and iw iJ == 1J, we obtain the unsteady force
Do F and moment DoM, as given in Appendix Y.2. After some further rearrangement of the
matrice s invo1ved, we find
{ Do!IP*L } { ž } { ž } { z }
Doml P* L 2 = -[Ma] jj - ([C w ] + [Cen tJ - [Ka] (j ,
(11.70)
where the overdot denotes d/dT, and
Do! = DoF eiw-r, Dom = DoM eiw-r, z = e eiw-r, (j = iJ eiw-r.
(11.71)
Graphs ofvarious elements ofthe added mass, damping and stiffness matrice s are given
by Inada & Hayama (1990a) as functions of the taper parameter a. In these diagrams,
the regions of Cwll + Cell < O and C w 22 + C e 22 < O are mapped, which are the
regions of the parameter space where flutter is possib1e for purely trans1ationa1 and pure1y
rotationa1 motions, respective1y. Simi1ar1y, regions where Kall < O and K a 22 < O are the
corresponding regions where divergence is possib1e. Samp1es of such diagrams are shown
in Figure 11.23. From the definition of a, it is noted that a > O and a < O represent
divergent and convergent channels, respective1y; hence,
1 + a > 1: divergent channe1;
1 + a < 1: convergent channel;
whi1e 1 + a = 1 represents a uniform channel.
For purely trans1ationa1 motions, it is clear from Figure 11.23(a) that, generally, both
sing1e-mode flutter and divergence are possib1e for divergent channe1s, It is a1so seen that
the minimum va1ue of a for flutter occurs when {3 = ILI4Ho  1.1, which can be
interpreted to mean that this occurs for a specific slendemess Hol L of the channel. For
both higher and 10wer {3, a has to be 1arger for flutter to be possib1e. A1so, for wT --+ 00,
no flutter is possib1e. Divergence, when wT --+ O, occurs if {3 > 10 or {3 < 0.2. The near1y
vertica1 dashed 1ines (for Kall = O) associated with wT = 1 and 00 are not connected
to 10ss of stability by divergence (which requires w = O), but to a decrease in the natural
frequency.
Flutter for the divergent channel is identified (Inada & Hayama 2000) as being related
to the time delay associated with w T in (Y.14) and hence to the osci1latory Reynolds
number - defined in (11.18). t By reference to equations (11.66) and (11.69), the mech-
anism of generation of negative damping in this geometry is also discussed. It is first noted
tThe oscillatory Reynolds number, usually denoted by {3, in this section is denoted as {3s, to avoid confusion
with {3 = I.L/4Ho as in equations (11.63),

1260 SLENDER STRUCTURES AND AXIAL FLOW
10 2
10 1
Kall >0
fJ
C wll + C ell > O
101
(a) 102
10 2
10 1
f3
10- 1
(b) 10- 2
10 1
f3
10- 1
(c) 1O2
10- 1 10 1
l+a
Figure 11.23 (a) Tbe regions where Cwl! + Cel! < O or Kall < O for in = 1.0, ex = O.
(b,c) Regions where C w 22 + C e 22 < O or K a 22 < O for in = 1.0, ex = O, for (b) Z f = 0,5 and
(c) Zf = 0.7, Boundary of C w ll + Cell = O: -, wT -+ O; -, wT = 1. Boundary of
Kall = O: ---, wT -+ O; - - -, wT = 1 and 00 (Inada & Hayama 1990a).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1261
that the inertia force [first term in braces in (11.66)] is proportional to 1 I h, and the resist-
ance force [second term in braces in (11.66)] to II h 3 ; hence, inertia effects are dominant
where h is 1arge, in the downstream part. Consider an increase in the gap due to motion; the
flow increases since the overal1 resistance decreases. At the downstream part of the chan-
ne1, inertia is dominant compared to resistance, and the flow rate cannot increase easi1y;
the pressure gradient is diminished, and the pressure in the downstream end of the channe1
increases ('the same effect as trying to c10se the exit, whi1e pouring in fluid at the in1eť). If,
on the other hand, the gap is reduced, the flow rate decreases, but again it does not decrease
readi1y at the exit because of the inertia; hence, the pressure decreases ('like trying to c10se
the in1et while sucking out the fluid at the exiť). Therefore, the pressure increases when the
gap increases, and it decreases when the gap decreases; pressure (force) and rate of change
of gap (velocity) are in phase, and energy flows into the strncture, that is, we have negative
damping. The converse applies in a convergent channe1, and we have a positive damping,
Figure l1.23(b) shows the possib1e instabi1ity region for pure1y rotational motions in
the case of 1 f = 0.5. F1utter is possib1e over almost the same region of (a, fJ) as in
Figure 11,23(a), but the region of possib1e divergence is considerab1y 1arger.
If the hinge is moved downstream, to 1 f = 0.7, it is seen in Figure l1.23(c) that flutter
is now possib1e a1so for convergent channe1s. However, in the 1atter region, the system
is a1so pro ne to deve10ping divergence; indeed, divergence is possib1e in this case over a
wide range of (a, fJ), considerab1y 1arger than in Figure l1.23(b).
Of course, in addition to sing1e-mode flutter, coup1ed-mode flutter is a1so possib1e,
but in that case a11 the e1ements in the matrices p1ay a ró1e, and the results are ohtained
numerica11y by solving the dynamical system as a who1e, that is, an equation of the form
([M] + [Ma]) U} + ([C] + [C w ] + [C e ]) U} + ([K] + [Ka]) {} = {O}, (11.72)
as described by Inada & Hayama (1990b), but using a different nondimensiona1ization
scheme. The presence of the de1ay terms in (Y.15) resu1ts not in a fourth-order, but a
sixth-order po1ynomia1 for the eigenfrequencies of the system. The system for which
these ca1culations have been done is somewhat different from that of Figure 11.22, to
correspond to one studied experimenta1ly by the same authors, shown in Figure 11.24. As
a result, al1 the fluid-re1ated matrices in (11.72) are twice those found in the foregoing,
The experiment [Figure 11.24(b)] invo1ved a tapered passage 100 mm wide and 100 mm
10ng, and a centra1 stee1 p1ate 0,8 mm thick, The in1et gap from p1ate to wall was fixed
at 1 mm, and the tapering varied, -0.6 < a < 9.0. Four p1ate-springs at the upstream
end provided the structura1 trans1ationa1, Kt, and rotationa1, Kr' stiffnesses of the system.
By changing the positions of these springs, the 10cation of the pivot, L f' and the circu1ar
frequencies of trans1ation Qt = (Kr! Ms)1/2 and rotation Qr = (Krlls)1/2 cou1d be
adjusted, Ms and ls being the structura1 mass and moment of inertia. The fit of this
p1ate in the channe1 was such as to render 1eakage from one subchanne1 to the other
negligib1e. The fluid was air, and for al1 experiments Re < 1000, that is, the flow was
laminar. Motions of the plate were sensed by eddy-current gap sensors at two streamwise
10cations. The structura1 natural frequencies and damping were measured in a widened
channel (28.5 mm), so as to minimize added mass effects. AU pertinentphysical parameters

1262
SLENDER STRUCTURES AND AXIAL FLOW
Fin
x
Ho
(a)
Lg
".....-r...._
.....- I --__
___.....- I
./'--- I
r<..
I '...... o
I
I
I
I
I
I
I ___---
1.:---....
....
-..
Tank --
'. Plate spring
.......-0
-....",; :
 Lk 1
. I
I
. I
I
".. I
--)
---
----
-_ I ,.,.
---.....1,................... Target
(b)
Gap sensor
Figure 11.24 (a) The system under consideration for comparison with experiments; (b) the
experimenta1 system (Inada & Hayama 1990b),
for the four experimental conditions, in which L f was the principal parameter varied, are
1isted in Inada & Hayama (l990b, table 1).
Typical results are shown in Figure 11.25. The dimensionless flow rate and oscil1atory
Reynolds number are defined by
Q Q
q = HoL Qr = HoL(Kr/ Is)1/2'
HoL Qr
/3s = .
v
(11.73)
Figure 11.25(a) shows the theoretical and experimental results for I f 0.32, that is,
with the pivot upstream of the axial mid-plate position. Theory predicts coupled-mode
flutter for both convergent and divergent channels, -0.7 < CI < 5. Interestingly, qcr is
diminished as the channel becomes more convergent, and is then increased sharply at
CI ::: -0,7 (I + CI ::: 0.3). Single-mode flutter is predicted for 5 < CI < 9. Agreement
between theory and experiment is generally very good - considering the complexity
of the system and the difficulty in the accurate measurement of all quantities required.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1263
1.40 1000
Single-mode
flutter
1.05
Stable
<:i: 0.70 500 Q)

0,35
(a) 0.0
1.2
0.9
<.> 0.6
'"
0.3
\ I
'-' ,,1 Unstable
Divergence
I I
Single-mode
flutter "'\
\ I Flutter

'-
Stable lij 'ii
I
o
1000
Q)
500 
(b) 0,0
1.40
1.05
Stable
<.> 0,70
'"
0.35
(c) 0,0
10- 1 1
l+a
o
1000
500 
o
10 1
Figure 11.25 Critical flow qcr and corresponding value of Re versus 1 + ex : - - -, divergence;
- - - , coupled-mode flutter; - , single-mode flutter: (a) for 1 f = 0.32, úJ r = 2,53;
(b) for 1 f = 0.50, úJ r = 2.09; (c) for 1 f = 0,68, úJ r = 2,53, Full symbols indicate a stable state;
open symbols an unstable one (Inada & Hayama 1990b).

1264
SLENDER STRUCTURES AND AXIAL FLOW
Nevertheless, the analytical model fails to capture two features of the observed behaviour:
(i) the sudden drop in qcr at 1 + a  3.5 and (ii) the second range of flutter (between 
and V points) for 0.35 < qcr < 0.5 approximately.
Figure 11.25(b) is for 1 f = 0.5. Theory predicts divergence for smalll + a, and single-
mode flutter for 1 + a  1.5 approximately. In the experiments, divergence was not
observed at all, while flutter was, and in excellent agreement with theory. Figure 11.25( c)
shows the results for 1 f = 0.68. Again, agreement between theory and experiment for
flutter is excellent. Divergence in this case was observed, but agreement with theory is not
as good, which is not unusual for divergence (cf, Chapters 3 and 8).
Additional results are shown in Inada & Hayama (1990b), comparing also theoretical
and experimental flutter frequencies (excellent agreement) and the form of the vibration
(mode shapes and the elť} amplitude ratio and phase), not given here for brevity.
In summary, these results firmly establish that flutter is widespread for divergent chan-
nels, but can also occur for convergent channels. Divergence, in the results presented,
is confined to divergent channels. The location of the hinge does not prevent instability,
although the values of qcr are about four times larger for 1 f = 0.32 than for 1 f = 0.5
and 0.68 for the main in stability range, and about twice as large for the secondary one.
By taking these results together with those of Mulcahy, one can consider a downstream
constriction as an extreme case of a convergent channel, which promotes stability. Par-
ticularly valuable, however, are Inada & Hayama's experiments, which lend weight and
support to the theoretical findings.
Using the same model, Inada & Hayama (1993) investigate the effect ofvarious dimen-
sionless parameters on stability, considering purely translational (transverse) motions.
Sample results for flutter are: (i) the value of qcr increases with the mass ratio (struc-
turalladded mass), 7I = M s /(2p L 3 I Ho);t (i i) qcr is insensitive to the mass-damping
parameter 7I8, 8 being the logarithmic decrement for damping, provided 7I8 is be10w
a certain value; (iii) flutter occurs only for small enough 7I when Re is increased from
10 3 to 10 4 . Similarly, it is found that (iv) divergence occurs at small qcr if7I is small, and
(v) it occurs over a wider range (1 + a, q) as lh = Hol L is reduced.
11.4.3 The Porcher & de Langre 'pressure loss' model
This model, referred to by its authors as the 'pressure loss method' , represents an amalgam
of the Hobson and Inada & Hayama models, but it is simpler in form and has been extended
to deal with systems of many degrees of freedom. The model has been presented in various
forms by Porcher & de Langre (1997) and Porcher (1994, 2000).
The system consists of two plates of infinite width, as shown in Figure 11.26, only one
ofwhich is free to move, such that H*(x, t) = H *(x) + h*(x, t), the asterisk denoting
a dimensional quantity throughout. Starting with the 2-D Navier-Stokes and continuity
equations and integrating over the channel thickness in a similar manner as in Inada &
Hayama (1990a,b), one eventually obtains the following dimensionless equations:
, I
H+(HU) =0,
H Ů + H p' + (y - 1)H ' U 2 + H U 2 y' + (2y - 1)H U U' + 2T = O,
(11.74)
tNote that Ji here is basically the inverse of that in Chapter 10, where it is the ratio of addedlstructural mass,

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1265
-00
.
2L
Figure 11.26 The system considered by Porcher & de Langre in their 'pressure loss' model
(Porcher & de Langre 1997; Porcher 2000),
where
x*
X=-,
L*
H*
H=-,
L*
U t*
t=
L* '
U*
U= _* '
U u
P*
P = - *2; (11.75)
pU u
u: is the mean upstream flow velocity, "C = ! Cf U 2 , and y = 1 for a unifonn velocity
profile, whi1e y = 6/5 for a parabolic one; the prime and overdot denote 8/8x and 8/8t,
ln contrast with previous fonnulations, here y may be spatially variable (hence y' t= O),
taking into account the lacal ve10city profile while integrating across the channel width,
y = (U 2 ) / (U)2; this differs from the fonnulation in (11.60).
The total dimensionless pressure, n(x, t) = P(x, t) + ! U 2 (x, t), upstream and
downstream is given by
n-oo = n(-L) + !Kl U 2 (-L), nL = noo + !K2 U 2 (L),
(11.76)
with K 1 and K2 being entrance and exit pressure 10ss coefficients,
These equations are linearized, taking H(x, t) = H (x) + h(x, t), U(x, t) = U (x) +
u( x, t), P(x, t) = P (x) + p(x, t), where h « H and similarly for u and p, and Kj =
Kj + kj[h(t)], i = 1,2, Then, the perturbation components are expressed as
h(x, t) = :::>n qJn(x) e iwt ,
n
u(x, t) = :::>n un(x)e iwt ,
n
p(x, t) = :::>n Pn , (x)e iwt
n
and the unsteady friction coefficients kl and k2 are expressed by the first-order terms of
Taylor expansions,
'""" 8 Kj iwt '""" iwt
kj(h) = L.. 8h (O)a n qJn(x)e = L..kj,nan e ,
n n

1266
SLENDER STRUCTURES AND AXIAL FLOW
Hence, for each modal component n, one obtains the following equations for the fluctuating
pressure and velocity:
( H un)' + (qJn U )' +iwqJn = O,
- - , -, - -z--
iwHun+HPn+qJn P +2CfUun+y'[qJnU +2HUu n ] (11.77)
,-z -,- --, --, -
+ (y - l)[qJn U + 2H U u n ] + (2y - 1)[H U u n + H U U n + qJn U U] = O,
with
- - 1 -z
1C n ,-oo = 1C n (-L) + KlU(-L)un(-L) + zkl,n U (-L),
- - l-Z
1C n (L) = 1C n ,+oo + Kz U(L) un(L) + zkz,n U (L).
(11.78)
Integrating equations (11.77) from O to x, the flow velocity U n (x) and pressure Pn (x) com-
ponents are obtained, with the constants of integration determined from (11.78). Hence,
the moda1 fluidelastic force, Fjn = f::L PnqJj dx is obtained, leading to
,, { z jn . jn jn
Fj =  -a n (-W Ma + JW C a + Ka )},
(11.79)
n
where M1 n , c1 n and K1 n are the fluid-dynamic added mass, damping and stiffness moda1
components.
This general form of the analysis may be simplified considerably for a uniform channel
and rigid-plate two-degree-of-freedom motions, such that
h(x,t)=e+xe.
(11.80)
Considering purely translationa1 and rotationa1 motions separately, some very interesting
results are obtained, summarized in Table 11.3, It is seen that in the tota11y inviscid form
of the model (second row of the table) only divergence is possible, in the rotationa1 mode,
while if wa11 friction is taken into account (first row), flutter in the translational mode is
a1so possible. The case with an abrupt pressure loss at the outlet admits only divergence,
while an abrupt pressure loss at the inlet is the worst possible case, as found by others also
and discussed throughout this chapter.
Another interesting set of ca1culations was conducted by Porcher (2000), in an attempt
to connect the type of in stability possible to the sign of the fluctuating velocity - in these
ca1culations further simplifying the system by taking kl = kz = O. The results are given in
Table 11.4. Hence, the following is established: (i) if the real part of the fluctuating velocity
is negative (positive) for a11 values of x, a divergence instability is possible (impossible);
(ii) if the imaginary part is negative (positive) for all values of x, a flutter in stability is
possible (impossible); (iii) if the real part or imaginary part changes of sign with the value
of x, the instability depends on the manner in which the fluctuating velocity evolves with x.
For comparison with experiment, in a cylindrical geometry, see Section l1.5,2(b).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1267
Table 11.3 Possible one-degree-of-freedom instabilities for the system of Figure II. 26,
for the simplified cases shown, The tick (.j) indicates 'yes' , and the dash (-) 'no';
while the question mark (?) means 'depending on the values of K 1 and K 2 ' (Porcher &
de Langre 1997),
System Particulars Translationa1 mode Rotationa1 mode
Divergence Flutter Divergence Flutter
'«M:,jj/J Cf i= O
) ./ ./
(j'ffj':)),}'»XI K , = K2 =0
V / ///// / / / / /1 Cf =0
) ./
1/ / / / / / / / / / / /1 K l= K 2=0
(/ / / / / / / / / / /1 Cf =0 ./ ./ ./ ./
)
t/ / / / / / / / / / / /1 K I i=0, K2 =0
v///////////'l- Cf =0 ./ ./
)
Á.
1/ / / / / / / / / / / /1 K 1 = O, K 2 i= O
(///////////t Cf =0 ./ ? ./ ?
)
t/ / / ////// / /) K I i= O, K 2 i= O
Table 11.4 Signs of the real and imaginary components of the
fiuctuating velocity for all -L < x < L (Porcher 2000),
System Translationa! mode Rotational mode
Rea! Imaginary Rea! Imaginary
Cf i= O, K 1 = K 2 = O >0 > O, <O > O, <O > O, <O
Cf=O, Kl =K2=0 O > O, <O > O, <O > O, <O
Cf = O, K I i= O, K2 = O <O >0, <O >0, <O > O, <O

1268
SLENDER STRUCTURES AND AXIAL FLOW
11.4.4 Other work
Nagakura & Kaneko (1993) extended Inada & Hayama's analysis from a rigid hinged
plate to a flexible cantilevered one, in a uniform narrow flow passage. The fluid mech-
anics model is effectively the same, but the damped beam equation replaces the lumped
parameter structural model of the rigid plate. Both theory and experiments have been
done, and sample results are shown in Figure 11.27. It is seen in Figure l1.27(a) that,
as the gap is diminished, flutter occurs in progressively higher modes. The same occurs
in Figure l1.27(b) as the mas s ratio (added/structural mass) f.l, = (pL2I H )/(pphp) is
increased - much as in Figure 10.40; Qo = (EI lm L 4)1/2 - cf. Section 2.1.3. In the
experiments, a copper plate was used, L = 200 mm, width b = 100 mm, plate thickness
hp = 0.2 mm, in air-flow. Agreement is seen to be very good.
Inada & Hayama (2000) a1so consider a flexible plate in a narrow passage, but with
c1amped ends. They examine stability, obtaining flutter in the third mode for the particular
parameters they consider. They a1so study wave propagation and obtain the pertinent
dispersion relation. Then, they exarnine the stability of progressive waves in the plate,
identifying two different types of in stability.
Fujita et al. (2000), using the same basic model as Inada & Hayama (1990a,b), exarnine,
among other things, the effect of inlet and exit loss coefficients, in and ex, for a uniform
channel. Typica1 result_are shown in Figure 11.28. For the particular set of parameters in
these ca1culations and H = 3 mm, it is seen in Figure l1.28(a) that flutter always occurs
first. For H = 6 mm, however, Figure 11.28(b) shows that either divergence (for sma11 ex)
or flutter (for 1arge ex) can occur first. For long enough bodies, only divergence is possible.
20 ]00
o 80 k
]5
],10   60 
N
e;
;:;, '-, 40
5 o o o 20 in
o Q o--
o o
o 2 4 6 8 10 o 2 4 6 8 10
(a) Ho (mm) Ho (mm)
20 100
O
Unstab]e
,, 10 a 50
00 O
O 10 2 O 10 2
O 10 O 10
(b) 11 11
Figure 11.27 (a) Critical flow velocity and frequency for flutter of a cantilevered plate in a narrow
flow passage, as a function of Ho. (b) Dimensionless critical flow velocity, Uer = Uer/QoL, and
frequency, Ú)er = 27rf/ Q o, as functions ofthe mass ratio f.l = (pL 2 / Ho)/(pph p ). Curves: theory;
data points: experiment (Nagakura & Kaneko 1993).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1269
1000
3000
900
;.. 2500
I 800
ti
I:;:) 2000
700
1500
600
0.0 0.2 04 0,6 0,8 1.0 0.0 0,2 0.4 0,6 08 1.0
l;ex l;ex
Figure 11.28 The effect of ex and H on stability for a rigid plate in a narrow channel; the critical
flow velocity is expres sed as a Reynolds number Re cr = Ucr H Iv: (a) for H = 3 mm, in = 0.39;
(b) for H = 6 mm, in = 0.39 (Fujita, Shintani & Yoshino 2(00).
Fina1ly, Kaneko & Hirota (1992) study an active feedback control scheme for suppres-
sing leakage-flow induced vibrations, using a piezoelectric bimorph actuator bonded to
the upstream end of the plate, showing that it can be very effective.
11.5 CVLlNDERS IN ANNULAR FLOWS
11.5.1 Hobson et a/. studies
Hobson's (1982) sellÚna1 study of annular-flow-induced instabilities had a defining influ-
ence on all subsequent work on this subject, including that on plate-like structures in l-D
flow (Section 11.4). Hobson's work was motivated by the annular-flow-related problems
in AGRs mentioned in Section 11.1.2.
The system under consideration is shown in Figure 11.29(a). It consists of a centrebody
in a narrow annulus, both unifonn except near the ends. Various end conditions are con-
sidered, as shown in Figure 11.29(b). What follows, is an amalgam of Hobson's (1982)
and Spurr & Hobson's (1984) work. The main assumptions are that (i) the annulus is very
thin (Hla « 1) where r = a is the mean radius, so that variations of fluid variables in
the radial direction can be neglected, (ii) the perturbation velocities are small compared
to the mean ax.ial flow velocity, allowing linearization, and (iii) the dynallÚc relationships
between pressure, velocity and annular clearance may be obtained by linearization of the
steady-state relationships.
The equations of continuity and momentum may be derived by considering the mass
and momentum fluxes into and out of a small annular element, and the frictional forces
on the walls, yielding
ah a(hu) 1 a(hv)
-+-+--=0,
at ax r ae
a a 1 a ah
-(p hu) + -[h(p + p u 2 )] + --(p huv) - p- + 2ix = O, t
at ax r ae ax
a a 1 a 2 p ah
-(phv) + - (phuv) + --[h(p + P v )] - -- + 2iB = o,
at ax r ae r ae
(11.81)
(11.82)
(11.83)
tu equation ( 11.82) is expanded, and (11,81) multiplied by p u is subtracted from it, one obtains the familiar
Navier-Stokes fonn; similarly for (11.83),

1270
SLENDER STRUCTURES AND AXIAL FLOW
//////////////////////
-[L-=---
77 / / / / / / / / / / / / / / / / / / / /7
L
(a)
3- ' /////////U_ ,
////////////////
/////////////////
/////////////////
Datum configuration
(
Downstream sea!
/////////////////
'4///////' U '
////////////////
ó b (
 d
(b)
Upstream sea!
" " , , / " " " " " " / " / / / / /""7/-
Diffuser
Figure 11.29 A cylindrical centrebody in annular flow: (a) system detinition; (b) different
contigurations, 'a', 'b', 'c' and 'ď, of inlet and outlet (Hobson 1982).
where u and v are the axia1 and circumferential flow velocities, p the fluid density, p the
pressure, and Tx and Tf} the frictional forces, defined by
Tx=T(ujY), Tf}=T(VjY), T=Cf!py 2 , y2=U 2 +v 2 ;
(11.84)
an average value for Cf is obtainable from pressure-gradient measurements in steady flow,
using H (d P j dx) = - Cf P U 2 , U being the steady flow velocity.
Assuming planar centrebody motions in the e = O plane, we can write
h (x, e, t) = H + h' (x) cos e e iQt ,
(11.85)
where the prime denotes the perturbed quantity. Hence, we expres s
u(x, e, t) = U + u'(x) cose e iQt ,
p(x, e, t) = P(x) + p'(x) cose e iQt ,
v(x,e,t) = v'(x)sinee iQt ,
(11.86)
TX=t"x+T, Tf}=Tf}+T.
Furthermore, unsteady skin friction terms are postulated to be obtainable by linearization
of (11.84), for example, T = Cf [!p y 2 (uj Y) J :::::: Cf P U u'; thus,
T = Cf P U u',
, l c u ,
Tf} = 2 f P v,
(11.87)
Substitution of (11.85)-(11.87) into (11.81)-(11.83), 1inearization, and elimination of
the steady flow equations, leads to a set of linear first-order equations in u'(x), v'(x)

ANNULAR- AND LEAKAGE-FLOW-INDDCED INSTABIUTIES 1271
and p'(x):
du' +  v' = _ ( U  + iQ ) h',
dx a Hdx H
( . d U ) ' pU, dp'
p lQ + 2U - + 2C f- u + -v +-
dx H a dx
( . U pU 2 d 1 d P ) ,
= - lQ P H + H dx + H dx h,
( . d U ) ' 1 ,
p lQ + U - + Cf - v - - p = O,
dx H a
(11.88)
in which a is the mean radiu s of the annulus.
Various fluid boundary conditions are considered at the in1et and out1et, as follows.
(i) Short loss-less inlet. The flow accelerates rapid1y over a very short distance from
a reservoir, where the pressure is Po and constant, up to the annulus inlet at x = O,
so that p + ! P (u 2 + v 2 ) = Po. If the incoming flow is irrotationa1, then v' (O) = O.
Hence, at the in1et
p'(O) + p U u'(O) = O.
(11.89)
(ii) A constricted inlet or outlet. At the throat of the constriction the mean width is
H* = J),H. It is assumed that h* varies with vibration of the centrebody, h(t) =
H + h'(t), such that h*(t) = H* + h'(t). A1so, by continuity, we have u* h* = uh.
By Bemoulli's equation, we may write p* + !p(u*2 + v*2) = const. Combining
the foregoing and linearizing, we have at x = O
, P U 2 ( h' h' u' ) I
p(O)=-- ---+- ,
J), H* J),H* J),U O
In a similar way for a constricted outlet, we have
(11.90)
u'(L) - [ U* h' J), ' ]
- H*(1 + J),) + (1 - J),2)U* P L'
(11.91)
(iii) Free discharge. In this case, the perturbation at x = L is simply
p'(L) = O.
(11.92)
(iv) Outlet variable-efficiency diffuser. A1though it is recognized that the perform-
ance of the diffuser is generally dependent on e as the centrebody is displaced, a
quasi-steady, quasi-one-dimensional approach is taken, in which the pressure recov-
ery from the annulus pressure p to the constant static pressure in the downstream
reservoir Poo is expressed as
1 2
Poo - P = 2 P u YJ(h),
(11.93)
where YJ(h) is the pressure recovery factor. Further, assuming
YJ(h) = 17 + YJ',
YJ' = [dYJ(h)Jdh]h',

1272
SLENDER STRUCTURES AND AXIAL FLOW
equation (11.93) is linearized to give
p'eL) = -1 p u 2 r/ - r] P U u'(L).
(11.94)
Furthennore, a diffuser perfonnance parameter 8, of order unity, may be defined
8 = [dr](h)jdh]H.
(11.95)
For any given configuration, 1] may be detennined from steady-state experiments
with the centrebody concentric; 8, however, must be measured from several ad hoc
experiments in which the centrebody is moved, so that the flow passage becomes
progressive1y more eccentric.
The linearized equations (11.88), together with the appropriate boundary conditions
upstream and downstream are sufficient to define the flow prob1em. They may be solved
by first eliminating v/ and then solving the resulting equations in u/ and p/ by a finite
difference scheme using central differences for all derivatives (Spun' & Hobson 1984). At
the boundaries, second-order accuracy may be retained by eliminating the axial ve10city
algebraically and using instead a second-order Poisson equation for p/, This results in a
b10ck tridiagonal matrix re1ating u/ and p/ at discrete axial10cations to the va1ues of h',
which may be solved by standard means.
Hobson (1982), however, presents analytical solutions t by neg1ecting Tx and TO in
(11.82) and (11.83) and simplifying h (x) = li. The solution consists of comp1ementary
and particu1ar parts,
3
u/ex) = Lili e aiX ,
i=1
3 '1"\
/  Ia-
v (x) = L.. Vi e aix - H h ,
i=1
3 pQ 2 a 2 _
p/(x) = L Pi e aix + h.
i=1
Substituting into (11.88) one obtains a characteristic equation for ai and hence three
solutions for Ui, Vi and Pi' The boundary conditions are simi1ar1y expres sed in tenns of
Ui and Pi'
In either fonn of the solution, the problem is completed by the modal equation for the
structure (the centrebody), associated with h(x, e, t) = h'(t) 1/r(x, e):
d 2 h' dh'
_ 2 + 2s Qs - + Q; h' = F' j Ms,
dt dt
(11.96)
where subscript s stands for 'structure', and
r L r 2n
F'(t) = a 1010 p/(x, e, t)1/f(x, e)de dt,
r L r 2n
Ms = a 1010 m(x, e)1/f2(x, e) de dx;
m(x, e) being the mass of the structure per unit annular area, For h' (t) = li exp(iQt),
F'(t) = F exp(iQt), the added mas s Ma and added (fluid-dynamic) damping a are
given by
- 2-
Ma = Re(F)j(Q h),
2Q a = -Im (F )j(Q MJ,).
(11.97)
t Actually, Hobson's analysis can a!so deal with shell-type deformations of the centrebody, taking the
azimuthal variations in (11.85) and (11.86) to be cos ne and sin ne, n being the circumferentia! wavenumber.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1273
Sample results of the dimensionless fluid-dynamic damping factor, ;, as a function of
the reduced flow velocity, U , defined by
a = (pa 3 j Ms)(aj H);,
u = U j(Qa),
(11.98)
are shown in Figures 11.30 and 11.31. For all the cases to be discussed, Hobson gives
analytical expressions for ;; for example, for an upstream constriction,
* Jr { 2M - 1 + M úJ2 (1 - sech 8) }
 = -(1- sech8) ,
a 2úJ 1 + (M úJ tanh 8)2
(11.99)
where úJ = Qaj U*, U* = U j M being the 'throať velocity and M is defined in conjunction
with (11.90), and 8 = Lja; for a downstream constriction,
* Jr { MúJ2 (1 - sech 8 - M 2 (1 - M)( 1 + sech 8) + 1 }
{ = -(1- sech8) .
a 2úJ 1 + (M úJtanh 8)2
(11.100)
Figure 11.30(a) shows the effect of an upstream constriction for" a short annulus,
8 = Lja = 1; a sma1ler M indicates a larger constriction, while M = 1 corresponds to
the datum configuration 'a' of Figure 11.29(b) (no constriction). It is seen that the datum
configuration remains stable with increasing U . The same is trne for other upstream con-
strictions with M :::: 0.5. For M < 0.5, however, negative damping is generated, increasingly
more pronounced for smaller M; the critical U at which ; < O, U C f, is correspondingly
lower. For a long annulus, 8 -+ 00, the results, Figure 11.30(b), are similar. However, the
effect is not uniform; for example, for M = 0.4, for the higher 8, U Cf is higher, but ; is
more strongly negative thereafter.
For a downstream constriction, however, the system remains stable for all M, as seen in
Figure 11.30( c). Thus, Mil1er's (1970) findings - see Section 11.1.2 - are fully repro-
duced, establishing more firmly still the desirability of placing constrictions downstream,
for stability.
Annuli with a datum-configuration inlet and a diffuser outlet are considered next. If
the diffuser efficiency is assumed not to vary with h, that is, /j = O, increasing efficiency
11 is found to reduce the positive damping of the datum configuration until, at 100%
pressure recovery, that is, 11 = 1, the aerodynamic damping vanishes. If the efficiency
deteriorates with h, for example /j = -I, even with 11 = 1 the aerodynamic damping
remains positive. If, on the other hand, efficiency improves with h, for example /j = I, the
system may become unstable for a high enough pressure recovery (high 11) at sufficiently
high U , as shown in Figure 11,31. Hence, a good diffuser may by itself be responsible for
in stability.

1274
SLENDER STRUCTURES AND AXIAL FLOW
0,8
0.4
*0 O
J.J> -0.4
-0,8
O
(a)
3.0
4,0
1.0
2,0
u
*0
J.J>
0,8
0.4
1.0
2.0
(c)
5,0
4,0
5,0
(b)
u
.u = 1.0 datum
3,0
4.0
5.0
u
Figure 11.30 Tbe tluid-dynamic damping factor ; versus reduced tlow velocity U , defined in
(11.98). (a,b) The effect of an upstream constriction, characterized by fL: (a) for 8 = 1; (b) for
8  00. (c) The effect of a downstream constriction. Note: a smal1 fL means a large constriction
(Hobson 1982).
0,8
0.4
*0
J.J>
o
-0.4
-0,8
o
1.0
2,0
3,0
5,0
4,0
u
Figure 11.31 The effect on the ftuid-dynamic damping of a downstream diffuser with a gradual1y
improving diffuser efficiency, 8 = 0,1 (Hobson 1982),

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1275
Force transducer
Inlet tube
Glue
To massive support and shaker
Figure 11.32 The apparatus used by Hobson et al. for the dynamics of a cylindrical centrebody
in annular flow (Spurr & Hobson 1984),
Some of these theoretical findings were tested by experiment s in the apparatus shown
in Figure 11,32. The centrebody is made of a 25.4 mm steel tube with 0,25 mm wall
thickness, purposely made light so that inertia would not overwhelm the fluid forces in
the measurements, The diametral gap was only 1 mm. The diffuser was of 6° half-angle,
but could be replaced by a straight section to reproduce the datum configuration, or by
diffusers of different half-angle. ·
Theory and experiment are compared in Figure 11.33: (a) for the effect of an upstream
constriction (no diffuser), and (b) and (c) for the effect of a diffuser (no upstream con-
striction). It is seen in Figure 11.33( a) that theory and experiment agree very well for the
datum configuration. With the upstream constriction, however, agreement is less good:
although the instability and qualitative observed behaviour are correctly predicted, theory
overpredicts the threshold,
In the experiment s on the effect of the diffuser (Spurr & Hobson 1984), the
force/displacement transfer function was measured directly, with an impedance head on
the shaker, using random noise excitation over a frequency range of 20-120 Hz. In the
results presented in Figure 11.33(b) and (c), g(w) = XaH /(a 2 /),.p), where Xa is the aero-
dynamic transfer function, obtained by subtracting the flow-off from the flow-on value, and
a 2 /),.p is a fraction of the force on the centrebody undergoing a vibration taking up the fuH
c1earance, /),.p being the pressure drop across the annulus. A reference flow velocity is then
defined by /),.p =  p Urf' whichis used toevaluate thereducedfrequency, w = Qa/ Uref.
The real part of g(w), Re(g), represents the fluid stiffness, and the imaginary part, Im (g),
the fluid damping.
It is seen in Figure 11.33(b) and (c) that for both diffuser half-angles of 4° and 6°,
Re(g) < O. For the force and displacement sign conventions utilized by Spurr & Hobson,
this signifies a positive fluid stiffness, hence a centralizing force on the centrebody. In
Figure 11.33(b), we also have Im(g) < O, which for the sign convention used means a
positive damping; in Figure 11.33( c), however, for the 6° diffuser, Im (g) > O, signifying

1276
SLENDER STRUCTURES AND AXIAL FLOW
20
10
x
"x
\
x, /,u = 0,18
\ IC.--x
x_x'"
'"
<:>
'" O
""
+

-10
-20
O
(a)
4
8 12
U
3,0
16
20
3.0
1.0
 O
 -1.0

#_(g)
--"---..-..-
1.0
O
-1.0
 #_(g)
..-.-...-....-
.-..-.-..-----
- 3,0 --
 --_.(;)-- -3.0
 ._--
-.....-...--
-5.0
O
40
80
120
160
-5.0
O
40
80
120
160
200
(b)
w
(c)
w
Figure 11.33 (a) Comparison between theory and experiment for the case of no constriction
(f.L = 1, datum) or an upstream constriction (f.L = 0.18); (Hobson 1982). (b,c)The rea1and imagin-
ary parts of the normalized force/displacement transfer function, g(w), versus reduced frequency,
w for (b) 4° diffuser half-angle, and (c) 6° half-angle (Spurr & Hobson 1984).
a negative aerodynamic damping, and hence the possibility of self-excited vibrations.
This, indeed, was what was observed in the experiments with varying flow.
Other results, for 0° and 2° diffusers show similar agreement as in Figure 11.33(b).
It should be remarked, however, that in the theoretical ca1culations the value of '1 used was
approximately double the value obtained by steady-state measurements. This straining
of parameters (cf. Section 5.8.1) does not invalidate the comparison, but indicates that
quantitative agreement is not as good as shown,
This work has been extended by Hobson (1984) to include the effects of arbitrary
structural mode shape and arbitrarily nonuniform annular geometry. Further work on the
effect of diffusers has been done by Parkin & Watson (1984), Parkin et a/. (1987) and
Hobson & Jedwab (1990). In the latter, it is pointed out that, provided the centrebody and
the channel are coaxial, empirical measurement ofloss and recovery factors associated with
sudden expansions permit the type of theory just described to deal with stability in such
configurations. Even this, however, is not feasible for eccentrical1y located centrebodies
in diffusers. The flow then becomes very complex. Intriguingly, for a 30° diffuser, a
Strouhal periodicity may exist in the flow, even in the absence of movement; with motion,
its continued existence is dependent on vibration amplitude and frequency in a complicated
manner (Hobson & Jedwab 1990). However, this is beyond the scope of this discussion.
Hobson's and Spurr & Hobson's findings have been reconfirmed by analysis and a set
ofvery nice water-flow experiments by Fujita & Ito (1992).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1277
11.5.2 Other work on rigid centrebody dynamics
It is reeognized, of eourse, that the fluid dynamies and fluid-stmeture interaetion are
fundamenta1ly similar, whether the eentrebody is rigid with just one or two degrees of
freedom or flexible. However, this separation of the work, here for rigid eentrebodies,
in Seetion 11.5.3 for flexible ones, and Seetions 11.5.4 and 11.5.5 for predominantly
numerieal methods, serves to group together like treatments and types of results,
11.5.2(a) Mateescu & Paídoussis' inviscid analytical model
A more rigorous analytiea1 model than Hobson's but also one ofredueed seope, is that of
Mateeseu & Pa'idoussis (1985, 1987), for inviscid and viseous flow, respectively,
ln the inviscid analysis, the system eonsidered is shown in Figure 11.34: an axially
symmetrie hinged rigid eentrebody of not neeessarily unifonn eross-seetion, eoaxially
loeated in a variable-area axially symmetrie duet. The axial variations in the annular
passage are presumed to be smooth, and therein lies the main limitation of this analysis,
as well as of its viseous eounterpart: abrupt eonstrietions and area ehanges, found to be so
important for stability eonsiderations (Seetions 11.4 and 11.5.1) are not modelled, though
eonvergent and divergent annular-passage eonfigurations are.
We define the upstream annular c1earanee by Ho and the eorresponding flow velocity by
Uo, the radius of the eentrebody by Rb(X) = a - Hb(X) and that of the duet by Rw(x) =
a + Hw(x), so that H(x) = Hb(X) + Hw(x), The latera1 displaeement ofthe eentrebody
axis is eo = (x - Lo)a(t), where Lo is the length OH in Figure 11.34, and
a(t) = ao e iQt
(11.101)
is the oseillatory angular motion of the eentrebody. We are here interested in detennining
the fluid-dynamic forees on the eentrebody, assoeiated with this motion.
-U o -.-------,
- U o ---------,------ _._,-
H(x)
L
Figure 11.34 Geometry of the annulus, showing the eentrebody and the duet, both ofaxia1ly
variable eross-seetion. The eentrebody oseil1ates about the hinge H (Mateeseu & Pai'doussis 1985).

1278
SLENDER STRUCTURES AND AXIAL FLOW
Assuming the flow to be incompressib1e, the velocity potential <I>(x, r, e, t) must
satisfy the Laplace equation, V 2 <I> = O, subject to the boundary conditions
a <I> I = ae r + a <I> I ( aer _ aHb ) + (  a<l» I  ae r
ar Rb at ax Rb ax ax r ae Rb r ae '
a<l> I a<l> I aHw a<l> I
ar Rw =  Rw'  X-+-CX) = UO,
where r is the radial coordinate, Rb stands for r = Rb and similarly for Rw, and
er(x, e, t) = eo(x, t)cose = (x - Lo)a(t)cose,
with a(t) as in (11.101), Introducing the dimensionless quantities
x
 =-,
a
r Uot Qa L Lo
r = -, -r = -, úJ = -, Z = -, Zo = -,
a a Uo a a
Hb Hw
hb = -, hw = -, h() = hb() + hw(),
a a
e r
Br = -,
a
and separating the flow into steady axisymmetric and unsteady components,
<I> (x, r, e, t) = Uoa[<t>s(, r) + <t>(, r, e, -r)],
the boundary value prob1em must satisfy
a 2 <t>s +   ( r a<t>s ) = O,
a2 r ar ar
a 2 <t> 1 a ( a<t» 1 a 2 <t>
a2 +  ar ra;: + r2 ae 2 = O,
subject to the boundary conditions
a<t>s I ( a<t>s a<t» I ahb
a;: r=l- h b = - a + a r=l- h b --a['
a<t> I aBr ( a<t>s a<t» I aBr ( 1 a<t» I 1 aB r
a;: r=l- h b =  + --a[ + a r=l- h b --a[ +  ae r=l- h b  ae'
a<t> 1
- -O
ar r=l+hw - ,
(11.102)
(11.103)
(11.104)
(11.105)
(11.106)
a<t>s I ( a<t>s a<t» I ahw
a;: r=!+hw = --a[ + a r=!+hw---af
a<t>s I = 1
a -+-CX) ,
a<t> I = O
a -+-CX) .
(11.107)
Denoting the steady axisymmetric pressure perturbation by Ps(, r) and the unsteady
perturbation by p(, r, e, -r) and using the unsteady Bemoulli equation, we have
1 2 [ a<t> 2 ]
Ps(, r) + p(, r, e, -r) = --p U o 2- + (V <t>s + V<t» - 1 .
2 a-r
(11.108)
The unsteady fluid-dynamic forces on the centrebody do not depend on Ps ( , r), because it
is axially symmetric and the flow steady. Henceforth, attentionisfocusedon p(, r, e, -r).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1279
To obtain an ana1ytica1 solution, equations (11.106)-( 11.108) are linearized, uti1izing for
enhanced accuracy a 10ca1linearization procedure. Defining the mean axia1 flow ve10city
at 10cation x (in dimensiona1 form) as
- 1 l a + Hw 1 2:n: a<I>
U(x) = -rdtJdr,
TC(Hw + Hb)(2a + Hw - Hb) a-Hb o ax
the 10ca1linearization is made with respect to the dimension1ess form of U (x),
U() = U (x)jUo = hojh(O,
(11.109)
where ho == h(O) = H(O)ja. At the same time, we introduce a change ofvariable,
z = Ci' - a)ja = r - 1,
(11.110a)
and the boundary condition for the unsteady flow component on the oscillating body may
be expres sed in the linearized form
a</> I I da [ a</>s ] J
- = -( -Zo) + a(T) - cos tJ.
az z=1- h b dT a z=1- h b
(11.11 Ob)
Taking into account the form of this boundary condition, we express
[ da A A ]
</>(, z, tJ, T) = dT </>1(,z)+a</>0(x,z) cos tJ,
(11.111)
in which a(T) = ao exp(iwT). Hence, the prob1em may now be formu1ated by the
linearized equations
2 A 2A A
a </>j a </>j a</>j A
a2 + az2 + ih"' - </>j = O,
j = O, 1,
(11.112)
in which the fact that z « 1 has been used, subject to
a<Í>o I ho
az Z=-hb - h() '
a<Í>o I - O
az z=h w - ,
a<Í>1 I
- =  -Zo,
az Z=-hb
a<Í>1 I
- -O
az z=h w - .
(11.113)
The linearized form of the perturbation pressure equation is
1 2 [ A J A J
2 d a A da A a</>1 a</>o
p(, z, tJ, T) = -p U o dT 2 </>1 + dT </>0 + U(O a + a U(O a cos tJ.
(11.114)
It is assumed that rotary motions of the centrebody are resisted by a rotationa1 spring
and dashpot, such that its equation of motion is
JH ex + Ca + Ka = M,
(11.115)

1280
SLENDER STRUCTURES AND AXIAL FLOW
JH being the moment of inertia about H, and M the moment of the pressure forces about H,
M = -a31l2Jr ( -/o)p(x, -h, O, .)s(O cos O dO d,
where it is recalled that p here is dimensiona1, and
1 - hb() { 1 - hb() ' } 1 - hb()
s() = [1 +h(O]1/2 1-  -/0 hb() ::: [1 +h()]1/2 (11.117)
accounts for the loca1 slope of the body surface, and O' = dO / d, Rendering (11.115)
dimensionless, we have
(11.116)
d 2 a da
(1 + (J Q2)----;Z + (c + (J Ql)- + (k + (J qo)a = O,
d. d.
(11.118)
where
(J = Jr pa 5 / JH,
c = Ca/JH Uo,
k = Ka 2 fJH uJ;
(11.119)
(J is the dimensionless added mass parameter for unconfined tlow, and q2, ql and qo are
the coefficients of added mass, damping and stiffness, respectively:
t ao
QO = - 10 ( -/o)s() U() a (, -hb) d,
q[ = - fo' (Ii -lo)s() [.p.(, -hb) + U ( /a[ (, -hb)] d,
Q2 = -11( -/o)S()l(, -hb)d.
(11.120)
A solution of the form
j(, z) = L fJk() Fjk(Z),
(11.121)
is considered for equation (11.112), so that
fj() + c fJk() = O, Fj(z) + Fjk(Z) - (1 + C)Fjk(Z) = O,
admitting solutions
fJk(O = A jk exp(ickO + Bjk exp( -iCk), Fjk(Z) = a jk exp(f3lk Z) + b jk exp( -f32kZ),
f31k = (f3k - 1), f32k = (f3k + 1), f3k = ) 5 + 4cr (11.122)
Next, U() = ho/h(O and  -/0 in (11.113) may be expres sed in terms oftruncated
Fourier expressions
N N
U() = LDokcOSCk + LEOksinCk,
k=O k=l
N N
 -/0 = L Dik COSCk + L Elk sinCk.
k=O k=l
(11.123)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1281
Thus, for example, for a eylindrieal eentrebody in a variable eross-seetion duet, that is
for h() = hw(), introducing these into (11.113) we obtain
N N
j(, -hb) = LDjkGk(h)eosCk + LEjk Gk(h)sinCk,
k=O k= I
(11.124)
-4 /3k 1
Gk(h) = -
/31- 1 [exp(/3k h ) - 1]
2
--
/3k + 1 '
/3k = (5 +4c;)I/2,
This method of solution is extended to deal with very short eentrebodies by the use of
the Heaviside funetion (Mateeseu & Paldoussis 1985). Also, for very slender bodies with
smooth axial variations in the annular passage, the sZender-body approximation may be
uti1ized, so that (11.112) simp1ifies to [čJ 2 /čJZ 2 + a /čJz - 1]j = O, which admits solutions
of the form
j(Z;) = Aj()e,BIZ + Bj()e,B2Z; /31 = !(/3 -1), /32 = -!(/3 + 1), /3 =./5.
(11.125)
The eoefficients Aj and Bj are detennined from (11.113), For example, for a ey1indrical
eentrebody in a variable eross-seetion duet, we obtain
O(-hb;) = U()G(h), I(-hb;) = (-Zo)G(h),
./5 1
G(h) = - - -(./5 - 1).
exp(./5h) - 1 2
(11.126)
ln the ea1culations to be presented, the general fonn of this theory is simplified by
eonsidering exclusively a ey1indrieal eentrebody in a variable eross-seetion annulus, sueh
that the annular clearanee varies either 1inearly,
h() == hw() = ho + hp,
hp = (hl - ho)/ Z,
(11.127)
where hl = h(I), or parabolical1y,
h() = ho + hp + hc2.
(11.128)
Typieal results showing the effeet of hp and hinge loeation Zo are shown in
Figure 11.35(a). It is seen that, for a unifonn annulus (h p = O), if the hinge is upstream
of the ey1inder mid-point (Zo < !), the fluid-dynamie damping is positive and the system
is stable; if, on the other hand, the hinge is downstream of the mid-point (Zo > !), the
damping is negative, indicating the possibi1ity of flutter - when c + a ql < O. The reason
for this very neat result is diseussed in subseetion (b).
It is also seen in Figure 11.35(a) that, for a eonvergent annulus (eurve 1, hp < O), the
system is stabi1ized, which is manifested both by larger ql for any Zo and the faet that
Zo > 0,5 is neeessary for a negative ql. A divergent annulus (eurves 3 and 4, hp > O),
however, has a destabilizing effeet. These effeets are strong, even for Ihp I = 0.05.

1282
SLENDER STRUCTURES AND AXIAL FLOW
10
10
5
ql °
-5
-10
0,00
(a)
0,25
0,50
'o
0.75
5
qo
°
-5
-0,05
(b)
° 0,05
hp
0.10
1.00 ql
-10
0,00
(c)
0,25
0,50
'o
0,75
1.00
Figure 11.35 Typieal results for a uniform ey1inder in a variable-seetion duet, for Z = 1 and
ho = 0,1. (a) Variation of the eoeffieient of fluid-dynamic damping, q), with hinge loeation
Zo : 1. hp = - 0.05 (eonvergent duet); 2, hp = ° (uniform duet); 3. hp = 0.05 (divergent duet);
4. hp = 0,10. (b) Variation of fluid-dynamie stiffness eoefficient, qO, with hp: 1.10 = 0.0; 2.10 = 0,5;
3, Zo = ] .0, (e) Effeet of wan eurvature on fluid-dynamie damping for hp = 0,05: 1. hc = 0.05;
2. hc = O; 3, hc = - 0.05 (Mateeseu & Pa'idoussis 1985),
Figure 11.35(b) shows the effect of hp and lo on the stiffness coefficient qO, lt is seen
that qO = O for a uniform passage (h p = O). However, for hp i= O, there generally is a
stiffness effect. In the figure, negative stiffness occurs (i) for lo = O (curve 1) and hp > O
(divergent annulus), and (ii) for lo = 1.0 (curve 3) and hp < O (convergent annulus), For
lo = O, the system is dynamically stable (q! > O), while for lo = 1, it is unstable. In either
case, the system may lose stability by divergence, depending on the values of (1, c and k,
Figure 11.35( c) shows the effect of curvature in a divergent annular passage on ql. lt is
seen that the effect is not very strong. A concave form, as viewed from the annulus side,
has a slight destabilizing effect.
Consider a numerical example, to get a physical feel of these results: a cylindrical
centrebody in water fl.ow in an annular passage with radiu s a = 50 mm and Ho = 5 mm,

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1283
so that ho = 0.10, smoothly divergent (h p = 0.5; half-angle = 2.86°). The centrebody is
supported at its mid-point (lo = 0.5); it is assumed that its natural frequency Qn = 60 rad/s
é::::: 9.55 Hz, and its structural damping factor is s = C/(2QnJH) = 0.005, or 0.5% of
the critical. Considering the fluid to be water, let (J = 0.2, a reasonable value. From
Figure 11.35(a), we find ql = -0.79. Flutter is therefore possible for C + (Jql S O or,
expressing this in dimensional tenns using (11.119) and noting that C / JH = 2s Qn,
for 2s Qna/UO + (Jql S O. This gives UO cr = 0.19 mis, which is quite small. For a
convergent annulus, however, the system would remain dynamically stable for all Uo and
s, since ql > O.
Next, consider the possibility of divergence. For lo = 0.5, this is impossible, for all hp,
since qo > O in Figure 11.35(b). Let us consider lo = 1.0 instead. For a divergent annulus
(h p > O), qo > O and hence again divergence is impossible. For the convergent case, with
hp = -0.05, one obtains qO é::::: -8. Hence, using (11.118) and (11.119), it is found that
Qa2/UJ+(JqO < Ofordivergence,orUOcr = Qna/«(Jlqo\)1/2 é::::: 2.4 mis. Foranarrow
annulus this is a rather high flow velocity for water flow, because of the associated large
pressure drop, although in this analysis viscous effects are ignored; besides, this system
is also dynamically unstable, with a much lower UO cr .
Finally, it is important for the reader to read subsection (f), on the effect of the precise
nature of the upstream and downstream end conditions.
11,5,2(b) The mechanism of instabi/ity
The mechanism of dynamic instability (flutter) has been elucidated by Pai'doussis (1987).
Considering the system to be slender, so that solution (11.126) holds trne, and further-
more the annular passage to be unifonn, so that h = ho, U (O = 1 and aJ>o/a = O, the
perturbation pressure (11.114) in dimensional fonn may be written as
[ d2a da ]
plx=-Hb = -pa (x - Lo) + 2U - G(H) cose.
dt dt
(11.129)
The two tenns in the square bracket are identified as an inertial (virtual mass) tenn and a
Coriolis tenn. The instability is controlled by this second tenn. By similarity to internal
flow in a pipe, as in Section 3.2.2, it is clear that, as the body rotates about the hinge,
the Coriolis-related forces will be in the same direction all along the body, (i) opposing
the motion on the part of the body downstream of the hinge, but (ii) aiding it upstream
of the hinge. Clearly, therefore, the criticallocation of the hinge is at lo = 1/2, so that if
lo > 1/2, the system is subject to negative fluid-dynamic damping. This can also be seen
in the expression for ql in (11.120); since G(H) = const < O, this arises simply because
16 ( -lo)d < O for lo > 1/2, and so qO < o.
ln the foregoing discussion, end-forces ofthe type discussed, for example, in conjunc-
tion with Figure 8.89 are ignored - because of the shape of the ends and the matching
rigid extensions up- and downstream of the centrebody. Here again, the reader is referred
to subsection (f).
It is remarked that in (11,129) there are no centrifugal-type tenns, as for example in the
pipe problem, since the body is rigid and the equivalent of flexure does not exist. In fact,
there are no stiffness tenns whatsoever in (11,129). This is discussed further in subsection
(f) and Section 11.5.4. If the annulus is nonuniform, however, stiffness tenns do arise.

1284
SLENDER STRUCTURES AND AXIAL FLOW
Considering (11.114) and (11.126) in the ease of a nonuniform duet, an additional tenn
appears, equal to [U()(acPo/a)]a, which gives a eontribution of U(x) U'(x) G(H) -
U 2 (x) G'(H) in the square braeket of (11.129), where ()' = d()/dx. In this ease, it is
more diffieult to see the effect on stability sinee in the integra1 for qo the integrand is
a funetion of. However, qO < O is dearly possible, sinee G(H) < O, G'(H) > O, and
henee the outeome depends on relative magnitudes of U(x) and G(H), their derivatives
and the location of the hinge point.
11.5.2(cJ An approxímate víscous theory for lamínar flow
The analytical model in subseetion (a) has been extended by Mateeseu & Pa'ídoussis
(1987) to deal, approximately, with laminar viseous flows. The system is simplified by
eonsidering a unifonn eylindriea1 eentrebody in a unifonn annulus, as in Figure 11.36,
where the different definition of a is noted vis-a-vis Figure 11.34,
For the flow in the annulus, we start with the Navier-Stokes and eontinuity equations.
For example, the first of the Navier -Stokes equations, assoeiated with the axial eomponent
of the flow, expres sed in dimensionless tenns by means of (11.104), is
av x v av x v av x v  av x _ _ ap 2h { a2vx a 2 v x  av x  a 2 v x }
a-c + x a + r ar + e r ae - a + Re a2 + ar 2 + r ar + r 2 ae 2 .
(11.130)
Eaeh of the three flow ve10city eomponents and the pressure are then split into potential
and viseous parts; for example,
vx(, r, e, -c) = vf(, r, e, -c) + v(, r; e, -c).
(11.131)
The superseript p stands for potentia1, and v for viseous, To obtain the potential solution,
the Navier-Stokes equations are redueed to Euler's equations (for Re --+ (0), sueh that
//
-u
az
H
L ----""j i' a
 )aw.-:- =-t -
-u
/ // / / / / / / // / / / / / / / / / / / / / // / / / / / / / / / / / / // / / / / / / / / // / / / // / / / /// / / // / /// // // .
L
Figure 11.36 A unifonn eylindríeal eentrebody oseil1ating about the hinge A, in a uniform
eylindríeal duet. Note that az = r - a here, or z = r - 1 (Mateeseu & PaYdoussis 1987).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1285
V x = 1 + vf, V r = vf, Ve = v:. Proceeding then as in subsection (a), one finds
A g(h - z)
<po(, z) = - g'(h) ,
A __g(h-z)_Z
<Pl(, z) - g'(h) ( o),
(11.132)
where z = r - 1, and
g(h - z) =  exp [ (q + l)(h - Z) ] +  exp [ -(q - l)(h - Z) ] ,
q+l 2 q-l 2
g'(h) = exp [(q + l)h] - exp [-(q - l)h] , q = .J5;
(11.133)
from these expressions, one can find vf, vf and v:; also, the perturbation pressure on the
centrebody (z = O) is
{ d2a da }
pP(, O, e, -r) = - (-Zo) d-r 2 + 2 d-r G(h) cos e,
g(h) q 1
G(h) = -- = - - -(q - 1).
g'(h) exp(qh) - 1 2
(11.134)
The potential components are then subtracted from the Navier-Stokes and continuity
equations, yie1ding the goveming equations for the viscous component of the flow. Thus,
in the case of equation (11.130), we have
ov ov ov 1 ov ( V ) ovf v ovf vi ovf
-+vx-+vr-+ve--+ v -1 -+v -+ve--
o-r o or r oe x o r or r oe
= _ opv + 2h { o2vx + o2v x +  ov x +  o2v x } . (11.135)
o Re o2 or 2 r or r 2 oe 2
Equation (11.135) is then linearized. It is further simp1ified for not very high Reynolds
numbers, whereby ov/o-r may be neglected compared with (2h/Re)(o2 vx / or 2). Also,
because of the narrowness of the annular passage, similar assumptions to those normally
made in boundary-layer theory are introduced, name1y that (i) the radial component of
viscous motion, v, is negligible, and (ii) axial and circumferential variations in V x and Ve
are small in comparison with radial variations. Accordingly, equation (11.135) and those
for the other components, are drastically simplified into the following set:
02v x
OZ2
Re opv
--
2h o'
02ve
OZ2
Re 1 opv
---
2h r oe'
opv
0,::::::-,
oz
(11.136)
where z = r - 1 « 1.
To proceed, let the average value of the dimensionless circumferential flow ve10city
across the annular gap be defined by
1 f 1+h
ve(, e, -r) = - ve(, r, e, -r) dr,
h 1
(11.137)

1286
SLENDER STRUCTURES AND AXIAL FLOW
z
x
x
Figure 11.37 Diagram showing transformation of coordinates and de6nition of the angle fJ
(Mateescu & Pai'doussis 1987).
and similarly for v x , Then, with the aid of Figure 11.37, the total mean velocity may be
expressed as
u V(, e, T) = U(v x cosf3 + ve sin(3),
(11.138)
where the angle f3 (, e, T) is given by
' f3 U ve - ( I:
SIn = -----= :::::: Ve 5, e, T),
UV
(11.139)
since U V :::::: U, Evaluating ve from the potential flow solution, we find
sinf3 = B(, T) sine,
1 gl(h) [ da ]
B(,T)::::::h g'(h) dT (-lo)+a ,
(11.140)
in which g' (h) is as in (11.133), and
( 2 ) 2 [ 1 ] ( 2 ) 2 [ 1 ] 16q
gl(h) = - exp -(q + 1)h - - exp --(q -1)h + 2 2 '
q + 1 2 q - 1 2 (q - 1)
Hence, in terms of the new set of coordinates defined in Figure 11.37, multiplying the first
of (11.11 Ob) by cos f3 and the second by sin f3 and combining, we obtain
a 2 v Re apv
az 2 = 2h ax '
(11.141)
where here V = V (z). Equations (11.139) and (11.141) may be interpreted as follows:
the mean velocity in the annulus, V, remains practically constant in magnitude, bm its
direction osci11ates in time through an angle f3, related to motions of the cylinder and
hence to the mainly circumferential fluid motions imparted to the fluid in the annulus, t
ln this simplified treatment, ve(, e, T) is determined approximately from potential
flow. Using equations (11.111) and (11.132), and proceeding as in Mateescu & Patdoussis
tThe similarity of this approximate treatment to that in Section 7.5.3 should be noted.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1287
(1987), one obtains
VeZ) = _ Redpv { !Z(h-Z) }
2h dX 2 '
dpv 24 1 12 2
- --- and i=-PU
dX h Re Re
(11.142)
Here i is the tangential shear stress on the cy1inder surface; its axial component makes
no contribution to the 1atera1 forces on the centrebody, but its tangential component
iO = i sin f3 does. Eventual1y, the moment on the centrebody is determined, via
M = -2a311 1 1f (-Io){p U2[pp(, O, O, i) + pV(, O, O, i)] cos O + iO sin O} dO d.
(11.143)
The potential and viscous components, MP and MV, respective1y, are found to be
1 I 2 [ A J ]
A d a A a<pl da
MP=pU 2 a 3 n 1 (-Io) <Pl(,0) di2 + <po(,O)+---af di d,
12 1 + h 1 1 [ da ]
MV = -p u 2 a 3 n- - - d Wl() + a wo() d,
Re h O i
_! gl(h) W _! gl(h) -I
wo() - h g'(h)' 1() - h g'(h) ( o).
(11.144)
Eventually, an equation of the form of (11.118) is obtained, where now q2, ql and qo, the
added mass, damping and stiffness coefficients, is each made up of an inviscid component
determined via (11.121 )-( 11.124) t and a viscous component determined as above.
Typical resu1ts for ql and qo are shown in Figure 11.38. It is seen that viscous effects
tend to stabi1ize the system in terms of flutter, by raising the values of ql; the curves no
10nger pass through 10 = 0.5, As expected, the effect becomes stronger as h becomes
smaller. Thus, for h = 0.05, we see that Zo > 0.5 by a considerab1e amount for
flutter to become possib1e. It is also seen that qo =1= O for the uniform annulus under
consideration if viscous effects are taken into account. For Zo:::: 0.5, negative stiffness is
obtained, showing that divergence becomes possib1e; the effect is stronger for the narrower
annu1i.
It is noticed that the values of ql in Figure 11.38(a) for Z = 5.0 are much 1arger than for
Z = 1.0 in Figure 11.38(b) - the 1atterbeing the same as in Figure 11.35. The same is trne
for qo. C1ear1y, as the length of the annulus increases, the fluid-dynamic forces increase
drastical1y, which is reasonab1e on physical grounds,
An examp1e of the evolution of the comp1ex eigenfrequency of a system with increasing
U is shown in Figure 11.39. It is seen that the system loses stability by flutter at a very low
flow velocity: U = 0.063 mis. As the flow is increased, the frequency of oscillation Re(Q)
diminishes as a resultofthe negativeqo [see Figure 11.38(a)]. Eventually, Re(w) = O while
tu the slender-body approximation is used for the potential flow, that is, equations (11,125) and (11.126),
the quantity in curly brackets in MP may be written as (hwl ()(d2a/d-r2) + 2hwo(n(da/d-r)}.

1288
750
SLENDER STRUCTURES AND AXIAL FLOW
15
ql
500
250
°
-250
-500
°
0,25
0,50
Zo
0,75
80
ql
5
°
-5
1.00
-10
°
0,75
0,25
0,50
Zo
6
4
40 2
qo °
-40
-80
°
0.25
0,50
Zo
0,75
(a) Z= 5,0
qo °
-2
-4
1.00
-6
°
1.00
0,25
0,50
Zo
0,75
(b) Z= 1.0
Figure 11.38 (a) Variation of the dimension1ess tluid-dynamic damping coefficient q I and stiffness
coefficient qO with hinge location Zo, for Z = 5.0. (b) The same for Z = 1.0. Curves (a): h = 0,05;
curves (b): h = 0.10; (c) curves (c): h = 0.15; - , including unsteady viscous effects; - - -,
potentia1 tlow theory (Mateescu & Pa'idoussis 1987).
Im(Q) < O, indicating a state of divergence, but its existence would have to be checked
by nonlinear theory or experiment.
Final1y, Table 11.5 shows the effect of the annular passage width, h, on stability. It is
c1ear that for relatively large h, viscous effects, though not negligible, are not very strong;

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILmES 1289
0.1 0,01
2 0,02
5 10 15 0,04
° 12 0.05 13
(Q) 0.063
0,08 .(Q)
0.10
 -4 &'
 

-0.2
-8
4.0
5,0  Values oť U (mis) 0,20
-12
5,77 -0.4
(a) (b)
Figure 11.39 (a) Argand diagram of the complex eigenfrequency Q (rad/s) as a function of the
flow velocity V (m/s) for a system in annular water-flow with a = 0.1, h = 0.05, I = 5,0, ID = 0.6,
and in vacuo Qn = 60 rad/s and t;s = 0,005. (b) Part of the Argand diagram blown up (Mateescu
& Pai'doussis 1987).
Table 11.5 The effect of h on the critical flow velocity, V er , for the
systemofFigure 11.39 (I = 5.0, ID = 0.6, a = 0.1; Qn = 6Orad/s
in vacuo, and t;s = 0.005); (Mateescu & Pa'idoussis 1987).
h Values oť V er (m/s)
Viscous theory Potentia! flow theory
0,05
0,10
0,15
0,0629
0,0271
0,0255
0.0060
0.0122
0.0187
for example, V er is raised by 36% for h = 0.15. For a very narrow annulus, however, for
example h = 0,05, V er is raised lO-fold by viscous effects,
11.5.2(d) An approximate viscous theory for turbulent flow
This analysis is extended to deal, approximately, with turbuZent annular flows by Mateescu
et oZ. (1989). The Navier-Stokes equations, in this case, again after subtracting the inviscid
(potentiaI flow) component, are
av t
- + (V. V)V t + [(vt - lx)' V] Vrp
at
",I t tp ",It p t t
+le-(Vo V r + ve V r ) -lr- Ve (Ve + ve) = -Vp +F,
r r
(11.145)

1290
SLENDER STRUCTURES AND AXIAL FLOW
where the superscript t stands for 'turbulenť and p for 'potential'. Making the same type
of simplifications as for laminar flow, the left -hand side of (11.145) is neglected, and hence
Ft ::::: V pt ::::: lx o( - v v; )/oz + le o( - v; v )/oz = -(-II P U 2 )OT t loz, (11.146)
where the primes denote the turbulent fluctuations, z = r - I, and T t is the turbulence
stress vector (tensor),
t 8 o(UV)
T=--,
a oz
where 8 is the eddy viscosity, and U V = lx U V x + le U Ve. Defining again the average
ve10cities across the annular gap, V , v x and Ve, introducing the angle f3, defined by
sin f3 = Ve I V , and determining Ve by potentia1 flow, we find
(11.147)
opt I OTt opt
-=---cosf3:::::-,
01; pU2 oz oX
opt I oTt opt
ae ::::: - pU2  sinf3 ::::: OX B(I;, T) sine,
opt
-=0
oz '
(11.148)
with B(I;, T) as in (11.140).
To proceed, a simplified model of turbulence is assumed, name1y the (I ln )-power law
for the wa11turbulent flow profile [see Section 2.2.1(f)]. Thus, UV lu, = C[(u, Iv)az]l/n,
where u, = (Tw Ip) 1/2 is the wall friction velocity and v the kinematic viscosity. Hence
U V lu, = [nl(n + 1)]C[(u,alv)(!h)]I/n, and applying the momentum equation to an
elemental volume de dX dh, we obtain p U 2 (opt loX) = - (21 h)Tw = - (21 h)p u;,
leading to
opt
oX
K I
=
-h Re 2 /(n+l) ,
[ ( n + I ) I ] 2n/(n+l)
K = 2 4 1 / n - -
n C
(11.149)
with n = 7 and C = 8.74, which give good results for Re :::: 10 5 (Schlichting 1960), and
K = 0.081. The pressure perturbation is then obtained by integration of the second of
equations (11.148),
t t K I
p = Po + h Re 2 /(n+l) B(I;, T) cose,
with dpb/dl; = -(K I h) Re- 2 /(n+1).
Typical results, for both the laminar and the turbulent versions of the theory, compared
with experiment are given in subsection (e).
(11.150)
11,5.2(e) Experiments and comparison with theory: uniform annuli
The experiments in question (Mateescu et al. 1988, 1989) were designed to test the theory
indirectly, but more fundamentally: rather than testing stability, measurements were made
of the unsteady pressure on the centrebody as it was forced to oscil1ate at a prescribed

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1291
Honeycomb screen
------------ M esh screen
: :
Fixed ogive
Possible hinge locations
Osci1lating centre-body
Fixed ogive
----- Mesh screen
Orifice Pla:?
I I
I 
Air from
blower
Figure 11,40 Sehematie diagram of the apparatus for a eylindrieal eentrebody in a eoaxial eyl-
indrieal duet, subjeeted to rotational oseillations about a hinge in an annular air-flow (Mateeseu,
Pai'doussis & Bélanger 1988),
variable frequeney and amp1itude, and the measured pressure was then eompared with the
theoretieally predieted one.
The apparatus is shown in Figure 11.40. The eentrebody was a hollow aluminum eyl-
inder fitted with eonvex spherieal end-eaps. Matehing eoneave ends were fitted on the
fixed upstream and downstream ogives, so that as the ey1inder oscillated about a hinge,
minimal disturbanee to the flow would result. The eentrebody was foreed to osei11ate via
an eleetromagnetie shaker (B&K 4801; exeiter head 4812). The unsteady pressure was
measured both on the eentrebody and the outer duet wall via semieonduetor differential
pressure transdueers (Ku1ite PTQH-360-1D) at several axia1 and eireumferentia110eations.
The eentrebody and annulus geometry were: L = 421 mm, 2a = 88.9 mm, H = 4.85 mm
(h = 0.109); Lo was varied: 115,210 and 306 mm.
The results are presented in terms of differential pressure amplitude,  ft, and the
phase angle, 1/1, re1ative to the osei11ation displaeement of the eentrebody. This latter was
measured via an aeeelerometer, and 1/1 was determined with the aid of an FFT signal
ana1yzer.
Typiea1 measured data are eompared with the potential and laminar viseous versions of
the theory in Figure 11.41 for U = 52.4 and 73.0 mis. It is seen that both versions of the

1292
120
80
'(;'
e::,
<Q"
'<:1
40
O
5
SLENDER STRUCTURES AND AXIAL FLOW
40
.
';;' -40
<>

<>

 -120
(a)
__ _ __ ._ _ '" -"''''i?'= ...-= _(b)
(a  ----ť>__''$_ U
25
45
j(Hz)
85
-200
5
65
25
45
j(Hz)
65
85
Figure 11.41 Unsteady pressure amplitude, I1P, and phase, 1/1, measured at X / L = 0.658, () =
0°, versus oscillation frequency, f, for Lo/ L = 0.499; (a) U = 52.4 mis and (b) U = 73.0 mis:
0, <>, experimental data; - - -, potential flow theory; -, laminar viscous theory (Mateescu
et a/. 1988),
theory are in reasonably good agreement with experiment for I:1p. However, the phase,
which is important in determining the fluid-dynamic damping, is much better predicted
by the viscous theory. Here it should be remarked that the two flow velocities in the tests
correspond to Re = 3.3 x 10 4 and 4.8 x 10 4 , which are c1early not in the laminar range.
Hence, it is perhaps surprising that laminar theory does so well!
The measurements in Figure 11.41 were taken not far from the centre of the centrebody
(x / L = 0.658). Agreement between theory and experiment deteriorated for measurements
c10ser to one of the ends, for example, at x / L = 0,824. Various attempts were made 'to
improve' the flow conditions at the end-caps, for example by fi1ling the 2.5 mm gap with
soft felt, but were not successful. More wil1 be said about this in the next subsection.
On the other hand, the measurements on the oscil1ating centrebody were found to be
sensibly identical to those made across the annular gap on the outer duet. This shows that
the theoretical assumption of negligible radial flow in the annulus is sound, and offers a
location for easier installation of the pressure sensors in later experiments.
The measurements were then compared with the turbulent flow version of the theory
(Mateescu et aj. 1989). Typical results are shown in Figure 11.42. The simplified viscous
theory used, although capable of predicting the pressure at different circumferentialloca-
tions around the circumference, can only give a single, average value for the phase angle.
Despite this, agreement between theory and experiment is excellent.
The laminar and turbulent versions of viscous theory are cross-compared in
Figure 11.43; it is seen that the difference between the two is smal1, with the amplitude
being a little larger according to the turbulent theory and the phase essentially identical.
Data from two sets of experiments straddle the two theories, though those labeled 'more
recenť are considered to be more reliable. af course, the difference between the two
theories may be larger for different geometries and is expected to grow with Re.
11.5.2(f) Sensitivity to boundary conditions
It is recalled that in the inviscid analysis of the uniform annular system of Figure 11.36,
the centrebody is predicted to lose stability by flutter [Section l1.5.2(a)]; moreover, the

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1293
40
40
';;;' -40
,(1)
(1)

(1)

 -120
-40
-120
-200
5
80
25
45
65
-200
85 5
120
25
45
65
85

'"
e::.
<Q.,
"<::]
40
80
40
o
5
25
45
j(Hz)
65
o
85 5
(b)
25
45
j(Hz)
65
85
(a)
Figure 11.42 Unsteady pressure amplitude, /).P, and phase, 1/1, measured at xl L = 0.591 and
various values of 0, versus the oscillation frequency, f, for LolL = 0.274(/0 = 2.59); (a) U =
52.4 mis and (b) U =73.0 mis. The symbols represent experimenta1 data and the lines are the
theoretical resu1ts obtained for turbulent viscous theory (Mateescu et al. 1989).
80 -40
 'bij 
p.. & -120
<;( 40 
"<::] 
O -200
5 25 54 65 85 5 25 54 65 85
j(Hz) j(Hz)
Figure 11.43 Laminar and turbulent viscous theory compared (Lo = 210 mm, x = 227 mm,
U = 52,4 mis, Re = 3,3 x 10 4 ): -, turbulent theory; - - -, laminar theory; 0, first set of
experimenta1 data; ., more recent measurements (Mateescu, Pai'doussis & Bélanger 1988, 1989).
absence of fluid-dynamic stiffness tenns [Section 11.5.2(b)] prec1udes 10ss of stability by
divergence. Neverthe1ess, when viscous effects are tak:en into account [Section 11.5.2(c)],
stiffness tenns do arise and they can become negative, rendering divergence possible,
although, in this case also, the dominant mechanism for 10ss of stability is flutter. This
was considered to app1y general1y, to al1 such systems, unti1 the early 1990s.

1294
SLENDER STRUCTURES AND AXIAL FLOW
Two developments have occurred subsequently, however, forcing a c10ser 100k at
the generality of these results, notably thanks to de Langre et al, (1992), as briefly
reviewed by Pa'idoussis (1997); refer also to de Langre & Granger (1995). The first
was the development of CFD codes (see Section 11.5.4), in which it is required that
the computational domain be 10nger than the oscil1ating body, and hence it is necessary
to specify the perturbation pressure beyond the oscil1ating body, across the end-gaps,
and on the immobi1e extensions at either end. The second development (described in
subsection (g) and in Section 11.5.4) was two new sets of experiments in which the
centrebody was fixed, while a portion of the outer containing pipe was forced to oscil-
late (see Figure 11.63), in transverse or rock:ing motion. Clearly, to ensure containment
of the flow, the oscillating and immobile segments of the outer pipe in this case have
to be physically connected, if only by a membranous tube Or bellows, Similarly, in the
CFD codes, continuity between the last mesh point on the oscil1ating cylinder (or tube)
to the first mesh point on the immobi1e extension thereof would result in a simi1ar type of
connection, even if it were physically absent - unless, of course, the model is made to
negate it.
Now, in the analyses of Section 11.5.2(a,b), the perturbation pressure is specified only
overthe osci11ating portion ; E {O, l}, while it remains unspecified beyond. t If, however,
perturbations vanish for the inviscid system of Figure 11.36,* if not at; = O and I, then at
-1* and Z + 1*, where Z* is an appropriate length, this amounts to saying that the system
is effectiveZy supported at these extremities - in accordance with the above discussion.
By analogy then to the pipe conveying fluid, as discussed in Section 3.4.1, this invis-
cid system is inherently conservative and should lose stability by divergence rather than
flutter!
The problem is reexamined by de Langre et al. (1992) by means of linearized potential
flow theory applied to a generally shaped centrebody, analytical1y and by means of a finite
element code, in particular for the system of Figure 11.36 and hence for rock:ing motions
oť a rigid centrebody. This was later extended by Bé1anger et al. (1993) to deal with
combined translational (lateral) and rocking motions. Two modifications to the foregoing
theory are introduced. In the first, the displacement of the cylinder for a system supported
at ; = O and  = 1 and hinged about  = Zo is specified to be
Xw(' e, T) = a(T)XR(;, e) + E(T)XT(' e),
(11.151)
where a(T) is the rotational and E(T) the translational amplitude, subscripts R and T
denote rotation and translation, and
XR(' e) = [(; -loH - cos e e r + sin e elJ) + a cos e e ][H(;) - H(; -I)],
(11.152)
XT(' e) = (- cose e r + sine elJ)[H() - H( -I)],
tHere  = x/a, and 'l' later, are as in Section 11.5,2(a).
* Aforteriori if felt is stuffed in the end-gaps [Section 11.5,2(d)].
 In a sense, the problem lies with Figures 11.34 and 11.36, both of which suggest that there is minimal free
space across the end gaps. Moreover, the analyses of Section 11.5,2(a) and (b) would remain unaltered if a
cylindrical membrane were added to connect the oscillating centrebody to the immobi1e extensions thereof, .

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1295
in which e r , eYJ and e are unit vectors in the outward radial, circumferential and axial direc-
tions, respectively,t The Heaviside functions, HO, ensure that nonzero displacement is
confined to the length of the oscil1ating cy linder. If the Heaviside functions are suppressed,
the Mateescu & Pai'doussis (1985) formulation is retrieved,
The second modification is to apply the nonpenetration boundary condition, not at the
mean position of the oscillating body, but at the moving boundary. Defining coordinates
 and 1] to be in the axial and circumferential principal directions on the surface of the
cylinder, n in the direction ofthe inward normal to the surface, and the corresponding unit
vectors as e, eYJ as before and n = e x eYJ = -e r , we can expres s
Xw = X e + XYJ eYJ + X n D,
v = V e + VYJ eYJ + Vn D,
(11.153)
where V is the mean velocity vector (in this case VYJ = Vn = O). Defining the perturbation
velocity by v, then V'D = ocpjon, cp being the velocity potential, and it is found that
ocp . V .
- = x n + I;' VI; X n + Xn(VI;,VI;) = X n + VI;'(X n VI;),
on
(11.154)
where
OX n OX n
VI; X n = o e + 01] eYJ'
o V o VI)
VI;,VI; = o + 01] ,
(11.155)
the latter of which is zero for a uniform cylindrical annulus. Here V:E is the surface nabla
operator, and V I; = V e + VYJ eYJ is the surface velocity - in the uniform annulus simply
V e, The important outcome of this improvement to the analysis is that stiffness terms
now arise in the fluid-dynamic force on the oscil1ating cylinder. Thus, if cp is defined by
2
'" [ , (1) (2) ]
cp = w aiCr:)CPi + aiCr:) CPi '
i=1
(11.156)
. (1) (1 ) (2) (2) .
where for I = 1, we have al Cr:) = a(r), CPI = CPR and CPI = CPR ' and for I = 2, we
have a2(r) = E(r), cpi I) = cpi l ) and cpi 2 ) = cp?), the perturbation pressure is found to be
I ocp (1) I ocp (2)
P = -cpl)a(r) - cp) + o: a(r) - o: a(r)
I 0-1,(1) I 0-1,(2)
- -I,(1)E(r) - -1,(2) +..:!.L E(r) - ..:!.L E ( r)
o/T o/T o o'
(I 1.157)
tNondimensionalization here is carried out in terms of an arbitrary characteristic length. 1*. Thus,
a = a*/ ľ, a* being the dimensional radius,  = x /1*, T = tU/ ľ, etc" where x, t and U are dimensiona1.

1296
SLENDER STRUCTURES AND AXIAL FLOW
X 10 3
2,5
2,0 - "
1.5
1.0
0,5
qc O
-0,5
-1.0
-1.5
-2.0
-2.5
0,0
"
"
(a)
0,1 0,2 0.3 0.4 0.5 0,6 0.7 0.8 0.9 1.0
X 10 3
0,2
-0,2
0.0 --------------------.....-----------------------
-0,4
qK
-0,6
-0.8
-1.0
-1.2
O O 0.1 0.2 0.3 0.4 0,5 0.6 0.7 0.8 0.9 1.0
(b) IOIl
Figure 11.44 Variation with Zo/ Z of (a) the fluid added damping and (b) the fluid added stiffness,
according to potential flow theory: - - -, ignoring end effects and applying nonpenetration condition
at the mean boundary position: -, taking end effects into account and applying nonpenetration
condition on the moving boundary (de Langre et aZ, 1992).
The flow potentials are solved subjeet to the following boundary eonditions:
04,0)
 = -( -lo)[H() -H( -l)]eose,
on
oljJ (2)
 = -[H() - H( -I) + (-10){8() - 8( -l)}] eose,
on
oljJ (1)
 = -[H() -H( -l)]eose,
on
(11.158)
oljJ(2)
 = -[8() -8( -l)]eose,
on
where 8 () is the Dirae delta.
Caleulations are then eondueted, (i) with these two modifieations taken into aceount, that
is, speeifieation of end effeets and boundary eonditions applied on the moving boundary, or
(ii) not [as in Mateeseu & PaIdoussis (1985)]; results are shown in Figure 11.44. It is seen

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1297
° °
0- -
-45 O
biJ - ,/ O O
"
::=.. -90 -45
"
CIO
'"
..c::
p., -135
----
(a) (c) ---
-18O -90
0,020 0,015
0.015 _O
" 0,010 O
."
2 O
ť. 0,010 O
 O D
---- "- 0,005
0,005 ,/ "-
(b) (d)
0,000 0,000
° 2 4 6 8 10 0,05 0,10 0,15 0,20 0.25
 w
Figure 11.45 Axial variation of (a) the phase and (b) the amplitude of the dimensionless pres-
sure according to potential flow theory for Zo/ Z = 1/2, e = 0°, O!o = 0.002, ú) = 0.19. (c,d). The
dimensionless pressure versus dimensionless frequency (for  = 7.8, Zo/ Z = 2.60, e = 0°; and
o!o = 0.0023). In both cases: - - -, theory ignoring end effects and applying boundary conditions at
mean position of the boundary; -, theory taking end effects into account and applying boundary
conditions on moving boundary; o, experiments (de Langre et aZ. 1992, Bélanger et aZ. 1993).
that the fluid damping term, qc [equivalent to ql in (11.118)], becomes zero when these
effects are taken into account, whereas ifthey are not (as for ql in Figures 11.35 and 11.38)
a positive or negative flow-induced damping is predicted. The reverse happens to the fluid
stiffness term, q K: when the aforementioned effects are accounted for, one obtains a
negative fluid-dynamic stiffness. Hence, the inviscid system is subject to divergence and
not to flutter, as expected by the foregoing discussion.
The new results of course apply without question to cases where the cylinder ends are
supported (e.g. clamped to immobile extensions.) They also apply to situations where
the ends are effectively connected to immobile extensions. In this regard, it should
be said that Mateescu & Pa'idoussis' (1985) analysis is not wrong. However, by not
specifying boundary conditions beyond the moving body, the problem solved does not
quite correspond to that in the experiments and that shown in Figure 11.36. This is rein-
forced by the results of Figure 11.45, for points not in the central portion of the oscillating
cylinder; here a('r) = ao exp(iú>T). It is seen that, when the aforementioned improve-
ments to the theory are introduced, agreement with experiment s improves, With regard to
Figure 11.45(b), it should be said that, after reaching maxima clo se to x = O and x = 9.5,
the pressure diminishes precipitously thereafter,
The variation with frequency is also better predicted by the modified theory, for points
near the ends ofthe oscillating cylinder, as seen in Figure 11.45(c) and (d), for instance-
even though viscous effects are ignored in this theory. It is of interest that it is the large
alterations in the predicted phase that are responsible for suppressing flutter,
It is recalled that all of the foregoing, illuminating and fundamentally important though
it may be, has been obtained with linearized inviscid theory, ff viscous theory is used, the

1298
SLENDER STRUCTURES AND AXIAL FLOW
Table 11.6 The effects of (i) specification of end-conditions and, for potential
flow, (ii) whetber tbe nonpenetration condition is specified on tbe moving boundary
(nonzero stiffness) or not (zero stiffness), on tbe stability of a uniform rigid centre-
body in uniform annular flow, executing rocking or lateral translational motions
(Pai:doussis 1997),
Motion Flow Effects (i) and Added Added Possible
(ii) specified? stiffness damping instability
Rocking Potential No Zero Nonzero Flutter
Potential Yes Nonzero Zero Divergence
Viscous No Nonzero Nonzero Divergence!flutter
Viscous Yes Nonzero Nonzero Divergence/flutter
Lateral Potential No Zero Zem
Potential Yes Nonzero Zero Divergence
Viscous Yes Nonzero Nonzero Divergence!flutter
dynamical behaviour is a1tered, yet again. As seen in Section 11.5,2(c), and as verified
by Hobson's experiments (Section 11.5.1), depending on tbe distribution of pressure loss
along tbe annulus, tbe system may lose stability by flutter.
Collecting all this information togetber, Table 11.6 may be constructed. The nota ti on
'divergence/flutter' simply means that botb are possible, but not necessarily in tbat order.
It should a1so be noted tbat, for viscous flow and rockiog motion, a1tbough tbe added
damping is nonzero in botb entries, it is quite different if tbe eods are presumed to be fixed
or otherwise, for in the former case the effect of tbe Coriolis-related terms is DulI.
Before closing, it ought to be mentioned that, in viscous tbeory, in addition to the
nonpenetration condition, ideally tbe no-s1ip condition on the surface of tbe cylinder
should also be applied on tbe moving boundary. Therefore, tbe no-s1ip condition is not
U w = xw, where U w = V w + V w is the total fluid velocity at tbe wal1 and Xw tbe wall
velocity. By a Taylor series expaosion, the no-s1ip condition on the moving bouodary is
Uo = Xw - xw,(V U)o,
(11.159)
where subscript O denotes quantities associated with tbe meao position of the cy1inder.
Depending on the type of analysis used, if one applies tbe simpler condition rather than
(11.159), some of the added damping terms may be disregarded (cf, Section 11.7).
11.5.2(g) TAXI experiments: nonuniform annuli
A major experimental programme was undertaken at CEN -Sac1ay in France in tbe 1990s,
involving tbe so-called TAXI apparatus (de Langre et al. 1994), shown schematica11y in
Figure 11.46(a). Tbe centrebody is 468 mm 10ng, 66 mm in diameter upstream and 40 mm
downstream, witb a 20° diffuser segment connecting the two cylindrical parts. Tbe inner
diameter of the outer duct is 70 mm, containiog water flow.
The central portion of tbe outer duct is free to osci1late, and is connected to immobile
extensions on either side via rubber bellows-like joints. t A simp1ified structura1 model of
tThe stiffness of these joints is affected by the internal pressure, and hence it varies with flow velocity;
special measurements were necessary to determine this functional relationship,

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1299
Moving part oftbe stmcture
n / 
i c ::::::' -- co' 
1 :::  uo*
Exl 1 1 U\ ..
(a)
I I I
-L 2 -x, -LI O
I
LI X,
I
L 2
(b)
25
 20
 /_Mode 1
,!3 15 /.
 .
 10 0::.-.............
h- o
5  '\0_0.. Flutter
Mode 2 a... o.... !
o
5 10
Flow rate (1/s)
O
O
(c)
3.0
2,0
 1.0
i!P 0,0
}-1.0
.S:! -2,0
gj -3,0
:g -4 O
'3 '
 -5,0
-6.0
z -7,0
0,0
(d)
---
,
_._._..-.._----
5,0 10.0 15,0 20,0 25.0 30.0
Flow velocity, ut (mis)
Figure 11.46 (a) Schematic of the TAXI apparatus; (b) simplified structural model; (c) typical
evolution of damping with flow, (d) Evolution of flow-induced damping with flow velocity for the
rotational mode: -.-, experiment; - - -, theory ofPorcher& de Langre (1997); -'-, level where
negative fluid-dynamic and positive structural damping cancel each other (de Langre, Porcher &
Axisa 1994; Porcher & de Langre 1997).
the system is shown in Figure 11.46(b). Two displacement transducers, located upstream
and downstream on the oscil1ating outer duct, al10w the separation oť any ftow-induced
motion into its two moda1 components, associated with latera1 translation and rotation
(rocking),
With increasing ftow, the system becomes unstable by ftutter. As seen in Figure 11.46( c),
the moda1 damping oť the translational mode (Mode 1) is a1ways positive. That oť the rota-
tional mode (Mode 2), however, eventual1y vanishes at a ftow rate oť about 8.51/s (litres/s),
indicating loss oť stability by ftutter at that point. The same is shown in Figure 11.46( d), t
where it is seen that when the ftuid-dynamic ('ftuid-elastic') damping reaches -4.5%,
cancelling the positive structural damping oť 4.5%, then ftutter ensues.
AIso shown in Figure 11.46(d) is the theoretical curve obtained by the theory oť
Section 11.4,3. It is recal1ed that, although developed on a paral1el-plate model, the final
tParts (c) and (d) of tbe figure are not wholly consistent with each otber, because or small differences in tbe
manner or processing and presenting tbe experimental results in different published versions oť this work.

1300
SLENDER STRUCTURES AND AXIAL FLOW
fonn is applieable at a modallevel to the system at hand, and henee for eaeh of the two
modes eoneemed. In the figure, it is seen that the evolution with flow of the fluidelastie
damping of the rotational mode is qualitatively similar to the experimental one, but the
eritieal flow veloeity is overpredicted eonsiderably. Other experiment s with nonunifonn
annuli are diseussed in subseetion (h) and in Seetions 11.5.4 and 11.5.5. In partieular, the
reader is referred to Seetion 11.5.5(e).
11.5.2(hJ Recent work
A theoretieal model, similar in spirit to Inada & Hayama's (1990a,b) but for an axisymmet-
rie system, is presented by Arai & Tajima (1998). The system eonsidered is a truneated
eonieal eentrebody in a eonieal passage, free to peďonn rotational oseillations about a
pivot loeated up- or downstream of the eentrebody. The angles of eentrebody and duet are
both variable, a1lowing the eonsideration ofunifonn and various eonvergent and divergent
annular geometries. This model, whieh takes viseous forees into aeeount, has been used
by Arai et al. (1998) to study the stability of the system, and the results are eompared with
experiment. A view of the apparatus used is shown in Figure 11.47(a).
(a)
0.08
0,04
A t::.C\,,,..,,
tr>c......" ,
..rf:rOr.  ','}, -;. 1::.t:A'
0-0-. -af.',a.....
A"i:/:A.t::....- Ob
'" Lo =-18 mm
O
0,0
"J>
(b)
.sr -0,04

gp 0.12

'"
c
0.08
4':1"'.'
,.t>'
1it.''A
.t>t>:
'h.'
-0.-0. ."
,.i:< L o =-18 mm
A
"r--o
" 00
,.' ba
0.04
0.0
(c)
-0,02
0,0
0.8 1,6
Q (x 10- 3 m 3 /s)
2.4
Figure 11.47 (a) View of the experimental apparatus used by Arai et al., showing in this
ease a truneated eonieal eentrebody in a eylindrieal duet; note the two possible loeations of the
pivot. (b) The damping ratio  versus the volumetric flow rate for a divergent eentrebody shape
(half-angle = 5°) in a eylindrieal duet, with exit-to-inlet area ratio Ar = 4,67 and in = 60 Hz: _,
, , . ., theory; 1::::., experiment with upstream pivot (Lo = -18 mm); o, experiment with downstream
pivot (Lo = 78 mm). (e) The same for a eonvergent eentrebody (half-angle = 12°) in a eonvergent
duet (half-angle = 15°), Ar = 5.26 and in = 38 Hz: symbol s the same as in (b); (Arai, Shiomi &
Tajima 1998).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1301
Typica1 measurements of the overa11 (structura1 plus fluid-dynamic) damping in
Figure 11.47(b) (c) are in phenomena11y good agreement with theoretica1 prediction. It
is seen that, when the pivot is upstream of the centrebody (Lo = -18 mm), the system
remains stab1e with increasing flow. lf the pivot is downstream (Lo = 78 mm), however,
the system becomes unstab1e by flutter at sufficient1y high flow rate.
Interesting1y, for a uniform cylindrica1 centrebody and duct (not shown), the system
loses stability by flutter, but at about 20% higher flow it becomes statica11y unstab1e,
touching the duct, and the frequency of oscillation vanishes. Thus, in this case, flutter is
succeeded by divergence.
This work has been taken further by Arai & Tajima (1999) and Li et al. (2002), see
Section 11.5.5(b), for combined transverse trans1ationa1 and rotationa1 oscillations.
An interesting re1ated system is studied by Tanaka et al. (2001): a train of interconnected
cy1inders (cf. Section 8.9.9), modelling a fast-moving train in a tunnel. The experimenta1
model is shown in Figure 11.48. The outer cross-section of each car is a truncated circ1e; the
cars are hollow, and interior-mounted springs constrain 1atera1 motion [Figure 11.48(b)].
The cars are interconnected by pin joints. Different shapes of nose and tai1 are used
interchangeab1y, ranging from a bullet-shaped one, to spheroida1 and square-cut b1unt
(cf. Figure 8,23 and 8.83). The experiment was re1ative1y sma11-sca1e; the car radiu s was
25 mm, with the rest to-sca1e as in Figure 11.48. Experiments were conducted in two
tunne1s, such that, if coaxial1y located, the annu1ar gap was either H = 4.5 or 14 mm
Flow
I st car
I
Nth car
Tank
Laser displacement I = 0 I
sensor ES
(a)
Digital recorder
Flow'meter
Floor
Bearing Pillar
pjn Floor Spring Bearing
Carm Carm+ I
Figure 11.48 (a) Schematic ofthe model train in a tunnel. (b) Cross-sectional views ofthe system,
showing a section through one of the cars, and the manner in which two cars are interconnected
(Tanaka et a/. 2001),

1302
SLENDER STRUCTURES AND AXIAL FLOW
(h = 0.18 or 0,56). However, eccentric arrangements were also tested, characterized by a
train-to-tunnel centre1ine separation of  H or  H.
In the first set of tests the system was coaxial and h = 0.56, with five cars, the most
streamlined nose (A) and varying tails (A, B, C or D) - see Figure 11.49(a). In this
case, the system remained stable in all cases except that with tail A, and the train vibrated
with small amplitude, as shown for instance in Figure 11.49(b). In the case of tail A,
an instabi1ity developed, as shown in Figure 11.49(c). Here it should be noted that the
amplitude can never become very large because of the interior restraining springs, unlike
the systems discussed in Sections 8,3.2 and 8.9.4.
In the second set of tests, h = 0.18. In this case, instabi1ity occurred for all tail-shapes,
With decreasing flow, a considerable hysteresis was displayed in all cases. A typica1 result
is shown in Figure 11.49( d), The corresponding mode shape of the osci11ation is shown in
Figure 11.49(e).
(a)
Type A
Type B
8
7
6 ,.....,....,..........
5 ......
4 .................
Frequency
'N 3
e;
Amplitude
....:.::.:::..::..::.::..:::::.::::'::::
C 2 ...
:::
<I)
6-
<I)
....

1 ..........
(b)
O
O
I 8
20
40
60
80
7' ....  .............
i 6 ............ ..1 ..... (f;....... ..
-< 5 ......... ., .., ,......". ., .....
4.. ....... . ... ..., .. Frequenéy.... ..........
3 .,....,.,........,.....".,....................,.....................
AmPli...l
..:...:.......:.....:..'..:::...:::::::.:::::...
(c) J)l ...JI
O
O
20
40 60
Flow velocity, U (mis)
80
Type C
Type D
10
100
:....
..../'- .-@
................................/....................................
Frequency
: ( )P]
..=-'
O
O 20 40 60 80
F10w velocity, U (mis)
100
I 
<I)
] O
'E, -1
e
-< -2
TypoB",U64mh  
(e)
O
2
Joint
3
4
5
100
Figure 11.49 (a) Different types of end shape used on the leading and trailing cars, (b) Amplitude
and frequency of oscillation for a five-car system and h = 0.56, with nose A and tail B, as a function
of U. (c) The same for a system with nose A and tail A. (d) The amplitude and frequency versus U
for a five-car system and h = 0.18, for nose A and tail B. (e) The mode shape for the system in (d)
at U = 64 mis (Tanaka et a/. 2001).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILmES 1303
Tests with eccentric arrangements in the wider tunnel were also conducted, again with
nose A and varying tails. The following interesting results were obtained: (i) for tails A, B
and C, there was a tendency for the trailing car, for which the amplitude of movement was
largest, to move towards the centre (towards the wider part of the annulus); (ii) for tail D
and eccentricity  H, the opposite occurred, with the trailing car moving towards the near-
wall, reaching a maximum static displacement of 2.5 mm; (iii) for tail D and eccentricity
 H, more complex behaviour was observed. An attempt was made to explain these obser-
vations in a similar manner as in Section 8.3,3(b), but here bringing to bear annular flow
considerations, for example, that the flow over a well-streamlined end corresponds to that
in an annular diffuser.
11.5.3 Flexible cylinders in annular flow
Hobson (1982) in his analysis deals in principle with flexible cylinders also, simply by
considering modal quantities which would arise in discretizing a distributed parameter
system; but he does not actually do any detailed work on such systems. Here we consider
some work specifically on flexible cylinders, either experimental or analytical, while
similar numerically based work is presented in Sections 11.5.4 and 11.5.5.
11,5.3(a) Uniform annuli
Experiments with flexible cylinders in annular flow have been done by Pai"doussis &
Pettigrew (1979) and the experimental results have been compared with Pai"doussis'
(1973a) theory - basically using equations (8.21) and (8.27a). The experimental system
is shown in Figure 11.50. The test cylinders were made of either silicone or polyurethane
rubber, specially cast as in Appendix D; their diameter was 15.2, 20.3 or 22.9 mm. The
channel had an internal diameter of 25.4 mm, so that three radial gaps H were possible:
5.10,2.55 and 1.25 mm (Hla = 0.67, 0.25 and 0.11, a being the cylinderradius).
ln early e;1(:periments, a fine steel cable was embedded along the cylinder central axis
to prevent excessive flow-induced axial extension. Later, this was replaced by a thin steel
ribbon, which also defined a preferential direction of motion. In some experiments, a
miniature accelerometer was embedded in the cylinder [Figure 11.50(a)], to allow other
than visual observation of the dynamics. Experiments were conducted either with water
flow or with air-water mixtures simulating two-phase flows, flowing upwards.
Excepting cantilevered cylinders with the bullet-shaped downstream end replaced by
a blunt one, all cylinders experienced fluidelastic instabilities in liquid flow. Generally,
the cylinders either fluttered in a well-defined mode, or buckled close to or against the
channel wall, touching it at a number of points depending on the mode number. A buckled
cantilevered cylinder is shown in Figure 11.50(c). Cylinders with a blunt end remained
stable over the whole range of available flow velocities.
ln some cases, divergence (buckling) and oscillations of similar mode shape alternated,
at the same flow velocity. In other cases, oscillations were superimposed on cylinders
wbich were buckled. Also, for cantilevered cylinders, the following phenomenon occurred
fairly often, particularly at high flows: the cylinderwas buckled in a specific mode, and only
the portion between the free end and the last point where it touched the wall would vibrate,
sometimes violently, rattling against the channel wall. The phenomena just described were
mostly steady. However, in some cases a buckled cylinder would sbift erratically from

1304
SLENDER STRUCTURES AND AXIAL FLOW
Top
Test
section
I wall>!
Bullet shaped 25.4 mm
downstream end r-;l
(lucite)
Spider
support
Ze = l.25D
Clamped
top end
Free end
Accelerometer
I
I
I
Steel ribbon
9,5 X 0,2 mm
Brass ribbon
13 x 0.125 mm
Pinned end I
(b)
(c)
Figure 11.50 Schematic of test cylinder, showing typical end conditions and methods of support:
(a) cantilevered cylinder; (b) other end conditions; bottom: pinned end; top: clamped, axia11y
sliding end; (c) photograph of a cantilevered cylinder buckled in the fourth mode (PaYdoussis &
Pettigrew 1979).
one side of the channel to the other; also, oscillating cy1inders would change mode, back
and forth. This unsteadiness, coupled to the small amp1itude of motion imposed by the
narrow flow channel, made it difficult at times to identify the goveming phenomenon and
the mode shape.
The behaviour of the system with increasing flow was quite complex, As the flow
velocity increased, there ensued a succession of instabilities of increasing mode number
and, in the case of flutter, of generally increasing frequency. In the case of silicone rubber
cy1inders, at the lowest flow at which the experiment could be run, the system was already
subject to some form of in stability,
The behaviour of the system in simulated two-phase flow was quite different. The cy1in-
der would vibrate, sometimes severely, in a broad-band random manner. This persisted at
all flow velocities, and no fluidelastic instabilities could be detected over the whole range
of simulated quality covered t - see Figure 8.28(b). Hence, these experiments will not be
discussed further.
Before presenting quantitative experimental results in water flow, the theoretical dynam-
ics is discussed briefly. Figure 11.51 shows the dimensionless complex eigenfrequencies
t Of course, instabilities may have occurred, but remained buried in the high background of turbulent
two-phase vibration; see Section 8,7,3.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILmES 1305
14 2
10 2
,-, 6 2

 4 2
---- -----------
...--
-. ...-...-..----.
-----------
"-
-.----..----
---------.......--
--.-.-'
2 2
.....
'"
".
,.'
I
I
I
.I
'./,--
,.... ...
I .
I
I
I
I
---....
,.-,
" .-
'-_/
o
4 2
2 2
,-,

 O
-2 2
_4 2
O 2 4
"-"
" ).
, .' \
,-"" .,
" ,
" I
I
6
8
10
12
14
u
Figure 11.51 Rea1 and imaginary components of the eigenfrequencies of the lowest six modes
of a canti1evered system in annu1ar flow for Bq = 1.0, f = 0.8, tJ = 0.469, X = 4.55, h = 4.0,
Y = 3,37, Xe = 0.02, Cb = 0.05, J-l = C = O (Pa:idoussis & Pettigrew 1979).
w as functions of the dimensionless flow velocity u - see equations (8,28a,b) - for a
cantilevered system with Hla = 0.25 (h == DI Dh = 4), Here, in contrast to the results
presented in Section 8.3, a great number of instabilities occur for u :::: 14, each at much
lower u than in unconfined flow. In this range of u, according to line ar theory, the system
becomes unstable six times in the first mode alone. Often, instabi1ities in several mode s
overlap; thus at u = 9.2, the system is unstable concurrently in its first, fifth and sixth
modes by flutter, and in its second mode by divergence.
Similar results are shown for a pinned-pinned cylinder in Figure 11.52 - showing
both the mode s originating fmm the right- and left-hand part s of the complex frequency
plane - cf, Figure 3.34 and associated discussion. lt is seen that the system is unstable
in at least one mode for u> 2.4, in at least two modes for u> 5,9, and so on (though only
modes I and 2 are shown).

1306
 4 2
 .......
O
 O

\S5 -22
SLENDER STRUCTURES AND AXIAL FLOW
12 2
8 2
-4 2
" ,
, '
:.....;! '\
, .
_------'" '0_'''''., _./._._._.
4 2
\ /
,,,--,,-'
2 2
.- ,/', ./
/ " / '
. / 1 -' ,/ \ , . \.1
/ j' . '""
i I 1st mode (+)
,....' I
-4 2
;-.....
.....
-6 2
O
2
4
8
u
10
12
14
16
6
Figure 11.52 Real and imaginary components of the eigenfrequencies of the fust and
second modes of a pinned-pinned system in annular flow for ECJ = 1.0, .B = 0.461, X = 4.56,
h = 4,0, Y = 6,33, Cb = 0.05, C = O, and hysteretic damping coefficient J-L = 0.066. Loci originat-
ing on negative Re(w) side of the complex frequency plane are marked (-); otherwise, (+), Also
shown is the fust-mode locus when J-L = ° (Pa'idoussis & Pettigrew 1979).
Now, in the case of unconfined tlow the instabilities occur sequential1y with little or no
overlapping, so that there was no difficulty in deciding the sequence of instabilities one
should expect to ob serve with increasing tlow. In the present problem, however, where two
ar more instabilities are predicted to occur concurrently, the question arises as to which
one, or which ones, will actua11y materialize; here, it is presumed that the instability with
the simplest modal fonn will be dominant, and this supposition will be put to the test when
comparing with observation.
It is recal1ed that in the foregoing discussion, as elsewhere in this book, the nomenclature
of 'first mode', 'second mode', etc. is purely nominal- indicating the locus of the mode
evolving continuously from the corresponding eigenfrequency at u = O. Far high enough
u, however, the modal form of the nth mode does not correspond to that of the nth
beam mode; the system being non-self-adjoint, normal modes do not exist. Thus, as
shown for example in Figure 11.53, motions associated with the 'first-mode' at u = 0.75
have a normal first beam-mode shape, but at u = 1.5 a significant second beam-mode
component is apparent, and at u = 5.75 the motion is more like that of a fourth beam-
mode. For cantilevers, transition from one modal shape to another a1ways occurs within

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1307
u = 0,75
(J) = 0-0.075 i
(a)
u = 2,75
(J) = 6,98-0.38 i
(b)
u = 1.50
(J) = 0-1.47 i
u=4.19
(J) = 7,93-0,06 i
u = 4,56
(J) = 0-0.42 i
u = 8.06
(J) = 0-0,04 i
u = 5,75
(J) = 0-0,63 i
u=I1.12
(J) = 17.3-0,28 i
Figure 11.53 Modal shapes with increasing flow velocity for the system of Figure 11.51
(a) nominal first-mode modal shapes; (b) nominal second-mode shapes (Pa'idoussis &
Pettigrew 1979).
eaeh of the instability zones, c10se to the point where Im (w) is minimum; thus, referring
to Figure 11.53(a), the third modal shape shown has a third-mode shape and the fourth a
fourth- mode shape, transition oeeurring at u :::::: 5.2, whieh eorresponds to a loeal minimum
ofIm(w), as may be verified from Figure 11.51; similarly with the first and seeond modal
shapes of Figure 11.53(b).
ln eomparing with experiment, however, in what follows, the mode number is taken to
eorrespond to the actual beam-mode shape; that is, the mode number is here equa1 to the
number of antinodes in the moda1 shape,
A eompendium of experimental results for the three annular gaps is shown in
Figure 11.54, eompared with theory. t There are various reasons why the results for eaeh
annular gap are not identieal for all ey 1inders. First, there is the effeet of the instrumentation
in some of the ey1inders; the aeeelerometers, although miniature, c1early a1ter the mass
and stiffness distribution of the ey1inders vis-a-vis the uninstrumented ones, Then, there
is the effeet of eylinder material and length, which are partly but not fully aeeounted for
by nondimensionalization of the ftow velocity, For example, apart from other parameters
that may be different, there is the following distinetion between the longer and the more
ftexible ey1inders, on the one hand, and the shorter and stiffer ones, on the other: a ftutter-
ing eylinder ofthe former group is more 1ike1y to develop a larger limit eyc1e and, henee,
toueh the ehanne1 wa11 in the eourse of its motions; onee this happens, a part of the eylinder
tFor cylinder I, for example, the dimensionless u is related to U (mis) by u = 3.16U.

1308
u
10
SLENDER STRUCTURES AND AXIAL FLOW
2,5 mm annulus
@-@
(í3) '\, @
@ I I T 1. )
19 I @ I
11 @, .   :@\}
') @:) \
@\l.    G)CD
""( 21 G)
G) 19 
@i
8
6
@
.
T'T.
,I..
i1
...
@
\I
..
T =
r?\ I .
\.0. : I
'\ I .\..
. 
\ I.V
@ 0
G) @ 1
0 
@ .
@\; I
i
r
16
14
12
10
u
8
6
4
2
o
Figure 11.54 Compendium of stability behaviour of cantilevered cylinders with increasing flow
velocity and comparison with theory. Cylinders I, 3, 6, 7, 13, 18, 19: silicone rubber; cylinders
10, II, 17, 20, 21, 22: polyurethane rubber; all cylinders 508 mm long, except 3, 13 and 17
which are 300 mm long approximately. B: buckling; O: osciIlation (flutter); B + O: buckling
with superimposed oscillation; B + TV: buckling with top (free-end) vibration, Theoretical para-
meters: (a) for 15.2 mm (diameter) cylinders, sCf = 1.0, .B = 0.472, X = 2,13, h = 1.5, Y = 6,54,
(b) for 20.3 mm cylinders, sCf = 1.0, .B = 0.469, X = 4,55, h = 4,0, Y = 3.37, (c) for
22.9 mm cylinders, sCf = 0.25, .B = 0.471, X = 9,53, h = 9,0, Y = 2,73; in all cases Cb = 0,05,
f = 0.8, Xe = 0.02, C = O (Pai"doussis & Pettigrew 1979).
1.3 mm annulus
@
@:;".
1': .
I.
.
4
@-
I
1
@
@", CD I;
CCD
/
Cylinder no,
I
.
.
.
.
.
I
t----< B or B + TV
....---10
-B+O
- B (theory)
. .. O (theory)
6
7
momentarily becomes supported in the clamped-pinned fashion and may tend to 'stick' to
the wall if, for the length involved, the clamped-pinned cylinder is unstable by divergence.
Finally, there is the difficulty of interpreting the mode number. Tbe modal shapes were not
always classical; while some of the antinodes were symmetrically located on either side
2
o
20
18
16
14
12
CD
5.1 mm MM,"" CD CD @J !
----cD  1 :r :: :  I
I
@ : i
@ I 
 . -
10
8
6
4
2
o
2
3 4 5
Mode number

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1309
of the channe1 centrelines, there were others that did not quite cross the centreline, If al1
this is considered, it is remarkable that al1 the experimental results collapse into re1atively
compact groups to the extent that they do in Figure 11.54, by mere nondimensionalization
of the flow velocities. It is equal1y gratifying that the theoretical predictions are fairly close
to the experimental ones. In some aspects, this agreement is clearly less than satisfactory,
for exarnple theoretical divergence not always corresponding to experimental buckling.
On reflection, however, considering all the foregoing and the total neglect of mechanical
interaction with the channel, agreement could hardly be much better.
One very clear, albeit obvious, conclusion is that, the narrower the annulus, the 'less
stable' is the system. For exarnple, for a cantilevered cylinder, at a flow velocity of 1 mis
the largest cylinder (narrowest annulus) is subject to fourth-mode instability, while for the
smallest cylinder (widest annulus) only to second-more instability.
Experimental and theoretical results for pinned-pinned and clarnped-clamped cylinders
are compared in Figure 11.55. Agreement is seen to be reasonable in this case also.
26
24
22
u 20
18
16
14
12
10
8
6
4
2
O
T
,
,
,
,
,
,
Clamped-clamped cylin ,
,
,
Ii ,
L
T
,
T .q
'I
, .
. ....
.
. .
.
. .
.
.
.
. .
.
! .
.
I .
II I
Pinned-pinned cylinder : 
1---1 B
I H ....-- o
. I ........ B + O
I . - B (theory)
... o (theory)
28
26
24
22
20
18
16
14
12
10
8 u
6
4
2
O
2
3 4 5
Mode number
6
7
Figure 11.55 The experimental regions of instability compared with theory for clamped-
clamped and pinned-pinned cylinders. Theoretica1 parameters (a) for pinned-pinned cylinder:
Bq = 1.0, f3 =0.461, X =4.56, IL =0.066, h =4.0, y =6.33, Cb =0.05, c=O; (b) for clamped-
clamped cylinder: the same except f3 = 0.467, Y = 6.24, IL = 0.062 (Pai'doussis & Pettigrew ] 979).

1310
SLENDER STRUCTURES AND AXIAL FLOW
u-
"nn
--- --
r H
I
--.I.. _--
"Hn.
-. 7ď---==--c--=- ---==-,:. e F-...-..- .-.
///.//h _------ Ir ;,........../...'.//.
u-
eo
x
L
Figure 11.56 A ftexible cylinder in annular ftow (Pa'idoussis et al. 1990).
ln practica1 terms, it is of considerable interest that an effective way of complete1y
eliminating instabilities of a cantilevered system is by making the downstream (free) end
blunt, rather than stream1ined. The system with both ends supported, however, remains
prone to in stability, and at flow ve10cities fully in the range of engineering applications
for relatively narrow annuli,
The bifurcational behaviour of the system is much richer than in unconfined flow, even
though it has neither been fully explored, nor has the ana1ysis been pursued by nonlinear
theory (in the manner of Section 8,12 for articulated systems). The dynamics promises to
be even more interesting for eccentric systems.
An important conc1usion that may be reached is that the theory for flexible cylinders in
confined flow of Section 8.2 may also be used for uniform annular configurations, provided
that the correct parameters for the viscous coefficients are used (see, e,g, Sections 2.2,3
and 11.3,3), and provided that the annulus is not too narrow; in the latter eventuality the
theory discussed next may be preferable.
An analytical model for very narrow annular configurations has been developed by
Pai'doussis et a/. (1990), adapting for a flexible cylinder Mateescu & Pai'doussis' (1985,
1987) work on rigid centrebodies (Section 11.5,2), The system under consideration, with
both ends supported, is shown in Figure 11,56, It consists of a cylinder of radiu s a, cross-
sectional area A, mass per unit length m, and flexural rigidity EI. Rigid semi-infinite
extensions at either end of, and of the same diameter as, the flexible cylinder are assumed
for analytical convenience. The uniform annular gap is H; the undisturbed flow velocity
is U, and the fluid density p,
The radial displacement in the azimuthal plane defined by () is
er(x, (), t) = eo(x, t) cos() = E(x) cos() e iQt ;
(11.160)
E (x) may be expressed in terms of the eigenfunctions Vrk (x) of the beam with the same
boundary conditions (c1amped-clamped) as the cylinder,
E(x) = LakVrk(X) = Lak[Vrlk(X) + Vr2k(X)),
k k
(11.161)
where Vrlk (X) = - cos fJkX + CYk sin fJkx, Vr2k (x) = cosh fJkx - CYk sinh fJkx and CYk =
(cosh fJkL - cos fJkL)/(sinh fJkL - sin fJkL), the fJkL being the corresponding beam
eigenvalues - cf, Section 2.1.3.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1311
Proceeding as in Section 8.2, the equation of motion may be written as
a4eo [ aPm l L ] a2eo
El-- --A(L-x)+ hdx-
ax 4 ax x ax 2
ap m aeo a2eo
- A +m at 2 = Fp + Fv, (11.162)
where it has been assumed that the downstream end can free1y slide axially; Fp (x, t)
is the unsteady inviscid (potential) fluid force on the cylinder per unit length, and Fv (x, t)
the corresponding lateral viscous force; Fdx) is the longitudinal steady viscous force
on the cylinder due to skin friction.
Proceeding as in Mateescu & PaYdoussis (1985), the perturbation pressure may be
derived from a velocity potential, expressed as
" A "Q
cp(x,r,(),t) = akCPk(x,r) cos()e 1 t.
k
After considerable manipulation, one finds (PaYdoussis et aZ. 1990)
Fp(x, t) = -pna 2 e iQt L ak( _Q2 Pk2 + iQ Pkl + PkO),
k
where each of the Pks is associated with the sth time derivative; for example, Pk2 is
associated with the inertial effects. The hs are found to be
(11.163)
(11.164)
2
h2 = - L Gjk o/jk,
j=l
2
Pk1 = -2U L Gjk o/jk'
j=l
2
2 2" '
PkO = -U f3 k (-I)JGjk o/jk,
j=l
(11.165)
with
G1k = [ -qk + ! tanh(qk h )] / [( qf- i) tanh(qk h )] ,
G { [cZ - ! tan(cZh)] / [(cZ 2 + i) tan(cZh)]
2k -
- [-q + i tanh(qh)] / [(ci - i) tanh(qh)]
for f3 2 a 2 > l
k 4'
for f3 2 a 2 < l.
k 4'
(11.166)
[ 5 2 2 ] 1/2 * [ 2 2 5 ] 1/2 [ 5 2 2 ] 1/2 H
qk = 4 + f3 k a ,c k = f3 k a - 4 ' q = 4 - f3k a , h - -
a
Similarly, following Mateescu & PaYdoussis (1987), one finds (PaYdoussis et aZ. 1990)
2 'Qt" - -
Fv = -pna e 1 ak(iQ Pk1 + Pko),
k
(11.167)
where
_ 12 2 + h 2
Pk1 = U -- "G'k Wk o/'k,
Re ah 2  J J J
j=l
2
- 2 12 2 + h " ,
PkO = U Re ah 2  Gjk "'Jk o/jk'
j=l
(11.168)

1312
SLENDER STRUCTURES AND AXIAL FLOW
with
l h -1
Wlk = -[COSh(qkn + Rl sinh(qkn]e{/2 d,
o 1 + 
{ l h -1
-[cos(c;n + Rz sin(Ckn]e{/2 d,
W2k = o 1 + 
h -1
[ -[COSh(Ckn + R2 sinh(Ckn]e{/2 d,
10 1 + 
for pfa 2 > i and < i, respectively; also,
r
 = - - 1,
a
qk sinh(qk h ) - ! COSh(qk h )
Rl = l '
-qk COSh(qk h ) + 2" sinh(qk h )
* C k sin(ckh) + ! cOS(Ck h )
R 2 = l '
C k cos(ckh) - 2 sin(ck h )
Furthermore, it is found that
-Ck sinh(Ckh) + ! cosh(Ckh)
R2 = .
Ck cosh(Ckh) - ! sinh(Ckh)
o Pm opv pU 2 24 pU 2
-A::: --A = ---A,
OX 01; a Re ah
24 2
FL = -p:n:aU
Re
(11.169)
Considering now also cases where the upstream end is free to slide axially (8 = 2), or
neither end can (8 = 1), as well as the case where only the downstream end can (8 = O),
the dimensionless equation of motion is
YJ"" - u 2 l(24/Re) (1 + h- l ) [( 1 - 18) -I;] YJ" - (1 - 2v)8(2 - 8)ITYJ"
+ u 2 (24/Re)(ll h) YJ' + ff = (LlaE/)4(F p + F v ), (11.170)
where
eo
YJ =-,
a
p:n: L 2
a=-,
m
x
1;= I:
P AL 2
IT=-
EI '
L h - H
l= -, -,
a a
( m ) 1/2 2
ú) = - QL ,
EI
_ ( EI ) l/2 .!...-
'l' - 2'
m L
( P:n:a 2 ) l/2
u= - UL
EI '
(11.171)
P being the pressurization atx = 1 L; primes and dots denote o( )/01; and o( )/01:, respect-
ively. It is stressed that these dimensionless parameters are not identical to those in
Section 8.2. In particular, h = Hla, here; whereas in Section 8.2, h = DI Dh = a 1 H.
First, in Figure 11.57, inviscid theory is compared with a version of the viscous theory
in which only the unsteady viscous effects are taken into account (i.e, pressurization,
pressure drop and sUĎace traction effects are excluded), A relatively large viscosity has
been assumed (/-L = 0,007 Pa s, typical for oH) to amplify the viscous effects, It is seen

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1313
50
,.....,
i 25

5,0
(a)
20
10
a
,.....,

5 O

-10
-20
0,0 2,5 5,0
(b) u
Figure 11.57 (a) Thc rcal and (b) imaginary componcnts ofthc nondimcnsional cigcnfrcqucncics
of thc lowcst thrcc modcs as functions of thc nondimcnsional flow vclocity, u : - - -, potcntial
flow thcory; - , viscous thcory, taking into account only thc unstcady viscous cffccts; for
L/a = 20, H/a = 0,1, a = 323,7, [EI/(pna 2 L 2 )]1/2 = 1.33 mis, and fluid dynamic viscosity
fL = O and 0,007 Pa s, rcspcctively, 0, First mode; ., second mode; ó, third mode (Pai'doussis,
Mateescu & Sim 1990),
that Re(w n ) is little affected; but as expected, Im(w n ) is considerably. Nevertheless,
the critical flow velocity for divergence is not raised too much by the unsteady viscous
effects: only from Ucd = 2.13 to 2.29, corresponding to points A and A', respectively, in
the Im(w n ) versus u plot. The coupled-mode flutter threshold, ucf, is raised even less-
see Table 11.7.
As seen in the table, however, once the steady viscous effects are taken into account, the
values of Ucd become quite different, and they depend on the precise boundary conditions.
Here, C means completely c1amped, and CS lateral1y c1amped but with unimpeded axial
sliding; thus, 'C-CS ends' means condition C at the upstream end and condition CS at the
downstream one.
Next, this analytical model (fulI viscous theory) is compared with that of Section 8.2
[here labeled as PaYdoussis (1973a)] in Figure 11.58, for a system with L / a = 20 (Le.,
e = L / D = 10) and h = H / a = 0.10, that is, for a not very s1ender system, in a not too
narrow annulus, this time with water flow. The agreement is quite reasonable, as may also
be ascertained in Table 11.8 (second line) when only the inviscid terms are considered.

1314
SLENDER STRUCTURES AND AXIAL FLOW
Table 11.7 The critical flow velocities for divergence, Ucd,
and coupled-mode flutter, ucf, for the system of Figure 11.56
(h=Hla=O.I); C-C, C-CS, etc. are defined in the text
(Pai'doussis et al. 1990).
Theory Critical values of u
Ucd ucf
Potential 2,13 3,50
With only unsteady
viscous forces 2,29 3,56
Complete theory
(C-CS ends) 2.49 3.80
(CS-C ends) 2.10 3.33
(C-C, n = 10) 2,30 3.57
(C-C, n = 50) 2,37 3,62
Table 11.8 Comparison of Ucd obtained with the potential-flow versions
of the present and earlier theory (Pai'doussis et al. 1990).
h Values of Ucd Percent discrepancy,
(= Lla) (= Hla) Present Pai'doussis based on last
theory (1973a) column
20 0.05 1.49 1.39 7.2
0,10 2,13 1.94 9,8
0,15 2,64 2.34 12.8
100 0.01 0,631 0,627 0.7
0.05 1.425 1.387 2.7
20
15
,-.,
i 10

5
5.0
2,5
,-.,
:
$ 00 -------
 I
I
I
\
-2.5 1
\
\
\
, ,
-5.0
2 3 4 O 2 3 4
u (b) u
(a)
Figure 11.58 Comparison between two theoretical models for Lla = 20,
h=Hla=O,lO, [EJI(pna 2 L 2 )]1/2= 1.3 mis and JL=0.OO115 Pa s: - , Pai'doussis
(1973a) model, essentia1ly that of Section 8.2; -, Pai'doussis, Mateescu & Sim (1990) model;
(Pai'doussis et aJ. 1990).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILlTlES 1315
As seen in the table, agreement is best for small Hla and large Lla. This is because
(i) the PaIdoussis et al. (1990) model presumes a very narrow annulus, an assumption
not made in Section 8.2 and Patdoussis (1973a), and (ii) the latter model is based on the
slender-body assumption, whereas the theory presented here is more generally accurate
in this respect; cf. Section 8,3.5 and Table 8.5.
The results presented so far are,for the relatively wide annulus of h = Hla = 0,10.
More interesting dynamical behaviour is displayed in the case of h = 0.075, as shown in
Figure 11.59. Because of the increased effect of viscosity, the first loop of the bifurcated
first mode never reaches I m «(VI) < 0, Hence, this system loses stability by coupled-mode
flutter (at point C'), which is a fundamentally important finding. As shown in Table 11.9,
however, once the steady viscous effects - which become stronger for this narrower
annulus than in the case of h = 0.10 - are taken into account, the system loses stability
by divergence. Nevertheless, for the CC and CS-C boundary conditions, even with steady
viscous effects, the system loses stability by coupled-mode flutter.
Loss of stability by coupled-mode flutter may appear to be perplexing because the
system is inherently conservative by virtue of the boundary conditions; however, it is not
really, because of the viscous forces. Evidently, if the nonconservative viscous effects
become sufficiently important - here for the smaller h - the system may lose stability
50

a'"
 25

5.0
(a)
10
5
 a
'"
 O

-5
-10
0,0 2,5 5,0
(b) u
Figure 11.59 (a) The real and (b) imaginary components of the nondimensional eigenfrequencies
of the lowest three modes as functions of the nondimensional flow velocity u, for potential flow for
the same system as in Figure 11.57, but with h = Hla = 0.075 : - - -, potential flow theory;
-, viscous theory, considering ooly unsteady viscous effects. 0, First mode; e, second mode;
/::", third mode (Pai'doussis et al. 1990).

1316
SLENDER STRUCTURES AND AXIAL FLOW
Table 11.9 The critical flow velocity, u c , for loss of stability
of the systems of Figure 11.57 and 11.59, showing the effect
of annular gap, h = Hla. Here, D denotes divergence (u c =
Ucd); F denotes coupled-mode flutter (u c = ucf); Pai:doussis
et al. 1990).
Theory Values or Uc
h = 0,1 h = 0,075
Potential 2,13 (D) 1.84 (D)
With only unsteady
viscous forces 2,29 (D) 3,34 (F)
Complete theory
(C-CS ends) 2.49 (D) 4.54 (D)
(CS-C ends) 2.10 (D) 2,97 (F)
(C-C, TI = 10) 2.30 (D) 3,35 (F)
(C-C, TI = 50) 2.37 (D) 3.41 (F)
Table 11.10 Critical flow velocities for systems with different dimensionless
annular gaps, h = Hla (fulI theory); in the case of C-C, TI = 50 (Pai:doussis
et a/. 1990).
End conditions h = Hla Uc Type or instability and mode, n
C-CS 0.15 2.78 Divergence, n = 1
C-CS 0,10 2.49 Divergence, n = 1
C-CS 0,075 4,54 Divergence, n = 2
C-CS 0,05 6.80 Autter, n = 3
CS-C 0,15 2,54 Divergence, n = 1
CS-C 0.10 2.10 Divergence, n = 1
CS-C 0.075 2.97 Autter, n = 2
CS-C 0.05 5.15 Autter, n = 3
C-C 0.15 2,76 Divergence, n = 1
C-C 0,10 2.37 Divergence, n = 1
C-C 0,075 3.41 Flutter, n = 2
C-C 0,05 5,97 Flutter, n = 3
by coupled-mode flutter. Similar behaviour has been reported by Grotberg & Reiss (1982),
for instance; see Section 7.9.
ln fact, as the system becomes more highly confined, it loses stability either by diver-
gence or flutter in the higher modes, whereas in less confined situations it typical1y loses
stability by divergence in the first mode - cf. Section 8.3 and Table 11.10.
This theory (Paldoussis et al. 1990) should be preferred for examining the dynamics of
cylinders in sufficiently narrow annuli (h = Hla < 0.1). A similar analytical model has
been developed by Fujita & Shintani (2001), in the spirit of the Inada & Hayama (1990a,b),
Porcher & de Langre (1997) and Fujita et al. (2000) mode1s discussed in Section 11.4,
but for flexib1e cylindrical systems; the main findings in the foregoing have thereby been
reconfirmed.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1317
11.5.3(b) Nonuniform annuli
All of the foregoing is for uniform annuli. A very interesting study on the effect of a
nonuniform annular configuration was conducted by Gorman et a/o (1987) in conjunction
with problems encountered in PWRs in France, specifically in the sheathed instrument-
ation tubes (doigts de gant), shown in Figure 11.60(a)0 These very slender tubes, about
705 mm in diameter and 6m 10ng, experienced vibration of large enough amplitude to
cause unacceptable impacting on the walls of the flow-containing structure, These tubes
are subjected to annular flow over their lowest 1,5 m approximately, Yet, calculations based
on the model of Section 8.2 showed that the critical flow velocity was higher than 45 mis,
while in the reactor the maximum flow velocity is 6 mis, and problems were experienced
1


:::
Po

S
o

Upper tube guide


:::
Po

S
'"
1
(a)
lnstrumentation
tube: 5,2 mm (i,d,)
7,5 mm (o,d,)
t
t
1 r W 12,7 mm (dia,)
I 15°
I
I ["""""T
i cm
i
i
I
I
i
i
I
i
I
i
I
I
I
I
i
 ,.- lnstrumentation tube
(b)
-
12'7 p WW
(dia) 15°
. I -Hl cm -j!f-2,Omm I
tj I T
! S
I u
I V')
i ....;
(c)
!
'"
I
3
Figure 11.60 (a) Schematic of an instrumentation tube installed in a nuc1ear reactor; (b) typical
conventional nozzle geometry; (c) different nozzle geometrie s tested in the full-scale model experi-
ments: 1, conventional conical nozzle; 2, split cylindrical nozzle; 3, cylindrical nozzle (Gorman,
Godon & Planchard 1987),

1318
SLENDER STRUCTURES AND AXIAL FLOW
Table 11.11 Experimental results from model testing of the instru-
mentation tubes, in annular passages terminated by different types of
nozzle [Figure I I.60(c)]. Maximum flow refers to the flow at which tbe
test was discontinued and for which tbe vibration level is given; B, C
and X are defined in the text (Gorman et a/. 1987),
Test geometry
Nozzle 1
No nozzle, uniform annulus
Nozzle 2,3, or grooved 1
Larger tube, nozzle 1
Larger tube, nozz1e 2
Maximum U (m/s)
Vibration level
4.5-5,5
5,0
7.5
5.5
8.5
X
B
B orC
X
C
before that! Severa1 mechanisms were suspected to be responsible, among them one due
to the diffuser-tenninated nozzle shown in Figure 11.60(a) and (b). In view of Hobson's
(1982, 1984), Spurr & Hobson's (1984) and Parkin & Watson's (1984) work described in
Section 11.5,1, it was known even then that diffusers are highly destabi1izing at the exit
end of annular flow regions.
An extensive experimental investigation was conducted with the aid of a full-scale
experimental apparatus, involving the fulI 6 m long tube (7.5 mm outer diameter), and
truncated versions thereof, The tube was subjected to annular flow over the lower 1.3 m
of its length, in a flow passage of 14 mm inner diameter, diminished to 12.7 mm in the
nozzle region, The results, given in greater detail by Gonnan et al. (1987), are discussed
here with the aid of Table 11.11. The va1ues of U given are flow velocities in the annulus
upstream of the nozzle. The vibration leve!s, B, C and X denote: B, vibration of insufficient
amplitude to cause contact with the wall of annular flow passage; C, stronger vibra-
tion, with possibly light contact; X, strong vibration with steady and severe impacting on
outer wa11.
With nozz1e I of Figure 11.60(c), which model s the nozzle used in the reactor
[Figure 11.60(b)], the vibration leve! increased with flow, eventua11y reaching level X,
whereupon the 'test was tenninated in order to avoid destruction of Test Facility' !
The nozzle was then replaced by a unifonn annulus, actually extending over 3.5 m,
rather than the usual!.3 m. Tbe vibration never exceeded level B; in this case, the test
was tenninated due to rupture of a feed line, resulting from the high pressure necessary
to overcome the larger pressure drop in the annulus. Tbis strengthened the suspicion that
the problem lay with the nozzle.
Then, the system was tested with nozz1es 2 and 3; in both cases, the vibration never
exceeded leve! B. When nozzle I was modified by machining eight grooves of the type
used in nozzle 2, again the level was acceptable, at level C. The next two tests were
conducted with a larger diameter (stiffer) tube, but this a1so means a narrower annulus.
When nozzle I was used, vibration leve! X was reached at U = 5,5 mis, For nozz1e 2,
however, even at U = 8.5 mis the vibration was quite mild.
This study provides a remarkable demonstration of how a minute detail in a design can
be responsible for big trouble in a large and expensive system. It also shows that, whereas
a system may be quite stable in unifonn annular flow, a diffuser section at the outlet can
reduce the critical flow by a large factor.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1319
a + 1
n I n
t-X=X n
I
1012 10 /2
I ! !


x
r Baftle Secondary
plate coolant
- -
Tube
(aj
(b)
L
Diffuser Jet
section section
1=1 0 1'=/ 0
I I I
. --_/
U oi _
P t
(c)
Stili sections
(L -/ 0 )/2
Stili sections
(L - 3/ 0 )/2
q g
K=(N+l)/2
Figure 11.61 (a) Diagram of one design of annular flow passage through a baffte plate;
(b) segmented cylinder model, showing the concentrated fluid-dynamic forces and moments act-
ing on each segment; (c) diagram showing the diffuser-shaped annular flow passage and the 'jet
expansion' zone just downstream (Yasuo & Pa'idoussis 1989).
Another system wherein a structure is subjected to high-velocity annular flow over small
segments of its length is that ofheat exchanger tubes in the regions where they go through
the baffle plates, as depicted (for a particular design) in Figure 11.61(a). The effect of the
shape of the annular passage for a heat exchanger tube, supposed to be subjected to axial
flow over its whole length, except for a segment where it is subjected to annular flow, has
been investigated by Yasuo & Pa"idoussis (1989).
In the analysis, the tube is subdivided into segments, as shown in Figure 11.61 (b), over
each of which the fluid dynamic forces are constant. t The effect ofaxial flow upstream
and downstream of the annulus is small and can be ignored, but the effect of added mass
and viscous damping are inc1uded, the latter in a globallogarithmic decrement. The fluid
forces are only considered in the annulus itself, supposed to be conical and to extend
over a single element, and in the 'jet section', where the flow is unconfined but has a
much higher flow velocity than downstream. To determine the flow-induced forces and
. moments in these two segment s, the Mateescu & Pai"doussis (1985) model, described in
Section 11.5.2(a), is used. A weakness of the model is that no axial boundary conditions
are applied at either end of the annulus; thus, entry and exit pressure losses or pressure
recovery, both steady and unsteady, are not modelled, Another weakness is that the fluid
forces are determined via an inviscid model. Nevertheless, the important contribution
tThe same could have been obtained by assuming variable forces and performing a Galerkin projection,

1320
SLENDER STRUCTURES AND AXIAL FLOW
5
1=Tan-ls
21 f3  .__._.-  q
hola = 2.0 
lo-+ !.
10

I""
o
0,00
0,05
0,10
0,15
slo/ho
Figure 11.62 Dimensionless critical tlow velocity for tlutter, U c f, for a tube going through a
diffuser-shaped opening in a baftle plate; s = tan ex, and TJ is the mass-damping parameter (Yasuo
& Pai:doussis 1989).
made by this simple model is the development of analytical c1osed-form expressions for
the critical flow velocities of divergence and flutter of the tube, for small diffuser angles ex.
Thus, for divergence, the critical value for the inlet flow velocity at the entrance of the
annular region, Uai = Ucd, is
u =  { £1 (  )} 1/2
cd L ma 21 0 '
(11.172)
where L and 10 are defined in Figure 11.62, and ma = p]'w 2 k is the added mas s per unit
length, with k = [(bo/a)2 + 1]/[(bo/a)2 - 1]; apart from the L/210 factor, this result is
the same as for uniform inviscid flow - cf. Section 8.3.1(c). The critical dimensionless
flow velocity is Ucd = 7T(L/210)1/2.
For flutter, it is shown that the system is stable in a convergent annulus, while it is
unstable in one of divergent shape. The dimensionless critical flow velocity for flutter is
given by
- [ Uai ]
Ucf= -
foD cf
M8
IJ = pD2 '
1 (1 + slo/bo)2[(bo/a)2 -1]2
27T(a/L) (slo/bo) (bo/a)2 IJ,
- ma { 10 }
M = m + T 1 + 2(k - 1) L '
(11.173)
k _ (bo/a)2 + 1
- (bo/a)2 - 1 '
where 10, a, bo and s (= tan ex) are defined in Figure 11.62, 8 is the logarithmic decrement
of damping, p the fluid density and D the tube diameter. In U cf , Uai is the flow ve loci ty
at the inlet of the annulus and fo is the first-mode natural frequency in stilI fluid. In M, m
is the tube mas s per unit length, and ma the corresponding added mass. In Figure 11.62,
U cf /IJ is plotted for varying slo/bo and bo/a.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1321
Consider as an example a heat exchanger tube with L = 2603 mm, Z = Zo = 19 mm,
ho = 17 mm, a = 11 mm and inner tube radiu s 10 mm, E = 2 x 10 11 N/m 2 , tube
density 8.0 x 10 3 and p = 10 3 kg/m 3 ; the effective tube damping ratio is taken to
be Š = 812n = 0.01. Equation (11.172) gives Ucd = 280 mis in the baffte and 110 mis
upstream, assuming an area ratio of 0.4: 1; for water flow, such flow velocities are not
practicably feasible. On the other hand, considering ex = 6°, U cf Iry = 319, which corres-
ponds to Uoi = 6.3 mis, and upstream of the baffte Uo = 2.5 mis. These are relatively low
flow velocities that could easily arise in heat exchangers,
Thus, although it cannot be claimed that a simple analytical model such as this is very
accurate, it succeeds in showing that a diffuser-shaped segment, extending over merely
0.7% of its length, is capable of rendering the system unstable; if, however, the baffle
holes are chamfered the other way, tube vibrations would be damped,
11.5.4 Numerical methods: prescribed motions
The theoretical methods reviewed in Section 11.5.3 for the dynamics of flexible cylinders in
annular flow have been mainly analytica1. Here some numerical and computational (CFD)
methods are discussed for obtaining the fluid-dynamic forces resulting from prescribed
motions of the inner or outer components of the annulus, while the fuHy coupled problem
of flow-induced motions is discussed in Section 11,5,5. Experiments designed to test the
theory are also described.
11.5.4(a) CFD methad ar Mateescu, Bé/anger and Pai'daussis
The basics of the method is described first, for it can be applied to many problems, other
than the annular flow configuration, for example driven cavity flow with a backward facing
step; see, for example, Mateescu cf aZ. (1991, 1994a,b,c), Bélanger (1991), de Langre cf aZ.
(1992).
We start with the incompressible Navier-Stokes and continuity equations, which in
cylindrical coordinates and in dimensionless form may be written as
av
-+G(V,p)=O,
af
au 1 a(rv) 1 aw
V.V=-+--+--=O
ax r ar r ae '
(11.174)
(11.175)
where, denoting by an asterisk the dimensional quantities, V = V* I U is the dimensionless
flow velocity, G(V, p) contains all but the time derivative terms, x = x* I h, r = r* I h
and e are the dimension1ess cylindrical coordinates, f = f* U I h is dimensionless time,
{u, v, w} = {u*, v*, w*, }I U are the dimension1ess velocity components in the axial,
radialandcircumferentialdirections, x, r ande, respectively, and p = (p*- p)/(p*U2),
U == U*, such that
v{:},
{ Gu(u, v, w, p) I
G(V, p) = Gv(u, v, w, p) ,
Gw(u, v, w, p)
(11.176)

1322
SLENDER STRUCTURES AND AXIAL FLOW
a(uu) 1 a(rvu) 1 a(wu) ap
Gu(u, v, w, p) = - + -- + -- +-
ax r ar r ae ax
1 [ a2u 1 a ( au ) 1 a 2 u ]
- Re ax 2 + -; ar r ar + r 2 ae 2 '
a(uv) 1 a(rvv) 1 a(wv) w 2 ap
Gv(u, v, w, p) = - + -- + -- - - +-
ax r ar r ae r ar
1 [ a2v 1 a ( av ) 1 a 2 v 2 aw v ]
- Re ax 2 + -; ar r ar + r 2 ae 2 - r 2 ae - r 2 '
a(uw) 1 a(rvw) 1 a(ww) vw ap
Gw(u,v,w,p)=+-; ar +;:-ae+--;:-+ ae
1 [ a 2 w 1 a ( aw ) 1 a 2 w 2 av W ]
- Re ax 2 + -; ar r a;: + r 2 ae 2 + r 2 ae - r 2 .
(11.177)
(11.178)
(11.179)
Here Re = U h / v =  Re Dh is the Reynolds number, Dh = 2h being the hydraulic diameter.
The temporal discretization of the Navier-Stokes equations is based on a three-time-
level implicit scheme (second-order accurate) in the form
3yn+l _ 4yn + yn-1
+ G n + 1 =0
2( ,
(11.180)
where the superscripts n = 1, n and n + 1 indicate three consecutive time levels, ( =
(n+1 _ (n = (n _ (n-1 represents the time step, and G n + 1 = G(yn+ 1, pn+ 1 ). The unsteady
flow solution is supposedly known at (n-l and (n, and the solution at time level (n+l has
to be obtained from
yn+l + aG n + 1 = Fn,
V. yn+1 = O,
(11.181)
(11.182)
where
a = (, F n = 1(4yn - yn-l). (11.183)
Equations (11.181) and (11.182) are subject to the usual no-slip and no-penetration
conditions at the fixed and moving boundaries,
ln order to advance the solution to the (real) time level (n+1, an iterative pseudo-
time relaxation procedure is used; a pseudo-time variable, -r, is introduced to define the
intermediate values of the velocity, V (-r), and pressure, j) (-r), between the real time levels
(n and ("+1. The relaxation procedure consists in the pseudo-time integration oť equations
(11.181) and (11.182), which are augmented by pseudo-time derivatives in the form
av v v
- + y + aG = F n ,
a-r
aj) v
8 - + v . y = O,
a-r
(11.184)
where 8 represents an artificia1 compressibility and G = G(V (-r), j)( -r». An implicit Euler
scheme [aj / a-r = (jv+1 - jV) /  -r, where j is V or j)] is used to seDÚ-discretize the

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1323
above equations in the pseudo-time interval /1 i = i ,,+ 1 - i" between i" and i ,,+ 1 ; then,
using the pseudo-time variations /1 V and /1p defined as
{ /1u } { U ,,+ 1 - U" }
/1 V = V,,+1 - V" = /1v = i),,+1 - i)" ,
/1w w"+1 - w"
/1p = p"+1 - p", (11.185)
the resulting equations are recast in delta form (Mateescu et a/o 1994a)
(I + /1i)/1 V + a/1i /1G = /1i(F n - V" - aG"),
/1p + (/1ij1J)V. (/1 V) = -(/1ij1J)V . V",
(11.186)
(11.187)
where
{ /1G } { 0"+1 - O" }
/1G = G"+1 - G" = /1G: = +1 -  '
/1G w G"+1 - G"
w w
in which /1G u , /1G v and /1G w are expressed using lagged nonlinearities, such as
(11.188)
o(U V /1u) 1 o(ri)v /1u) 1 o(W V /1u) o (/1p)
/1G u = ox + -; or + -; oe + 
_  [ o2(/1U) +   ( r O(/1U» ) +  02(/1U) ] .
Re ox2 r dr dr r 2 de 2
(11.189)
The initial values in this pseudo-time integration are the values of the flow variables
at rea1 time level t n , that is, for v = 1, V 1 = vn and p 1 = pn o The new values of the
local velocity of the moving boundary at t n + 1 are imposed as boundary conditions at the
beginning of the pseudo-time integration and are kept constant, along with F n , until
this procedure converges (say at v = K) and the corresponding pseudo-time variations,
/1 V = V K + 1 - V K and /1p = pH1 - pK, become smal1er than a specified margin; when
convergence is achieved, dVjdi and djJjOi vanish in equations (11.184), which thus
reduce to (11.181) and (11.182), and the new solution at the real time level t n + 1 is found
as V n + 1 = V K and pn+1 = pK.
An alternating direction implicit (ADI) method based on an approximate factorization
(Hirsch 1990; Soh 1987) is used to separate the numerical integration of equations (11,186)
and (11.187) in the X-, r- and e-directionso Trus is done by introducing the intermediate
variables /1u , /1v , /1w and /1p which are solved in the x-sweep defined by the equations
_ [ d(U V /1u ) d( /1p ) 1 d 2 ( /1U ) ]  
(1 + /1i)/1U + a/1i + - - - 2 = /1i(F: - U V - aG),
dX dX Re dX
_ [ d(U V /1v ) 1 02( /1V ) ]  
/1v+a/1i dX - Re dx2 =/1i(F-vv-aG), (11.190)
_ [ d(U V /1w ) 1 d 2 (/1W ) ]  
/1w + a/1i - - 2 = /1i(F - w" - aG),
OX Re dX

1324
SLENDER STRUCTURES AND AXIAL FLOW
- i o(u) i v
P+8=-8V.V ,
(11.191)
where F n = 1 ( 4u n - u n - l ) Fn = 1 ( 4v n - vn-I ) and F n = 1 ( 4w n - w n - l ) . then
u 3 'v 3 w 3 "
by introducing the intennediate variables , s;",  and P which are solved in the
r-sweep defined by the equations t
u + ai [ O(r1J v ) _  ( r O(U» )] = u ,
r or Re or or
(1  ai [ o(rV V K'v) o( p )  v 
+i)V+- +r--w w
r or or
_ {  ( r O(K'V» ) _ v }] = v , (11.192)
Re or or r
 ai [ o(rVV) v  1  1 o ( o(» )] -
w+- +v w+----- r- =w,
r or Re r Re or or
 i 1 o(rv) -
P+ -- = P,
8 r or
and fina11y u, v, w and P are solved subsequently in the O-sweep defined by
u + ai [ O(u.,v u) _   a2(u) ] = ,
r 00 Re r 00 2
v + ai [ O(u.,v v) _  { 02(v) _ 2 o(w) }] = K'v, (11.193)
r 00 Re r 00 2 00
ai [ O(u.,v w) o(p) 1 1 { 02(w) o(v) }] 
(1 + i)W + ----;- 00 + ae - Re  00 2 + 2ae = w,
i 1 o(w) 
P+8=P'
The spatial differential operators in the above equations are discretized by centra1 differ-
encing on a staggered grid, in which u, v, w and p are defined at different points; for a
3-D cell (i, j, k) centred at (xť, rf. OD where Pi,j,k is defined, the grid points for the
nondimensiona1 velocity components Ui-l,j,k, Ui,j,k. Vi,j-l,k, Vi,j,k and Wi,j,k-l, Wi,j,k
are defined at the centroids of the faces perpendicular to the x-, r- and O-directions,
respectively. For a better spatial resolution, stretched grids based on hyperbolic stretching
tThe r-sweep and 6-sweep equations as given here correspond to the form developed by Mekanik (1994)
and Mateescu et a/. (1996) - see subsection (b). They are more correct and computationally more convenient
for a staggered grid than those originally given in Mateescu et a/. (1994a), in that some tenns which vary with
r have properly been moved from the 6-sweep to the r-sweep equations.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1325
functions have been used in this analysis to concentrate more grid points in the regions of
higher velocity gradients, such as near the oscillating or fixed boundaries.
For each sweep, the fourth equation is used to eliminate the pressure from the other
equations, and, after several further algebraic manipulations, the problem is finally reduced
to the solution of decoupled scalar tridiagonal systems, one system for the variations in
each velocity component and each sweep; further detail s on the decoupling procedure and
the spatial discretization are given by Mateescu et al. (1994a).
The high computational efficiency of this method is largely due to the reduction of
the problem to the solution of decoupled scalar tridiagonal systems, which can easily be
obtained numerically by scalar tridiagonal matrix inversion.
Consider now the system ofFigure 11.63, where in this case it is the outer boundary ofthe
annulus whieh oscillates while the centrebody remains immobile - although the reverse is
as easy to do. The fluid in the annulus is either quiescent or a fully developed laminar flow.
The Reynolds number is based on the hydraulic diameter, Dh = 2h : ReDh = 2hU jv.
ln the quiescent-fluid case, the whole of the outer cylinder is supposed to oscillate (2-D
case), while in laminar flow (3-D case) only the central part, oflength l*. In both cases, the
cylinder executes prescribed small-amplitude harmonie oscillations in the () = O plane,
he(t) = -h£coswt,
(11.194)
where h£ is the dimensional amplitude. The boundary conditions, in nondimensional form,
are in this case
u=v=w=O
(11.195)
on the inner cy linder (r * = a) and the fixed parts of the outer cy linder, and
u = O,
v = écose,
w = -é sin e
(11.196)
on the moving parts of the outer cylinder, where é = de jdt = w£ sin wt. Conditions
(11.196) become approximate boundary conditions, based on the assumption of
sma11-amplitude oscil1ation, where they are applied at the mean position ofthe osci1lating
1 1* j he(t)
_::::: ::::: ::: :::::::::::::: :::::::: ::: ::::::::: ::::. l.
U . (h t
__'__'__'_'__'__'___'_'_'_ ..__._ ._._._._._,__._._,_.__.J=
I.
U
.
..... --...... .... ................................................ -- -.. .............................
.................... ........................................................... ..............................................
I.
1*
o
I
1*
o
I
Figure 11.63 Geometry of the annular flow with an oscillating outer boundary over the cent-
ral portion of length /* (3-D case), showing also the immobile portions of length /0' upstream
and downstream, included in the computational domain (Mateescu et a/, I 994b).

1326
SLENDER STRUCTURES AND AXIAL FLOW
outer cylinder (r* = a + h) on a fixed grid, as in this analysis, By considering a truncated
Taylor expansion of the fluid velocity in the vicinity of the moving cylinder, one can obtain
improved boundary conditions on the mean position of the oscillating cylinder
Uo = Xw - Xw' (VU)o,
(11.197)
where Xw is the wal1 displacement from the mean position and Xw the corresponding
velocity, U is the mean velocity, and the subscript O denotes the mean position of the
cylinder (the boundary of the mesh). This replaces (11.196), which in this notation is
U w = xw. Hence, this means that a more accurate set of boundary conditions are:
ulr=rw = e[oUolor]r=r w cos O, vlr=rw = écosO, wlr=r w = -é sin 0,
(11.198)
where [oUolor]r=r w represents the velocity gradient, at the outer wa1l, ofthe steady lam-
inar flow in the absence of any oscillations; rw = (a + h) I h. In these improved boundary
conditions [with an error of order 8 2 , where 8 is as defined in equation (11.194)], only the
first condition is different from the approximate one. The unsteady fluid-dynamic forces
calculated with the improved boundary conditions (11.198) for the 3-D case will depend
also on the nondimensional displacement, e (not only on the displacement velocity, é), a
dependence that is missing in (11.196); this could be important in the analysis of fluid-
elastic instabilities of divergence type, occurring at vanishingly smal1 frequency, W, which
depend on the stiffness component of the fluid forces. However, for smal1 amplitudes and
higher frequencies, it is found that the difference in the fluid-dynamic forces obtained by
the two sets of boundary conditions (11.196) or (11.198) is rather smal}, and hence in the
first setof computations (Mateescu et al. 1994b,c), equations (11.196) are used. Neverthe-
less, as is discussed in Section 11.5.5, there are important differences at the fundamental
level.
Final1y, the artificial compressibility is chosen via 8 = (4 q 2 t)-1, where q = (u 2 +
v 2 + w 2 )1/2 is a representative velocity and t the time step in the real time integration.
Furthermore,  T = C x I A+, where C is the Courant-Friedrichs-Levy number, equal
to 30--40 in these calculations, x is a representative value of mesh spacing, and A+ =
aq + [(aq)2 + aI8]1/2 is a characteristic propagation velocity, with a = 2tI3, as in
(11.183) [see Mateescu et al, (1994a)].
Typical results for the 2-D problem, strictly belonging to Section 11.3,3, are shown in
Figure 11,64, The oscillatory Reynolds, or Stokes, number f3 is defined by f3 = Qh 2 Iv,
with h as in Figure 11.63 and Q the radian frequency of oscillation. Results for f3 = 300
are given in Mateescu et al. (1994a), and they look similar, except that they are smoother;
also, results for the perturbation pressure are given. An interesting result is that if a Crank-
Nicolson scheme is utilized instead of the ADI scheme described in the foregoing, strong,
spurious (numerical1y induced) oscillations are found in the pressure, of higher frequency
than those of the oscillating cylinder.
For the 3-D problem, results have been obtained for a laminar flow entering the annulus
with the following velocity profile:
(r; - rl) ln(rln) - (r 2 - rl) ln(rolrj)
u=2 ,
(r; + rl) ln(rolrj) - (r; - rl)
v = w = O,
(11.199)

10,0
ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1327
9,8
9,6
r
9.4
9,2
9,0
-1.5
(a)
1.5
1.0
0,5
w 0,0
O,5
-1.0
-1.5
°
(b)
-1.0
-0,5
n = 11
/
0,0
0,5
1.5
n = 13
30
60
1.0
w
90
e (deg)
120
150
180
Figure 11.64 Typieal (a) radial variations (at () = 45°) and (b) eireumferential variations (at
r = 9.75) ofthe nondimensional eireumferential veloeity eomponent, w, aeross the annular gap for
five real time levels within the harmonie eycle, t n = 2nn/19, and for oseillatory Reynolds number
f3 = 3000, (Mateeseu eta!, 1994b).
where rj and ro are the inner and outer radii of the annulus. Typical results are
shown in Figure 11.65 obtained by (i) the fuH 3-D nonlinear analysis and (ii) a com-
putationaHy more efficient but approximate hybrid time-integration solution based on
azimutha1 Fourier expansions, developed in Mateescu et aZ. (1994c). It is seen that,
even with one term in this Fourier expansion, the effectively linearized solution is
quite close to that of the fuH nonlinear solution. These results have been obtained for
rj = a / h = 9 and r o = (a + h) / h = 10, Z = Z* / h = 100, and Zo = Zo / h = 20 (i.e. the calcu-
lation domain is - 20 < x < 120, where x = x* / h), the asterisk denoting the dimensional
quantities.

1328
SLENDER STRUCTURES AND AXIAL FLOW
50
0iJ O
"

<>.
.....
o
"
'"

f -50
a
-100
0,6 --
<>.
.....
o 0.4
"
"C
€
;:o. a
8
<:
0,2
0,0
-20
o
20
40
60
80
100
120
x
Figure 11.65 Typical axial variations of the nondimensional unsteady pressure amplitude and
phase angle in the 3-D unsteady flow in a uniform annular passage, at r = 9.94 and () = 7.5°,
for ReDh = 100 and for two osci11atory Reynolds numbers: (a) f3 = 5; (b) f3 = 10. Compar-
ison between: -, hybrid solution for N = I; - - -, fuU 3-D nonlinear solution (Mateescu
et a/. 1994c).
The method has also been extended to deal with discontinuous annular passages of the
type shown in Figure 11.66, involving an annular backstep in the inner cy1inder, Typical
results for the perturbation pressure are shown in Figure 11.67.
11,5.4(b) CFD method based on time-dependent coordinate transformation
The dynamics of the same general system as that shown in Figures 11.63 and 11,66 has
been re-examined (Mateescu et a/, 1996; Mekanik 1994) by refining the analysis such
that the boundary conditions are applied on the moving boundary. This is done by a time-
dependent coordinate transformation via which a new fixed-grid computational domain is
obtained and the system equations are transformed.
For a stationary centrebody and an oscillating outer cylinder, for example, as in
Figure 11.63, the time-dependent coordinate transformation is
z = (r - ri)/ H«(), t),
()=(), X=X, t=t,
(11.200)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1329
hrh
o hr j
u
I 1;= hlo I
'II t t
J hr
- 
x
hrd
hrjd
1* = hl
1 1;=hI O I
III .. III
Figure 11.66 Geometrical representation of the nonuniform annular flow passage with an annular
backstep in the inner cylinder, which is fixed. The central portion, of length l* = lh, of the outer
cylinder executes transverse translational oscillations in the longitudinal plane e = O (Mateescu
et al. 1994c).
OD O
"


""
o
"
CI)
"
..c: -50
A..
-100

 0,15
"
"'"
,€
J 0,10
50
0,25
0,20
0,05
0,00
-20
O
20
x
40
60
Figure 11.67 Typical axial variations of the nondimensional unsteady pressure amplitude and
phase angle for 3-D unsteady flow in a nonuniform annular passage with a backstep, at r = 9.94
and e = 7.5°, for (a) ReDh = 100 and f3 = 5, and (b) ReDh = 250 and f3 = 10. Compar-
ison between: -, hybrid solution for N = 1; - - -, full 3-D nonlinear solution (Mateescu
et al. 1994c).

1330
SLENDER STRUCTURES AND AXIAL FLOW
where H(O, t), the nondimensional annular clearance during oscillatory motion, IS
defined by
H(O, t) = j (ri + 1)2 - e 2 (t)sin 2 0 +e(t)cosO -ri.
(11.201)
ln this fixed computational domain, the moving outer boundary becomes fixed at Z = 1,
and the Navier-Stokes and continuity equations (11.174) and (11.175) are transfonned to
av av
ar + ZC(O, t) az + Q(V, p) = O,
au 1 [ a(rv) aw aw ]
VV=-+- A-+ZB-+- =0,
ax r az az ao
(11.202)
(11.203)
where the components of the vector Q(V, p), corresponding to the convective, pressure
and viscous terms, become
a(uu) 1 [ a(rvu) a(wu) a(wu) ]
Q ( U v w p) =-+- A-+-+ZB-
u , " ax r az ao az
au ap 1
+ ZC- + - - -F(u),
az ax Re
a(uv) 1 [ a(rvv) a(wv) a(wv) 2 ]
Q ( U v w p) =-+- A- + -+ZB--w
v , " ax r az ao az
av ap 1 [ 1 ( aw aw )]
+ZC-+A--- F(v)-- 2-+2ZB-+v
az az Re r 2 ao az '
(11.204)
a(uw) 1 [ a(rvw) a(ww) a(ww) ap B ap ]
Q ( U v w p) =-+- A + -+ZB-+vw+-+Z -
w , " ax r az ao az ao az
aw 1 [ 1 ( av av )]
+ZC-- - F(w)+- 2- +2ZB--w
az Re r 2 ao az '
in which
az 1
A---
- ar - H(O,t)'
1 a z de e a H
C=---=- ,
z ae dt H(O, t) ae
1 az 1 aH
B-------
- Z ao - H(O, t) ao '
1 a 2 z
D = Z ao 2 ' r = ri + ZH(O, t),
(11.205)
a 2 1 [ a2 a
F(q) = ; + 2" (r 2 A + z2 B2)----{ + (r A 2 + ZB 2 + ZD)--.!i
ax r az az
a 2 q a 2 q ]
+ ao2 + 2ZB az ao '

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1331
The time integration scheme, the pseudo-time relaxation procedure and the ADI scheme
for the spatial part of the solution are quite similar to those described in Section 1I,5.4(a)
and hence are not given here [see Mateescu et al. (1996)].
As in the previous subsection, the outer cylinder executes transverse translational
osci11ations, he(t) = - he cos wt, where w and t are dimensionless. The following
dimensionless parameters are used:
x=x*jh, r=r*jh, t=w*t*, w=w*(t*jt), {u,v,w} = {u*,v*,w*}jUref,
V = V* j Uref, P = (p* - po)j(p*Ur)' f3 = w*h 2 jv, ReDh = 2hU jv = 2Re,
(11.206)
where starred quantities are dimensional, and so is h; Uref = U or w*h for cases with
or without axial flow, respectively; ReDh is the Reynolds number based on the hydraulic
diameter Dh = 2h, and f3 is the oscil1atory Reynolds (or Stokes) number.
The 2-D results for a quiescent fluid in the annulus and infinitely long centrebody
and outer cylinder given in Figure 11.68 show the effect of amplitude on the perturbation
pressure according to (i) the mean-position analysis and (ii) this time-dependent coordinate
transformation analysis. It is seen that, up to e = 0,05, the two results remain c1ose. For
e > 0.05, however, that is, for amplitudes larger than 5% of the annular gap h, the mean-
position analysis seriously underestimates Im (p), and hence the phaselag between motion
and pressure, which plays an important róle in determining stability.
4
3

 2
I
O
O 0,04 0.08 0.12 0,16
0,8
.
0.6
 .
l' 0.4

0,2
0,05 0,1 0.15
Amplitude. E
Figure 11.68 Influence of the dimensionless amplitude of oscillation, £, on the real and imaginary
components of the dimensionless unsteady pressure, p = p(Z, (), t)/ exp(iúJt); 2-D analysis for
f3 = 300, ri = 4.5, ro = 5 and () = 7.5 0 : O, mean-position analysis;., coordinate-transformation
analysis (Mateescu, Mekanik & Paldoussis 1996).

1332
15
SLENDER STRUCTURES AND AXIAL FLOW
0.4
2-D case
O
-50
-100
100 150 -100 -50 O 50 100 150 200
x
(b)
<lJ
....
f;J 10
'"
2
Q..
>..
-c
'"

 5
:::>
111-1.'1"1'1&"."-" I..
2-D case
o
200
100
bi)
<lJ

<lJ
'"
'"
..Q
I>-;
With backstep
O
-100
-200
-50
O
50
x
(a)
0.3
0,2
0,1
0.0
100
50
Figure 11.69 Typical axial variations of the amplitude and phase of the unsteady pressure, p, for
Ti = 9, To = 10, Z = 100, Za = 60, E = 0,1 and e = 7.5°. In (a) f3 = 50; in (b) f3 = 5(w = 0,1),
ReDh = 100 : - - -, mean-position ana1ysis; -, coordinate-transformation analysis (Mateescu
et aZ, 1996).
Typical variations of the unsteady pressure for 3-D geometries with and without flow
are shown in Figure 11.69. Without axial flow, we see a large reduction in pat the location
of the backstep, x = 50, vis-a- vis the case without a backstep. As expected, the maximum
value of p for the 3-D configuration is lower than that for the 2-D case, With axial flow, we
see a higher peak in the perturbation pressure (clo se to x = O) for the stepped geometry,
as compared to the uniform one, The much different scales in p for the two cases, with
and without flow, reflect the different Uref, as discussed below equations (11.206).
11.5.4(c) Experiments
Some experiment s were conducted to test the theory in the apparatus shown in Figure 11.70,
in air-flow by Sim (1991) and Mekanik (1994). Various schemes for sealing the outer
cylinder, between moving and immobile parts, were used. The same type of centrebody
as in the experiments of Section 11.5.2(e) was used, but in this case the system was
horizontal. The phase between the acceleration of the oscil1ating outer cylinder, sensed
by an accelerometer, and the pressure on the centrebody was measured directly via an
FFT analyzer, Agreement with the theoretical results, as shown in Figure 11.71 for lateral
motions, is good, The comparison in Figure 11.71 is limited to smal1 amplitude motions.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1333
Stationary, cylindrical
centre-body
L x  Microphone pressure transducers
Airto
blower
I.
(a)
Orifice plate
(b)
Sponge
rf-1

LLJ
(i)
-Dn
J 
----ULJ
J
(ii)
(iii)
Figure 11.70 (a) Schematic ofthe apparatus used; (b) different sea1ing arrangements between the
moving and fixed parts of the outer cylinder: (i) sponge-felt gaskets between flanges; (ii) c1ose-
fitting flanges; (iii) flanges with rubbing contact (on1y for U = O) (Sim 1991; Mekanik 1994).
2

::J
'" 1.5
'"
!t
>.
] 0.5
'"
<:
;::J
O 50 100 150 O 50 100 150
x x
ISO 150
100 100
Oj) 50 i 50 /'
:!! O ' O .....,..
O>
 -50...... .' -50/./"
f
-100 -100
-150 -ISO
-20 O 20 40 60 80 100 120 -20 O 20 40 60 80 100 120
(a) x (b) x
Figure 11.71 Typica1resu1ts fortheunsteadypressure p(x) forrj = 4,785, ro = 5.785, 1= 100,
10 = 60, é: = 0.05375, Z = 0.127 and () = 7,5°: (a) in axia1 flow, with ReDh = 2900, ú) = 0.416
(f = 36 Hz, f3 = 1208); (b) in quiescent fluid, for f3 = 1020 (f = 30.4 Hz, ú) = 1): ' . . " mean-
position ana1ysis; -, coordinate transformation ana1ysis, ., experiment (Mateescu et al. 1996).
Some experiments were a1so conducted for rocking motions and a typical result is
shown in Figure 11,72. In this case, theoretical results for the coordinate transformation
method are not available for comparison, but nevertheless agreement with the mean-
position analysis is very good. t
t Some theoretical results for the backstep geometry using the coordinate transfonnation method are also
available; see Mekanik (1994, section 6.7.2).

1334
SLENDER STRUCTURES AND AXIAL FLOW

, 4
i5..
 3
 2

€1
"

í9 O
;:J -80 -40 O
150
100
Oj)
" 50
::2-
"
'" O
"
..c:
"'"
-50
-100
40 80 120 160 -20 O 20 40 60 80 100 120
x x
Figure 11.72 Typical results for the unsteady pressure amplitude and phase for rocking motion
of the outer cylinder about the hinge (at x/Z = 0.81) in quiescent fluid, for E: = 0.1075 and
f = 30.4 Hz: -, mean-position analysis; D, c>, experiment (Mekanik 1994).
It should be said that, in this case also, agreement with theory is good, only for axial
locations sufficiently removed from the extremities of the moving outer cylinder - cf,
Section Il.5.2(e,f).
This CFD method converges well for Re Dh ::: 3000 approximately. Beyond that, numer-
ical problems ha ve been encountered; in any case, however, for such Reynolds numbers,
turbulence modeling would need to be introduced.
11.5.4(d) Spectral collocation and hybrid spectral methods
The spectra1 collocation method in the absence of a mean annular flow has been intro-
duced in Section 11.3,3(a). The method for an annular flow without oscillation of the
boundaries is developed in Mateescu et a/, (1994e). Here, the method for the case with
an annular flow and an oscillating inner boundary is discussed, following Mateescu et a/.
(1995a,b ).
Consider two concentric (coaxial) cylinders of radii a and ba, where b > 1, subjected
to fully developed flow velocity U*(r) = UoU(r), where Uo is the mean axial velocity
and U (r) = 2[ (b 2 - 1) ln r - (r 2 - 1) ln b ]/[(b 2 + 1) ln b - (b 2 - 1)] represents the radial
profile function of the axial velocity, in which r = r* I a is the nondimensiona1 radial
coordinate (Mateescu et a/, 1995a).
The central portion of length /* = a/ of the inner cylinder executes, in the plane () = O,
axially variable transverse oscillations with respect to the rest of the outer and inner
cylinders which are fixed, defined by
ae(x, t) = aeE(x) cos Qt = Re{aeE(x) exp(iQt)},
(11.207)
where Re{} denotes the real part, e is the nondimensional amplitude relative to the
radius a (assumed small) at xl/ = 1/2, Q the radian frequency ofthe osci11ations, and x
the nondimensional axial coordinate (also relative to the inner radius, x = x* la); E(x)
denotes the axial mode of oscillations, represented, for example, by the eigenfunctions of
a c1amped-clamped beam - see equations (11.161).
For the unsteady viscous incompressible flow generated by the osci11ations, the first of
the time-dependent Navier-Stokes equations, in the cylindrical coordinates ax, ar and (),

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1335
is given by
oV x l [ oVx o(UoU+v x ) veovx l O(P+P) ]
-+- (UoU+Vx)-+V r +--+-
ot a ox or r oe p ox
=  [ o2 vx +! ( O(UOU + VX) ) +  o2Vx ] ; (11.208)
a 2 ox 2 r or or r 2 oe 2
the other two are the same as (11.8) and (11.9), but with (U oU + ux)(ovr/ox) and
(UoU + ux)(ove / ox), respectively, addedin the first term in square brackets, and respect-
ively, o2v r /ox 2 and o2ve/ox2 in the right-hand-side square brackets; similarly, the
continuity equation is (11.10), but with the term r(ovx/ox) added. Here, V x , V r and
ve are the axial, radial and circumferential components of the perturbation velocity and P
the corresponding pressure; P (x) is the mean pressure, presumed to be known.
These governing equations are subject to the boundary conditions
V x = V r = Ve = O,
(11.209)
on the fixed parts of the cylinders, and
V x =0,
V r = aécose,
Ve = -aé sine,
(11.210)
on the oscillating centra1 portion of the inner cylinder, where é = de/dt = iQe(x, t).
It is recognized that the first of conditions (11.210) is only approximate, and that a more
precise formulation would be V x = -eUo(oU /or)!r=l cos e, but it is argued that for high
enough frequency and low enough amplitude this refinement is negligible.
For solving this unsteady 3-D flow problem, the annular space (ar, e) at any x = const.
plane is transformed into a rectangular computationa1 domain (Z, e), obtained by the
transformation
r-l
Z=1-2, e=e, h=b-l; (11.211)
cf, equations (11.13) and (11.14) and Figure l1.12(b), although in this case, since the
cylinders are concentric, h = const. instead of h(e),
The Navier-Stokes and continuity equations are transformed accordingly; for example,
the first Navier-Stokes equation becomes
oV x + ! [ Uo ( U oVx _ Vr OU ) +  Op ]
ot a OX h oZ p ox
=  { o2vx  [ o2 vx _ ...[jj 0v x D o2vx ] } ( 11.212 )
a 2 ox 2 + h 2 oZ2 oZ + oe 2
where D = h 2 /[2+(1- Z)h]2. In the computationa1domain, the followingpartial spectral
expansions are considered for the unsteady pressure and velocity components:
m
V x = aS1e ie int LUj(x)Tj(Z) cose,
j=O
m
Ve = aS1€ie int L Wj(x)Tj(Z) sine,
j=o
m
V r = aS1€i e int L Vj(x)Tj(Z) cos e,
j=O
m-2
P = pa 2 S1 2 e e int L Pj (x) Tj (Z) cose,
j=o
(11.213)

1336
SLENDER STRUCTURES AND AXIAL FLOW
where Tj (Z) represents Chebyshev polynomials of order j, andUj(x), Vj(x), Wj(x) and
P j (x) are unspecified functions of x. These semispectral expansions are used instead oť
the full ones, so as to avoid a substantial increase in the size of the fulI matrices associated
with the 3-D spectral colIocation approach, which might become nearly singular and
produce numerical difficulties. Thus, uti1izing (11.213), the Navier-Stokes and continuity
equations may be expres sed in the fonn
m
L {Uj(X) Tj (Z) + (41 h 2 )Uj(x)[Tj'(Z) - .fDTj(Z) - DTj(Z)]
j=O
-Re[U(Z)Uj(x) - (21 h)Vj (x)U'(Z)]Tj (Z) - i,8[U j (x) - Pj (x)]Tj (Z) } = O,
m
L {Vj (x) Tj (Z) + (41 h 2 )[Vj(x)(Tj'(Z) - .fDTj(Z» - 2D(Vj(x) - Wj (x»T j (Z)]
j=O
-Re U(Z)Vj (x) Tj (Z) - i,8[Vj (x) Tj (Z) + (21 h)Pj(x)Tj(Z)]} = O, (11.214)
m
L {Wj(x)Tj(Z) + (41 h 2 )[W j (x)(Tj'(Z) - .fDTj(Z» - 2D(W j (x) + Vj (x»Tj (Z)]
j=O
-Re U(Z)Wj (x) Tj (Z) - i,8[Wj(x) - .fD(21 h)Pj (x)]Tj (Z) } = O,
m
L {Uj (x) Tj (Z) - (21 h)[Vj(x)Tj(Z) - .fD(Vj(x) - Wj(X»Tj(Z)]} = O,
j=O
where Re = UoalIJ and ,8 = Qa 2 Iv are the Reynolds and the oscil1atory Reynolds
(or Stokes) numbers, and where Pj(x) and T;(Z) denote the derivatives with respect
to x and Z, respectively, and U'(Z) = (-21 h)[dU Idr]r=1+h(1-Z)/2.
ln order to avoid numerical difficulties caused by a large size of fulIy populated matrices,
encountered in a comp1ete 3-D spectral collocation approach, the axial derivatives are
discretized by a finite difference approach. Hence, the physical domain is discretized into
a number of grid points Pkj (Xk, r j), obtained at the intersection of planes of constant Xk
with cylinders of constant r j. In the computational domain, the corresponding grid points
(x K, Z J) fonn a rectangular grid, where K E {1, n} and J E {I, m - I}.
For axial discretization, equations (11.214) may be expres sed in the generic form
Fj'(x) - Re U(Z)Fj(x) + £j(x, Z) = O,
(11.215)
where the first two tenns, of interest for the axial discretization, represent the diffusion
and convection tenns associated with Fj(x), which can successively be Uj(x), Vj(x) or
Wj (x). Equation (11.215) has to be discretized at the grid points XK and ZJ. This is done
via a mixed central-upwind finite difference scheme (Spalding 1972):
Fj'(XK) - Re U(ZJ)Fj(XK) = aK+l,JFj,K+l + aK-l,JFj,K-l - aK,JFj,K,
(11.216)
whereaK+l,J = [2-(1+o)ReK+l,J ]j(D.xK+ID.), aK-l,J = [2+(1-o)ReK,J ]/(D.XK D.),
D. = D.xK+D.xK+l, aK,J = aK+l,J+aK-l,J, ReK,J = UoU(ZJ )aD.xK Iv, Fj,K = Fj(XK);

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1337
ReK,J represents the mesh Reyno1ds number at the grid point (XK, ZJ), whi1e 8 is ° or-1
according1y as ReK+l,J is smaller orlarger than 2. When ReK+l,J < 2(8 = O), equation
(11.216) reduces to a central differencing scheme for both the diffusion and convection
terms; this type of discretization 1eads to well-converged numerica1 soIutions. However,
this mesh Reyno1ds number requirement is often too restrictive, requiring a very fine grid;
this restriction can be re1axed by considering ReK+l,J > 2, with 8 = - 1 to prevent the
coefficient aK+l,J becoming negative. In this case, equation (11.216) corresponds to an
upwind differencing scheme for the convective term and central differences for the dif-
fusion term; this upwind differencing introduces an artificia1 dissipation which stabi1izes
the numerica1 scheme for higher mesh Reyno1ds numbers.
The mixed finite difference discretization (11.216) for the axia1 derivatives and a co1-
10cation approach in the radia1 direction are used to discretize the Navier-Stokes and
continuity equations at every grid point (x K, Z J) in the form
m
I)aK-l,JUj,K-l - (aK,J + bJ)Uj.K + aK+l,JUj,K+l + cJ Vj,K]
j=O
m-2
+ L[gJ,K-l Pj,K-l + gJ,K Pj.K + gJ,K+l Pj,K+tJ = °
j=o
(11.217)
for the first equation, and simi1ar forms for the other three; al1 the coefficients of
the unknowns Uj.K, Vj,K, Wj,K, and Pj,K are determined from equations (11.214).
These equations are subject to the boundary conditions imposed at the collocation points
(XK, Z = ::!:l):
m
L {Uj,K' Vj,K, Wj,K}T Tj (Zlb) = {O,O,O}T,
j=o
(11.218)
m
L {Uj,K, Vj,K, Wj,K}T Tj(Zmb) = {O, E(XK), -E(XK)}T,
j=o
respective1y, on the fixed parts of the inner cy1inder (Z lb = 1) and on the outer cy1inder
(Z lb = - 1), and on the moving part of the inner cylinder (Zmb = 1). The full set of
the equations, such as (11.217), for K E {l, N} and J E {l, m - I} and the boundary
conditions (11.218) for K E {I, N} togetherform an aIgebraic systemof equations, which
can be expressed in the matrix form as
A:F = B,
(11.219)
where:F = [Uj.K, Vj,K, Wj,K, Pj,K]T represents the vectorofthe unknown coefficients
ofthe ve10city components and unsteady pressure (with j E {O, m} and K E {l, n}), 13 is
the known right-hand side of the equations (nonzero only for the boundary conditions on
the oscillating cylinders), and A is a block-tridiagonaI matrix ofthe system. The numericaI
soIution for the unknown coefficients Uj,K, Vj,K, Wj.K and Pj,K is obtained using an
LV decomposition technique.

1338
SLENDER STRUCTURES AND AXIAL FLOW
The unsteady fluid dynamic force acting on the cy1inder at axia110cation ax is
i 211' [( J.Lavr ) J.L ( ave Ve 1avr ) .
F(x,t) = a -p+2-- cos O -- -+-+-- smO
O a ar a ar r r ao
J.L ( av x avr ) ac ] d
+ - - + - - O,
a ar ax ax r=l
(11.220)
where the last term in the square brackets makes no contribution to the over-
all force. The unsteady component of the force may be expressed as F(x, t) =
7Tpa 3 Q2 e EGI)ft(x) exp(iwt), and the nondimensiona1 fluid dynamic force F(x) is
m { . [ 8 m-I ] }
F(x) = -  Pj(x) + Ji Wj(x) - Vj(x) + hc '  q (Vq(x) - Wq(x)) ,
)=0 ) q=)+l
(11.221)
where Co = 2, c j = 1 for j 2': 1, q = j + 1 and q + j = odd.
This method has been further improved and applied to cases of eccentric cy1inders by
Mateescu ct al. (1995b).
Typica1 results for prescribed motions of the centrebody, which is a c1amped-c1amped
cylindrica1 beam between semi-infinite rigid extensions, are shown in Figure 11.73, for
both the concentric and eccentric systems. Re Dh is the Reyno1ds number based on the
hydraulic diameter, Dh = 2a(b - 1), The form ofthe graphs for Rc{F} versus xl I shows
that the oscillation is in the first beam mode. The quasi-sinusoida1 form of Im{F} for
Re or Re Dh i= O is characteristic of the presence of a travel1ing wave component in
the oscillation; when Re Dh = O, then Im {F} is s1ightly negative over the whole length,
O<xll<1.
It is noted that with increasing tJ, essentially increasing frequency, both the rea1 and
imaginary parts of the force are reduced, for both systems. A1so, for the eccentric system
it is seen in Figure 11. 73(b) that increasing the Reyno1ds number, Re Dh' resu1ts in 1arger
va1ues of the fluid-dynamic force.
Part1y to va1idate this theory, a 'semi-analytica1 model' has a1so been constructed (Sim
1991; Mateescu ct al. 1995a), in which the viscous forces are approximated ana1ytically
following the same procedure as in Mateescu & Pa'idoussis (1987), Section 11.5.2(c), but
using the exact spectra1 solution for the unsteady potentia1 flow rather than an approximate
potentia1 solution. Thus, this semi-analytica1 model represents an improvement over the
ear1ier ana1ytica1 method of Section 11.5.2(c).
An interesting comparison is presented in Figure 11.74 between various versions of
the theory: (a) between the semi-ana1ytica1 model and the ear1ier ana1ytica1 one, and
(b) between the semi -ana1ytica1 model and the hybrid method presented in this section, ln
(a) the two resu1ts are quite c10se, whereas in (b) it is seen that the ear1ier theory (suitable
for very narrow annuli), underestimates the more accurate results obtained by the hybrid
spectrallfinite difference method.
11.5.5 Numerical methods: dynamics
This section dea1s with numerica1 studies of the dynamics and stability of cy1indrica1
structures in annu1ar flow; supporting experiments are a1so reviewed, Clear1y, therefore,

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1339


<::;,.
 2
;:::: 4

<::;,. 2
I
'f3 O
-2
6
4
o
6
-4
-O 2 O O 0,2 0.4 0,6
(a) xii
6

 3
<::;,.

O
6
 O
<

-6
12
(b)
Figure 11.73 Influence of {3, and in (b) of ReDh also, on the real and imaginary parts of the
dimensionless fluid-dynamic force ft produced by first-beam-mode oscillation of tbe clamped-
clamped centrebody, for b = 1.25, / = 15. (a) Concentric system, Re == Uoa/v = 1252 : 0, {3 =
500; !:::,., {3 = 5000; 0, {3 = 10000, (Mateescu et a/, 1995a), (b) Eccentric system, erel = 0.4;
(i) oscillations in tbe plane of symmetry: e. ReDh == 2a(b - I)Uo/v = O, {3 = 500; ., ReDh =
1250, {3 = 500; ., ReDh = 1250, {3 = 5000; _, ReDh = 2500, {3 = 500; (ii) oscillations normal
to tbe plane of symmetry: !:::,., ReDh = 1250, {3 = 500 (Mateescu et al. 1995b).
125
100
E 75
<
 50
25
<"
00
4
E 2
<
 O
I
(a)
-2
000
025
050
xii
075
25
1.00
15
0.25
(b)
050
x/l
075
100
Figure 11.74 Comparison of the unsteady dimensionless fluid-dynamic force, ft, computed
according to various tbeoretical models, for / = 15 and {3 = 5000. (a) b = 1.1, Re = 2000:-,
semi-analytical model of Mateescu, Paidoussis & Sim (1995a); - - -, Mateescu & Pai'doussis
(1987) analytical model. (b) b = 1.05, Re = 3000 : -e-, hybrid spectral method ofthis section;
-, semi-analytical model (Mateescu et al. 1995a).

1340
SLENDER STRUCTURES AND AXIAL FLOW
the work discussed here is dosely re1ated to that in Section 11.5.4. It is also related, in
terms of the physica1 systems considered, to the predominantly ana1ytica1 work presented
in Sections 11.5.2 and 11.5.3.
11.5,5(a) The Bélanger et al. model
A computationa1 method for the dynamics of cy1indrica1 structures subjected to laminar
annu1ar flows by simultaneous integration of the Navier-Stokes and structura1 equa-
tions has initially been presented by Pai'doussis et al. (l992b). It has subsequently been
improved, notab1y by app1ying the nonpenetration condition on the moving boundary
(thus retrieving the fluid-dynamic stiffness terms), in Bé1anger et al. (1994) and further
discussed for 1arge added mass by Bé1anger et al. (1995); the following is based on these
1atter two papers.
The method is based on that of Section 11.5.4(a) and is app1ied to geometrie s such as
those in Figure 11.75. Thus, the Navier-Stokes and continuity equations for a 1aminar
flow of incompressible fluid are written in nondimensional form as
au
at + G(U, p) = O,
v.U=o,
(11.222)
40H
t.:::::::::::::::. :. :. :. :. :. :. :. :. :. :. :::. :. :. :. :. :. :. :. :. :. :. :. :. :. :. :. :. :. :. :. :. I
.1
I .
.1
10
10
(a)
I 40H
: ---------------------___________________ J
-----------------------------------------
E(I)
!
-ř-
(b)
10
.1
Figure 11.75 The annular geometrie s used to study the stability ofthe vibrating central portion of
the outer cylinder: (a) geometry with unifonTI annular passage (of width H in this figure; geometry
of width 2H, not shown, was also considered); (b) 'backstep' geometry, with a step-change in the
annular passage width (Bélanger et al. 1994).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1341
where
1
G(U, p) = V. uu + Vp - -V 2 U.
Re
(11.223)
Re is the Reyno1ds number, U the velocity vector of the fluid, and p the pressure. Taking H
and U to be the characteristic 1ength and ve10city, and denoting by an asterisk same-symbo1
dimensiona1 quantities, the following nondimensiona1ization scheme has been used:
U*
U=-,
U
p*
p = pU2 '
Ht*
t=-,
V
v = HV*,
VH
Re = -. (11.224)
v
Assuming the solution for the fluid to be known at time 1evels t k , k f= n, where the time
increment is given by /j.t = t n - t n - 1 , the following two second-order discretizations for
the time derivative,
(  r+1/2 =
8U n + 1 / 2 _ 9U n + U n - 1
3/j.t
(  r+1
3U n + 1 _ 4U n + U n - 1
2/j.t
(11.225)
may be used to advance the solution from t n to t n + 1 / 2 or from t n to t n + 1 , respectively;
both wi11 actua11y be required in the integration scheme adopted for the structure, to be
introduced shortly. The semidiscretized momentum equation and the continuity equation
become, after introducing (11.225) in (11.222) and rearranging,
U n + 1 / 2 + (3/j.t/8)G n + 1 / 2 = (9U n _ U n - 1 )/8,
U n + 1 + (2/j.t/3)G n + 1 = (4U n _ U n - 1 )/3,
v . U n + 1 / 2 = O,
V . U n + 1 = O
,
(11.226)
(11.227)
where G n + 1 / 2 = G(un+1/2, pn+1/2), G n + 1 = G(U n + 1 , pn+1). [To be abso1utely clear,
here and in what follows, the superscript n + 1/2 is n + i.] Solutions to the implicit systems
of nonhomogeneous equations (11.226) or (11.227), respectively, for (U n + 1 / 2 , pn+1/2) or
(U n + 1 , pn+1), may then proceed using the methods described in Section 11.5.4(a).
No-s1ip boundary conditions are imposed in solving equation (11.226) or (11.227). This
implies that the ve10city of the wall must be known at t n + 1/2. However, it is noted that in
a numerica1 fluid-structure procedure, the disp1acement of the wall must a1so be known in
advance to solve (11.226), as it serves in redefining the fluid domain in which the solution
is defined at t n + 1 / 2 . Hence, for the present purposes, we consider that the fluid solution
at t n + 1 / 2 and, in particu1ar, the fluid force pn+1/2 acting on the structure are functions of
the disp1acement and velocity ofthe wa11s at t n + 1 / 2 , that is, Fn+1/2 == p(sn+1/2, é n + 1 / 2 )
where genera11y s and é are some moda1 wa11 disp1acement and velocitř' respectively.
The considerations of this paragraph a1so app1y in going from t n to t n + when solving
equation (11.227).
The mechanica1 system consists of a portion of the outer cylinder that is free to vibrate
transversely as a rigid body in one plane (Figure 11.75), its disp1acement being denoted
by s*(t*). This cy1inder is restrained by a spring K and a dashpot C; it is of1ength 40H,
radiu s 10 H and mass M, and is de1imited upstream and downstream by two fixed cy1inders

1342
SLENDER STRUCTURES AND AXIAL FLOW
of radius 10H and length 10 = 40H. The inner cylinder, of radiu s 9H, is stationary.
Hence, the equation of motion of the vibrating cylinder may be written in dimensionless
fonn as
£(t) = -2ú)n£(t) - ú)e(t) + a F(e, k),
(11.228)
where
e*
e=-,
H
ú) = (K H == ú)* H
n VMU nU'
C
 = 2 ../KM '
(11.229)
pH 3
a=-
M'
F*
F= ,
pU2 H2
A dot denotes a time derivative, Ú)n and  are the dimensionless natura1 frequency and the
reduced damping of the system, and a is a factor weighting the relative contribution of
the fluid and mechanica1 forces, ln keeping with the discussion above, the fluid-coupling
force exerted on the cylinder, F(e, k), has been expressed as a function of e and £, which
is symbolic of the way in which it is obtained numerically; however, in fact, F is a1so
a function of £. In the calculations to be presented C = O is taken ( = O); so, only
flow-induced damping, if any, is considered, for clarity.
Assuming e n , t n and sn to be known at time t n , a time-marching solution is effected
by means of a second-order Runge-Kutta integration scheme involving the following
predictor-corrector sequence.
Predictor step:
£n+1/2 = £n + (Mj2)tn, £n+1/2 = £n + (Mj2)sn,
Fn+1/2 = F(e n + 1 / 2 , t n + 1 / 2 ), e n + 1 / 2 = -2ú)ntn+l/2
_ú)en+1/2 + a Fn+1/2,
(11.230)
Corrector step:
e n + 1 = £n + /j.t t n + 1 / 2 ,
t n + 1 = t n + /j.t e n + 1 / 2 ,
Fn+l = F(en+1, £n+1),
£n+1 = -2ú)ntn+1 _ ú)en+l + a Fn+1.
(11.231)
As seen in the predictor step, intennediate va1ues for displacement and velocity are first
determined at the intermediate time level t n + 1 / 2 = t n + /j.t /2; these are e n + 1 / 2 and t n + 1 / 2
which, once so obtained, serve in the ca1culation of the boundary conditions required to
integrate the Navier-Stokes equations, The latter can then be advanced to t n + 1/2, indend-
ently ofthe structural problem, allowing for the ca1culation ofthe fluid force, F n + 1 / ,The
acceleration, sn+1/2, ofthe stmcture is then obtained from the last equation in (11.230),
and the solution can proceed on to the corrector step which is solved in the same manner
for the variables e n + 1 , t n + 1 , F n + 1 and sn+1. '

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1343
0.3
Tn
0.2
 .....
 0.1 "
'"
0,0
-0,1
(a)
0.10
Tn
0,)0
Tn
0,05
\
\
\
\
\
\ Ij
\ li
\':.:.....
...............
/.'-- ....--.:.""'..
\\ ' .::.." ",
\\_//
........
0.00
-0.05
-0,10
(c)
03
0.05
0,2
Tn

 0,00
'"
01
-0,05
0.0
\
\
\
\
"
/
/
/
/
..
.. ..-
........---
----
-0.10
O 2
(b)
4
6
t*(s)
8
10
O.1
12 O 2
(d)
4
6 8
t*(s)
10
12
Figure 11.76 Displacement of the vibrating outer cylinder as a function of time. (a,b) The
effect of increasing U, (a) for a uniform geometry and (b) for the backstep geometry: - ,
Re = 100, úJ n = 0,12; - - -, Re = 200, úJ n = 0.06; ..., Re = 300, úJ n = 0,04, (c,d) The effect
of the velocity 'profile' on the dynamics: (c) -, Re= 100, úJ n =0,12, uniform geometry
of width H; - - -, Re = 100, úJ n = 0,12, stepped geometry; '" , Re = 200, úJ n = 0,24, uni-
form geometry width 2H; (d) - , Re=300, úJ n =0.04, uniform geometry of width H;
- - -, Re = 300, úJ n = 0,04, stepped geometry (Bélanger et al. 1994),
Typical results are shown in Figure 11.76(a,b), showing the effect ofincreasing the flow
velocity; here it is noted that, because of expressions (11.229), increasing U results both in
larger Re and smaller W n . It is seen in Figure 11. 76(a) that the uniform system is stable and
overdamped at Re = 100 and 200, but unstable by divergence for Re = 300. The system
with a back-step geometry, however, remains stable and underdamped [Figure 11.76(b)];
this reflects the fact that, over half of the cylinder, the flow velocity is half of the value
upstream.
The effect of the shape ofthe annulus is further explored in Figure 11.76(c) and (d). In
(c) the mean upstream flow velocity is the same, but in one case (dotted line) the reduced
diameter segment extends over the whole of the moving outer part (annular width 2H,
Re = 200); in that case, the system behaves very much like that with the step further
downstream, The most highly damped case is that of uniform width H, because of the
larger added mass.
The results in Figure 11.76 are for M / Ma = 2, where Ma is the dimensional added
mass. It is possible, however, for narrow enough annuli and hollow centrebodies, for this
ratio to be much smaller. It has been found (Bélanger et al. 1995) that, when M / Ma < 1,
the time-marching scheme just described becomes numerically unstable. A procedure has
been devised to fix this anomaly. It consists of adding an estimate of the added mass to
the structural mass, and then subtracting the equivalent inertia force from the right-hand

1344
SLENDER STRUCTURES AND AXIAL FLOW
side of equation (11.228), which thus becomes
(1 + a Ma)'i(t) = -2ú)nt(t) - ú)s(t) + a ft(s, t, e),
ft(s, 8, e) = F(s, t) + Mae,
(11.232)
where Ma is an estimate of the actual dimensionless added mass M; / p H 3 . This is
referred to as the 'fluid inertia compensated scheme' (PIC), as opposed to the original
'non-compensated scheme' (NPlC). The predictor/corrector steps, equations (11.230)
and (11.231), according to PIC are now modified, as follows.
Predictor step
sn+l/2 and t n + 1 same as before; ft n + 1 / 2 = F(sn+l/2, t n + 1 / 2 ) + Ma e n + 1 / 2 ,
e n + 1 / 2 = [1/(1 + a Ma)][-2 Ú)n t n + 1 / 2 - ú) sn+l/2 + a ft n + 1 / 2 ].
(11.233)
Corrector step
sn+l and t n + 1 / 2 same as before; ft n + 1 = F(sn+l, t n + 1 ) + Ma e n + 1 ,
e n + 1 = [1/(1 + a Ma)][-2 Ú)n t n + 1 - ú) sn+l + a ftn+l].
(11.234)
Here F(sn+l/2, t n + 1 / 2 ) and F(sn+l, t n + 1 ) are obtained as before from the Navier-Stokes
equations, whereas Maen+1/2 and Maš n + 1 are calculated by considering the following
expressions for the acceleration, used, respectively, in the predictor and the corrector step:
e n + 1 / 2 = (8t n + 1 / 2 _ 9t n + t n - 1 )/(3 M),
sn+1 = (3t n + 1 _ 4t n + t n - 1 )/(2 t),
(11.235)
which are consistent with expressions (11.225) for the fluid acceleration. This new scheme
has been found to stabilize the numerical in stability.
Typicalresults are shown in Figure 11.77, ln (a), it is seen thatfor M / Ma = 2, theresu1ts
by the PIC and NPlC schemes are essentially identical. In (b), however, it is seen that for
M / Ma = 2/3, whereas the PIC scheme predicts the system to be stable, the NPlC scheme
predicts it to be unstable by divergence - a spurious result due to numerical in stability,
Lest the reader think that alt such systems are stable, another calculation by the FIC
scheme is also shown in Figure 11.77(b), for M / Ma = 2/9, where we see a divergence
instability.
It is noted that, in all the results presented, the only fonn of in stability is divergence. This
appears to be trne for lateral translational motions, though flutter is possible for rotational
(rocking) motions, as seen in the following two subsections.
Additional work, comparing the mean-position (MP) analysis as in the foregoing, and
time-dependent coordinate transformation (TDCT) analysis of Section 11.5.4(b) has been
done by Mekanik (1994): (i) for lateral motions of unifonn and backstep geometries
(by both analyses); (ii) for lateral motions of diffuser geometries (by MP analysis), and
(iii) for rocking motions of unifonn geometries (by MP analysis). In these, yet unpublished
calculations, confined to Re = 200 and Ú)n = 0.1 with Ú)n as in (11.229), the systems
considered remain stable, yet differences between the MP and TDCT resu1ts can be large.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1345
0,10
0,05
S 0,00
'"
-0,05
-0,10
°
(a)
0,3
....,/./...//
0,1 ....':.::.
..
0,2
0,0 ---- ----- -
-0,1 I
5 10 15 20 ° 5 10 15 20
t (b) t
Figure 11.77 Displacement versus time for Re = 50, úJ n = 0.12, for various M / Ma values.
(a) M/Ma = 2: -, NPIC results; - - -, PIC results, (b) M/Ma = 2/3: -, NPIC results;
- - -, PIC results; M / Ma = 2/9: ' , ., FIC results (Bélanger et aZ. 1995),
600
 500

, 400
 300
Oj
g 200
<
100
O
O
0

- A 
-
2 3 4
Flow velocity (mis)
5
Figure 11.78 Typical experimental result for a rigid, flexibly supported cylinder, in a 6° diffuser,
showing the occurrence of flutter involving generally orbital, irregular motions; 1 gal = 1 cm/s 2
(Pujita et aZ. 1994).
11.5.5(b) Other, semi-empirical models
Several other studies have been conducted, generally in turhulent ftow, with abrupt changes
or with diffuser segments in the annular passage, necessitating some degree of empiricism
in the model.
Dne such study is by Fujita et aZ. (1994), who reexamined essentially the problem first
studied by Hobson (Section 11.5.1) both theoretically, in a finite element framework, and
experimentally. CFD modelling is used for some parts ofthe study for Re up to 30 000. The
CFD technique used is similar to that described in Section 11.3.3(b), but more complex,
also making use of the k-s model. In the experiments, a rigid rod, ftexibly supported by
thinner long struts on either side, was used in annular water ftow. A typical example is
shown in Figure 11.78, showing the occurrence of ftutter at U = 3.6 mis, This compares
with a theoretical value of U  5 mis, which for this type of problem is not bad, except
for the worrying feature that prediction is not conservative,

1346
SLENDER STRUCTURES AND AXIAL FLOW
10
8
,-.,
 6
t! 4
C1
2
°0
(a)
10
8
,-., 6

- 4
ci
2
°
0,0
(c)
'
'...
....,- ---
----------
.
.
-
-
2 3
cp (deg)
4
Icp =401
------------
----
-----
-
,
0,1
0,2
0,3
0.4
e
10
8
6
4
2
°
5 °
(b)
10
8
6
4
0.5
2
°
0,0
(d)
2 3
cp (deg)
4
5
ICP=40
.
.
--- :.-- --.- -- T--- ..---_
-
0.1
.
0.2
0.4
0,5
0,3
e
Figure 11.79 Critical flow rate, Qcr, for flutter of a rigid cylindrical eentrebody, free to undergo
translational and rotational motions in a tapered (diffuser) duet: (a,b) the effeet oftaper angle, fP, for
two eylinders with different eentre of mass; (e,d) the effeet of eeeentrieity e (nondimensionalized
by Ho) for fP = 4°, for the same two eylinders. - , _, Theory and experiment eorresponding to
the first eigenvalue; - - -, ., theory and experiment for the seeond eigenvalue (Li cf a/, 2002).
A model very similar to those by Porcher & de Langre (1997), Seetion 11.4.3, and
Arai & Tajima (1999), Section 11.5,2(h), has been developed by Li cf al. (2002) and
solved numerically by a colloeation method. They a1so conducted experiments. Some
typica1 results, comparing prediction with experiment are shown in Figure 11.79, It is
seen that agreement is excellent for concentric configurations and very good for eccentric
ones, Jt is of great interest that for the coaxia1 system, when ({J = O (uniform cylindrica1
annulus) the system is stable; however, even a very slightly divergent annulus is sufficient
to cause the system to flutter; moreover, a1though increasing ({J from 1 ° to 2° reduces Qcr
significant1y, Qcr is not greatly changed for ({J > 2°, as seen in Figure 11.79(a) and (b), It
is also seen [Figure 11.79(c) and (d)] that, at least for such small va1ues of ({J, the effect of
eceentricity in the position of the centrebody can be sma1l.
11.5.5(c) The MOCCA code
The MOCCA code is based on a linearized unsteady model for the fluid-structure inter-
action between an annular flow and a vibrating structure (the centrebody or outer duet).
This model dea1s with an incompressible viscous fluid flow at high Reynolds number in
a finite-length annular region ofaxia1ly variable cross-section. It has been developed by
Electricité de France, Chatou, as an extension of their N3S code, which is a CFD code
for 3-D turbulent flows (us ing the finite element method and a k-s turbulence model) in
contact with motionless structures; MOCCA, on the other hand, can deal with moving
boundaries. Unlike the model s presented in subsection (a), in this model the flow can be
turbulent. The basic philosophy behind it is similar to Hobson's (Section 11.5.1), but it is

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1347
(a)
y
x
Centre-body in its
initial position
(b)
Figure 11.80 The annular system under consideration and the coordinate system used for the
MOCCA model (Perotin & Granger 1997).
more general and sophisticated, and less empirical. Here this code and its application to a
specific problem are presented, following dosely Perotin & Granger (1997).
The system under consideration is shown in Figure 11.80; although the analysis is
developed for an oscil1ating centrebody, it can equally apply to an oscil1ating duct. The
basic equations are the same as (11.222) and (11.223), ar
div V = O,
av 1 1
- + v. V V = - - V p + - div T,
at p p
(11.236)
where V is the flow velocity, p the pressure, p the fluid density, and T the stress tensor,
For turbulent flow, using the k-s model, the stress tensor may be expressed as T = (f.L +
f.Lt)(VV + VV T ), where f.Lt is the eddy viscosity [Section 2.2.1(£)]. The wan boundary
conditions, specifically the nonpenetration and no-slip conditions, are
V'D = Vw'D and T'D = T*(V*), with T*'D = O;
(11.237)
the first is retained as such, but the second is replaced by a nonlinear equation relating the
wan shear stress, T*, to the flow velocity outside the boundary layer, V* (Gunzburger &
Nicolaides 1993).
The method of integral relations, originally fonnulated by Dorodnitsyn (Holt 1984;
Zwil1inger 1989) is then used. The basic idea ofthis method is to integrate the goveming
partial differential equations, with respect to one of the independent variables. The original
equations can thus be replaced by a system of equations containing independent variables
reduced by one. To this end, the unknowns are linearly expanded using arbitrary interpol-
ation functions (polynomials, trigonometric expressions, etc.) of the integration variable.
Solving the system then reduces to identifying the coefficients of the above-mentioned

1348
SLENDER STRUCTURES AND AXIAL FLOW
r = r f u r w
r w: Control surface associated
with the inner and outer
cylindrical structures
r2 D r f : Control surface in the
fluid domain
Figure 11.81 Control volume Q and control surfaces r w and rf used in the method of integral
relations (Perotin & Granger 1997).
expansion. The integration domain may be divided into several subdomains. This method
allows us to take into account the boundary conditions in the integrated fonnulation, as
well as possible boundary motions.
When applied to the system at hand this method is equivalent to integrating equa-
tions (11.236) over the control volume Q shown in Figure 11.81; r f denotes the control
surface in the fluid domain, and r w the control surfaces associated with the inner and outer
cylindrical structures. Gauss' divergence theorem and boundary conditions (11.237) then
yield:
mass conservation
[ V.ndy = - [ Vw.ndy;
Jrf(t) Jrw(t)
(11.238)
linear momentum conservation
[ av dw +[ V(V.n)dy+[ £'ndy-[ 2. í . ndy
JQ(t) at Jrf(t) JrfUrw(t) p Jr} (f) P
= [ 2. í *dY- [ Vw(Vw.n)dy. (11.239)
Jrw(t) P Jrw(t)
ln cylindrical coordinates, equation (11.238) and the z-component of (11.239) are:
 [ (2 r d r ] +  [ (2 Ve d r ] +  [ (2 r V z d r ] = 0,
at Jr) ao Jr) az Jr)
 [f r 2 r V z d r ] +  [ (2 Ve V z d r ] +  [f r 2 r V z 2 d r ]
at r] ao Jr] az r]
1 a [f r 2 ] 1 ar! [ 1 ar2 I
+ -- prdr + -TI-P - -r2-p
P az rl p az rj P az r2
1 a [f r 2 ] 1 a [f r 2 ]
+ - ao 'Cez dr + - a r 'C zz dr ;
P r] P Z r,
(11.240)
1 * 1 1 * I
= -r2 'C z + - TI 'C z
P r2 P r]
(11.241 )

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1349
a similar equation to (11.241) is obtained for the e -component. The radial component, as
in foregoing analyses, is neglected.
It is next assumed that the vibratory amp1itudes are small, such that a linear stability
analysis can be performed. Thus, it assumed that
V(r, e, z, t) = V(r, e, z) + u(r, e, z, t), per, e, z, t) = per, e, z) + per, e, z, t),
(11.242)
with lul « I U I and lpi « l P I. In this expression, (u, p) represent the motion-induced
velocity and pressure perturbations, while (V, P) denote mean turbulent flow velocity and
pressure fields associated to a fixed fluid domain which has sti1l to be determined.
To proceed further, we transform the time-varying physical domain, V(t) = [rl (t),
r2 (t)] x [O, 2n] x [O, L] into the fixed computational domain, Vo = [-1, + 1] x [O, 2n] x
[O, L]. This is achieved by defining the variable 1], via
h
r = ro + 1]-,
2
(11.243)
with ro =  (rl + r2), h = r2 - rl; 1] 1ies between -1 and + 1 when r varies between
rl and r2. Consequently, substituting equation (11.243) into (11.240) and (11.241) et seq,
yields, in terms of variables (1], e, z), an equivalent set of equations associated with the
fixed computational domain V o . For example, equation (11.240) becomes:
a (roh) a ( h j I ) a ( h j I ( h ) )
- + - - Ve d1] + - - ro + 1]- V z d1] = O,
at ae 2 -1 az 2 -1 2
(11.244)
with similar expressions for the linear momentum conservation equations.
Taking (11.242) into account, the new set of conservation equations can be decom-
posed into a set of first -order equations for the mean flow, and one of second-order for the
motion-induced perturbations. For example, equation (11.244) yields:
first order (steady part)
 [ H J I Ue d1] ] +  [ H J I ( RO + 1] H ) U z d1] ] = O;
ae 2 -1 az 2 -1 2
(11.245)
second order (unsteady part)
_ aií - aro H a [ 1 ]
Ro- + H- + -- j ued1]
at at 2 ae -1
+  [ H J I ( ro + 1] ) U z d1] + H J I ( R O + 1] H ) uzd1] ]
az 2 -1 2 2 -1 2
+ :0 [HlI V Od"] + :z [Hli ( Ro +"  )V'd"] o; (1[24Q)
in these equations, roce, z, t) = R o(z) + roce, z, t), h(e, z, t) = H( z) + ií(e, z, t).
A similar set of equations are obtained from the transformed form of equation (11.241) and
its e -counterpart.

1350
SLENDER STRUCTURES AND AXIAL FLOW
The first-order equations are solved by the N3S or any other similar code. Before
solving the perturbation equations, they are further simplified for narrow annular gaps, as
follows: (i) the influence of u r is neglected as compared to ue and uz, and similarly for the
corresponding integrals from r1 to r2; (ii) the radial variation of each velocity and pressure
perturbation component is assumed to be identical to its first-order counterpart; for UB, the
r-variation is supposed to be identical to the U z and U z profile. Hence, equation (11.246)
may be rewritten as
- - [ 1 ] +
-aro - ah H aUe
H - + Ro - + - 1 cjJz d1J -
at at 2 -1 ae
+ :z [ r; L ( H (ro + ") + h ( Ro + "  ) ) 'd"]
[ - 1 ( - ) ]
a H - H _+
+ az 2i1 RO+1J 2 cjJ z d1J U z
[ H 1 +1 ( H ) ] au+
+ - Ro + 1J - cjJz d1J  = o,
2 -1 2 az
(11.247)
where
Ve(1J, e, z, t) = [ut(e, z, t)]cjJz(1J, z), V z (1J, e, z, t) = [ U ; (z) + ui(e, z, t)]cjJz(1J, z),
cjJz being the velocity distribution, and similarly for V r and p. Similar expressions for the
linear momentum equations may be obtained and are given in Perotin & Granger (1997).
Conceming the viscous and turbulent stresses, a simplified approach is taken. Viscous
and turbulent stresses inside the fluid domain are neglected in the second-order linear
momentum conservation equations, and frictional shear stresses on walls are modeled via
a quasi-steady expansion of the usual skin friction law for pipe-flow (Idelchik 1986),
T* = -ip CflVI V,
(11.248)
ln this expression V is the flow velocity outside the boundary layer and Cf is the friction
coefficient on the wall, which depends on Reynolds number, Re = 2 H IVI/v. Expanding
Cf into a steady and an unsteady part, equation (11.248) yields
-* O -* 1 C I U I -
T e =, Te = -2 P F ue,
'f; = -ip CF I U I Uz , i; = -ip [CF u U z + CFIUI U Z + ČF IUI U z ],
(11.249)
where C F represents the friction coefficient resulting from the mean turbulent flow and ČF
is the part of the friction coefficient due to the motion-induced flow perturbation; I U I =
[U; + U; ]1/2 and u = (U dlU!) uz' Let CF be expres sed as CF = CFo Re m (Idelchik

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1351
1986), with m = m ( Re ) and C Fo = C Fo ( Re ); CF can be deduced from the expression for
C F by making a quasi-steady approximation of cf. We thus obtain
CF = m CFo ReRe m - 1 = m C F [ Uzl I U I 2 ] u z .
(11.250)
As the system of perturbation equations is linear, a Laplace transformation is introduced,
replacing time by the transform variable s. Also, assuming that motion takes place in the
(x, z)-plane and taking advantage ofaxisymmetry, we can express
h(f), z, s) = h+(z) cosf) q(s),
Ťo(f),z,s) = r(j(z) cosf)q(s),
where h+ and r(j are deduced from the mode shape of the vibrating structure, and q(s)
is the modal amplitude of vibration. Hence, the mass conservation equation (11.247) and
the corresponding momentum equations may be written in terms of
ut(f),z,s) = ut (z, s) sinf)q(s), ui(f),z,s) =ui(z,s) cosf)q(s);
p+(f), z, s) = p+(z, s) cosf) q(s). (11.251)
Thus, eventually one obtains a set of three ODEs which may be written as
a
-{Y} + [A]{Y} + {B} = {O}, {Y} = {ut, ui, p+}T,
az
where [A] and {B} are complex and depend, the former on the mean flow variables only,
and the latter also on the particular mode shape. Once the inlet and outlet boundary
conditions are derived, these equations are solved by a fully implicit first-order finite
difference scheme.
The first two inlet-outlet boundary conditions are obtained from the conservation of
the kinetic energy within two control volumes located outside the annular region; for
example, Figure 11.82 shows the control volume for the inlet, modelled as a streamtube.
Thus, we have
(11.252)
tI (p+ PV2)V'ndY = t2 (p+ PV2)V'ndY +Cdi ti (PV2) V.ndy,
(11.253)
fo (impenneable wall)
f 1: Upstream end
f 2 : Inlet surface element
ofthe annular region
r2
h,r 1
dO \ "
,,'
Figure 11.82 Streamtube control volume used in deriving the inlet boundary condition (Perotin &
Granger 1997).

1352
SLENDER STRUCTURES AND AXIAL FLOW
for i = 2 here (i = 1 for the outlet case). At second order, this leads to
f, [ft + p(l + CDlo) U, U, + p ČDmI U I2] U, ( RO + O  ) d.  O fon  O;
L [ft + p(l- CDoo,) U, u, - p ČDood U I 2 ] U , (RO + O  ) dO = O for z  L.
(11.254 )
ln these expressions, C D in and C D out are the steady parts of the singular 10ss coefficients at
the inlet and outlet of the annular region, and CD in and CD out the corresponding unsteady
components,
The third boundary condition is obtained by assuming a sufficiently regular flow
upstream of the annular passage, such that
ue = O
for z = O.
(11,255)
We now consider motions of the inner cylinder of the form
N
rl(e, z, s) = Lr{;(z) coseqi(S),
i=1
(11.256)
where qi (s) represents the ith moda1 amplitude of vibration and r{;(z) is the mode shape
associated with the ith normal mode ofthe centrebody, The motion-induced pressure fields
are determined in terms of equations (11.251), with
N
{ut(z, s), ui(z, s), p+(z, s)} = L {ulJi(z, s), u(z, s), P/(z, s)},
i=1
(11.257)
where U()i, Uzi and Pi are deduced from solution of the fluid problem associated with
N
- "" +
hi(e,Z,S)=  -rli(z) coseqi(S),
i=1
N
_ ",,1 +
roi(e, Z, s) =  Zr li (z) cose qi(S),
i=1
(11.258)
Taking equation (11.249) into account, the generalized fluide1astic force associated with
the jth normal mode, Qj, can finally be expressed as
N
Qj(s) = L Bij (s) qi(S),
i=1
(11.259)
with
- l L [ + 1 - - + ] +.
Bij(s) = -Jr TI Pi <Pp + -p CFIUlu ei <Pz r 1j dz,
o 2 17=-1

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1353
Bi} denotes the generalized fluidelastic force associated with the jth mode, induced by a
unitary displacement of the ith mode. In matrix form, [B] can be viewed as the transfer-
function matrix relating the modal responses to the genera1ized fluidelastic forces. For
each mode of vibration, the modal mass Ml t , damping qt, and stiffness Kl t , of the
structure in stilI air are supposed to be known. Introducing
Mi} = Mt oi}, Ci} = qt oi}, Ki} = K i st oi},
(11.260)
where oi} is the Kronecker delta, the coupled fluid-structure system can thus be
characterized, in matrix form, by the folIowing equations:
[[M]S' + [C]s + IKI- [B(.,)]] LJ = o.
(11.261)
where it should be noted that [B] also depends on the flow velocity. For a given .flow
velocity, the solution of this generalized eigenvalue problem leads to the natural cir-
cular frequencies (Vi and the damping coefficients i, i = 1, ,.. , N, of the coupled
fluid-structure system.
This model has been tested against data from the TAXI experiment, described in
Section 11.5.2(g); special preliminary CFD ca1culations were necessary for determining
CD in and CDout quasi-staticalIy. The experimental data are compared with the MOCCA
predictions in Figure 11.83(a) and (c), and with Hobson's model in (b). It is obvious that
the MOCCA code is very successful, indeed; agreement with experiment is phenomen-
alIy good, and much better than that achieved by Hobson's model, and better than by the
Porcher & de Langre model (Figure 11.46). An apt question is: exact1y why?
The most important differences between the Hobson and MOCCA models have been
identified by Perotin & Granger as folIows. (i) Hobson's model assumes that the outer
cylinder consists of a series of constant diameter pipes separated by sudden expansions
or contractions, the centrebody being of constant diameter. At the junctions of the annu-
lar sections where the flow area changes, inter-element compatability conditions have to
be specified, obtained from quasi-steady approximations of the mass and kinetic energy
conservation equations which involve loss or pressure-recovery coefficients. The use of
such a quasi-steady modelIing within the time-varying annular region is highly question-
able. This procedure has not been retained in the MOCCA formulation, which handles
naturalIy the axial variation of the annular section without requiring its approximation
by piecewise constant functions. (ii) The equations used by Hobson as a starting point
of his model can be obtained by integrating equations (11.236) from rI to r2, which is
not tantamount to integrating over the control volume shown in Figure 11.81. The latter
procedure has the advantage of satisfying exact1y mass conservation in any case. Trus does
not hold true for Hobson's method, except if the annular section does not vary axialIy;
in particular, for the comparison with the TAXI experiments, the axial flow velocity is
underestimated by 20% with Hobson's model, downstream of the diffuser. (iii) Hobson
implicitly assumes that the pressure and velocity profiles along the r direction are constant
and equal to 1. He also considers, in each constant-diameter pipe, the folIowing simplified
mean turbulent velocity and pressure fields: Ur (r, z) = O, Uz (r, z) = const = U , and
- - - - -2-
P(r,z) = P(z)withdPjdz = const = -pCF(U jH). Although this approximation

1354
21
18
 15

'-.J>
'5 12
'u
S
"
o
u 9
01)
s::
t 6
Q
3
O
O
(b)
SLENDER STRUCTURES AND AXIAL FLOW
30
'N'
S 25
;>,
u
5

o-
" 20
<!:
"
u
a
s 15
'"

10
o
(a)
a a a [J [J
____9-----------------....__
---
[J
.
.
.
.
3 6
Flow rate (m 3 /s)
9 X 10- 3
21
.
18
.
15
.
12
",
" .
[J. 
'
"
....---
9
,
,
,
,
,
,
0'\0
"'
..........
,
,
''o
'.
a a
6
a
[J
3
a
o
O
(c)
9 X \0-3
9 X 10- 3
3 6
Flow rate (m 3 /s)
3 6
Flow rate (m 3 /s)
Figure 11.83 Comparison of MOCCA and Hobson models to TAXI experimental data. (a) Res-
onant frequency, MOCCA model versus experiment; (b) damping coefficient versus flow rate,
Hobson's model versus experiment; (c) damping coefficient versus flow rate, MOCCA model versus
experiment. - , _, Theoretical and experimental results for translational mode; - - -, D, same
for rotational mode (Perotin & Granger 1997),
is justified for annular passages of constant section, it may be inaccurate in the case of
nonunifonn annular flows. The use of a CFD code to compute the mean turbulent flow
overcomes this problem.
Fina1ly, it is noted that the same steady and unsteady singular loss coefficients have
been used in both the Hobson and MOCCA model s for the comparisons in Figure 11.83;
although in Hobson's model they would have to be estimated empirically. An additional
advantage of the MOCCA model is that they can be predicted without empiricism,
Concerning the Porcher & de Langre model, although some of the Hobson assumptions
have been refined, for example, item (iii), some have not, and this accounts for the re1ative1y
less successful predictions for this particular system. In this respect, however, it ought to

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1355
be added that, like all CFD-type models, MOCCA does not provide an understanding of
how the system becomes unstable, something that Hobson's and Porcher & de Langre's
simplified models do. Hence, these models are all complementary, and each is valuable
in its own way.
11.6 ROTATING CVLlNDERS IN DENSE FLUID
The physics of the whirling of shafts, a c1assical problem in dynamics, is c10sely related
to that of many of the gyroscopic systems treated in this two-volume book. Moreover, it is
intimately related to problems of rotofluidelasticity, involving structures in swirling flows
(Section 7.6) or rotating structures inducing swirl on the sUITounding fluid (Section 10.3.10
and here).
The dynamics of rotating shafts has received a great deal of attention for a long time,
and there exists a plethora of papers and books on the subject, as well as on the effect
of lubricated bearings, labyrinth seals, and so on. Several books have appeared recently,
for example by Chi1ds (1993), Ehrich (1993), Kriimer (1993), Lalanne & Ferraris (1998),
Szeri (1998), Guran et a/. (1999) and Adams (2001) - see Antunes (2002a). There is a
c10se relation of this topic to that of rotating annular Taylor-Couette flow, which has been
studied very extensively - see, forexample, Taylor (1923), Coles (1965), Drazin & Reid
(1981), and Andereck et a/, (1986) - with new extensions constantly adding new facets
to it, for example by considering altemate rotations of the cylinders and two immiscible
liquids [see, e.g., Chaiken et al. (1987), Atobe & Funakoshi (1994), Baier & Graham
(1998,2000)].
In this book, we are, of course, interested in fluid-structure interaction and in keeping
with the rest of topics covered, mostly in the high rather than low Reynolds number range.
Hence, we shall not deal specifically with the dynamics of shafts in fluid-film bearings and
seals, where extreme fluid confinement results in the dominance of viscous over inertial
fluid forces; typically, the Stokes number, S = w H 2 Iv < 1 for such systems, w being
the radian frequency of osci1lation and H the thickness of the fluid- filled annulus. Rather,
we shall concentrate on cases of more moderate confinement, when inertial forces are
important. The pioneering work on this topic, for the larger S, has been done by Black
(1969), Fritz (1970) and Hirs (1973). It has been extended and refined in a series of
studies by Antunes and associates (Axisa & Antunes 1992; Antunes et al. 1992, 1996,
1999; Grunenwald et a/, 1996; Moreira et a/, 2000, 2003). The discussion in this section
is based on this work, as well as on a masterful summary, higWy recommended to the
interested reader, by Antunes (2oo2a).
The premier example of a rotor in a moderately confined annulus (with Hl R  0.1) is
the shaft of the coolant pump of pool-type Liquid Metal Fast Breeder Reactors (LMFBRs),
running within the liquid-metal-filled outer sleeve. For example, Figure 11.84 shows the
primary coolant pump of the Superphénix LMFBR.
11.6.1 A mechanical analogy
Following Antunes (2002a), we first consider a rotating mechanical system which displays
a remarkab1e number of features similar to those of the fluid-coupled system. This is
presented here in an encapsulated formo

1356
SLENDER STRUCTURES AND AXIAL FLOW
Pump body
Pump motor
support strneture
Resilient suspension
ring
;, o 4 Roof slab
o O o
o o O Sealing
0- o
Biologieal
o {) shielding
'o 00
.':.: ::-'J
 ". ....
:;,/? Argon
':.(: -" ":,::":.;.
Shaft
Sleeve
Sodium
Suetion plenum
Isolation valve
Impel!er
Hydrostatie
beanng
Diseharge bowl
Leak-tight bal!
joint
Figure 11.84 A primary coolant pump oť the Superphénix LMFBR; the shaft is '" 7 m long
(Axisa 1993).
The system is shown in Figure 11.85. t Its equations of motion are found to be
[ M+ O m O ] { X } [ CX 2MQ ] { x }
M + m ji + -2MQ C y Ý
+ [ K x - 2MQ2 cQ ] { X } { O }
-cQ Ky - 2MQ2 y = ° '
(11.262)
whereC x = Cx+c, C y = Cy+c, Kx = Kx+k, Ky = Ky+k. Twofeaturesareimmediately
remarked: (i) the damping matrix is skew symmetric, and (ii) the rotating dampers c
contribute both to the damping and stiffness matrices.
tA ehange in the notation has been introdueed to bring out similarities with the work in Section 3.4.1,

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1357
y
C x
x' \Q
Kx
x
Ky c y
Figure 11,85 Flexibly-mounted rotating masses within a flexibly-mounted immobile and rigid
ring (Antunes 2002a).
ln the case where C x = C y = c = 0, Kx = Ky = K, defining
x
 = R'
y
17 = R '
T= ( K ) 1/2 t
M+m '
M
f3= M+m '
Q
w=
(K/M)1/2'
(11.263)
equation (11.262) simplifies to
!J + [-2;'/2 w 2P2w] !J + [1- :w 2 I _02W2] !J  !J, (l1.2M)
whereby the similarity to equation (3.92) becomes obvious. Bere the dimensionless rota-
tional speed (ij plays the same rOle as the dimensionless flow velocity u; f3 is equivalent
to the ratio of mass conveyed to total mass of the pipe, per unit length. We recognize
the Corio1is gyroscopic terms in the skew-symmetric damping matrix, coupling motion
in the two directions, and the centrifugal terms in the stiffness matrix, proportional to (ij2,
tending to diminish the system stiffness.
Defining the complex eigenvalues of the system by An = O'n + iw n , n = 1, 2, solutions
of equation (11.262) are obtained and presented in diagrams showing the reduced modal
damping, -O'n/wQ, versus the reduced spin, Q/wQ, with f3 as a parameter; here WQ =
[K /(M + m)]1/2,
Typical results in the form of so-called Campbell diagrams, are shown in Figure 11.86,
ln (a), we see the classical case for the nondissipative system. For Q/wQ = 0, the sys-
tem has a repeated frequency, Wl/WQ = W2/WQ = 1, of course. For Q/wQ > O, the

1358
1.5
SLENDER STRUCTURES AND AXIAL FLOW
1.5
0.5
<=> <=> Forward
 
" c. O
a I Backward Forward
0,5 -0.5
-1
-1.5
(a) 0.5 1.5 O 0,5 1.5
2 2
1.8 1.5
1.6
1.4
1.2 0,5
<=> <=>
 1.0  O
a" c. Backward
0.8 I
-0.5
0,6
-1
0.4
0.2 -1.5
/
(b) O / 1.5 -2
O 0.5 2 O 0.5 1.5 2
2 2
1.8 1.5
1.6
1.4
1.2 / 0,5
<=> / '"
 / 
" 1.0 / c. O
a Backward / / Backward
I
0,8 / -0.5
/
0,6 "
/ 1
0.4 /
/ -15
0,2 /
/
O / -2
(c) O 0.5 1 1.5 2 O 0.5 I 1.5 2
Q/wo Q/wo
Figure 11.86 Campbell diagrams for the system of Figure 11.85 for f3 = M / (M + m) = 1:
(a) C x = C y = c = O, y = Kx/Ky = 1; (b) /;sta = 0.1, /;rot = O, Y = 1; (c) same as (a) but with
y = 0.4 (Antunes 2002a).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1359
two frequencies split apart, Detennining the eigenvectors of the system, it is shown
that the top branch is associated with motions in the direction of rotation, {x, y}T ex
{cos (Vt, sin (Vt} T; this is the forward whirl mode, also known as the direct mode; whir1-
ing in this case takes place in the direction of rotation. The lower branch, involving
{x, y} T ex {cos (Vt, - sin (Vt} T, is the backward whirl mode, a1so known as the retrograde
mode. Eventua1ly, the backward mode vanishes and is immediately converted into a second
forward mode. For sufficiently high ('}./(Vo, the two modes coa1esce, giving rise to flutter,
as seen in the damping diagram on the right. Such dynamica1 behaviour has been noted
throughout this book, especially in Chapters 10 and 11.
ln Figure l1.86(b), we see the effect of 'stationary' damping, where sta = C /[2(M +
m)wo]withC = C x = CY' whi1erotatingdampingc = O. Itisseenthatthecoa1escence of
the two forward modes does not occur, but neverthe1ess one of the modes develops negative
damping; hence, sing1e- as opposed to coup1ed-mode flutter arises. It is of interest that if
c =1= Obut C = O, one sti11 obtains coa1escence of the two forward modes as in (a), but
offthe (-O'n/(Vo)-axis; eventually one branch becomes associated with negative damping.
This could be thought to represent the difference between 'internaľ and 'externaľ damping
of the system; refer to Antunes (2002a).
Figure 11. 86( c) shows the effect of asymmetric stiffness, quantified by y = K x / K y' It
is seen that the main effect is to produce a range of ('}. / (vo over which the frequency of the
backward mode is zero, such that the system in unstable by divergence, prior to regaining
stability and eventually becoming unstable by flutter,
11.6.2 Rotor in fluid: linear analysis
We consider a cross-section of a rotor eccentrically located in an annulus filled with a dense
fluid, as shown in Figure 11.87(a). For the control volume shown, following Antunes et al.
y
y
x
x
(a)
Figure 11.87 Geometry of the fluid annulus formed by the rotor and a stationary outer cylinder,
showing (a) an elementary control volume and (b) the resultant fluid-dynamic force (Antunes
et al. 1996).

1360
SLENDER STRUCTURES AND AXIAL FLOW
(1996) and Antunes (2002a), the continuity and momentum equations may be written as
 ff { p dV + f ( p V .n dS = o,
at lev les
 ff { V Pdv+ f { VpV.ndS=LFi'
at lev les '
I
(11.265)
(11.266)
where ev is the control volume and es the corresponding boundary surface, V is the
velocity of the fluid crossing es, and n is the outward normal; Fi are the forces acting on
the fluid in ev. For incompressible fluid and neglecting any flow in the z-direction, these
equations reduce to
ah 1 a
- + --(hu) = o
at R ao '
[ a 1 a 2 ] h ap
p -(hu) + --(hu) +T +T + -- =0
at R ae r s R ae '
(11.267)
(11.268)
where h(O, t) is the variable fluid gap, u(O, t) and p(O, t) are the gap-averaged values
of the tangential flow velocity u (r, e, t) and pressure p (r, e, t) in accordance with the
so-cal1ed 'bulk-flow' formulation,
1 l h ((},t)
u(O, t) = - u(r, O, t) dr,
h(e,t) °
1 (h((},t)
p(O, t) = h(O, t) 10 p(r, o, t)dr;
Tr and Ts are the shear stresses on the rotor and 'stator', given by
Tr = -p(QR - u)2 fr,
1 2 f
Ts = Zp u s,
(11.269)
in which fr and fs are empirical friction coefficients (Hirs 1973) of the form
f = aRe b ,
(11.270)
where Re depends on the flow-wal1 relative velocity; a and b are empirical coefficients
depending on 8 = H / R, Re and roughness, H being the mean depth of the annulus, or the
depth for zero eccentricity [Wendt (1933), Antunes (2002a); for instance, for 80/..[8 <
Re < 10 4 , f = 1.84[8(1 + 8)]°,25 Re-O,5],
Next, we consider smal1 amplifude motions, such that
h(e, t) = ho (O) + hl (O, t), u(O, t) = uo(O) + Ul (O, t), p(e, t) = po + Pl(e, t).
(11.271)
Substituting in (11.267) and (11.268), we obtain: (i) zero-order nonlinear equations for
the steady (mean) flow
a
-(houo) = o,
ao
apo u aho auo Rp [ 2 2 ]
- ao = p ho ae + 2p Uo ao + 2ho fs Uo - fr (QR - uo) ,
(11.272)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1361
where the static gap is known from geometry,
[ ] 1/2
ho«() = (R + H)2 + (eo sin()2 - R - eo cos()  H(1- 8 COS(),
(11.273 )
in which 8 = eol R; and (ii) first-order, linearized equations for the perturbations, due to
vibratory motions about the static position,
ahl 1 a
- + --(ho Ul +uohd = o,
at R a()
OPl 1 opo Uo Ohl OU 1 Uo oho
-- = --hl + Rp-- + Rp- + 2P--Ul
o() ho o() ho ot ot ho o()
u6 Ohl ouo Uo ouo OUl
+p-- + 2p-Ul +2p--hl +2p u o-
ho o() o() ho o() o()
R
+p- [fs Uo + fr(fl,R - Uo)] Ul.
ho
( 11.274)
Antunes et al, (1996) first obtain solutions of (11.272), thus determining the steady flow
velocity uo«() and pressure po«(), and hence the steady forces on the rotor, ln any case,
Uo and po need to be known prior to solving equations (11.274), which is the next task.
ln fact, there is a difficulty in obtaining a general solution of equation (11.274). This is
discussed in Section 11,6.4, Here, suffice it to say that this difficulty does not arise in two
particular cases: (i) for concentric configurations; (ii) for frictionless eccentric configura-
tions (i.e. for fr = fs = O). The one case of an eccentric rotorwith fr = fs =1= O presented
in Figure 11.89 was obtained by approximate means, as a1so discussed in Section 11.6.4.
Here, presuming that the perturbation pressure Pl «(), t) has been deterrnined without
restrictions, the forces on the rotor are evaluated through
F1(t) = -R i: Pl «(), t) cos() d(),
Fr (t) = - R i: Pl «(), t) sin () d(), (11.275)
Defining hl «(), t) = -x(t) cos () - y(t) sin (), we have
{ Ff(t) } { X } { X } { X }
Fr (t) = -[M] ji - [C] Y - [K] Y .
(11.276)
If the fluid friction effects are neglected, the elements of the matrice s are:
Mxx = Myy = Ma(2/8 2 )[1 - (1 - 8 2 )1/2], M xy = Myx = O,
C xx = C yy = O, C xy = -C yx = Mafl" Kxy = Kyx = O,
Kxx = -M a (fl,2/4)(1 - 8 2 )-1/2, Kyy = -M a (fl,2/4)(1 - 8 2 )1/2,
(11.277)
where 8 = eol H is the dimensionless static eccentricity and Ma = P Jr R 2 18 is the ref-
erence added mass per unit length; 8 = Hl R, The similarity in strncture of the matrices
in (11.276) and those of the mechanical system in (11.262) and (11.264) for ex = C y = c
is very close.

1362
IS.... 1.2
"r
<L>
13
=
<L>
I
:g
o
::E
0.4
1'<:1.... 0,2
..o
.9
 O
"O
:g -0,2
o
::E
-0.4
-0,6
O
(a)
SLENDER STRUCTURES AND AXIAL FLOW
1.4
1.4
1.2
0.8
0,6
04
0,2
O
/' 0.8
2/ /'
0,6
/
;1 04
I 0,2
O
0.2
-0.4
-0.6
3.5 O
(b)
.. ;.... j-.-..-....
L... ...j.. .__.
: I
...... t-........j--.. ...
, ,
...-- .:1.-"'" .....
1; ! .-..J.:,...'
;- ....j;/ 1
-i , -t " ...... >,j" .j.,
""0 , ' 1 ,2/'! I
. 'ľ<'1--"'-  -"";-'1 j !
, ... _.._.....,..- .... "';' ...! .-
j "'" ; I
:! I I I
, .j- . .+--.-.--+, .\ ./.
.............1 --"-r -.- +  ..-.... ./-:.::..L. .-.
. .---j .-.. - "'--2 t . j
I __..-1
.......- I
- ----T- -- - .._- . --!
I
-- ..--- ---Í
I
..I.
;
.;
0,6
.. . t-
i
. j
I
1-. ,
,..
1-
I
0,5
II
21
I
;
;
I
0.5 1 1.5 2 2,5 3
Reduced spinning velocity, Q
3,5
1.5 2 2,5 3
Reduced spinning velocity, li
Figure 11.88 Evolution of the dimensionless modal frequency ev j and damping dj, respectively
Wj /W r and -aj /w r , W r being the dry rotor natural frequency, as functions ofthe reduced spinning
velocity?2 = /wr, for a concentric system (E' = O) and P = Ma/(Ma +M r ) = 2/3: (a)undamped
system; (b) damped system, fr = J, = 0.0 1. The forward mode is marked as '1', and the backward
one as '2' (Antunes et al. 1996).
Combining equation (11.276) with the equation of motion of the rotor itself, we
can write
([Mr] + [Mj(e))) {} + ([C r ] + [Cj(Q))) e} + ([Kr] + [Kj(e, Q))) {;} = {O},
(11.278)
for the equations of the system, where the subscripts r and f stand for rotor and fluid;
[Mr], [C r ] and [Kr] are diagonal matrice s with elements Mr, 2M r w r í;r and Mrw;,
respectively, supposing a rigid rotor on flexible isotropic supports, in which W r is the
in vacuo structural frequency and í;r the corresponding damping ratio. This equation is
solved by standard means, yielding the two eigenfrequencies Wj and damping ratios í;j.
Figure 11.88(a) and (b) shows typical results for the concentric system, undamped and
damped. The similarity to those forthe mechanical system, Figure 11.86(a) and (b), is quite
striking. t The fact that the frequency at Q = Q / Wr = O is (ij j = W j / Wr ::: 0.6, j = 1, 2,
tHere (see eaption), as in equations (11.263), (3.71) and (8,28a), we use {J = Ma/(Ma + Mr) throughout,
which has the niee property of O < {J < I, whereas Antunes et a/. in this and other papers use {J == {JAnt =
Ma/Mr; henee 1/{JAnt = 1 + I/{J,

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1363
18"" 0.8
",
'"
.0
<:::
'"
5-
'"
-=
';;j
"O
 0,2
o
QA
0,6
0.4
0,8
0,2
I",,""
gp 0,2
'I O
"O
:g -0,2
o
::s -0.4
-0.6
O 0,5 1.5 2 2.5
(a) Reduced spinning velocity, Q
0,8
0,6
0.4
0.2
B
O
-0,2 :z
-0.4
3 O 0.5 1.5 2 2,5 3
(b) Reduced spinning velocity, li
B
Figure 11.89 Evolution of Wj and dj with  for an eccentric system (8 = 0,7) and f3 = 2/3:
(a) undamped system; (b) damped system, fr = fs = 0,01. Modes F and Bare, respectively,
the forward and backward whirl modes; Z is the zero-frequency locus (Antunes et al. 1996;
Antunes 2002b),
(rather than unity) is due to added-mass effects. Moreover, for the damped system, it is
seen that (single-mode) flutter in this case occurs at Q c:::: 2, considerab1y lower than for
the undamped system. Hence, damping actually destabilizes the system, which is simi1ar
to what happens in fluid-conveying pipes (Section 3.5.3).
Figure 11.89(a) and (b) shows simi1ar diagrams for an eccentric system without and with
damping (fr = fs = 0.02). Again, the simi1arity ofthe undamped case here (diagrams on
the left) to that in Figure 11.86(c) is striking. It is clear from re1ations (11.277) that for the
eccentric system the stiffness is asymmetric, which corresponds to a mechanical system
with y = Kx / K y i= 1. The undamped system loses stability by divergence at Q c:::: 1.2 and
then by flutter at Q c:::: 2.2. Damping in this case causes the divergence to be suppressed, t
though for a smaller damping level (fr = fs = 0.01 or smaller), it wou1d still occur.
Autter, on the other hand, occurs at considerab1y lower Q: for the damped system at
Q c:::: 1.75 as compared to Q c:::: 2,2 for the undamped one. Here the reader is referred to
Section 11.6.4 for a discussion of the approximate method used to calcu1ate Figure 11.89(b)
and for a more rigorous analysis.
t As pointed out by Antunes (2002b), some of the numerical results with damping presented in Antunes
et a/, (1996) are incorrect, tending to exaggerate the effect of fluid friction on the dynamics - though they are
qualitatively correct.

1364
SLENDER STRUCTURES AND AXIAL FLOW
The effect of eccentricity is to reduce the critical Q for flutter; for example, for the
undamped system and f3 = 3/4, Q = 2.3 for c = O and Q = 1.8 for c = 0.8, the reduction
occurring smoothly (Antunes et at. 1992) - see Figure 11.92.
In this analysis, any coherent structures arising from rotation of the shaft have been
ignored, although for small enough Reynolds (Re) and Taylor (Ta) numbers t such struc-
tures are expected, at least for a nonvibrating shaft. Thus, for Re < 4000 and Ta < 41,
c1assical Couette 2-D vortices should exist in the flow; and, for Ta > 41, 3-D Taylor vor-
tices (Tritton 1988; Antunes et al. 1999). The argument can be made here that, for high
enough Re, the shaft vibration disturbs and perhaps destroys such structures, but in any
case taking them into account in an analysis such as this is too fonnidable a task.
11.6.3 Experiments
Some early experiments have been conducted by Fritz (1970) in a similar apparatus to
those described in what follows, but were of limited scope.
An extensive set of experiment s was conducted by Antunes et at. (1992) in an apparatus
consisting of a vertical shaft (extemal radiu s R = 75 mm, and mass Mr = 50 kg) in a water-
filled annulus (H = 11 mm) rotating with speeds up to 600 rpm. The shaft was held through
conical bearings at the upper end, and support stiffness was provided by struts linking the
rotor support-plate to the main structure, bolted to the floor. The system could be excited
by a shaker providing a random excitation ofband-limited white noise. The response was
sen sed by accelerometers and inductive displacement transducers. Real-time monitoring
via a Fff spectral analyzer and post-processing by computer pennitted identification of
modal parameters and response characteristics. Good results were obtained for the splitting
of the frequency between the forward and backward modes, in excellent agreement with
theory, as well as reasonable agreement in the corresponding damping ratios, but the
apparatus did not allow exploring the dynamics beyond the point where ev of backward
whirl mode is predicted to vanish (at QA in Figure 11.89). Moreover, difficulties were
encountered with surface effects: a flat liquid surface could not be maintained for Q >
200 rpm, because of waves and frothing due to air entrainment.
A new, improved but similar apparatus was built, therefore, with a more slender rotor
(L = 250 mm, R =47.0 or43.5 mm, Mr = 6.9 or7.0kg) inannuli of H = 3.2 and6.7 mm,
capable ofhigher rotational speeds, up to 3000 rpm (Grunenwald et at. 1996). A special
device, a large-gap labyrinth seal, minimized wave effects and eliminated surface-related
ventilation.
A typical result, showing the onset of flutter are shown in Figure 11.90. The instabil-
ity is associated with the forward mode, as predicted by theory. The increase in r.m.s.
displacement at the critical speed is quite sharp, as also seen in the x-y plots.
Typical Campbell-type diagrams for a concentric and two eccentric systems are shown
in Figure 11.91, where they are compared with the Antunes et at. (1996) theory.
Agreement between theory and experiment for the concentric case, Figure 11.91(a), is
fairly good in every way. The only qualitative feature not reproduced is the theoretically
predicted vanishing of the backward-mode frequency prior to its transfonnation into a
forward mode. This behaviour is similar to that discussed in Chapters 3-5, related to
divergence in pipes conveying fluid (see e.g. Figure 3.26), and is at least partly due to
tHere, Re = 2H V Iv, where V is an average ftow velocity ofthe orderofQR, and Ta = Re y'HI R .

I I
0,8
" o
" 06
8
"
1;1 Expenmental I -
]. 04 instability O
'6 velocity \J '"
O; 02
8
 -I
-
O
O 500 1000 1500 -I O -I O
Spinning velocity (rpm) (b) x (mm) (c) x (mm)
(a)
Figure 11.90 (a) The on set of flutter for a concentric configuration; (b) x-y plot of the rotor
response at the onset of flutter (in the forward mode), at Q = 1250 rpm; (c) x-y plot for Q
1350 rpm (Grunenwald et a/, 1996).
1.5
o o o o o
I
;>:
u
=

"
.:::
'"
'"
"
"2
 0.5
=
"
8
ti
"
+ ,/
,
,
,
+/
+/
+ ,
+ ,
,
/
,
,
,
,
o
O
O
2
(a)
1.5
I
tf
=
"
6-
"
.:::
'"
'"
"
"2
 0.5
"
8
ti
,,' o
,
...............,.:
",1 ...............:
. ,
" ....
x
x
x
,
,
,
I
(b)
O
O
1.5
le
;>:
u
=
"
6-
"
.:::
'"
'"
"
"2
 05
=
"
8
ti
-.,Je ..
-..... X'
;'1'....._)(
.... ....._.....x.....
I' .........
.... .......,
O
O
(c)
Dimensionless spinning velocity, Q
2
02
I"",
oj)
"
. 0,1

'"
'"
"
"2
 O
=
"
8
ti
-0.1
O
0,2
I"",
oj)
=
5. O I

'"
'"
"
"2
. O
=
"
8
i5
-O I
O
0,2
I"",
oj)
=
j 0,1
'"
'"
"
"2
 O
=
"
8
ti
I
Experimental.:
instability :
+
+
+
+
I
Experimental 
instability :
2
A
/// \
/ \
, \
/// '.....",'
, X
,
I
I
I
I
I
I
o A
x
2
I
,
I
I
I
I
----i
--- o \
\
I
I
I
I
,
,
I
I
I
/
2
-O 1
O
2
Dimensionless spinning velocity, Q
Figure 11.91 Dimensionless modal frequencies, (ij j, and damping, d j, versus Q for (a) a con-
centric system, (b) an eccentric system (8 = 0.6, Hl R = 0.07), and (c) an eccentric system in
a wider annulus (8 = 0.6, Hl R = 0.15): - , - - -, theory; o, experimental, first forward
mode; x, backward mode; +, second forward mode; in (b) the theoretical results are for 8 = 0.7
(Grunenwald et al. 1996).

1366
SLENDER STRUCTURES AND AXIAL FLOW
3.5
.
3,0
10] 2,5

"g 2.0
<)
>
--g 1.5
u
.g
 1.0
0.5
0.0
0,0
0.2
0.4
Eccentricity, E
0,6
0,8
Figure 11.92 Theoretical stability chart with some experimental results superposed for H / R =
0,15, Experimental in stability threshold (flutter): ., in the forward mode; ., in the backward mode
(Grunenwald et aZ, 1996).
nonlinear effects. t The critical spin velocity for whirl is associated with the forward mode
in both theory and experiment; it is underpredicted in this case by rv 15 %.
Agreement is similarly good for the eccentric case in the narrow annulus (H / R = 0.07)
shown in Figure l1.9l(b), where it is recalled that the theoretical results are only approx-
imate, as mentioned in the previous section (and they are for £ = 0.7, whereas in the
experiments c = 0.6), The critical whirl is again associated with the forward mode, in
this case in excellent quantitative agreement with theory.
The dynamical behaviour for the eccentric case in the wider annulus (H / R = 0,15)
in Figure l1.9l(c) is considerably different. Most importantly, in the experiments, whirl
in this case developed in the backward mode. The approximate theory is ambivalent,
showing a loop of the backward mode developing divergence over a small range of Q ,
and the forward mode fluttering (whirling) thereafter. In this case, theory overpredicts
the critical whirl speed by rv 20%.
Figure 11.92 summarizes the dynamics for different eccentricities in the wider annu-
lus. The flutter instabilities for small £ are associated with the forward mode, correctly
predicted by theory but quantitatively underestimated by rv 15%. For higher c, flutter is
associated with the backward mode; it occurs at much lower Q and agrees best with the
theoretically predicted thresholds for divergence. This discrepancy is discussed further in
the next section.
11.6.4 A more general linear theory
As already mentioned, when discussing solution of equations (11.274), a difficulty arises
in the general case of an eccentric rotor when frictional effects are taken into account.
Specifically, integration of the first of equations (11.274) gives an equation for Ul «(), t)
involving an integration constant c(t). For (i) concentric and (ii) eccentric frictionless
tThe reader is a1so referred to the discussion of Figure 11.93 in Section 11.6.4; improved linear theory
achieves much better prediction, without even accounting for nonlinear effects.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1367
systems, this constant may be eliminated and the solution is obtained without difficulty.
Unfortunately, however, in the aforementioned general case, there is no way of extracting
an explicit solution for c(t), Hence, in general, Pl«(), t) and the ftuid-dynamic forces
Fí (t) and Fr (t) involve the unspecified c(t) and its time derivative c(t); therein lies the
difficulty. In the theoretical computations by Antunes et al. (1996) for eccentric systems
with friction taken into account, the following device has been used: the inviscid forces are
computed for the eccentric frictionless system, while the frictional ones from an equivalent
concentric one. This applies to the theoretical results of Figures 11.89(b), 11.91 (b) and (c)
and 11.92, which therefore can only be regarded as approximate.
The problem has been circumvented in a very neat way by Moreira et al, (2003); see
a1so Antunes (2002a). First, after Ul «(), t) is substituted into the second of equations
(11.274), aplj()() is not integrated, since altematively to (11.275), the forces may be
obtained directly through
I Jr a « () t )
Fí (t) = R Pl , sin () d(),
-Jr a()
I Jr apl«(),t)
Fr (t) = R cos() d(),
-Jr a()
(11.279)
while azimuthal continuity of the pressure field may be assured via
I Jr apl«(), t) d() = O, Vt.
-Jr ao
(11.280)
Proceeding then as in the sequel to (11.275), one obtains from (11.279) and (11.280) three
second-order equations, in x and y and its derivatives, and in c and c. Then, the offend-
ing unknown c is simply enlisted into the solution vector, t which henceforth becomes
{x(t), y(t), c(t)}T, The three equations are then transformed into a first-order set of five
equations (cf, Section 2.1.2) involving the vector {x, y, x, y, c}T, which are solved by
standard means.
As shown by Moreira et al. (2003), c(t) has a physical meaning, Computing the average
tangentia1 ftow rate, the average tangential ftow velocity is found to be u(t) = Co + c(t);
hence, Co = K Q R is the zero-order average tangential ftow velocity and c(t) the ftuctu-
ating first-order one. For frictionless eccentric systems (fr = fs = O) or concentric ones
with friction (8 = O), c(t) may be detennined explicit1y in terms of x(t) and y(t), and
hence in the above the additiona1 equation for c(t) may be dispensed with; for frictionless
concentric systems (fr = fs = O and 8 = O), c(t) = O.
Figure 11.93 shows results computed with (a) the approximate solution of Antunes et al.
(1996) and (b) this new, more rigorous solution. The differences are most profound in the
modal damping. As seen in Figure 11.93(b), the dj curves are interlaced, and the dynamics
is more intricate, More importantly, (i) the divergence is confined to a much smaller range
of Q , (ii) ftutter occurs in the backward mode, and (iii) it does so at considerably lower
ftow (in this case at Q = 1.2 rather than 1.7), Clearly all three facets of this predicted
behaviour are in much better agreement with the experimental data of Figures 11.91(c)
and 11.92.
tA course of action intermediate between making a virtue out oj necessity and if you can 't beat them,
join them!

1368
SLENDER STRUCTURES AND AXIAL FLOW
>.
u
o:
<!)
::o
O"'
<!)
<.!::

o
e
.."
<!)
u
::o
.."

0,8
0,6
0.4
0,2
°
0,6
0.4
0,2
B
0,8
Ol)
o:
.
os
.."

o
 °
<tJ
<!)
 -0,2
o
O;
 -0.4
e
i5
°
0,5
1 1.5 2 2,5
Reduced spinning velocity
0,2
0,8
0.6
0.4
0,2
°
-0,2
-0.4
°
(b)
0,5
I 1.5 2 2,5
Reduced spinning velocity
(a)
Figure 11.93 Dimensionless modal frequency, Wj, and damping, d j, versus dimensionless spin-
ning velocity Q for an eccentric system (s = 0.7) with f3 = 2/3, and with friction taken into account
(fr = fs = 0.005): (a) according to the approximate analysis; (b) according to the general analysis,
Here F and B denote forward- and backward-mode loci, while Z the zero-frequency locus (Moreira
et al. 2003).
Extensive calcu1ations for other parameters may be found in Antunes (2002a), including
stability maps t for H / R versus Q , s versus Q, y versus Q, etc.
11.6.5 Nonlinear effects
It has been found (Grunenwa1d et al, 1996) that 1inear theoreti.ca1 predíctions deteriorate at
high spinning ve10cítíes and 1arge eccentricities. Indeed, when the rotor becomes unstab1e,
the amplítude of motion is of the order of magnítude of the annu1ar gap, and under such
círcumstances nonlínear effects can hard1y be ignored.
A prelímínary nonlínear study of a simp1ified fonn of the system, name1y for planar
motions of the rotor, has been conducted by Antunes et al. (1999). Solutions of the
nonlínear equations (11.267) and (11.268) are obtaíned by first integrati.ng the continuity
equation to obtain u(O, t) and then expressing ap (O, t)/ao as a Fourier series, thereby
obtaíning a non1inear differentia1 equation for F X (t), Time integration is done by a high-
order Runge-Kutta scheme,
Calcu1ations have been conducted for a concentric system with H = 6 mm (H / R =
0.14). Some resu1ts for the rotor disp1acement X (t) and the dimension1ess tangentia1
tThis writer is one of the fortunate few to have seen these maps in co/Dur, as in the original submission. Alas,
in the published book they were rendered in black and white. This is almost as bad as sacrilegiously republishing
Yann Arthus-Bertranďs 'EarthfromAbove' (1999, New York: Harry N. Abrams) with the stunning, colour
photographs replaced by b&w ones!

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1369
X 10- 3
2
g
 O

-1
-2
0,51

 0.49
0.48
0.47
O 2
(a)
0.010
0,005
4 6
Tirne (s)
0.5
0.4
0.3
0,2
0.1
O
8 10 O 2 4 6 8 10
(b) Tirne (s)
Figure 11.94 Vibratory p1anar response of a rotor, X (t), and the dimension1ess tangentia1 flow
ve1ocity, K (t), for (a) Q = 100 rpm and (b) 800 rpm (Antunes et aZ. 1999).
velocity K(t) = c(t)j(f?H) are shown in Figure 11.94. It is seen that, at low spin velocities
(f? = 1 00 rpm), X (t) < 2 mm; the response is Gaussian, and the corresponding PSD (not
shown here) has a peak at tbe in-fluid natural frequency of the rotor. At f? = 800 rpm,
however, a little higher than the onset of divergence (f? = 750 rpm), the amplitude is of
the order of the annular gap, and the response is no longer Gaussian. Moreover, at the
lower f?, except for the spikes, K (t) ::::: 0.5 which is the value predicted by the linearized
analysis. At the higher f?, however, the large rotor amp1itudes drastical1y interfere with
the co-rotating flow, and K (t) occasional1y reaches zero. It has also been found that
nonlinear theory is able to capture one feature of the experimental behaviour not predicted
by the linear theory: that the rotor frequency does not vanish as the system goes through
divergence. Hence, it is clear that, close to and beyond the instability threshold, nonlinear
theory is required for prediction.
The analysis is extended for 2-D (orbital) motions of the cross-section of the shaft by
Moreira et aZ. (2000). As in Section 11.6.4, one ends up with five first-order equations,
in tbis case nonlinear. Excitation is provided by a band-limited random Gaussian force.
Typical results for two eccentric systems in a narrow and a wider annulus (H jR = 0.068
and 0.154, respectively) are shown in Figure 11.95(a) and (b), where they are compared
with linear theory and experiments. The nonlinear data are plotted in different diagrams
(on top) from experimental data (below) in each case, to avoid overlapping. This in itself
bespeaks of the closeness between non1inear model prediction and experiment, which is
indeed remarkable.
Further calculations of kurtosis, skewness, and response amplitude histograms may
be found in Moreira et aZ. (2000). Moreover, a very wide variety of orbital motions is
obtained, as shown for example in Figure 11.96 for the wider annular geometry.
ln conclusion, the nonlinear analysis is capable ofpredicting most salient features ofthe
observed behaviour. The few remaining discrepancies are attributed to 3-D flow effects,
related to the axial changes induced by end effects (unavoidable in experimental rigs),
which are obviously not accounted for in the flow model.

1370 SLENDER STRUCTURES AND AXIAL FLOW
10 10
8 o o 8
'N o
;; 6
>,
5
g.
...

2 2
O O
O 200 400 600 800 1000 O 100 200 300 400 500 600
10 10
8 8
'N
;; 6 D 6
>,
u
C
.,
50


2 2
.
O
O 200 400 600 800 1000 100 200 300 400 500 600
15
10
c
I
ci)
SI 5


-5 -5
O 200 400 600 800 1000 O 100 200 300 400 500 600
15 5
10
c lil
I
 5 O
'e..
e
'"

-5 -5
O 200 400 600 800 1000 O 100 200 300 400 500 600
(a) Spinning velocity (r,p,m,) (b) Spinning velocity (r,p,m,)
Figure 11.95 Comparison of moda1 frequencies and damping according to linear (line s ) and
nonlinear theory (points) in the top diagrams in each case, and linear theory (lines) and experiments
(points) below, with varying Q, for two eccentric systems (8 = 0.6): (a) for H / R = 0.068; (b) for
H / R = 0.154 (Moreira et al. 2000).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1371
Or,p,m, 100 r.p,m. 200 r,p,m,
0,2 0,5
0,2
0,1 0.1 *
I o o  o
:...
-0,1 -0,1
-0,2
-02 -0.5
38 4.0 4,2 3.5 4,0 4.5 4,0 42 4.4
0.3 300rp.m 0.5 400 r.p.m 2 450 r,p,m,
0.2 0.4
I 0.1  0.3 
:...
O
O 0.2
-0,1 0.1 -1
4.4 4,6 48 4,9 5,0 5,1 5,2 4 5 6
X (mm) X (mm) X (mm)
Figure 11.96 Orbital pattems of motion at various Q for the wider annular system of
Figure 11.95(b); (Moreira et a/. 2000).
11.7 COAXIAL SHELLS
A great deal of work has been done on the vibration of coaxial shells coupled by a quiescent
dense fluid in the annulus, as reviewed in Sections 11.3.5 and 11.3.6. For very thin, not very
long shells and not very narrow annuli, the inertial (added-mass) effect is so strong, that
in engineering applications it is sometimes the on ly effect taken into account, even if the
fluid is flowing. Then, using empirical damping data, it is possible to obtain approximate
estimates of the vibration due to, for example, pump pulsations or turbulence excitation.
Sometimes this is masked, if the analysis is begun with modal equations, thus bypassing
the discussion of how the dependence of modal frequencies and damping ratios on the
mean flow is obtained. Neverthe1ess, it is this dependence that is the most difficult to
calculate; this is the subject ofthis section.
ln this regard, it is opportune to discuss here the feasibility of using one of the standard
finite-element codes, for example, NASTRAN, ANSYS, ADINA, to solve the whole
problem. Although such codes have long been used to successfully predict inertialloading,
that is the effect of a quiescent annular fluid [see, e.g. Chung et a/. (1981)], they have
utterly failed - so far - to predict the dynamics with flowing fluid. t
11.7.1 Coaxial shells with supported ends
The dynamics of coaxial shells subjected to axial flow in the inner one and in contact with
a quiescent annular fluid has been discussed in Section 7.5, Here, the analysis is extended
tl n time, newer versions ofthese codes will no doubt succeed. However, according to Amabili (2002), a good
Ph.O, student of his has already spent a year fruitlessly trying to make one of these codes 'work' for a flowing
fluid in or over a shell; Lakis (2002) had similar experience, using another such code, Oespite their advertised
'mutliphysics' capabilities, these codes cannot yet handle single or coaxia1 shells coupled to fluid flow,

1372
SLENDER STRUCTURES AND AXIAL FLOW
U
I
ho
Figure 11.97 Two coaxial shells subject to both internal and annular ftow, with velocities Ui and
U o , respectively (Paidoussis et al. 1984a).
to the case of a flowing annular fluid, or when both internal and annular fluids are flowing,
for shells with both ends supported. Work on the subject has been conducted by Weppelink
(1979), Pai'doussis et aZ. (1984a, 1985, 1990) and El Chebair et aZ. (1989, 1990).
11.7.1 (a) Inviscid model
The first study on the subject is by Weppelink (1979) t who, among several other things,
treats the dynamics of a shell in a coaxial cylinder subject to internal and annular flows,
using the Morley-Koiter shell equations (Section 7.2.1) and a potential flow model, obtain-
ing solutions both by the Rayleigh-Ritz and a finite-element method. This remarkable
study was motivated by a specific app1ication: the thermal shield of the sodium-steam
heat exchanger of the SNR - 300 reactor; hence, many of the ca1culations presented are for
this particular system, involving a fairly wide annulus (the ratio (b -a)/a = 0,168 :::: 1/6,
band a being the outer (rigid cylinder) and inner (baffled flexible shell) radii]. For counter-
flow in the inner shell and annulus, the reduction in the critical flow ve1ocity, vis-a-vis inner
flow only, is of the order of 0.5 for simply supported, clamped-clamped and cantilevered
shells.
A more extensive study has been conducted by Pai'doussis et aZ. (1984a), generally for
two flexible shells and compressible fluid flow in either the internal shell or the annulus,
or in both; see Figure 11.97. Shell motions are described by Fliigge's equations, while the
fluid forces are obtained by potential flow theory and formulated in the Fourier-transform
genera1ized-force framework. Indeed, the formulation for internal flow has already been
given in Section 7.3.1 (a), equations (7.69)-(7 .82b). Some of it is repeated here, along with
the analysis for the annular flow, to achieve a unified presentation.
The governing shell equations may be written as
{£ii' £i' £jJ = {O, O, qt}, {£io' £o' £jo} = {O, O, q},
(11.281)
where, for instance, £ii = £1 (Ui, Vi, Wi) - y.(a 2 ui/at 2 ), £20 = £2(U o , vo, W o ) -
Yo(a 2 vo/at 2 ), in which £1, £2 and £3 are as in equation (7.29), with the subscripts
tWeppelink's (1979) study, a M.Sc. Thesis, apart from a couple of conference presentations, has never been
published in archival joumals,

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1373
i and o denoting the interior and outer shells; q{ = Vi qj/(Psjhj) and q = Yo qo/(Psoho),
with qj = (Pj - Po)lr=a and qo = Polr=b since hj « a, ho « b, a and b being the interior
and outer shell radii. The fluid outside the outer shell is presumed to be of low enough
density for its effect on the dynamics to be ignored. The shell extremities at x = O and
L are fully clamped, thus obeying equations (7.3), and they are attached to semi-infinite
rigid extensions (baffles) upstream and downstream.
The equations of motion are solved by Ga1erkin's method, such that
f Uj } 00 { AmnCOSne(aa/ax) }
Vj = z= Bmn sin ne <Pm (x) eiQt ,
Wj m=l C mn cos ne
{ uo } 00 { Dmn cos ne (ba/aX) }
V o = z= Emn sin ne <Pm (x) ejt,
W o m=l Fmn cos ne
(11.282)
(11.283)
where Amn to Fmn are constants to be determined, and <Pm (x) are appropriate comparison
functions, here the eigenfunctions of a c1amped-clamped beam.
The interna1 and annular flows satisfy equation (7.69) and the nonpenetration relation-
ships - see equation (7.35) - on the surface of the shells. The perturbation velocity
potentials may be written in the form
<l>j(x, r, e, t) = 1/1in(X, r) cosne ejt,
<l>o(x,r,e,t) = Vron(x,r)cosneejt,
(11.284)
where Vrjn and Vron must be determined in the manner described in Section 7.3.1(a).
One eventually obtains
* iUj(Kj - li) z= 00 *
Vrjn(a, r) = {E n @, r) <pm(a)CmnL
AjL
m=l
* iUo(K o - li)  * _ *
Vron(a, r) =  {F n @, r) <pm(a) C mn + Gn(a, r)<Pm(a)FmnL
AoL
m=l
where the asterisk here denotes the Fourier transform - cf. equation (7.72) - and
ln (Ajr) 1(Aob)Kn(Aor) - In(Aor)K(Aob)
E f(i r ) - - F f(i r ) -
n\....' - 1(Aja)' n\....' - 1(Aob)K(Aoa) - 1(Aoa)K(Aob)'
In(Aor)K(Aoa) - 1(Aoa)Kn(Aor)
G n @, r) = I' b I I K' b '
n(A o )Kn(Aoa) - ln (Aoa) n(Ao )
(11.285)
(11.286)
in which
li = aL, Kj = QL/Uj, Mj = Ui/q, Af = [ii2 - M?@ - Kj)2]/L2,
Ko=QL/U o , Mo=Uo/co, A=[li2_M;'(Ii-Ko)2]/L2; (11.287)
a is the Fourier variable, I(z) == dln(z)/dz and similarly for K(z); q is the speed of
sound and Mj the Mach number for the internal flow, and Co and Mo for the annular flow.

1374
SLENDER STRUCTURES AND AXIAL FLOW
Then, using equations (7.37), relating the perturbation pressures Pi and Po to the velocity
potentials cI>i and cI>0 as in (11.284), we obtain
Jii (Ci, r, e) = -i Pi (Ui! L)(Ki - a)1/ti (a, r) cos ne,
p(a, r, e) = -i Po(U o / L)(K o - a)1/tn(a, r) cos ne,
where
(11.288)
Jii(Ci, r, e) = L i: Pi(g, r, e)eia dg, Pi = p/x, r, e)e iQt ,
and similarly for Po' The perturbation pressures enter the equations of motion as qi =
(Pi - Po)lr=a exp(iS1t), qo = Polr=b exp(iS1t). Hence, taking the inverse transform of
(11.288) and using (11.285), the following expressions are obtained:
00
qi = L Qmn (g) cos ne e iQt ,
m=1
00
qo = L Rmn (g) cos ne e iQt ,
m=1
(11.289)
P 'U 2 C 1 00 ( K' (i\2
Qmn(g) = I i n 1- UJ En(a, a)4>:n(a)e-ia da
2n L -00 )..i L
_ PoU;C mn 1 00 (Ko - a)2 F ( - ) -1.* ( - ) -ia d -
2 na, a 'f'm a e a
2n L -00 )..oL
P U2F 1 00 ( K _a ) 2 ,
_ o o mn o G ( - ) -1.* ( řV'\ -Ia d -
2 na, a 'f'm a Je a,
2n L -00 )..oL
P U2C 1 00 ( K (i\2
Rmn () = o o 2 mn o - u J Fn (a, b )4>:n (a)e -ia da
2n L -00 )..oL
+ Po U; mn 1 00 (Ko - a)2 Gn(a, b)4>:n(a)e-ia da.
2n L -00 )..oL
In the analysis, the following dimensionless quantities are used:
U i = Ui!Ui, U 0= Uo/U o , fJi = (Pia)/(Psihi), fJo = (Pob)/(Psoh o ),
8i=a/L, 80=b/L, 8r=a/b, Ur=Ui!U o , Pr=Po/Pi, (11.292)
C mn = Cmn/L, Fmn = Fmn/L, S1i = S1a[psi(1- vf)/EiJ1/2,
S1 0 = S1b [Pso(1 - v)/ Eo r/ 2 ,
Ui = {Ei![Psi(1 - vf)]}1/2, U o = {Eo/[Pso(1 - v)]}1/2. (11.293)
The dimensionless generalized fluid forces may now be defined by
= Yi 1 1
Qkmn (g) = ---:-- h ' L 4>k(g) Qmn(g) dg,
PSI I o
-  1 1
Rkmn (g) = : L 4>k(g) Rmn(g) dg,
Pso o o
(11.290)
(11.291)
(11.294)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1375
which, in view of equations (11.290)-(11.293), may be written as
I -2 2
= {J"8"U" 00 (K" - a)
Qkmn =  1 1 En(a,8i)Hkm(a)da
2n -00 Ai L
p. u 2 1 00 ( _ ) 2 ]
fJi 8 iPr o Ko - a _ _ _ -
- U2 Fn(a, 8i)Hkm(a) da C mn
2n r -00 AoL
I P. U - 2 1 00 ( _ ) 2 ]
fJi 8 iPr o Ko - a _ _ _ -
- U2 Gn(a, 8i)Hkm(a) da F mn,
2n r -00 AoL
I P. u 2 1 00 ( _ ) 2 ]
= fJ0 8 0 o Ko - a _ _ _ -
Rkmn = Fn(a, 8 o )Hkm(a) da C mn
2n -00 AoL
I P. U 2 1 00 ( _ ) 2 ]
fJo80 o Ko - a -
+ Gn(a, 8 o )Hkm(Ji) da F mn,
2n -00 AoL
Hkm(a) = 11 4Jm(l;) e ia $ dg 11 4Jk(g)e- ia $ dg.
(11.295)
(11.296)
(11.297)
It should be recalled here that g = xl L and that Ai L and AoL are dimensionless.
Equations (11.282), (11.283) for the displacements and equations (11.289) for
the pressure are then substituted into (11.281) and nondimensionalized (Amn
Amnl L, .,., F m n = Fmnl L), so that they become functions of Am n, B mn, ..., F m n and,
implicitly or explicitly, of the frequency Qo or, in cases where the inner she11 only is
flexible, of the frequency Qi .
In the case of incompressible fluids, considerable simplification is possible by reformu-
lating the problem, rather than simply taking Mi = Mo --+ O in the foregoing, as described
in Appendix Z.1.
- -
Evaluation of the generalized forces Qkmn and Rkmn involves a set of complicated
integrations with respect to a, which must be performed with care due to the presence of
singularities; refer to Appendix Z.2.
The calculations to be presented in what follows are for so-called 'stee1-water' systems,
that is, stee1 shells and water both in the annulus and in the inner she1l, with the following
parameters:
b = 1 ()() mm, h = 0.50 mm,
Lib = 10, v = 0.30,
Ui = 5308 mis,
Pr = 1.0;
-gap system
(11,298)
alb = 2/3, hla = 7.5 x 10- 3 , {Ji = 17.09,
lo -gap system
alb = 10/11, hla = 5.5 x 10- 3 , {Ji = 23.30;
the notation 'y-gap' signifies that (b - a)la = y.

1376
SLENDER STRUCTURES AND AXIAL FLOW
2,00
Do ] ' '::n::' , ' '
1.75
m 3
N

150
x
I Cf 1.25
;.:;
u
"
g.
'" 1.00
<J:::
'"
'"
'"
"2
o
'on
"
Š
i5
F
0,75
B E
0.5 1.0 1 5 2.0 2.5 3,0
Dimensionless annular flow veloeity, Do x 10 2
0,50
0.25
o
O
Figure 11.98 The real dimensionless eigenfrequencies Qi (n = 3; m = I, 2, 3) of the to-gap
with c1amped-clamped steel-water system [vide equations (11.298)] as functions of the di men-
sionless annular flow velocity U o ; the fluid in the inner shell is stagnant and the outer shell rigid
(Pai:doussis et a/. 1984a).
We first consider systems with the outer shell replaced by a rigid cylinder. A typical
frequency versus flow diagram for the l -gap steel-water c1amped-clamped system for
internal flow and a stagnant annular fluid is given in Figure 7.56 in Section 7,5,2. The
frequency versus flow diagram for the same system but with the inner shell filled with
stagnant fluid and annular flow is shown in Figure 11,98, The dynamical behaviour of the
system at low and medium U o is similar to that for interna I flow: the system loses stability
by divergence at point A, is restabilized at point Band then, through coalescence of the first-
and second-mode loci, becomes subject to coupled-mode flutter at point C. At higher flows,
however, the dynamical behaviour in this case becomes more intricate. Point D marks
the occurrence of another restabilization, in which the complex combined first/second-
mode eigenfrequencies become dissociated into two distinct real eigenfrequencies: the
frequency of one mode (presumed to be the renascent first mode) is sharply diminished
with increasing U o, leading to another divergence at point E; the frequency of the other
rises sharply and, through coalescence with the third axial mode, results in coupled-mode
flutter at point F.
For this particular system, the critical flow velocities for n = 3 are lower than those for
n = 2 and n > 3, t These values are tabulated in Tab1e 11.12, where they are compared
with the corresponding va1ues of internal flow. As expected, the stronger coupling due to
tFor the 1-gap system, however, one obtains the same entic a] va1ues, TJ o = 0,0289, for both n = 2 and 3,

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1377
Table 11.12 The lowest critical flow velocities for divergence and
coupled-mode flutter of the 1o-gap clamped-clamped steel-water sys-
tem conveying internal or annular flow, or both (n = 3); the outer shell is
rigid (Pai'doussis ef a/. 1984a),
Flow status
Critica1 flow velocities, ff i or ff o
Divergence
Flutter
Intema1 flow (U i i=- 0, U o = O)
- -
Combined flow (Ui i=- O, U o = 0,010)
Annular flow (ff i = 0, ff o i=- O)
0,0255
0,0173
0.0136
0,0314
0,0288
0.0185
5
N
::; 4
x
1 CIo
;>;
5
'"
O"

'"
'"
"
=
o
;;;
c:
Š
iS m I
m=2
m=1
00
3
m=2
2
m=3
Uj +
v
m=3
23456789
Dimensionless internal flow velocity, U j x 10 2
10
11
Figure 11.99 The real dimensionless eigenfrequencies [20 of the 1o-gap damped-clamped steel-
water system as functions of the dimensionless inner flow velocity Ui (n = 3; m = 1,2, 3); open
symbols: antisymmetric modes; dosed symbols: symmetric modes. Both shells are flexible and the
fluid in the annulus is stagnant (Pa'idoussis ef a/. 1984a).
the narrowness of the annulus makes the critical values for annular flow lower than for
interna! flow, Also included in the table is a case with both annular and internal flows
and a fixed value of U o i= O. In this case, the critical values are intermediate; for this
conservative system, the effect of the two flows is wholly synergistic - cf, Figure 8,57
for flow in and around cylinders (Section 8,6.2).
Next, the more interesting case of both shells flexible is considered. In this case, we
have two sets of modes for each n: one with deformation of the two shells azimuthally in
phase (as in Figure 2,7, for instance), and the other with the modes in antiphase. A typical
frequency versus flow diagram for internalflow is shown in Figure 11.99. It is seen that it
is the antisymmetric m = 1 mode that is associated with the lowest critical flow velocity;
the higher fluid accelerations in this case generate the largest couplíng and added mass,
hence lower frequencies and critical flows - cf. Section 9.3.3. Several features of the
dynamícs are worth commenting upon as follows.

1378 SLENDER STRUCTURES AND AXIAL FLOW
2,5
o:
II)
E=
II) II)
u,.<::
'" CI)
t O
.- <=
-o <=
ta;;:
 o

-2,5
2,5
o:
II)
E=
II) II)
..;j n
]. O -
:.a
_ o
"'.....
'6 o
'"

-2.5
O 0,20 0,40 0,60 0,80 1.00 O 0,20 0.40 0,60 0,80 1.00
x/L x/L
(a) (b)
2.5
o:
II)
E=
II) II)
u,.<::
'" CI)
 O
.- <=
-o <=
- .-
"'.....
'g o

-2,5
2,5
o:
II)
E=
II) II)
u,.<::
'" CI)
í5...... O
CI) II)
g
"'.....
'6 o
'"

-2,5
O 0,20 0.40 0,60 0,80 1.00 O 0,20 0.40 0,60 0.80 1.00
x/L x/L
(c) (d)
Figure 11.100 Axial modal shapes of the inner and outer shells, characterized by Wj and W a ,
respectively, for the antisymmetric (n = 3, m = 1)-mode of the to-gap steel-water system of
Figure 11.99, at (a) Ui = 0.010, (b) Ui = 0.020, (c) U i = 0.0253 (just prior to the threshold
of divergence), and (d) Uj = 0.030 (after divergence); the annular fluid is stagnant (Ua = O);
(Pa'idoussis et al. 1984a).
(i) The critical flow velocities for divergence in Figure 11.99 are essentia11y identical
to those in Figure 7,56 where the outer shell is replaced by a rigid cylinder. The
reason for this is explained with the aid of the moda1 plots of Figure 11.100. It
is seen in Figure 11.100(a) that at V i = 0.010 the axial mode shapes are almost
c1assical and azimuthally antisymmetric, the latter as indicated by the different
signs for Wj and Wo at the beginning and throughout the half cyc1e of osci11ation

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1379
shown. In Figure 11.100(b) and (c) it is seen that the vibration of the outer shell
is diminished dramatically, and ceases entirely as the system becomes unstable by
divergence, as shown in Figure 11.1 OO( d); this reflects the diminution of the modal
frequency, as divergence is approached, and hence of the attendant accelerations
of the inner shell in that mode; hence, the fluid accelerations in the annulus tend to
zero, which effectively decouples the outer shell from the inner one. Divergence
occurs at the same value of U i, whether the outer shell is rigid or flexible.
(ii) Strong travelling wave components are present in some of the modes at higher val-
ues of Uj in this gyroscopic conservative system, as may be seen in Figure 11.101,
for flow velocities beyond the onset of coupled-mode flutter. This arises by the
merging of the antisymmetric m = 1 and m = 2 loci (Figure 11.99); however, as
shown in Figure 11.101, the resultant mode shapes are azimuthally symmetric.
(iii) Similarly nonclassical behaviour is displayed by the other modes ofthe system. For
example, in Figure 11.102 it is seen that at U i = 0.010 the antisymmetric m = 3
mode is nearly classical, but trave1ling wave components are nevertheless clearly
discernible, At Uj = 0.0365 [Figure 11.102(b)], on the other hand, when the locus
of this mode is closest to that of the symmetric m = 1 mode (Figure 11.99), the
character of the mode changes dramatically: it becomes symmetric also, and the
outer shell vibrates as if it were in its first axial mode, whilst the inner one vibrates
as if it were in a degenerate third mode. At U j = 0.050 [Figure 11.102(c)], the
modal shapes of both shells resemble that of the first axial mode; the amplitude
of motion of the outer shell is unaccountably relatively larger than that of the
inner one.
2.5
;::
<1)
s::=
<1) <1)
..;;
 t o
'6 e

 o

-2.5
2,5
;::
<1)
s::=
<1) <1)
..;;
].  o

_ o
"'.....
 o

- 2.5 o
(a)
0,20 0.40 0.60 0.80 ),00
x/L
o
(b)
0,20 0.40 0.60 0,80 1.00
x/L
Figure 11.101 Axial modal shapes of the inner and outer shells, characterized by Wj and w o ,
respectively, for the antisymmetric (n = 3, m = 1)-mode of the 1o-gap steel-water system of
Figure 11.99 while fluttering, at (a) V i = 0.050, (b) Vi = 0.090; the annular fluid is stagnant
(Pai:doussis ef a/. 1984a).

'"
N
lJ:Jqs l:JUU!jO
u:Jw:J:J1!jds!p j1!!p1!lI
'"
N
I
'"
N
lJ:Jqs l:Jno JO
u:Jw:J:J1!jds!p j1!!p1!lI
o

o
00
O
o
\O
O

><
o
'<t
O
o
N
O

O 
o

o
00
O
o
\O
O

'"
o
'<t
O
o
N
O
o é:
o

o
00
O
o
\O
O

'"
o
'<t
O
o
N
O

V)0 
N
I
o .
"O '"
O '"
s 6
A"O
<') :;
II
E 
<') c
115
:: '"
'-' '"
u ,-
'E "O
 's
sl:!:
S !a
=
'E 
'"
-š-š
... ..
.£
;>.
o
, II
u ._
&I
"'---

ó"O
;:> 
"0\1')
'O

>. o,...:.,
 II
 ._ 00
,I 
 e:
ót
q
'0
 II
..c ,-
I
 ---

o ....
"O '"

... .
-
c-
, 
..s 5Í)
"O
"O
s
ťJ 
 >.
"O '"
g ...
s:; £
'<'i '"
. 
<ť:
£
M '"
= 
"1 co
,..; ,
-I
-š
"O

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1381
(iv) After the system has 10st stability in the symmetric m = 1 mode, at Vi = 0.0406,
it remains unstable in the 10west three modes, but the other three modes do not
become implicated in any instabilities, their frequencies tending to asymptotes
on the right-hand side of Figure 11.99. A possible explanation is provided by
Figure 11.103, showing the modal shapes for the symmetric m = 2 mode. With
increasing flow, this mode as well as the symmetric m = 3 mode do not become
altered as much as the others with increasing V i ; they remain symmetric and
they retain their quasi-classical demeanour, although the outer shell amplitudes
become relatively larger than those of the inner shell [Figure 11.103(b)], in these
cases also. This may be the reason for the three highest frequencies tending to
asymptotic values of high V i : at these flow velocities the outer shell is vibrating
essentially independently of the inner one in these modes.
We next consider the dynamics with annu/ar fiow (and both shells flexible), A typical
frequency versus flow diagram is shown in Figure 11.104. In this case, the system with a
flexible outer shell is less stable than if it is rigid, as seen by comparison with Figure 11.99,
and as tabulated in Table 11.13. This is clearly due to the fact that the flowing fluid comes
into direct contact with, and 'excites' directly, both shells - whereas in the previous
case the outer shell could become excited only through indirect inertial coupling with the
stagnant annular fluid.
The overall dynamical behaviour is more intricate in this case than for internal flow.
Divergence in the antisymmetric m = 1 mode occurs at U o = 0.00958; then, coupled-
mode flutter occurs in conjunction with the m = 2 antisymmetric mode at U 0 = 0.0136,
This is followed by renewed divergence at point E and coupled-mode flutter at point F -
the latter involving the second antisymmetric and fint symmetric modes. Beyond point
G, the involvement of the symmetric m = 1 mode in the flutter instability ceases, when
it is replaced by the antisymmetric m = 3 mode. Moreover, in this case of annular flow,
all modes eventually become unstable, the symmetric ones becoming associated with
divergence and flutter at high values of V o , The fact that instabilities are possible in alt
modes no doubt reflects the direct excitation ofboth shells by the annular flow, as discussed
earlier.
Modal shapes are available for these calculations as well, but are not presented here
in the interests of brevity; see Chan (1984). The general character of the modal shapes is
Table 11.13 The lowest critical flow velocities
for divergence of the fo-gap clamped-clamped
steel-water system (n = 3) subjected to either
internal or annular flow (Paí'doussis et a/. 1984a).
Flow status Critical flow velocities, U i or 7J o
Only inner shell
flexible
Both shells
flexible
Internal flow
Annular flow
0.0255
0.0136
0.0255
0.00958

1382
2.5
SLENDER STRUCTURES AND AXIAL FLOW
i:i
"
S==
" "
u..c:
.. '"
  O
'i3 f@
c;t:
'i3 o
..
p::: -2.5
2,5
i:i
S==
" "
u..c:
.. '"
  O
g
0;"'"
'i3 o

-2.5
O
(a)
0,20
0.40 0,60 0,80 1,00
x/L

O
(b)
0.40 0,60 0,80 1.00
x/L
0.20
Figure 11.103 Axial mode shapes of the inner and outer shells, characterized by Wi and w o ,
respectively, for the symmetric (n = 3, m = 2)-mode of the -to-gap steel-water system of
Figure 11.99 at (a) U i = 0,010 and (b) U i = 0.090; the annular fluid is stagnant (Pai'doussis et a/,
1984a).
5
m=3
N
o 4
x
Ict
:>;
u
,::
"
;::!
cT
"
ti::
'"
'"
"
-a
,S

"
S
O
3
m=2
2
m=3
i (

....---0
'-F
;;E
2 3 4 5 678
Dimensionless annular flow velocity, U o X 10 2
Figure 11.104 The real dimensionless eigenfrequencies Do ofthe l -gap clamped--clamped steel-
water system as functions of the dimensionless annular flow velocity U o (n = 3; m = 1,2,3),
Open symbols: antisymmetric modes; clo sed symbols: symmetric modes. Both shells are flexible
and the fluid in the inner shell is stagnant (Pai'doussis et a/. 1984a),

ANNULAR- AND LEAKAGE-R..OW-INDUCED INSTABIUTIES 1383
Table 11.14 Effect of compressibility on
divergence of the !-gap clamped-clamped
steel-air system with variable annular ftow and a
constant, small inner ftow, for n = 2 (Pai'doussis
eta/,1984a).
Mach numbers Critical ff o for divergence
M Mo m = 1 m=2 m=3
1
0.001 0,001 0,802 1.10 1.47
0.025 0.500 0.797 1.08 1.44
simi1ar to that shown in Figures 11.100-11,103, with the following significant differences:
(i) both shells a1ways participate in the vibration; (ii) symmetric modes remain symmetric,
and antisymmetric modes continue being antisymmetric as V o is increased.
A simp1ified model for predicting the critica1 flow ve10cities for on1y one of the shell
flexib1e by Horáček (1993) is presented in Section 11.7.3.
Finally, some ca1culations have been conducted with compressžbleflows, These ca1cu1a-
tions are much more comp1ex, sometimes running into numerica1 difficu1ties. Hence they
have been confined to annu1ar flow with a rigid outer cy1inder; further, to avoid difficu1ties
with Ki becoming singu1ar at Vi = O, a small constantinterna1 flow is taken (V i = 0.005).
Moreover, the method of solution deve10ped by Pa'idoussis ef al. (1984a) is only va1id
provided the eigenfrequencies are rea1, hence on1y up to the thresho1d of divergence.
Some typica1 resu1ts for a stee1-air system are given in Tab1e 11.14, where the resu1ts
for Mo = 10- 3 are compared to those for Mo = 0.50. t It is seen that the effect is
perp1exing1y, and disappointing1y, small: ofthe order of 1 %. The effect on the frequencies
is a1so small, but becomes more important for the 1arger m. Indeed, the sma11ness of the
effect of compressibi1ity in these resu1ts is due to the fact that L / a is re1ative1y 1arge, and
hence the va1ue of n associated with divergence in small; for 1ess slender shells, though
such ca1cu1ations have not been done by Pa'idoussis ef al. (1984a), the effect is not so
sma11.
A simi1ar ana1ysis, in a finite-e1ement framework for dea1ing with shells ofaxia11y
variab1e stiffness, has been done by Fujita ef al. (1993). Some resu1ts are discussed in
Section 11.7.2(b).
The important overa11 conc1usion from this study is that the instability thresho1ds for
coaxia1 shells subject to annu1ar flow are considerab1y 10wer than those invo1ving a sing1e
shell subject to interna1 or externa1 flow (as in Chapter 7). For stee1 shells, simi1ar to those
ana1ysed here, conveying water, the dimensiona1 critica1 flow ve10cities for the case of a
sing1e shell were typically over 100 mis, which is c1ear1y beyond the range of practica1
interest in industria1 systems. On the other hand, for coaxia1 shells conveying fluid in
both the inner and annu1ar spaces, the critical flow ve10cities are of the order of tens of
metre s per second: that is, they reach into the range of nonnal industria1 app1ications.
Consequently, the study of viscous and nonlinear effects, in the subsections that follow,
is of practical import, as welI as academic interest.
tl n these calculations, while U o is varied, Mo is artificially kept constant.

1384
SLENDER STRUCTURES AND AXIAL FLOW
11. 7. 1 (b) Steady víscous effects
Steady viscous effects have already been considered for internal flow in a solitary she11, in
Section 7.5.2, The equations ofmotion in the case of coaxial shells are: equations (7.147)
for the inner she11, and
" '2 2
Llo(U o , Vo, W o ) + qlo U o + q2o(V + W o ) + q3o(U' - W o ) = Yo(a Uo/at ),
L2o(U o , Vo, W o ) + qlo V + q3o(V' + w) = Yo(a 2 v o /at 2 ), (11.299)
'" [ 2 2 ]
L3o(U o , VO, w o ) - qlo W o - q3o(U o - v + w') = -Yo a wo/at - qo/ Psh
for the outer one, where ( )' = b a( )/ax, as opposed to ( )' = a a( )/ax, band a being
the radii of the outer and inner shells, and {qlo' q2o, q3o} as given by equation (7.148).
Other symbols are as in Section 7.5.2; subscript o [o in subsection (a)] denotes the outer
shell.
The next task is to determine the basic loads in this case, fp xi' Prd and fpxo' Pro}'
both sets having the form of equations (7,145). To do this, the averaged pressure field in
the annulus must be determined in the same way as for internal flow. Proceeding as in
Section 7.5.2 - see also Appendix Z.3 - the pressure in the annular region is found to be
_ { 2b Po } 2 2 l r 1 ( 2 2 )
Po(x,r)-- b2-r Uwox-PoUro+Po a;: Uoo-U ro dr+Po(O,a),
(11.300)
where r m is the radius at which U o is a maximum (Appendix Z.3); U wo and Uwi (to be
used later) are the shear flow velocities of the annular flow on the outer and inner she11,
respectively, defined by
2 _ dUo I _ T woo .
U wo - - 1Jo- d = -,
r r=b Po
2 dUo I
U wi = 1J o- d
r r=a
Twoi
(11.301)
Po
With the pressure in the inner and annular regions determined, equations (7.155) and
(11.300), it is now easy to obtain the basic 10ads on the shells.
On the inner shell, the radial basic 10ad is Pri = Pi (x, a) - Po (x , a); utilizing (11.300)
evaluated at r = a and (7,156), it may be expressed as
- { 2Pi 2 [ 2b ] 2 }
Pri = ----;;U ri + b 2 _ r Po U wo x + Pi(O, a) - Po(O, a),
(11.302)
where the fact that u?:o must vanish at the wan has been used. The axial basic 10ad is
Pxi = Twi + Twoi. Hence, through equations (7,152) and (11.301), one obtains
Pxi = Pi U;i + Po U;oi'
(11.303)
On the outer shell, the radial basicload is Pro = Po (X, b) - Pe, where Pe is the uniform
static pressure of the fluid surrounding the outer shell; hence, utilizing equation (11.300),
one obtains
- _ { 2b } 2 l b 1 ( 2 2 )
Pro - - b 2 _ r Po Uwox + Po a ;: U oo - u ro dr + Po (O, a) - Pe, (11.304)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1385
The axial basic load is simply equal to Twoo, or
- U 2
Pxo = Po 7:00'
(11.305)
The quantities Uú, Uwi, U wo are evaluated from empirical data in a similar manner as
in Section 7.5.2 and as given in detail in Paldoussis et al, (1985, appendix 2); in particular
[ b2 _ a2 ] 1/2
r m = 21n(bja) ,
-2 1 { b2 - r } 2
U wo ="8 b(b _ a) fo U o '
(11,306)
where fo is a friction coefficient, obtained in the same way as Ji for internal flow. The
integral in (11.304) is shown to be negligible. Therefore, if the pressures in the inner,
annular and outer fiuids are assumed to be equal at x = L, then
2Pi 2 { 2b Po } 2
Pi(X)  -Uú(L - x) + Pe, Po(X)  2 2 Uroo(L - x) + Pe,
a b - r m
- '" 2Pi 2 { 2b Po } 2 _ '" { 2b Po } 2
Pri - -Uú(L - x) - b 2 2 Uroo(L - x), Pro - 2 2 Uroo(L - X).
a - r m b - r m
(11.307)
The method of solution then proceeds in the same way as in subsection (a), with the
perturbation pressures presumed to be the same as given by inviscid theory. The system
parameters used in the c alc ulations ar e those in (11.298).
A typical diagram of Qo versus U o is given in Paldoussis et al. (1985, figure 3); it is
qua1itatively similar to that in Figure 11,104, but the two highest frequencies continue to
increase with U o even at U o = 0.12, Quantitatively, however, the critical flow velocities
for annular flow are considerably lower than forinviscid flow, as may be seen in Table 11.15
for (i) the outer shell rigid (i.e. replaced by a rigid cy1inder) and (ii) both shells flexible;
the opposite is trne for internal flow,
The explanation for this may be grasped easily, in terms of the pressurization of the
flowing fluid required to overcome frictional pressure drop, and of the attendant differences
Table 11.15 Comparison of the dimensionless critical flow velocities for
annular and intemal flow, U o and U i, respectively, for inviscid and viscous
flows in the to-gap clamped-clamped system (n = 3); 'C-m.' stands for
coupled-mode (Pai:doussis et al. 1985).
Flow Phenomenon Critical values oHli x 10 2 or TJ o x 10 2
Inner shell flexible Both shells flexible
Viscous Inviscid Viscous Inviscid
Annular Divergence 0,36 1.36 0,36 0,96
flow only C-m flutter 1.04 1.85 1.31 1.36
Inner Divergence 4.14 2.55 4.14 2,55
flow only C-m flutter 4,86 3,14 4.82 3.13

1386
SLENDER STRUCTURES AND AXIAL FLOW
Table 11.16 The effects of additional pressurization and of a wider
annulus on stability of two coaxial shells with clamped ends subject to
annular flow (n = 3). The inner shell is filled with stagnant fluid. The
units of Pi(L) and Po(L) are kPa (Pa'idoussis et a!, 1985),
System
1
1(j-gap
Pressures
TJ o x 10 2 for divergence
Pi(L) = P(}(L) = Pe
Pi(L) = Pe, Po(L) = Pe + 50
Pi (L) = Po(L) = Pe + 50
P i (L) = Po(L) = Pe
0.36
0,21
0,36
0,85
I
rgap
in pressure on the two sides of the inner she11 - see equation (11.307). Thus, in the
case of annular flow, the pressure on the annulus side of the flexible she11 is higher than
on the interior side for O::: x < L. Now, since external pressurization tends to collapse
(destabilize) the she1l, it is clear that the system would lose stability due to flow at lower
U o for viscous flow than for inviscid flow (where the transmural mean pressure difference
is null). The situation for internal viscous flow is exactly the opposite (see Section 7.5.2).
Additional external (annular) pressurization makes the system even 1ess stable, as seen
in Tab1e 11.16; for check:ing purposes, if both internal and annular fluids are equally
pressurized, the effect is cancelled out.
AIso shown in Table 11.16 is the critical ve10city for a wider annulus (the ! -gap system);
naturally, it is higher than for the lo -gap system. However, viscous effects in this case
are equally important as for narrower annuli: U o = 2.06 x 10- 2 for invsicid flow versus
0.85 x 10- 2 for viscous flow; the percentage reduction is similar to that for the l -gap
system.
These results demonstrate the importance of steady viscous effects associated with
the effective pressurization resulting with pressure drop. The magnitude of the effect
may appear surprisingly large. In this regard, it is found that, in the absence of flow,
a uniform pressurization of 36,3 kPa in the annulus is sufficient to collarse the inner
she11 if the outer one is rigid. On the other hand, for divergence of the 10 -gap system
with flow, the pressurization at x = O is 146.4 kPa at x = O and zero at x = L, with
a linear variation in-between. Thus, the order of magnitude of the mean pressurization
in the two cases is similar. Of course, in this particular set of calculations Lib = 10,
which is rather large. The effect would be smaller for lower Lib. These results are a1so
particular to Pí(L) = Po(L) = Pe (in most cases). The effect would be different if, say,
Pí (O) = Po(O) = Pe. ff both flows were present simultaneously, viscous effects could
either enhance or depress stability, depending on the geometry and whether the two flows
are concurrent or countercurrent (Pa"idoussis et al. 1992).
Calculations on the effect of some of the main system parameters, n am e1y Lla, hla
and (b - a)la, on the most critical annular flow ve10city (i.e. the lowest U oc for any n),
denoted by U; c, in the case of a rigid outer cylinder have been conducted by Pa'idoussis
et al. (1992). They are shown in Figure 11.105. It is seen in Figure 11.105(a) that, even
for Lla = 5, the difference between inviscid and viscous results is large - though the
percentage difference is smaller than for larger L I a. As for internal flow - see Figures 7.12
and 7.19 - U; c is associated with larger n for smaller LI a; more so, in the case of viscous

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1387
3
n=4
2
N
o
X
*0
IQ
6
O
(a) O Ua 25
4 3
2
3
3
2
N
o
x 2
I}
4

O O
O 5 10 15 20 25 0,0 0.4 0.5
(b) hla x 10 3 (c)
Figure 11.105 The overall (lowest) dimensionless critical annular flow velocity, U: c , for a
clamped-clamped shell in a rigid coaxial outer cylinder, and a stagnant inner fluid (Pi = Po).
(a) The effect of Lla for the fo-gap system; (b) the effect of hla for the fo-gap system; (c) the
effect of gap; default parameters as in (11.298). The values of n associated with each U: c are indic-
ated in the figures, o, Inviscid flow; Ll, with steady viscous effects taken into account (Pai:doussis
et a/. 1992a),
ftow. The results in Figure 11.105(b) on the effect of h / a are totally expected, So are those
in Figure 11.l05(c) on the effect of gap; furthennore, it is seen that U: e tends towards
reaching asymptotes as (b - a)/a becomes Iarge.
11.7. 1 (c) Unsteady viscous effects
The general treatment of unsteady viscous effects for internaI ftow has been given in
Section 7.5.4. The problem has aIso been studied for annular ftows (EI Chebair et al.
1990). The analysis follows the same procedure as for internaI ftow, and it wil1 not be
given here in the interests of brevity.
As seen in Table 11.17, taking unsteady viscous effects into accout for the 1 -gap
steeI-water system has no effect on U o e for divergence; steady viscous effects, however,
as expected, do have an important effect. Taking unsteady viscous effects into account

1388
SLENDER STRUCTURES AND AXIAL FLOW
Table 11.17 The dimensionless critical annular flow velocity U oe for n = 3,
according to inviscid theory, and taking unsteady or steady viscous effects
into account, or both, for the ťo- and ,Óo -gap simply supported shell systems
(El Chebair et a/. 1990),
Unsteady inviscid theory
U nsteady viscous effects
Steady viscous effects
Unsteady and steady viscous effects
I
10 -gap
1.03 x 10- 2
1.03 x 10- 2
2.35 x 10- 3
2.35 x 10- 3
I
lOO- gap
3,1 x 10- 3
8,0 x 10- 3
4.95 x 10- 4
5.0 x 10- 4
for the 16o -gap system t has a substantia1 stabilizing effect ( U o e = 8 X 10- 3 versus
3.1 x 10- 3 ), However, this effect is a1most totally swamped by the steady viscous effects.
AU the results in Tab1e 11.17 are for systems with L j b = 10. For 1ess slender systems,
unsteady viscous effects might be more pronounced.
11.7. 1 (d) Nonlinear effects
The dynamics of shells in a rigid coaxia1 cylinder, subject to annu1ar ftow has been studied
by Amabili et aZ, (2001). As in Section 7.8.2, Donne11's nonlinear shallow shell theory is
used, with a potentia1 ftow model for the annu1ar ftow.
Proceeding as in Pai'doussis & Denise (1972) and Section 7.2,3, and considering a
compressib1e fluid, the equiva1ent of (7.45) in this case for the pressure on the suďace of
the she11 is
K(Ab) ln (Aa) - 1(Ab) Kn(A.a) ( a a ) 2
Plr=a = P A[K(Ab) 1(Aa) _ 1(Ab) K(Aa)] at + U ax W,
(11.308)
according to this travelling wave type solution, where A 2 = a 2 - M 2 [a + (QjU)]2,
a = mn j L for the mth axia1 mode number, ln and Kn are the modified Besse1 functions
of the first and second kind, Mis the Mach number, and Q the oscillation radian frequency,
If the Fourier transform method is used instead, as in subsection (a), the suďace pressure
for w = A(t) sin(mnxjL) cos(nO) is given by
L 1 00 m[l-(-1)me-iaL] K(Ab)ln(Aa)-I(Ab)Kn(Aa)
plr=a = P"2 cos(nO) -00 m 2 n 2 - a 2 L2 A[K(Ab) 1(Aa) - 1(Ab) K(Aa)]
x [A(t) + 2ia U A(t) - a 2 U 2 A(t)] e iax da. (11.309)
The solution procedure for the coup1ed system is comp1ete1y ana1ogous to that described
in Section 7,8.2 and will not be given here in detail. The moda1 expansion for w (x, O, t)
is as in equation (7 .254b).
tThe parameters for the tt-gap system are as in (11.298), The ,6o -gap system has the same Lib, and
alb = 99/100, hla = 5,05 x 10- 3 and {Ji = 25.36.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1389
Typical results showing the variation of the An 1 and A n 2 modal components with flow
velocity U are shown in Figure 11.106, for system parameters given in the caption and
n = 3; a more complete set of results is given in Amabili et aZ. (2001). Of course,
Donnell's shallow shell theory is not very accurate for such low n, but these calcula-
tions were done to compare with experimental data - see subsection (e). Branch 2 in
Figure 11.l06(b) bifurcates from branch 1 (undeformed configuration) at U = 81.8 mis,
which is the threshold of instability predicted by linear theory. So does branch 3 in
Figure 1 1. 106(a) , at U = 115 mis. Both bifurcations are static and they are strongly
subcritical.
ln connection with branch 3, it is particularly important to note that the system, given
enough perturbation, may develop divergence for U as low as U = 25.5 mis. Progressively
smaller perturbations are necessary for divergence to be precipitated as U is increased.
The same applies to branch 2 for 35.7 mis < U < 81.8 mis. Hence, for U > 25.5 mis, at
least two (four starting from 35.7 mis) equilibrium positions coexist with the undeformed
configuration of the shell. Given enough perturbation, divergence is possible at flows as
low as one-third the value predicted by linear theory!
15
10
5
 o
..
'<:
-5
-10
-15
o
(a)
15
10
5
 O
'<:
-5
-10
-15
O
(b)
I
""o
............
4 / /
,
,
,
........................::::::::::: }f»-...---
-------- \..
20
40
60
80
100
120
,
/
-"-...
..--.........
--"'-..
.- -
--..o...--
...-....---
.-'
- .....
,
,
,
,
,
.
20
40 60 80 100
Flow velocity, U (mis)
120
Figure 11.106 Amplitude of nonoscillatory solutions versus flow velocity for an elastomer shell
conveying air for n = 3 (a = 24,7 mm, Lla = 5.5, hla = 0.05, bla = 1.25, E = 2.43 MPa,
v = 0.47, Ps = 1220kg/m 3 , P = 1.25kg/m 3 ): -, stable branches; ---, unstablebranches.
(a) Amplitude of the m = 1 mode, A3,tl h; (b) that of the m = 2 mode, A3,21 h (Amabili et aZ,
2001).

1390
SLENDER STRUCTURES AND AXIAL FLOW
6
 5
,<:M" 4 Branch 3
:::
o
:::
.g 3 Branch 2
:.a
:::
o
" 2
]
] Stable
o
20 30 40 50 60 70 80 90
F\ow velocity, U (mis)
Figure 11.107 Basins oť attraction oť the undisturbed equilibrium and the biťurcated solutions,
showing minimum amplitude oť disturbances oť A3,! mode shape necessary to precipitate divergence
(Amabili et al, 2001),
The amount of perturbation necessary to induce divergence depends on its fonn. A
typica1 resu1t for perturbations of the fonn ofthe A3,l component is shown in Figure 11.107.
It is seen that the perturbations necessary are not unreasonably 1arge, being of the order
of 1h to 3h.
As in the case of interna1 flow, no secondary bifurcations 1eading to flutter have been
found, suggesting that the coupled-mode flutter predicted by linear theory may in fact not
materia1ize,
The importance of these results cannot be overemphasized, especially the strongly
subcritica1 divergence. lt is not yet known how qua1itative1y genera1 this resu1t is over the
who1e parameter range and when viscous effects are taken into account, but its practica1
importance is se1f-evident: designers and operators need worry that the actua1 critica1 flow
velocity, a1ready in the practica1 engineering range, may, in fact, be very substantially
lower than predicted by linear theory; and this, without considering imperfections.
11.7. 1 (e) Experiments
Experiments have been conducted by E1 Chebair et al. (1989). The apparatus used, for
experiments with e1astomer shells and air-flow, is shown in Figure 11,108. To correspond
best with the theory in subsection (b), the pressure of the stagnant inner air was made
equa1 to that at the annu1ar flow at x = L, as shown in Figure 11.108(b). She11 motions
were sensed by two fibre-optic (MTI KD-100 fotonic) sensors, azimuthally separated
by 140 0 , allowing identification of the dominant circumferentia1 wavenumber n. The
signals, proces sed by a dua1-channel FFf ana1yser al10wed separation of the different
moda1 components.
General1y, the second and third circumferentia1 modes, n = 2, 3 (and, in both cases,
the first axia1 one, m = 1) of the shell were most prominently excited by the turbu1ent
flow. The amp1itude of vibration was very sma11, typically 1ess than 0.04 mm (measured
at mid-span of the she11) for annular flow ve1ocities, U, in the range O-50 mis,
As the flow ve10city was increased, the frequency of both modes, n = 2, 3, was
diminished, as shown for a typical case in Figure 11-,109. In this particu1ar case, divergence
occurred at U = 45.5 mis, and the shell buckled in the n = 2 mode. The amplitude of

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1391
Airout
Honeycomb
Table
Screen 4
Screen I
Screen 3
Screen 2
(a)
Airin
(compressed air supply)
(b)
Figure 11.108 (a) Schematic diagram of the apparatus for clamped-clamped elastomer shells
subjected to annular air-flow, (b) Arrangement for ensuring that the mean pressure of the inner
stagnant fluid, PA, is equal to that of the mean static pressure in the annular flow at x = L, PE
(El Chebair et al. 1989).
the buckling was very large, and the two sides of the she1l in its central portion actually
touched, as shown in Figure 11,11 O( a). In some cases, as the ftow velocity was increased
further, the buckled shape was transfonned quite abruptly from n = 2 to n = 3, ln other
cases, however, typically for the smaller values of LI a, buckling occurred first in the
n = 3 mode, as shown in Figure 11,11 O(b).
As seen in Figure 11.109, agreement in the frequencies is reasonable for the dominant
mode (in this case n = 2, m = 1) at low ftows. The critical ftow velocity, however,
indicated as a vertical chain-dotted line, is overestimated by linear theory (the theory of
subsection (b), taking steady viscous effects into account); in fact, theory predicts that the
system should lose stability in the n = 3 mode, at a slightly smaller value of Ve than the
n = 2 mode - in which it loses stability in the experiment.
A summary of some of the results obtained in these experiments and comparison with
theory for (i) the critical circumferential mode, ne, and (i i) the critical ftow velocity,
Ve, for divergence are given in Table 11.18. It is seen that although both theory and
experiment predict that ne increases with diminishing Lla and (b - a)la, the agreement
is not complete; in this regard, the effect of imperfections should be invoked. In particular,
the state of zero extemally imposed tension and moments on the shell, as assumed in the
theory, is very difficult to achieve experimentally.

1392
SLENDER STRUCTURES AND AXIAL FLOW
45
5
40 --0..................
........'O
35
30
"'.
e; 25
:>,
u
"
"
S- 20


15
10
o
O 10 20 30 40 50 60 70
Flow velocity (mis)
Figure 11.109 Variation of frequency with flow velocity of the two principal modes excited
by the annular flow (see Figure 11.108): - - - , experiment; -, theory of subsection (b);
L/a = 6, (b -a)/a = 0.25, PA = Ps (EIChebairetal.1989).
8
@
(a)
(b)
(c)
Figure 11.110 Photographs of the buckled shell in (a) n = 2 mode, (b) n = 3 mode, with the
cross-sectional shape of the shell sketched below in each of the two cases (EI Chebair et al. 1989),
(c) The post-divergence shape of the shell as predicted by nonlinear theory for n = 3, m = 1 at
U = 30 mis (Amabili et al. 2001).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1393
Table 11.18 Comparison between experiment and the
theory of subsection (b) for a clamped-clamped shdl sub-
ject to annular air-flow (a = 24,7 mm, hla = 0.05, E =
2.43 MPa, v = 0.47, Ps = 1220 kg/m 3 ); (El Chebair etal.
1989).
(i) The circumferential mode for divergence, ne
L/a (b - a)/b Experiment Theory
5,5 0.25 3 3
6.0 0.25 2 3
6.5 0.25 2 3
7.0 0.25 2 2
6,0 0.10 3 3
6,0 0.50 2 2
(ii) The critical flow velocity, U e (mis); L/a = 6
(b - a)/b PA. PB Experiment ne Theory
0.10 PA = PB 29.7 2 32,0
0.25 PA = PB 45.3 2 58,0
PA =0 29,6 2 41.4
PA = PB 48.6 3 55,5
0.50 PA = PB 59.2 2 79.3
Table 11.19 Comparison of linear and nonlinear invis-
cid and linear viscous theories with experiment for diver-
gence shown in Figure l1.1lO(b) and (c); (Pai'doussis
et al. 1989; Amabili et al. 2001).
Theory/experiment
Critical flow velocity, U e (mis)
Linear inviscid
Linear viscous
Experiment
Nonlinear inviscid
81.8
55,5
48,6
25.5 < U e < 81.8
The linear theory consistently overestimates Ve, by 8 to 40%, Partly, this discrepancy is
undoubted1y due to the fact that the divergence is subcritica1, as predicted by the nonlinear
(inviscid) theory of subsection (d); imperfections, however, must also p1ay a role.
No flutter of the shell was ever encountered as the flow was increased to values very
substantially above that at the thresho1d of buckling, in contradiction to the predictions
of 1inear theory, and in agreement to those of non1inear theory. For the experiment of
Figures 11.1 09 and 11.11 O(b), the critica1 flow velocities are compared in Tab1e 11.19,
Unfortunately, at present, no theoretical results are avai1able for non1inear theory with
viscous effects taken into account, though the predictions by inviscid nonlinear theory
are qualitatively promising. The theoretically computed buckled shape of the shell in
Figure 1 1.1 lO(c ) certain1y looks very much like the experimental one.

1394 SLENDER STRUCTURES AND AXIAL FLOW
11.7.2 Coaxial cantilevered shells
ln this section, an analytical model for this system, taking into account steady viscous
effects is described first, followed by a discussion of experiments and then of a CFD
model, taking also unsteady viscous effects into account. Most of the work on this topic
has been done by Pa'idoussis et a/. (1991, 1992a), Fujita et a/. (1993) and Nguyen et al.
(1993a,b,1994).
11. 7.2(a) Analytical model of Pai'doussis, Nguyen & Misra
This model deals with the dynarrůcs of a cantilevered shell in a coaxia1 rigid cylinder
subjected to annular flow, ar to both annular and internal flow (Pa'idoussis et a/. 1991).
Steady viscous forces, due to pressurization and surface traction are taken into account,
while the perturbation pressure is computed by inviscid theory, as in Section 11.7.1 (b).
The framework of this analytical model is quite sirrůlar to that in Section 11.7.1 (b),
except that in this case an outflow model is incorporated; hence extensions to the cantilever
beam eigenfunctions are used for the axia1 variations in fluid displacement beyond the free
end, as in Section 7,3.2. The analysis, having been given in one form or another in other
parts of the book, is not given here in detail, but a summary may be found in Appendix ZA.
A typical Argand diagram for a 1 -gap steel-water system with stagnant inner fluid and
parameters as in (11.298) is shown in Figure 11.111 for n = 2, for both purely inviscid
theory and with steady viscous effects taken into account.
6.0
11m = 1
o m=2
. m=3
4,0
468
,0/& (D;) x 10 3
Figure 11.111 Typica1 Argand diagram for the dimensionless eigenfrequencies ni of the ťD-gap
cantilevered system with annular flow (and quiescent fluid in the inner shell), as the dimensionless
flow velocity U o is varied; system parameters are defined in equations (11.298). These ca1culations
are for n = 2, m = 1,2,3; - - - - , inviscid theory; -, with steady viscous effects taken
into account (Páidoussis et a!, 1991).
M
o 2,0
x 10
---
19:
 34
0,0 ..----038
O
14
15 10 15,2 - - - - Inviscid theory
-2.0 15
14 16 18 - Inviscid and steady viscous
-4.0
O
2
10
12
14

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1395
According to inviscid theory, the system loses stability by single-mode ftutter in the
second mode (m = 2) at U 0 = U o e ::::: 3.4 X 10- 2 . The evolution of the first-mode (m = 1)
locus is not shown beyond U o = 0.7 x 10- 2 , but suffice it to say that this mode remains
stable. The following observations have been made: (i) for this particular system, the
overall critical ftow velocity, U: e , is actuaUy associated with n = 6 (and m = I). (ii) the
same applies for wider gaps (1/5, lilO), but then stability is first lost by divergence,
followed by restabilization, and then ftutter in the same mode; for narrower gaps, sta-
bility  lost directly by ftutter; (iii) for any given n, as the gap becomes narrower, so
does U oe.
The inftuence of steady viscous effects on the stability of the system is profound, as
may been seen in Figure 11.1I1 and Table 11.20, for the 1 -gap system. Figure I 1.1 II
shows that this system, which according to inviscid theory loses stability by ftutter ( U o e :::::
3.4 X 10- 2 , m = 2), now loses stability by divergence at a ftow velocity approximately
ten times smaller ( U o e ::::: 0.34 x 10- 2 , m = 1), followed at considerably higher ftow
velocity ( U o e ::::: 1.5 X 10- 2 ) by coupled-mode ftutter.
The values of U o e for different n are shown in Table 11.20, where for the moment the
results on the far right of the table (M #- O) are ignored. It is seen that, for each n, U o e
according to the viscous theory is one order of magnitude less than that obtained via inviscid
theory - except for n = I, for which the U oe are sensibly the same, similar to the case
of internal ftow. With viscous theory, there is a local minimum U o e for n = 3, similar to
that for inviscid theory, as shown in the middle column of Table 11.20; however, as n is
increased further (n 2: 5), so does U o e become smaller, to at least n = 8. Mathematically,
Table 11.20 Critical annular flow velocities, U o e, for Ya-gap cantilevered system, for different n,
according to the two versions of the theory: inviscid and viscous; 'S.m.' stands for single mode,
'c.m.' for coupled-mode. The identification of the mode number m at instability for some of the
results with dissipation (J-L 1= O) was not clear-cut; hence, no value of m is given. The accuracy of
computed U oe is ::1::0.00002 (Paidoussis etal. 1991).
n Inviscid theory (f-( = O) Viscous theory (f-( = O) Viscous theory (f-( = 5 x 10- 3 )
- 2 Instability type - 2 Instabili ty type - 2 Instability type
U oe x 10 U oe X 10 U oe x 10
1 6,672 S.m. flutter (m = 4) 6.656 S,m, flutter (m = 4) 6.508 S.m. flutter (m = 4)
2 3.429 S.m. flutter (m = 3) 0.361 Divergence (m = 1) 0.361 Divergence (m = I)
1.51 C,m, flutter (m = 1) 1.51 c.m. flutter (m = I)
3 2,461 S,m, flutter (m = 3) 0,288 Divergence (m = 1) O,284 a Divergence (m = 1)
1.02 C,m. flutter (m = 2) 1.02 c'm, flutter (m = I)
4 2,534 S,m, flutter (m = 4) 0,294 Divergence (m = I) 0,86 S,m, flutter
1.08 C,m. flutter (m = I)
5 2,559 S,m flutter (m = 3) 0,306 c.m. flutter (m = I) 0,80 c'm, flutter
6 2,130 a Divergence (m = I) 0,208 C,m. flutter (m = I) 0,87 c'm, flutter
2,284 c'm, flutter (m = I)
7 2,650 Divergence (m = I) 0.140 C.m. flutter (m = I) 1.73 c'm, flutter
3,000 C,m, flutter (m = 1)
8 3.526 Divergence (m = I) 0.107 c.m. flutter (m = I) > 1.0 c.m. flutter
4.179 c'm, flutter (m = I)
a Lowest critical flow.

1396
SLENDER STRUCTURES AND AXIAL FLOW
this may be identified with the effect of the viscous force q30 = [(1 - v 2 )a Pr/ Eh],
which is the dominant pressurization tenn; it is seen from the second and third equations
(7.147) that q3 is associated with first and second derivatives of vand w with respect to (),
thus leading to tenns proportiona1 to n and n 2 ; hence, the destabi1izing effect ofthe steady
viscous forces increases with n, apparently without limit. Physica11y, however, this result
is unreasonable; as known from experiments, in stability occurs at a reasonab1y low n -
in the experiment s of subsection (b) for n = 2 to 4.
The key to this paradoxica1, nonphysica1 resu1t has been identified to be re1ated to
neg1ect of both structura1 and unsteady viscous dissipative effects, If they are inc1uded,
for examp1e, via a hysteretic model [by rep1acing E by E(1 + f.l, i), presuming osci11atory
motion - cf. equation (3.72)], as seen in the rightmost co1umns ofTab1e 11.20 the behav-
iour becomes more physica11y reasonab1e: (i) divergence occurs at U o e = 0.361 X 10- 2
and 0,284 X 10- 2 for n = 2 and 3, respective1y; (ii) then flutter at U o e = 0.80 X
10- 2 for n = 5 (or for n = 4 if f.l, = 5 X 10- 1 , instead of f.l, = 5 X 10- 3 ); see a1so
Figure 11.112.
In connection with Tab1e 11.20, it is noted that what is here referred to as coup1ed-mode
flutter arises either by coa1escence of modes with different m or from the two branches of
the same mode (from the +Re( Q i) and -Re (Q i) sides ofthe plane) as in Figure 11.111,
for instance. It is a1so noted that there are some entries for coup1ed-mode flutter not
preceded by divergence. In that case, the coa1escence of the two modes (or branches of
the same) occurs on the +Im( Q i )-axis, where the system is stab1e; the locus ofthat mode
then leaves the axis and crosses to instability, just as it wou1d for sing1e-mode flutter.
Indeed, such entries cou1d simp1y have been denoted as 'flutter'.
As in the case of interna1 flow, U oe for n = 1 and f.l, =1= O is in fact lower than for the
undamped system (Table 11.20).
The effects of L / a, h / a and (b - a) / a on stability are shown in Figure 11.113. For the
reasons a1ready given, these resu1ts have been obtained with a small amount of dissipation
added: in the case of part (a) according to the hysteretic model with f.l, = 5 x 10- 3 , and in
the case ofparts (b,c) according to a viscoe1astic model [replacing E by E(l +iX Q )] with
X = 5 x 10- 6 s. A1though these resu1ts are usefu1, the trends disp1ayed are qualitative1y
not unexpected and hence do not call for any discussion.
It is of interest to point out that the reduction of critica1 flow ve10city with annu1ar gap
according to inviscid theory is simi1ar for canti1evered shells and shells with supported
ends - contrary to the argument s presented in Pai'doussis et al. (1991, section 6.3). Thus,
comparing a 160 -gap to a l -gap system, a reduction of the order of 3 is obtained for both
sets of boundary conditions. t The destabilizing forces for both divergence and flutter are
proportiona1 to the added mass, which increases with diminishing gap by the same amount
in both cases.
Finally, the stability of the system being subjected to both interna1 and annu1ar flow
concurrently, for simplicity for inviscid flow, is illustrated in Figure 11.114. It is obvious
that the effect on stability of the two flows simultaneous1y is not as simple as in the case
of shells supported at both ends, where the two flows act pure1y additive1y, The difference
lies in the latter system being inherently conservative, whereas the cantilevered system is
inherently nonconservative.
An inviscid model has a1so been constructed by Fujita et a/. (1993) in a FEM framework.
tComparing the data by El Chebair et a/. (1990, table S) and by Paidoussis et a/. (1991, figure 9),

'"
:=:
x 4
I::}
8
6
D I' = 5 X /O-I
A I' = 5 X /0-2
01'=0
2
o
O
2 4 6
Circumferentia1 mode, n
/O
Figure 11.112 The effect of structural damping (variable f.L) on stability ofthe ťD-gap cantilevered
system with annular flow according to viscous theory (with steady viscous forces taken into account),
showing the emergence of a minimum U oc for flutter, when f.L ::ft O, at n = 5, - - -- ,Divergence
boundary; -, flutter boundary (Pai'doussis et al. 1991),
6
4
'"
:=:
x
I
2
O
O
(a)
4
 3
x
125 2
O
O
(b)
n=5
4
4
4
7
4 5 4 4
.-------1______ 2 _______2
/O 15 20
Ua
25
30
4
4
"2
 3
X
I 2
,O" .A 2
/3 /
,/ ..u'/
,/0' _/- 3
5 -.a-
42=:>/3
5 10 15 20
hla X 10 3
25
JY
- 3
/O 15
hla X /03
00
(c)
25
20
Figure 11.113 The overall critical (for any n) dimensionless flow velocity, U; c' for annular flow
in the cantilevered system, with the associated circumferential mode n shown for each data point:
- - --, divergence boundary; -, flutterboundary. (a) The effectof Lla for (b-a)la = 1110,
(b) The effect of hla and Lla for (b - a)la = 1110: 0, Lla = 5; , Lla = 10, (c) The effect
of hla and dimensionless gap for Lla = 10: 0, (b - a)la = 115; , (b - a)la = ! , The inner
fluid is stagnant, and a small amount of damping is assumed in the shell (PaYdoussis et al. 1991,
1992a),

1398
SLENDER STRUCTURES AND AXIAL FLOW
5
4
N 3
o
-
x
Ic
2
2
fY; x 10 2
3
4
Figure 11.114 Stability map for the shell simu1taneously subjected to internal and annular fiows,
for the -to-gap cantilevered system and n = 3; resu1ts obtained by inviscid theory (PaYdoussis
et a/. 1991),
Table 11.21 The effect of gap size on the
critical fiow velocity for tlutter of cantilevered
shells, U oe , and the corresponding circumferen-
tial mode for L / a = 6 (El Chebair et a/. 1989).
(b - a)/a
0,10
0,25
0.50
U oe (mis)
ne
37.1
51.1
55.3
4
4
3
11,7.2(b) Experiments
Experiments for cantilevered shells have been conducted by El Chebaír et a/. (1989)
and Nguyen et a/. (1993b) utilizing the same apparatus as for shells with clamped
ends (Figure 11.108), the same type of elastomer shells conveying aír, and very similar
instrumentation.
The El Chebaír et a/. experiments were only preliminary, no theory for cantilevered
shells being avaílable at the time. The parameters oť the shell used are as in the heading
of Table 11.18, The system lost stability in the n = 4 or n = 3 mode, depending on the
gap and Lja, by ftutter of large amplitude - with the inner sides of the shell coming in
contact, and the outer sides touching the outer cylinder wan, Some results are given in
Table 11.21.

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1399
Figure 11.115 Photograph of the free-end cross-section of cantilevered elastomer shell subjected
to annular air-flow, fluttering with n = 2; L/a = 8, (b - a)/a = 1/4 (Nguyen et at. 1993b),
A more extensive experimental programme was that ofNguyen ef aZ. (1993b), The shell
parameters in this case are a little different: hla = 0.055, a = 24.8 mm, E = 2,82 MPa,
Ps = 1160 kg/m 3 , v = 0.47 and hysteretic damping coefficient f.J., = 0.0195; the main
differences are in the values of hla, which is about 10% larger, and E, which is about
15% higher than in the EI Chebair ef aZ. experiments.
In this case, both divergence and ftutter have been observed, the former in only a few
cases, while the latter in all cases.
For gla == (b - a)/b = lilO and Lla = 8, divergence and ftutter were observed
sequentially as U o was increased, at U oe = 24.1 and 25.4 mis, respectively, both in the
n = 2 mode, In the case of divergence, a stationary ovalling of the shell was observed,
As the ftow was increased, the shell started oscillating with large amplitude, as shown in
Figure 11.115. For g I a = 1110 and LI a ::: 7.5 as well as for all cases with g I a = 1/4 and
1/2, only ftutter was observed. In most cases, the test was discontinued shortly after the
onset of ftutter. This is because, after a few cyc1es of oscillation, the shell started ftapping
violently against the inner wall of the outer plexiglas cylinder.
In all but one case, the instabi1ities observed were associated with n = 2, The exception
was one of several nominally identical shells with Lla = 5.5 and gla = lilO, for which
ftutter with n = 3 was observed.
The vibration measured by the fibre-optic sensors, analysed by the FFf analyser and
properly interpreted, allowed the identification of modal components and the generation
of plots such as those shown in Figure 11.116. The system, in this case, lost stability
by ftutter at U oe  40 mis, whereas the theory of subsection (a) predicts divergence at
U oe  50 mis, followed by ftutter at a higher ftow.
Theoretical and experimental stability data from all these experiments are compared
in Figure 11.117, Were it not for the discrepancy as to the type of instabi1ity for the
larger Lla (divergence in the theory and ftutter in the experiments), agreement in the

80
-r
m 2
-
- ..... .. m= 1
¥ 60
>,
 40
"
'"
<:T

 20
o
O
(a)
10 20 30 40 50 60
Ua (mis)
80 m=4
N"60 m=3
e; m-2
>,
 40
"
6-
 20

o
O
(c)
10
80
60
..
40
20 m=2
m=1
O 10 40 60
O 20 30 50
(b) Ua (mis)
20 30 40
Ua (mis)
50
60
Figure 11.116 Predicted and measured frequencies of flow-induced vibration of a cantilevered
shell as functions ofthe annular flow velocity for (a) n = I, (b) n = 2, (c) n = 3; (b -a)/a = 1/4,
L/a = 8: -, theory of Section 11.7.2(a); _, experiment (Nguyen et al. 1993b).
W 100
60
,----.. .X...!'....!.._. ť
F/

:! 40
*
::::,0
F
F
.
20
(a) g/a = 1/10
0 4
5
6
7
8
120
80
---ť-... F F
, ._.._-
F:: O
F
F
I
F
.
.
F
.
60
40
20
(b) g/a = 1/4
9
0 4
9
5
6
7
8
90 F

:! 60
*
::::,0
30
(c) g/a = 1/2
0 4 5 6 7 8 9
L/a
Figure 11.117 Overall (i.e. lowest for any n) critical annular flow velocity for flutter or divergence,
U;c' of cantilevered shell as a function of L/a, for (a) g/a = 1/10, (b) g/a = 1/4, (c) g/a = 1/2,
where g = b - a; also shown is the threshold of flutter in cases where stability is first lost by
divergence. - , - - - - , Theory of Section 11.7.2(a); _, experiment (flutter); o experiment
(divergence). F signifies flutter; D signifies divergence (Nguyen et al. 1993b).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1401
Table 11.22 Critical annular flow velocities, Uoe, from the
cantilevered shell experiments of Fujita et a/, (1993); D and F
stand for divergence and flutter.
Lla hla (b - a)la U oe (mis) ne D/F
6 5 x \0-3 5.0 x \0-2 15 3 D
6 8 x \0-3 3,7 x 10- 2 55 3 D
11.5 8 x \0-3 2,7 x \0-2 4] 4 D
74 4 F
thresholds of instability is judged to be reasonably good. In this regard, there are at
least two weaknesses in the experiments: the shells are not sufficiently thin for comfort-
able comparison with thin shell theory, and they are very flexible, and hence prone to
imperfections (e.g. at the supported end). Weaknesses in the theory are that it is linear
and it neglects unsteady viscous effects, The latter deficiency is addressed in the next
subsection,
Experiments were also conducted by Fujita et al, (1993), using PVC shells with a =
100 mm, L/a = 6 or 11.5, hla = 5 x 10- 3 to 10- 2 and (b - a)/a = 2,7 x 10- 2 to
5 x 10- 2 , thus with much thinner shells and much narrower annuli than in the foregoing
experiments. The fluid was air. A dummy cylinder was placed above the shell, but not
touching it, to regularize the flow beyond the free end. Seven optical transducers were
positioned at 45° intervals around the circumference.
Both divergence and flutter have been observed, the latter only for the longer shells,
The results on stability are summarized in Table 11.22. Typical results are also shown in
Figure 11.118.
The experimenta1 instabi1ity thresholds have been compared to the authors' own inviscid
flow theory, As expected, in view of the smallness of the gap, agreement is poor, since
steady viscous effects in this case must be very important indeed.
11. 7.2(c) CFD model of Nguyen, Pai'doussis & Misra
All the theoretical model s described so far for shells subjected to internal or annular flow
are inherently approximate since the perturbation pressure is determined by inviscid flow
theory, and then viscous effects, steady and unsteady, are taken into account separately.
However, in the model developed by Nguyen et aZ, (1994) to be described here, the flow
perturbations are obtained via a CFD solution of the linearized unsteady Navier-Stokes
equations, which are solved using the finite-difference-based time-marching technique
with artificial compressibility described in 11.5.4(a) and 11.5.5(a).
(i) GenemZ formulation. The system considered involves a cylindrical she1l in a con-
centric rigid outer cylinder, subject to incompressible annular flow. On the inner side, the
shell is filled with stagnant fluid of small density. The shell is of length L, free at x = L
and with a semi-infinite extension at x .:S O. The equations of motion are the first set of

1402
SLENDER STRUCTURES AND AXIAL FLOW
200
'N' 150
e;
o --- N= I
a -.- N=2
t:> '--N=3
<> ---N=4
O--'N=5
o
o
8
>,
u
<=
"
5-
Ji
 100 '<>...,
!: ............
"s -.o
..g 1>- ...00..... ....."
 "'''''Q''
___ '" 'IX, 
50 """"" '" <> 'e, <>, f;
'" '\.. ,
<.. \ '
\ o \
Divergence
8
--
-....
....
....
...
08 ................0
Q ...,
,
,
,
I
Flutter
(b)
o
O 10 20 30 40 50 60 70
(a) Annulus flow velocity,  (m/s)
Figure 11.118 (a) Dominant components of vibration for a PVC shell subjected to annular air-
flow (Lla = 11.5, hla = 8 x 10-3, g la = 0,027); (b) the shell at divergence (Fujita et a/. 1993).
equations (11.281):
t' ( ) " ( . ) ( .. ' ) ( a2u qx ) O
L I u, V, w + ql u + q2 v + w + q3 u - W - Y  - - = ,
al Psh
t' "... ( a 2 v qe )
L2(U,V,W)+qlv +q3(V +w)-y -- =0,
al Psh
t' ", . .. ( a2w qr ) O
L3(U,V,W)-qlw -q3(U -v +w )+y al2 - Psh = ,
(11.310)
in which the steady viscous forces ql, q2 and q3 are incorporated as in Section 7.5.2, and
determined as in Section 11.7.1 (b); qx, qe and qr represent the unsteady fluid forces acting
on the she1l, associated with the subscripted quantities. Internal dissipation in the shell is
incorporated according to a viscoelastic model, thus replacing E by E [1 + X (a / a l)] in y.
The standard cantilevered shell boundary conditions apply (Section 7.2.1).
The Navier-Stokes and continuity equations for the viscous annular fluid are as in
equations (11.174) and (11.175), which linearized are given by
{ aVr avr } dv ( aVr ) 1 ap
-+U(r)- =2- - ---
al ax dr ar p ar
{ I a ( avr ) Vr 1 a 2 v r a 2 v r 2 aVe }
+ v -; ar r ar - r2 + r2 ao2 + ax2 - r 2 ao '

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1403
{ ave + U(r) aV e } = dl) ( av e +! av r ) _  ap
ar ax dr ar r ae pr ae
{ I a ( ave ) Ve 1 a2ve a 2 ve 2 aV r }
+v -- r- --+--+-+--
r ar ar r 2 r 2 ae 2 ax 2 r 2 ae '
(11.311 )
{ avx av x dU } dv ( av x av r ) 1 ap
-+U(r)-+-v r =- -+- ---
ar ax dr dr ar ax p ax
{ I a ( av x ) 1 a 2 v x a 2 v x }
+v -- r- +--+-
r ar ar r 2 ae 2 ax 2 '
and
1 a 1 ave av x
--(rv r ) + -- + - = O.
r ar r ae ax
(11.312)
It is noted that {u, v, w} used in Section 11.5.4( a) has been rep1aced by { V X , v r , ve} here, to
avoid conťusion with the shell disp1acements. To obtain these equations, the ftow ve10city
fie1d has been assumed to be made up oť steady and perturbation components, so that
V x = U (r) + v x , V r = V r and Ve = Ve; the turbu1ent tota1 pressure, see equation (2.106),
is Pt = P + p. The ftow has been assumed to be genera11y turbu1ent; hence, v = V m + Vt as
in Section 2,2.1(f), made up oť a mo1ecu1ar and a turbu1ent (eddy viscosity) component.
The eva1uation oť Vt is dependent on the particu1ar turbulence model emp10yed. In the
present ana1ysis, Prandtľs mixing 1ength model (see Appendix Z.5) is adopted; its use is
justifiab1e in light oť the computationa1 difficulties in the prob1em under consideration.
Because oť the axisymmetric and ťully deve10ped nature of the ftow, Vt == Vt (r) is taken
and hence v == v(r) as well; the ťull expression for v(r) is given in Appendix Z.5. It is
noted that equations (11.311) are stilI va1id iťthe ftow is 1aminar, in which case dv/dr = O
since v = v m .
Equations (11.311) and (11.312) are subject to the no-slip condition at the interfaces
with the cy1inders; at the outer surface of the ftexib1e shel1, r = a + !h, the no-s1ip
condition is. approximated by the following equations:
V x = au/ar,
Ve = av/ar,
v r = aw/ar,
(11.313)
where u, vand w are the axia1, circumťerentia1 and radia1 disp1acements of the shell,
respective1y; at the inner surface of the rigid cylinder, r = b,
V x = ve = V r = O.
(11.314)
It is seen that v r , Ve, V x and p are ťully specified by equations (11.311}-(11.314) and, in
princip1e, can be solved for, Once these perturbations are detennined, the unsteady ťorces

1404
SLENDER STRUCTURES AND AXIAL FLOW
(or stresses) on the sheU are given by
{ aVr av x } I { a ( ve ) 1 aV r } I
qx =U rx =pv -+- , qe =Ure =pv r- - +-- ,
ax ar ss ar r r ao ss
qr =U rr = { _P+2 P V avr } I ' (11.315)
ar ss
where 'ss' denotes that the expression(s) should be evaluated at the sheU surface. In
equations (11.315), the stagnant fluid within the sheU is assumed to have a negligibly
small inertial effect on the dynamics of the system; as a result, the unsteady forces are
simply the unsteady viscous stresses due to the annular flow, Such a situation would arise
with a liquid annular flow and air as the stagnant fluid within.
(ii) Method oj solution. For the purpose of solving equations (11.310), the shell dis-
placements u, vand w, as weU as the unsteady viscous forces qx, qe and qr, are expres sed
in the foUowing functional forms:
{ U(X,O,t) } 00 { U(X,t)cosno } { qx(X,O,t) } 00 { qx(X,t)cosno }
v(x,O,t) = L v(x,t)sinnO , qe(x,O,t) = L qe(x,t)sinnO ,
w(x,O,t) n=1 w (x,t)cosnO qr(x,O,t) n=1 qr(x,t)cosnO
(11.316)
where n is the circumferential wavenumber.
For nondimensionalization, in this and subsequent sections, the following dimensionless
parameters are defined:
 = x/ L,
w = w /L,
1] = r/L, Si = a/L, So = biL,
qx = yqx/ PshL, qe = yqe/ PshL,
íi = u/L, V = VIL,
qr = yqr/ PshL,
(11.317)
Expressions (11.316) are now substituted into equations (11.310) and the boundary con-
ditions, resulting in infinite series of sin nO and cos nO. Since sin nO and cos nO form an
orthogonal set, their coefficients in the series must varrish separate1y; thus, in terms of the
dimensionless parameters in (11.317), equations (11.310) may be written as
r ( A A A A A A ) (a 2A /a 2 ) A O
.1--1 u, v, w; q1, q2, q3 - Y u t + qx = ,
r ( A A A A A A ) (a 2 A /a 2 ) A O
.1--2 u, v, w; q1, q2, q3 - Y v t + qe = ,
(11.318)
r ( A A A A A A ) (a 2 A /a 2 ) A O
.1--3 u, v, w; q1, q2, q3 + Y w t - qr = ,
subject to the boundary conditions:
íi = v = w = aw/a = O
Ri(u, V, w) = O,
i=I,2,3,4
at  = O,
at  = 1,
(11.319)
(11.320)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1405
where the latter are dimensionless forms of (7.4a)-{7.4d), For example, the third of
(11.318) and the last of (11.320) are
[ a ] { a\i [ kn2 ] au kn a 2 í) a 4 w
1+ X - _ke3_ + v - -(1- v) e.- - -(3 - v)£2_ + ní) + ke 1 -
at I a3 2 I a 2 I a2 I a4
a 2  } { a 2  ( a  ) }
22 W 2 2  2 W U A 2A
-2kn £i a2 + [I + k(n - 1) ] W - qI £i a2 + q3 £i a - nv - n W
{ a2 A }
+ Y at - qr = 0,
a 2  2 a  a 3  a 
2U n A n v 3 W 2 W
- £. - - -(1 - v)u - -(3 - v)e.- + e, - - (2 - v)n £,- = 0,
I a2 2 2 I a I a3 I a
(11.321)
(11.322)
It may be seen that the only independent spatial variable in equations (11.318)-( 11.320)
is . To solve these equations with the finite difference method requires that the shell be
represented by (N + I) nodal points, evenly spaced in the axial direction; node ° is at
 = 0, the clamped end of the shell, and node N is at  = 1, the unsupported end. Any
node j is associated with three unknowns Uj, Vj and Wj, representing, respectively, the
axial, circumferential and radial displacements of the shell at  =  j. Equations (11.318)
are then written for al1 nodes i, 1 :S i :S N, with nodes I and N being given more attention
because they are affected by the boundary conditions.
At node 0, the boundary conditions (11.319) could be rewritten as
uo = vo = wo = 0,
(11.323)
and if the first derivative of w with respect to  is replaced by the O[(Ll)2]-accurate
central difference representation, then
(1/2Ll)(WI - W-I) = ° or W-I = WI.
(11.324)
where node -1 is fictitious and is symmetrical with node 1 about node O; Ll is the spacing
between any two successive nodes, determined by Ll = II N = j+1 - i.
At node N, the boundary conditions (11.320) are imposed, thereby increasing the num-
ber of equations to (3N + 4), whereas there are only 3N unknowns associated with the
N nodal points. To increase the number of unknowns, another fictitious node (N + 1)
beyond the free end of the shell is introduced, the number of unknowns thus becom-
ing (3N + 3); nevertheless, the number oť equations is still one higher than that of the
unknowns. This difficulty may be resolved by realizing that, for node N (andfor node N
alone), it is possible to combine equation (11.321) and boundary condition (11,322) into a
single equation, hence reducing the total number of equations by one. Thus, differentiating
equation (11.322) with respect to  and multiplying it by k£i, after some rearrangement
the following equation is obtained:
a 3  k 2 o  k a 2  a 4  0 2  a 2 
-k£?-(1-v)£'-.!!.._..!!..(3-v)£?+ke1-2kn2 £? = -kvn 2 £?.
I 03 2 I a 2 I 02 I a4 I a2 I a2
( 11.325)

1406
SLENDER STRUCTURES AND AXIAL FLOW
It is seen that the tenns on the left side of (11.325) are aZl present in the first pair of braces
in equation (11.321); substituting the right side of (11.325) for these tenns sirnp1ifies
equation (11.321) to
(1 + X :t ) {vei  + nv - kvn 2 cf  + [1 + k(n 2 - 1)2] Úl}
_ { ql e? a 2 ÚJ + q3 ( e' au _ nv _ n2ÚJ ) } + { y a2ÚJ - q } = O,
1 a2 1 a at 2 r
(11.326)
Thus, discretized about node N are the following six equations: the first two of (11.318),
the first three of (11.320), and (11.326),
In the present analysis, a !uZly implicit scherne, known to be unconditiona1ly stable,
is adopted to carry out nurnerical tirne integration; al1 equations involved are evaluated
at the new tirne level (n + 1), where uj+l, vj+l and ÚJj+1 are nodal unknowns to be
solved for. Consequently, all first and second tirne derivatives are approxirnated by the
ťJ[(t)2]-accurate backward difference representations, namely
(af/at)n+l = (1/2t)[3r+1 - 4r + r- 1 ] + O[(t)2],
(a 2 f/at 2 )n+l = [1/(t)2][2r+l - 5r + 4r- 1 - r- 2 ] + ťJ[(t)2],
(11.327)
where I is a nodal variable, and t is the tirne step taken forthe nurnerical tirne integration.
On the other hand, for inner nodes i, such that 2  i  (N - 1), all spatial derivatives
are approxirnated by the O[()2]-accurate central difference representations:
(af/aOi = (1/2OLfi+l - Ii-tJ + ťJ[()2],
(a 2 f/ae)i = [1/()2][Ji+l - 2/i + Ji-tJ + ťJ[()2],
(a 3 f/a3)i = [l/2()3][Ji+2 - 2/i+l + 2Ji-l - Ji-2] + ťJ[()2],
(a 4 f/a4)i = [1/(04][/i+2 - 4/i+l + 6Ji - 4/i-l + Ji-2] + ťJ[()2].
(11.328)
For node i = 1, in the first two of (11.328), uo, vo and ÚJo are provided by the first three
clarnped-end boundary conditions. In the last two of (11.328), however, U-I, V-I and
Úl-I are also required, but only Úl-I is known frorn (11.324); the problern is circurnvented
by the use of a forward difference representation for the third derivatives of U and v, so
that U-I and V-I are not needed. Similar provisions are rnade in conjunction with node
i = N, as described in Nguyen et al, (1994).
Once all the (3N + 3) equations have been replaced by their finite difference
representations, they are assernbled into the global rnatrix equation of the fonn
[A]{X n + 1 } = {Rn+l},
(11.329)
where [A] is a 3(N + 1) x 3(N + 1) constant coefficient rnatrix; {X n +1} is a 3(N + 1)
vector containing unknown nodal displacernents at the tirne level (n + 1), that is,
I An+ 1
u.
{X n +1}T = ,.. ' I
An+l An+l J T
Vi W i
at n  de i ' . , ,
for 1  i  (N + 1);

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1407
{Rn+l} is a 3(N + 1) vector which is a function of {Qn+l}, {X n }, {X n - I } and {X n - 2 },
where
y
at node i, 1 :5 i :5 N
] T
O O O
'---'
at node (N + 1)
{Q"+I}T  {-..
, (tJx)7+ 1 (tJe)7+ 1 (tJ r )7+ 1 ,
ln the above expression, the zeros at node (N + 1) represent the right sides of the first
three of equations (11.320). The e1ements of {Qn+ I} are determined in what follows.
As expected, this time integration procedure is not self-starting; for the first time step
(n = O), the vectors {X°}, {X-I} and {X-2}, corresponding to t = O, -D.t and -2(D.t),
respectively, are needed in orderto detennine {Xl} according to equation (11.329). These
vectors are evaluated using the theory of subsection (a), as described in Nguyen et aJ.
(1994).
(iii) Solution oj the Navier-Stokes equations. Proceeding now to solve the 1inearized
Navier-Stokes and continuity equations, we first express the variables in the fonn
{ vr(x, O, r, t) } { vr(x, r, t) cos no }
ve(x, O, r, t) _  ve(x, r, t)sinnO
vx(x, O, r, t) -  vx(x, r, t) cos nO '
) n-l
p(x, O, r, t - p(x, r, t) cos nO
( 11.330)
which are then substituted into equations (11.311) and (11.312), yielding infinite series,
identical1y equal to zero, of sin nO and of cos nO. For the reasons mentioned earlier, the
coefficients of sin nO (or cos nO) in the series must be equal to zem, resulting in
dV
- + G(v, p) = O,
dt
Gv(v) = O;
(11.331)
for example, the vr-equation and the continuity equation are:
_ _ _ _ dV r 1 dp dv dV r
Gr(vr, Ve, v x , p) = U(r)---;;- + --;- - 2- d -;-
uX p ur r ur
{ d2vr 1 d ( dvr ) (1 + n 2 L 2n _ }
-v -+-- r- - v --Ve
dX 2 r dr dr r 2 r r 2 '
_ _ _ 1 d _ n _ dV x
Gv(v r , Ve, v x ) = --(rv r ) + -Ve + -.
r dr r dX
The numerical method used for the solution of these equations is based on the
method developed by Soh & Goodrich (1988) and is very similar to that described in
Section 11.5.4(a). Hence, it is not given here in detail, but is outlined in Appendix Z.6.
It is a finite-difference time-marching solution with artificial compressibility and pseudo-
time. Integrations are perfonned in pseudo-time using the ADI scheme, carried out on a
staggered grid, as shown in Figure Z.I.
Therefore, here we shall presume that a solution has been obtained and hence tJx, tJe
and tJr have been determined via (11.315)-(11.317), Hence, a solution of (11.329) can be
obtained, which gives the shell disp1acements at time leve1 (n + 1), et seq. for later times.

1408
SLENDER STRUCTURES AND AXIAL FLOW
(iv) Remarks on the moving boundary. These remarks are of importance, in a context
transcending this particular problem; they are related to the discussion in 11.5,2(t).
Since shell motions are assumed to be sufficiently small, the width of the annular gap
may be considered to vary neither with time t nor with  (uniform gap); nondimensionally,
it is taken to be (£0 - £i), for which the shell is motionless. Hence, the resulting staggered
mesh becomes fixed throughout the physical time integration, In fact, however, when the
shell is in motion, the flow velocities U (1]) at some of the grid points that were initially in
contact with the deformed shell surface may not always be zero, due to a rapid increase
in U (1]) in the vicinity of the shell wall.
In early stages of developing the theory, the use of this fixed grid led to results that were
virtually identical, both qualitatively and quantitatively, to those obtained with the theory
of subsection (a), but without any unsteady fluidforces included, Hence, the only type
of instability obtained was divergence, which is evidently due to the flow pressurization
required to push the fluid through the annular gap. This indicates that the unsteady forces
are completely eliminated by the use of a fixed grid. On the other hand, the 'real' grid
should continuously deform with the moving shell, such that they always remain in contact;
this would require the continual recalculation of (i) the coordinates of, and (ii) U (1]) at, all
spatiallocations of the flow variables, as the time integration progresses, Hence, all the
coefficients ofthe tridiagonal systems of1inear algebraic equations of Appendix Z.6 would
need to be re-evaluated at the beginning of all iterations within every physical time step,
for a large number of time steps. Consequently, the amount of computing time associated
with the use of the real grid would be increased enormously and, at the time this work was
conducted, prohibitively.
In the calculations to be presented, what may be considered as an approximation to the
real grid has been adopted. This approximation mak:es use of the fixed grid; however, at
the beginning of every iteration within each physical time step, the flow velocities U(1])
at all the grid points initially in contact with the motionless shell wall are updated, For
a given i, U(1]) is determined from the velocity profile defined on [£i + UJi, £0] where
UJi is the instantaneous radial displacement of the shell at  = i. This approximation has
been found to recover the unsteady fluid forces.
(v) Calculations and results. In general, the optimum values of the pseudo-time step j}. T
and artificial compressibility 8 vary from one physical time step to another, In practice,
however, the optimum values of j}. T and 8 found for the first time step, in which the program
had been run on the McGill mainframe computer, were used for all subsequent time steps
when the program was run on a CRAY X-MP 2/8 computer. As a result, convergence
was achieved with as few as 35 pseudo-time steps in the first physical time step, and as
manyas 150 pseudo-time steps in subsequent physical time steps. For each configuration
of the system considered, the program was run for a total of 960 physical time step s for
a given n and m. Typical times for a computer run on the CRAY were 1600-2400 CPU
seconds.
The results to be presented have been run for parameters corresponding to the Nguyen
et al, (1993b) experiments, given in subsection (b): L/a = 8, hla = 0,0552, a =
24.84mmandvariableb,E = 2.82 MPa,Poissonratio11 = 0.47, Ps = 1.16xl0 3 kg/m 3 ,
X = O,0195/Q, U = 55.93 mis, P = 1.205 kg/m 3 , 11m = 15,18 X 10- 6 m 2 /s,
Regardless of how the shell was excited initially, if the average flow velocity was below
some critical value, shell motions would die out with time due to the presence of material

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1409
100
 50
-
II
"'-J>
 O
'"
o
X
<;S: -50
-100
O
(a)
200
 100
-
II
"'-J>

'"
o
-
X
<;S: -100
-200
O
(b)
160
320 480 640
Time step number
800
960
160
320 480 640
Time step number
800
960
Figure 11.119 The dimensionless radial displacement IV of the cantilevered shell at  = 1 as a
function of time: (a) for the -to-gap system at U = 0.45 (Re = 1.1 X 104), showing the onset of
divergence; (b) for the -gap system at ff = 1.00 (Re = 9,8 x 104), showing flutter (Nguyen et al.
1994).
and fluid damping. However, if the flow velocity were greater than a critical value, shell
motions would become continually larger, signifying a loss of stability. In the present
work, both types ofinstability were observed: divergence (static instability), as illustrated
in Figure 11.119(a) for the lo -gap system, and flutter as in Figure 11.119(b) for the
i-gap system. The shell was initially excited in the m = n = 2 mode in the case of
Figure 11.119(a) and in the m = 1 and n = 2 mode in the case of Figure 11.119(b).
The relative importance of unsteady viscous effects on the stability of the system
may be assessed by plotting U; for various annular gap widths gja = (b - a)ja.
Figure 11.120 gives the results of this theory, together with (i) those obtained by the
theory of subsection (a) where unsteady forces are calculated by inviscid theory, thus
neglecting unsteady viscous effects, and (ii) experimental data. It is seen that, with steady
viscous effects included, the values of U ; and associated type ofinstability (divergence or
flutter) are better predicted. Nevertheless, theory predicts divergence (D) for gja = 0.25,
whereas flutter (.) was observed in the experiments. This aspect has not been fully
explored yet.

1410
SLENDER STRUCTURES AND AXIAL FLOW
1.8
100
1.5
0,6
D
...... 80
D F ............"'... F
,,," F
D ......... 60 
........... :"',
'\....'\.'" =-
............... F *
.......... 40 ::J
"\..,: D
...
1.2
I'S 0,9
0,3
D
20
0,0
0,0
02
0.4
0,6
0,8
o
1.0
g/a
Figure 11.120 The effect of annular gap on the overall critical flow velocity, U; , for L/a = 8:
-, inviscid theory with only steady viscous effects included; \\\ \, CFD model; in both cases,
D denotes that theory predicts loss of stability by divergence, and F by flutter, Experiments: o,
divergence (only at g/a = 0.1);., flutter (Nguyen eta!, 1994),
It is of interest that unsteady viscous effects generally destabi1ize the system; hence
their neglect is not advisable. The results obtained by this theory are certainly promising,
and further work along these 1ines is warranted now that the computational tools have
been so much improved.
11.7.3 Wave propagation and simplified stability theory
Wave propagation in shells subjected to annular flow is no different, in principle, from
that of shells containing internal flow (Section 7.4). In fact, the El Chebair et al. (1990)
analysis discussed in Section 11.7.1(c) uti1izes a wave-type solution, adapted to finite-
length simply supported shells, although the results have not been plotted in the traditional
frequency versus wavenumber form (cf. Figure 7.53). A simi1ar analysis, with the results
plotted in the traditional way, has been made by Horáček & Zolotarev (1991 b), using
the Goldenveiser-Novozhi1ov shell equations and considering annular flow of an inviscid
acoustic medium. In the interests of brevity, this analysis is not discussed here in detail.
Suffice it to say that the method and solutions are similar to Horáček & Zolotarev's
(1991a), discussed in Sec ti on 7.4. Instead, Horáček's (1993) simp1ified analysis ofwave
propagation is discussed, which also provides some very nice, simple expressions for the
stability of coaxial systems subjected to either annular or intemal flow.
The system considered is a single flexible shell of radius a, generally with two coaxial
rigid cylinders: one inside, of radius bi, and one outside, of radiu s b e . Thus, one has two
annular flows: an internal, and an externa1 one. The other parameters are as heretofore
but, to remember this important difference, the subscripts i and e are used, rather than i
and o. Clearly, by taking bi = O one retrieves the full internal flow case. The fluid forces
are obtained via potential flow theory, but with mean pressurization taken approximately
into account. She11 motions are described by Vol'mir's simp1ified semi-membrane theory
(cf. Section 7.3.4). A single equation is involved which, similarly to Donnelľs applies
well only for very thin shells and higher circumferential wavenumbers n; after re-writing

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABILITIES 1411
it in a fonn closer to that of equations (7.1) and (7.29), it is given by
a 4 ( a2 ) 2 a4w Pa a 6 w
k- -+1 w+(1-v 2 )a 4 ---(1-v 2 )-
ae 4 ae 2 ax 4 Eh ae 6
a 2 ( 2 a 2 a 2 ) 2 1 a 4 qr
+ y at2 a ax 2 + ae 2 w = Psh Y ae 4 '
where k = /2 (h/a)2, w is the radial displacement, P = Pi(!L) - Pe(!L) is the steady
hydrodynamic pressure difference across the shell at x = ! L, Y = Ps a 2 (1 - v 2 ) / E, and
qr = Pi - Pe is the perturbation pressure difference as in (7.29).
By assuming solutions of the fonn
(11.332)
w = Wo exp[i(Qt - ax - ne)],
(11.333)
and proceeding as in Section 7.2.3, one eventually obtains
[ a2w a 2 w
qr = -a (Pi Ii - Pe Ie) at 2 + 2(Pi Ii Vi - Pe Ie Ve) axat
2 2 a 2 w ]
+ (Pi Ii Vi - Pe Ie Ve ) ax 2 '
(11.334)
where
. _ 7, _ [ ln (aa) ] { I - [ 1(abi)Kn(aa)]j[K(abi)ln(aa)] }
II =.l.1 (n, a) - ,
aa I(aa) 1 - [1(abi)K(aa)]j[K(abi)I(aa)]
7 _ 7 _ [ Kn(aa) ] { I - [ K(abe)ln(aa)]/[I(abe)Kn(aa)] }
.l.e = .l.e(n, a) - ,
aa K(aa) 1 - [K(abe)I(aa)]/[I(abe)K(aa)]
(11.335)
in which ln and Kn are Bessel functions in the usual notation, and I (z) = dl n (z) /dz,
K(z) = dKn(z)/dz.
For lang waves, such that aa ---+ O, abi ---+ O, ab e ---+ O, using the approximations
(Abramowitz & Stegun 1970)
1 ( z ) n
In(z)  n! 2"
1 ( 2 ) n
and Kn(z)  "l(n - 1)!  '
expressions (11.335) reduce to
1 { I + (b i /a)2n }
Ii  -[1 - (aa)2/(2n(n + 1)] 2 ;
n 1 - (bila) n
1 { I + (a/b e )2n }
Ie  --[1 - (aa)2/(2n(n - 1»] 2 .
n 1-(a/b e )n
It is clear from (11.336), and a1so fcom (11.335), that if bi = O or b e ---+ 00, one retrieves
the expressions for fulI interna1 flow or unconfined external flow, respectively.
(11.336)

1412
SLENDER STRUCTURES AND AXIAL FLOW
Now, substituting equations 01.333) and 01.334) into 01.332), a10ng with the
dimensionless parameters
a = aa, Q = Qa J psO- v 2 )jE, !-ti = piajPsh ,
P = O - v 2 )paj Eh, V i = Vi / J Ej[psO - v 2 )],
!-te = Peaj Psh,
V e = Ve / J Ej[psO - v 2 )],
01.337)
yields the characteristic equation
kn 4 (n 2 _ 1)2 + a40 - v 2 ) + P n 6 _ Q 2 (a2 + n 2 )2 = n 4 [ (!-ti Ii - !-te Ie ) Q2
- - _ - -2 -2 -2
- 2(!-ti Ii V i - !-te Ie Ve)a Q + (!-ti Ii V i -!-te Ie Ve)a ]. (11.338)
Fina11y, introducing the in vacuo natura1 frequency, Qo , defined by
-2 (1 - v 2 )a4 + k(n 2 - 1)2 n 4 + pé
Q = (11.339)
o (a2 + n 2 p ,
and for the coefficients of added mass of the internal (Ii) and external (Ie) fluid the new
positive variables
It = !-ti Ii n 4 j(a 2 + n 2 )2, I; = -!-teIen4j(a2 + n 2 )2, 01.340)
the characteristic equation 01.338) takes the simple fonn:
Q2 0 + It +I;')-2a Q (It Vi + I;' V e)-[ Q -(It V ; + I;' V; )a 2 ] = O. (11.341)
Typical results for the 1 -gap steel-water system defined by parameters (11.298),
also considered by Pai'doussis et al, (1984a) and El Chebair et al. (990) [see Section
11.7.1(a,c)], are shown in Figure 11.121. Two variants are considered in each case: with
the fluid on the other side of the shell stagnant, or replaced by vacuum. The divergence
flow ve10city is the same, whether the fluid is stagnant or replaced by vacuum, for the
reasons already elucidated. The flutter flow velocity, on the other hand, occurring where
the two branches of the curve coalesce, is much smaller in the presence of stagnant fluid
on the other side of the shell rather than vacuum, because of the increased added mass.
Indeed, it is easy to find simple expressions for divergence and flutter directly
from (11.341). Thus,for divergence, with interna1 and externa1 flows, one obtains
- Qo 1
VO D -
I - Ci JI[
- Q o 1
and VeD = a .fIf '
(11.342)
For flutter, with internal flow ( V i =1= O, Ve = O) and externa1 flow ( V i = O, V e =1= O),
_ Q [ 1 + I'! + I* ] 1/2 _ Q [ 1 + I'! + I* ] 1/2
V, =  I e and V =  I e
IF a It(1 + I;) eF a I;(1 +In
Results from these expressions are compared to the values given by El Chebair et al.
for divergence and flutter of a finite-length simply supported shell by considering a =
mn:aj L, with m = 1 (i,e, a = 0.283). The results are found to be very c1ose, as seen in
Table 11.23. Clearly, therefore, for sufficiently 10ng wave1engths, Horáček's analysis is
very usefuI indeed.
(11.343)

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1413
-4
O
(a)
16
12
8
...,
o
x 4
10:
O
-4
O
(b)
16
12
8
'O
X
10:
---------------,
Do __ ...- -- ,,;"'" ,,;"'" J
/' "
/' /
/' /
/' /
,/' /
Q=aU /
'/
....-
...
....
-
-
-
-
-
-
-
-
-
/'
-
-
/'-
/'
/'
,/
/'
/
/
/
/
"
"
//
-
U iD
8
16
24 32
Vi X 10 3
40
48
56
------- - ------- ---------- --
Q ,,,..,..............."'" I
o ................................... ,//
,.'" / ,/
... -- -
..----- // ..---
... /' -
,.,.- - ----
... Q=aU -
",'" / e..-
,. /" /--
,/ -
-
-
-
...
",,'"
8
]6
24 32
Ve X 10 3
40
48
56
Figure 11.121 Dimensionless frequency, O, versus dimensionless internal or external flow velo-
city, Vi or U e, for the fo-gap simply supported steel-water system, with parameters as in (11.298).
(a) Internal flow: - - - , with vacuum (I e = O) outside the shell; - , with stagnant fluid
(Ve = O) outside the shell. (b) External flow: - - - , with vacuum (Ii = O) inside the shell; -,
with stagnant fluid (Vi = O) inside the shell (Horáček 1993).
Table 11.23 Comparison of the critical flow velocities for divergence and coupled-
mode flutter from El Chebair et al. (1990) and Horáček (1993),
Internal flow
External flow
Values for divergence, UiD or UeD
El Chebair et al, Horáček
Values for flutter, Ui F or U e F
EI Chebair et al.
Horáček
2.18 x 10- 2
2,09 x 10- 2
1.94 x 10- 2
1.03 x 10- 2
1,93 x 10- 2
1.02 x 10- 2
2.20 x 10- 2
2.10 x 10- 2

1414
SLENDER STRUCTURES AND AXIAL FLOW
11.7.4 Practical experiences
Some of the spectacular problems related to nuc1ear-reactor thermal shields
(Section 11.2.1) involved cylindrical shells subjected to annular flow. Another, equally
spectacular therma1 shie1d instability, this one of a French Fast Breeder Reactor (FBR)
Superphénix (SPX-l), involved a free surface and more complicated dynamics: a
cy1indrical-shell weir with the falling of a cylindrica1 nappe over the weir (Ai'ta & Gibert
1986; Aita et al. 1986, 1987; Lacroix & Viollet 1987). A schematic of the system, with
the flow over the weir shown by a bent arrow at the upper right, is shown in Figure 11,122.
An excellent abbreviated description of the phenomenon is given by Axisa (1993), and
this is adopted almost verbatim in what follows.
The shell in question is made of stainless steel, with a = 10.14 m, h = 27 mm; the fluid
is 1iquid sodium. Excessive vibrations were recorded by strain gauges which were installed
at various 10cations on the reactor vessel and bafftes for vibration monitoring purposes.
Vibration was nearly periodic, with a typica1 frequency of about 0.3-0.4 Hz; however, the
time signa1 displayed important 10w frequency amplitude modulation, The problem was
investigated at the CEN-Sac1ay laboratories of CEA (Commissariat a l'Energie Atomique),
Roof slab
  1:1 .. G o to> ti;; c> C' o o o o c o Q o <:' o o o o c:-
o 0'0 o. Co o o o o o o Co o o o c c
o í) t.ooooooC'c-0oo.oo
Core cover
plug
Intermediate collector:
inter redan space
Cold collector
Core support structure
Figure 11.122 Schematic of the Superphénix intemals, showing the upward flow cooling the
main pressure vessel from below the core, up to the crested weir of the thermal baffle shell (Axisa
1993).

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1415
Unstable
overflow
Step 1
Step 2
Step 3
t
Fluid
impact
 CD
CD
 0 1
r<D
II
Figure 11.123 The scenario of the fluidelastic instability of the Superphénix shell-weir: 1. lateral
motion of the weir; 2, perturbed flow and sloshing at the weir collector; 3. perturbed flow and
sloshing at the restitution collector; 4. falling fluid nappe; 5. perturbed inlet flow (Axisa 1993).
based on idealized mock-ups, Shortly after the first laboratory tests, Gibert and AHa were
successful in proposing a satisfactory nonlinear analytical model to explain the essential
features of the phenomenon, which is only briefly summarized here.
The source of energy feeding the observed self-excited vibrations is gravity, as schem-
atically sketched in Figure 11.123, The fal1ing sodium passing over the weir of the main
vessel cooling flow impinges on the free level in the restitution colIector, thus producing a
fluctuating load on the flexible weir shelI, Vibration of the weir produces a possible feed-
back mechanism for controlling the flow rate over the weir. The phenomenon is further
complicated by the fact that, at the low frequencies of interest here, sloshing modes can
be coupled to the structural modes of the weir. The model assumes that the steady weir
flow rate q is controlIed by the velocity of the gravity waves (critical discharge flow),
I K gl/2 h 3 / 2
q(t, e) =
O
for h > O,
for h < O,
(11.344)
where h(t, e) is the fluctuating depth of liquid above the weir at time t and azimuthal
angle e; h < O means that the upstream free level is below the weir edge, As it impinges
on the restitution colIector free surface, the falling liquid produces a flow-rate source,
Q=q(t-T,e),
(11.345)
where T is the time of falI of the liquid. The impinging liquid also produces a pressure
source related to the velocity Vf of the fa11ing liquid,
p = pg z + (VtleR)Q,
(11.346)
where  P is the pressure at the level z below the free level, and e R the lateral width
or gap of the restitution collector; equation (11.346) may be derived by integrating the
momentum equation of an elemental fluid volume of height z and width e R.

1416
SLENDER STRUCTURES AND AXIAL FLOW
Figure 11.124 Photograph of the i-scale 'EPOC' model of the crested-weir flutter (n = 6) tested
at LNH of Electricité de France (Lacroix & Viollet 1987),
Equations (11.345) and (11.346) may be used to describe the fluidelastic coup1ing in the
system. Then, modal analysis may be used to derive linear stability maps in conjunction
with the main system parameters: the coolant flow rate and the height of the falI, Two types
of flutter in stability have been found: (i) a coupled-mode flutter involving the coupling
between two sloshing modes ofthe colIectors separated by the weir; (ii) a sing1e-mode flut-
ter, associated with negative damping of one of the fluid-coup1ed shelI modes. Nonlinear
analysis (Mta ef al. 1987) yielded the amplitude of the limit cyc1e, t
Experiments were also conducted at Laboratoire National ďHydraulique (LNH) of
Electricité de France on l-scale and fulI-scale models with ambient water flow, observing
fluidelastic simi1itude to the reactor system. A 'slow mode', involving n = 4, 5 and 6 was
observed, depending on conditions, as il1ustrated in Figure 11.124; a 'fast mode' was also
observed, which was more irregular, with breaking of the wave-crests,:j:
Fortunately, it was possible to cure the weir instabi1ity of Superphénix without any
change in the design, by properly adjusting the flow rate of the main-vessel coolant. This
was made possible by adjusting the flow rate at the bottom ofthe core assemblies. However,
further study of the phenomenon has continued; see, for example, Fujita ef al. (1996);
Tanaka ef al. (2000).
tl n retrospect, the phenomenon is fundamentally similar to, though vastly more complex than, that described
in conjunction with Figure 11.2, used as an introduction to the topic of annular-flow-induced instabilities,
+The author was fortunate enough to see a demonstration of the phenomenon in both the i-scale and full-
scale mock-ups at LNH, The sight ofthe enormous aluminum shells ('thin' only because ofthe large diameter)
oscil1ating at large amplitudes was quite awesome!

ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1417
Figure 11.125 Residua1 deforrnation of the CEN-Saclay cantilevered shell subjected to mainly
annular but also internal flow, after buckling (Gibert 1999; de Langre 1999),
Another practical experience is related to an experiment conducted in the 1970s at the
CEN-Sac1ay Laboratories of Commissariat a ľEnergie Atomique (CEA) in France, mod-
e11ing an internal component ofFast Breeder Reactors of the Superphénix type. It invo1ved
a canti1evered cylindrical shel1 (h :::::: 0.5 mm, hla:::::: 5 x 10- 3 , Lla :::::: 10) in annu1ar
water-flow (dimension1ess gap gla :::::: 5 x 10- 2 ), with some interna1 flow as well, enter-
ing the shel1 through the perforated upstream end; the downstream end was open and free
(de Langre 1999; Gibert 1999).
The dynamical behaviour with increasing flow was quite interesting, The shel1 vibrated
with increasing amplitude, unti1 at a certain point (U :::::: 7 mis) a divergence instability
occurred and the shel1 buckled in the n = 3 mode (m = 1 or 2); the amplitude was
1arge enough to cause p1astic deformation. An after-test photograph of the buckled shel1
is shown in Figure 11.125. Thus, un1ike the elastomer canti1evered-shell experiments
discussed in Section 11,7.2, it is c1ear that the occurrence of flow-induced instability may
actual1y destroy a metal shell.
A case invo1ving the heat shield of an aircraft engine after-burner is reported by Ziada
et aZ, (1988). The heat shield is a thin, corrugated and slightly conical shel1, separating
the hot gases on the engine side from the co1d housing outside; it is subjected simultan-
eous1y to internal and annu1ar flows, as shown in Figure 11.126; a damaged shel1 is also
shown.
Model tests at simi1ar Mach and Reyno1ds numbers as in the prototype showed that
the cu1prit was c1ear1y the high annu1ar flow: a c1ear flutter instability developed at Ua ::::::
190 mis (whichcompares to Ua = 172 mis maximum-powerflow in the prototype). When
internal flow was added, the effect was stabilizing (cf. Figure 11.114; the pressurization,
neg1ected in that figure, wou1d amplify the effect). Some results on the onset of flutter for
a constant inner flow (Mi = 0.6) and varying annu1ar flow are shown in Figure 11.127(a)
indicating that the narrower the annu1us, the lower is the critical flow velocity.
The addition of damping (by a damping foi1) had an interesting effect: the system
developed divergence prior to flutter, as shown in Figure 11.127(b) - cf. Section 7.9 and
Grotberg & Reiss (1984).
The cure eventual1y adopted was to reduce the annu1ar flow over the 1ast part (sector 5)
of the shel1, by adding some flow resistance, achieved by bending the troughs of the
corrugations at the 1eading edge slightly outwards, as shown in Figure 11.128 by the

1418


SLENDER STRUCTURES AND AXIAL FLOW


P=2.15bar


IA


'"
:E -
 .+
-.3 _--:",g
e --

e -..3




:+ 1 I
.+ I
I


2


P = 1.64 bar


5


,&;
<=
o
.

u
o




::;I


.+






I A


Heat shield
Housing


Section A
A


(a)


(b)


Figure 11.126 (a) Geometry of the after-bumer, showing fixation points (- , . ) and fuel injection
locations (+ + +); (b) typical damage of the heat shield (Ziada et a/. 1988).


300


300

 M i =O,6 200 O
'"
o;




 200 N o-, t-
.- '"
1:
 ,.., '-O Divergence Flutter
N N

 II
'" "<J
"O
.B 100

 100
<=
.g


oD
:>
O O 1.2
O 0,1 0.2 0.3 0.4 O 0,2 0.4 0,6 0,8 1.0
(a) Dynamichead,
poVo2(bar) (b) Pressure drop (bar)


Figure 11.127 (a) Total r.m.s. vibration amplitude versus the dynamic head of the annular flow
for severa1 values of the gap width, g, and for a constant Mach number of the inner flow, Mi = 0.6.
(b) The effect of increased damping on the vibration for g = 1.82 mm and Mi = O: o, without the
damping foil; .Il, with the damping foil (Ziada et a/, 1988).




ANNULAR- AND LEAKAGE-FLOW-INDUCED INSTABIUTIES 1419


...JA


3


Sector 4


5




Sector 4


Side view of detail A


Figure 11.128 Schematic of the origina1 (-) and modified (- - -) leading edge of sector 5
of the heat-shield shell (Ziada ef aZ, 1988).


dashed line. This mi1d resistance allows the major part of the cooling flow to continue in
the normal direction. The rest goes into the main jet, along the inner surface of the she11,
and hence the desired cooling effect is not 10s1. The result was that the annular flow under
operating conditions became smaller than the critical. This countermeasure solved the
prob1em in the prototype as well.
Thus, this was a c1ear case of annu1ar-flow-induced instability. Note that the optimum
10cation for the flow resistance wou1d have been at the downstream end. However, work
on 1eakage-flow instabi1ities was not too wide1y known at the time and, in any case,
downstream 10cation wou1d have been practically difficult; in the event, the upstream flow
resistance in this case caused no prob1ems.


11.8 MISCELLANEOUS APPLlCATIONS


There are numerous engineering systems invo1ving annu1ar flows, sometimes in conjunc-
tion with interna1 flow as well. Some of these have given rise to major prob1ems, a number
of which have been discussed in Sections 11.2 and 11.7.4, notab1y re1ated to nuc1ear
reactors.
Here, examp1es of engineering systems with annu1ar flow are simp1y listed, as follows:
(i) nuc1ear reactor thermal shie1ds (Sections 11.2.1 and 11.7.4);
(i i) heat exchanger therma1 shie1ds (Pai'doussis & Lak:is 1972);
(iii) oi1 and gas drilling and production systems [Figure 3.18; see also Luu (1983)];
(iv) tibre spinning (Papanastasiou ct aZ. 2000);
(v) wire coating (Poddubny & Saltanov 1991; Papanastasiou ct aZ. 2000; Kaneko
ct aZ. 2002);
(vi) 1ubrication in wire drawing (Christopherson & Nay10r 1955);
(vii) thread/annu1ar-flow systems for surgica1 imp1ants (Frei ct aZ. 2000);
(viii) 'lubricated pipe1ining', invo1ving the flow of a fluid with annu1ar fluid 1ubrication
(Ooms ct aZ. 1984; Bai ct aZ. 1992);
(ix) transportation of capsu1es in pipe1ines (Feng ct aZ. 1995);




1420


SLENDERSTRUCTURESANDAJaALFLOW


(x) rotary pump shafts (Section 11.6);
(xi) labyrinth seals and similar applications [Section 11.6; Brown & Leong (1984)];
(xii) aircraft engine heat shields (Section 11.7.4).
(xiii) beams trave1ling in a tube, mode1ling high-speed trains in tunnels (Sugimoto
1996; Sugimoto & Kugo 2001).
In each case, typically one representative reference is given, The interested reader may
use this as one end of Ariadne's c1ue of thread to wend his way along the paper trail to
the desired end. In several of these engineering applications there is scarcely a mention
of 'problems' . However, in view of the proclivity of annular- and leakage- ftow systems to
instabilities (Section 11.1.1), much must have remained unreported in the open literature!




EPILOGUE


Although not covering absolutely everything, this two-volume book treats all important
aspects offluid-structure interactions involving axial flow in, around, and between slender
structures: pipes, shells, cy1inders, strings, plates and foils. Indeed, even for aspects not
specifically treated, sufficient related discussion and some references point the way to
more complete treatments in the archival1iterature, to help the interested reader,
The second volume took considerably longer time to complete and be pub1ished than
anticipated, As a result, and partly due to the very appearance of Volume 1, many papers
related to its subject matter have been published in the interim. To make this two-volume
set as complete as possible, some of these new references are given in Appendix Q,
together with some published prior to 1998 but mis sed out when writing Volume 1, t as
well as some corrigenda.
Writing this book has been a labour of love. It is hoped that, by bringing all this materia1
together, this book wi1l be as useful to the research community as its writer has toiled to
make it be.


tTo prove that this is not such an esoteric subject after al1 (thanks to Emmanuel de Langre), a 1958 reference
is a1so cited for an image in one of the Tintin books: flutter of canti1evered pipes conveying fluid has been in
the popular culture for as long as its academic study!


1421




Appendix L
Matrices in Hybrid
Analytical/Finite-Element Method
of Lakis et a/.


L. 1 MA TRICES FOR A CYLlNDRICAL SHELL IN VACUO


Here are given the matrice s R, A, Q and P required in the analysis of the shell in vacuo,
associated with equations (7.110), (7.112), (7.1l4a,b) and (7.115) of Section 7.3.3(a). The
reader is also referred to Lakis & Pai:doussis (1972a,b) and Lakis & Neagu (1997).
The elements on each row of the 3 x 8 matrix R are given by


R(1, i) = aj e Aix / a , R(2, i) = é ix / a , R(3, i) = {3j é ix / a ,


i = 1,2,..,,8,
(L.l)


a being the mean radius of the shell. The Aj are the roots of the biquartic discussed beneath
equation (7.109), and the aj and {3j are defined in the same paragraph; the Aj, ai and {3j
are all complex.
The elements of A, an 8 x 8 matrix, are given by


A(1, i) = aj, A(2, i) = 1, A(3, i) = A;ja, A(4, i) = (Ji,
A'
A(5, i) = aj é i1 / a , A(6, i) = é i1 / a , A(7, i) = -.!.. é i1 / a , A(8, i) = (Jj é i1 / a ;
a


i = 1"",8,


(L.2)


where I is the length of the shell element.
The elements of matrix Q, of order 6 x 8, are given by


Q(1, i) = ai(A;ja) é;x/a, Q(2, i) = (1la)(n{3j + 1)e A ;x/a,
Q(3, i) = (1la)({JiAj - naj) eA;x/a, Q(4, i) = _(A;ja)2 e A;x/a,
Q(5, i) = (11 a 2 ) (n 2 + (Jjn) eA;x/a, Q(6, i) = (11 a 2 ) (2nAi +
 (Ji Aj + !naj) é ix / a ;
i = 1,2, . .. ,8,


(L.3)


1423




1424


SLENDER STRUCTURES AND AXIAL FLOW


The elasticity matrix for an isotropic shell is given by


p=


D*
vD*
O
O
O
O


O O
O O
O O
D vD
vD D
O O


vD*
D*
O
O
O
O


O
O
io - v)D*
O
O
O


where D = Eh 3 /[12(1 - v 2 )] and D* = Eh/(1 _ v 2 ).


O
O
O
O
O
io - v)D


(LA)


L.2 MATRICES ASSOCIATED WITH FLUID FLOW IN
A CYLlNDRICAL SHELL


The exp1icit fonn of the 8 x 8 matrices S F, D F and G F in equation (7.121 a) is detailed
here, as follows (Lakis & Laveau 1991):
SF(p,q) = 7r [-8iYi 2 Zq,i1pq,i +8eYe 2 Zq,e1pq,e] '
DF(p, q) = 2iA q 7r [-8iYi U
 Zq,i lpq,i + 8eYe U; Zq,e lpq,e] , (L.S)
G F(p, q) = -A
7r [-8JJ
2 Zq,i lpq,i + 8e U; 2 Zq,e lpq,e] ,


p, q = 1, . ..,8, where subscripts i and e denote interna1 and external flow, respecůve1y,
whi1e the nonsubscripted i = R; Zq,i and Zq,e are given by equations (7.119a,b), and



 _ Pu a u
Ou - ,
Pslhl


-* U u
U =-,
u Ua


a u
Yu=-,
aul


2 PO, 1; 1)
Ua = ,
Pslhl


(L.6)


in which u = i for interna1 and u = e for externa1 flow; PsI is the density, hl the thickness,
and a u I the radiu s of the first element of the she1!. P (1, 1; 1) is the first element of the
e1asŮcity matrix (LA) for the first shell element.
The quantities lpq,u in (L.S) are shorthand notations for


t . 1 exp[i(A p +A q )(lj/a u )]-l
lpq,u = i(A p + A q )
(tj -li)/a u


li and I j being the values of x at nodes i and j, respectively.


if Ap + Aq i= O,
if Ap + Aq = O,


(L.7)


L.3 MATRICES ASSOCIATED WITH FLUID FLOW IN
A CONICAL SHELL


The conical system is shown in Figure L.1. It is understood, of course, that in the case of
interna11y flowing fluid, the conica1 shell has to be a truncated one, but the ana1ysis can
proceed in genera1 as if it were a complete cone.




HYBRID ANALYTICAL/FINITE-ELEMENT METHOD


1425


R 2


Figure L.I Geometry of the surface of a conical shell of half-angle a e , and detinition of some of
the parameters used in the analysis and of the coordinate system (x, ex, O).


As in Section 4,2, the flow velocity is not constant but, assuming an incompressible
fluid, the volumetric flow rate (Jr x 2 sin 2 £l c ) U = const; hence, we define U x 2 = Q =
const. The analysis is carried out in a system of spherica1 polar coordinates (x, a, O),
defined in Figure L.I. t Thus, the Laplacian of the velocity potential <I> for an inviscid
incompressible fluid in this case is (Lakis et al. 1992)


a 2 <1> 2 a<l> 1 a 2 <1> 1 a <I> 1 a 2 <1>
'\72<1> = - + - - + - + - + - - = 0, (L 8)
ax 2 x ax x 2 sin 2 a a0 2 x 2 tan a a£l x 2 a£l 2 .


This could readily be extended to a compressible fluid, in which case the right-hand side
is equal to


1 { a2<1> 2Q a 2 <1> Q2 a 2 <1> }
c 2 at 2 +
 axat +7 ax 2 '


but in this case a new Q would be invol ved, expressing the constancy of the mass flow rate.


tLakis et al. (1992) refer to these as conical coordinates, but they are not the usual conical ones; they are
really spherical polar, with x playing the role of the radius,




1426


SLENDER STRUCTURES AND AXIAL FLOW


The equiva1ent of equations (7.37) and (7.35), the latter being the impenneability
condition, in this case are:


[ a<l> Q a<l> ]
p=-p -+--
at x 2 ax C/=a'
(VC/)C/=a =
 a a <l> I = [ a a W +
 a a W ] ,
x ot C/=a t x x C/=a
where a = a c - tan- 1 [h/(2x)] is the half-angle of the inner-wall opening ofthe shell.
The displacement w, nonnal to the wa11, is taken to be


(L.9)


(L.1O )


8
w(x,(},t) = LCq/,.q-lcosneeií2t,
q=l
in which I is an arbitrary reference length, used for nondimensionalization, Assuming next


y = ../x/l ,


(L.ll )


8
<I>(x, e, ot, t) = L Rq (ot)Sq (x, e, t),
q=l
substituting into (L.I O), and the resultant equation into (L.I I ) and then into (L.8), yields
/I I I n 2
Rq(ot) + - Rq(ot) -
 Rq(ot) = O. (L.12)
tan ot SIn ot
Using the Frobenius method (Kom & Kom 1968) we obtain the following approximate
solution:


8
" [ aw Q aw ]
<I>(x,e,ot,t) =
 kx - + 2- ,
at x ax C/ - a
q=l -


(L.13)


where


I I +
a2 + (5n + 7) na 4 + n(n + 4)(5n + 1) a 6 }
a 12 1440 51840
k =;; I (n + 2) 2 (n + 4)(5n + 7) 4 (n + 4)(n + 6)(5n + 1) 6 .
+ 12 a + 1440 a + 51840 a
Fina11y, substituting into equation (L.9) we obtain


8 I .. Q . Q2 (A
 - 6A q + 5) I
p = -pk L XW q + x2 AqWq + 7 4 W q ,
q=l
where W q is the radial displacement of the shell and W q and W q are its first and second
time-derivatives.
Introducing equation (L.ll) into (L.14), performing the matrix operations required
by the finite element method and integrating with respect to x and (}, we obtain the
inertia, Coriolis and centrifuga1 force matrices, mf, cf and k f, respectively, defined as
in equation (7.12Ia). The e1ements of SF, DF and GF in this case, however, are given as
follows,


(L.14)




HYBRID ANALYTICAL/FINITE-ELEMENT METHOD


1427


L.3.1 Elements of SF


For Am + An = -4:


SF(m, n) = -npk 13 sin a ln (ljl li), (L.15a)
where a = a c - tan- I [hl(2x)], h being the shell thickness, and li and lj are the values
of the x -coordinate at nodes i and j;
for Am + An =1= -4:


SF(m, n) = _ 2npk sin a { ex p [ ( Am + An + 4 ln lj ) _ ( A m + An - 2 ) ln I J
(Am +An +4) 2 2
_ exp [ ( A m +
n + 4 ) ln li _ ( A m +
n - 2 ) ln I J } ; (L.15b)


I is a scaling length for y = (x I I)I/2, used in the analysis.


L.3.2 Elements of D F


For Am + An = 2:


DF(m, n) = -npk QA n sin a ln (lj I li) ;


(L.16a)


for Am + An =1= 2:


DF(m, n) =


2npkQA n sin a { ex p [ ( Am + An - 2 ) ln (
 )J
(Am + An - 2) 2 li
_ exp [ ( A m +
n - 2 ) ln (
 ) J } .


(L.16b)


L.3.3 Elements of G F


For Am + An = 8:


G F (m, n) = -
 pk 73 2 sin a (A
 - 6Am + 5) ln (t ) ;
for Am + An =1= 8:


(L.17a)


n 2 ( Am - 6A n + 5 )
GF(m, n) = - -pk Q sina
2 Am + An - 8
x {ex p [ ( A m +
n - 8 ) ln I j _ ( Am +
n - 2 ) ln I J
_ exp [ ( A m +
n - 8 ) ln li _ ( A m +
n - 2 ) ln I J }. (L.17b)




Appendix M
Anisotropic Shells


The constitutive model for a general anisotropic cylindrical shell may be defined by


{ :; } = [ g

 g


Tx() Q16 Q26


Q16 ] { Sx }
Q26 S() ,
Q66 Yx()


(M.I)


with the Qij depending on the exact manner in which the shell is constructed. Thus, for
a composite shell, the ply layout (e.g. angles and symmetry vis-o-vis the neutral axis) is
important. For an isotropic shell, this reduces to (Holm-Christensen 1990)


{Qll, Q12, Q16, Q22, Q26, Q66} = 1
 v 2 { I, v, 0, I, 0,
(l - V)} . (M.2)


For an orthotropic shell in which there is symmetry about the neutral axis (the lamination-
or ply-Iayout is 'balanced'), one has Q 16 = Q26 = 0, and so there is no coupling between
bending-stretching and torsion of the shell.
In this scheme ofthings, for the orthotropic shell, in Fliigge's equations (7.1) we have


£1 (u, v, w) =QllU" + Q12(w' + v") + Q66(U" + v")
+ k [Q66(U" + w''') - QI1 w"'] ,
£2(U, v, w) =QI2U" + Q22(V" + w') + Q66(U" + v")
+ k [Q66(3v" - 3w"') - Q12W"'] , (M.3)
£3(U, v, w) =Q12U' + Q22(V' + w) + k [Qll (w"" - u"') + QI2(2w"" - v"')
+ Q22(W.... + 2w" + w) + Q66(4w"" - 3v'" + u''')],


where the dots and primes are as in equation (7.2). A more general form of these equations,
with Q16 and Q26 i= O, is given by Holm-Christensen (1990).
A more usual manner of expressing the constitutive relation for an orthotropic shell is
to do so in terms of the stress and couple resultants, Ni and Mi, respectively, related to
the stress components O'i by (Bert & Chen 1975a,b)


j h/2
{Ni, Md == {I, Z}O'i dz,
-h/2


(MA)


1428




ANISOTROPIC SHELLS


1429


where z is the distance from the middle surface; thus,


I NI } [ All
Z
 = (l/a) AJ2
I MI } [ Dll
Z
 = (l/a)2 DJ2


1

 g ] I v.: W } ,
O A66 u' + v'
g

 g ] { v'
W
.. } .
O D66 v' - 2w "


(M.5)


(M.6)


ln (M.5), the Aij are the elements of the stretching stiffness matrix, while in (M.6) Dij
are those of the bending stiffness matrix, ln more general cases where bending and
stretching are coupled, there is also a so-called "bending-stretching" stiffness matrix
[e.g. Ugura1 & Cheng (1968)] with element s Bij, which gets involved in the stress-
strain relations and hence in the dynamics. The reader is referred to Ugural & Cheng
(1968), Bert & Egle (1969), Jain (1974), Bert & Chen (1975a,b), Alam& Asnani (1984),
Bicos & Springer (1989) and to references therein for a more genera1 discussion of the
dynamics of anisotropic shells,
For a homogeneous isotropic material, All = A22 = Eh/(1 - v 2 ), Al2 = VAll,
A66 = !(1 - V)All, and Dij = l2 h2 Aij.
It may be worth mentioning here that in the hybrid finite-element method of Lakis
et aZ. (Selmane & Lakis 1997a.b), for an anisotropic shell the matrix P of equation (L.4)
becomes


Pll Pl2 O Pl4 PIS O
P2l P22 O P24 P2S O
P= O O P33 O O P36 (M.7)
P41 P42 O P44 P4S O
PSI PS2 O PS4 PSS O
O O P63 O O P66




Appendix N
Effect of Viscosity in
Rotofluidelasticity


ln this appendix, some of the details of the matched asymptotic expansion uti1ized by
Howard et a/. (1972) are given, eventually leading to an expression for the wall pressure
fluctuation in a shell containing fluid in steady rotation and vanishingly smal1 axial flow
velocity (Ro = O).
We start with the dimensionless perturbation equation (7.220) of Section 7,6.4 and seek
an "inner" (physically outer) boundary layer solution assuming


{Vr, VB, V x } = {VrO + E 1 / 2 Vrl +... , VBO + E 1 / 2 VBl +... , VxO + E 1 / 2 V x l + ...},
A A E l / 2 A (N I)
P = PO + Pl + . .. , .


where PO, PI, vro, etc., are all assumed to be functions of the stretched variable



 = (1 - ,.)E- 1 / 2 ,


(N.2)


and all to be of order ij = w / a, the dimensionless radial amplitude of shell deformation.
From equation (7.2ISa) - throughout with Ro = 0- and the corresponding continuity
equation,


1 d AA in A . A

---;;:(rVr) - 7VB -lKV x = O,
r dr r


(N.3)


it follows that Po and vro are independent of
 to zero order, so that the following equations
are obtained for voo and vxo:


d2voo . A A . A A
d
2 - lWVBO = -In PO + 2v r o,
d 2 v x o . A A . A
d
2 - lWVxO = -lK Po,


(N.4)


which can be solved easily, giving


VOO =
 (npo - 2iV r o) (I - e-A
) ,
W
A K A (I -A
 ) ( . A ) I / 2
VxO = -;;- PO - eS, ).. = lW .
W


(N.S)


1430




EFFECT OF VISCOSITY IN ROTOFLUIDELASTlCITY


1431


Then, substituting (N.!) into (N.3) and omitting higher-order terms gives
dVrl ,
 .


- = -lnVeo -lKVxO + Vro,
d



(N.6)


which may be integrated to give



 -i [ 2 2
 2 '
 ] ( " 1 -A.
 ) ,,
 C
Vrl = W (n + K )PO + lnVrO ,,+):e +" VrO + ,


(N.7)


where C is a constant.
The next step is to match the interior, inviscid core solution, here labelled "outer" and
denoted by superscript o, and the boundary layer solution, labelled "inner" and denoted
by a superscript i. The outer solution may be regarded as the lowest order of the viscous
solution for E -+ O; hence, to both order O and E 1/2 this flow is inviscid, since the viscous
terms enter only to order E - see equation (7.220), Thus,


pO = pg + O(E) = AgJn(ar) + O(E).


(N,8)


From pg, the perturbation velocity components are given directly by equations (7.215a-<:).
Hence, by lowest-order matching,



i
O (I)
oi
Po = Po = Po '



i
o (1)
oi
vro = vro = vro'


(N.9)


where r = 1 indicates the wall.
To order E 1 / 2 the following inner representation is obtained:
v
 = v
b + E 1 / 2 {
i [(n 2 + K 2 )pgi + 2inv
bJ (
+
e-A.
) +
v
b + C
} + O(E),
(N,lO)
For matching,
 is replaced by (l-r) / E 1/2 and we expand for small E 1/2. The exponential
terms drop out, leading to the following outer representation for the inner solution:
v
i = v
b + (1- r) {
i [(n 2 + K 2 )pgi + 2inv
bJ + v
b} + E 1 / 2 C
 + O(E).


(N.11 )


This now has to be compared to the corresponding inner representation of the outer solu-
tion, equations (7.215a-c) for Ro = O. To the order considered, replacement of r by
1 - E1/2
 and expansion in terms of E 1 / 2 works similarly as a Taylor expansion of Pg
about r = 1 with only the first two terms retained, that is,


" ( dV O )
VOl = VOl - ( 1 - r ) ---2Q + O ( E )
r rO dr ' '
r=l


(N.12)


From equation (7,215a), one can obtain dvro/dr, which wil1 involve the second derivative
of p, obtainable from equation (7,218) in terms of p and d p / dr; the latter may be expressed
in terms of v r and p by means of (7.215a), resulting in
v
i = v
b + (1 - r) (
i ) {[ (n 2 + K 2 )pgi + 2inv
b ] + v
b} + O(E).


(N ,13)




1432


SLENDER STRUCTURES AND AXIAL FLOW


Hence, by comparing with (N.H), it is found that Cf = o.
We next apply the boundary condition at the mean position of the oscillating cylinder,


vr(l) = iw
.


(N.14)


Substituting equation (N.1O) therein with cf = o leads to


. A A 01 ' iEl/2 [( 2 2) A OI " 2 . A OI ' ]
l W 1J = V - - n + K P + lnV
rO WA O rO .


(N.15)


This can then be used to detennine Ag in (N,8), Calculating v
b from (7.215a), one obtains


-(4 - w2)

Ag = ( 2nEl/2 ) [ 2n ] a2El/2 .
1 + A aJ
(a) - ""7Jn(a) - 2 (n 2 + K 2 )J n (a)
WA W K A


(N.16)


From this expression, it is clear that the viscous terms wil1 be important only in the
neighbourhood oť the eigenfrequencies for inertial osci1lations, given by W = w i , obtained
by solutions of the equation


I 2n
aJn(a) - ""7Jn(a) = O.
W


(N.17)


Therefore, to the lowest order in E 1 / 2 the term 2n E 1 / 2 /WA may be neglected, leading
fina1ly to the expression for the dimensionless wa11 pressure fluctuation amplitude,


A (4-w2)

Pw = [ 2 2 2 ] ,
2n aJ n -l(a) a (n +K) 1/2
n + -- + E
W Jn(a) K2(iw)I/2


(N.18)


where it is reca11ed that a 2 = [(4 - w 2 )/w 2 ] K 2 .




Appendix o
Nonlinear Motions of a Shell
Conveying Fluid


0.1 THE PARTICULAR SOLUTION, Fp


Tbe fulI form of equation (7.263) is given by


Fp = Ci (t) cos(nO) sin(nx I L) + C2(t) sin(nO) sin(n x I L) + C3(t) cos(nO) sin(2n x I L)
+ q(t) sin(nO) sin(2n xl L) + cs(t) sin(nx I L) + C6(t) sin(3n x I L)
+ C7(t) sin(5nxl L) + C8(t) cos(nxl L) + C9(t) cos(2nxl L) + ClO(t) cos(3nxl L)
+ Cll (t) cos(4n x I L) + Ci2(t) cos(nO) + C13(t) cos(nO) cos(n x I L)
+ CI4(t) cos(nO) cos(2n x I L) + CIS(t) cos(nO) cos(3nx I L)
+ CI6(t) cos(nO) cos(4nx I L) + Ci7(t) cos(nO) cos(5nxl L)
+ CI8(t) cos(nO) cos(6nxl L) + CI9(t) cos(nO) cos(7nxl L) + C20(t) sin(nO)
+ C21 (t) sin(nO) cos(nxl L) + C22(t) sin(nO) cos(2nxl L)
+ C23(t) sin(nO) cos(3nxl L) + C24(t) sin (nO) cos(4nxl L)
+ C2S(t) sin(nO) cos(5n x I L) + C26(t) sin(nO) cos(6n x I L)
+ C27(t) sin(nO) cos(7n xl L) + C28(t) cos(2nO) + C29(t) cos(2nO) cos(n x I L)
+ C30(t) cos(2nO) cos(3nx I L) + C31 (t) sin(2nO) + C32(t) sin(2nO) cos(nxl L)
+C33(t)sin(2nO)cos(3nxIL), (0.1)


where the functions Cj(t), i = 1, ...,33, are as follows:


CI (t) = AnlklO + e)2, C2(t) = BnlklO + e)2, C3(t) = A n 2(4k)/(4 + e)2,
C4(t) = B n 2(4k)/(4 + e)2, cs(t) = AOlk, C6(t) = A03k19, q(t) = Aoskl25,
22-
C8(t) = -(AnI A n2 + Bnl B n2)kl4, C9(t) = (Ani + B n2 )kI32,
- 2 2-
CIO (t) = (An I A n 2 + Bn I B n 2)k 136, Cll (t) = (A n2 + B n2 )k 1 1 28,
Ci2(t) = -Anl A Ol k ql2, C13(t) = -A n 2(AoI + 9A03) (P<) 10 + e)2,


1433




1434


SLENDER STRUCTURES AND AXIAL FLOW


Cl4(t) = Ani (AOl - 9A03) (PC) /(4 + e)2,
( 1- ) 2
C1S(t) = A n 2(AoI - 25Aos) '2k /(9 + e) ,
( 1- )/ 2 ( 1- ) 2
CI6(t) = Ani (9A03 - 25Aos) '2k (16 + e), C17(t) = 9An2A03 '2k /(25 + e) ,
Cls(t) = 25AnlAOS (pc) /(36+e)2, Cl9(t) = 25An2AOS (pc) /(49+e)2,
( 1- ) 2
C20(t) = -BnI A Ol k q/2, C21(t) = -B n 2(AoI + 9A03) '2 k /(1 + e) ,
( 1- ) 2
C22(t) = Bnl (AoI - 9A03) '2k /(4 + e) ,
( 1- ) 2
C23(t) = B n 2(AOl - 25Aos) '2k /(9 + e) , (0.2)
C24(t) = B n l(9A03 - 25Aos) (PC) /(16 + e)2, C2S(t) = 9Bn2A03 (F) /(25 + e)2,
( 1- )/ 2 ( 1- ) 2
C26(t) = 25BnlAOS '2k (36 + e), C27(t) = 25Bn2AOS '2k /(49 + e) ,
C2S(t) = (-A
I + B
I - 4A
2 + 4B
2)kq/32,
( 1- ) 2
C29(t) = -9(A n l A n 2 - Bnl B n 2) 4 k /(1 + 4e) ,
( 1- ) 2
C30(t) = (An1 A n2 - B n 1B n 2) 4k /(9 + 4e) ,
C31 (t) = -(Ani Bnl + 4An2Bn2)kq /16,
( 1- ) 2
C32(t) = -9(A n IB n 2 + A n 2 B nl) 4k /(1 + 4e) ,
( 1- ) 2
C33(t) = (An1 B n2 + A n 2 B nl) 4k /(9 + 4e) ,


in which


EhL 2
k--
- JT 2 a '


_ n 2
k=k-,
a


e = (
) 2 (
 ) 2 q = (
t :
 '


(0.3)


and al1 the Ani etc. are understood to be Ani (t) etc.


0.2 THE DISCRETIZED EQUATIONS OF MOTION


The discretized equations of motion resulting fmm equation (7.267) in the case of the
Paidoussis & Denise model are given by the following system of seven equations. The




NONLINEAR MOTIONS OF A SHELL CONVEYING FLUID 1435


first one is


.. . ( 2 U 2 p1'(2 I n (Tra/L) )
Anl(t) +2
nIWnIAnl(t) + W nl - 2mII
(1'(a/L) Anl(t)
4UpLI n (2Tra/L) . 3 2 2
- A n 2(t) +hIAnl(t) +hIAnl(t)Bnl(t) +h2 A nl(t)A n2 (t)
3mII
(21'(a/L)
+ h3Anl (t)B
2(t) + h4An2(t)Bnl (t)B n 2(t) + hsAnl (t)AQl (t) + h6Anl (t)AQ3(t)
+ h7 A nl (t)Aos(t) + hsA n l(t)A51 (t) + h9 A nl (t)A53(t) + hlOAnl (t)A5s(t)
+ hllAnl (t)AQl (t)A03(t) + hl2Anl (t)AQ3(t)Aos(t) = O, (0.4)


where


1'(L [ ( Tr2 n2 ) 2 ( Eh1'(4 ) ( 1'(2 n2 ) -2 ]
W;I= 2m l D L2 + a 2 + a 2 L4 L2 + a 2 '
1 h 2 In (1'(a/L) ch1'(L
m l - - p 1'( L + P L r I -
- 2 s 2I
(1'(a/L),.,n - 4W n lml'


and h i, i = 1, . . . , 12, are coefficients depending on geometry, material properties and n
that arise from projections ofthe part of equation (7,248) involving the stress function F;
D = Eh 3 /[12(1 - v 2 )]. The second equation has the same structure as the first, but with
the AnI and A n 2 replaced by Bnl and B n 2, and vice versa.
The third equation is


.. , ( 2 U 2 p1'(2 I n (21'(a/L) )
A n 2(t) + 2
n2Wn2An2(t) + W n 2 - m2I
(2Tra/L) A n 2(t)
8U pLl n (1'(a/ L) , 3 2 2
+ Anl(t) +kIA n2 (t) +kIA n 2(t)B n2 (t) +k2 A n2(t)A nl (t)
3m2I
(1'(a/L)
+ k3An2(t)B
1 (t) + k4Anl (t)Bnl (t)B n 2(t) + ksA n 2(t)Aol (t) + k6An2(t)AQ3(t)
+ k7An2(t)AOs(t) + ksA n 2(t)A61 (t) + k9An2(t)A53(t) + klOAn2 (t)A5s (t)
+ kll A n 2(t)Aol (t)AQ3(t) + k12An2(t)AOI (t)Aos(t) = O, (0.5)


where


2 =!!!:.... [ ( 41'(2 n2 ) 2 ( 16EhTr4 ) ( 4Tr2 n2 ) -2 ]
W n 2 2 D L 2 + 2 + 2 L 4 L 2 + 2 '
m2 a a a
1 h 2 In (21'(a/L) ch1'(L
m 2 - - P 1'( L + P L r 2 -
- 2 s 4I
(21'(a/L),.,n - 4W n 2 m 2'


and ki, i = 1,..., 12, are appropriate coefficients. The fourth equation is obtainable
from the third in the same way as the second from the first. The fifth, sixth and seventh




1436


SLENDER STRUCTURES AND AXIAL FLOW


equations are:


.. . ( 2 U 2 p 7r2I o(7ra/ L» )
AOJ(t) + 2
OIWOJAOJ(t) + WO I - mOJI
(7ra/L) AQl(t)
+ IIAoI (t) [A
l (t) + B;l (t)] + 12AOI (t) [A
2(t) + B;2(t)] + 13 [A
l (t) + B;l (t)]
+ 14 [A
2(t) + B;2(t)] + IsAQ3(t) [A
l (t) + B;l (t)]
+ 16AQ3(t) [A
2(t) + B;2(t)] + hAos(t) [A
2(t) + B;2(t)] = O, (0.6)


.. , ( 2 3U 2 p7r 2 4J(37ra/ L» )
AQ3(t) + 2
Q3WQ3AQ3(t) + wQ3 - mQ3I
(37ra/L) AQ3(t)
+ nI AQ3(t) [ A
l (t) + B;l (t)] + n2 A 03(t) [A
2(t) + B;2(t)]
+ n3 [A
l (t) + B;l (t)] + n4 [A
2(t) + B;2(t)] + nsAOI (t) [A
l (t) + B;l (t)]
+ n6 A OI (t) [A
2(t) + B
2(t)] + n7Aos(t) [A
l (1) + B;l (t)] = O, (0.7)


.. . ( 2 5U 2P 7r 2 1 0 (57ra/L» )
Aos(t) + 2
oswosAos(t) + Wos - mosl
(57ra/L) Aos(t)
+ PI Aos(t) [ A
l (t) + B;l (t)] + p2AOS(t) [ A
2(t) + B
2(t)]
+ P3 [A
l (t) + B;l (t)] + P4 [A
2(t) + B;2(t)] + PSAOI (t) [A;2(t) + B;;Z(t)]
+ P6A03(t) [A
l (t) + B;l (t)] = O, (0.8)


where


24J(7ra/ L)
mOl = P s h7r L + pL 1
(7ra/ L) ,
2 1 0(37ra/L)
mQ3 = P s h7r L + pL 3
(37ra/ L) '
24J(57rafL)
m OS = P h7r L + P L
s 5
(57raf L)'


2 7r L ( D7r 4 Eh )
WOl=- - L 4 +2 '
mOl a
2 _ 7rL ( 81D7r 4 Eh )
w03 - - L 4 + 2 '
m03 a
2 _ 7r L ( 625D7r 4 Eh )
wos - - L 4 + 2 '
mOs a


ch7r L

Ol =
2WOJmOl
ch7r L

03 =
2W03 m 03
ch7r L

os =
2wosmos


and li, ni and Pi, i = 1, . . . , 7, are coefficients similar to ki.




Appendix P
Flutter of Collapsible Tubes
Conveying Fluid


P.1 BIBLlOGRAPHY


An enonnous amount of work has been done on this subject, much too much to even
consider reviewing properly in a book such as this, not specializing in biomechanics.
Nevertheless, it was thought to be usefu1 to provide a list of some of the most import-
ant studies from 1969 on. Thus, the interested reader, armed with the discussion in
Section 7.9.5 and with the aid of this list, can 1aunch a more in-depth study ofthe subject.


Year


Title and reference


1969


Author(s)
CONRAD, W.A


KATZ, AI., CHEN, Y. &
MORENO, AH.


1970


UR, A & GORDON, M.


1971


GRIFFITHS, D.J.


1973


BROWER, R.W. &
NOORDERGRAAF, A


1975


GRlFFITHS, D.J,


BROWER, R.W. &
SCHOLTEN, C.


1977


PEDLEY, T.J.


WEAVER, D.S. &
PAIDOUSSIS, M.P.


Pressure-flow re1ationships in collapsib1e
tubes. lEEE Trans. Biomed. Eng, 16,
284-295,
Flow through a collapsib1e tube:
experimental analysis and mathematical
model. Biophysical J. 9, 1261-1279.
Origin of Korotkoff sounds. Amer.
J, Physiol. 218, 524-529.
Hydrodynarnics of male rnicturition. Med.
Bio. Eng. 9,581-596.
Pressure-flow characteristics of collapsib1e
tubes: a reconci1iation of seerning1y
contradictory results. Ann. Biomed. Eng. 1,
333-355.
Negative-resistance effects in flow through
collapsib1e tubes. Parts 1-3, Med, Biol.
Eng. 13, 785-802.
Experimental evidence on the mechanism
of instabi1ity of flow in collapsib1e vesse1s.
Med, Biol. Eng. 13, 839-845,
Puhnonary fluid dynarnics. Annu. Rev.
Fluid Mech. 9, 229-274.
On col1apse and flutler phenomena in thin
tubes conveying fluid. J. Sound Wb. 50,
117-132.


1437




1438 SLENDER STRUCTURES AND AXIAL FLOW
Year Author(s) Title and reference
GRIFFITHS, D.J, Oscillations in the outflow from a
collapsible tube. Med. Biol. Eng. Comput.
15, 357-362.
SCHOENDORFER, D.W. & The collapsib1e tube as a prosthetic vocal
SHAPIRO, AH. source. Proc. San Diego Biomed. Symp. 16,
349-356.
MATSUZAKI, Y. & Stability analysis of straight and buckled
FUNG, Y.c. two-dimensional channels conveying an
incompressible flow. J. Appl. Mech. 44,
548-552.
1978 BONIS, M. & RIBREAU, C. Étude de quelques propriétés de
l'écoulement dans une conduite collabable.
La Houille Blanche 33, 165-173.
CONRAD, W.A, Note on the oscillations of collapsible
COHEN, M.L. & tubes. Med. Biol. Eng. Comput, 16,
McQUEEN, D.M. 211-214.
1979 KAMM, R.D. & Unsteady flow in a collapsible tube
SHAPIRO, AH. subjected to external pressure or body
forces. J, Fluid Mech. 95, 1-78.
BONIS, M. Ecoulement visqueux pennanent dans un
tube collabable elliptique. These de
doctorat d'Etat, Université de Technologie
de Compiegne, France.
MATSUZAKI, Y. & Nonlinear stability analysis of a
FUNG, Y.-C. two-dimensiona1 model of an elastic tube
conveying a compressible flow. J. Appl.
Mech. 46,31-36.
PAIDOUSSIS, M,P' Discussion to Matsuzaki & Fung. J. Appl.
Mech. 46,963-964.
1980 GROTBERG, J.B. & Fluid-dynamic flapping of a collapsible
DAVIS, S.H. channel: sound generation and flow
limitation. J. Biomechanics 13, 219-230.
PEDLEY, T.J. The Fluid Mechanics oj Large Blood
Vessels, Cambridge University Press.
1981 McCLURKEN et al. Steady supercritical flow in collapsible
tubes. Part 2: theoretical studies, J, Fluid
Mech. 109,391-415.
1982 GROTBERG, J.B. & A subsonic flutter anomaly. J. Sound Vib.
REISS, E.L. 80, 444-446,
BERTRAM, C.D. & A mathematical model of unsteady
PEDLEY, T.J. collapsible tube behaviour. J. Biomechanics
15, 39-50.




FLUTTER OF COLLAPSIBLE TUBES CONVEYING FLUID 1439


Year


Tit1e and reference


Author(s)
BERTRAM, C.D.


1983


COWLEY, S.J.


STEPHANOFF, KD. et al.


1984


GROTBERG, J.B. &
REISS, E.L.
OHBA, K et al.


GAVRIELY, N. et al.


1985


WEBSTER, P.M. et al.


OHBA, K et al,


1986


CANCELLI, C. &
PEDLEY, T.J,
GROTBERG, J.B. &
SHEE, T.R.
BERTRAM, C.D.


SAKURAI, A. & OHBA, K


BARCLAY, W.H. &
THALAYASINGAM, S.


WILSON, T.A.,
RODARTE, J.R. &
BUTLER, J.P.


MATSUZAKl, Y.


Two modes of instabi1ity in a thick-walled
collapsib1e tube conveying a flow.
J. Biomechanics 15, 223-224.
On the wavetrains associated with e1astic
jumps on fluid-filled e1astic tubes.
Q. J. Mech. Appl, Math. 36, 289-312.
Fluid flow a10ng a channe1 with an
asymmetric oscillating constriction. Nature
305,692-695.
Subsonic flapping flutter. J. Sound Vib. 92,
349-361.
Self-excited osci11ation of flow in a
collapsib1e tube. Tech. Rep, Kansai Univ.
No. 25,
Measurement and theory of wheezing
breath sounds. J, Appl. Physiol. 57,
481-492.
Wa11 motion in expiratory flow limitation.
J Appl. Physiol, 59, 1304-1312.
Se1f-excited osci11ation of flow in
collapsib1e tube. II. Ana1ysis and
experiment on initiation of oscillation.
Tech. Rep. Kansai Univ, No. 26.
A separated-flow model for collapsib1e-tube
osci11ations. J. Fluid Mech. 157,375-404.
Compressib1e-flow channe1 flutter. J. Fluid
Mech. 159, 175-193.


Unstab1e equi1ibrium behaviour in
collapsib1e tubes. J. Biomechanics 19,
61-69.
Se1f-excited osci11ation of flow in
collapsib1e tube. III. A resonant oscillation
of air flow. Tech. Rep. Kansai Univ. No. 28.
Self-excited osci11ations in thin walled
collapsib1e tubes. Med. Biol. Eng. Comput.
24, 482-487.
Wave-speed and viscous flow limitation.
Handbook of Physiology (eds P.T. Macklem
& J. Mead), pp. 55-61. Bethesda, MD:
American Physio1ogica1 Society.
Self-excited oscillation of a collapsib1e tube
conveying fluid. In Frontiers in
Biomechanics (eds G. Schmid-Sch6nbein,
S. Woo and B. Zweifach), pp. 342-350.
New York: Springer-Verlag.




1440


SLENDER STRUCTURES AND AXIAL FLOW


Year


Tit1e and reference


1987


Author(s)
KAMM, R.D.


1989


GROTBERG, J.B. &
GAVRIELY, N.


KAMM, R.D. &
PEDLEY, T.J.


JENSEN, O.E. &
PEDLEY, T.J.
BERTRAM, C.D.,
RAYMOND, C.J. &
BUTCRER, KS.A.


GAVRIELY, N. et al.


ORBA, K, SAKURAI, A. &
OKA,J.


MORGAN, P. &
PARKER, KR.


ORBA, K, et al.


1990


JENSEN, O.E.


1991


BERTRAM, C.D.,
RAYMOND, C.J. &
PEDLEY, T.J.
WALSR, C.,
SULLIVAN, P.A. &
RANSEN, J.S.
BERTRAM, C.D.,
RAYMOND, C.J. &
PEDLEY, T.J.


BERKE, G.S. et al.


F10w through collapsible tubes. In
Handbook oj Bioengineering (eds
R. Skalak & S. Chien), pp. 23.1-23.19.
New York: McGraw-Hill.
F1utter in collapsible tubes: a theoretical
model of wheezes. J. Appl. Physiol. 66,
2262-2273.
F10w in collapsible tubes: a brief
review. ASME J. Biomech. Eng. 111,
177-179.
The existence of steady flow in a collapsed
tube. J. Fluid Mech. 206, 339-374.
Oscil1ations in a collapsed-tube analog of
the brachial artery under a
sphygmomanometer cuff. ASME
J. Biomech. Eng. 111, 185-191.
F1utter in flow-1imited collapsible tubes: a
mechanism for generation of wheezes.
J. Appl. Physiol. 66,2251-2261.
Self-excited oscil1ation of flow in
collapsible tube. IV. Laser Doppler
measurement of local flow field. Tech. Rep.
Kansai Univ., No. 31.
A mathematical model of flow through a
collapsible tube. I. Model and steady flow
results. J. Biomechanics 22, 1263-1270.
F10w oscillation and wave propagation in
collapsible tube. In Progress and New
Directions oj Biomechanics (eds Y.C. Fung
et al. ), pp. 213-227. Tokyo: Mita Press.
Instabi1ities of flow in a collapsed tube.
J. Fluid Mech. 220, 623-659.
Mapping of instabi1ities for flow through
collapsed tubes of differing length.
J. Fluids Struct. 4, 125-153.


Subcritical flutter in collapsible tube flow:
a model of expiratory flow in the trachea.
ASME J. Biomech. Eng. 113, 21-26.
App1ication of non1inear dynamics
concepts to the analysis of self-excited
oscil1ations of a collapsible tube conveying
fluid. J. Fluids Struct. 5,391-426.
Experimental evidence in the in vivo canine
for the collapsible tube model of phonation.
J. Acoust. Soc. Am. 89, 1358-1363.




FLUTTER OF COLLAPSIBLE TUBES CONVEYING FLUID 1441


Year Author(s) Tit1e and reference
1992 JENSEN, O.E. Chaotic osci11ations in a simple
collapsible-tube model. ASME J. Biomeeh.
Eng. 114,55-59.
BERTRAM, C.D. & A collapsible-tube osci11ator is not readily
BUTCHER, K.S,A. enslaved to an external resonator. J. Fluids
Struet. 6, 163-180.
PEDLEY, T.J. Longitudina1 tension variation in
collapsible channels: a new mechanism for
the breakdown of steady flow, ASME
J. Biomeeh. Eng. 114,60-67.
YAMANE, T. & ORITA, T. Self-excited osci11ations with and without
supercritical flow in collapsible tubes. Proe.
7th lnternational Conferenee on Biomed.,
Engineering, Singapore, pp. 502-504.
1994 BERTRAM, C.D., Prediction and measurement of the
SHEPPEARD, M.D. & area-distance profile of collapsed tubes
JENSEN, O.E. during self-excited osci11ation. J. Fluids
Struet. 8, 637-660.
GROTBERG, J.B. Pulmonary flow and transport phenomena.
Annual Rev. Fluid Mech, 26,529-571.
1995 BERTRAM, C.D. Dynamica1 system ana1yses of aperiodic
flow-induced oscillations of a collapsible
tube. J. de Physique 5, 2101-2116.
WAGNER, C.D. & Nonlinear chaotic dynamics of arterial
PERSSON, P.B. blood pressure and renal blood flow. Am.
J. Physiol. 268, H621-H627.
BERTRAM, C.D. The dynamics of collapsible tubes. In
Biologieal Fluid Dynamies (eds
c.P. Ellington & T.J, Pedley), pp. 253-264.
Cambridge (UK): The Company of
Biologists Ltd.
WALSH, C. F1utter in one-dimensiona1 collapsible
tubes.1. Fluids Struet. 9, 393-408.
BERTRAM, C.D. & Area and pressure profiles for collapsible
GODBOLE, S.A. tube oscillations of three types. 1. Fluids
Struet. 9,257-277.
1996 ARMITSTEAD, J.P., A study of the bifurcation behaviour of a
BERTRAM, C.D. & model of flow through a collapsible tube.
JENSEN, O.E. Bul/. Math. Biol. 58,611-641.
MATSUZAKI, Y. & Numerical ana1ysis of flow in a collapsible
SElKE, K. vessel based on unsteady and quasi-steady
flow theories. In Computational
Biomeehanies (eds K. Hayashi &
H. Ishikawa), pp. 185-198. Tokyo:
Springer - Verlag.
1997 LAROSE, P.G. & F1utter and long-wave instabilities in
GROTBERG, J.B. compliant channels conveying developing
flows. J. Fluid Mech. 331, 37-58.




1442


SLENDER STRUCTURES AND AXIAL FLOW


Year


Author(s)
DAVIES, C. &
CARPENTER, P.W.


1998


HVANG, L.


LVO, X,y' & PEDLEY, T.J.


PEDLEY, T.J. & LVO, x.y,


HAYASHI, S" HAYASE, T.
& KAWAMURA, H.


1999


KOVNANIS, K. &
MATHlOVLAKIS, D.S.


BERTRAM, C.D. &
CASTLES, R.J.


BAVER, T.T. et al.


TANG, D. et al.


BROOK, B.S.,
FALLE, S.A.E.G. &
PEDLEY, T.J.


HAYASHI, S. et al,


2000


LVO, X.Y. & PEDLEY, T.J.


BERTRAM, C.D.


BERTRAM, C.D. &
CHEN, W.


Title and reference


Instabi1ities in a plane channel flow
between comp1iant walls. J. Fluid Mech.
352, 205-243.
Reversa1 of the Bemoulli effect and
channe1 flutter. J. Fluids Struct. 12,
131-151.
The effects of wa11 inertia on flow in a two
dimensiona1 collapsib1e channel. J. Fluid
Mech. 363,253-280,
Model1ing flow and oscillations in
col1apsible tubes. Theor. Comput. Fluid
Dyn. 10, 277-294.
Numerica1 ana1ysis for stability and
self-excited oscillation in collapsib1e tube
flow, ASME J. Biomech. Eng, 120,
468-475,
Experimenta1 flow study within a
self-oscillating col1apsible tube, J. Fluids
Struct, 13,61-73.
F10w 1imitation in unifonn thick-walled
collapsib1e tubes. J. Fluids Struct. 13,
399-418.
Forced expiratory wheezes in a patient
with dynamica1 expiratory narrowing of a
centra1 airways and an osci11ating pattem
of the flow-vo1ume curve. Respiration 66,
163-166.
A non1inear axisymmetric model in
fluid-wa11 interactions for steady viscous
flow in stenotic elastic tubes. J. Biomech,
Eng. 121,494-501.
Numerica1 solutions for unsteady
gravity-driven flows in collapsib1e tubes:
evo1ution and roll-wave instabi1ity of a
steady state. J. Fluid Mech, 396, 223-256,
Dynamic characteristics of collapsible tube
flow. JSME lnt. J., Series C, 42, 689-696.
Multip1e solutions and flow 1imitation in
collapsible channel flows. 1. Fluid Mech.
420,301-324.
Simp1e resistive mode1s for steady flow
through unifonn and tapered-stiffness
col1apsible tubes. J. Fluids Struct. 14,
593-612.
Aqueous flow limitation in a
tapered-stiffness colIapsible tube. J. Fluids
Struct, 14, 1195-1214,




FLUTfER OF COLLAPSIBLE TUBES CONVEYING FLUID 1443


Year


Tit1e and reference


Author(s)
BERTRAM, C.D. & SHE, J.


2001


HVANG, L.


2002


SHlM, E.B. & KAMM, R.D.


CAl, Z.x. & LVO, X, Y.


2003


BERTRAM, C.D. &
ELLlOTT, N.S.J.


Chaotic and mode-locked interactions
between flow-induced collapsib1e-tube
osci11ation and pulsatile upstream forcing.
ln Stoehastie & Chaotie Dynamies in the
Lakes (eds D.S, Broomhead et al.),
pp. 510-516. Me1vi11e: American Institute
of Physics.
Viscous flutter of a finite e1astic membrane
in Poiseui11e flow. J. Fluids Struet. 15,
1061-1088.
Numerical simulation of steady flow in a
compliant tube or channel with tapered wa11
thickness. J. Fluids Struet. 16, 1009-1027.
Self-excited osci11ations in a fluid-beam
collapsib1e channel. Paper IMECE
2002-39025, FS/, AE & FN+N Symposium,
(ed. M.P. Paldoussis et al. ) New Orleans,
LA. New York: ASME.
F1ow-rate limitation in a uniform
thin-walled collapsib1e tube, with
comparison to a uniform thick-walled tube
and a tube of tapering thickness. J. Fluids
Struet. 17,541-559.


Considering experimental papers first, some of the most important are by Conrad et al. t
(1969, 1978), Griffiths (1975, 1977), Brower & Scholten (1975), Bonis & Ribreau (1978),
Weaver & Paldoussis (1977), Bertram et al. (1982, 1990, 1991, 1994, 1995, 1999,2003),
Ohba et al. (1984, 1985, 1989), Sakurai & Ohba (1986), Webster et al. (1985), Gavriely
et al. (1984, 1989), Bertram & She (2000) and Bertram & Elliott (2003). Many of these
papers have substantial analytical components,
Important analytical contributions have been made by Conrad (1969), Kamm & Shapiro
(1979), Weaver & Paldoussis (1977), Matsuzaki et al. (1977, 1986), Grotberg et al. (1980,
1982, 1984, 1985, 1989), Stephanoff et al. (1983), Cancelli & Pedley (1985), Jensen &
Pedley (1989), Jensen (1990, 1992), Armitstead et al. (1996), Walsh et al. (1991, 1995),
LaRose & Grotberg (1997), Luo & Pedley (1998, 2000), Huang (1998, 2001), Hayashi
et al. (1998, 1999), Shim & Kamm (2002).


P.2 ANAL VTICAL MODELS


Here, an outline of some of the model s is given, generally in greater detail than in
Section 7.9,5.


tUnusually, in these two paragraphs, et a/, is used to refer also to the ensemble of papers involving the lead
author, for brevity,




1444 SLENDER STRUCTURES AND AXIAL FLOW
P.2.1 Conraďs (1969) lumped-parameter model
This is a heuristic model and, being particularly simple, is given in toto in Section 7,9,5.
P.2.2 Bertram & Pedley's lumped-parameter model
The summary given by Annitstead et al. (1996) is adopted here. All variables are normal-
ized with respect to Ao = Jr ro2, ro being the radiu s of the uncollapsed cross-sectional
area of the tube, the fluid velocity uo' and the fluid density p*; thus, Po = p*u 0 2 , Qo =
uoAo, the resistance RO' = p*u o / Ao, the inertance I; = p*r o / Ao, and so on, All
starred quantities are dimensional, whereas unstarred ones denote their dimensionless
counterparts.
Denoting the area of the narrowest part of the tube (the point of maximal collapse) by
A *, the dimesionless area is A = A * / Ao. Hence, an average area of the collapsed section
of the tube, A is defined by
- I
A = 2:(1 + A)
and an approximate average velocity, U, by
(P,I)
I -
U=2:(UI+U2)/A
(P.2)
where Ul and U2 are the dimensionless flow velocities in the rigid sections upstream and
downstream, respectively, where A = I. Hence the conservation oj mass may be expressed
in terms of the fluid volume accumulating in the collapsed area of the tube and the flows
in and out, that is,
Ul - uA = IA
where I is the length of the collapsible tube. Downstream, mass conservation gives
(P.3)
U2 - uA = O.
(PA)
Assuming that the flow is laminar, the flow in the inlet and outlet pipes may be
characterized by
Pu - PI = RIUI + hŮl,
P2 = R2 U 2 + /zŮ2
(P,5)
where Pu is the pressure far upstream, and subscripts 1 and 2 denote the values at the
upstream and downstream rigid sections, respectively.
The energy in the upstream part oJthe collapsed tube is considered next. It is assumed
that energy loss is concentrated in the last quarter of the collapsible tube and that it is
represented by the Poiseuille resistance function R (A). Therefore, in the collapsed section
upstream of the point of maximal collapse or constriction,
PI + ui - (p + u2) = iIR(A)u + lu,
(P.7)
where R(A) = 2Rk/A for A::: 1 (circulartube) and R(A) = Rk(1 + I/A2) for A < 1
(approximately e1liptical tube) and Rk = 4J.L* /(p*uor o )'

FLUTTER OF COLLAPSIBLE TUB ES CONVEYING FLUID 1445
Downstream of the constriction, the fluid is assumed to emerge as a separated jet, result-
ing in energy dissipation and 1ess than full pressure recovery, The quasisteady momentum
balance equation is
P - P2 = - J(A)u, (P.8)
where J(A) represents the head loss, and is given by J(A) = (A -1 - 1) for A < 1, and
i (A -2 - 1) for A :::: 1 or if the flow is reversed (U2 < O).
The following form of the tube law is used:
{ Pk(1 - A- 3 / 2 )
P - Pe = P(A) = 100Pk(A - 1)
for A < 1,
for A :::: 1.
(P,9)
Reduction oj the equations is then undertaken, yielding
. 2(U1 - U2) , P(A) + Pe + J(A)u - R2 U 2
A = , U2 =
l h
ú - A + 1 [ _ _ P A _ R U uf _ u _ lR(A)U2
1 - l + (A + 1)11 Pu Pe () 1 1 + 2 2A2 4A
2(uf - u) lÚ 2 ] .
+ (A + 1)2 + A + 1 '
(P,lO )
the three ODEs are given in functional form in the main text.
P.2.3 The Cancelli & Pedley (1985) model
Here, the simplified and modified version of the model, as given by Jensen (1992) is
adopted.
A different nondimensionalization scheme is used in this case, as follows. The appro-
priate length scale is L = (T*D/K;)1/2 and the time scale i = L/c, where T* is the
longitudinal tension, D the diameter of the uncollaped tube, K; the constant in the tube
law [equation (7.278a)], and c = (K;/ p*)1/2. Hence,
x*
x = --:;-,
L
A*
a=*,
Ao
p*
P = K* '
p
u*
u= *'
Co
L*
A = --;::-,
L
(P.lI )
where A = !n D2; it is thus noted that the tube length is scaled with respect to a tension
parameter.
With this scheme, the mass and momentum conservation equations are
a,t + (ua),x = O,
u,t + uU,x = -P,x - d,
(P.12)
where the symbols have the usual meaning, d (x, t) denotes energy dissipation, and the
subscripts denote derivatives. The dissipation is mainly due to flow separation at the
constriction and it may be represented by
d = (X - l)u U,x ;
(P.13 )

1446
SLENDER STRUCTURES AND AXIAL FLOW
x = 1 upstream of the eonstrietion, where the flow is fulIy attaehed and aeeelerating, and
O < X < 1 (X = 0.2 typiealIy) downstream. The point of separation, X, is taken to be the
point where the flow starts to deeelerate,
U,x(X, t) = O.
(P.14)
A modified tube law is adopted, where
P - Pe = P(a) - !a,xx,
(P.15)
where the seeond eomponent is associated with longitudinal tension in the tube walI -
more fulIy !a,xx(1 + ! a: x )-3/2; the tube length has been sealed so that tension has a
dimensionless magnitude of unity.
The boundary eonditions specify that the areas and pressures at the ends of the
eolIapsible tube mateh those in the rigid extensions upstream and downstream, Thus,
a(O, t) = a(A, t) = I,
p(O, t) = Pu - !u 2 (0, t) - RIU 2 (0, t) - II U,t(O, t),
p(A, t) = R2U2(A, t) + /z U,t(Alt),
(P.16)
in which Ri = R; A02 1 p*, Ii = 1;* A * 1 p* L are the resistanee and the inertanee upstream
(i = I) and downstream (i = 2); refer to Bertram ct a1. (1994, appendix A),
P.2.4 The Grotberg & Reiss (1984) 2-D model
The dimensionless equations are obtained by defining the length seale b*, the half-height
of the interplate ehannel, and the time seale c* Ib*, where c* = (£*b* 1 p*)1/2, £* being
the elastanee of the walIs and p* the fluid density. Then,
'[)*
'[)=-
£*b*4 '
f*b*
2f=-,
c*
g* I-tb*
2 g =-
* '
c
p*
P=,
P c
x*
x=-
b*'
W*
W=-,
b*
c*
t = -t,
b*
U*
s=-
* '
c
f*
y=-;,
g
p;h*
I-t = p*b* '
<t>*
<t>=-
b*c*'
Q *b*
Q=-,
c*
(P.17)
where '[)* = Ey h*3/[12(1 - v 2 )], Ey being Young's modulus, v the Poisson ratio and
h * the plate thickness. t The dimensionless equations eorresponding to (7,292)-(7.294)
are
V 2 <t> = O, <t>,t + !(V<I»2 + 2f<l> + P = O,
l x + A
I-t w,tt + 2g W,t + '[)w,xxxx + W - d w,xx x (w,x)2 dx + P - Pe = O,
(P,18)
tNote that I.t here is the inverse of what it is in most of Chapter 7. In Section 7.9.5(b.iii), it is denoted as I.t*,
but here the asterisk is not used to avoid confusion,

FLUTIER OF COLLAPSIBLE TUBES CONVEYING FLUID 1447
in which d = V* j (6)" * b*2 h *2 £*) and ).. = ).. * j b* , ).. * being the wavelength of deforma-
tion of the wal1/channel. The impermeability condition on the lower wal1 is
W,t + cI>,x W,x - cI>,z = O.
(P.19)
For symmetric or antisymmetric motion of the parallel plates,
cI>,z =0
or
cI> = O,
(P,20)
respectively, at Z = O,
For travelling wave solutions, we have {w, cI>} oe exp{ i [kx - Q t]}, where k = 21f j).. is
the dimensionless wavenumber, The condition for flutter is taken to be Lm ( Q) = O, so that
Q == Re( Q ), Then,if Q isexpressedas Q = akS,itisfoundthata = Y(Y+J.,Lk tanhk)-l
and
[ (1 + Bk 4 )tanhk ] 1/2
S(k) =
J.,L a 2 k 2 tanh k + k(1 - a)2
(P.21)
for symmetric motions,
P.2.5 The Luo &. Pedley (1998) 2-D model
In this case, flow velocity nondimensionalization is with respect to u = q* jb, where q*
is the flow rate and b the rigid-wal1 separation, time with respect to bM U o ' stresses with
p*U 0 2 , and tension with p*Uo2b; the Reynolds number is Re = p*UobjJ.,L*. Hence,
the conservation of mass and Navier-Stokes equations (Section 2.2.1) may be written as
Ui i = O,
Ui,t + Uj Ui,j ::::: -P,i +Re- 1 Ui,jj'
(p.22)
At inlet, on AB in Figure 7.110, for a Poiseui1le flow,
Ul = 6y(1 - y), U2 = O,
(P.23 )
y being the vertical dimensionless coordinate; other appropriate conditions apply for the
outlet and the upper and lower walls. Moreover, denoting the dimensionless displacement
by I}, curvature by K, temsion by T, normal fluid stress on the membrane by O'n, and the
extemal pressure by Pe, the membrane equation is
J.,L I},tt -K T - O'n + Pe = O,
(P.24 )
where J.,L = (p; h*)j(p* b*), p; being the membrane density and h* its thickness.

Appendix Q
Viscous Force Coefficients for
Cylinders in Axial Flow
Various forms of possible dependence of the normal and longitudinal force coefficients,
FN /(!p D u 2 ) and FL/(!p D u 2 ), on the angle of inclination of the cylinder to the
flow, i, have been proposed by Taylor (1952a) as shown in Figure Q,l.
Here we shall adopt the linearized forms already presented in Section 8.2.2, namely
equations (8,13), and discuss plausible values for the coefficients CN, CT, C and Cb,
Q.1 THE LONGITUDINAL VISCOUS COEFFICIENT CT
Perhaps this is the coefficient on which most information may be found.
Hoemer (1965) compiles data from Relf & Powell (1917), also utilized by Taylor,
and from other sources as well, for inclined cables, wires and cylinders, and eventually
generates lines of best fit in remarkable agreement with the data. Specifically, in con-
junction with equation (8.11) he finds CDp = 1.1 and Cf == CT = 0.02. This suggests
CT = (4/7T) CT = 0.025.
Hannoyer (1977), following Hoemer's approach, expresses FL = ! P U 2 7T D cf' so
that C T = 7T Cf' From elementary boundary layer theory, utilizing the Prandtl-Schlichting
FN/ pDU2 F L / pDU2 Type of roughness
2 I ) , I ( . (J
(a) C Dp sin j + 4 Re-' sin' j 5.4 Re-' cos j sin' j
(b) C Dp sin 2 j + Cf sin j Cf cos j ---
' ,
/) )<"'\5::- ;j) ) (/' \\ /  )  \I'\I
(c) Cf sin j Cf cos j

(d) C Dp sin 2 j Cf cos j 
Figure Q.l Types of roughness with corresponding force formulae for FN and FL, in the normal
and longitudinal direction, respectively (Taylor 1952a),
1448

VISCOUS FORCE COEFFICIENTS, CYLINDERS IN AXlAL FLOW 1449
skin friction formu1a for fiat plates and noting that the correction proposed by Hoemer for
fiow over a cylinder is negligible for turbulent boundary layers (in this case Re ::.::: 2 x 106),
the value of c; = 4 X 10- 3 is found to apply. This yields CT = 0.013, and hence
CT = 0.016.
Ni & Hansen (1978) use therelationship C; = (total drag)/ (p Uk 11: DL), in which
U M is the maximum fiow velocity in the test-section, related to the mean velocity U
by UjUM ::.::: 1.14 (Knudsen & Katz 1958). Ni & Hansen present measured values of
C; versus Reynolds number (7.2 x 10 4 < ReD < 1.1 x 105; ReD = U Djv); the test
values are found to cluster around C; = 3.0 x 10- 3 . This leads to CT = 11:(1.14)2 C; =
0.012, and hence CT = 0.015. More recently, Sudarsan ef a/. (1997) measured FL =
 (pAj D) U 2 CT atReD = 0.3 x 10 4 t04x 10 4 and obtainedaverage valuesof CT = 0.013
for nylon tubes and CT = 0.0475 for rubber tubes, the difference being attributed to
different surface roughness.
Dowling (1988a) uses the expression h = p 11: a U 2 C and C = 2.5 X 10-3; hence,
CT = 11: C = 0.008, and CT = 0.010. Lee & Kennedy (1985) use CT = 0.008, On
the other hand, based on an extensive set of measurements, Chen & Wambsganss' (1972)
range is 0.011 < CT < 0.019,
It may be concluded that a reasonable range for CT is
0.010 < CT < 0,025.
(Q.l)
It should be noted that, if the Hoemer (1965) approach is followed, then, for a turbulent
boundary layer over a fiat plate, Cf = O,044j(Red 1 / 6 , where ReL = ReD(Lj D); as
noted before, the correction for the fact that it is a cylinder rather than a fiat plate is
negligible. The important thing is that Cf decreases with increasing length. Moreover,
thOUgh8 = LjD::.::: 25forPai'doussis'experiments,forinstance,itcaneasilybe8 = 1000
or more for towed arrays. Does it mean that Cf wou1d decrease accordingly, and that the
boundary layer thickness would increase to something quite enormous? The answer is:
no, t If the boundary layer grows too much, then any unsteadiness in the fiow or small
motion of the cylinder will cause its being shed at that point, whereupon a "new" boundary
layer would start forming. *
Q.2 THE NORMAL VISCOUS COEFFICIENT CN
In Hoemer's compilation, CN is necessarily equal to CT, and so it is in Taylor's formulation
for roughness of the type shown in Figure Q.1(b) and (c). However, the consensus of
researchers' views appears to be that, generally, CN i= CT, at least for solitary cylinders.
Pai'doussis (1966b) uses CN = CT ::.::: 0.04 and 0.02 for comparison with his experiments
(8 ::.::: 24), while Pai'doussis (1973a) uses 0.04 and 0.01 approximately - specifically
8CN = 8CT = 1 and 0,25,
For form of roughness of the general type in Figure Q.1(b), Ortloff & Ives (1969)
suggest that the physically realizable range for CN is govemed by! ;::; CN JCT ;::; 2, where
CN JCT = 2 holds for smooth cylinders and CN JCT =  for very rough ones. Thus, for
tVery useful discussions with A.P, Dowling of Cambridge University in Aug-Sep 1995 are gratefully
acknowledged.
*For impulsively started cylinders, see Kawaguti (1961).

1450
29,5
SLENDER STRUCTURES AND AXIAL FLOW
00
29,0 o
N
Č
'..:::::
;.:; 28.5
'-'
c:
OJ
50
 28.0

Z 27,5
Test I-A
27.0
O
20
40
(a)
0,050
0,045
0,040
0,035
oV>
Ó
. 0,030
Oj)
c:
' 0,025

 0,020
::E
0,015
0.010
20
40
(b)
00 o o
o
o
o o
Test 1-B
60 O
- Theory
o Expenment
80
20
40 60
o
20
40
60
80 100
0-
0.
'i'
O
a
o
jAA
J AA
a A
A
?
A
;
V
o Test 1-B
o
j,7
'}
/
A"é
AO
20 40 60 80 O 20 40
Mean axial f10w velocity, U (ftJs)
60 80 100
Figure Q.2 (a) Fundamental natural frequency, JI, and (b) corresponding modal damping ratio,
1;'1, versus the flow velocity (1 ft/s = 0.3048 mis) for c1amped-clamped, axisymmetrically con-
fined metal cylinders in water flow (h = 1, !,1 from left to right panels), In (b) the damping
was evaluated by: o, the bandwidth method; !:::", the magnification factor method (constant fOfce);
., the magnification factor method (constant displacement); -, theory with CN for best fit
(Chen & Wambsganss 1972),
60 O

VISCOUS FORCE COEFFICIENTS, CYLINDERS IN AXIAL FLOW 1451
the range of CT in (Q.l), for smooth cylinders one obtains CN = 0.020-0.050, while for
rough ones CN = 0.005-0.013.
Some very careful experiments have been conducted by Chen & Wambsganss (1972),
using c1amped-clamped and cantilevered rods in axial water-flow, excited electromagnet-
ica11y, with the vibration being sensed optically. Typical results t are shown in Figure Q.2.
Based on their experiment s, CN is found to range between 0.025 < CN < 0.103.
Lee & Kennedy (1985) use CN = 0.20, considerably higher than others. On the other
hand, Triantafyllou & Chryssostomidis (1985) use CN = CT = 0.006, while Dowling
(1988a) use s CN ICT = 0.25 and 0.75, which means CN = 0.003 and 0.008, a great deal
smaller than those used by others.
Hence, the total range of values is quite enormous: 0.003 < CN < 0.200 - a1most a
ratio of 100:1. Here it is noted that the very 10w values (but also the highest) are used for
application to towed systems and hence generally very large 8 = LID. It is difficult to
come up with a recommendation here, but a reasonable suggested range could be
0.005 < CN < 0.040; (Q.2)
thus, for the sma11est CN and CT in (Q.l) and (Q.2), we have cNlcT = ; while for the
corresponding highest values, we have CN ICT = 1.6, not too far from 2.
Q.3 THE ZERO-FLOW NORMAL COEFFICIENT CD
It is reca11ed that C = (4/n)(MI E1)1/2 LCD, in which CD is not dimensionless; it has
the units ofvelocity - cf. u = (MI E1)1/2U L, which is used in what follows.
The drag coefficient for a 10ng cylinder oscillating in quiescent viscous fluid has been
discussed in Section 2.2.3. In particular, via equation (2.153), the viscous force per unit
length is found to be given by F = -Cd Q P A(dz/dt), where z is the displacement, Q
the radian frequency, p the fluid density, A the cylinder cross-sectional area, and Cd the
drag coefficient. This latter may either be found from Figure 2.9 reproduced from Chen
ct al. (1976), or from the approximate fonu of (2.157) obtained by Batchelor (1967),
namely Cd = 2.Ji1,JS, in which S = Q R 2 Iv, R being the radiu s of the cylinder and
v the kinematic viscosity. Comparing F above to the form used for CD in equation (8.13)
we have CD = (n 12)D Q Cd.
Consider now an elastomer cylinder of the type used in the author's experiments, in
which D = 25.4 mm, Q  2n x 4 rad/s. Taking v  1 x 10- 6 m 2 /s for water flow, one
obtains Cd = 0.044; hence, in this case CD  0.044 mis. Now, invoking the similarity
of the nondimensional expressions for C and u, we have C = (4/n) (UIU)CD. In the
experiment s, ul U "-' 0(1), and so C  0.06, which is not necessarily negligible with
respect to u CN for very sma11 u - see penultimate tenu in equation (8.27b).
For more substantial cylinders, say made of metal, 15 mm in diameter and a frequency
of 30 Hz, one finds S  10 600 and Cd = 0.028, and therefore C D  0.124. For metal
cylinders, however, u I U "-' 1 120 or even 1/100. Hence, in such cases C  0.008 or less,
which is negligible when compared with u C N, especial1y for u > 1.
Final1y, for damping in two-phase mixtures, the reader is referred to Pettigrew & Taylor
(1993) or Pettigrew & Knowles (1997).
tNote that the manner of analysis of their data makes the results insensitive to the error in the equation of
motion from Pai:doussis (1966a),

1452
SLENDER STRUCTURES AND AXIAL FLOW
Q.4 THE BASE DRAG COEFFICIENT Cb
An excellent discussion of the mechanisms of 2-D and 3-D base drag is given by Maull
(1978), with a great dea1 of physica1 insight. In the introduction, he observes that "The
literature of separated flows is very large, and when it is studied the conclusion that could
easily be reached is that the world is two-dimensiona1", This remains largely trne today!
It is noted that, conventionally, Cb is defined as (Drag)j(! pA U 2 ), where A is some
cross-sectiona1 area, typica11y the area of the cylindrica1 forebody in situations such as
that considered here, Hence, ca1ling this C;;, the Cb in equation (8,14) is related to it
by Cb = C;;. Therefore, Cb = Cb = C;;; that is, the tabulated values oj base
drag coefficient are those to be used directly in the dimensionless equation oj motion,
equations (8,27 a,b).
Hoerner (1965) proposes that
C;; = 0.029j(C f_b)1/2,
(Q.3)
where Cf-b is the skin-friction drag on the forebody, thus relating C;; to boundary layer
growth on the forebody. Hoerner describes a "jet-pump" mechanism, via which the wake
pressure at a blunt end is reduced by the tubular jet around it; this effect is diminished by
the "insulating" effect of the separated boundary layer. t
Thus, in view of equation (Q.3), for a forebody with C f-b = 0,02, we have Ci; = 0,21,
considerably smaller than C;; = 1,17 which applies to a hemispherica1 end with no
forebody,
Tables of base drag coefficients for various base- and forebody-shapes are given by
Hoerner (1965, chapter III), IdeI' chik (1966, section 10) and Blevins (1984, chapter 10).
Based on this information, a reasonable range for Cb is
0,1 < Cb < 0.7,
(Q.4)
but the choice of a suitable va1ue depends on both the length af the cylindrica1 forebody
and the base shape,
tOf course, if the system oscillates transversely, the boundary layer will be affected, and so will the base
drag coefficient. However, virtually all that is available relates to base drag on stationary objects.

Appendix R
Response to Arbitrary Force
Field of a Cylinder in Axial Flow
The equation of motion is
£- [1J(, T)] = f(, T), (R.I)
where £-[1J(, T)] represents the left-hand side of (8.27a) or (8.27b), and f(, T) the
dimensionless force field; if the latter is generated by pressure perturbations on the cy linder
surface, then
(2Jr
f(,T) = (L 3 jEI) 10 p(x,e,t)cose Rde,
where R is the cylinder radius.
The cylinder is supposed to be supported according to one of the standard set ofboundary
conditions, and so equation (R.l) may be discretized via Galerkin's method, utilizing
(R.2)
1J(, T) = L <Pr()qr(T),
(R.3)
r
where <Pr() are eigenfunctions of a beam with the same boundary conditions as the
cylinder. Thus, one obtains a set of equations which may be written in matrix form:
Mij + Cq + Kq = Q,
(R.4 )
where q = {ql, Q2,..., qn}T, the elements ofM, C and K are given by (8.33), and the
element s ofQ are Qs = IOl<ps()f(, T) d.
In the previous discussion it has been presumed that the boundary conditions are such
that the associated beam eigenva1ue problem is self-adjoint. If this is not trne, then the
method of Section 8.4.2 must be used, involving also the adjoint eigenfunctions 1/fs, which
together with <Ps form a biorthogonal set. Eventually, however, an equation of the form
of (R.4) is obtained.
Here, it is noted that M is diagonal, but C and K are not, and, moreover, they are not
symmetric. Accordingly, equation (R.4) represents not only a coupled set of equations
but one that cannot be decoupled easily. The asymmetry in this system is entirely due
to the presence of fluid forces; for example, in the case of C, due to the Coriolis terms
2xf31/2ui}r Id <Ps<p;d. An approximate method for decoupling equations (R.4) has been
given in Section 8.4.2. Here a more general method is presented.
Equation (R.4) is first reduced to first-order form,
Rz + Ez = F,
(R.5)
1453

1454
SLENDER STRUCTURES AND AXIAL FLOW
where B, E and F are partitioned matrice s involving M, C, K and Q (see equations (2.16)
and Section 2.1.2), and z = {q, q}T; B is general1y nonsymmetric and detB =1= O. For the
homogeneous form of this equation, and with solutions of the form
z = eJ.t'u,
the eigenvalue problem
(BJ-t +E)u = O
(R.6)
is obtained; the adjoint form of equation (R.6) is
(B T J-t + ET)v = 0,
(R.7)
It is presumed that the 2n eigenvalues of (R.6) and (R.7) have been determined and that
they are distinct. Hence, the spectra1 matrix J.t and the modal matrice s A and N associated
with equations (R.6) and (R.7), respectively, can be determined. Since the J-t are distinct,
one must then have NTBA = diag and NTEA = diag, an easi1y proven resu1t.
Reverting now to the nonhomogeneous equation (R.5) and introducing
z=A
(R.8)
one obtains
BAt +EA =F,
and pre-multiplying by NT one has
Jt + L = NTF == 41;
(R.9)
in view of the biorthogonality property discussed above, J and L are diagona1 and hence
the system has been decoup1ed.
The response to a deterministic farce fie1d may now be obtained from (R.9) and has the
form
j(T) = j(O)eJ1i' + aj l' e J1iC ,-h) Fj(h) dh. (R.1O)
Then, by using equations (R.8), (R.5) and (R.3), the response may be eva1uated in terms
of the original dimension1ess coordinate rJ(, T).
ln cases where the force field is random, but stationary and ergodic, one may proceed
as follows. The space-time cross-correlation function of the response is defined as
1 i T /2
1/frJ(' o, TO) = rJ(, T)rJ( + O, T + TO) = lim - rJ(, T) rJ( + O, T + TO) dT,
T.....oo T -T/2
which, in view of equations (R.3) and (R.5), may be written as
1 T /2 n n
1/frJ(,O,TO)= lim T i LL<Pj()<Pj(+O)Zn+j(T)Zn+j(T+TO)dT,
T.....oo -T/2. .
I ]
(R.1l)

RANDOM-FORCE RESPONSE OF A CYLINDER IN AXIAL FLOW 1455
where n is the order of equation (R.4). This equation may be manipulated further in terms
of Fourier transforms to yield
n n 1 1 00
1/r'1(' O, TO) = L L 4>i() 4>j( + o) lim - Zn+i(-W)Zn+j(w)e iwro dw,
, . Too2nT -00
I J
(R.12)
where Z(w) is the Fourier transform of Z(T).
Next, the Fourier transforms of equations (R.8) and (R.9) are, respectively,
2n
Zi(W) = L AirZr(W) and (Lr + iwJr)Zr(w) = <l>r(w),
r=l
(R.B)
where Zr(w) is the Fourier transform of r(T), and Air are the elements of A. Also, it is
convenient to define
Hr(w) = (Lr + iw Jr)-l.
Substituting equations (R.B) and (R.14) into (R.12) gives
(R.14 )
n n 2n 2n
1/r'1(' O, To) = L L 4>i() 4>j( + O) L L An+i,r An+j,s
i j r s
1 1 00 .
x lim - Hr(-w) Hs(w) <l>r(-W) <l>s(w)eIWT:O dw.
Too 2nT -00
(R.15)
The generalized excitation cross-correlation spectral density is defined as
, 1
G<t>rs(w) = hm -<I>r(-w) <l>s(w)
Too T
1 n n
= lim - "" N;n+k N;n+ p Qk(-W) Qp(w)
T-+oo T  , ,
k p
1 n n 1 1
= lim T L L N;n+k N;n+p {' ( 4>k() 4>p( + o)
T-+oo kP' '1010
x f(, -w)f( + o, w)d d( + o), (R.16)
where the definitions of equations (R.9), (R.5) and (R.4) have been utilized;
<I>(w) , Q(w), f(, w) are the Fourier transforms of <I>(T), Q(T) and f(, T), respectively,
and N! are the elements of NT.
IJ
Equation (R,15) may now be written in the form
1 n n
1/r'1(' O, TO) = 2n L L 4>i() 4>j( + O)
i j
2n 2n 00
x LLAn+i,rAn+j,s 1 Hr(-w) Hs(w) G<t>n(w)eiWT:°dw, (R.1?)
r s -00

1456
SLENDER STRUCTURES AND AXIAL FLOW
Finally, upon setting O = TO = 0, the mean-square response is obtained as
1 n n
1J2(, T) = - LL(jJi()(jJJ()
2n
i J
2n 2n 00
X LLAn+i,r A n +J,s l Hr(-w) H s (w)G<1>,,(w)dw.
r s -00
(R,18)

Appendix S
The Equations of Motion of
Articulated Cylinders in
Axial Flow
S.1 SELF-PROPELLED SYSTEM
Proceeding as outlined in Section 8.9,9,t the generalized forces Qy and Qm, associated
with equations (8.207) integrated from zero to cjl and equations (8.205), (8.206), (8.208)
and (8,209), for virtua1 displacements 8y and 8()m, respectively, are detennined first. Then,
the equations ofmotion for a neutrally buoyant system of N cylinders (M == pA = m),
derived via the Lagrange equations (Section 2.1.1), using also T = Tf + Te and V = Ve,
are as follows:
N j-I N N
2m s 1 ji + 2m 1 2 L C j L Ck ek + m 1 2 L C] ej + m Ul L C jej
j=1 k=1 j=1 j=1
( N j-I N N )
1 CN 2 ,1 Z Z .
+ 2 Dm U s 1 )i + 1 f; C j E Ck Ok + 2 1 f; C j Oj + Ul f; C j Oj
N ( N )
1 CT 2 .'
-2 D mU lf;C j Oj+(1-ft)mU Y+lf;C j Oj+UON
. 1 Z 1 Z ( Sl )
+(1-fn)mU(y+U()I)--qmU ()N+-mU Cb+CT- ()1=0,
2 2 D
(S.I)
where s = L7=1 Cj, with Cj as defined by (8.201), and
N ( i-I ) { i-I ]
2mZ z f;Ci {; Ck (hm + Ci (jim ji + Z {; Ck ek
N ( 1 i-I 1 ) N ( i-I 1 )
+ 2mZ 3 ?; C] "l {; Ck (jkm + "?/i (jim e i + mU Z2?; Cj E Ck (jkm + "lci (jjm é i
tDetails may be found in Hennan (1975),
1457

1458
SLENDER STRUCTURES AND AXIAL FLOW
N ( j-I ) N
'z . 1 z z '
-mUZcmy-mUZ f;C j 8 jm f;q(h -"lmUZ f;8 j m C j Oj
N N
- m U 2 Z L 8jm Cj Oj + k L(Oj+l - Oj)(8j+l,m - 8jm)
j=1 j=1
[ m-l
+  CN m U Z2 c 2 Y . + Z3 c 2 '" C é + Z3 c 3 é + Z2 c 2 U O
2D 2 m 2 mkk 3 mm 2 mm
k=1
( N ) N j-l
+ Z2 Cm L C j Y + Z3 Cm L C j L Ck é k
j=m+l j=m+l k=1
N N ]
1 3 2 . 2
+"lZ Cm L CjOj+UZ Cm L CjOj
j=m+l j=m+l
-  Cr m U 2 Z 2 C [  c' O' - (  C , ) O ]
2D m  J J  J m
j=m+l j=m+l
+ (1 - ft)m U Z Cm ( y + Z t Cj é j + U ON ) - Cb m U 2 Z Cm (ON - Om) = O,
J=1
(S.2)
where subscript m = 1".., N; the total number of articulations, here denoted with N,
should not be confused with the subscript in CN (which stands for 'nonna1'),
S.2 TOWED SYSTEM
Proceeding in a sirnilar way, the equations of motion for the towed system are given by
N j-l N N
2" 2 '" "'.. 2 "'Z.. ""
2m s ZZtr Otr + 2m Z Ztr  C j  q Ok + m Z Ztr  C j Oj + m U ZZtr  C j Oj
j=1 k=1 j=1 j=1
( N j-l N N )
1 CN '2 ,1 2 2 .
+2VmUZtr sZZtrOtr+ Z f;Cjf;q O k+ 2 Z f;CjOj+UZf;CjOj
-   m U2ZZtr ( t Cj Oj - s Ot) + (1 - ft)m U Ztrfztr é tr + Z t Cj é j + U ON )
J=I} \ J=1
( . ) 1 2
+ (1 - fn)m U Ztr Ztr Otr + U 01 - -Cb mU Ztr(ON - Otr) = O, (S.3)
2

ARTICULATED CYLINDERS IN AXIAL FLOW
1459
where s = 'Lf=l Cj, and
2m 1'1" ECj (E C k 8 ,m + Cj8jm) O"
+ 2m I' ECj (E Ck 8 km + Cj8jm ) (E CkOk)
+ 2m I' Ec; (ECk8,m + Cj 8jm) Oj
+m U I' E Cj (E Ck 8'm + Cj 8jm) tJ)
- m U li" Cm tJ" - mU 1 2 E Cj8jm (E Ck tJk) - m U I' E8jm c; tJ j
N N-l
- m U 2 1 L 8 jm Cj e j + k L (ej+l - ej)(8j+l,m - 8jm)
j=l j=l
1 CN [ 1 2 2' 1 2 2 L . 1 3 3 ' 1 2 2
+ --m U -1 1tr Cm e tr + -1 Cm m-l Ck ek + -1 cme m + -1 cmU e m
2D 2 2 3 2
k=l
+ 121tr Cm ( t C j ) etr
j=m+l
3 ;.-.  . 13 N 2' 2
+1 Cm  CjCkek+ 2 1 Cm L cjej+U1 Cm
j=m+l k=l j=m+l
-  CT m U 2 1 2 C [ ;.-. C ,e, - ( ;.-. C , ) e ]
2D m ]] ] m
j=m+l j=m+l
+ (1 - !t)m Ul ( Cm 1tr e tr + 1 Cm t Cj ej + Cm U eN )
j=m+l
1 2
- -Cb m U 1 Cm (eN - e m ) = O,
2
where 1tr is the towrope length and e tr the associated angle - Figure 8.91(b).
t Cje j ]
j=m+l
m=I,...,N,
(SA)
S.3 SYSTEM WITH A GRIPPED NOSE
The equations of motion for a system with a gripped nose, that is, a canti1evered system
in axial flow may be obtained from (SA) by eliminating all tenns involving e tr , e tr and
e tr and changing the lower limits in the summations from 1 to 2, save for the summations
involving the springs.

Appendix T
The Nonlinear Equations of
Motion of a Cylinder in
Axial Flow
ln this appendix, some parts of the derivations required in Sections 8.10.2 and 8,10.5 are
given in detail, as well as the equations of motion if the centreline is taken to be extensible.
T.1 THE INVISCID HVDRODVNAMIC FORCE FA(X, T)
The aim here is to outline the derivation of equation (8.235). We start with the expression
for the velocity potential, equation (8.232), that is,
</J(X, Y, Z, T) = U X + </Jo(X, Y, Z) + </JI (X, Y, Z, T), (T.I)
where U X is due to the mean flow, </Jo to variations in the body cross-section (here none),
and </JI to motion of the body, and the pressure equation (8.233),
o</J 1 2 1 2
P - Poo = -Pat - '2p(V</J) + '2pU . (T.2)
Then, using the relationships for derivatives in the Eulerian and in the Lagrangian
coordinates,
o a ou o ov o au av o o 5
OX = oX - oX ax - oX oy + ox oX oy + (E),
o o o a
ay oy ' Oz - oZ '
o o ou o ov o ou ov o o 5
at = oT - oT ax - oT OY + oT OX oy + (E),
(T.3)
we obtain
( a</J ou a</J av a</J au av a</J )
P - Poo = -P aT - aT OX - aT ay + aT ax ay
1 ( o</J ou a</J av a</J au av a</J ) 2
- '2 P ax - oX oX - ax ay + ax ax ay
_ P ( a</J ) 2 _ P ( a</J ) 2 + pU2. (TA)
2 aY 2 az 2
1460

NONLINEAR EQUATIONS FOR A CYLINDER IN AXIAL FLOW 1461
Substituting now equation (T.1) in (TA), we have
P-P. =_ [ acfJI _!!!: ( u acfJo acfJI ) _ ( _!!!: )( acfJo acfJI )]
00 P aT aT + ax + ax aT aT ax ay + ay
_ [( l-!!!: )( U acfJo acfJI ) _ ( _!!!: )( acfJo acfJI )] 2
2 P ax + ax + ax ax ax ax ay + ay
( ) 2 ) 2
1 acfJo acfJI 1 acfJo acfJI 1 2
- zP ay + ay - zP ( az + az + ZpU .
(T.5)
After many manipulations and truncation to O(E 4 ), we obtain an expression of the form
P - Poo = (P - Poo)o + (P - Pooh + (P - Pooh,
(T,6)
where (i) (P - Poo)o is the pressure distribution in steady flow past the undeformed
motionless body, (ii) (P - Pooh is the pressure distribution due to steady motion of
the cylinder through fluid at rest, and (iii) (P - Pooh is the remainder of the pressure
distribution.
It is found (Lighthil11960; Lopes et al. 1999) that
1 I acfJo ( a cfJ o ) ( acfJo ) ( acfJo ) ] 5
(P _ Poo)o = Po = __P 2U _ + _ 2 + _ 2 + _ 2 + O(E );
2 ax ax ay az
(T.7)
I au av acfJI au av acfJI 2 au 1 2 ( au )
(P - Pooh = P2 = -p -U aT - aT ay + aT ax ay - U ax + ZU ax 2
2U  acfJI _ U  acfJI  (  ) 2 ( acfJI ) 2
+ ax ax ay ax ay + 2 ax ay
+ ( a cfJ I ) 2 +  ( a cfJ I ) 2 ] + O(E 5 ). (T.8)
2 ay 2 az
Neither of these two pressure distributions contributes to a net force on the body. Net
forces result from the unsteady part of the motion of the body and from variations of the
cross-section with X, that is, the pressure distribution (P - Pooh == PI. Subtracting (T.7)
and (T.8) from (T.5) yields the equations for PI. Furthermore, since cfJo is due to variations
of the body cross-section with X, this may further be simplified by taking cfJo = O, yie1ding
PI = -p f {  + [ U ( 1- !!!: ) - (  + U )]  } cfJI
aT ax aT ax ax
 ( acfJI ) 2 _  acfJI acfJI } O E 5 .
+ 2 ax ax ay ax + ( )
(T.9)

1462
SLENDER STRUCTURES AND AXIAL FLOW
Once PI is detennined, the latera1 force, or lift, per unit length in the Y -direction may
be obtained:
FA (X, T) = 1. PI(-dZ)
fs x
= -p 1. { [  + ( V ( I -  ) - (  + V  ))  ] <PI
fs x aT ax aT ax ax
( ) 2 }
I a<pI a v a<pl a<pI 5
+- - ---- (-dZ)+O(E)
2 ax ax ay ax '
(T.1O )
where the minus sign in front of dZ enters to reconcile the coordinate axes used here with
Lighthi1ľs, and Sx is the circumference of the cy1inder.
The next step is to detennine <PI, which must satisfy the 2-D Laplace equation, subject to
the conditions given beneath equation (8.232); for example, the non-penetration condition,
expres sed in cylindrical coordinates, is
a<pI ( av ) 2 ( a<pI . a<PI ) ( av av au av
- +cosO - cos 0- -smO- +cosO -- - V- +--
ar ax ar rao aT ax aT ax
+ 2V   -  a<PI ) + O(E 5 ) = O at r = R, (T.11)
ax ax ax ax
while on the channel wall we have
a<pI I = O.
ar
r=Ro
(T.12 )
T.1.1 The linear expression for the lift
It is instructive to solve the 1inear problem first, as an introduction to the more difficult
non1inear one. Solutions to the general2-D Laplace equation V 2 S(r, O) = O are of the
form
S(r, O) = (A + B lnr)(C + DO) + f {Enrn cos nO + Fnrn sin nO
n=O
+ Gnr- n cos nO + Hnr-n sin nO },
where A, B, C and D and En, Fn, G n and Hn are functionsof X, subjecttothe appropriate
boundary and regularity conditions.
Simp1ifying (T.I I ) to first order, the boundary conditions to be satisfied, that is, those
given below (8,232) are
a<PI I =0, a<PI I = ( +V ) CoSO,
ar r=Ro ar r=R aT ax (T.14)
<PI (r, O, X, T) = <PI (r, 0+ 21l', X, T), <PI (r, O, X, T) = <PI (r, -O, X, T).
(T.13 )

NONLINEAR EQUATIONS FOR A CYLINDER IN AXIAL FLOW 1463
According to these boundary conditions and considering the form of the solution for the
2-D Laplace equation (T.13), we can expres s CPI in the form
CPI (r, (), X, T) = V(X, T)<I>(r, (), X), (T.lS)
where V(X, T) = [(avjaT) + U(avjaX)] is the 1inear re1ative fluid-body ve1ocity. lt
is noted that <I> is also a solution of the 2-D Laplace equation, but with the following
boundary conditions:
(a<I>jar) Ir=Ro = O,
<I>(r, (), X) = <I>(r, () + br, X),
(a<I>jar) Ir=R = cos(),
<I> (r, (), X) = <I> (r, -(), X) .
(T.16)
The potential <I> is of the form (T.13) with the coefficients A, B, ..., Hn being func-
tions of X. The fourth condition (T.16) implies that D = O, and the second leads to
BC = O, n = 1, £1 - GII R 2 = 1 and Fl - HII R 2 = O. Hence, <I> may be written as
<I>(r, (), X) = A' + [( 1 +  ) r + l ] cos() + ( : r + l ) sin(), (T.l?)
where A' = AC (in the event that B = O and C i= O). Then, applying the first of conditions
(T.16), we obtain Hl = O and Gl = -R5R2 j(R5 - R2), which leads to
<I>(r, (), X) = [( 1 - 2 R5 2 ) r - 5R2 2 ] cos(), (T.18)
Ro - R r (Ro - R )
where the constant A' has been suppressed, since the potentia1 can only be determined to
within a constant. Notice that the third condition (T.16) is automatically satisfied. t
Next, truncating equation (T.1O) to first order, we obtain the linear expression of the
inviscid hydrodynamic force,
FA(X, T) = -p (  + U  ) J CPl(r, (), X, T)(-dZ). (T.19)
aT ax Ys x
Then, using (T.lS) and (T.18), the integral in (T.19) becomes
[2n V(X, T) ( R - 2 :5 R 2 ) cos ()(-R cos ()d()
10 Ro - R
= ( R + R: ) V(X, T)JTR 2 = xV A,
Ro - R
where X is the virtual mass coefficient, and A is the cross-sectional area of the cylinder.
The inviscid hydrodynamic force is then given by the fami1iar expression
FA(X, T) = - (  + U  ) [MV(X, T)],
aT ax
where M = XpA is the virtua1 (added) mas s per unit length.
(T.20)
tFor a slender cylinder with R/ L « 1, we have <I>(r, e, X) = 0(<,), and hence, it is verified a posteriori
that 4>1 = V(X. T) <I> (r, e, X) is of second order (V being of first order), which is an assumption made in the
analysis in conjunction with equation (T,l).

1464
SLENDER STRUCTURES AND AXIAL FLOW
T.1.2 The lift expression, correct to O( E4)
Let us now consider the nonlinear case. In this case, we define the potential </JI by
</JI (r, e, X, T) = V(X, T)et>(r, e, X) + \II(r, e, X, T),
(T.2I)
where the potential \II is the non1inear part of </JI, correct to fourth order, which a1so
satisfies a 2-D Laplace equation. Again, \II needs to be 2JT-periodic and even with respect
to e. Conceming the two other boundary conditions, the condition on the outer channel is
simply
a \II 1 _ O
ar r=Ro - ,
(T.22)
whereas the condition of non-penetration at r = R requires additiona1 manipulations.
Substituting (T.21) into (T.ll), and truncating the expression obtained to fourth order
yields
[ aet> a\ll ] ( av ) 2 ( aet> aet»
V(X, T)- + - +cose - V(X, T) cose- -sine-
ar ar ax ar rae
[ auav au av av av ] 5
+cose -V(X T)+-+2U-----et> +V(E )==0
, aTaX ax ax ax ax '
which, according to the second condition (T.16), can be reduced to
a \II I ( au av au av )
- =- --+2U-- cose
ar r=R aT ax ax ax
( av ) 2 ( aet> aet> )
- V - cose- - sine- cose
ax ar rae
av a V 5
+ --et>cose + V(E ).
axax
Then, replacing et> by (T,18) at r = R, we obtain
a \II 1 ( au av au av ) ( b ) ( av ) 2
- = - - - + 2U - - cos e - a - - V - cos e
ar r=R aT ax ax ax 2 ax
b ( av ) 2 av a V R
+ -V - cos3e + --(a +b)-(1 +cos2e) +V(E 5 ),
2 ax ax ax 2
(T.23)

NONLINEAR EQUATIONS FOR A CYLINDER IN AXIAL FLOW 1465
where a = -R 2 /(R5 - R2) and b = -R5/(R5 - R 2 ). Consequent1y, considering
conditions (T.22), (T.23), and the form ofthe solutions (T.13), we fina11y obtain
[ ( au av au av ) ( b ) ( av ) 2 ] ( R2 )
\{I(r, e, X) = - aT ax + 2U ax ax - a - 2" V ax ar + b---;:- cose
+ V (  ) 2 [( _ R4 ) r3 _ R 4 RZ ] cos3e
6 ax RZ - R6 r 3 RZ - R6
+  av (a + b)!!. [( R3 ) r2 _ R 3 Rci ] cos2e
ax ax 4 Rci - R4 r 2 Rci - R4
av av R 2
+ --(a + b)-lnr + O(E 5 ), (T.24)
ax ax 2
where the 1ast term satisfies condition (T.22) on1y for Ro 1arge, that is, R/ Ro « 1.
Actually, this is not an excessive requirement: one can verify that a ratio of 0.1 for R/ Ro,
which is indeed re1ative1y 1arge compared to R/ L '" 0.025 for a slender body, 1eads to
[(a + b)R2/2Ro] '" 0.05R (the coefficient of the derivative of the 1ast term with respect
to r at r = Ro). Hence, apart from specia1 cases where the outer channe1 is very close to
the cy1inder, we may conclude tha.tJhi&-approximation is reasonab1e.
Substituting now the expression for </JI into (T.1 O) and truncating at fourth order yie1ds
FA(X T) = -P 1. {[  + ( U ( 1-  ) - (  + U ))  ] V<I>
, fs x aT ax aT ax ax
( a a ) 1 ( av ) 2 2 av av a<l> ) 5
+ -+U- \{I+- - <I> -V---<I> (-dZ)+O(E).
aT ax 2 ax ax ax ay
(T.25)
Then, substituting <I> by (T.18) and \{I by (T.24), with special attention to the integra-
tion of products of cosines, equation (T.25) gives us the inviscid hydrodynarnic force,
equation (8.236).
T.2 TAPERED FREE-END BOUNDARY CONDITlONS
The variation of the Lagrangian for the ogival free end is
l t2 l t2 1 L
o .cdt = - [PcA(s)(x ox + ji oy) - PcA(s)gox] ds dt,
tI tI L-l
(T.26)
Pc being the ogival end density. It is convenient to re-write this in terms of virtual
displacements in the longitudinal and transverse directions, ou L and ou N, respective1y,
{ ox } = [ C?S el
oy Slil el
-sine l ] { OUL } ,
COSel OUN
(T.27)

1466
SLENDER STRUCTURES AND AXIAL FLOW
leading to
] t2 ] t2
o £ dt = - m [)i(sin elOUL + cos elOUN) + x(cos elOUL - sin elOUN)] Se dt
t) t)
] t 2
+ mgse(COSelOUL - SineloUN)dt + O(E 3 ),
t)
(T.28)
in which Se = (1/ A)fZ_zA(s) ds. Since X is of second order, this equation reduces to
] t 2 1 t2
O £dt = - [mYSeOUN - mgse(OUL - y'OUN)] dt + O(E 3 ).
t) ti
(T.29)
Considering next the virtual work by all the fluid forces acting on the tapering end
(Figure T.l), and following (8.249), we have
] t2 ] t2 j L
oWdt = {[ -F px + h COSel + (FA + FN) sineI] OX
t) t) L-Z
+ [F py + FL sinel - (FA + FN)COSeI] oY} dsdt,
(T.30)
F py

F L
PcAg
PcAy
Figure T.! The end-piece at the free end of the cylinder showing the forces acting on it; Pc is the
density of the cylinder and of the end-piece, The notation in this figure has been simplified; most
of the "forces" shown, for example FL, are really forces per unit length and should be understood
to stand for ff-I h ds, etc. (Lopes et a/, 2002),

NONLlNEAR EQUATIONS FOR A CYLlNDER IN AXIAL FLOW 1467
which, in view of (T.27), may be written as
1 t2 1 t2 1 L
8Wdt = {[ -F px cos(h + F py sin 01 + h] 8UL
tI tI L-I
+ [FpxsinOl + FpyCOS01- (FA + F N )]8uN} dsdt,
(T. 31)
ln this expression, F N and F L are not constant, since the diameter is a function of s,
and simi1arly F px , F py and FA involve the cross-sectional area, also varying with s.
Thus, noting that 01, y' and 8y are constant for L - I < s < L and that yl! = O, we
can write
----
1 t2 8Wdt = 1 t2 { -COSOl {L F px 8uLds +sinOl {L F py 8uLds
ti tI J L-I J L-I
+ {L h8uL ds + sino l l L Fpx8uN ds + COSOl l L Fpy8uN ds
J L-I L-I L-I
- f {L FA8uN ds - l L FN8uN dS } dt, (T.32)
J L-l L-l
in which the parameter f (O :::: f :::: 1) has been introduced in the terms involving FA,
since the ideal inviscid hydrodynamic force wil1 generally not materialize fully over the
ogival end - see Section 8.2.4.
ln (T,32) the simplified expressions for the various forces - leading to a linear final
boundary condition - are
F px = -p[dA(s)/ ds] + O(E 2 ),
, [ 1 D 2 (s) ]
F py = Y --pU 2 CT- + pgA(s) + p(dA(s)/ds) + O(E 3 ),
2 Dh
FN = !pD(s)UCN(Y + Uy') + O(E 2 ),
2
1 2 2
h = "2 PD (s)U CT + O(E ), (T.33)
FA = Xp(j + U)7')A(s) + XpU(y + Uy')[dA(s)/ ds] + O(E 3 ),
COSOl = 1 + O(E 2 ), sinOl = y' + O(E 3 ),
having also made use of equations (8,241) and (8.242).

1468
SLENDER STRUCTURES AND AXIAL FLOW
Substituting (T.33) into (T.32), and combining with (T.29) as in (8.224), after several
manipulations and simplifications one obtains
8 1 t2 £dt + 1 t2 8Wdt
t) t)
= It 2 { ( -(pA)IL-1 + PDU2CThSe + PDU2CTSe + (m - PA)gSe) } 8UL dt
1 t2 [ 1 2' , 1 "
+ - -pDU CThy Se - (m - pA)gy Se - -pDUCN(y + Uy )Se
tj 2 2
- [J M()i + UY') + my] Se + 1 MU(y + Uy') }UN dt + O(E 3 ), (T.34)
where
1 l L 1 l L
Se = - A(s) ds, Se = - D(s) ds,
A L D L
and where it is understood that A = A IL-l, D = DIL-l' y' = y' IL-l, Y = ylL-1 '
h = D/Dh, M = XpA.
The first term in (T.34), in curly brackets, represents a small addendum to the axial
pressure-tension terms in the equation of motion, rather than contributing to the bound-
ary conditions. The second term, in square brackets, is associated with the transverse
shear boundary condition. Accordingly, the boundary conditions at S = L - 1 are
(Lopes et al. 1999a):
(T.35)
- Ely'" + [I M(y + UY') + my] Se - 1 MU(y + Uy') + (m - pA)gy' Se
+ iPDU2CThy'se + iPDUCN(y + Uy')se = O, y" = O; (T,36)
These are equations (8.253) of the main text.
T.3 AXIALL Y EXTENSIBLE CYLlNDERS
If the cy1inder is supported at both ends, or if the inextensibility condition is not invoked
even for a cantilevered cy1inder, 8X and 8s are no longer equal; by defining 8 as the axial
strain along the centreline of the cylinder, one may relate them through the condition:
OX 1
os 1 + 8 '
(T.3?)
1 +e(X) = ( :; Y + ( :; Y
For a cy1inder undergoing planar motion, extensible or inextensible, the curvature is
K = oe/os, where in this case e is defined by
(T.38)
ox/oX
cos e = ,
1 + s(X)
oy /OX
sin e =
1 + s(X) .
(T.39)

NONLINEAR EQUATIONS FOR A CYLINDER IN AXIAL FLOW 1469
ln tenns of the X co-ordinate, the curvature for an extensible centre1ine is
ae ax I ae
K=--=--;
ax as I + £ ax
ae I [ a2y ax ay a 2 x ]
ax = (1 + £)2 ax2 ax - ax ax2 '
(T.40)
The kinetic energy of the cy linder is
Tc = m 1 L (ů 2 + v 2 ) dX,
where u and v are as in Figure 8.100. The potentia1 energy is
(T.4!)
I 1 L ( T ) 2 I 1 L 1 L
Vc=-EA -+£ dX+-EI (1+£)2 K 2dX-mg (X+u)dX,
2 O EA 2 O o
(T.42)
in which, because of (T.40), the second integrand may be simplified to (ae I a X)2 dX.
Following Pai:doussis (1973a), the axia1 variation of the tension Tis given by t
aT ( I 2 )
ih' = - 2 PDU CT +mg ,
or, since aT lax = (aT lax)(1 + U'),
(T.43a)
aT ( I 2 ) I
- = - -pDU CT + mg (1 + u).
ax 2
(T.43b)
Integrating from X to L, we obtain
T(X) = (!PDU 2 CT + mg) [L - X + u(L)(1 - 8) - u] + T(L),
(T.44 )
in which T(L), the tension at the downstream end, may be expressed as
- I [ I ( D ) ] L
T(L) = T8 + _pD2U2Cb(1- 8) - -pDU 2 CT 1+- + mg -8;
2 2 Dh 2
(T.45)
here 8 = O signifies that the downstream end is free to slide axia1ly (or wholly free), while
8 = I if the supports do not a1low axia1 extension. In the latter case (8 = 1), there may
be an extemally imposed unifonn tension T ; while if the downstream end is sliding or
free (8 = O), there may be a base drag at that end, equal to !pD2U2Cbo Hence, the fina1
expression for T(X) is
T(X) = T8 + (PDU2CT + mg) [L (I - 8) - X + u(L)(1- 8) - u]
I 2 DL 1 22
- -pDU CT--8 + -pD U Cb(1 - 8). (T.46)
2 Dh 2 2
tThis simple approximation for í.JT jí.Jx is acceptable, only because it matches that for í.Jpjí.Jx,
equation (8,241), which is used here also; this latter cannot readily be refined for h 'i- O.

1470
SLENDER STRUCTURES AND AXIAL FLOW
Proceeding similarly, starting with equation (8.241), one obtains the following
expression for p(X):
Ap(X) = Ap(L) + ( !PDU2CT - pgA ) [L - X + u(L)(1 - 8) - u] + O(E 4 ),
2 Dh
(TA 7)
with
Ap(L) = [(1-2V)PA+pgA]8,
(TA8)
where p is the mean pressure (pressurization) measured at X = i L and v is the Poisson
ratio - see Paldoussis (1973a).
Reca1ling that u "'-' O(E 2 ) and v "'-' O(E), and using the second equation (TAO),
(oe /ox)2 = v,,2(1 - 2u ' - 2v '2 ) - 2v ' v"u" + O(E 5 ), is obtained. Moreover, E: is given
by (T.38), name1y E: = u' + iV'2 - iV'4 - iU I V '2 + O(E 5 ). After extensive manipulations,
we find
l t2 l t 1 L {
8 Vsdt = - EA (u" + v'v") - El (v"v lll + Vi v"")
tI tI o
[ I 2 2 ( - 1 2 D L ) ] I "
+ -pD u Cb(1- 8) + T - -pDU CT-- 8 v v
2 2 Dh 2
( 1 2 ) [ I 1 12 ( I ) I II ] }
- mg + "2PDU CT 1 + u -"2v + L - X - "2L8 v v 8u dX dt
+ 11 t % L { _ E A (VI u" + v" u' +  v I2 V") + E 1 Vlili - E 1 (3u lll v" + 4u" VIII
+ 2U ' V IllI + v'u"" + 2v,,3 + 2v '2 V"" + 8v ' v"V Ill ) + (mg + PDU2CT)
[ I 1 13 (L X I L8) ( " I" '" 3 12 II )
x V -"2v + - -"2 -v + v u + v u +"2v v
II ] [ 1 2 2 ( - 1 2 D L ) ]
+ (u - u(L)) v + "2 pD U Cb(1 - 8) + T - "2 PDU CT Dh "2 8
x (-v" + v'u" + v"u ' + VI2v") }8VdX dt,
8 I t2 Tc dt = -m l t2 [L (u8u + ii8v) dX dt,
ti t) 10
(TA9)
(T.50)
where primes and dots denote derivatives with respect to X and t.

NONLINEAR EQUATIONS FOR A CYLINDER IN AXIAL FLOW 1471
Proceeding similarly as for the canti1evered cy1inder, for an extensible cy1inder (usua11y
one with supported ends), we obtain
1 t2 1 t 1 L { I ( 1 v 2 vv' )
8Wdt=- --pDU 2 CT l--Z---v'2
tj ti O 2 2 U U
( ) 1 ( vv' )
- M vv' + 2Uv'v' + U 2 v'v" - "2PDU2CN li + v,2
( 1 2 D ) [ , 1'2 , II ]
- "2 pDU CT Dh - pgA -u +"2 v - (L - X)v v
_ pDU2CD v' ( v'lv'l + v'lvl + vlv'l + vlvl )
2 p U U2
+ [pgL8 + (1 - 2V)P] Av'v" }8U dX dt
_ [t%L {MV(1 _ v,2) + 2MUv' (1 _ v'2) + MU 2 v" (1 _ v'2)
- Mv' (u + 3uů' + 2U 2 u") - M (4UU' + 2ů + vv') (v' + Uv")
1 ( 1 "2 . '2 )
2 , V V "vv '3
- -pDU (CT - CN) v - -- - u v - - - v
2 2 U2 U
( 1 2 D ) { ' 1'3 "
+ -pDU CT- - pgA v - -v + [u - u(L)(1 - 8)] v
2 Dh 2
- (L - X) (v" - u"v' - u'v" - v'2v") }
- [pgL8 + (1 - 2V)P] A (v" - u" v' - u'v" - v'2v")
1 2 ( V u'v ŮV 1 V 3 )
+-pDU CN -+-+----
2 U U U2 2 U3
1 ( v'lvl + vlv'l vi vi ) }
+ "2PDU2CDP v'lv'l + U + U2 8vdX dt + O(E 5 ).
(T.51 )
The fina1 equations of motion are given by
mu - M (v v' + 2U v' v' + U 2 v" v') - EA (u" + v' v") - E I (v" VIII + v' VIII')
1 2 ( V v' '2 ) 1 . 2
- "2 pDU (CN - CT) li + v + '4PDCTV

1472
SLENDER STRUCTURES AND AXIAL FLOW
1 2 ( D ) { ' 1'2 [( 1 ) ] , ff }
+ 2 pDU CT 1 + Dh u - 2 v + 1 - 2<5 L - X v v
_ !pDU2CD v' ( v'lv'l + v'lvl + vlv'l + VIVI )
2 p U U2
+ [PD2U2Cb(1 - <5) + T <5 + (1 - 2V)A]J<5] v'v"
+(mg-pgA){U'- v2 +[(I-<5)L-X]V'V"}+O(ES)=O, (T.52)
(m + M)ii + 2MUv' (1 - Iv'2) + MU 2 v" (1 - vf2)
- Mv' (ii + 3uú' + 2U 2 u") - Miiv,2
- M (4UU' + 2ú + vv') (v' + Uv")
+ [PD2U2Cb(1 - <5) + T<5 + (1 - 2V)AP<5] (-v" + v'u" + v"u' + v'2v")
1 2 ( v' l v l + V l v' l V I V I) ( 3 2 )
+ -pDU C v'lv'l + + - - EA u"v' + u'v" + -V' V"
2 Dp U U2 2
+ E I v",' - E I ( 8v' v" v", + v' u"" + 2v'2 v "" + 2v,,3 + 2u' v",' + 4u" v", + 3u'" v'')
1 2 ( D ) { I '3 " [( 1 ) ]
+ 2 pDU CT 1 + Dh -2 v + [u - u(L)(1 - <5)] V + 1 - 2<5 L - X
x (-v" + v'u" + v"u' + v'2v") }
1 2 ( '3', vv f2 1 V 2 V' )
-2 pDU (CN-CT) v +vu +U+2 U2
1 2 ( D ) , 1 2 ( ÍJ u'v úv 1 ÍJ3 )
+ 2 pDU CN + CT Dh v + 2 pDU CN U + li + U2 - 2 U3
+ (mg - pgA) {v' - v'3 + [u - u(L)(1 - <5)] v"
+ [(1 - <5) L - X] (-V" + v'u" + v"u' + v'2v") } + O(E S ) = 0, (T.53)
Some further infonnation on the derivation of these equations, and results obtained with
them, may be found in Modarres-Sadeghi et al. (2003).

NONLINEAR EQUATIONS FOR A CYLINDER IN AXIAL FLOW 1473
T.4 CORRECTION TO EQUATIONS FOR
A FLUID-CONVEYING PIPE WITH
SUPPORTED ENDS
The equations of motion for this system are given by equations (5.36a,b). It is immediately
obvious that there is something inconsistent, when it is realized that there is a zero-order
term in the first equation. These equations have never been used by either the author or
his colleagues to obtain solutions, and hence this error has remained undetected.
The derivation of these equations, originally in Semler et a/. (1994), is correct up to a
certain point; however, there is a problem in the final equations, arising from the derivation
of the externally applied tension To and pressurization P. In the derivation, To and P are
taken to be constant, whereas they are functions of X (xo in Chapter 5). Indeed, from
equation (3,36),
(To - P) = (To - P)IX=L - [MŮ - (M + m)g] (L - X),
(T.54 )
leading to
a '
ax (To - P) = MU - (M + m)g. (T,55)
Using then expressions (T.54) and (T.55) in the subsequent derivations, one eventu-
ally obtains the dimensionless nonlinear equations of motion of an extensible cylinder
conveying fluid:
ft + 2U /'fiiť + U 2 /1' + Ů /'fiu' - A/1' - (v"l1" + 1111111)
+ (ů/'fi - y) [1f2 - (1 - )v'I1'] + (r - A - n) v'v" = o,
iY + 2UF{Jtf + U 2 11' - (r - ll) v" + 11'" + yl1
+ (r - A - ll) (/1'v' + /111' + v'211')
- (3u lll l1' + 4/1'11" + 2u'l1 111 + v'u"" + 211'3 + 2v,2 v "" + 8v'I1'v lll )
+ (ů/'fi - y) [l1u' + v'3 - (1 -) (-11' + /1'v' + u'l1' + v'2v")] = O,
(T.57)
(T.56)
where u = ul L, v = viL, the dimensionless tension r = T(L)L 2 I EI, the axial flexib-
ility A = EAL 2 I EI, and the pressure at the downstream end n = P(L)L 2 I EI. These
equations should replace (5.46a,b); note that, in Chapter 5, w is used instead of v.

Appendix U
Added mass matrix by FEM
For incompressible fluids, the goveming equation for dynamic pressure is V 2 p = O.
Moreover, ateach solid surface bounding the fluid, the condition op/on = -p a n mustbe
satisfied, where op/on denotes the pressure gradient and a n the acceleration, both nonnal
to the surface, and p is the fluid density. For the 2-D case considered here, these two
equations may be written as
a 2 p/az 2 + a 2 p/ai = 0,
lzap/az + lyop/ay = -p(lza z + lya y ),
(U,I)
(U,2)
where lz and ly are the direction cosines of the surface nonnal.
Following Suss et al, (1976) and Pa'idoussis et al. (1977), the finite element fonnulation
used is based on the minimization of the functional
h HL[ e )' + e )'] dA+ 1>a n P ds.
(U.3)
where A is the area of the fluid in the cross-section, and C is the contour of the fluid
boundaries. By varying \II with respect to pand its z- and y-derivatives, and transfonning
the resulting expression by use of the divergence theorem, the first variation of \II may be
stated as
J1 [ a2p a2p ] i [ ap ]
8\11=- + 8pdA+ -+pa n 8pds.
A az ay C* an
Minimization of \II with respect to variations in p gives the requirement a \II /a(8p) = 0,
which leads to equations (U. 1) and (U ,2) as the Euler equations of the variational principle.
ln the finite element calculation, the cylinder-cluster cum fluid-container cross-sectioD
is represented by a set of boundaries consisting of connected straight-line segments, and
by triangular elements within the fluid region (Figure U.l), Within each element, pressure
is assumed to vary linearly, so that
(U.4 )
p(z, y) = a1 + a2Z + a3Y,
ap/az = a2,
ap/ay = a3,
(U.S)
The constants aj are expres sed in tenns of the Dodal coordinates and nodal pressures
(Figure U.1), in the fonn
{a} = [d]{p},
where
[ 1 Zl
[d] = 1 Z2
1 Z3
] -1
Y1
Y2
Y3
(U.6)
1474

ADDED MASS MATRIX BY FEM
1475
Y
P3 (z3' Y3)
A

Pl (Zl' YI) P2 (Z2' Y2)
Z
Figure U.l Typical mesh used for the added mass computation by the finite element method, and
nodal coordinates and pressures associated with one finite element (Pai:doussis et aZ. 1977),
Thus, the derivatives ap/oz and op/oy may be expressed within the qth element in terms
of the nodal pressures {p} as
{ OP/oz }
op/oy q = [D]q {p}q ,
where
[D] = [ d 21
q d31
d22
d32
d 23 ]
d33 .
q
(U.7)
The area integration in equation (U.3) may be expressed as the sum of integrals over
the element areas as
Hl[ ( : )' + G )'] clA    li, W )' + G )}Aq, (U,8)
where the summation is taken over all the elements. By making use of equation (U.7), the
integration over the qth element may be written approximately as
[( ) 2 ( ) 2 ]
1 ap ap 1 T T. 1 T
Z f i q oz + oy dAq  ZAq{p}q [D]q [D]q{p}q = Z{p}q [h]q{p}q;
(U.9)
[h]q is referred to as the element hydrodynamic influence matrix for the qth element.
The contour integration in equation (U.3) is carried out in a piecewise-linear fashion
by averaging the nodal pressures at the end-points of each line segment on the boundaries
(Figure U.l):
l iT
C P ajpdsj  zp aj Lj(pl + P2) = {g}j{P}boundary,
J
(U. 10)
where aj is the outward normal acceleration of the jth boundary segment, and where {g}J
is given by
{g} J = 1 p aj L j {I, I}.
(U. lI)

1476
SLENDER STRUCTURES AND AXIAL FLOW
Summing the contribution from the entire system of elements and boundary segments,
and expressing the unknown pressure field in terms of a global vector of nodal pressures,
{p}global, the finite element approximation to the functional \{I is given by
I T T
\{I:::: 2 {P}global[H]{P}global + {P}global {G}.
(U. 12)
Minimizing \{I by taking variations with respect to {p }global results in a set of simultaneous
linear algebraic equations for {p }global, namely,
o\{l/O{OP}global = [H]{P}global + {G} = {O}.
(U.13)
The global hydrodynamic inftuence matrix [H], assembled from the element inftuence
matrices [h]k, is singular; it must be constrained by setting the pressure at one node equal
to a reference value, to provide a linearly independent set of equations.
The term {G} in equation (U.13) expresses the boundary accelerations which are to be
applied to the system, When a unit acceleration is applied to each cylinder in the system in
tum, first in the z-direction, then in the y-direction, a total of 2K acceleration conditions
results, where K is the number of cylinders in the system. Solving equations (U.13)
separately for each acceleration condition t yields a system of nodal pressures, from which
the resultant z and y components of force per unit length on each cylinder are obtained by
integration over the surface of each cylinder. Thus, for the ith cylinder,
[2:rr: [2:rr:
F: = 10 plz Ri de, F; = 10 ply Ri de,
(U.14)
where Ri is the radiu s of the ith cylinder. Equation (U. 10), which is the piecewise-linear
approximation to the contour integral in equation (U.3), may be used for calculating
these resultant forces when one component of acceleration at a time has been used in the
acceleration boundary conditions (either a z = 1 or a y = 1). For example, denoting by
{G}zi the vector {G} of equation (U.13) resulting from a unit z-acceleration of the ith
cylinder, and by {P}zi the pressure field found by subsequent solution of equation (U,13),
the z-component of force on the ith cylinder is then given simply as
Fzi,zi = (l/p) {G}; {P}zi, (U,15)
where division by p removes the dependence on density in equation (U, 10). Proceeding
similarly, each of the elements of the added mass matrix may be found; thus for the element
representing the r (z or y) direction component on cylinder i due to a unit acceleration in
the s (z or y) direction of cylinder j is
Mri,sj = O/n p Rf>Fri,sj,
(U.16)
with Fri,sj determined as in (U.15).
The added mas s matrices for a number of typical systems involving three and nine
cylinders have been computed by this FEM and then compared with those obtained by the
"c1assica1 method" of Section 9,2.2 (Suss et al. 1976). In terms of time, calculations by
FEM take about 20 times longer than the c1assical method. However, as already stated,
the FEM approach is much more versatile for anything other than idealized geometries.
tThe matrix [H] is formed on ly once, as the system geometry remains fixed in all cases.

Appendix V
The Viscous Coupling Matrix
The details for this analysis may be found in Lin (1987b), After expressing the solution
in tenns of two comp1ex velocity potentials in equation (9.36), a set of transfonnations in
the spirit of Section 9,2.2 allows the application of the boundary conditions to the solution
and the detennination of the constants aim to dim.
In the final result, equation (9.38), the forces in tenns ofthe cylinder velocity amplitudes
WI and VI or in terms of acce1erations WI and ih in the z- and y-direction, respectively,
may be written as
I Fi ) K [
z _ 'r. R z -iru L ail
- -l P Jr ,e
Fi I Til
Y 1=1
K
(Jil ] { WI } == P Jr RZ '" [ a il
f3il VI I  Til
1=1
(Jil ] { I } ,
f3il VI
(V. I )
where wI= wl exp(-iQt), vI=I![exp(-iQt), and the overdot denotes dO/dt;
ail, (Jil, Til and f3il are the dimensionless coefficients of the hydrodynamic force, given by
I ::; } = { =;::: } + H(1)(k Ri) { ii:11 }
Til aill ! -Ai1l
f3il (Ji 11 - i 11
+ t* 'f,m(-R j /Rij)m+1 [{ : } Cm+l + { ::; } sm+l ]
j=1 m=O Jmi Jmi
(Jjml f3 jmi
-. E(-l)mJl(kRi)Hl(kRij) [{ ;:: } Cm+l + { ::r7: } Sm+J
Jml nJml J
(V.2)
where C m +! = cos(m + l)1/fij and Sm+1 = sin(m + 1)1/fij; the asterisk denotes that the
summation exc1udes the ith tenn. The constants aiml etc. are detennined by solving the
1477

1478
SLENDER STRUCTURES AND AXIAL FLOW
following equations:
o q O qHl)(k Rd [a i " a i "]
q O qHI)(k Ri) O Tiql l3iql
-q O Ri Hl)' (k Ri) O Aiql 'Eiql
O -q O RiHI)' (k Ri) Jiql niql
[  Smq
K 00
+ ?; * E  ::
 Smq
- C mq
 Smq
 Smq
- C mq
TJ Smq
TJ C mq
Ri  C mq
Ri  Smq
-TJ C mq ] [ otjml
TJ Smq Tj ml
Ri  Smq Ajml
-Ri  C mq Jjml
aj ml ]
13 jmi
'Ejml
njml
[ 1 O ]
O -1
= OilOlq  I ;
q = 0,1,...,00; i, j = 1,2, .,. ,K,
(V.3)
where
Smq = sin(m + q)o/ij, C mq = cos(m + q)o/ij,
(_l)m(m + q - I)! Rj+l Rq-l
- TJ=q(-l)mJq(kRi)Hq(kRij),
- (q - l)!(m - I)! R+q ,
 = (_l)m J(k Ri) Hq(k Rij), J(k Tj) = dJq(k ri)jdTj,
Hl)l(krd = dHl)(kri)jdri' (VA)
The soIution of (V.3), representing an infinite set of coupled aIgebraic equations, involves
first truncating m and q to say M, so that we have 4K M Iinear aIgebraic equations, from
which otiml, aiml, l3iml, Aiml, Jilm and nimi may be determined numerically.
It is noted that otil, ail, Ti I and l3il are complex, and hence equation (V. 1) may be
written as
{;iJ = P d;  ([ :i ;]{ : } + Q [ :t ;t] { : }) , (V.5)
where R denotes the reaI component and I the imaginary one for each coefficient.
An example of the reaI and imaginary parts of the hydrodynamic force coefficients
for an equilateraI-triangle three-cylinder c1uster with G c = 0,1, Q = 20n rad/s in
unconfined water at 20°C is given below (Lin 1987b), where [Yi] and [Yi] are the square

THE VISCOUS COUPLING MATRIX 1479
matrices in (Y.5):
1.8515 -1.1046 0.1778 0.2348 0.2695 -1.0099
-1.1046 1.8515 0.1778 -0,2695 -0,2348 1.0099
[Yi] = 0,1778 0.1778 1.4449 -0.4709 0.4709 O
0.2348 -0.2695 -0.4709 1.5804 0.6053 -0,6772
0.2695 -0.2348 0.4709 0.6053 1.5804 -0.6772
-1.0099 1.0099 O -0.6772 -0.6772 1.9870
(Y.6)
0.1547 -0.1237 0.0050 0,0489 0.0483 -0.1227
-0.1237 0.1547 0.0050 -0.0483 -0.0489 0.1227
[Yi] = 0.0050 0.0050 0.0700 -0.0260 0.0260 O
0.0489 -0.0483 -0.0260 0.0982 0.0479 -0.0808
0.0483 -0.0489 0.0260 0.0479 0.0982 -0.0808
-0.1227 0.1227 O -0.0808 -0.0808 0.1829

Appendix W
Spring Constants Used
the Calculations of
Section 9.3.5
.
ln
Tables W.I-W.3, from Pai'doussis et al. (1983a), give the dimensionless spring stiffnesses
used in the calculations of Figures 9,31-9.35. In these tables, as [cij], [kij], etc" are
symmetric matrices, only the upper half af the matrix elements need be given.
Table W.l The spring stiffnesses used in some of the calculations. For Figure 9.31 (a),
k II = 4,77 x 10 4 , k; I = 1.13 x 10 3 , and clI is variable; for some of the calculations for
Figure 9.32, ClI = 1.73 x 10 2 , k;1 = 2.36 x 10- 2 klI, andklI is variable.
i, j: 1, 1 1,2 1,3 2,2 2,3 3,3
5Cij/CII: 5 8 5 8 5 5
5C'.jCII: 5 9 6 9 6 7
IJ
11 kij / kll: 11 5 5 10 5 10
13k'.jkll: 13 6.3 6.3 12 6.3 12
IJ
Table W.2 The spring stiffnesses used in the calculations of Figure 9,31(b);
kll = 3.46 x 10- 4 , k;j = kU' and ClI is variable,
i, j: 1, 1 1,2 1,3 2,2 2,3 3,3
4cij/clI: 4 7 4 7 4 4
4<j/CII: 5 8 5 8 5 5
4k i j/k ll : 4 5 4 5 4 4
1480

SPRING CONSTANTS USED IN THE CALCULATIONS OF SECTION 9.3.5 1481
Table W.3 The spring stiffnesses used in some of the calculations of Figure 9.32:
Cli = 1.73 x 10 2 , k;1 = 2.54 x 10- 2 kll andkll isvariable.
i, j: 1, 1 1,2 1,3 2,2 2,3 3,3
5cij/C[\: 5 8 5 8 5 5
5 Ci .j C[ I : 5 9 6 9 6 7
ZJ
15 kij / k\\: 15 9 9 14 9 14
19k ' .jkll: 19 11 11 18 11 18
ZJ

Appendix X
Flow-Induced Vibration of
Clusters: Some Expressions
in Detail
X.1 CURVE-FITTED EXPRESSIONS FOR Kij,()(w)
The integrations in equation (9.103) have been carried out numerica11y by Curling (1982)
and the results have been curved fitted as follows:
(i) for points on the same cylinder,
K.. ( w ) = allSRI ( 1 + e-a3ISRI ) .
l},() 1 S 2 '
+a2 R
Kij,(}(w) = O;
rCi) = s(j), i = j,
rCi) =s(j), i =l=j;
(X.1a)
(ii) for points on a pair of cylinders (Curling & Pa'idoussis 2003; Gagnon &
Pai"doussis 1994a),
Kij,(}(w) = Mij (o/r(i)r(j)J zrs (w) + [Nij - Mij(o/r(i)r(j)]Jyrs(w), (X.1b)
R(Q/2n) 1 ..(ff (w/2n)
SR= =-- ,
V c 2 8 U c
Mij (o/r(i)r(j» = Trig(i, o/r(i)r(j) )Trig(j, o/r(i)r(j»
Ni' = 1 1 if i, j :::: K or i, j > K
} O otherwise,
J = b18;}ISRI ( 1 + e-b3IsRI )
zrs 1 b c2 S 2
+ 2 8 rs R
and a simi1arexpression for J yrs ; where, as inequation (9.97), rCi) = i, Trig(i, e) =
cose for i .:::: K, and rCi) = i - K and Trig(i, e) = sine for i > K.
These re1ationships ho1d for -50 :::: SR :::: 50, 2.0 :::: 8rs = Rrs/ R :::: 6.0;
Rrs is the centre-to-centre distance between cylinders r and s. The constants
are: (al,a2,a3) = (38.34,40,61,19,09); (bl,b2,b3) = (-33.80,44.34,0.0628)
rCi) =1= sej),
i,j=1,2,.,.,2K,
where
(X.2)
(X.3)
(X.4 )
1482

FLOW-INDUCED VIBRATlON OF CLUSTERS
1483
and (CI, C2) = (-1.735,0.6295) for lzrs; (b1, b2, b3) = (52.18,140.7,0,03942)
and (Cl, C2) = (-1.345,1.210) for lyrs. The expressions for lzrs and lyrs are
given in graphical fonn in Curling & Paldoussis (2003).
X.2 EXPRESSIONS FOR Kln.;(W)
Here,
Kln.(W) = -A(-z*, i, n) - I(z*, n, 0.0) x B(-z*, i, n) + A(z, i, n)
+e-ZI(-Z,n, 1.0) x B(z,i,n); i,n = 1,2,...,N, (X.5)
where
fL wfJ1/2
z = -(b + i 2n) = -(b + i2n); i = .J=1,
U e br U c
1 { Qln bl n C[n /ln }
A(Z,i,n) = 1 'A.4/4 -+2'+3+4 '
-nz Z Z Z Z
</Jn() 1 d</Jn() 1 d 2 </Jn(O 1 d3</Jn()
I(z,n,) =  - Z2  + z3 d2 - z4 d3 '
_ 1 1 { Z ) )} .
B(z,i,n) - 4 4 4 4 e I(z,i, 1.0 -I(z,i,O.O ,
(1- 'A.dz ) (1 - 'A.nlz )
(X.6a)
(X.6b)
(X. 6c )
(X.6d)
b in (X.6a) is defined in equation (9.98). In equation (X,6b), Qln through fln are the
weighted residuals given by
Qln = 11</JI()</Jn()d,
(I d2</Jn ()
C[n = 10 </JI () d2 d,
bl n = 11 </JI () d</J;) d,
 _ (I A. ( i: ) d3</Jn() di:
Jln-l o 'f'l<,; d3 <,;.
(X.7)
For the case where f LI Ve = w/(2n u e ) = O, equation (9.105) simplifies to
Kln,(W) = 11 dl 11 d2</JI(1)</Jn(2).
(X,8)
(For the orthononnal pinned-pinned boundary conditions, this equals zero when either i
or n is even, and 8/(imr 2 ) when both i and n are odd.)

Appendix V
Some Details for 1-D Leakage
Flow Analyses
Y.1 MULCAHY'S (1988) 1-d.o.f. ANALYSIS
The expressions for TI and T2 of equation (11.46) are:
TI = [-R2 R4 + RI Rs + RI R3 V}]/[R + Ri V}],
T2 = [R2 Rs + (R2 R3 + RI R4) V}]/[R + Ri V}],
(Y.1)
where Vs = 1/ws = U s/(QL) is the reduced ve1ocity, and
R2 = f3R YR + f3L n,
R3 = [KD f31(f3I - f3D) + Ku f3f;(f3] - f3u)]
+ [KR f3(2f3I - 3f3R) + KL f3E (2f3I - 3f3d] + f3i kI,
R4 = YR[KD f31- f3R(f3I - f3R) + KR f3]
- n[Ku f3f; + f3LCf3I - f3d + KL f3E],
Rs = [f3R y1- f3L d],
(Y.2)
in which, for example, f3 R = U s 1 U R with U s defined by (11.44), and YR = L R 1 L.
The expressions for CD and CMjK in equation (11.48) are
CD = (LI H v)(LI H s)Vs[-SI T2 + S2 Ti + S4],
- - 2
CMjK = (LIHv)(LIHs)[-S2 T2 + Ss + (S3 - Sl TdVs], (Y.3)
-"\I -.\.A.......L..LL J .J'{L..iJ(JIL,o.Á..I............
Sl = [KD f31 + f3(KR - 1)] YR - [Ku f3f; + f3E(K L + 1)] YL
S2 = (f3R y1- f3L yb,
1484

SOME DETAILS FOR l-D LEAKAGE FLOW ANALYSES 1485
S3 = [KD f3b(f3[ - f3D) - f3(f31 - f3R) + KR f3(2f31 - 3f3R)] YR
- [KU 13& (131 - f3u )f31 (131 - f3d +  KL f31 (2131 - 3f3d] YL,
S4 = [KD f3b - f3R 131 + KR f3] Y + [KU 13& + f3L 131 + KL f3I] yf,
Ss = (f3R Y + f3L yl); (Y.4)
these expressions do not depend on Vs.
Using the expressions for CM/K in (Y.3) in order to evaluate (11.51), it is found that
the following condition must be satisfied for divergence:
S3 Rl > Sl R3.
(Y.5)
V.2 INADA & HAVAMA'S (1990a) ANALVSIS
With the aid of equations (11.63), the steady force F and moment M about the pivot
(x = l f) are found to be
F Pin in (13 - ex)
----
LP* p* 2 2(1 + ex) ,
M ( Pin in ) ( 1 )
L2 P* = P* - "2 2 - l f
_ (13 - ex) [  { -21n(1 + ex) + ex(ex 2 + ex + 2) } _ l f ] .
4ex 3 1 + ex 2(1 + ex)
(Y.6)
(Y.7)
Following the steps outlined after equations (11.69), the dimensionless unsteady
pressure distribution along the channel, p(x), is found to be
p(x) = -{iwl] (x) + 2(f3y - ex)l}(x) + in}q(O)
+ {iwlll (x) + 2(f3y - ex) 113 (x) + 2!z(x)} ě + {3(f3 - ex) 14 (x) + in}e
+ [iw Hhl (x) -l f 111 (X)} + 2(f3y - ex) Hh3(X) -l f 1I3(X)}
+ 2{1I2(x) -l f !z(X)}}j + [3(13 - ex){114(X) -l f 14(X)} + l}(x) - l Ain]ď,
(Y.8)
where ě == iwe, ď = iwď, and lmn(x) is defined by
m
l y xm 1 m' m!
J x = dx = - -1 -J I 'x
mn( ) [h( )]n m L() 0'( _ ')' n-J( ),
o x ex j=O J. m J.
(Y.9)

1486
SLENDER STRUCTURES AND AXIAL FLOW
m and n being positive integers. Note that In (O) and J mn (O) become null. Expressing a1so
q (O) in terms of e, 1J, ě and .(J, by substituting x = 1 in (Y.8) and considering the boundary
conditions for p(l) in (11.69), we obtain
( O ) =  ( 1 + Bl ) , + B2
q , e . e
ft lwft + r 1wIl + r
!lzl- Z fltl ( B4 ) . Bs
+ 1+ . 1J+ , 
ft lwft + r lwft + r
where
(Y.10)
( ft J13 ) ft h ex ( ft )
Bl = 2(f3y - a) - - 13 + 2- - in - - 1 - - ,
J11 J11 h 2 (1) J11
ex
B2 = 3(13 - a)I4 + in + h 3 (1) '
( l J 23 -Z f J 13 ) J 12 -Z f 1 2
B4 = 2(f3y - a) 2 II - 13 + 2 ft
!lzl- Z fltl !lzl- Z f lt1
ex I ft ( 1 ) )
- .n + - - - Z f - 1
I h2(l) !lzl- Z f J 11 2 '
ex
Bs = 3(13 - a)(J14 -Zf 14) + 13 -Zf in + h 3 (1) (1 -Zf),
ex
r = 2(f3y - a)I3 + in + h 2 (1) '
in which In == In (1) and J mn == J mn (1) for simplicity; it is recalled that
In(x) = lX [h(x)r n dx.
Then, integrating p(x) and (x - Zf)P(X) from x = O to 1, after (Y.10) has been
substituted into (Y.8), we obtain the unsteady force  F and moment M:
(Y.1l)
(Y.12)
{ ::;: 2 } = - (iw[m a ] + [ca] + 1 + \WT [Cb]) {  }
- ([k a ] + 1 + 1 iWT [k b ]) {  } ,
where the e1ements of all the matrice s are given exp1icidy by lnada & Hayama (1990a,
appendix B). It is seen that there are components in F and M with a 90° time de1ay
from e, 1J, ě and .(J, dependent on
wT= Qtoft = _ pLI1IiQQI . (Y.14)
r (pQ/ Ho){2(f3y - a)I3 + in + ex/ h2(1)} Q
(Y.13 )
The numerator of (Y.14) is the abso1ute va1ue of the inertia force for osci11ations of amp-
1itude  Q and frequency Q, and the denominator is the increase in fluid resistance

SOME DETAILS FOR l-D LEAKAGE FLOW ANALYSES
1487
for a flow increment Q. Thus, if wT « 1, viscous effects are dominant, and the
quasi-steady fluid force is obtained. If wT » 1, however, inertial fluid flow effects
predominante, and the inlet flow rate q(O) responds only to the 'squeeze effecť, that
is, q(O) = (Jl1/h)ě + [(hl -lf Jl1) Ih] tJ, in which the delayed effects become
very small.
If the matrices are separated into real and imaginary parts and combined via
Cbmn T Cbmn
M -m - ' C -c + '
a,mn - a,mn 1 + w2 T2 ' w,mn - a,mn 1 + w2 T2 '
C - - kb,mnT K - k kb,mn
e,mn - 1 + w2 T' a,mn - a,mn + 1 + w2 T2 '
(Y.15)
for m = 1,2, n = 1,2, equation (Y.13) may be written as
{ :;:;= 2 } = -(iw [Ma] + [CwD {  } - (iw [C e ] + [KaD {  } .
(Y.16)
Then, with the aid of (11.71), the final fonn (11.70) of the equation of motion may be
obtained.

Appendix Z
Coaxial Shells: Some Details
Z.1 GENERALlZED FORCES; INVISCID
INCOMPRESSIBLE FLOWS
Proceeding in the same manner as in Section 1 1.7. 1 (a), but assuming the fluid to
be incompressible ab initio, tbe following expressions replace equations (11.295) and
(11.296):
= -2 ( (1) - (1) - ) - ( (2) - (2) - )
Qkmn = Qo qkmn C mn + skmn F mn + 2Qo qkmn C mn + skmn F mn
( (3) - (3) - )
+ qkmn C mn + skmn F mn ,
= -2 ( (1) - (1) - ) - ( (2) - (2) - )
Rkmn = Qo r kmn C mn + t kmn F mn + 2Qo r kmn C mn + t kmn F mn
( (3) - (3) - )
+ r kmn C mn + t kmn F mn ,
(Z. I)
and
-j-l
(j) _ ( _ l) Hl {Ji Si Vi 1 00 E h ( - " ) -j-2 H ( ;;;"\ d -
qkmn - 2 (U ) 3-j n a, SI a km aj a
7í r 10 0 -00
-j-l
ti, S' P V 1 00
j 1-'1 I roh -j-2 __
+(-1) 3' Fn(a,Si)a Hkm(a)da,
27í U2 S - J -00
r o
-j-l
, R, S. P V 1 00
(J) _ ( - l) j 1-' 1 I r o G A ( - . ) -j-2 H ( ;;;"\ d -
Skmn - 3 " n a, SI a km aj a,
27í U2 S - J -00
r o
, ti v j - 1 1 00
(j) _ ( _ l) j+l 1-' 0 o p, h ( - ) -j-2 H (;;;"\ d -
r kmn - 2-' n a, 10 0 a km\aJ a,
27í 10 0 J -00
-j-l 00
t(j) = ( _l ) Hl {Jo V o 1 O ( -či S ) -či j - 2 H k ( fi\ d-či
kmn 2- in, o m u J ,
27í 10 0 -00
(Z.2)
with j = I, 2, 3, and En, Fn and On as defined by equations (11.286), but with a replacing
both Ai and Ao'
1488

COAXIAL SHELLS: SOME DETAILS
1489
Z.2 NUMERICAL EVALUATlON OF THE GENERALlZED
INVISCID FORCES
ln principle, the integrals in equations (11.295) and (11.296) could be evaluated e1egantly
by contour integration. However, in practice, several difficulties are encountered, the
major one being the addition and subtraction of very large numbers, while the tinal result
is small. Hence, this method has been abandoned in favour of direct numerical integration.
From equations (11.287), one may obtain the zeros of AjL and AoL as a 1,2 = :I:: Kj Mj/
(1 :I:: Mj) for AjL and a1,2 = :I:: Ko Mo/(1 :I:: Mo) for AoL. Since Mj and Mo are real, and
for this conservative system so are Kj and Ko up to onset of divergence, these zeros 1ie on
the Re(a)-axis. These singularities are in fact branch points, and to deal with them, the
integration path is moditied as in Figure Z.l in order to bypass them.
Next, we consider the roots of I (z) = O, where z = Aj a; they are either imaginary
or zero. Upon denoting the jth root as Zj and using the detinition of Aj in (11.287), the
singular points in the a-plane may be calculated as
{ 2 } 1/2
aj = - [Kj M j 2 /(1 - M j 2 )] :I:: i [- z](L/a)2 /(1 - M?)] - [Kj Mi/(1 - Mh]
(Z.3)
If
[ - z](L/a)2 /(1 - M j 2 )] > [Kj Mj/(1 - M j 2 ) r '
(ZA)
equation (Z.3) yields complex conjugate pairs. Otherwise, the aj are real numbers. It
tums out that, within the range of the reduced frequency Kj of the system concemed and
for small Mach numbers, inequality (ZA) app1ies. The real a-axis, is, therefore, free of
zeros ofl(Aja).
Fina1ly, a computer subroutine has been used to determine the zeros of
[I (Aob) K (Aoa) - I (Aoa) K (Aob)], whereby it has been found that, for typical system
parameters, the zeros 1ie c10se to the Lm (a)-axis; hence this term presents no difticulties
for integration on the Re(a)-axis.
It is evident, therefore, that the only singularities of the various integrands that lie
on the Re(a)-axis are the branch points of AjL or AoL, as the case may be. Consider
as an example the tirst integral in equation (11.296) and the path of integration shown in
Figure Z.l. On the semi-circ1e around the branch point a 1, a -a 1 = £ e jll , da = i £ e ill de,
andAoL = [(a-al)(a-a2)]1/2  [£eill(cil-a2)]1/2 as£ -+ O. Theintegralaround
the semi-circ1e is then given by
1 0 ( - ) 2
Ko - oq . iII
. 1/2 F n (al, £o)Hkm(cil)l£e de.
n [£e 11l (al -(2)]
Since it may be shown that both Fn (cil, £0) and Hkm (cil) are tinite, the above integral,
which is proportional to £1/2, vanishes as £ -+ O, Similarly, the integration along the
semi-circ1e around the other singular point, a2, makes no contribution. These conc1usions
may be extended to other integra1s in the genera1ized fluid force terms.

1490
SLENDER STRUCTURES AND AXIAL FLOW
dnv (a)
r Semi-circle of radius E
a2
al
C>(a)
L- Branch lines ---'
Figure Z.I The path of integration along the Re(a)-axis in the complex a-plane, for the deter-
mination of the generalized fluid forces of equations (11.295) and (11.296), showing singularities
at a] and a2 (Pa'idoussis et a/, 1984a),
The two-point Gaussian quadrature scheme is employed to accomplish the numerical
integration. Examination of the integrands has shown that their values decrease asymp-
totically as lal becomes large. Thus, the range of a for quadrature can be truncated to
(-50,50) and 100 points used.
Further details of the numerical method may be found in Chan (1984) and Pa'idoussis
cf al. (1984a).
Z.3 THE MEAN ANNULAR VISCOUS PRESSURE FIELD
Procee ding in the same manner as for inner flow (Section 7.5.2), it may be shown that
U ro UBo = O everywhere. Then, defining by r m the radius where U o is a maximum, one
obtains
dUo/dr = O and u xo u ro = O,
moreover, on the shell surfaces,
at r = r m ;
(Z.5)
U xo u ro = Uo = O
at r = a and r = b.
(Z.6)
Hence, proceeding in the way leading to equation (7.153), one obtains
r a Po
C2(X) = --,
2po ax
 ap o { b2 - r } _ _u 2
2po ax b - 7:00'
(Z.7)
and
Po { 2b } 2
- = - b 2 _ 2 UrooX + C4(r),
Po r m
where U;oo is defined in equations (11.301).
Comparing equations (Z,8) and (7. 150b), the functions C3(X) and C4(r) are defined,
leading eventually to equation (11.300).
(Z.8)

COAXIAL SHELLS: SOME DETAILS
1491
It is also necessary to relate U;oo and U;oi of equations (11.301), The second
equation (Z.7) has been obtained by applying the boundary equations (Z.6) at r = b;
if, instead, those at r = a are applied, then
 o Po { r - a 2 } = _U 2 ,
2po ox a WI'
from which the following relationships are easily obtained:
b{ 2 2 }
2 r m - a 2
U wi = -;; b2 _ r U wo '
b { r2 - a 2 }
Twoi = -;; b _ r Twoo,
(Z,9)
Z.4 MODEL FOR CANTILEVERED SHELL WITH
ANNULAR FLOW
The outer cyIinder being rigid, the equations of motion are the first of (11.281), and the
fonn of solution is that of equations (11.282). The method of solution then proceeds in
the same manner as in Section 11.7.1, but here also considering an outflow model. Hence,
the axial variation of the pressure perturbations, described in tenns of the cantilever beam
eigenfunctions 4Jm (x) for O ::: x ::: L, is extended by the outflow model, r m (x) for
L < x ::: L', where L' "'-' O(3L) typically.
Hence, defining the dimensionless generaIized forces as in (11.294), they are given by
= P .U C 1 00 I ! Q. ] { I ( a£' )}
Qkm = I ?m = 1 -Ui a 2  _ 1 {Hkm(a) + Nkm(a)} da
2n: qi -00 IX £i In (IX £i)
Po U 1 I ! Qi - ] { -
_ j 00 = --=- _ Uoa 2 Wln(Ci, £i)C m
2n: qi -00 ex £0 Qr
+W2n(a, £i) F m } {Hkm(a) + Nkm(a)}da, (Z. 10)
where
r;v I(Ci£o)Kn(a£) -In(a£)K(Ci£o)
Wl ,IX £)-
n , - I(Ci£o)K(a£i)-I(a£i)K(Ci£o)'
In (Ci £) K(Ci £i) - I (a £i) Kn(a £)
W2n(Ci, £) = I - I I I - ;
In (ex £0) Kn (Ci £i) - In (Ci £i) Kn (ex £0)
(Z. I I)
Nkm(a) is defined exactly as Hkm(a) in (11.297) but with rm(), the outflow func-
tion, instead of 4Jm (). In the above equations, the dimensionless parameters are defined

1492
SLENDER STRUCTURES AND AXIAL FLOW
differently from Section 11.7.1 and hence are given below:
- Ui
Ui= U i '
r
£=-
L'
- Am
Am=-,
L
- Em
Em=-,
L
in which
- U o
U o = U o '
a
£,--
1- L'
- Bm
Bm=-,
L
- Fm
Fm=-,
L
[ ] 1/2
Ei
Ui= 2 '
Psi (1 - Vi )
Psi h i L
qi=
Yi
Eihi L
a 2 (1 - Vf) ,
- Qa
Q,--
1- Ui'
Qi
Qr = =-,
Qo
- Qb
Qo = U o '
b
£0=-'
L
L'
Z=-
L'
(Z.12)
Ci=aL,
x
= L '
- Cm
C m =-,
L
- Dm
Dm=-,
L
[ E ] 1/2
U o = o 2 '
Pso(1- Vo)
(Z. 13 )
Pso ho L
qo=
Yo
Eo ho L
- b 2 (1- v)'
Equation (Z. tO) may be written explicitly as a quadratic function of Qi ,
= { (1) - (1) - } -2 { (2) - (2) - } - { (3) - (3) - }
Qkm= qkm Cm + r km Fm Qi + 2 qkm Cm + r km Fm Qi + qkm Cm + r km Fm ,
(Z.14 )
where
, ' U2 1 U J-1 ] 1 00 { I ( - , ) }
(J) = ( _I ) J+1!!:..........L  CiJ-2 n a£1 { H ( Ci ) + N ( Ci )} dCi
qkm 2 -, 3-J I' C ,) km km
n ql £i -00 n a£1
U 2 1 U J-1 ] 1 00
+(_I)J PO}! o 3 . CiJ-2 W1n (Ci£i){Hkm(Ci)+Nkm(a)}da,
2n qi (£oQr) -J -00
(Z.15)
. U 2 1 U J-1 ] 1 00
r = (-1)J Po }! o 3- ' a J - 2 W2n(a£j) {Hkm(a) + Nkm(a)} dCi,
2nqi (£oQr) J -00
j = 1,2,3.
The functional forms for a flexible outer shell, instead of a rigid cylinder are also given in
Pai:doussis et aZ. (1991), but not here, since no results for that case are available.
For the steady viscous forces, the same relationships as (11.302), (11.303) and (11.307)
apply, but in this case it is presumed that the annular and inner fluid static pressures are
aZways equal at x = L.

COAXIAL SHELLS: SOME DETAILS
1493
The method of solution is a1so similar to that used in Section 11.7.1, except that in this
case, because the beam eigenfunctions do not fuHy satisfy the shell boundary conditions
at the free end, the extended form of Ga1erkin's method is used. Thus, the equations of
motion and boundary conditions are satisfied in the mean. The appropriate variationa1
statement is
8E = 1 21f l L { i [,Cii 8Ui + ,CZi 8Vi - .qi 8w;]
+ o [£Io 8u o + £20 8v o - £;0 8w o ] } dx dB
_1 21f {'Di [Rli8Ui + (1- vi)R2i8vi +ki(R3i8w; +R 4i 8W i)]
+ 'Do [RIO 8u o + (1 - V o )R20 8v o + k o (R30 8w + R40 8W o )]} IX=L dB = O,
(Z.16)
where the £*s are defined by the left-hand side of equations (7.147) and (11.299) re-
written so that the right-hand sides equal zero, and the Rs by (7.4a-d), and 'Di =
Ei h;j(1 - vf), 'Do = Eo h o l(1 - v';); cf. equation (7.97). The upstream boundary
conditions are automaticaHy satisfied by the choice of admissible functions.
Z.5 VELOCITY PROFILE AND VISCOSITY IN
ANNULAR FLOW
For laminar flow, the ve10city profile and the radius at which the flow velocity is maximum,
r m , are given by (Knudsen & Katz 1958)
_ [ b 2 - r 2 - 2rln(blr) ]
U(r)-2U a 2 2 2 '
a + b - 2r m
b2 _ a 2 } 1/2
r m = { 21n(bla) ,
(Z.17)
where Ua is the average flow ve10city, defined as the ratio of the vo1umetric flow rate to
the cross-sectional area of the annulus.
For turbulent flow, however, in the absence of better data, Knudsen & Katz's (1958)
empirical data are used:
U ( r ) = { 1. 14U a [(r - a)/(r m - a)]O,102
1. 14U a [(b - r)/(b - r m )]O,142
for a .::: r .::: r m,
for r m < r .::: b;
(Z.18)
r m is detennined from the multilinear representation ofBrighton & Jones' (1964) experi-
mental results. Although (Z.18) may not be valid for Re < 10 4 (based on the hydraulic
diameter), they should be adequate at the critical flow velocities of interest, for which
7 x 10 3 < Re < 3 x 10 5 approximately.
We next outline the method for detennining Vt (r). According to Prandtl's mixing length
hypothesis, Vt = 121(dU Idy)l, where 1 is Prandtl's mixing length and y the coordinate
measured perpendicularly from the wall. An expression for 1 I R for smooth pipes of radius

1494
SLENDER STRUCTURES AND AXIAL FLOW
R is given by equation (2.101). In the absence of equivalent data for annuli, this expression
is adapted for annular flows (Nguyen 1992; appendix E,2), eventually leading to
v (r) = { (rm - a)2 { r m  a r (  )
t -(b _ r m )2 {  } 2 ( dU )
b - r m dr
for a :::: r :::: r m,
(Z.19)
for r m < r :::: b,
Z.6 SOLUTION OF NAVIER-STOKES EQUATIONS
Consider the annular ftow between a ftexible shell and a rigid outer cylindrical tube. The
equations to be solved are equations (11.331), To be consistent with the fully implicit
scheme employed for time derivatives in the analysis of shell motions, the two-point
backward difference appmximation is now used to discretize the first of equations (11.331)
in physical time
(yn+l _ yn)/l::1t + G(yn+l, pn+l) = O,
which has first-order accuracy in time, Since the continuity equation,
equation (11.331), should be satisfied at every time step, then
Gv(yn+l) = O.
(Z.20)
the second
(Z.21)
Then, by means of the new notations
n+l -n+l -n
V =v -v,
n+l -n+l -n
p =p -p,
(Z.22)
equations (Z,20) and (Z.21) may be rewritten as
v n + 1 + I::1tG(v n + 1 , pn+l) = -l::1tG(yn, pn),
Gv(v n + 1 ) = O,
(Z.23)
since G () and Gv() are linear operators.
For the solution of the first of equations (Z.23) satisfying the zem-divergence con-
straint (the second equation) on the perturbation velocity, a continuous auxiliary system
in pseudo-time and involving an artificial compressibility is introduced:
ov  A A n n
a.. + v + I::1t G(v, p) = -l::1t G(V , P ),
ap 
8- + Gv(V) = O,
a..
(Z.24)
where .. is a pseudo-time which should be distinguished fmm the physical time t, 8 is the
artificial compressibility coefficient, and v = y* - yn, p = p* - pn; here the asterisk
denotes a transient value in pseudo-time. It is seen fmm equations (Z.24) that, as the
steady state is reached in pseudo-time, ov / a.. and ap / a.. are both virtually zem; hence,
vand p become v n + 1 and pn+l, respectively. Therefore, the solution of (Z.23) in each
physical time step is practically the same as the steady-state solution of (Z,24) in pseudo-
time. Equations (Z.24) therefore have no physical meaning until reaching steady state
in pseudo-time, As these equations are eventual1y discretized in pseudo-time, 8 and the
pseudo-time step 1::1.. must be appmpriately chosen to ensure that the steady-state solution
is always achieved in pseudo-time at every physica1 time level. It should also be mentioned

COAXIAL SHELLS: SOME DETAILS
1495
that [, and  'l' have no effect whatsoever on the stability of the solution obtained in the
physical time domain.
Equations (2,24) may be written in a more compact form as
aIl
- + (Ar + Ax + Ae) II = R,
a'l'
(2.25)
where R is kept constant at its value at the physical time level n, and
rr{},
{ -ex GrCv, pn) }
R = -ex Ge(v, [in)
-ex GAv, [in) ,
O
ex = t,
and
[ 1_2 dV ..Q..._!!...Q... ( r..Q... )
a dr ar r ar ar
O
Ar = ex O
1 ( 1 a )
cdi r + ar
a a 2
U(r)- - v-
ax ax 2
O
O
dv a v a ( a )
- dr ar - r ar r ar
O
O
O
O
dv a v a ( a )
- dr ar - r ar r ar
O
1 a ]
p r
O '
O
Ax =ex
2vn
71
o 0 2
U(r)- - v-
ox ax 2
O
O
dv a
- dr OX
O
Ae =ex
v(1+n2)
---;:z-
!! ( dV + 2V )
r dr r
dU
dr
O
O O
1 + v(1+n2) O
a ---;:z-
O vn 2
-;:z
1 n O
ali r
O
O
1 a a 2
- + U (r)- - v-
a OX ox 2
1 a
ali OX
O
O
1 o
p ax
O
O
_..!l...
pr
O
O
[, is defined in (2.24). Equation (2.25) may be integrated in pseudo-time through two
intennediate steps denoted by * and **:
I1* - I1 k k
+ ArIl* + (Ax + Ae) II = R,
'l'
I1** I1 k
- A I1 * A ** A k
+ r + x II + e II = R,
'l'
I1k+1 _ I1 k
+ ArIl* + AxIl** + AeIlk+1 = R.
'l'
These equations may be written in compact "delta" form:
(I + 'l' Ar)l] = 'l' [R - (Ar + Ax + Ae)I1 k ],
(I + 'l' Ax)ň = ft, (I + 'l' Ae)I1 = ň,
(2.26)

1496 SLENDER STRUCTURES AND AXIAL FLOW
r- - --.,. Computational domain - - --, qL
O:'V OI'V 
j=M 'V O 'V O 'V O 'V O 'V O 'V O 'V O 'V I O
Rigid ..!::Í:J;       T  I T
boundary I
j=M-I 'VOl 'V O 'V O 'V O 'V O 'V O 'V O 'V O 'V °1'V O
I
TI T T T T T T TI T
I I
j=j 'V OI'V O 'V O 'V O 'V O 'V O 'V O 'V O 'V OI'V O
TI T T T T ij T T T TI T
I I
j=2 'V OI'V O 'V O 'V O 'V O 'V O 'V O 'V O 'V OI'V O
Moving 4         I T
I
boundary
j=1 'V ?I'V O 'V O 'V O 'V O 'V O 'V O 'V O 'V OI'V O
2 3 i i+ I N N+I N+2
Flow in  N  Flow out
O I I 2 . i-I i . . . N-l
Flexible  . . . . . . . f
shell
Clamped end Free end
Figure Z.2 A schematic of the arrangement of (i) flow variables on a staggered mesh, and (ii) nodes
on the flexible shell: T, V r ; 'V, V x ; o, P, Ve, where V denotes either v or V, and similarly for P; .,
nodes on the shell (Nguyen ef a/, 1994),
where ft = 11* - II k , ft = 11** - II k , II = lIk+l - II k , I is the 4 x 4 identity
matrix, and k denotes a certain pseudo-time step. The method of solution represented
by equations (Z.26), commonly referred to as the ADI (A1temating Direction Implicit)
scheme, is carried out on a staggered grid shown in Figure Z.2. This grid is in fact
stretched in the radial (r or 1J) direction, with a large number of grid points in the vicinity
of the physica1 boundaries. The distribution of these points is determined by a two-sided,
hyperbolic tangent stretching function. The three equations (Z.26) are referred to as the
r-, x- and B-sweeps, respective1y. In the staggered grid, v r is coup1ed with ft during the
r-sweep. Similarly, V x and Ve are coup1ed with ft in the x- and B-sweeps, respectively.
Since the physical system (Z.23) and the pseudo-time system (Z.24) are both initia1
boundary value problems, initia1 and boundary conditions must be specified. Details of
this and of the imp1ementation of the r-, x- and B-sweeps may be found in Nguyen cf al.
(1994) and Nguyen (1992).

Appendix Q
Recent Publications Related to
Volume 1
As stated in the Epilogue, some references on the dynamics of pipes conveying fluid
and related topics which have appeared since 1998 (when Volume I was published) are
given here, together with some older ones missed out. To help the reader, these refer-
ences are grouped in categories. This Appendix concludes with a list of corrigenda to
Volume l.
Q.1 FUNDAMENTALS AND RELATED TOPICS
Of particular importance is the work on convective and absolute instabilities and on local
and global instabilities by Kulikovskii and by de Langre and associates, cited here and
discussed briefly in Section 10,2.3. AIso of great importance is the extensive discussion
on energy transfer and wave propagation in translating continua in general, and pipes
conveying fluid in particular, in Lee & Mote (I 997a,b),
Finally, an unexpected controversy has erupted on the existence and/or successful real-
ization of follower forces t and the experiments by Sugiyama et al. discussed in Volume I
(pp. 67 and 170). Starting with Koiter's (1996) sally, the Sugiyama et al. (1999) rebuttal
and Elishakoff's (2000) repartee, the controversy has mushroomed - though the rocket-
based realization of a follower force seems sound enough - culminating in the priceless
Ugly Duckling paper by Sugiyama et al. (2002).
ANDERSON, R. & BEJAN, A. (1983) Buckling of a turbulentjet surrounded by a highly
flexible duct. Phys, Fluids 26,3193-3200.
BENAROYA, H., WEI, T., KUCHNICKI, S, & DONG, P. (2003) Extended Hamilton's
principle for fluid-structure interaction, IMechE 1. Multi-body Dynamics 217 (Pt K),
153-170,
BOLOTIN, V.v. (1999) Dynamic instabilities in mechanics of structures. Appl. Mech.
Rev, 52, RI-R9.
DE LANGRE, E, (2000) and (2002) ln Vol. 2 References.
DE LANGRE, E. & OUVRARD, A.E. (1999) ln Vol. 2 References.
DOARÉ, O. & DE LANGRE, E. (2002a,b,c) ln Vol. 2 References.
ELlSHAKOFF, I. (2000) Essay about so called follower forces. Dept Mech. Engineering
Report, Florida AtIantic University, Boca Raton, FL, U.S.A.
tlt is recalIed that in a cantilevered pipe they unquestionably exist, but are inextricably associated with
Coriolis forces along the pipe.
1497

1498
SLENDER STRUCTURES AND AXIAL FLOW
KOITER, W,T. (1996) Unrealistic follower forces. J. Sound Vib. 194,636-638.
KOUNADIS, A.N. Nonclassical stability problems: in stability of slender tubes under
pressure, J. Aerospace Engg. 14, 6-11.
KULlKOVSKII, AG. (1993) On the stability loss ofweakly non-uniform flows in exten-
ded regions, The formation of transverse oscillations of a tube conveying a fluid. J.
Appl. Maths Mech. 57, 851-856.
KULlKOVSKII, AG. & SHlKINA, I.S. (1988) On the bending vibrations of a long tube
with moving fluid. lzvest. Akad. Nauk Armyanskoi" CCP 41,31-39.
LANGTHJEM, M.A & SUGIYAMA, Y. (2000) Dynamic stability of columns subjected
to follower loads: a survey. J. Sound Vib, 238, 809-851.
LEE, S,-Y. & MOTE, C.D. Jr (1997a,b) ln Vol. 2 References.
SUGlYAMA, y" LANGTHJEM, M.A. & RYU, B.-J, (1999) Realistic follower forces,
J. Sound Vib, 225, 779-782.
SUGlYAMA, Y., RYU, S,-U. & LANGTHJEM, M.A (2002) Beck's column as the Ugly
Duckling. J. Sound Vib. 254,407-410,
VERICHEV, S.N. & METRIKINE, A V, (2002) Instability of a bogie moving on a flexibly
supported Timoshenko beam. J. Sound Vib. 253, 653-668.
2.2 PIPES WITH SUPPORTED ENDS
From the following list, of special note are the papers by Chucheepsakul & Monprapussom
(2000) and Sorokin & Terentiev (2003), which bring novel elements into the analysis of
such systems vis-a-vis what is found in Volume 1.
CHUCHEEPSAKUL, S. & MONPRAPUSSORN, T. (2000) Divergence instabil-
ity of variable-arc-Iength elastica pipes transporting fluid. J. Fluids Struct, 14,
895-916.
DOZDOV, AD. (1997) Stability of a viscoelastic pipe filled with a moving fluid. ZAMM
77,689-700.
GLA VARDANOV, V.B. & ATANACKOVIC, T,M. (2000) Stability of a pipe through which
a string is pulled lnt. J. Non-Linear Mech. 35, 7-20.
KANG, M.G. (2000) The influence of rotary inertia of concentrated masses on the natural
vibrations of fluid-conveying pipes. J. Sound Vib. 238, 179-187.
MAALAWI, K.Y. & ZIADA, M.A (2002) On the static in stability of flexible pipes
conveying fluid. J. Fluids Strnct. 16, 685-690.
OZ, H.R. (2001) Non-linear vibrations and stability analysis oftensioned pipes conveying
fluid with variable velocity. lnt. J. Non-Linear Mech. 36, 1031-1039.
SOROKIN, S.v. & TERENTlEV, AV. (2003) Nonlinear statics and dynamics of a simply
supported nonuniform tube conveying an incompressible inviscid fluid. J. Fluids Struct.
17,415-431.
VEDULA, L. & NAMACHCHlVAYA, N. (2002) Stochastic stability oflinear gyroscopic
systems: application to pipes conveying fluid. ASME Paper IMECE2002-39024.
ZHANG, Y,L., GORMAN, D.G. & REESE, J.M. (1999) Analysis ofthe vibrationofpipes
conveying fluid. Proc. lnstn Mech. Engrs, 213, Pt C, 849-860.
ZHAO, F., WANG, Zh., FENG, Zh. & LlU, H. (2001) Stability analysis of Maxwell
viscoelastic pipes conveying fluid with both ends simply supported. Appl. Math. &
Mech. (China) 22, 1436-1445.

RECENT PUBLICATIONS RELATED TO VOLUME 1
1499
Q.3 CANTILEVERED PIPES
Work on this inherently nonconservative system continued to be done with unre1enting
pace and ever-expanding scope. New facets of behaviour continue to surprise and de1ight
the researcher - further to what is covered in Vo1ume 1.
BORGLUND, D. (1998) On the optima1 design ofpipes conveying fluid. J. Fluids Struct.
12, 353-365.
DENISOV, K.P. & KHITRIK, V.L. (2001) A note on the stability ofpipes conveying fluid.
J. Sound Vib. 244,904-914.
DOARÉ, O. & DE LANGRE, E. (2002b,c) ln Vol. 2 References.
DZHUPANOV, V.A. (1998) Systematic review on the modes of a canti1evered pipe con-
veying fluid and 1ying on a multiparametric resisting medium. J. Theor. & Appl. Mech.
(Bu1garia) 28, 34-53.
ELlSHAKOFF, I. & IMPOLLONIA, N. (2001) Does a partia1 e1astic foundation increase
the flutter ve10city of a pipe conveying fluid? J. Appl. Mech. 68,206-212.
FREDRIKSSON, M.H., BORGLUND, D. & NORDMARK, A.B. (1999) Experiments on
the onset of impacting motion using a pipe conveying fluid. Nonl. Dyn. 19,261-271.
GURAN, A. & ATANACKOVIC, T.M. (1998) Fluid conveying pipe with shear and
compressibi1ity. Eur. J. Mech., A/Solids 17,121-137.
IMPOLLONIA, N. & ELlSHAKOFF, I. (2000) Effect of e1astic foundations on divergence
and flutter of an articu1ated pipe conveying fluid. J. Fluids Struct. 14, 559-573.
JIN, J.-D. (1997) Stability and chaotic motions of a restrained pipe conveying fluid.
J. Sound Vib. 208,427-439.
JIN, J.-D. & ZOU, G.S. (2003) Bifurcations and chaotic motions in the autonomous system
of a restrained pipe conveying fluid. J. Sound Vib. 260, 783-805.
LANGTHJEM, M.A. & SUGIYAMA, Y. (1999) Vibration and stability ana1ysis of
canti1evered two-pipe systems conveying different fluids. J. Fluids Struct. 13,251-268.
MANABE, T., TOSAKA, N. & HONMA, T. (1999) Dynamic stability ana1ysis of flow-
conveying pipe with two 1umped masses by integra1 equation method. In Boundary
Element Techniques (ed M.R A1iabadi), pp. 385-394. London: Queen Mary and
Westfie1d College; a1so in PDPTA'OO lnťl Conf (ed H.R. Arabnia), pp. 2839-2845;
Las Vegas: CSREA Press.
MONETTE, C. & PETTIGREW, M.J. (2004) Fluide1astic instabi1ity of flexib1e tubes
subjected to two-phase interna1 flow. J. Fluids Struct. (to appear).
PANUSSIS, D.A. & DlMAROGONAS, A.D. (2000) Linear in-plane and out-of-p1ane
1ateral vibrations of a horizonta11y rotating fluid-tube cantilever. J. Fluids Struct. 14,
1-24.
PRZYBYLOWICZ, P.M. (2000) Active damping of trans verse vibration in pipes convey-
ing fluid subjected to kinematic excitation. Z4MM81, S215-S216,
RYU, S.-U., SUGlYAMA, Y. & RYU, B.-J. (2002) Eigenva1ue branches and modes for
flutter of cantilevered pipes conveying fluid. Comp, & Struct. 80, 1231-1241.
SÁLLSTROM, J.H. (1994) Stability optimization of beams conveying fluid or carrying
other axia11y moving materia1s. Struct. Optimiz. 7,219-226.
SARKAR, A. & PAIDOUSSIS, M.P. (2003) A canti1ever conveying fluid: coherent modes
versus beam modes. lnt. J. Non-Linear Mech. (in press).
SARKAR, A. & PAIDOUSSIS, M.P, (2003) A compact 1imit-cyc1e osci11ation model of
a cantilever conveying fluid. J. Fluids Struct. 17,525-539.

1500
SLENDER STRUCTURES AND AXIAL FLOW
WANG, Zh., ZHAO, F., FENG, Zh. & LlU, H. (2000) The dynamic behaviors of vis-
coelastic pipes conveying fluid with the Kelvin model. Acta Mech. So/. Siniea 13,
262-270.
YOSHIZAWA, M" SUZUKl, T., TAKYANAGI, M, & HASHlMOTO, K. (1998) Nonlin-
ear latera1 vibration of a vertical fluid-conveying pipe with end mass, JSME Int, J" Ser.
C, 41, 144-153.
Q.4 PIPES WITH PULSATING FLOW; PARAMETRIC
RESONANCES
Interesting new twists to an "old" problem are provided by Jensen (1999) and Gulyayev &
Tolbatov (2002), The effect of pressure and flow pulsations in realistic pipeline systems
has been studied extensive1y by Hayama and his associates and Fujita & Tanaka.
FUJITA, K. & TANAKA, M. (1990) A coup1ed vibration analysis of piping systems
with contained fluid. In FZow-/nduced Vibration - 1990 (eds S,S. Chen, K. Fujita &
M,K. Au-Yang), PVP-Vol. 189, pp, 227-232, New York: ASME,
GORMAN, D.G., REESE, J.M. & ZHANG, Y.L. (2000) Vibration of a flexible pipe
conveying viscous pulsating fluid flow, J. Sound Vib. 230,379-392,
GULYAYEV, v.I. & TOLBATOV, E. Yu. (2002) Forced and self-excited vibrations of
pipes containing mobile boiling fluid clots. J. Sound Vib. 257,425-437.
HAYAMA, S, & MATSUMOTO, M, (1987) A study on the vibrations of pipelines caused
by interna1 pulsating flows. In FZow-lndueed Vibrations -1987 (eds M.K. Au- Yang &
S,S. Chen), PVP-Vol. 122, pp. 111-116. New York: ASME.
HAYAMA, S. & NAGAI, M. (1991) A study on the vibrations of pipelines caused by
internal pulsating flows. Proc. Asia-Pacific Vibr. Conf, Monash U., Australia, Vol. 2,
pp,10.28-10.33,
JENSEN, J.S. (1999) Articulated pipes conveying fluid pulsating with high frequency.
Non/. Dyn. 19, 171-191.
SOROKIN, S.v. & TERENTIEV, A.V. (2003) Cited in Section Q.2.
Q.5 RISERS AND OCEAN MINING SYSTEMS
Of course, a great deal of work has been and continues to be done on the dynam-
ics of risers. Here only papers concerned with the effect of internal flow are cited,
as well as work on aspirating pipes and ocean rnining, further to the discussion in
Section 4.3.
CUl, G. & ASO, K. (1998) The optimum stepped-pipe string with vibration absorbers for
mining manganese nodu1es, to reduce its maximum axial stress. ASME J. Vibr. Aeoust.
120,917-921.
GUO,H,Y" WANG,S.Q., WU,J.N. & LlU,D.F. (2000) Dynarniccharacteristics ofmarine
risers conveying fluid. China Oeean Engg. 14, 153-160.
PAIDOUSSIS, M.P, (1999) Aspirating pipes do not flutter at infinitesimally small flow.
J. FZuids Struet. 13,419-425.
PRAMILA, A. (1992) Undamped canti1evered pipes aspirating fluid may be stable.
Rekenteiden Mekaniikka 25,3-14,

RECENT PUBLICATIONS RELATED TO VOLUME 1
1501
WILSON, J.F. & BIGGERS, S.B. (1974) Responses of submerged, inclined pipelines
conveying mass. ASME J. Engg Ind. 96, 1141-1146.
n.6 DIVERS SYSTEMS AND APPLlCA TlONS
Here, papers on sundry systems are listed, among them papers on bellows without flow
which, because of the interest in added mas s effects, may a1so be of interest to readers of
this book; also, work on applications, for example on Coriolis mas s flow meters.
AUERBACH, O., MOSER, M. & KENNER, T. (2002) Pumping by beating on a tube.
ASME Paper IMECE2002-39035.
BELHADJ, A., CHEESEWRIGHT, R. & CLARK, c. (2000) The simulation of Coriolis
meter response to pulsating flow using a general purpose F.E. code. J, Fluids Struet. 14,
613-634.
CHEESEWRIGHT, R. & CLARK, C. (1998) The effect of flow pulsations on Coriolis
mass flow meters. J. Fluids Struet. 12, 1025-1039.
FUJITA, K., !TO, T., KOHNO, N. & NUNOKAWA, K. (1993) Study on the axia1-flow-
induced vibration of coil springs. J. Fluids Struet. 7, 689-705.
HERGÉ (1958) Les Aventures de Tintin: Coke en Stoek, pp, 44. Paris: Casterman.
JAKUBAUSKAS, v.F. & WEAVER, D.S. (1998) Transverse vibrations of bellows
expansionjoints. Parts I and II, J. Fluids Struet, 12,445-456 and 457-473.
JAKUBAUS KAS , v.F. & WEAVER, D.S. (1999) Transverse natural frequencies and
flow-induced vibrations of double bellows expansion joints. J, Fluids Struet. 13,
461-479.
TERENTIEV, A.v. (2(01) Nonlinear dynamics of two-degrees-of-freedom fluid convey-
ing elasto-visco-plastic model system. Nonl. Dyn, 26,273-295.
TREIBER, G., TSCHEUSCHNER, H.-D. & MŮLLER, H.-P. (1980) Dampfung von
Druckschwingungen von Weizenteig bei der Rohrforderung. Tei1 II. Riiker & Konditor
28,245-247.
WANG, X. & BLOOM, F. (1999) Dynamics of a submerged and inclined concentric pipe
system with internal and external flows. J. Fluids Struet. 13,443-460,
WANG, X. & BLOOM, F. (2001) Stability issues of concentric pipes containing steady
and pulsatile flows, 1. Fluids Struet, 15, 1137-1152.
n.7 CORRIGENDA TO VOLUME 1
p. 173, below eq. (3.131): K = 2n/A.
p. 200, Fig. 4.2: replace Fex by Fen.
p. 209, line 2: replace J-t by J-td.
p. 284, eq. (5.21), second line: replace y" by z".
p. 287, eqs (5.36a,b): corrected in Appendix TA.
p. 433, equations below (6.86): the integrand on the right-hand side of [1]] should be
[4>3]''' T [4>3]'''.
p. 490, second of eqs (F.33): the left-hand side should be , not 'TJ.
Readers are encouraged to communicate to the author further errata, if any; at
mary,fiorilli@mcgi11.ca and michael.paidoussis@mcgi11.ca

References t


ABKOWITZ, M.A. (1969) Stability and Motion Control oj Ocean Vehicles. Cambridge,
Massachusetts: MIT Press.
ABLOW, C.M, & SCHECHTER, S. (1983) Numerical simulation of undersea cable
dynamics. Ocean Engineering 10, 443-457,
ABRAMSON, H.N., CHU, W,H, & KANA, D,D. (1966) Some studies of nonlinear lateral
sloshing in rigid containers. Joumal oj Applied Mechanics 33, 777-784,
ABRAMOWITZ, M. & STEGUN, I. (eds) (1970) HandbookoJMathematical Functions,
9th edition, New York: Dover Pub1ications.
ADAMOVIC, D. (1999) Osci11ations of a slender cylinder in axial flow. B.Eng. Honours
Thesis, Department ofMechanical Engineering, McGi11 University, Montréal, Canada.
ADAMS, ML (2001) Rotating Machinery Vibration: Jrom Analysis to Troubleshooting.
New York: Marcel Dekker.
AITA, S, & GIBERT, R.J. (1986) Fluid-elastic instability in a flexible weir: a theoretical
model. In Flow-Induced Vibration - 1986 (eds S.S. Chen, J,c. Simonis & Y.S. Shin),
PVP-Vol. 104, pp. 51-57. New York: ASME.
AITA, S" TlGEOT, y" BERTAUT, C. & SERPANTIE, J.P. (1986) Fluid-elastic in stability
in a flexible weir: experimenta1 observations. In Flow-Induced Vibration -1986 (eds
S.S. Chen, J,C. Simonis & Y.S. Shin), PVP-Vol. 104, pp. 41-49. New York: ASME.
AITA, S" BENASSIS, T., GIBERT, R.J. & GILBAUD, D. (1987) Fluid-elastic in stability
of a flexible weir: non-linear analysis. Proceedings oj lntemational ConJerence on
Flow-/nduced Vibrations, Bowness-on-Windermere, England, pp. 317-327,
AITKEN, J. (1878) An account of some experiment s on rigidity produced by centrifugal
force. Philosophical Magazine, Series V, 5,81-105.
ALAM, N, & ASNANI, N.T. (1984) Vibration and damping analysis of a multilayered
cy1indrical shell, Part I: theoretical analysis. A/AA Joumal22, 803-810,
ALLEN, J,J, & SMITS, A.J. (2001) Energy harvesting eel. Joumal oj Fluids and
Structures 15, 629-640.
ALSTR0M, p" EGUÍLUZ, V.M. & COLDING-J0RGENSEN, M. (1999) Instability and
'sausage-string' appearance in blood vessels during high blood pressure, Physical
Review Letters 82, 1995-1999.
ALTSTADT, E" GRUNWALD, G., LIEWERS, P., SCHUMANN, P. & WEISS, F.-P.
(1991) Condition monitoring and diagnostic engineering management. In Proceedings
oj COMADEM 91, pp. 440-450. Bristol: Adam Hilger.
AMABILI, M, (1996) Effect of finite fluid depth on the hydroelastic vibrations of circular
and annular plates. Joumal oj Sound and Vibration 193, 909-925.
AMABILI, M. (1997a) Flexural vibration of cylindrical shells partially coupled with
external and internal fluids. ASME Joumal oj Vibration and Acoustics 119, 476-484.


t AU MERL Reports, e,g, Pai:doussis & Denise (1970) are available on-line via www.library.mcgi11.ca/
pseVsubguide/mecheng.htm


1503




1504
SLENDER STRUCTURES AND AXIAL FLOW
AMABILI, M. (1997b) Ritz method and substructuring in the study of vibration with
strong fluid-structure interaction. Joumal oj Fluids and Structures 5, 507-523.
AMABILI, M, (2001) Vibrations of circular plates resting on a sloshing liquid: solution
of the fulIy coupled problem. Joumal oj Sound and Vibration 245, 261-283.
AMABILI, M. (2002) Private communication (20 June 2002).
AMABILI, M. & DALPIAZ, G. (1995) Breathing vibrations of a horizontal circular cyl-
indrical tank shell, partially filIed with liquid. ASME Journal oJVibration and Acoustics
117,187-191.
AMABILI, M. & GARZIERA, R. (2000a) Vibrations of circular cylindrical shelIs with
nonuniform constraints, elastic bed and added mass. Part I: empty and fluid- filIed shelIs.
Joumal oj Fluids and Structures 14, 669-690,
AMABILI, M, & GARZIERA, R. (2000b) Misuratore di portata a vibrazione di guscio con
masse aggiunte (Vibrating shell flowmeter with added masses), Italian patent PR2000
A 000038 (pending), See also patent #01314676 granted in 2002.
AMABILI, M. & GARZIERA, R. (2002a) Vibrations of circular cy1indrical shelIs with
nonuniform constraints, elastic bed and added mass. Part II: ShelIs containing or
immersed in axial flow. Joumal oj Fluids and Structures 16, 31-51.
AMABILI, M. & GARZIERA, R. (2002b) Vibrations of circular cylindrical shelIs with
nonuniform constraints, elastic bed and added mass. Part III: steady viscous effects on
shelIs conveying fluid. Joumal oj Fluids and Structures 16, 795-809.
AMABILI, M. & KWAK, M.K. (1996) Free vibration of circular plates coupled with
liquids: revising the Lamb problem, Joumal oj Fluids and Structures 10, 743-761.
AMABILI, M. & PAIDOUSSIS, M.P. (2003) Review of studies on geometrically nonlinear
vibrations and dynamics of circular cylindrical shelIs and panels, with and without
fluid-structure interaction, Applied Mechanics Reviews 56,349-381.
AMABILI, M. & PELLlCANO, F. (2001) Nonlinear supersonic flutter of circular
cylindrical shelIs, AlAA Joumal39, 564-573,
AMABILI, M" PAIDOUSSIS, M.P. & LAKIS, A.A. (1998a) Vibrations of partially
filIed cylindrical tanks with ring-stiffeners and flexible bottom. Joumal oj Sound and
Vibration 13,259-299.
AMABILI, M., PELLlCANO, F. & PAIDOUSSIS, M.P. (1998b) Nonlinear vibrations
of simply supported circular cylindrical shelIs, coupled to quiescent fluid. Joumal oj
Fluids and Structures 12, 883-918.
AMABILI, M., PELLlCANO, F. & PAIDOUSSIS, M.P. (1999a) Nonlinear dynamics and
stability of circular cylindrical shelIs containing flowing fluid, I. Stability. Joumal oj
Sound and Vibration 225,655-699.
AMABILI, M., PELLlCANO, F. & PAIDOUSSIS, M.P. (1999b) Nonlinear dynamics
and stability of circular cylindrical shelIs containing flowing fluid. II, Large amplitude
vibrations without flow. Joumal oJSound and Vibration 228,1103-1124.
AMABILI, M., PELLlCANO, F. & PAIDOUSSIS, M.P. (1999c) Addendum to 'Nonlinear
vibrations of simply supported, circular cylindrical shelIs, coupled to quiescent fluid' .
Joumal oj Fluids and Structures 13, 785-788.
AMABILI, M., PELLlCANO, F. & PAIDOUSSIS, M.P. (2000a) Nonlinear dynamics and
stability of circular cylindrical shelIs containing flowing fluid. III. Truncation effect
without flow and experiments. Joumal oj Sound and Vibration 237, 617-640.
AMABILI, M" PELLlCANO, F. & PAIDOUSSIS, M.P. (2000b) Nonlinear dynamics
and stability of circular cy1indrical shells containing flowing fluid. IV. Large-amplitude
vibrations with flow, Joumal oj Sound and Vibration 237,641-666.

REFERENCES
1505
AMABILI, M., PELLlCANO, F. & VAKAKIS, A.F. (2000c) Nonlinear vibrations and
multiple resonances of fluid-filled circular shells, Part 1: Equations of motion and
numerical results. ASME Journal oJ Víbration and Acoustics 122, 346-354.
AMABILI, M., PELLlCANO, F. & VAKAKIS, A.F. (2000d) Nonlinear vibrations and
multip1e resonances of fluid-filled, circular shells, Part II: perturbation ana1ysis. ASME
Journal oJ Víbration andAcoustics 122, 355-364.
AMABILI, M., PELLlCANO, F. & PAlDOUSSIS, M,P. (2001) Nonlinear stability of
circular cylindrica1 shells in annular and unbounded axial flow. Journal oj Applied
Mechanics 68, 827-834.
AMABILI, M., GARZIERA, R. & NEGRI, A. (2002) Experimental study on large-
amplitude vibrations of water-filled circular cylindrica1 shells. Journal oj Fluids and
Structures 16, 213-227,
ANDERECK, C.D., LIU, S.S. & SWINNEY, H.L. (1986) The Rayleigh-Tayorinstabilíty.
Journal oj Fluid Mechanics 164, 155-183.
ANDERSON, J.M., STREITLlEN, K., BARRETT, D.S. & TRIANTAFYLLOU, M.S.
(1998) Oscillating foi1s of high propulsive efficiency. Journal oj Fluid Mechanics 360,
41-72,
ANDERSON, R. & BEJAN, A. (1983) Buckling of a turbu1entjet surrounded by a highly
flexib1e duct. Physics oj Fluids 26,3193-3200.
ANDERSON, W.J, (1973) Dynamic in stability of a cable in incompressible flow. AlAA
JournaI11,587-588,
ANDRONOV, A. A. , VITT, A.A. & KHAIKIN, S.E. (1966) Theory oj Oscillators.
New York: Dover,
ANONYMOUS (1965) Dunlop Dracones. Dunlop Rubber Co. Ltd and Dracone
Developments Ltd, Faw1ey, Hants, UK.
ANONYMOUS (1975) Browns Ferry Nuclear Plant Units 1 & 2 safety analysis report for
plant modifications to eliminate significant in-core vibrations. General Electric Report
NEDC-21091.
ANONYMOUS (1976) U.S, NRC Safety Eva1uation Report, Suppl. No. 8, July.
ANONYMOUS (1977) lťs in the bag! Hennes lnternational, Autumn 1997, p. 121.
ANONYMOUS, (1993) Flexa. Technische Gebiideaustriistung Holdinggesellschaft
GmbH, Leipsig, Germany, and Serendip BV, Arnhem, The Netherlands.
ANONYMOUS (1998a) Dunlop Dracone Barges, Manchester, UK.: Dunlop Ltd (date
arbitrary, for citation only).
ANONYMOUS (1998b) The Dracone Barge Operating Manual, lssue 3. Manchester,
UK.: Dunlop Ltd (date arbitrary).
ANONYMOUS (1998c) Aquarius Water Transportation and Storage Bags. London:
Aquarius Water Trading & Transportation Co. Ltd. (date arbitrary),
ANONYMOUS (1998d) Geco-Prakla Marine Seismic Data Acquisition. Paris: Schlum-
berger Geco-Prakla.
ANTUNES, J. (2002a) Dynamics of rotor/flow coupled system. In Flow lnduced
Vibrations (ed. P. Anagnostopoulos), Chapter 5, Southampton: WIT Press.
ANTUNES, J. (2002b) Private communication (21 Dec 2002).
ANTUNES, J., AXISA, F. & HAREUX, F. (1992) Flexural vibrations ofrotors immersed
in dense fluids. Part II: experiments. Journal oj Fluids and Structures 6,23-38,
ANTUNES, J., AXISA, F. & GRUNENWALD, T. (1996) Dynamics ofrotors immersed
in eccentric annular flow. Part 1: theory. Journal oj Fluids and Structures 10, 893-918.

1506
SLENDER STRUCTURES AND AXIAL FLOW
ANTUNES, J., MENDES, J" MOREIRA, M. & GRUNENWALD, T. (1999) A theoretica1
model for non1inear p1anar motions of rotors under fluid eonfinement. Journal o! Fluids
and Structures 13, 103-126.
ANWAR, H.O. (1967) Inflatab1e dams. ASCE Journal o! the Hydraulics Division 93,
99-119.
ARAI, M. & TAnMA, K. (1998) Leakage-flow-indueed vibrations of an axisymmetrie
body. Part 1: Ana1ysis of the moment acting on an axisymmetrie body for rotationa1
motion. JSME lnternational Journal41, 347-354,
ARAI, M. & TAJIMA, K. (1999) Leakage-flow-indueed vibrations of an axisymmetrie
body. Proceedings oJthe Asia-Pacijic Vibration ConJerence, Vol. 2, pp. 991-996.
ARAI, M., SHIOMI, H. & TAJIMA, K. (1998) Leakage-flow-indueed vibrations of an
axisymmetrie body. Part 2: Theories and experiment s on the stability of an axisymmetrie
body for one degree-of-freedom oť rotationa1 motion. JSME lnternational Journal41,
355-363.
ARMITSTEAD,J.P., BERTRAM, C,D. &JENSEN, O.E. (1996) A studyofthe bifureation
behaviour of a model of flow through a eollapsib1e tube. Bulletin o! Mathematical
Biology 58, 611-641.
ARNOLD, R.N. & WARBURTON, G.B. (1949) Flexura1 vibrations of thin eylindriea1
shells having freely supported ends. Proceedings oJthe Royal Society oJLondon A197,
238-256.
ATOBE, T. & FUNAKOSHI, M. (1994) Chaotie motion of fluid particles due to the
a1temate rotations of two eeeentrie ey1inders. Journal o! the Physical Society o! Japan
63, 1738-1753.
ATLURI, S, (1972) A perturbation ana1ysis of non-linear free flexura1 vibrations of a
cireu1ar eylindriea1 shell. lnternational Journal o! Solids and Structures 8, 549-569.
AUGU, N. (1999) Nonlinear dynamics of a eylinder subjeeted to an axia1 extema1
flow. Rapport de DEA, Département de Méeanique, Eeo1e Centra1e de Nantes and
Department of Meehaniea1 Engineering, MeGi11 University.
AU- YANG, M,K. (1976) Free vibration of fluid-eoup1ed eoaxia1 eylindrical shells of
different 1engths. Journal o! Applied Mechanics 43, 480-484.
AU-YANG, M.K. (1978) Natura1 frequencies of eylindriea1 shells and pane1s in vacuum
and in a fluid. Journal oJSound and Vibration 57,341-355.
AU- YANG, M.K. (1979) Pump-indueed aeoustic pressure distribution in an annu1ar eavity
bounded by rigid walls, Journal o! Sound and Vibration 62, 577-591.
AU-YANG, M.K. (1981) Norma1 modes of a eoup1ed fluid-shell system. In lnteract-
ive Fluid-Structural Dynamic Problems in Power Engineering (eds M.K. Au-Yang &
F.J. Moody), PVP-Vol. 46, pp. 105-118. New York: ASME.
AU-YANG, M.K. (1984) Dynamics of coup1ed fluid-shells. In Advances in Fluid-
Structure lnteractions - 1984 (eds G.c. Everstine & M,K. Au-Yang), PVP-Vol. 78,
pp. 185-203. New York: ASME.
AU- YANG, M,K. (1986a) Dynamics of coup1ed fluid-shells. ASME Journal o! Vibration,
Acoustics, Stress, and Reliability in Design 108, 339-347.
AU- YANG, M.K. (l986b) Turbu1ent buffeting of multispan tube bund1e. ASME Journal
oJVibration, Stress, and Reliability in Design 108, 150-154.
AU-YANG, M.K. (ed,) (1993) TechnologyJor the '90s. New York: ASME.
AU- YANG, M,K. (1999) Common mistakes and miseoneeptions in flow-indueed vibration
and coup1ed fluid-strneture dynamies ana1ysis. ASME, PVP Professional Deve10pment
PubIieation PVPD-29.

REFERENCES
1507
AU-YANG, M.K (2001) Flow-/nduced Vibration oj Power and Process Plant Compon-
ents. New York: ASME Press.
AU- YANG, M.K & CONNELLY, W.H. (1977) A computerized method for flow-induced
random vibration analysis of nuc1ear reactor internals. Nuclear Engineering and Design
42, 257-263.
AWREJCEWICZ, J., KRYSKO, v.A. & KRYSKO, A.V. (2002) Spatio-temporal chaos
and solitons exhibited by von Kármán model. /nternational Journal oj Bifurcation and
Chaos 12, 1465-1513.
AXELRAD, D.R. (1993) Stochastic Mechanics oj Discrete Media. Berlin: Springer-
Verlag.
AXISA, F. (1993) Flow-induced vibration of nuclear system components. In Technology
Jor the '90s (ed. M.K Au-Yang), pp. 899-956. New York: ASME.
AXISA, F. (2001) Modélisation des systemes mécaniques. Tome 4: Vibrations sous
écoulements. Paris: Hermes.
AXISA, F. & ANTUNES, J. (1992) Flexural vibrations of rotors immersed in dense fluids.
Part I: theory. Joumal oj Fluids and Structures 6, 3-21.
AXISA, F., ANTUNES, J. & VILLARD, B. (1988) Overview ofnumerical methods for
predicting flow-induced vibration. ASME Journal oj Pressure Vessel Technology 110,
6-14.
BAl, R., CHEN, K & JOSEPH, D.D. (1992) Lubricated pipelining: stability of core-
annular flow. Part 5: Experiments and comparison with theory. Joumal oj Fluid
Mechanics 240, 97-132.
BAIER, G. & GRAHAM, M.D. (1998) Two-fluid Taylor-Couette flow: experiments and
linear theory for immiscible liquids between corotating cylinders. Physics oj Fluids 10,
3045-3055.
BAIER, G. & GRAHAM, M.D. (2000) Two-fluid Taylor-Couette flow with countercurrent
axial flow: linear theory for immiscible liquids between corotating cylinders. Physics
oj Fluids 12, 294-303.
BAIRSTOW, L., RELF, E.F. & JONES, R. (1915) The stability of kite baloons:
mathematica1 investigation, Aeronautical Research Council (UK) (ARe) R&M 208.
BAKEWELL, H.P. Jr. (1964) Narrow-band investigations of the longitudinal space-time
correlation function in turbulent airflow. Journal oj the Acoustical Society oj America
36, 146-148.
BAKEWELL, H.P. Jr. (1968) Turbulent wall-pressure fluctuations on a body of revolution.
Joumal oJthe Acoustical Society oj America 43, 1358-1363.
BAKEWELL, H.P. Jr., CAREY, G.F., LIBUHA, J.J., SCHLOEMER, H.H. & VON
WINKLE, W.A. (1962) Wall pressure fluctuations in turbulent pipe flow. United States
Navy Underwater Sound Laboratory Report No. 559.
BALENDRA, T., ANG, KK., PARAMASIVAM, P. & LEE, S.L. (1982) Free vibra-
tion analysis of cylindricalliquid storage tanks. /ntemational Journal oj Mechanical
Engineering Science 24,47-59.
BALSA, T.F. (1977) Potential flow interactions in an array of cylinders in cross-flow.
Joumal oj Sound and Vibration 50, 285-303.
BARK, F.H., MEIJER, P,S. & COHEN, H.l. (1978) Spin up of a rapidly rotating gas.
Physics oj Fluids 21, 531-539.
BARTHOLOMEW, G.A. & TUNNICLIFFE, P.R. (eds) (1966) The AECL study for an
intense neutron generator - Technical details. Report AECL-2600, Atomic Energy of
Canada, Chalk River, Ontario, Canada.

1508
SLENDER STRUCTURES AND AXIAL FLOW
BASILE, D., FAURÉ, J. & OHLMER, E, (1968) Experimenta1 study of various fuel rod
model s in parallel flow. Nuclear Engineering and Design 7, 517-534.
BATCHELOR, G.K (1967) An lntroduction to Fluid Dynamics. Cambridge: Cambridge
University Press.
BATCHELOR, G.K & GILL, A.E. (1962) Analysis of the stability ofaxi-symmetric jets.
Journal oj Fluid Mechanics 14, 529-551.
BATHE, KJ. (1996) Finite Element Procedures in Engineering Analysis. Englewood
Cliffs, N,J.: Prentice-Hall.
BAUER, H,F. (1971) Hydroelastic vibrations of a unifonnly rotating infinitely long circular
cylindrical container. Acta Mechanica 12, 307-326.
BAUER, H.F. & KOMATSU, K (1994) Coupled frequencies of a hydroelastic system of
an elastic two-dimensional sector-shell and frictionless liquid in zero-gravity, Journal
oj Fluids and Structures 8,817-831.
BAUER, H.F. & KOMATSU, K (1998) Vibration of a hydroelastic system consisting of
a sector shell and viscous liquid in zero gravity. Journal oj Fluids and Structures 12,
367-385.
BEALE, S,B. (1993) Potential flow in in-line and staggered tube banks. NRC Technical
Report IME-CRE- TR -006, N ational Research Council of Canada, Ottawa, Canada.
BEALE, S.B, & SPALDING, D.B. (1999) A numerical study of unsteady fluid flow in
in-line and staggered tube banks. Journal oj Fluids and Structures 13, 723-754.
BEHDINAN, K & TABARROK, B. (1997) Dynamics of flexible sliding beams -
non-linear analysis. Part II: transient response. Journal oj Sound and Vibration 208,
541-565,
BEHDINAN, K & TABARROK, B. (1998) A finite element fonnulation for sliding beams.
Part I. lnternational JournalJor Numerical Methods in Engineering 43, 1309-1333.
BEHDINAN, K, STYLIANOU, M,C. & TABARROK, B. (1997) Dynamics of flex-
ible sliding beams - non-linear analysis, Part I: Fonnulation. Journal oj Sound and
Vibration 208, 517-539.
BEHDINAN, K, STYLIANOU, M.C. & TABARROK, B. (1998) Sliding beams. Part II:
time integration. lnternational Journal Jor Numerical Methods in Engineering 43,
1335-1363.
BEHRINGER, K, KOSÁLY, G., Kosné, Lj., MlTEFF, L., SEIFRITZ, W. &
VARADI, G. (1977) Neutron noise analysis in a Boiling Water Reactor. Eidg. Institut
ftir Reaktorforschung Wiirenlingen, Schweiz, EIR-Bericht Nr, 328,
BEJAN, A. (1982) The meandering fall ofpaper ribbons, Physics oj Fluids 25,741-742.
BEJAN, A. (1987) Buckling flows: a new frontier in Fluid Mechanics. In Annual Review
oj Numerical Fluid Mechanics and Heat TransJer (ed, T.C. Chawla), pp. 262-304.
Washington: Hemisphere.
BÉLANGER, F. (1991) A numerical method for confined unsteady flows related to fluid-
structure interactions. Ph.D. Thesis, Department of Mechanical Engineering, McGill
University, Montreal, QC, Canada.
BÉLANGER, F., DE LANGRE, E. & PORCHER, G. (1993) Linearized potential flow
theory and its application to fluidelastic forces. Unpublished work.
BÉLANGER, F., DE LANGRE, E., AXISA, F., PAIDOUSSIS, M.P. & MATEESCU, D.
(1994) Dynamics of coaxial cylinders in laminar annular flow by simultaneous integ-
ration of the Navier-Stokes and structural equations. Journal oj Fluids and Structures
8, 747-770.
BÉLANGER, F., PAlDOUSSIS, M.P. & DE LANGRE, E. (1995) Time-marching analysis
of fluid-coupled systems with large added mass. AlAA Journal 33, 752-755.

REFERENCES
1509
BELETSKlI, V, V. & LEVIN, E,M. (1993) Dynamics oj Space Tether Systems. Advances
in the Astronautical Sciences 83, San Diego: Univelt Inc., American Astronautics
Society.
BENJAMIN, T.B. (1960) Effects of a flexible boundary on hydrodynamic stability, Journal
oj Fluid Mechanics 9,513-532.
BENJAMIN, T.B. (1961) Dynamics of a system of articulated pipes conveying fluid-
I. Theory. Proceedings oJthe Royal Society A261, 457-486.
BENJAMIN, T,B. (1962) Theory of the vortex breakdown phenomenon. Journal oj Fluid
Mechanics 14, 593-629,
BENJAMIN, T.B. (1963) The threefold c1assification of disturbances in flexible surfaces
bounding inviscid flows. Journal oj Fluid Mechanics 16, 436-450.
BENSOUSSAN, A., LlONS, J.L. & PAPANICOLAOU, G. (1978) Asymptotic Analysis
oj Periodic Structures. Amsterdam: North-Holland.
BERGÉ, P., POMEAU, Y. & VIDAL, c. (1984) Order within Chaos. New York: John
Wiley & Sons.
BERGER, B.S, & PALMER, M.E. (1977) Transient motion of a cyHndrical shell of
arbitrary cross-section in an acoustic medium. Journal oj Applied Mechanics 44,
482-486.
BÉROT, F. & PESEUX, B. (1998) Vibro-acoustic behaviour of submerged cylindrical
shells: analytica! formulation and numerica! model. Journal oj Fluids and Structures
12,959-1003. Errata 14, 151.
BERRY, lG. & REISSNER, E. (1958) The effect of an internal compressible fluid column
on the breathing vibrations of a thin pressurized cylindrica! shell. Journal oj the
Aeronautical Sciences 25, 288-294.
BERS, A, (1983) Space-time evolution ofplasma instabilities - absolute and convective.
InHandbookoJPlasmaPhysics (edsM. Rosenbluth&R. Sagdeev), Vol. 1, pp,451-517.
Amsterdam: North-Holland.
BERT, C.W, (1998) Private communication (Nov, Dec 1998).
BERT, C,W. & CHEN, T.L.C. (1975a) Stability of composite-material tubing contain-
ing flowing fluid. Proceedings oj 12th Annual Meeting oj the Society oj Engineering
Science, pp. 809-818.
BERT, C.W, & CHEN, T.L.C. (1975b) Wave propagation in fluid-conveying piping con-
structed of composite material. ASME Journal oj Pressure Vessel Technology 97,
178-184,
BERT, C,W, & EGLE, D.M. (1969) Dynamics of composite, sandwich, and stiffened
shell-type structures, Journal oj Spacecraft and Rockets 6, 1345-1361,
BERTRAM, C.D. (1987) The effects ofwal1 thickness, axial strain and end proximity on
the pressure-area relation of collapsible tubes. Journal oj Biomechanics 20, 863-876.
BERTRAM, C.D. (1995a) The dynamics of collapsible tubes. In Biological Fluid Dynam-
ics (eds C.P. El1ington & T.J. Pedley), pp, 253-264, Cambridge (UK): The Company
of Biologists Ltd.
BERTRAM, C.D. (1995b) Dynamical system analyses of aperiodic flow-induced osci1la-
tions of a collapsible tube. Journal de Physique 5,2101-2116.
BERTRAM, C.D, (1999) Private communication (12 Feb 1999).
BERTRAM, C.D. (2000a) Simp1e resistive model s for steady flow through uniform and
tapered-stiffness collapsible tubes. Journal oj Fluids and Structures 14, 593-612.
BERTRAM, C,D. (2000b/2001) Private communications (15 Feb 2000, 6 Aug 2(01).
BERTRAM, C.D, & CASTLES, R.J. (1999) Flow limitation in uniform thick-wal1ed
collapsible tubes. Journal oj Fluids and Structures 13, 399-418.

1510
SLENDER STRUCTURES AND AXIAL FLOW
BERTRAM, CD. & CHEN, W. (2000) Aqueous flow limitation in a tapered stiffness
col1apsible tube, Journal oj Fluids and Structures 14, 1195-1214.
BERTRAM, CD. & ELLlOTT, N.SJ. (2003) Flow-rate limitation in a uniform thin-wa11ed
col1apsib1e tube, with comparison to a uniform thick - wa11ed tube and a tube of tapering
thickness. Journal oj Fluids and Structures 17, 541-559,
BERTRAM, C.D. & PEDLEY, T.J, (1982) A mathematica1 model ofunsteady collapsible
tube behaviour. Journal oj Biomechanics 15, 39-50.
BERTRAM, C.D. & SHE, J. (2000) Chaotic and mode-locked interactions between flow-
induced collapsible-tube osci11ation with pulsati1e upstream forcing. In Stochastic
and Chaotic Dynamics in Lakes (eds D.S. Broomhead, E.A. Luchinskaya, P.V.E.
McClintock & T, Mullin), pp, 510-516. Melvi1le, NY: American Institute ofPhysics.
BERTRAM, C.D., RAYMOND, C.J. & PEDLEY, T,J. (1990) Mapping of instabilities for
flow through collapsed tubes of differing length. Journal oj Fluids and Structures 4,
125-153.
BERTRAM, CD., RAYMOND, C.J. & PEDLEY, T.J. (1991) Application of nonlinear
dynamics concepts to the ana1ysis of self-excited osci11ations of a collapsib1e tube
conveying fluid. Journal oj Fluids and Structures 5, 391-426. ,
BERTRAM, CD., SHEPPEARD, M.D. & JENSEN, O.E. (1994) Prediction and meas-
urement of the area-distance profile of collapsed tubes during self-excited osci11ation.
Journal oj Fluids and Structures 8, 637-660.
BESANc;ON, P. (1978) Dynamics of a c1uster of pipes conveying fluid in bounded axia1
flow. M.Eng. Thesis, Department of Mechanical Engineering, McGi11 University,
Montréa1, QC, Canada.
BHATTACHARYYA, S.K., VENDHAN, CP. & SUDARSAN, K. (2000) The finite ele-
ment method for hydroelastic instability of underwater towed cylindrical structures.
Journal oj Sound and Vibration 237, 119-143.
BHAVE, S.K., ACHAREKAR, C.L., MURTHY, C.B.N. & GOYAL, S.K. (1987)
Vibrations in penstocks. Water Powerand Dam Construction 39, 41-43.
BICOS, A.S, & SPRlNGER, G.S. (1989) Ana1ysis offree damped vibration of1aminated
composite plates and shells. lnternational Journal oJSolids and Structures 25, 129-149.
BISHOP, RE.D. & FAWZY, I. (1976) Free and forced oscillation of a vertical tube con-
taining a flowing fluid. Philosophical Transactions oJthe Royal Society (London) 284,
1-47.
BISHOP, RE.D, & JOHNSON, D,C, (1960) The Mechanics oj Vibration. Cambridge:
Cambridge University Press,
BISHOP, RE.D. & PRICE, W.G. (1979) Hydrodynamics oJShips. Cambridge: Cambridge
University Press,
BISPLlNGHOFF, RL. & ASHLEY, H. (1962) Princip les oj Aeroelasticity. New York:
John WHey & Sons.
BISPLlNGHOFF, RL., ASHLEY, H. & HALFMAN, RL. (1955) Aeroelasticity.
Cambridge, MA: Addison-Wesley; also 1996, New York: Dover,
BLACK, H.F. (1969) Effects ofhydraulic forces on annular pressure seals in the vibrations
of centrifugal pump rotors. IMechE Journal oj Mechanical Engineering Science 11,
206-213.
BLAKE, RW. (1983) Fish Locomotion. Cambridge: Cambridge University Press.
BLAKE, R & LlGHTHILL, J, (1990) Biofluiddynamics of ba1istiform and gymnotiform
locomotion. Part 1, Journal oj Fluid Mechanics 212, 183-207.

REFERENCES
1511
BLAKE, W.L. (1986) Mechanics oj Flow-Induced Sound and Vibration, Vo1ume II.
London: Academic Press.
BLAND, S,R, RHYNE, RH. & PIERCE, H.B. (1967) Study of flow-induced vibrations
of a p1ate in narrow channe1s. ASME Journal oj Engineering Jor lndustry 89,824-830,
BLEVINS, R,D, (1979) Formulas oj Natural Frequency and Mode Shape. New York: Van
Nostrand Reinho1d.
BLEVINS, RD, (1979) Aow-induced vibration in nuc1ear reactors: a review. Progress in
Nuclear Energy 4,24-49.
BLEVINS, R.D. (1984) Applied Fluid Dynamic Handbook, New York: Van Nostrand
Reinho1d.
BLEVINS, RD. (1990) Flow-lnduced Vibration, 2nd edition. New York: Van Nostrand
Reinhold,
BOHM, G,J, (1965) Natura! vibration of reactor intema1s. Nuclear Science and Engin-
eering 22, 143-152,
BOLOTIN, V.v. (1956) Vibrations and stability of an e1astic cy1indrica1 shell in supersonic
flow (in Russian). lnzhenernyi Sbornik 24, 3-46.
BOLOTIN, V.v. (1963) Nonconservative Problems oj the Theory oj Elastic Stability.
London: Pergamon.
BOLOTlN, V.V. (1964) The Dynamic Stability oj Elastic Systems. San Francisco:
Holden Day.
BOLOTIN, V.v. (1979) Random Vibrations oj Elastic Systems (in Russian), Moscow:
Nauka.
BOLOTIN, V.v., GRlSHKO, A.A., KOUNADIS, A.N. & GANTES, C,J. (1998) Non-
1inear panel flutter in remote post -critica1 domains, lnternational Journal oj Non-Linear
Mechanics 33, 753-764,
BONIS, M. (1979) Ecoulement visqueux permanent dans un tube collabable el1iptique.
These de doctora1 d'État, Université de Technologie de Compiegne, France.
BONIS, M. & RffiREAU, C. (1978) Étude de que1ques propriétés de l'écou1ement dans
une conduite collabab1e. La Houille Blanche 33,165-173,
BONIS, M, & RffiREAU, C. (1981) Wave speed in noncircu1ar collapsib1e ducts. ASME
Journal oj Biomechanical Engineering 103, 27-31.
BOTEZ, RM, & PAIDOUSSIS, M.P. (1996) Dynamics of an articu1ated cylinder in annu-
lar flow via non-linear and partially 1inearized versions of an ana1ytica1 model. Journal
oj Sound and Vibration 192, 645-668.
BOTMAN, M. (1965) Propu1sion by undulating p1ates, Journal oj Aircraft 2,456-462,
BOWERS, G. & HORVAY, G, (1975) Forced vibrations of a shel1 inside a narrow water
annu1us. Nuclear Engineering and Design 34,221-231.
BOYARSHlNA, L.G, (1984) Resonance effects in the nonlinear vibrations of cylindrica1
shells containing a liquid. Soviet Applied Mechanics 20,765-770.
BOYARSHlNA, L.G. (1988) Nonlinear wave modes of an elastic cylindrica1 shel1 par-
tially filled with a liquid under conditions of resonance. Soviet Applied Mechanics 24,
528-534,
BRAZIER-SMITH, P.R & SCOTT, J.F. (1984) Stability of fluid flow in the presence of
a compliant surface, Wave Motion 6, 547-560,
BREE, J" HALLIWELL, A.R. & O'DONOGHUE, T. (1989) Snaking offloating marine
oi1 hose attached to SPM buoy. ASCE Journal oj Engineering Mechanics 115, 265-284.
BRENNEN, C, (1974) An osci11atíng-boundary-1ayer theory for ciliary propu1sion.
Journal oj Fluid Mechanics 65, 799-824,

1512
SLENDER STRUCTURES AND AXIAL FLOW
BRENNEN, C. & WINET, C. (1977) Fluid mechanics of propulsion by cilia and flagella.
Annual Review oj Fluid Mechanics 9, 339-398.
BREPSON, R. & LÉON, P. (1974) Vibrations induced by von Kánnán vortex trai1 in guide
vane bends. In Flow-lnduced Structural Vibrations (ed, E. Naudascher), pp, 318-332.
Berlin: Springer- Verlag.
BRIGHTON, J.A. & JONES, J.B. (1964) Fully deve10ped turbu1ent flow in annuli. ASME
Journal oj Basic Engineering 86, 835-844.
BROCHARD, J., COMBESCURE, A., QUEVAL, J.c., AILLAUD, P. & BARTHELET, B.
(1987) Dynamic buck1ing offluid-coupled cylinders, Transactions 8th SMiRT, Brusse1s,
pp. 243-249.
BROWER, R.W. (1970) Pressure-flow characteristics of collapsib1e tubes. Ph.D. Thesis,
University of Pennsy1vania, Phi1ade1phia, PA, USA.
BROWER, R.w. & NOORDEGRAAF, A. (1973) Pressure-flow characteristics of co1-
1apsib1e tubes: a reconci1iation of seenůng1y contradictory results. Annals oj Biomedical
Engineering 1, 333-355.
BROWN, MJ. & ELLlOTT, L. (1988) Two-dimensiona1 dynamic ana1ysis of a floating
hose string. Applied Ocean Research 10, 20-34.
BROWN, R.D. & LEONG, Y.M.M.S. (1984) Experimenta1 measurement of1atera1 force in
a mode11abyrinth and the effect on rotor stability. Proceedings ConJerence on Vibrations
in Rotating Machinery, pp. 215-222, Paper C296/84. London: IMechE.
BROWN, S.J. (1982) Hydrodynamic response of fluid coup1ed cy1inders: simp1ified
damping and inertia coefficients. Welding Research Council Bulletin; Bulletin 281.
BROWN, S.J. & CHU, M. (1983) A case study of the seisnůc response of fluid-coup1ed-
flexib1e cylinders. Experimental Mechanics 23, 270-281.
BRUSILOVSKII, A.D., MEL'NIKOVA, L.M. & SHVEIKO, lu. lu. (1966) Vibrations and
stability of a cy1indrica1 shell in a gas flow. lnzhenernyi Zhurnal, Mekhanika Tverdogo
Tela, No, 1,67-73.
BRUSH, D,O, & ALMROTH, B.O. (1975) Buckling oJBars, Plates and Shel/s. New York:
McGraw Hi11.
BRYAN, G,H. (1890) On the beats in the vibrations of a revo1ving cylinder or bell.
Proceedings oJthe Cambridge Philosophical Society 7, 101-111.
BRYANT, L.W., BROWN, W.S. & SWEETlNG, N.E. (1942) Collected research on
stability of kites and towed gliders. Aeronautica1 Research Council (UK) (ARe)
R&M 2303.
BSCHORR, O. (1988) Propu1sive mechanisms in animal 10comotion. Aeronautical
Journal 92, 84-90.
BUCKLEY, TJ., MUGGERIDGE, D.B. & DONG, C.W.N. (1976) Dynamics of fluid-
conveying fiber-reinforced pipe. Research Report, Faculty ofEngineering and AppHed
Science, Memorial University of Newfound1and, St. John's, Newfound1and, Canada.
BUDIANSKY, B. & SANDERS, J.L. Jr. (1963) On the 'besť first order 1inear shell
theory. In Progress in Applied Mechanics - The Prager Anniversary Volume, Vol. 192,
pp. 129-140. Greenwich, CT: MacMillan, Inc.
BUGGlSCH, H.W. (1982) Nonlinear waves in gas-filled elastic tubes. In Proceedings
lUTAM Symposium on Nonlinear DeJonnation Waves (eds U. Nigul & J. Engelbrecht),
Ta1inn, pp. 101-114.
mrtvoL, V.N. (1975) Kolebaniia Ustoi'chivost DeJonniruemuykh System v Zhidkosti
(Vibration and Stability ofFlexib1e Systems in Flow). Kiev: Naukova Dumka.

REFERENCE S
1513
BULL, M.K. & NORTON, M.P. (1980) The proximity of coincidence and acoustic cut-
off frequencies in relation to acoustic radiation from pipes with distributed internal
turbulent flow. Journal oj Sound and Vibration 69, 1-11.
BURDESS, J.S. (1986) The dynamics of a thin piezoelectric cylinder gyroscope.
Proceedings oj IMechE 200(C4), 271-281.
BURGREEN, D., BYRNES,J.J. &BENFORADO,D,M, (1958) Vibration ofrods induced
by water in para!lel flow, Transactions ASME 80,991-1003.
CALKlNS, D.E. (1979) Hydrodynamic analysis of a high-speed marine towed system.
JournaloJHydronautics 13, 10-19.
CANcELLI, C. & PEDLEY, T.J. (1985) A separated-flow model for collapsible-tube
oscillations, Journal oj Fluid Mechanics 157, 375-404.
CANUTO, c., HUSSAINI, M.Y., QUARTERONI, A & ZHANG, T.A (1988) Spectral
Methods in Fluid Dynamics. Berlin: Springer- Verlag.
CAPODANNO, P. (1995) Mathematical study ofthe small oscillations of a liquid co1umn
in a symmetrical container under zero gravity, Journal oj Fluids and Structures 9,
773-786.
CARLlNG, J., WILLlAMS, T.L. & BOWTELL, G, (1998) Self-propelled anguil1iform
swimming: simultaneous solution ofthe two-dimensiona! Navier-Stokes equations and
Newton's laws of motion, The Journal oj Experimental Biology 201, 3143-3166.
CARLUCCI, L.N. (1977) Hydrodynamic mass and fluid damping of rod bundles vibrat-
ing in confined water and air-water mixtures. Transactions oj 4th lnternational
ConJerence on Structural M echanics in Reactor Technology (SM iRT), Paper No. D3/11,
San Francisco, CA, USA
CARLUCCI, L.N, (1980) Damping and hydrodynamic mass of a cylinder in simulated
two-phase flow. ASME Journal oj Mechanical Design 102, 597-602.
CARLUCCI, L.N, & BROWN, J.D. (1983) Experimenta! studies of damping and hydro-
dynamic mass of a cylinder in confined two-phase flow. ASME Journal oj Vibration,
Stress, and Reliability in Design 105, 83-89.
CARPENTER, P.W, (1984) The effect of a boundary 1ayer on the hydroe1astic in stability
of infinitely long p1ates, Journal oJSound and Vibration 93, 461-464.
CARPENTER, P.W, & GARRAD, AD. (1985) The hydrodynamic stability offlows over
Kramer-type compliant surfaces, Part 1: Tollmien-Schlichting instabi1ities. Journal oj
Fluid Mechanics 155, 465-510.
CARPENTER, P.w. & GARRAD, AD. (1986) The hydrodynamic stability offlows over
Kramer-type compliant surfaces, Part 2: flow-induced surface instabilities, Journal oj
Fluid Mechanics 170, 199-232.
CARPENTER, P.W, & MORRIS, P,J. (1990) The effect of anisotropic wall compliance
on boundary layer stability and transition. Journal oj Fluid Mechanics 218, 171-223.
CARPENTER, P.w., DAVIES, C. & LUCEY, AD, (2000) Hydrodynamic and compliant
walls: does the dolphin have a secret? Current Science 79, 758-765.
CARPENTER, P.W., LUCEY, A.D. & DAVIES, C. (2001) Progress on the use of compliant
",,,11,, for l:Jmif1ar-flow control. Journal oj AircraJt 38,504-512.
CASARELLA, M.J, & PARSONS, M. (1970) Cable systems under hydrodynamic loading.
Marine Technology Society Journal 4, 27-44.
CEDOLlN, L., HASSID, A, ROSSINI, T. & SOLIERI, R. (1971) Vibrations induced by
two-phase (gas + liquid) coolant flow in the power channels of a pressure tube type
nuc1ear reactor. Proceedings 1st lnternational ConJerence on Structural Mechanics in
Reactor Technology (SMiRT), Paper E4/5, Berlin, Germany.

1514
SLENDER STRUCTURES AND AXIAL FLOW
CESARI, F. & CURIONI, S. (1971) Buckling instability in tubes subject to interna! and
externa! axia1 fluid flow. In Proceedings 4th ConJerence on Dimensioning, pp. 301-311.
Budapest: Hungarian Academy of Science,
CHAlKEN, J., CHU, C.K., TABOR, M. & TAN, Q.M. (1987) Lagrangian turbulence and
spatia1 complexity in a Stokes flow. The Physics oj Fluids 30,687-694,
CHAKRABORTY, G., MALLlK, A.K. & HATWAL, H. (1999) Non-1inear vibration of
trave1ling beam. 1nternational Journal oj Non-Linear Mechanics 34, 655-670.
CHAN, S.P. (1984) Dynamics and stability of two coaxia! cy1indrica! shells convey-
ing fluid. M.Eng. Thesis, Department of Mechanica1 Engineering, McGill University,
Montrea1, Québec, Canada.
CHANDIRAMANI, N.K., PLAUT, R.H. & LlBRESCU, L.L (1995) Non-1inear flutter of
a buckled shear-deformable composite panel in a high-supersonic flow. 1nternational
Journal oJNon-Linear Mechanics 30, 149-167.
CHANG, W.K., PILlPCHUK, V, & IBRAHIM, R.A. (1997) Fluid flow-induced nonlinear
vibration of suspended cables. Nonlinear Dynamics 14, 377-406.
CHANG, Y.B. & MORETTI, P.M, (1991) Interaction offluttering webs with surrounding
air. TAPPI Journal74(3), 231-236.
CHANG, Y.B. & MORETTI, P.M. (2000) Flow-induced vibration of free edges of thin
films, ln Flow 1nduced Vibration (eds S. Ziada & T. Staubli), pp. 801-810. Rotterdam:
Ba1kema.
CHANG, Y.B. & MORETTI, P.M. (2002) Flow-induced vibration of free edges of thin
fi1ms. Journal oj Fluids and Structures 16, 989-1008,
CHANG, Y.B., CHO, C.H. & MORETTI, P.M. (1999) Edge flutter. In Proceedings ASME
Noise Control and Acoustics Division - 1999, NCA- Vol. 26, pp. 413-423. New York:
ASME.
CHANG, Y.B., FOX, SJ., LlLLEY, D.G. & MORETTI, P.M. (1991) Aerodynamics of
moving be1ts, tapes, and webs. In Machinery Dynamics and Element Vibrations, DE-
Vol. 36, pp. 33-40. New York: ASME.
CHAPNIK, B. & CURRlE, LG. (2000) Noise reduction using finite-length flexible
segments. ASME Journal oj Vibration and Acoustics 122, 94-108.
CHARRIER, J.-M. & HANNOYER, M. (1974) Inflated structures: air-springs and floats,
1nternational Journal oj Mechanical Science 16, 735-756.
CHEN, J.C. & BABCOCK, C.D. (1975) Nonlinear vibration of cy1indrica! shells. A1AA
Journal13,868-876.
CHEN, S,H. & CHEN, S.S. (1996) Chaotic vibration in fluid-stiffness-controlled
instabi1ity of a tube row in crossflow.Journal oj Applied Mechanics 63, 487-492.
CHEN, S.S, (1973) ln-plane vibration of continuous curved beams. Nuclear Engineering
and Design 25, 413-431.
CHEN, S.S. (1974) Para!le1 flow-induced vibrations and instabi1ities of cylindrica!
structures. Shock and Vibration Digest 6( 1 O), 1-11.
CHEN, S,S. (1975a) Dynamic responses of two para!le1 circu1ar cylinders in a 1iquid.
ASME Journal oj Pressure Vessel Technology 97, 78-83.
CHEN, S.S. (1975b) Vibrations of a row of circu1ar cylinders in a 1iquid. ASME Journal
oj Engineering Jor 1ndustry 97, 1212-1218. ,
CHEN, S.S, (1975c) Vibration of nuc1ear fue1 bundles. Nuclear Engineering and Design
35,399-422.
CHEN, S.S. (1977a) Dynamics of heat exchanger tube banks. ASME Journal oj Fluids
Engineering 99, 462-469.

REFERENCES
1515
CHEN, S,S, (1977b) Flow-induced vibration of circular cylindrical structures. Part I:
Stationary fluids and parallel flow. Shock and Vibration Digest 9(10),21-27,
CHEN, S.S. (1978) Crossflow-induced vibrations of heat exchanger tube banks. Nuclear
Engineering and Design 47,67-86.
CHEN, S.S. (1981) Fluid damping for circular cylindrica1 structures, Nuclear Engineering
and Design 63,81-100.
CHEN, S.S. (1987) Flow-lnduced Vibration oj Circular Structures, Washington: Hemi-
sphere.
CHEN, S.S. & JENDRZEJCZYK, J.A. (1978) Experiments on fluidelastic vibration of
cantilevered tube bundles. ASME Journal oj Mechanical Design 100, 540-548,
CHEN, S,S, & ROSENBERG, G.S. (1974) Free vibrations of fluid-conveying cylindrical
shells, ASME Journal oj Engineering Jor lndustry 96, 420-426.
CHEN, S.S, & ROSENBERG, G.S. (1975) Dynamics of a coupled shell-fluid system.
Nuclear Engineering and Design 32, 302-310.
CHEN, S.S. & WAMBSGANSS, M.W, (1970) Response of a flexible rod to near-field
flow noise, ln Proceedings Conference on Flow-lnduced Vibrations in Reactor System
Components. Argonne National Laboratory Report ANL-7685, pp. 5-31,
CHEN, S.S. & WAMBSGANSS, M.W. (1972) Para11el-flow induced vibration of fuel
rods. Nuclear Engineering and Design 18, 253-278.
CHEN, S,S., WAMBSGANSS, M,W, & JENDRZEJCZYK, J,A. (1976) Added mass and
damping of a vibrating rod in confined viscous fluids. Journal oj Applied Mechanics
43,325-329,
CHEN, S,S" JENDRZEJCZYK, J.A. & WAMBSGANSS, M.W. (1985) Dynamics of
tubes in fluid with tube-baffle interaction, ASME Journal oj Pressure Vessel Technology
107,7-17,
CHEN, T,L.C. & BERT, C.W, (1977a) Dynamic stability of isotropic or composite-
material cylindrical shells containing swirling fluid flow. Journal oj Applied Mechanics
44, 112-116; errata: 44, 513.
CHEN, T.L.C. & BERT, C.W. (1977b) Wave propagation in isotropic- or composite-
material piping conveying swirling liquid. Nuclear Engineering and Design 42,
247-255.
CHEN, Y.N. (1970a) Turbulence-induced instability of fuel rods in parallel flow. Sulzer
Technical Review, Research number, 1970.
CHEN, Y,N, (1970b) Flow-induced vibrations in tube bundle heat exchangers with cross
and para1lel flow. Part 1: parallel flow. In Flow-lnduced Vibration in Heat Exchangers
(ed. D,D. Reift), pp. 57-66. New York: ASME.
CHENG, E. & DOWELL, E.H. (1979) A theoretical analysis of nonlinear effects on
the flutter and divergence of a tube conveying fluid. Flow-lnduced Vibrations (eds
S.S. Chen and M.D. Bemstein), pp. 65-81. New York: ASME.
CHENG,J.-Y.,PEDLEY, T.J. &ALTRINGHAM,J,D. (1998)Acontinuousdynamic beam
model for swimming fish. Philosophical Transactions oj the Royal Society (London)
B353,981-997.
CHEUNG, K. & ZHOU, D. (2002) Hydroelastic vibration of a circular container bottom
plate using the Ga1erkin method. Journal oj Fluids and Structures 16, 561-580.
CHIBA, M. (1992) Non1inear hydroelastic vibration of a cylindrical tank with an elastic
bottom, containing liquid. Part I: experiment. Journal oj Fluids and Structures 6,
181-206,

1516
SLENDER STRUCTURES AND AXIAL FLOW
CHIBA, M, (1993a) Non-1inear hydroe1astie vibration of a eanti1ever eylindriea! tank. I
Experiment (empty ease). lnternational Journal oJNon-Linear Mechanics 28,591-599.
CHIBA, M. (1993b) Non-1inear hydroe1astie vibration of a eanti1ever eylindrica! tank. II
Experiment (liquid-filled ease). lnternational Journal oj Non-Linear Mechanics 28,
601-612.
CHIBA, M, (1993e) Experimenta! studies on a non1inear hydroe1astie vibration of a
c1amped eylindriea! tank partia!ly filled with liquid. ASME Journal oj Pressure Vessel
Technology 115, 381-388.
CHIBA, M, (1993d) Non1inear hydroe1astie vibration of a ey1indriea! tank with an e1astie
bottom, eontaining 1iquid. Part II: linear axisymmetrie vibration ana1ysis. Journal oj
Fluids and Structures 7,57-73.
CHIBA, M. (1996) Nonlinear hydroe1astie vibration of a ey1indriea1 tank with an e1astie
bottom, eontaining 1iquid, Part III: non-1inear ana!ysis with Ritz averaging method.
lnternational Journal oj Non-Linear Mechanics 31, 155-165.
CHIBA, M, & ABE, K. (1999) Nonlinear hydroe1astie vibration of a ey1indriea! tank
with an e1astie bottom eontaining 1iquid - ana!ysis using harmonie ba!anee method.
Thin- Walled Structures 34, 233-260.
CHIBA, T. & KOBAYASHI, N. (1983) Comparison of experimenta! and theoretiea! vibra-
tion results for fluid-eoup1ed eo-axia! ey1inder. Proceedings 7th SMiRT, Chieago, Paper
B8/9, pp. 507-514. Amsterdam: North Holland.
CHlLDS, D, (1993) Turbomachinery Rotordynamics Phenomena, Modeling andAnalysis.
New York: John Wi1ey & Sons,
CHlLUKURI, R. (1987) Added mas s and damping for ey1inder vibrations within a eon-
fined fluid using deforming finite e1ements. ASME Journal oj Fluids Engineering 109,
283-288,
CHOMAZ, J.M., HUERRE, P. & REDEKOPP, L.G. (1988) Bifureations to loea! and
globa! modes in spatia11y deve10ping flows. Physical Review Letters 60, 25-28.
CHOO, Y,I. & CASARELLA, M.J. (1972) Configuration of a tow1ine attaehed to a vehic1e
moving in a eireu1ar path, Journal oJHydronautics 6,51-57.
CHOO, Y.I. & CASARELLA, M.J. (1973) A survey of ana!ytiea! methods for dynamie
simu1ation of eab1e- body systems. Journal oj Hydronautics 7, 137-144.
CHOPRA, M.G. (1974) Hydromeehanies of1unate tai1 swimming propu1sion, Journal oj
Fluid Mechanics 64,375-391.
CHOPRA, M,G. (1976) Large amplitude 1unate tai1 theory of fish loeomotion. Journal oj
Fluid Mechanics 74, 161-182.
CHOPRA, M,G, (1977) Hydromeehanies onunate tai1 swimming. Part 2. Journal oJ Fluid
Mechanics 79, 49-69.
CHORIN, A.J. (1973) Numeriea1 study of slightly viseous flow, Journal oj Fluid
Mechanics 57, 785-796,
CHRlSTOPHERSON, D,G, &NAYLOR, H. (1955) Promotionoffluid1ubrieationin wire
drawing. Proceedings IMechE 169, 643-653,
CHU, H.-N. (1961) Influenee of1arge amplitudes on flexura1 vibrations of a thin eireu1ar
ey1indrieal she1l. Journal oj Aerospace Science 28,602-609.
CHU, M., BROWN, S., LIEB, B. & LESTlGI, 1. (1979) An experimental study of gap
and thiekness influenee on the vibration response and damping of flexible fluid-eoup1ed
eoaxial eylinders. In Dynamics oj Fluid-Structure Systems in the Energy lndustry (eds
M.K. Au-Yang & S. Brown), PYP-Yol. 39, pp. 1-19. New York: ASME.

REFERENCES
1517
CHU, M,K, & BROWN, S, (1981) Experiments on the dynamic behavioroffluid-coupled
concentric cyJinders. Experimental Mechanics 21, 129-137.
CHU, W. & KANA, D. (1967) A theory for non1inear transverse vibrations of a partially
filled e1astic tank. AlAA Journal5, 1828-1835.
CHUNG, H, & CHEN, S,S. (1977) Vibration of a group of circu1ar cy1inders in confined
fluid, Journal oj Applied Mechanics 44,213-217.
CHUNG, H., TURULA, P., MULCAHY, T.M. & JENRZEJCYK, J,A. (1981) Ana!ysis
of a cylindrlcal she1l vibrating in a cylindrical fluid region. Nuclear Engineering and
Design 63, 109-120,
CHWANG, A.T. & WU, T.y. (1974) Hydromechanics of low-Reynolds-number flow,
Part 1, Rotation ofaxisymmetric pro1ate bodies. Journal oj Fluid Mechanics 63,
607-622,
CHWANG, A.T., WINET, H. & WU, T.Y. (1974) A theoretica! mechanism of spirochete
locomotion, Journal oj Mechanochemical Cell Motility 3,69-76.
CIROVIC, S., WALSH, C. & FRASER, W,D, (2002) Wave propagation in a system of
coaxia! tubes filled with incompressible media: a model of pulse transmission in the
intercrania1 arteries. Journal oj Fluids and Structures 16, 1029-1049.
CLASEN, p, (1972) Die 'hydrodynamische Masse' eines Rohrs in einem Rohrbundel.
Forschung im lngenieurwesen 38, 33-37.
CLlNCH, J.M. (1967) Study ofvibrations induced in thin-walled pipes under varying flow
conditions. NASA Report CR-86659,
CLlNCH, J ,M, (1969) Measurements of the wall pressure field at the surface of a smooth-
wa!1ed pipecontaining turbulent waterflow. Journal oJSound and Vibration 9,398-420.
CLINCH, J,M. (1970) Prediction and measurement of vibrations induced in thin-wa11ed
pipes by the passage of interna! turbulent water flow. Journal oj Sound and Vibration
12, 429-451,
COLES, D.E. (1965) Transition in circular Couette flow. Journal oj Fluid Mechanics 21,
385-425,
COLLlER, J.G. (1981) Convection, Boiling, and Condensation, 2nd edition. New York:
McGraw-Hi11.
CONCA, c., PLANCHARD, J. & VANNINATHAN, M. (1990) Limits of the reson-
ance spectrum of tube arrays immersed in a fluid. Journal oj Fluids and Structures 4,
541-558; corrigenda 1991, 5, 601.
CONCA, J" PLANCHARD, J., THOMAS, B. & V ANNINATHAN, M. (1994) Problemes
Mathématiques en Couplage Fluide-Structure. Applications aux Faisceaux Tubulaires.
Paris: Eyrolles.
CONCA, 1., PLANCHARD, J. & VANNINATHAN, M. (1995) Fluids and Periodic
Structures. Chistester: John Wiley and Paris: Masson.
CONRAD, W.A. (1969) Pressure-flow relationships in collapsible tubes. IEEE Transac-
tions oj Biomedical Engineering 16, 284-295.
CONRAD, W.A., COHEN, M.L. & McQUEEN, D.M. (1978) Note on the oscillations of
collapsible tubes. Medical & Biological Engineering & Computing 16, 211-214.
CORCOS, G,M. (1963) Resolution of pressure in turbulence, Journal oj the Acoustical
Society oj America 35, 192-199.
CORCOS, G.M, & LlEPMANN, H.W. (1956) On the contribution of turbulent boundary
1ayers to the noise inside a fuselage. NACA Report TM-1420.
CORKE, T,C. & KUSEK, S,M, (1993) Resonance in axisymmetric jets with controlled
he1ica1-mode input. Journal oj Fluid Mechanics 249, 307-336,

1518
SLENDER STRUCTURES AND AXIAL FLOW
CORR, J,E, (1970) Big Rock Point vibration analysis. In Proceedings oj the ConJer-
ence on Flow-Induced Vibrations in Reactor System Components (eds G.S. Rosenberg,
M.w' Wambsganss & RP. Carter), ANL-7686, pp. 272-289, Argonne National
Laboratory, Argonne, IL, USA.
CORTELEZZI, L., PONG, A. & PAlDOUSSIS, M.P. (2003) F1utter of rotating she11s with
a co-rotating axia! flow. Joumal oj Applied Mechanics (in press).
COTTlS, M,G, (1968) On the dynamic response of an orthotropic finite shell to an arbitrary
pressure field. Joumal oj Sound and Vibration 7, 31-38.
COTTIS, M.G. & JASONIDES, J,G. (1964) The response of a finite thin cylindrica! she11
to random pressure fie1ds. In Proceedings 2nd lntemational ConJerence on Acoustical
Fatigue in Aerospace Structures, USAF Materia!s Laboratory, Dayton, OH, USA,
pp. 185-212.
COULOMB, D, (2000) Nonlinear dynamics of a pinned-pinned cylinder subjected to
an axial extema! flow. Unpublished report, McGi11 University and École Centra!e de
Nantes.
COWLEY, S.J. (1982) Elastic jumps on fluid-filled elastic tubes. Joumal oj Fluid
Mechanics 116, 459-473,
COWLEY, S.J. (1983) On the wavetrains associated with e1astic jumps on fluid-filled
e1astic tubes. Quarterly Joumal oj Mechanics and Applied Mathematics 36, 289-312,
CRANDALL, S.H. & MARK, w'D. (1963) Random Vibration in Mechanical Systems.
New York: Academic Press.
CRIGHTON, D.G. & GASTER, M. (1976) Stability of slowly diverging jet flow. Joumal
oJ Fluid Mechanics 77,397-413.
CRlGHTON, D.G. & OSWELL, J.E. (1991) Fluid 10ading with mean flow, L Response of
an e1astic plate to 10ca!ized excitation, Philosophical Transactions oj the Royal Society
oj London A335, 557-592.
CUMMINGS, G.E. & BRANDT, H. (1979) Dynamic response of a cylindrica! she11 in a
potentia! fluid, Journal oj Applied Mechanics 46, 772-778,
CURLlNG, LI. R (1982) Theory of vibration of c1usters of cylinders in axia! flow, M.Eng.
Thesis, DepartmentofMechanica! Engineering, McGi11 University, Montrea!, Québec,
Canada,
CURLING, LI. R (1990) Measurements of wa!1 pressure fluctuations on cylinders in a
bundle in turbulent axia! flow. Ph,D, Thesis, Department of Mechanical Engineering,
McGi11 University, Montreal, Québec, Canada.
CURLING, LI. R (2001) Private communications (19 Feb and 12Ju12001; 10 Jan 2003).
CURLING, L.R & PAlDOUSSIS, M.P, (1992a) Measurements and characterization of
wa!l-pressure fluctuations on cylinders in a bundle in turbu1entaxia! flow. Part 1: spectra!
characteristics. Joumal oj Sound and Vibration 157, 405-433.
CURLING, L.R & PAlDOUSSIS, M,P, (1992b) Measurements and characterization of
walI-pressure fluctuations on cylinders in a bundle in turbulent axial flow, Part 2:
tempora! characteristics. Journal oj Sound and Vibration 157,435-449.
CURLlNG, LI. R & PAIDOUSSIS, M.P. (2003) Analyses for random flow-induced vibra-
tion of cylindrical structures subjected to turbulent axial flow. Joumal oj Sound and
Vibration 264, 795-833,
CZAYKOWSKI, G.T., TADJBAKHSH, LG, & TU, Y.-O. (1966) Stability offlexible tapes
in parallel flow. IBM Joumall0, 220-224.
D' ANGELO, C. & MOTE, C.D. Jr. (1993a) Natura! frequencies of a thin disk c1amped
by thick collars with friction at the contacting surfaces, spinning at high rotation speed.
Joumal oJSound and Vibration 168, 1-14.

REFERENCES
1519
D' ANGELO, C. & MOTE, C.D. Jr. (1993b) Aerodynamica11y excited vibration and tlutter
of a thin diskrotating at supercritica1 speed, Journal oJSound and Vibration 168, 15-30.
DALTON, C, (1980) Inertia coefficients forriser configurations. ASME Journal oJEnergy
Resources Technology 102, 197-202.
DALTON, C, & HELFlNSTINE, R.A. (1971) Potentia1 tlow past a group of circu1ar
cylinders. ASME Journal oj Basic Engineering 93, 636-642.
DATTA, S.K. & GOTTENBERG, w.G. (1975) Instability of an e1astic strip hanging in an
airstream. Journal oj Applied Mechanics 42, 195-198.
DAUBERT,A., PUGNET,L., BOULOT,F., WARLUZEL, A. & BAROUILLET, A. (1971)
Analyse des principa1es instabilités hydrauliques et des prob1emes hydroé1astiques dans
les réacteurs a eau pressurisée. La Houille Blanche 26, 447-450,
DAUBERT, A., PUGNET, L. & BOULOT, F. (1974) On globa1 hydrau1ic in stability in
the vesse1 of a pressurized water reactor. In Flow-/nduced Structural Vibrations (ed.
E, Naudascher), pp. 75-80, Berlin: Springer- Verlag,
DAVID, T,S. & SRINIVASAN, AV (1974) Plutter of coaxia1 cylindrica1 shells in an
incompressib1e axisymmetric tlow, A/AA Journal12, 1631-1635,
DA VIES, A.L. (1965) The speed of sound in mixtures of water and steam. UKAEA Atomic
Energy Establishment Winfrith (Dorset) Report AEEW-M452.
DAVIES, T,M, (1998) Private communication (ApriI1998),
DAVIS, D.C, & KIM, G. (1991) Design against hydrodynamic instabilities in flat-p1ate
type fuel element assemblies. Transactions 11th SMiRT, Vol. J, pp. 105-109,
DAVIS, D.C. & SCARTON, H.A. (1985) Plow-induced p1astic collapse of stacked fuel
p1ates. Nuclear Engineering and Design 85, 193-200.
DE LANGRE, E. (1999) Private communication (17 Sep. 1999).
DE LANGRE, E. (2000) Onde s variqueuses abso1ument instables dans un canal élastique.
Comptes Rendus, Académie des Sciences de Paris, Série lIb, 328, 61-65.
DE LANGRE, E. (2002) Abso1utely unstable waves in inviscid hydroe1astic systems,
Journal oj Sound and Vibration 256, 299-317.
DE LANGRE, E, & GRANGER, S. (1995) Coup1age structure-écou1ement axia1 confiné,
Revue Franraise de Mécanique, No. 1,49-55.
DE LANGRE, E. & OUVRARD, A.E. (1999) Abso1ute and convective bending instabil-
ities in tluid-conveying pipes. Journal oj Fluids and Structures 13, 663-680.
DE LANGRE, E., BÉLANGER, F. & PORCHER, G. (1992) Linearized potentia1 tlow
theory applied to the calcu1ation of tluidelastic forces in annu1ar tlow configura-
tions, Proceedings Symposium on Flow-/nduced Vibration and Noise, Vol. 5 (eds
M.P. Pai"doussis & M.K. Au-Yang), PVP-Vol. 244, pp. 187-204, New York: ASME,
DE LANGRE, E" PORCHER, G. & AXISA, F. (1994) An experiment on tluidelastic
instability in a confined annu1ar tlow, Proceedings 2nd /nternational ConJerence on
Engineering Aero-Hydroelasticity, pp, 26-31, Pilsen, Czech Republic.
DeLAURIER, J,D. (1972) A stability analysis for tethered aerodynamically shaped
balloons, Journal oj AircraJt 9, 646-651.
DEN HARTOG, I.P. (1956) Mechanical Vibrations, 4th edition. New York: McGraw-Hill.
DEN HARTOG, J.P, (1970) The mechanism oftube vibrations. In Flow-/nduced Vibration
in Heat Exchangers (ed. D.D. Reift), p, 78. New York: ASME,
DENG, Z.-F. & POPELAR, C.H. (1972) Dynamic stability of a cylindrical shell in an
acoustic medium. Journal oJthe Acoustical Society oj America 52, 1430-1436.
DENISE, J.-P, (1971) Flutter of cylindrical shells conveying tluid. M,Eng. Thesis,
Department of Mechanica1 Engineering, McGill University, Montreal, Québec,
Canada,

1520
SLENDER STRUCTURES AND AXIAL FLOW
DIMENTBERG, M.P. (1991) A stochastic model of parametric excitation of a straight
pipe due to slug flow of a two-phase fluid. Proceedings oj IMechE 'Flow lnduced
Vibrations', pp. 207-209.
DMC/FMC (1998) New radial mode Coriolis mass flowmeters. Direct Measurement
Corporation, Longmont, CO, USA.
DOAN, R.L. (1958) The Engineering Test Reactor: a status report. Nucleonics 16, 102-
105.
DOARÉ, O. & DE LANGRE, E. (2002a) Local and global instabiHty of fluid-conveying
pipes on elastic foundations. Journal oj Fluids and Structures 16, 1-14.
DOARÉ, O. & DE LANGRE, E. (2002b) The flow-induced in stability of long hanging
pipes. European Journal oj Mechanics A/Solids 21,857-867.
DOARÉ, O. & DE LANGRE, E. (2002c) Effect of length on the instability of hanging
pipes. PaperIMECE2002- 39018, FSI, AE & FIV + N Symposium (ed. M.P. Pa'idoussis),
New Orleans, LA, USA. New York: ASME.
DOEDEL, E.J. & KERNÉVES, J.P. (1986) AUTO: software for continuation and
bifurcation problems in ordinary differential equations. AppHed Mathematics
Report, Ca1ifornia Institute of Technology, Pasadena, CA, USA (procurable from
doedel@cs.concordia.ca).
DONE, G.T.S. & SIMPSON, A. (1977) Dynamic stability of certain conservative and
non-conservative systems. IMechE Journal oj Mechanical Engineering Science 19,
251-263.
DONG, W.N. & MUGGERIDGE, D.B. (1975) Dynamics of a fluid carrying fiber-
reinforced pipe. In Proceedings Canadian Congress oj Applied Mechanics,
CANCAM 75, Fredericton, N.B., Canada, pp. 351-352.
DONNELL, L.H. (1933) Stability of thin-walled tubes under torsion. NACA Report 479.
DONNELL, L.H. (1976) Beams, Plates and Shells. New York: McGraw Hi11.
DOSSO, S.E. & RIEDEL, M. (2001) Array element locaHzation for towed marine seismic
arrays. Journal oJthe Acoustical Society oj America 110,955-966.
DOTSON, K.W., BAKER, R.L. & SAKO, B.H. (2000) Launch vehicle buffeting with
aeroelastic coupHng effects. Journal oj Fluids and Structures 14, 1145-1171.
DOWELL, E.H. (1966a) Flutter of infinitely long plates and shells. Part I: plate. AlAA
Journal4,1370-1377.
DOWELL, E.H. (1966b) Flutter of infinitely long plates and shells. Part II: cylindrical
shell. AlAA Journal4, 1510-1518.
DOWELL, E.H. (1966c) Nonlinear osci11ations of a fluttering plate (I). AlAA Journal4,
1267-1275.
DOWELL, E.H. (1967a) Nonlinear osci11ations of a fluttering plate. II. AlAA Journal5,
1856-1862.
DOWELL, E.H. (1967b) On the nonlinear flexural vibrations of rings. AlAA Journal 5,
1508-1509.
DOWELL, E.H. (1970) Panel flutter: a review of the aeroelastic stability of plates and
shells. AlAA Journal8, 385-399.
DOWELL, E.H. (1971) Generalized aerodynamic forces on a flexible plate undergoing
transient motion in a shear flow with an application to panel flutter. AlAA Journal 9,
834-841.
DOWELL, E.H. (1975) Aeroelasticity oj Plates and Shells. Leyden: Noordhoff Interna-
tional Publishing.

REFERENCES
1521
DOWELL, E.H. (1977) Comments on 'Collapse and flutter phenomena in thin tubes
conveying fluid' by D.S. Weaver & M.P. Pa'idoussis, and authors' response. Journal oj
Sound and Vibration 55, 312-314,
DOWELL, E.H. (1978) Comments on non-linear flexura! vibrations of a cylindrica! shell.
Journal oj Sound and Vibration 60, 596-597.
DOWELL, E.H. (1980) Flutter of plates and shells in practice. Practical Experiences with
Flow-Induced Vibrations, (eds E. Naudascher & D. Rockwell), pp. 160-174. Berlin:
Springer - Verlag,
DOWELL, E.R. (1982) Flutter of a buckled plate as an example of chaotic motion of a
deterministic autonomous system. Journal oj Sound and Vibration 85, 333-344.
DOWELL, E.H. (1984) Observation and evolution of chaos for an autonomous system.
Journal oj Applied Mechanics Sl, 664-673.
DOWELL, E.R. (1987) Personal communication on the handling of boundary conditions
on a moving shell wall in viscous theory.
DOWELL, E.H. (1995) Private communication to L. Ruang (12 Jul1995),
DOWELL, E.H. (1998) Comments on the nonlinear vibrations of cylindrical shells,
Journal oj Fluids and Structures 12, 1087-1089,
DOWELL, E.H. (2002) Private communication (4 Jan 2(02).
DOWELL, E.R. & ILGAMOV, M. (1988) Studies in Nonlinear Aeroelasticity. New York:
Springer- Verlag,
DOWELL, E.R. & VENTRES, C.S, (1968) Modal equations for the nonlinear flexural
vibrations of a cylindrica! shell. lnternational Journal oj Solids and Structures 4,
975-991.
DOWELL, E.R. & VENTRES, C.S. (1970) Flutter of low aspect ratio panels. AlAA
Journal8, 1162-1164.
DOWELL, E.R. & VENTRES, C.S. (1998) Modal equations for the nonlinear flexural
vibrations of a cylindrical shell, Revised and corrected, 20 June 1998, in cooperation
with DeMan Tang. Duke University, School of Engineering Report 98-1.
DOWELL, E.H. & WIDNALL, S,E. (1966a) Generalized aerodynamic forces on
oscillating cylindrical shell: subsonic and supersonic flow. AlAA Journal4, 607-610.
DOWELL, E.R. & WIDNALL, S.E. (1966b) Generalized aerodynamic forces on
oscillating cylindrical shell. Quarterly oj Applied Mathematics 24, 1-17.
DOWELL, E.R., SRINIVASAN, AV., McLEAN, J.D. & AMBROSE, J. (1974)
Aeroelastic stability of cylindrical shells subjected to a rotating flow. AlAA Journal
12, 1644-1651.
DOWELL, E.H. (ed.), CRAWLEY, E.F., CURTISS Jr., R.C., PETERS, D.A, SCANLAN,
R.R. & SISTO, F. (1995) A Modern Course in Aeroelasticity, 3rd edition. Dordrecht:
Kluwer Academic Publishers.
DOWLING, AP. (1988a) The dynamics of towed flexible cylinders. Part 1: neutrally
buoyant elements, Journal oj Fluid Mechanics 187, 507-532.
DOWLING, AP. (1988b) The dynamics of towed flexible cylinders. Part 2: negatively
buoyant elements. Journal oj Fluid Mechanics 187, 533-571.
DOWNER, J.D. & PARK, K.e. (1993) Formulation and solution ofthe inverse spaghetti
problem: application to beam deployment dynamics. AlAA Journal 31, 339-347,
DRAGOS, L. (1962) Sur le mouvement d'un fluide visqueux dans un tube élastique.
Comptes rendus de ľAcademie de Sciences de Paris 254, 417-422.
DRAZIN, P.G. & JOHNSON, R.S. (1989) Solitons: An lntroduction, Cambridge:
Cambridge University Press.

1522
SLENDER STRUCTURES AND AXIAL FLOW
DRAZIN, P.G. & REJD, W.H. (1981) Hydrodynamic Stability. Cambridge: Cambridge
University Press.
DRUCKER, E.G. & LAUDER, G,V. (1999) Locomotor forces on swimming fish: three-
dimensional vortex wake dynamics quantified using digital particle image velocimetry.
The Journal oj Experimental Biology 202,2393-2412.
DU BUAT, P.L.G. (1786) Principes ďHydraulique, vérifiés par un grand nombre
ďexpériences, 2nd edition, Vol. 2, pp. 226-259. Paris: lmprimerie de Monsieur.
DUGUNDJI, J., DOWELL, E. & PERKlNS, B. (1963) Subsonic fiutter of panels on
continuous elastic foundations. AlAA Journall, 1146-1154.
DURAND, W.F. (1936) Aerodynamic Theory, Vol. 6, Division R, Chapter 5, Berlin:
J, Springer; also 1963, New York: Dover.
DWIGHT, H,B. (1961) Tables oj lntegrals and Other Mathematical Data, 4th edition.
New York: Macmillan Publishing.
DYER, 1. (1959) Response of plates to a decaying and convecting pressure field. Journal
oj Acoustical Society oj America 31, 922-928.
EAGLESON, P.S" NOUTSOPOULOS, G.K. & DAILY, J.W. (1964) The nature of
self-excitation in the fiow-induced vibration of fiat plates, ASME Journal oj Basic
Engineering 86, 599-606.
EATOCK TAYLOR, R. & OHKUSU, M. (eds) (2000) Specia1 lssue on Marine
Hydroelasticity, Journal oj Fluids and Structures 14, 943-993.
EHRICH, F.F. (1993) Handbook oJRotordynamics. London: Applied Science Publishers.
EIFLER, W. & NDSING, R. (1967) Experimental investigation of velocity distribution
and flow resistance in a triangular array of parallel rods. Nuclear Engineering and
Design 5, 22-42,
EL BAROUDI, M,Y., LUDWIG, G.R. & RIBNER, H.S. (1963) An experimental investig-
ation of turbulence-excited panel vibration and noise (boundary-layer noise). AGARD
Report 465.
EL CHEBAIR, A. (1988) Theoretical and experimental study of internal and annular fiow
induced instabilities of cylindrical shells. Ph.D. Thesis, Department of Mechanica1
Engineering, McGil1 University, Montreal, Québec, Canada.
EL CHEBAIR, A., PAIDOUSSIS, M.P, & MISRA, A.K. (1989) Experimenta1 study of
annular fiow-induced instabilities of cylindrical shells. Journal oj Fluids and Structures
3,349-364.
EL CHEBAIR, A" MISRA, A,K. & PAIDOUSSIS, M.P. (1990) Theoretical study of
unsteady viscous forces on inner- and annular-flow-induced instabilities of cylindrical
shells. Journal oj Sound and Vibration 138,457-478.
EL NASCHIE, M,S, (1990) Stress, Stability and Chaos in Structural Engineering: an
Energy Approach. London: McGraw- Hill.
EL SHAFEI, A. & CRANDALL, S.H. (1991) Fluid inertia forces in squeeze film dampers.
In Rotating Machinery and Vehicle Dynamics, DE-Vol. 35, pp, 219-228. New York:
ASME.
ELlSHAKOFF, LB, (1975) Turbulent fiow-excited vibrations of a shallow cylindrica1
shell. AlAA Journal13, 1179-1182.
ELLEN, C,H. (1972) Stability of c1amped rectangular plates in uniform subsonic flow.
AlAA Journall0, 1716-1717.
ELLEN, C,H. (1973) The stability of simply supported rectangular surfaces in uniform
subsonic fiow. Journal oj Applied Mechanics 40, 68-72.

REFERENCES
1523
ELLEN, C.H. (1977a) The stability of an iso1ated rectangu1ar surface embedded in uniform
subsonic flow. Journal oj Applied Mechanics 44, 201-206.
ELLEN, C.H. (1977b) The non-1inear stability of pane1s in incompressib1e flow. Journal
oJSound and Vibration 54,117-121.
ELLlNGTON, c.P. & PEDLEY, TJ. (1995) Biological Fluid Dynamics. Cambridge: The
Company of Bio10gists Ltd.
ELYADA, D. (1989) C10sed-form approach to rocket-vehic1es aeroe1astic divergence.
AlAA Journal oj SpacecraJt 26, 95-102.
EPSTEIN, R.J., SRINIVASAN, R. & DOWELL, E.H. (1995) F1utter of an infinite1y 10ng
panel in a duct. AlAA Journal33, 109-115.
ERDOGAN, E & GUPTA, G.D. (1972) On the numerica1 solution of singu1ar integra1
equations. Quarterly Journal oj Applied Mathematics 29, 525-534.
ERINGEN, A.c. (1957) Response ofbeams and p1ates to random 10ads. Journal oj Applied
Mechanics 24, 46-53.
ESMONDE, H., FlTZPATRICK, J.A., RICE, H.J. & AXISA, E (1992) Modelling and
identification of non1inear squeeze film dynamics. Journal oj Fluids and Structures 6,
223-248.
ESPINOSA, EM. & GALLEGO-JUÁREZ, A.G. (1984) On the resonance frequencies of
uniform, circu1ar free-edge p1ates. Journal oj Sound and Vibration 94, 217-222.
ETKIN, B. (1998) Stability of a towed body. Journal oj Aircraft 35, 197-205.
EVANGELlNOS, c., LUCOR, D., SU, C.-H. & KARNIADAKIS, G.E. (2002) F1ow-
induced vibrations of non-linear cab1es. Part 1: mode1s and a1gorithms. lnternational
JournalJor Numerical Methods in Engineering 55, 535-556.
EVENSEN, D.A. (1963) Some observations on the non1inear vibration of thin cy1indrical
shells. AlAA Journall, 2857-2858.
EVENSEN, D.A. (1964) Non-1inear flexura1 vibrations of thin circu1ar rings. Ph.D. Thesis,
Ca1ifomia Institute of Technology, Pasadena, CA, USA.
EVENSEN, D.A. (1966) Nonlinear flexura1 vibrations of thin circu1ar rings. Journal oj
Applied Mechanics 33, 553-560.
EVENSEN, D.A. (1967) Nonlinear flexura1 vibrations of thin-walled circu1ar cy1inders.
NASA TN D-4090.
EVENSEN, D.A. (1968) Non1inear vibrations of an infinite1y 10ng cylindrica1 shell. AlAA
Journal6,1401-1403.
EVENSEN, D.A. (1974) Non1inear vibrations of circu1ar cy1indrica1 shells. In Thin
Walled Structures: Theory, Experiment and Design (eds yc. Fung & KE. Sech1er),
pp. 133-155. Eng1ewood C1iffs, NJ: Prentice-Ha11.
EVENSEN, D.A. (1977) Comment on 'Large amp1itude asymmetric vibrations of some
thin shells of revo1ution' . Journal oj Sound and Vibration 52, 453--454.
EVENSEN, D.A. (1978a) Author's rep1y. Journal oj Sound and Vibration 56, 305-308.
EVENSEN, D.A. (1978b) Author's rep1y. Journal oj Sound and Vibration 59,297-298.
EVENSEN, D.A. (1999) Non1inear vibrations of cy1indrica1 shells - 10gica1 rationa1e.
Journal oj Fluids and Structures 13, 161-164.
EVENSEN, D.A. & OLSON, M.D. (1967) Non1inear flutter of a circu1ar cylindrica1 shell
in supersonic flow. NASA TN D-4265.
EVENSEN, D.A. & OLSON, M.D. (1968) Circumferentially travel1ing wave flutter of a
circu1ar cylindrical she11. AlAA Journal6, 1522-1527.
FAHY, E (1985) Sound and Structural Vibration: Radiation, Transmission and Response.
London: Academic Press.

1524
SLENDER STRUCTURES AND AXIAL FLOW
FAHY, EJ. & PRETLOVE, AJ. (1967) Acoustic forces on a flexib1e panel which is part
of a duct carrying airflow. Joumal oJSound and Vibration 5,302-316.
FELlPPA, C.A & OHAYON, R. (1990) Mixed variational formu1ation of finite ele-
ment analysis of acoustoelastic/s1osh fluid-structure interaction. Joumal oj Fluids and
Structures 4,35-57.
FENG, J., HUANG, P.Y. & JOSEPH, D.D. (1995) Dynamic simu1ation of the motion of
capsu1es in pipelines. Joumal oj Fluid Mechanics 286,201-227.
FlRTH, D. (1977) Acoustic vibration of structures in liquid. Shock & Vibration Digest
9,3-7.
FLÚGGE, W. (1960) Stresses in Shells; 1973, 2nd edition. Berlin: Springer-Verlag.
FOALE, S., THOMPSON, J.M.T. & McROBIE, EA (1998) Numerical dimension-
reduction methods for non-linear shell vibrations. Joumal oj Sound and Vibration
215,527-545.
FOMICHEV, M.S., EMEL'YANOV, I. Ya. & BERZINA, B.E (1973) An investigation of
the hydrodynamics, vibration, damage and destruction of rods in vertical tubes with
various entry conditions of the stream. Fluid Mechanics - Soviet Research 2,89-97.
FORRESTAL, M.J. & HERRMANN, G. (1965) Response of a submerged cylindrical
shell to an axial1y propagating step wave. Joumal oj Applied Mechanics 32,788-792.
FORSBERG, K. (1965) A review of analytical methods used to determine the moda1
characteristics of cy1indrical shells. Lockheed Missi1es and Space Co. Report 6-75-
65-25. A1so, 1966, NASA CR-613.
FORSBERG, K. (1969) Axisyrnmetric and beam-type vibrations ofthin cylindrical shells.
AlAA Joumal7, 221-227.
FORSCHlNG, H.W. (1974) Fundamentals oj Aeroelasticity (in German). Berlin:
Springer- Verlag.
FRANCE, E.R (1973) Response of fuel pins to acoustic excitation in the presence of
flow within an Advanced Gas-coo1ed Reactor. Proceedings lntemational Symposium
on Vibration Problems in lndustry, Paper No. 215. UKAEA and NPL, UK.
FREI, c., LÚSCHER, P. & INTERMANTEL, E. (2000) Thread-annu1ar flow in vertical
pipes. Joumal oj Fluid Mechanics 410, 185-210.
FRIEDMAN, M.H. (ed.) (1992) Special Issue on Hemodynamics and the Arterial Wall.
ASME Joumal oj Biomechanical Engineering 114(3), 273-320.
FRIEDMANN, P.P. (1998) Private cornmunication (17 Nov. 1998).
FRITZ, R (1970) The effects of an annu1ar fluid on the vibrations of a long rotor. Part 1:
Theory. Part 2: Experiments. ASME Joumal oj Basic Engineering 92, 923-927 and
930-937.
FRITZ, R.J. (1972) The effect of liquids on the dynamic motions of immersed solids,
ASME Joumal oj Basic Engineering 94, 167-173.
FRITZ, RJ. & KlSS, E. (1966) The vibration of a canti1evered cylinder sUlTounded
by an annu1ar fluid. General E1ectric Company Report KAPL-M-6539, Schenectady,
NY, USA
FU, Y.M. & CHIA, c.Y. (1993) Non-linear vibration and postbuck1ing of general1y
1aminated circu1ar cylindrical thick shells with non-uniform boundary conditions.
lntemational Joumal oj Non-Linear Mechanics 28, 313-327.
FUJlTA, K. (1986) Vibration analysis of fluid-coup1ed two coaxial axisymmetric shells
containing fluid. Bulletin oJthe JSME 29, N248, 516-524.
FUJlTA, K. & ITO, T. (1992) Study of 1eakage flow-induced vibration of an axisym-
metric cylindrical rod due to axial flow. In Axial and Annular Flow-lnduced Vibration
and lnstabilities (eds M.P, Pai'doussis & M.K. Au-Yang), PVP-Vol. 244, pp. 33-43.
New York: ASME.

REFERENCES
1525
FUJITA, F. & SHINTANI, A (2001) Axia!leakage flow-induced vibration of the e1astic
rod as the axisymmetric continuous flexib1e beam. ASME Journal oj Pressure Vessel
Technology 123, 421-428.
FUJITA, K., ITO, T., KODAMA, T., TANAKA, N., KOGA, T. & EGUCHI, Y. (1992)
Experimenta! study on flow-induced vibration of a flexib1e weir due to fluid discharge,
In Fluid-Structure Vibration and Sloshing -1992 (eds D.C. Ma, K. Fujita & J. Tani),
PVP-Vol. 232, pp. 71-76. New York: ASME.
FUJITA, K., lTO, T. & BABA, K. (1993) Study of the unstab1e vibration of cy1-
indrica1 shells due to 1eakage flow. Proceedings lnternational ConJerence on Aero-
Hydroelasticity (ICAHE'93), pp, 286-295, Beijing, China.
FUJITA, K., lTO, T., KAWATA, Y. & !ZUMI, H. (1994) Axia11eakage flow-induced
vibration of a 10ng flexib1e rod with small gaps. In Flow-Induced Vibration - 1994
(eds M,K. Au-Yang & K. Fujita), PVP-Vol. 273, pp. 133-143. New York: ASME,
FUJlTA, K., lTO, T., KODAMA, T., EGUCHI, Y. & YAMAMOTO, K. (1996) Flow-
induced vibration of a flexib1e weir due to fluid discharge; effect of weir stiffness.
Journal oj Fluids and Structures 10, 79-98.
FUJITA, K., SHINTANI, A & YOSHINO, T. (2000) Vibrationa! behavior of a rigid body
supported by a damper-spring system in a narrow passage with fluid. In Flow lnduced
Vibration (eds S. Ziada & T. Staub1i), pp. 829-836. Rotterdam: AA Ba1kema.
FUJlTA, K., SHINTANI, A & ONO, M. (2001) Axia!leakage flow-induced vibration of
a thin cy1indrica! shell. In Fluid Structure lnteraction 2001 (eds S.K. Chakrabarti &
C.A Brebbia), pp. 129-142. Southampton: WIT Press.
FUKUYAMA, M., NAKAGAWA, M., ISHIHAMA, K., HAGIWARA, Y., TOYODA, Y.
& AKIYAMA, H. (1999) Dynamic buckling experiment s of fluid-structure-coup1ed
co-axia! thin cy1inder. Nuclear Engineering and Design 188, 13-26.
FULLER, c.R. & FAHY, F.J. (1982) Characteristics of wave propagation and energy
distributions in cy1indrica! e1astic shells filled with fluid. Journal oJSound and Vibration
81,501-518.
FUNG, R.-F., WANG, P.-H. & LEE, M,-J, (1998) Nonlinear vibration ana!ysis oftraveling
string with time-dependent 1ength by finite-e1ement method. Journal oj the Chinese
lnstitute oj Engineers 21, 109-117.
FUNG, Y.C. (1955) An lntroduction to the Theory oj Aeroelasticity. New York: John
WHey & Sons; a1so 1993, New York: Dover,
FUNG, Y.C. (1958) On two-dimensiona! panel flutter. Journal oJthe Aeronautical Sciences
25, 145-160.
FUNG, Y.C. (1966) Interation of mechanica! and aeroe1astic instabi1ities of a circu1ar
cylindrica! shell. In Dynamic Stability oj Structures, pp. 267-284. Oxford: Pergamon
Press.
FUNG, Y.T. (1983) Asymmetric effects on vibrations of a cylinder in turbu1ent para!le1
flows. Nuclear Science and Enginee ring 85, 178-187.
GAD-EL-HAK, M. (1986) Boundary 1ayer interactions with compliant coatings: an
overview. Applied Mechanics Reviews 39,511-524.
GAD-EL-HAK, M. (1987) Comp1iant coatings research: a guide to the experimenta!ist.
Journal oj Fluids and Structures 1, 55-70.
GAD-EL-HAK, M. (2002) Comp1iant coatings for drag reduction, Progress in Aerospace
Sciences 38, 77-99,
GAGNON, J,O, (1989) Fluid coup1ing and response characteristics of cylinder c1usters in
axia! flow. Ph.D. Thesis, Department of Mechanica! Engineering, McGill University,
Montrea1, Québec, Canada.

1526
SLENDER STRUCTURES AND AXIAL FLOW
GAGNON, J.O. (1994) Unpub1ished experimenta1 data; see Gagnon & Paidoussis (1994a).
GAGNON, J.O. & PAIDOUSSIS, M.P, (1994a) Fluid coup1ing characteristics and vibra-
tion of cyHnder c1usters in axia1 flow. Part I: theory. Journal oj Fluids and Structures
8,257-291.
GAGNON, J.O. & PAIDOUSSIS, M.P. (1994b) Fluid coupHng characteristics and vibra-
tion of cyHnder c1usters in axia1 flow. Part II: experiments. Journal oj Fluids and
Structures 8, 293-324.
GANAPATHI, M. & VARADAN, T.K. (1995) Non1inear free flexura! vibrations of
1aminated circu1ar cyHndrica! shells. Composite Structures 30,33-49.
GANAPATHI, M. & VARADAN, T.K. (1996) Large amp1itude vibrations of circu1ar
cyHndrica! shells. Journal oj Sound and Vibration 192, 1-14.
GANS, R.F. (1975) On the stability of shear flow in a rotating gas. Journal oj Fluid
Mechanics 68, 403-412.
GANS, RF. (1976) On Poiseui11e-Hke flow inarotating gas. ReportFFA AU-751, Part 27,
Flygtekniska Fórsóksanstalten (The Aeronautica! Research Institute of Sweden),
Stockholm.
GARNER, G, & CHANDRA, S. (1984) Fluid-structure coupHng between a finite cy1inder
and a confined fluid. Journal oj Applied Mechanics 51, 857-862.
GASTER, M. (1968) Growth of disturbances in both space and time, Physics oj Fluids
11,723-727,
GIBERT, RJ. (1988) Vibrations des Structures. Paris: Eyrolles.
GIBERT, R-J. (1999) Private communication. (5 Nov 1999).
GILLlS, G.B. (1996) Undu1atory locomotion in e10ngate aquatic vertebrates: angui11iform
swimming since Sir James Gray. American Zoologist 36,656-665.
GILLlS, G.B, (1997) Angui11iform locomotion in an e10ngate sa!amander (siren intenne-
dia): effects of speed on axia1 undu1atory movements. The Journal oj Experimental
Biology 200, 767-784.
GILLlS, G.B. (1998) Environmenta! effects on the undu1atory locomotion in the American
ee1 anguilla rostrata: kinematics in water and 1and. The Journal oj Experimental
Biology 201, 949-961.
GlNSBERG, J.H. (1973) Large amp1itude forced vibrations of simp1y supported thin
cyHndrica1 shells, Journal oj Applied Mechanics 40,471-477.
GlNSBERG, J.H. (1974) NonHnear axisymmetric free vibration in simp1y supported
cy1indrica1 shells. Journal oj Applied Mechanics 41, 310-311.
GLADWELL, G.M.L. & VIJAY, D.K. (1975) Natura! frequencies of free finite-1ength
circu1ar cy1inders. Journal oj Sound and Vibration 42,387-397.
GLAUERT, H. (1930) The stability of a body towed by a Hght wire. Aeronautica1 Research
Counci! (UK), (ARC) R&M 1312.
GON(:ALVES, P.B. & BATlSTA, R.C. (1988) Non-1inear vibration ana1ysis offluid-filled
cy1indrica1 shells. Journal oJSound and Vibration 127,133-143.
GON(:ALVES, P.B. & RAMOS, N.R.S.S. (1996) Free vibration ana1ysis of cy1indrica!
tanks partial1y filled with 1iquid. Journal oj Sound and Vibration 195, 429-444.
GON(:ALVES, P.B, & RAMOS, N.R.S.S. (1997) Numerica1 method for vibration ana!ysis
of cy1indrica1 shells. ASCE Journal oj Engineering Mechanics 123, 544-550.
GORDNIER, RB. & VISBAL, M.R. (2002) Deve10pment of a three-dimensiona1 vis-
cous aeroe1astic solver for non1inear panel flutter. Journal oj Fluids and Structures 16,
497-527.

REFERENCES
1527
GORMAN, D.J, (1969) The role ofturbu1ence in the vibration ofreactorfuel elements in
liquid flow, Atomic Energy of Canada Report AECL-3371.
GORMAN, D.J. (1971) Ana1ytical and experimental investigation of the vibration of
cylindrical reactor fuel elements in two-phase parallel flow. Nuclear Science and
Engineering 44, 277-290.
GORMAN, D.J. (1975a) Experimental and analytical study ofliquid and two-phase flow
induced vibration in reactor fuel bundles. ASME Paper 75-PVP-52.
GORMAN, D.l (1975b) Free Vibration Analysis oj Beams and Shafts. New York: John
Wiley & Sons.
GORMAN, D.J. & PLANCHARD, J. (1988) Masse ajoutée ďun faisceau tubulaire dans
le cas de grands déplacements. EDF Bulletin de la Direction des Etudes et Recherches,
Série C, No. 1, pp. 39-47.
GORMAN, D.J., GODON, J.L. & PLANCHARD, J. (1987) Analytical and experimental
study of the vibratory response of a flexible tube subjected to extemal annular flow
part way a10ng its length. Proceedings lntemational Conference on Flow lnduced
Vibrations (ed. R. King), pp. 339-347, Bowness-on-Windermere, UK Cranfield:
BHRA.
GOTTLlEB, D. & ORSZAG, S.A. (1977) Numerical Analysis oj Spectral Methods:
Theory and Applications. Phi1adelphia, PA: SIAM-CBMS.
GOULD, L.P. (1985) Finite Element Analysis oj Shells oj Revolution. London: Pitman
Publications Ltd.
GOYDER, H.G.D. (1992) Private communication, (25 Sep. 1992).
GRAFF, K.E (1975) Wave Motion oj Elastic Solids. New York: Dover.
GRAY, J, (1968) Animal Locomotion. London: Weidenfeld & Nicolson,
GREENSPAN, H. (1968) The Theory oj Rotating Fluids. Cambridge: Cambridge
University Press.
GREENSPON, J.E. (1960) Vibrations of a thick-wa1led cylindrical sheU- comparison
of exact theory with approximate theories. Joumal oj Acoustical Society oj America
32,571-578.
GREENSPON, J.E. (1961) Vibrations of cross-stiffened and sandwich p1ates with app1ic-
ation to underwater sound radiators. Joumal oj the Acoustical Society oj America 33,
1485-1497.
GREGORY, R.w. & PAlDOUSSIS, M.P. (1966) Unstable osci1lation of tubu1ar canti-
levers conveying fluid - I. Theory, Proceedings oJthe Royal Society (London) A 293,
512-527.
GRIFFITHS, D.J. (1977) Oscillations in the outflow from a collapsible tube. Medical &
Biological Engineering & Computing 15, 357-362.
GRIGORIEV, J.V (1978) Stability of a driU tube co1umn with an initial curvature in
the axial stream (in Russian). Joumal oj Bauman Moscow State Technical University:
Mashinostronye 5,23-28.
GRIGORYEV, U.V, GUSKOV, A.M. & SVETLlSKY, VA. (1979) Rod stability in axial
flow of viscous fluid. Prikladnaia Mekhanika 15, 67-73 (in Russian).
GRONINGER, R.D. & KANE, U. (1963) Plow induced deflections ofpara11e1 flat plates.
Nuclear Science and Engineering 16, 218-226.
GROSENBAUGH, M,A. (1998) Private communication (Nov 1998).
GROTBERG, J.B. (1994) Pulmonary flow and transport phenomena. Annual Review oj
Fluid Mechanics 26,529-571.

1528
SLENDER STRUCTURES AND AXIAL FLOW
GROTBERG, J.B. & DA VIS, S .H. (1980) Fluid-dynamic flapping of a collapsib1e channel:
sound generation and flow limitation. Journal oj Biomechanics 13, 219-230.
GROTBERG, J.B. & REISS, E,L. (1982) A subsonic flutter anomaly. Journal oj Sound
and Vibration 80, 444-446.
GROfBERG, J.B. & REISS, E.L. (1984) Subsonic flapping flutter. Journal oJSound and
Vibration 92, 349-361.
GROfBERG, J.B. & SHEE, T.R (1985) Compressible-flow channel flutter. Journal oj
Fluid Mechanics 159, 175-193.
GRUNENWALD, T., AXISA, F., BENNETT, G. & ANTUNES, J. (1996) Dynamics of
rotors immersed in eccentric annular flow, Part 2: experiments. Journal oj Fluids and
Structures 10, 919-944.
GRUNWALD, G. & HENNIG, K. (1977) Untersuchung stromungsinduzierter pendel-
schwingungen. Berichte de Tajung Stochastische Schwingungen und Zuverlassigkeit,
Helft II, Berlin,
GRUNWALD, G, & MULLER, E. (1984) Schwingungen eines Regelstabes in einem
Stromungskanal. Zeitschrift fiir Angewandte Mathematik und Mechanik 64, M454-456.
GUERNIER, C. (1978) Vibrations induced by flow in thin shells, Unpublished report,
Department of Mechanical Engineering, McGill University.
GUNZBURGER, M.D. & NICOLAIDES, R.A (1993) lncompressible Computational
Fluid Dynamics. Cambridge: Cambridge University Press.
GUO, C.Q. & PAIDOUSSIS, M.P. (2000a) Stability of rectangular plates with free-edges
in two-dimensional inviscid channel flow. Journal oj Applied M echanics 67, 171-176.
GUO, C.Q. & PAIDOUSSIS, M.P. (2000b) Analysis of hydroelastic instabilities of rect-
angular parallel-plate assemblies. ASME Journal oj Pressure Vessel Technology 122,
502-507.
GUPTA, RK. & HUTCHINSON, G.L. (1989) Solid-water interaction in liquid storage
tanks. Journal oj Sound and Vibration 135, 357-374.
GURAN, A, BAJAJ, A, ISHIDA, y" PERKINS, N., D'ELEUTERIO, G. & PIERRE, C.
(1999) Stability oj Gyroscopic Systems, Series on Stability, Vibration and Control of
Systems, Vol. A 2. Singapore: World Scientific.
HAGEDORN, P. (1981) Non-Linear Oscillations. Oxford: Clarendon.
HAMY, N. (1971) The Trebron Sea Chain System. Trebron Holdings Ltd., Montreal,
Québec, Canada.
HAN, AD. & YANG, T.Y. (1983) Nonlinearpanel flutterusinghigh-ordertriangularfinite
e1ements. AlAA Journal21, 1453-1461.
HAN, Y. & ROGERS, RJ. (2001a) Nonlinear fluid forces in cylindrical squeeze films.
Part I: short and long 1engths. Journal oj Fluids and Structures 15, 151-169.
HAN, Y. & ROGERS, RJ, (2001b) Nonlinear fluid forces in cylindrical squeeze films.
Part II: finite lengths. Journal oj Fluids and Structures 15, 171-206.
HANCOCK, H. (1958) Elliptic lntegrals. New York: Dover.
HANNOYER, M.J. (1972) A solution to line ar differential equations in the field of dynam-
ics of continuous systems. Mechanical Engineering Research Laboratories Report,
MERL 72-5, McGill University, Montréal, Canada. t
HANNOYER, M.J. (1977) Instabilities of s1ender, tapered tubular beams induced by
internal and external axial flow. Ph.D. Thesis, Department of Mechanical Engineering,
McGill University, Montreal, Québec, Canada.
t AU MERL Reports are available in digita! form from www.library.mcgill.ca/psel/subguide/mecheng.htm.

REFERENCES
1529
HANNOYER, MJ. & PAIDOUSSIS, M.P. (1978) Instabilities of tubular beams simul-
taneously subjeeted to internal and external axial flows. ASME Journal oj Mechanical
Design 100, 328-336.
HANNOYER, MJ. & PAIDOUSSIS, M.P. (1979a) Dynamies of slender tapered beams
with internal or external axial flow, Part 1: Theory, Journal oj Applied Mechanics 46,
45-51.
HANNOYER, MJ, & PAIDOUSSIS, M.P. (1979b) Dynamies of slender tapered beams
with internal or external axial flow. Part 2: Experiments. Journal oj Applied Mechanics
46,52-57.
HANSEN, M.H. (1999) Aeroelastieity and dynamies of spinning disks. Ph.D. Thesis,
Teehnieal University of Denmark, Lyngby. DCAMM Report No. S85, Lyngby,
Denmark.
HAN SEN, M,H., RAMAN, A. & MOTE, C.D., Jr. (1999) Estimation ofnon-eonservative
aerodynamie pressure leading to flutter of spinning disks. DCAMM Report No. 605,
Teehnieal University ofDenmark, Lyngby.
HANSEN, M.H., RAMAN, A. & MOTE, C.D., Jr. (2001) Estimation of noneonservat-
ive aerodynamie pressure leading to flutter of spinning disks. Journal oj Fluids and
Structures 15, 39-57.
HANSEN, R.J. & NI, c.c. (1976) An experimental study of flow-indueed motions of
flexible eables and eylinders aligned with flow direetion. ASME Paper 76- WA/FE-15.
HANSEN, RJ. & NI, C.C. (1979) Flow-indueed motions of a flexible eable aligned with
flow direetion. lEEE Journal oJOceanic Engineering OE-4, 152-156.
HARA, F. (1975a) Experimental study of the vibrations of a fuel pin model in parallel
two-phase flow, Transactions 3rd SMiRT, London, Vol. 1, Paper D2/3.
HARA, F. (1975b) A theory on the vibrations of a fuel pin model in parallel two-phase
flow. Transactions 3rd SMiRT, London, Vol. 1, Paper D2/4.
HARA, F. (1979) A theoretieal analysis of two-phase flow/fuel pin struetural dynamieal
interaetions. Transactions 5th SMiRT, Berlin, Part B, Paper B5/9.
HARA, F. (1980) Two-phase flow indueed parametrie vibrations in struetural systems-
pipes and nuelear fuel pins, Report ofthe Institute oflndustrial Scienee of the University
ofTokyo, Vol. 28, No. 4, Serial No, 183.
HARA, F. & KOHGO, O. (1986) Numerieal approaeh to added mas s and damping
of a vibrating cireular eylinder in a two-phase bubble fluid. In Computational
Mechanics'86, pp. VII.256-VII-260.
HARA, F. & YAMASHITA, T. (1978) Parallel two-phase flow indueed vibrations in fuel
pin model. Journal oj Nuclear Science and Technology 15, 346-354.
HAREL, P. & PERULLI, M. (1971) Influenee of a stationary uniform axial flow on
propagation of aeoustie modes of vibration in eyHndrieal duet. Journal oj Sound and
Vibration 15, 455-474.
HARRIS, R.W. & HOLLAND, P.G, (1971) Response of a eylindrieal eantilever to axial
air/water flow. Proceedings 1st lnternational ConJerence on Structural Mechanics in
Reactor Technology (SMiRT), Paper E3/6, Berlin, Germany,
HASLINGER, K.H., MARTIN, M.L. & STEININGER, D.A. (1990) Experimental ehar-
aeterization of fluid and squeeze film effeets in heat exehanger tube supports. Journal
oj Fluids and Structures 4, 605-629.
HAUGHTON, D.M, & OGDEN, R.W. (1979) Bifureation ofinflated eireularey1inders of
elastie material under axialloading, I. Membrane theory for thin-wal1ed tubes. II. Exaet
theory for thiek-wal1ed tubes. Journal oj the Mechanics and Physics oj Solids 27,
179-212 and 489-512.

1530
SLENDER STRUCTURES AND AXIAL FLOW
HAWTHORNE, W.R. (1961) The ear1y deve10pment of the Dracone flexib1e barge.
Proceedings oj the lnstitution oj Mechanical Engineers 175, 52-83.
HAYASHI, H" KODAMA, Y., SASAKI, S. & FUKANO, T. (1999) Wake vortices and
turbu1ent noise from two dimensiona1 bodies in flow. Proceedings 3rd lnternational
ConJerence on Engineering Aero-Hydroelasticity, Prague, pp. 187-192. Prague: Czech
Technica1 University.
HAYASHI, S., HAYASE, T, & KAWAMURA, H. (1998) Numerica1 ana1ysis of stability
and se1f-excited osci11ation in collasib1e tube flow. ASME Journal oj Biomechanical
Engineering 120, 468-475.
HEIL, M. (1996) The stability of cy1indrica1 shells conveying viscous flow. Journal oj
Fluids and Structures 10, 173-196.
HEIL, M. & PEDLEY, T.J. (1995) Large axisymmetric deformation of a cy1indrica1 she11
conveying a viscous flow. Journal oj Fluids and Structures 9, 237-256.
HEIL, M. & PEDLEY, T.J. (1996) Large post-buckling deformations of cy1indrica1 shells
conveying viscous flow. Journal oj Fluids and Structures 10, 565-599.
HELWIG, P,C, (1993) Private communication, Energy, Mines & Resources, Ottawa,
Canada K1A 0E4 (11 February 1993).
HENNIG, K. & GRUNWALD, G. (1984) Treatment of flow-induced pendu1um osci11a-
tions. Kernenergie 27,286-291.
HENNIG, K., PLATEN, E. & GRUNWALD, G. (1980) Treatment of flow-induced
pendu1um osci11ations. In Practical Experiences with Flow-lnduced Vibrations
(eds E, Naudascher & D. Rockwell), pp. 140-143. Berlin: Springer-Verlag.
HERMAN, B. (1975) Dynamics of articu1ated cylinders moving in fluid. B.Eng. Honours
Thesis, Department ofMechanica1 Engineering, McGi11 University, Montrea1, Canada.
HICKS, W.M. (1879) On the motion of two cy1inders in a fluid. Quarterly Journal oj Pure
and Applied Mathematics 16, 113-140 and 193-219.
HlGUCHI, K. & DOWELL, E.H, (1992) Effect of structura1 damping on flutter of p1ates
with a follower force, AlAA Journal 30, 820-825.
HlLDEBRAND, EB. (1960) Methods oj Applied Mathematics. Enge1wood Cliffs:
Prentice- Hall.
HINE, M.J. (1973a) Acoustically induced vibrations of slender rods in a cylindrica1 duct.
Journal oj Applied Mechanics 40, 459-463.
HlNE, M.J. (1973b) Acoustically induced vibrations of slender rods in a cy1indrical duct
with paralle1 flow. The Journal oJthe Acoustical Society oj America 53,665-668.
HlRS, G.G. (1973) A bu1k-flow theory for turbulence in 1ubricant fi1ms. ASME Journal
oj Lubrication Technology 95, 137-146.
HlRSCH, C. (1990) Numerica1 computation of interna1 and externa1 flow. Computational
MethodsJor lnviscid and Viscous Flows, Vol. 2. Chichester: John Wi1ey.
HOBSON, D,E. (1982) F1uid-e1astic instabi1ities caused by flow in an annu1us.
Proceedings oj 3rd lnternational ConJerence on Vibration in Nuclear Plant, Keswick,
pp. 440-463. London: BNES.
HOBSON, D.E. (1984) Instabilities of the AGR assembly during on-load refueling.
Proceedings Symposium on Flow-lnduced Vibrations, Vol. 4 (eds M.P. Pai'doussis &
M.K. Au- Yang), pp. 25-39. New York: ASME.
HOBSON, D.E. & JEDWAB, M. (1990) Investigations ofthe effect of eccentricity on the
unsteady fluid forces on the centrebody of an annular diffuser. Journal oj Fluids and
Structures 4, 155-169.

REFERENCES
1531
HOERNER, S.F. (1965) Fluid Dynamic Drag, 2nd edition, Brick Town, NJ: Hoerner Fluid
Dynamics.
HOERNER, S.F. (1985) Fluid Dynamic Lift. Albuquerque: Hoerner Fluid Dynamics.
HOFF, N.J. (1955) The accuracy of Donnell's equations. Joumal oj Applied Mechanics
22, 239-334,
HOLM-CHRISTENSEN, O. (1990) Instability of flexible tubes with orthotropic con-
stitution when exposed to internal fluid flow. DCAMM Report No. 407, Technical
University of Denmark, Lyngby.
HOLMES, P.J. (1977) Bifurcations to divergence and flutter in flow-induced osci1lations:
a finite-dimensional analysis. Joumal oJSound and Vibration 53, 471-503.
HOLMES, P.J. (1978) Pipes supported at both ends cannot flutter, Journal oj Applied
Mechanics 45,619-622,
HOLT, M. (1984) Numerical Methods in Fluid Dynamics. New York: Springer-Verlag.
HOOPER, J.D. & REHME, K. (1984) Large-scale structural effects in developed turbulent
flow through closely-spaced rod arrays. Journal oj Fluid Mechanics 145, 305-337.
HORÁČEK, J, (1985) The estimation of the dynamic response of cylindrical shells to a
random pressure field. Acta Technica ČSAV 33, 708-716.
HO RÁČEK, J. (1986) The computation of the dynamic response of cylindrical shells in
a turbulent flow. Acta Technica ČSAV 34, 214-229.
HO RÁČEK, J. (1989) Modely fluktuací tlaku na stěně v turbulentním proudu. Strojnícky
Čaopis 40,223-237.
HORACEK, J. (1993) Approximate theory of annular flow-induced instabilities of
cylindrical shells. Joumal oj Fluids and Structures 7, 123-135.
HO RÁČEK, J. & KRUNTCHEVA, M. (1997) Finite element modeling of the damping
of cylindrical shells vibrating in contact with an annular fluid region. Journal oj Sound
and Vibration 203, 723-727.
HO RÁČEK, J. & SIMERSKÁ, C. (1994a) Eigenvibration and stability of an elastically
supported plate in a channel conveying fluid. Zeitschrift fUr angewandte Mathematik
und Mechanik 74, T57- T58.
HORÁČEK, J. & SIMERSKÁ, C. (1994b) Dynamic behaviour of a flat body in
a rectangular channel. Proceedings 2nd lntemational ConJerence on Engineering
Aero-Hydroelasticity, Pilsen, Czech Republic, pp. 74-79.
HORÁČEK, J. & ZOLOTAREV, I. (1981) Stability and bending wave propagation in a
lonvcylindrical shell. Strojnícky Časopis 32,687-700 (in Czech).
HO RACEK, J. & ZOLOTAREV, I. (1983) Natural frequency, damping and stability of
long cy1indrical shells with material damping. Strojnícky Časopis 34, 189-203 (in
Czech).
HO RÁČEK, J. & ZOLOTAREV, I. (1984) Influence offixing the edges of a shell conveying
fluid on its dynamic characteristics. Soviet Applied Mechanics 20, 756--765.
HORÁČEK, J. & ZOLOTAREV, I. (1991a) Acoustic-structural coupling of vibrating
cylindrica1 shells with flowing fluid. Joumal oj Fluids and Structures 5,487-501,
HORÁČEK, J. & ZOLOTAREV, I. (1991b) Frequency-modal characteristics of cylindrical
shell with annular flow of a light fluid. Proceedings 8th World Congress on the Theory
oj Machines and Mechanisms, pp, 1503-1506, Prague, Czechoslovakia.
HORÁČEK, J. & ZOLOTAREV, I. (2002) Free vibration and stability of cylindrical shells
in interaction with flowing fluid. In NATO CLG Grant Report No. PST.CLF.977350
(eds F. Pellicano, Y. Mikh1in & I. Zolotarev), pp. 45-82. Prague: Institute of
Thermomechanics (ISBN 80-85918-76-5).

1532
SLENDER STRUCTURES AND AXIAL FLOW
HORÁČEK, J" TRNKA, J., VESELY, J. & GORMAN, D,G, (1995) Vibration analysis
of eylindrieal shells in eontaet with an annular fluid region. Engineering Structures 17,
714-724.
HORN, W" BARR, G., CARTER, L. & STEARMAN, R (1974) Reeent eontributions to
experiments on eylindrieal shell panel flutter, A/AA Journal12, 1481-1490,
HOSAKA, H. & CRANDALL, S.H. (1992) Self-exeited vibrations of a flexible disk
rotating on an air film above a flat surfaee. Acta Mechanica 3 [suppl], 115-127.
HOUGH, S.S. (1896) The rotation of an elastie spheroid. Philosophical Transactions oj
the Royal Society (London) A 187, 319-344,
HOURIGAN, K., THOMPSON, M.C. & TAN, B.T. (2001) Self-sustained osci1lations in
flows around 10ng blunt plates. Journal oj Fluids and Structures 15, 387-398.
HOWARD, L.N. & GUPTA, A.S. (1962) On the hydrodynamie and hydromagnetic
stability of swir1ing flows. Journal oj Fluid Mechanics 14, 463-476.
HOWARD, L., LANDAHL, M. & MASLOWE, S. (1972) Instabilities of rotating flows.
Progress Report, The Aeronautieal Researeh Institute of Sweden, DSR73742, 1
January-31 Oetober 1972. MIT, Cambridge, MA, USA.
HOWE, M.S. (1998) Acoustics oj Fluid-Structure /nteractions. Cambridge: Cambridge
University Press.
HSIEH, J.-C., PLAUT, RR. & YUCEL, O. (1989) Vibrations of inextensible eylindrieal
membrane inflated with liquid. Journal oj Fluids and Structures 3, 151-163.
HUANG, L. (1995a) F1utter of eantilevered plates in axial flow. Journal oj Fluids and
Structures 9, 127-147.
HUANG, L. (1995b) Mechaniea1 modeling of palata1 snoring. Journal oj the Acoustical
Society oj America 97, 3642-3648.
HUANG, L. (1995e) Private eommunieation to E.H. Dowell (19 Jun 1995).
HUANG, L. (1998) Reversal ofthe Bemoulli effeet and ehannel flutter. Journal oj Fluids
and Structures 12, 131-151.
HUANG, L. (1999) A theoretieal study of duet noise eontrol by flexible panels. Journal
oJthe Acoustical Society oj America 106, 1801-1809.
HUANG, L. (2001 a) Viseous flutter of a finite elastie membrane in Poiseui11e flow. Journal
oj Fluids and Structures 15, 1061-1088.
HUANG, L. (2001b) A theoretieal study of passive eontrol of duet noise using panels of
varying eomplianee. Journal oJthe Acoustical Society oj America 109, 2805-2814,
HUANG, L. (2002) Private eommunieation (27 Jan 2002).
HUANG, L. & FFOWCS WlLLIAMS, J,E, (1999) Neuromeehaniea1 interaetion in human
snoring and upper airway obstruetion, Journal oj Applied Physiology 86, 1759-1963;
see also pp, 1757-1758.
HUANG, L., CHOY, Y.S., SO, RM.C. & CHONG, T.L. (2000) Experimental study of
sound propagation in a flexible duet. Journal oJthe Acoustical Society oj America 108,
624-631,
HUERRE, P. (2000) Open shear flow instabilities. In Perspectives in Fluid Dynamics (eds
G.K. Batehelor, H.K. Moffat & M.G. Worster). Cambridge: Cambridge University
Press.
HUERRE, P. & MONKEWITZ, P.A. (1985) Absolute and eonveetive instabilities in free
shear layers, Journal oj Fluid Mechanics 159, 151-168.
HUERRE, P. & MONKEWlTZ, P.A. (1990) Loeal and globa1 instabilities in spatia11y
developing flows. Annual Review oj Fluid Mechanics 22,473-537,

REFERENCES
1533
HULTGREN, L. (1976) The stability ofaxia1 flow in a rotating gas. Report FFA AU-
751, Part 29, F1ygtekniska Fbrsbksansta1ten (The Aeronautica1 Research Institute of
Sweden), Stockholm.
HULTGREN, L. (1978) Stability ofaxisymmetric gas flows in a rapid1y rotating cy1-
indrica1 container. Report FFA AU-751, Part 37, Flygtekniska Fbrsbksansta1ten (The
Aeronautica1 Research Institute oť Sweden), Stockho1m.
HUTCHISON, H, (2001) Basic drives. ASME Mechanical Engineering 123(12), 51-55.
HWANG, J.Y. & THORPE, J.F. (1973) Further investigation of stability oť a p1ate in a
rectangu1ar flow channel. ASME Paper 73-DET-66.
IBNOU-ZAHIR, M. (1984) Prob1eme de va1eurs propres avec conditions aux limites non
10ca1es. Ph,D. Thesis, L'Université Pierre et Marie Curie, Paris, France.
IBRAHIM, R.A, ORONO, P.O. & MADABOOSI, S.R. (1990) Stochastic flutter oť a
panel subjected to random in-plane forces. Part 1. two-mode interaction. AlAA Journal
28, 694-702.
IBRAHIM, R.A, PILIPCHUK, V,N, & lKEDA, T. (2001) Recent advances in liquid
sloshing dynamics. Applied Mechanics Reviews 54, 133-199.
IDEL'CHIK, LE. (1966) Handbook oj Hydraulic Resistance, Translated from Russian,
AEC-tr-6630. Springfield, VA: United States Department oť Commerce, AIso 1986,
New York: Hemisphere.
lKEDA, T., HIRAYAMA, T. & NAKAGAWA, N. (1998) Nonlinear vibrations of a struc-
ture caused by water sloshing in a cylindrical tank, JSME lnternational Journal, Series
C, 41, 639-651.
ILGAMOV, M.A, TANG, D.M. & DOWELL, E.H, (1994) Flutter and forced response
oť a cantHevered pipe: the influence oť internal pressure and nozzle discharge. Journal
oj Fluids and Structures 8, 139-156.
INADA, F. & HAYAMA, S. (1988) A study on leakage-flow-induced vibrations (fluid-
dynamic ťorces acting on the walls oť a one-dimensional, narrow passage). JSME
lnternational Journal, Series III, 31, 39-47.
INADA, F. & HAYAMA, S. (1990a) A study oť leakage-flow-induced vibrations. Part 1.
Fluid dynamic ťorces and moments acting on the walls of a narrow tapered passage.
Journal oj Fluids and Structures 4,395-412.
INADA, F. & HAYAMA, S. (1990b) A study of leakage-flow-induced vibrations. Part 2:
stability analysis and experiments for two-degree-of-ťreedom systems combining
translational and rotationa1 motions. Journal oj Fluids and Structures 4, 413-428,
INADA, F. & HAYAMA, S, (1993) Leakage-flow-induced vibrations in a narrow tapered
passage (The effect of nondimensional parameters), Proceedings Asia-Pacific Vibration
ConJerence'93, Kitakyushu, Japan, pp. 242-247.
INADA, F. & HAYAMA, S. (2000) Mechanisms of leakage-flow-induced vibrations -
Single-degree-of-freedom and continuous systems. In Flow lnduced Vibration (eds
S, Ziada & T. Staubli), pp. 837-844. Rotterdam: AA Ba1kema.
INCE, E.L. (1956) lntegration oj Ordinary Differential Equations. Edinburgh: Oliver &
Boyd.
ISHII, T. (1965) Aeroelastic instabilities of simply supported panels in subsonic flow.
AIAA paper 65-772.
IWATSUBU, T. (1981) Dynamical relation between self-excited vibration and fish
propulsion mechanism. Memoires oť the Faculty oť Engineering, Kobe University,
No. 27, 1-14.

1534
SLENDER STRUCTURES AND AXIAL FLOW
JAHN, T.L. & VOTfA, J.J. (1972) Locomotion of protozoa, Annual Review oj Fluid
Mechanics 4, 93-116.
JAIN, RK (1974) Vibration of fluid- filled, orthotropic cyHndrical shells. Journal oj Sound
and Vibration 37, 379-388.
JEDWAB, M, (1987) F10w in annular diffusers. Ph.D. Thesis, Cambridge University,
Cambridge, UK
JEFFREY, N.E. (1968) Influence of design features on underwater towed system stability.
Journal oj Hydronautics 2,205-213.
JENSEN, O.E. (1990) Instabilities of flow in a collapsed tube. Journal oj Fluid Mechanics
220,623-659.
JENSEN, O,E, (1992) Chaotic oscillations in a simple collapsible-tube model. ASME
Journal oj Biomechanical Engineering 114, 55-59.
JENSEN, O.E. & PEDLEY, T.J. (1989) The existence of steady flow in a collapsed tube,
Journal oj Fluid Mechanics 206, 339-374.
JEONG, K-H, (1998) Natura1 frequencies and mode shapes of two coaxia1 cyHndrica1
shells coupled with bounded fluid. Journal oJSound and Vibration 215,105-124.
JOHANSSON, E.B. (1960) Hydraulic in stability of reactor para11el-plate fuel assembHes.
Nuclear Engineering & Science ConJerence, New York, Paper No. 57. New York:
Engineers Joint Council; also, General Electric Report KAPL-M-EJ-9 (1959).
JOHNS, D.J. (1966) CyHndrical shell flutter - theory and experiment. Journal oj the
Royal Aeronautical Society 70, 533.
JOHNS, D,J, (1969) A panel flutter review. AGARD Manual on Aeroelasticity, Part III,
Chapter 7, May 1969,
JONES, R (1927) Distribution of norma! pressures on a prolate spheroid, Aeronautica!
Research Counci1 (UK) (ARC) R&M 1061.
JONES, R.T, (1945) Properties of 10w-aspect ratio pointed wings at speeds below and
above the speed of sound. NACA Report 835.
JONSSON, V,K & SPARROW, E.M. (1966) Experiments on turbulent-flow phenomena
in eccentric annular ducts. Journal oJ Fluid Mechanics 25, 65-86.
JUNGER, M.C. (1955) The effect of a surrounding fluid on pressure waves in a fluid- filled
elastic tube, Journal oj Applied Mechanics 22,227-231.
JUNGER, M.C, & FEIT, D. (1986) Sound, Structures and Their lnteraction. Cambridge,
MA: MIT Press.
WNGER, M,C. & GARRELICK, J.M, (1984) Multiple moda1 resonances of thin cyl-
indrica1 shells vibrating in an acoustic medium. Journal oj the Acoustical Society oj
America 75, 1380-1382.
KADLEC, J, & OHLMER, E. (1971) On the reproductivity of the parallel-flow induced
vibration offuel pins, Nuclear Engineering and Design 17, 355-360.
KALANTZIS, S., MODI, Y.J., PRADHAN, S. & MISRA, A.K (1996) On the dynam-
ics and control of N-body tethered systems. AIAA/AAS Astrodynamics Specialist
ConJerence, San Diego, USA, Paper AIAA-96-357 I-CP.
KALAYCIOGLU, S. & MISRA, A.K (1991) Approximate solutions for vibrations
of deploying appendages. AlAA Journal oj Guidance, Control, and Dynamics 14,
287-293.
KAMM, RD. (1987) Flow through collapsible tubes. In Handbook oj Bioengineering
(eds R, Skalak & S. Chien), pp. 23.1-23,19, New York: McGraw-Hi1l,
KAMM, R.D. & PEDLEY, T.J. (eds) (1989a) Specia! issue on collapsible tubes. ASME
Journal oj Biomechanical Engineering 111(3), 177-260,

REFERENCE S
1535
KAMM, R.D. & PEDLEY, T.J. (1989b) F10w in collapsib1e tubes: a brief review, ASME
Joumal oJBiomeehanieal Engineering 111,177-179.
KAMM, RD. & SHAPIRO, A.H. (1979) Unsteady flow in a collapsib1e tube subjected to
extema! pressure or body forces. Joumal oj Fluid Meehanies 95, 1-78.
KAMM, RD., PATEL, N.R. & ELAD, D. (1993) On the effect of flow-induced flutter on
flow rate during a forced vita! capacity maneuver, FASEB Joumal7, A11.
KANA, D.D. (1966) Interaction between 1iquid propellants and the e1astic structure. In
The Dynamie BehavioroJLiquids in Moving Containers (ed. H.N, Abramson), NASA-
SP-106, Washington, DC: Govemment Printing Office, pp. 303-352.
KANAZAWA, RM, & BORESI, A.P. (1970) Ca!cu1ation of the response of rods to
boundary-1ayer pressure fluctuations. In Proeeedings ConJerence on Flow-Indueed
Vibrations in Reaetor System Components. Argonne Nationa! Laboratory Report
ANL-7685, pp. 47-63.
KANEKO, S. & HlROTA. K. (1992) A study on active contro1 of 1eakage flow induced
vibrations. Proeeedings I st Intemational ConJerenee on Motion and Vibration Control,
Yokohama, Japan, pp, 1135-1140.
KANEKO, S" OZAKI, y, & W ATANABE, T. (2002) Optica! fiber cab1e vibrations induced
by 1eakage-flow during coating process. Proeeedings 5th Intemational Symposium on
FSI, AE & FlV+N, (eds M.P. Pa'idoussis et al.), New Orleans, LA, USA New York:
ASME.
KANTOR, B. Ya. & KVITNITSKlI, M.P. (1988) The plane prob1em of hydroe1astic
stability for a hinge-supported p1ate. PMM Joumal oj Mathematies and Meehanies 52,
611-615.
KARAGlOZIS, K.N., GRINEVICH, E., PAIDOUSSIS, M,P" MISRA, A.K. &
AMABILI, M. (2003) Stability and non-linear dynamics of c1amped circu1ar cy1indrica!
shells containing flowing fluid. Proeeedings IUTAM Symposium on Integrated Model-
ing oJJully Coupled Fluid-Strueture Interaetions (eds H. Benaroya & T. Wei), Rutgers
University, USA.
KÁRMÁN, T. von (1930) Ca!cu1ation of pressure distributions on airship hulls, NACA
TM 574.
KARIMBAER, T,D., KOROLER, V,P. & EPANOV, V.G. (1974) Experimenta1 investiga-
tion of frequencies and forms of natura! vibration of glass-fibre reinforced p1astic shells
(in Russian). problemy Proehnosti 6,31-33.
KARPEL, M" YANIV, S. & LlVSHITS, D.S. (1998) Integrated solution forcomputationa!
static aeroe1asticity of rockets. AIAA Joumal oj Spaeeeraft and Roekets 35,612-618,
KATO, N., KODA, S. & TAKAHASHI, T. (1986) Motions of underwater towed system.
Proeeedings 5th Offshore Meehanies and Aretie Engineering Symposium, Vol. III,
pp. 426-433. New York: ASME,
KATZ, J. & PLOTKlN, A. (1991) Low-Speed Aerodynamies: Jrom Wing Theory to Panel
Methods. New York: McGraw-Hi1l.
KA W AGUTI, M. (1961) Effects of the ground and a tunne1 on the frictiona! drag of high
speed trains. Quarterly Report oj Railway Teehnieal Research Institute oj Japan 2,
No. 4,1-6.
KEANE, A.J. & PRICE, W.G. (1997) Statistieal Energy Analysis: An Overview, with
Applieations in Struetural Dynamies. Cambridge: Cambridge University Press.
KELLY, H.R, RENTZ, A.W, & SIEKMANN, J, (1964) Experimenta1 studies on the
motion of a flexib1e hydrofoil. Joumal oj Fluid Meehanies 19, 30-48.

1536
SLENDER STRUCTURES AND AXIAL FLOW
KELVIN, Lord, (1910) On the precessional motion of a liquid (liquid gyrostats); On an
experimental illustration of minimum energy. Mathematical and Physical Papers 4,
129-134 and 183-185.
KEMPNER, J, (1955) Remarks on Donnelľs equations, Journal of Applied Mechanics
22,117-118.
KENNEDY, RM. (1987) Forced vibration of a thin flexible cylinder in viscous flow.
Journal of Sound and Vibration 115, 189-201.
KENNEDY, C,C. & PANCU, C.P.D. (1947) Use ofvectors in vibration measurements and
analysis. Journal ofthe Aeronautical Sciences 14, 603-625.
KlM, KC. & KlM, J.S. (1978) The effect of the boundary condition on the added mass
of a rectangular plate. Journal of the Society of Naval Architects of Korea 15, I-II
(in Korean),
KlM, KC., KIM, J.S. & LEE, H,Y. (1979) An experimenta1 study on the elastic vibration
of plates in contact with water. Journal of the Society of Naval Architects of Korea 16,
1-7 (in Korean),
KlMURA, S. & BEJAN, A. (1983) The buckling of a verticalliquid column. ASME
Journal of Fluids Engineering 105, 469-473.
KlVITY, Y. & COLLlNS, R (1974) Non1inear wave propagation in visco-elastic tubes:
application to aortic rupture. Journal ofBiomechanics 7,67-76.
KLEIN, C, (1981) The effect of randomly varying added mas s on the dynamics of a flexible
cylinder in two-phase axially flowing fluid. M.Eng. Thesis, Department of Mechanical
Engineering, McGill University, Montrea1, QC, Canada.
KNOWLTON, EP. & STARLING, E.H. (1912) Tbe influence ofvariations in temperature
and blood pressure on the peďormance of the isolated mamma1ian heart. Journal of
Physiology 44, 206-219.
KNUDSEN, J.N. & KATZ, D.L. (1958) Fluid Dynamics and Heat Transfer, 6th edition.
New York: McGraw-Hill.
KO, P.L. (1993) Wear due to flow-induced vibration. In Technology for the '90s (ed.
M.K Au- Yang), Part III, Chapter 9. New York: ASME.
KO, P,L. (1997) Wear of power plant components due to impact and sliding, Applied
Mechanics Reviews :;0, 387-411.
KO, P.L. & ROGERS, RJ. (1981) Analytica1 and experimental studies of tube/support
interaction in multi-span heat exchanger tubes. Nuclear Engineering and Design 65,
399-409,
KOBAYASHI, S. (1962) 1Wo-dimensional panel flutter. I. Simply supported panel.
Transactions Japan Society for Aeronautical and Space Sciences 5, 90-102,
KOBAYASHI, Y. & LEISSA, A.W. (1995) Large amplitude free vibration of thick sha1low
shells supported by shear diaphragms, /nternational Journal of Non-Linear Mechanics
30, 57-66,
KOCHUPILLAI, J., GANESAN, N. & PADMANABHAN, C, (2002) A semi-analytical
coupled finite element formulaion for shells conveying fluids. Computers and Structures
80, 271-286.
KOIVUROVA, H. & PRAMILA, A. (1997) Nonlinear vibration ofaxially moving
membrane by finite element method. Computational Mechanics 20,573-581.
KOIVUROVA, H. & SALONEN, E.-M. (1999) Comments on non-linear formulations for
trave11ing string and beam problems. Journal of Sound and Vibration 225, 845-856.
KOLKMAN, P.A. (1980) Development ofvibration-free gate design: 1earning from exper-
ience and theory, ln Practical Experiences with Flow-/nduced Vibrations, pp, 351-385.
Berlin: Springer- Verlag.

REFERENCES
1537
KORN, G.A. & KORN, TM, (1956) Electronic Analog Computers. New York: McGraw-
Hm.
KORN, G.A. & KORN, TM. (1968) Mathematical HandbookJor Scientists and Engineers,
2nd edition. New York: McGraw HiU.
KORNECKl, A. (1967) Buckling of truncated conica1 shells under uniform static pressure.
A/AA Joumal 5,2099-2101.
KORNECKl, A. (1970) TraveHng wave-type flutter of infinite elastic p1ates. AlAA Joumal
8, 1342-1344.
KORNECKl, A. (1971a) Travelling wave type flutter of flat pane1s in inviscid flow (259
pages, 124 references), Parts I and II. Publ. No. 132 and 133, Agricultura1 Engineering
Faculty, Technion-lsrae1 Institute of Technology, Haifa, Israel.
KORNECKl, A. (1971b) A note on beam-type vibrations of circu1ar cy1indrica1 shells.
Joumal oj Sound and Vibration 14, 1-6.
KORNECKI, A. (1974) Static and dynamic instabiHty of pane1s and cylindrica1 shells in
subsonic potentia1 flow. Joumal oj Sound and Vibration 32, 251-263,
KORNECKl, A. (1978) Aeroe1astic and hydroe1astic instabi1ities of infinite1y 10ng p1ates,
I. SM Archives 3,381-440.
KORNECKI, A. (1979) Aeroelastic and hydroelastic instabi1ities of infinitely 10ng p1ates,
II. SM Archives 4,241-345.
KORNECKl, A., DOWELL, E.H. & O'BRIEN, J. (1976) On the aeroe1astic instabil-
ity of two-dimensiona1 pane1s in uniform incompressib1e flow, Joumal oj Sound and
Vibration 47, 163-178.
KRÁMER, E. (1993) Dynamics oj Rotors and Foundations. BerHn: Springer-Verlag.
KRAJCINOVIC, D. (1974) Vibration of two coaxia1 cy1indrica1 shells containing fluid.
Nuc/ear Engineering and Design 30, 242-248.
KRASNOPOLSKAYA, TS. & VAN HEDST, G,J.F. (1996) Wave pattern formu1ation in
a fluid annu1us with a radially vibrating inner cy1inder. Joumal oj Fluids Mechanics
328, 229-252.
KRAUS, H. (1967) Thin Elastic Shells. New York: John Wiley & Sons.
KRISTIANSEN, D. & EGELAND, O, (1999) Nonlinear oscillations in Corio1is-based
gyroscopes. Nonlinear Dynamics 19, 193-235.
KUBENKO, Y.D. & KOVAL'CHUK, P.S. (1998) Non1inear prob1ems of the vibration of
thin shells (review). lntemational Applied Mechanics 34,703-728.
KUBENKO, V.D. & KOVAL'CHUK, P.S. (2000) Non1inear prob1ems of the dynam-
ics of e1astic sheUs partia11y filled with a 1iquid. Prikladnaya Mekhanika 36, 3-34;
(in Russian). EngHsh version in lnternational Applied Mechanics 36, 421-448.
KUBENKO, Y.D., KOVAL'CHUK, P.S. & KRASNOPOL'S KAYA, T.S. (1982) Effect
of initia1 camber on natura1 nonlinear vibrations of cy1indrica1 sheUs. Soviet Applied
Mechanics 18, 34-39.
KÚSSNER, H.G. & SHWARZ, I. (1941) The oscillating wing with aerodynamica11y
ba1anced e1evator. NACA Technical Memorandum 99.
KUTlN, J. & BAJSlé, I. (1999) Characteristics of the shell-type CorioHs flowmeter.
Journal oj Sound and Vibration 228, 227-242.
KWAK, M.K. (1996) Hydroelastic vibration of rectangular plates. Joumal oj Applied
Mechanics 63, 110-115.
KWAK, M.K. (1997) Hydroe1astic vibration of circu1ar p1ates. Journal oj Sound and
Vibration 201,293-303.

1538
SLENDER STRUCTURES AND AXIAL FLOW
KWAK, M.K. & KIM, K.e. (1991) Axisymmetric vibration of circu1ar p1ates in contact
with water, Journal oJSound and Vibration 146, 381-389.
KYRIAKIDES, S. & CHANG, Y.e. (1990) On the inflation of a 10ng e1astic tube in the
presence ofaxia110ad. lnternational Journal oj Solids and Structures 26, 975-991.
LACROIX, e. & VIOLLET, P.L. (1987) Hydro-e1astic study of instabiHty of a crested
weir to vibrations. E1ectricité de France Report HE/44/86-24, Chatou, France.
LAI, Y.-e. & CHOW, e.-Y. (1973) Stability of a rotating thin e1astic tube containing a
fluid flow. ZeitschriftJiir angewandte Mathematik und Mechanik 53,511-517.
LAING, c.R., McROBIE, A & THOMPSON, J.M.T. (1999) The post-processed Ga1erkin
method appHed to non-Hnear shell vibrations. Dynamics and Stability oj Systems 14,
163-181.
LAITINEN, P.O. (1967) Cab1e-towed underwater body design. Report No. 1452, United
States Navy E1ectronics Laboratory, San Diego, CA, USA
LAITONE, E. V. (1947) Tbe subsonic flow about a body of revo1ution. Quarterly oj Applied
Mathematics 5, 227-234.
LAKIS, AA. (1971) Free vibration and response to random pressure fie1d of non-uniform
cy1indrica1 shells. Ph.D. Thesis, McGi11 University, Montrea1, Québec, Canada.
LAKIS, AA (1976) Theoretica1 model of cylindrica1 structures containing flowing fluids.
ln Proceedings Second lnternational Symposium oj Finite Element Methods in Flow
Problems, Santa Margherita Ligure, Italy.
LAKIS, AA (1978) Numerica1 ana1ysis of wall pressure fluctuations in two-phase flow.
Proceedings 5th lnternational ConJerence on Numerical Methods in Laminar and
Turbulent Flow, University College, Swansea, UK, pp. 757-767.
LAKIS, A,A (2002) Private communication (28 Jun 2002).
LAKIS, AA & DORÉ, R. (1974) Dynamic ana1ysis of anistropic thin cylindrica1 shells
subjected to boundary-1ayer induced random pressure fie1ds. Eco1e Po1ytechnique de
Montréa1 Report 74-R-27, Montréal, QC, Canada.
LAKIS, AA & LAVEAU, A (1991) Non-linear dynamic ana1ysis of anisotropic cy1-
indrica1 shells containing a flowing fluid, lnternational Journal oj Solids and Structures
28, 1079-1094.
LAKIS, AA & NEAGU, S. (1997) Free surface effects on the dynamics of cylindrica1
shells partia11y filled with 1iquid. Journal oj Sound and Vibration 207, 175-205.
LAKIS, AA & OURICHE, H. (1986) Dynamic ana1ysis of anisotropic conica1 shells.
Report EPM/RT-86/36, Beo1e Po1ytechnique de Montréa1, Montréal, QC, Canada.
LAKIS, AA. & PAIDOUSSIS, M.P. (1971) Free vibration of cylindrical shells partially
filled with liquid. Journal oj Sound and Vibration 19, 1-15.
LAKIS, AA & PAIDOUSSIS, M.P. (1972a) Dynamic ana1ysis ofaxially non-uniform
thin cylindrica1 shells. lMechE Journal oj Mechanical Engineering Science 14, 49-71.
LAKIS, AA & PAIDOUSSIS, M.P. (1972b) Prediction of the response of a cy1indrica1
shell to arbitrary or boundary-1ayer-induced random pressure fie1ds. Journal oj Sound
and Vibration 25, 1-27.
LAKIS, AA & SELMANE, A (2000) Hybrid finite element ana1ysis of 1arge amp1itude
vibration of orthotropic open and c10sed cylindrica1 shells subjected to a flowing fluid.
Nuc/ear Engineering and Design 196, 1-15.
LAKIS, AA & SINNO, M. (1992) Free vibration ofaxisymmetric and beam-like cyl-
indrical shells partia1ly filled with liquid. lnternational Journal oj Numerical Methods
in Engineering 33, 235-268.

REFERENCE S
1539
LAKIS, A.A, MOHAMED, S.S. & ROUSSELET, J. (1978) Turbu1ent two phase flow
loop faci1ity for predicting wal1-pressure fluctuations and shell response. Proceedings
24th lntemational lnstrumentation Symposium, A1buquerque, NM, USA, pp. 267-274.
Pittsburgh, PA: ISA
LAKIS, AA, VAN DYKE, P. & OURICHE, H. (1992) Dynamic analysis of anisotropic
fluid-filled conical shells. Joumal oj Fluids and Structures 6, 135-162.
LAKIS, A.A., SELMANE, A. & TOLEDANO, A (1998) Non-1inear free vibration ana-
1ysis of 1aminated orthotropic cylindrica1 shells. lntemational Joumal oj Mechanical
Science 40,27-49.
LAKIS, A.A., PAIDOUSSIS, M.P. & DUPUIS, C. (2000) Dynamic stability of fluid-
conveying cylindrical shells using a hybrid finite element approach. In Flow-lnduced
Vibration (eds S. Ziada & T. Staubli), pp. 343-348. Rotterdam: A.A. Ba1kema.
LALANNE, M. & FERRARIS, G. (1998) Rotordynamics Prediction in Engineering.
Chichester: John Wi1ey.
LAMB, H. (1898) On the ve10city of sound in a tube as affected by the e1asticity of the
walls. Manchester Memoirs (Proceedings oJthe Manchester Literary and Philosophical
Society) 42, No. 9.
LAMB, H. (1921) On the vibrations ofan e1astic plate in contact with water. Proceedings
oJthe Royal Society (London) A98, 205-216.
LAMB, Sir Horace (1957) Hydrodynamics, 6th edition. Cambridge: Cambridge University
Press.
LANDAHL, M. T. (1962) On the stability of a 1aminar incompressib1e boundary 1ayer over
a flexible surface. Joumal oj Fluid Mechanics 13, 609-632.
LANDAU, L.D. & LlFSHITZ, E.M. (1959) Fluid Mechanics. Reading, MA: Addison-
Wes1ey.
LANGDON, R.M. (1982) The vibrating cy1inder gyro. The Marconi Review, Fourth
Quarter,231-249.
LANGTHJEM, M.A. & OLHOFF, N. (2003) Modal expansion of the perturbation ve10city
potentia1 for a canti1evered fluid-conveying cylindrical she1L Joumal oj Fluids and
Structures 17, 147-161.
LAROSE, P.G. & GROTBERG, J.B. (1997) F1utter and long-wave instabi1ities in
comp1iant channe1s conveying deve10ping flows. Joumal oj Fluid Mechanics 331,
37-58.
LATORRE, R. & ZACHARIADIS, P. (1983) Feasibi1ity and economics of fluid cargo
transport by submerged flexib1e barges towed by a tugboat. Proceedings lntema-
tional Symposium on Ship Hydrodynamics and Energy Saving, E1 Pardo, Spain,
pp. X3-1-X3-24. Madrid: Cana1 de Experiencias.
LAUFER, J. (1953) The structure of turbulence in fully developed pipe flow. NACA
Technical Note 2954.
LEE, D. & KENNEDY, R.M. (1985) A numerical treatment of a mixed type dynamic
motion equation arising from a towed acoustic antenna in the ocean. Computers &
Mathematics with Applications 11, 807-816.
LEE, S.Y. & MOTE, C.D., Jr. (1997a) A generalized treatment of the energetics of trans-
1ating continua. Part I. Strings and second order tensioned pipes. Joumal oj Sound and
Vibration 204, 717-734.
LEE, S. Y. & MOTE, C.D., Jr. (1997b) A generalized treatment of the energetics of translat-
ing continua. Part II. Beams and fluid conveying pipes.Joumal oj Sound and Vibration
204, 735-753.

1540
SLENDER STRUCTURES AND AXIAL FLOW
LEE, T.S. (1981) Stabi1iy ana!ysis ofthe Ort1off-Ives equation, loumal oJFluid Mechanics
110, 293-295.
LEIBOVITCH, S. & STEWARTSON, K. (1983) A sufficient condition for the instability
of co1umnar vortices, Joumal oj Fluid Mechanics 126, 335-356.
LEISSA, A.w. (1969) Vibration of p1ates. NASA Report SP-160. Reprinted in book fonn
by the Acoustica! Society of America in 1993.
LEISSA, A.W. (1973) Vibration of shells. NASA Report SP-288, Reprinted in book fonn
by the Acoustica1 Society of America in 1993.
LELU, G. (1975) Computer mode11ing of the dynamic behaviour of a string of fue1 bund1es
in axia1 flow, Part B: response to genera!ized forcing and further deve1opments. Atomic
Energy of Canada Ltd. Report CRNL-1339. .
LEONARD, R. W. & HEDGEPETH, J ,M. (1957) On panel flutter and divergence of infin-
itely long unstiffened and ring-stiffened thin-wa11ed circu1ar cylinders. NACA Report
TR-1302.
LEVIN, L. & MILAN, D. (1973) Coup1ed breathing vibrations of two thin cy1indrica1
coaxia! shells in fluid. lntemational Symposium on Vibration Problems in lndustry,
Keswick, UK, Paper No. 616.
LEVIN, L., MILAN, D. & PAYAN, G. (1971) Recherches sur 1a réponse de coques
cylindriques a des excitations hydrodynamiques, La Houille Blanche 26,505-512.
U, D.- W., KANEKO, S. & HAYAMA, S. (2002) A study on annu1ar 1eakage-flow-induced
vibrations. Joumal oj Fluids and Structures 16, 909-930.
UANG, P.N. (1990) Thenna! acoustic wave propagation within a slightly compress-
ib1e viscous fluid-filled impermeab1e cy1indrica! e1astic tube. Ph.D. Thesis, Rensselaer
Po1ytechnic Institute, Troy, NY, USA.
LIANG, P.N. (1998) Private communications (Dec 1998 and Jan 1999),
LIANG, P,N, & SCARTON, H.A. (1994) Three-dimensiona1mode shapes forhigherorder
circumferentia! thermoe1astic waves in an annu1ar e1astic cylinder. Joumal oj Sound
and Vibration 177, 121-135.
UANG, P.N. & SCARTON, H.A. (1996) Attenuation of higher order circumferentia!
thennoacoustic waves in viscous fluid lines. Joumal oj Sound and Vibration 193,
1099-1113.
UANG, P.N, & SCARTON, H.A. (2002) Coincidence of the thennoe1astic and thenno-
viscous acoustic waves in fluid-filled elastic tubes. Joumal oJSound and Vibration 250,
541-565.
UBAl, A. & SIMMONDS, lG. (1998) Nonlinear Theory oj Elastic Shells, 2nd ed.
Cambridge: Cambridge University Press.
LIBRESCU, L. (1965) Aeroelastic stability of orthotropic heterogeneous thin pane1s in
the vicinity of the flutter critica1 boundary. Part I. Simp1y supported panels. Joumal de
Mécanique 4, 51-76.
UBRESCU, L. (1967) Aeroelastic stability of orthotropic heterogeneous thin panels in
the vicinity of the flutter critical boundary, Part II. Joumal de Mécanique 6, 133-152.
LIBRESCU, L. (1975) Elastostatics and Kinetics oj Anisotropic and Heterogeneous Shell-
Type Structures. Leiden: Noordhoff.
UGHTHILL, MJ. (1960a) Note on the swimming of slender fish. Journal oj Fluid
Mechanics 9,305-317.
UGHTHILL, M,J. (1960b) Mathematics and aeronautics. Joumal oJthe RoyalAeronaut-
ical Society 64, 375-394.

REFERENCES
1541
LlGHTHlLL, M.J. (1969) Hydromechanics oť aquatic anima! propu1sion, Annual Review
oj Fluid Mechanics 1, 413-446.
LlGHTHILL, M.J. (1970a) How do fishes swim? Endeavour 29, 77-83.
LlGHTHILL, M.J, (1970b) Aquatic anima1 propu1sion oť high hydromechanica1 effi-
ciency, Journal oj Fluid Mechanics 44,265-301.
LlGHTHILL, M,J, (1971) Large-amp1itude e10ngated-body theory oť fish 10comotion.
Proceedings oJthe Royal Society (London) B179, 125-138,
UGHTHILL, M.J. (1975) Mathematical Biofluiddynamics. Phi1ade1phia, PA: SIAM,
UGHTHILL, M.J. (1978) Waves in Fluids. Cambridge: Cambridge University Press.
LlGHTHILL, Sir J. (1990a) Biofluiddynamics oť balistiťorm and gymnotiťonn 10co-
motion, Part 2. The pressure distribution arising in two-dimensiona1 irrotationa! flow
ťrom genera! symmetrica1 motion oť a flexib1e flat p1ate nonna! to itselť. Journal oj
Fluid Mechanics 213, 1-10.
UGHTHILL, Sir J. (1990b) Biofluiddynamics oť balistiťonn and gymnotiťonn 10co-
motion. Part 3. Momentum enhancement in the presence oť a body oť e1liptic
cross-section. Journal oj Fluid Mechanics 213, 11-20.
UGHTHILL, Sir J. (1990c) Biofluiddynamics oť balistiťonn and gymnotiťonn
10comotion. Part 4. Short-wave1ength limitations on momentum enhancement. Journal
oj Fluid Mechanics 213,21-28.
LIN, H.C. & CHEN, S.S. (1977) Acoustically induced vibrations oť circu1ar cy1indrica1
rods. Journal oj Sound and Vibration 51, 89-96.
UN, L., ALURU, N.R., FORSTER, F.K. & ZHANG, X. (eds) (1998) Micro-Electro-
Mechanical Systems (ME MS) -1998, DSC-Vol. 66, New York: ASME,
LIN, T,C. & MORGAN, G,W, (1956a) A study oť axisymmetric vibrations oť cylindrica!
shells as affected by rotatory inertia and transverse shear, Journal oj Applied Mechanics
23, 255-261.
LIN, T.C. & MORGAN, G,W (1956b) Wave propagation through fluid contained in a
cylindrica1 e1astic shell. Journal oJthe Acoustical Society oj America 2S, 1165-1176.
UN, WH. (1984) Buffeting oť a s1ender circu1ar beam in axia1 turbu1ent flows. A1AA
JournaI22,690-695.
LIN, W.H, (1987a) Response oť a circu1ar rod to an axia1 turbu1ent flow. In Flow-1nduced
Vibrations-1987(edsM.K. Au-Yang&S,S. Chen), PVP-Vol. 122, pp. 81-86. ASME:
New York.
UN, WH. (1987b) Hydrodynamic ťorces on mu1tip1e circu1ar cy1inders osci11ating in a
viscous incompressib1e fluid, Zeitscrift fiir angewandte Mathematik und Mechanik 67,
487-501.
LIN, W.H. & CHEN, S.S. (1981) On added mass and radiation damping oť rod bund1es
osci11ating in compressib1e fluids. Journal oj Sound and Vibration 76, 441-453.
LIN, WH., & JENDRZEJCZYK, J.A. (1984) Hydroe1astic response oť a circu1ar tube in
eccentric annu1ar flows. Journal oj Sound and Vibration 94, 289-298.
UN, WH. & RAPTIS, A.C. (1986) Acoustic resonances in cylinder bund1es osci11ating
in a compressib1e fluid. ASME Journal oj Pressure Vessel Technology lOS, 197-201.
UN, WH., WAMBSGANSS, M.W & JENDRZEJCYK, J.A. (1981) Wall pressure
fluctuations within a seven-rod array. General E1ectric Report GEAP-24375.
LINDE, M, (1999) Private communication [address: Flygtekniska Forsoksanstalten (FFA) ,
Box 11021, SE-161 11 Bromma, Sweden] (6 Apri1 and 20 May 1999).
LINDHOLM, U.S., KANA, D.D. & ABRAMSON, H.N. (1962) Breathing vibrations oť
a circular cylindrica! shd1 with an internal fluid. Journal oj Aeronautical Science 29,
1052-1059,

1542
SLENDER STRUCTURES AND AXIAL FLOW
LIU, M. & GORMAN, D.G. (1996) Predictions of the vibratory characteristics of
cylindrical structures with presence of single-phase flow - I. Theoretical study;
II. Empirical modal analysis. Engineering Structures 18, 437-446 and 447-458.
LIU, P,J., RAKHEJA, S. & AHMED, A.K.W. (1998) Dynamic rollover threshold of artic-
ulated freight vehicles. Heavy Vehicle Systems, lntemational Joumal oJVehicle Design
5, 300-322.
LIVOLANT, M. & JEANPIERRE, F. (1978) Vibrational analysis of Superphenix internal
shells. Nuclear Engineering and Design 47, 289-296.
LOPES, J.-L., PAIDOUSSIS, M.P. & SEMLER, C. (1999a) Nonlinear equations of
a cylinder in steady axial flow. Mechanical Engineering Research Laboratories,
Report MERL-99-1, Department of Mechanical Engineering, McGill University,
Montreal, Québec, Canada.
LOPES, J.-L., PAIDOUSSIS, M.P, & SEMLER, C. (1999b) Nonlinear dynamics of
a cylinder in steady axial flow. Mechanical Engineering Research Laboratories,
Report MERL-99-2, Department of Mechanical Engineering, McGill University,
Montreal, Québec, Canada.
LOPES, J,-L., PAIDOUSSIS, M.P. & SEMLER, C. (2002) Linear and nonlinear dynamics
of cantilevered cylinders in axial flow. Part 2: The equations of motion. Joumal oj Fluids
and Structures 16, 715-737.
LOVE, A.E.H. (1927) A Treatise on the Mathematical Theory oj Elasticity, 4th edition
Cambridge: Cambridge University Press.
LOVEDAY, P.W. & ROGERS, C.A, (1998) Free vibration of elastically supported tmn
cylinders inc1uding gyroscopic effects, Joumal oJSound and Vibration 217, 547-562.
LOWE, TW. & PEDLEY, TJ. (1995) Computation of Stokes flow in a channel with a
collapsible segment. Joumal oj Fluids and Structures 9,885-905.
LU, Y. & ROGERS, R.J. (1995) Instantaneous squeeze-film force between a heat
exchanger tube and a support plate for arbitrary tube motion. Joumal oj Fluids and
Structures 9, 835-860,
LUCEY, A.D. & CARPENTER, P.W. (1992) A numerical simulation ofthe interaction of
a compliant wall and inviscid flow. Joumal oj Fluid Mechanics 234, 121-146.
LUCEY, A.D. & CARPENTER, P.W. (1993) On the difference between the hydroelastic
instability of infinite and very long compliant panels. Joumal oj Sound and Vibration
163, 176-181.
LUCEY, A.D" CAFOLLA, A.J., CARPENTER, P.W. & YANG, M, (1997) The nonlinear
hydroelastic behaviour of flexible walls. Joumal oj Fluids and Structures 11, 717-744,
LUCOR, D., EVANGELINOS, C., IMAS, L. & KARNIADAKIS, G.E. (2002) Flow-
induced vibrations of non-linear cables. Part 2: simulations. lnternational Joumal Jar
Numerical Methods in Engineering 55, 557-571.
LUNDGREN, T.S., SETHNA, P.R. & BAJAJ, A.K. (1979) Stability boundaries for flow
induced motions of tubes with an inc1ined terminal nozzle. Joumal oj Sound and
Vibration 64,553-571.
LUO, X.Y. & PEDLEY, TJ. (1995) A numerical simulation of steady flow in a 2-D
collapsible channel. Joumal oj Fluids and Structures 9,149-174,
LUO, X.Y. & PEDLEY, TJ. (1998) The effects ofwa11 inertia on flow in a two dimensional
collapsible channel. Joumal oj Fluid Mechanics 363, 253-280.
LUO, X,y' & PEDLEY, T.J. (2000) Multiple solutions and flow limitation in collapsible
channel flows. Joumal oj Fluid Mechanics 420, 301-324.

REFERENCE S
1543
LUU, T.P. (1983) On the dynamics of three systems involving tubular beams eonvey-
ing fluid. M.Eng. Thesis, Department of Meehanical Engineering, MeGi11 University,
Montreal, Québee, Canada.
LYON, R.H. (1973) Statistical Energy Analysis oj Dynamical Systems. Cambridge, MA:
MIT Press.
LYON, R.H. & DE JONG, R.G. (1995) Theory and Application oj Statistical Energy
Analysis. Boston: Butterworth-Heinemann.
LYON, R.H. & MAIDANIK, G. (1964) Review of some reeent researeh on noise and
struetural vibration. NASA TN D-2266,
MAGRAB, RB. & BURROUGHS, C. (1972) Foreed harmonie and random vibrations
of eoneentrie eylindriea! shells immersed in aeoustic fluids. Journal oj the Acoustical
Society oj America 52, 858-864,
MANGELSDORF, C.P. (1971) Koiter's modified energy funetional for cireular eyHndrieal
shells. AlAA Journal9, 2089-2099.
MANSFIELD, L. & SIMMONDS, J.G. (1987) The reverse spaghetti prob1em: drooping
motion of an elastiea issuing from a horizontal guide. Journal oj Applied Mechanics
54,147-150.
MARK, W.D. (1985) Aero/hydrodynamie stability of e1astieally supported plates in nar-
row ehannels with upstream barriers preventing flow redistribution. ASME Journal oj
Vibration, Stress, and Reliability in Design 107, 319-328.
MARKUŠ, Š. (1988) The Mechanics oj Vibrations oj Cylindrical Shells. Amsterdam:
Elsevier.
MARYNOWSKI, K. & KAPITANIAK, T. (2002) Kelvin- Voigt versus Biirgers internal
damping in modeHng ofaxially moving viseoelastic web. lnternational Journal oj
Non-Linear Mechanics 37, 1147-1161.
MAS LOWE , S.A. (1974) InstabiHty of rigidly rotating flows to non-axisymmetrie
disturbanees. Journal oj Fluid Mechanics 64,307-317.
MASLOWE, S, (1999) Private eommunieation (12 Jan 1999),
MAS ON, V.A., PETIIGREW, M,J., LELLI, G., KATES, L. & REIMER, E. (1978)
Dynamies of nuclear fuel ehannels: eomputer mode11ing and assoeiated studies. Atomie
Energy of Canada Report AECL-5976, Chalk River, Ontario, Canada.
MASSEY, B.S. (1979) Mechanics oj Fluids, 4th edition. New York: Van Nostrand
Reinhold.
MATEESCU, A.D. (1984) Pluid-elastic vibrations and stability of eyHndrieal shells
eonveying axial flow. M.Eng. Thesis, MeGi11 University, Montrea!, Québec, Canada.
MATEESCU, D. & PAlDOUSSIS, M.P. (1985) The unsteady potential flow in an axially
variab1e annulus and its effects on the dynamies of the osci11ating rigid eentre-body.
Journal oj Fluids Engineering 107, 421-427.
MATEESCU, D. & PAIDOUSSIS, M.P. (1987) Unsteady viscous effeets on the annular-
flow-indueed instabilities of a rigid eyHndrieal body in a narrow duet. Journal oj Fluids
and Structures 1, 197-215.
MATEESCU, D., PAIDOUSSIS & BÉLANGER, F. (1988) Unsteady pressure measure-
ments on an oseillating cylinderin narrow annular flow. Journal oj Fluids and Structures
2,615-628.
MATEESCU, D., PAIDOUSSIS, M.P. & BÉLANGER, F. (1989) A theoretieal model eom-
pared with experiments for the unsteady pressure on a eyHnder osei11ating in turbulent
annular flow. Journal oj Sound and Vibration 135, 487-498.

1544
SLENDER STRUCTURES AND AXIAL FLOW
MATEESCU, D., PAIDOUSSIS, M.P, & BÉLANGER, F. (1991) Computationa! solu-
tions based on a finite difference formu1ation for unsteady interna! flows. AIAA Paper
91-07-24.
MATEESCU, D., PAIDOUSSIS, M.P. & BÉLANGER, F. (l994a) A time-integration
method using artificia1 compressibility for unsteady viscous flows. Journal oj Sound
and Vibration 177, 197-205.
MATEESCU, D., PAIDOUSSIS, M.P. & BÉLANGER, F. (l994b) Unsteady annu1ar vis-
cous flows between oscillating cylinders. Part 1. Computationa1 solutions based on a
time-integration method. Journal oj Fluids and Structures 8, 489-507.
MATEESCU, D., PAIDOUSSIS, M.P, & BÉLANGER, F. (1994c) Unsteady annu1ar
viscous flows between oscillating cy1inders. Part n. A Hybrid time-integration solu-
tion based on azimutha1 Fourier expansions for configurations with annu1ar backsteps.
Journal oj Fluids and Structures 8, 509-527.
MATEESCU, D., PAIDOUSSIS, M.P, & SIM, W.-G. (1994d) Spectra1 solutions for
unsteady annu1ar flows between eccentric cy1inders induced by transverse osci11ations.
Journal oj Sound and Vibration 177, 635-649.
MATEESCU, D., PAIDOUSSIS, M.P, & SIM, W.-G. (1994e) A spectra1 collocation
method for confined unsteady flows with oscillating boundaries. Journal oj Fluids
and Structures 8, 157-181.
MATEESCU, D., PAIDOUSSIS, M.P. & SIM, W.-G. (1995a) Three-dimensiona1 viscous
flows between concentric cy1inders executing axially variable osci11ations: a hybrid
spectra1/finite difference solution. Journal oj Applied Mechanics 62,667-673,
MATEESCU, D., POTTIER, T., PEROTIN, L. & GRANGER, S. (1995b) Three-
dimensiona! unsteady viscous flows between oscil1ating eccentric cylinders by an
enhanced hybrid spectra1 method. Journal oj Fluids and Structures 9,671-695.
MATEESCU, D., MEKANIK, A. & PAIDOUSSIS, M.P. (1996) Ana1ysis oť 2-D and
3-D unsteady annu1ar flows with osci11ating doundaries, based on a time-dependent
coordinate transformation, Journal oj Fluids and Structures 10, 57-77.
MATSUZAKI, Y. (1973) Natura1 vibration and flutter of cylindrically curved pane1s. AlAA
Journalll,771-772.
MATSUZAKI, Y. (1981) Reexamination of stability of a two-dimensiona1 finite panel
exposed to an incompressib1e flow. Journal oj Applied Mechanics 48, 472-478.
MATSUZAKI, Y. & FUNG, Y.C. (1977a) Unsteady fluid dynamic forces on a simp1y-
supported circular cy1inder of finite length conveying a flow. Journal oj Sound and
Vibration 54, 317-330.
MATSUZAKI, Y. & FUNG, Y.C. (1977b) Stability ana1ysis of straight and buckled two-
dimensiona1 channe1s conveying an incompressib1e flow. Journal oj Applied Mechanics
44, 548-552.
MATSUZAKI, Y. & KOBAYASHI, S. (1969a) An ana1ytica1 study of the nonlinear flex-
ura1 vibration of thin circu1ar cy1indrica1 shells. Journal oj the Japan Society Jor
Aeronautical and Space Sciences 17, 308-315 (in Japanese).
MATSUZAKI, Y. & KOBAYASHI, S. (l969b) A theoretical and experimenta1 study of
the non1inear flexura1 vibration of thin circu1ar cy1indrica1 shells with c1amped ends.
Transactions oJthe Japan Society Jor Aeronautical and Space Sciences 12,55-62.
MATSUZAKI, Y. & KOBAYASHI, S. (1971) Unsteady supersonic aerodynamic forces
on an osci11ating circu1ar cylindrical she11. AlAA Journal9, 358-362.
MATSUZAKI, Y. & SElKE, K. (1996) Numerical ana1ysis of flow in a collapsib1e vesse1
based on unsteady and quasi-steady flow theories. In Computational Biomechanics (eds
K. Hayashi & H. Ishikawa), pp, 185-198. Tokyo: Springer-Verlag.

REFERENCES
1545
MATSUZAKI, y, & UEDA, T. (1980) Reexamination ofunsteady fluid dynamic forces on
a two-dimensiona! finite p1ate at smal1 Mach numbers. Joumal oj Applied Mechanics
47,720-724.
MAULL, D,J. (1978) Mechanisms of two and three-dimensiona1 base drag. In
Aerodynamic Drag Mechanisms oj Bluf! Bodies and Road Vehicles (eds G. Sovran,
T. More1 & W.T. Mason, Jr), pp. 137-159. New York: P1enum Press.
MAZÚCH, T., HORÁČEK, J., TRNKA, J. & VESELÝ, J. (1996) Natura! modes of a
thin c1amped-free steel cylindrica1 storage tank partially filled with water: FEM and
measurement. Joumal oj Sound and Vibration 193, 669-690,
McDONALD, D.A. (1974) Blood Flow in Arteries, 2nd edition. London: Edward Arno1d.
McLACHLAN, N. (1964) Theory and Applications oj Mathieu Functions. New York:
Dover Publications.
McLEAN, J.D. & DOWELL, E.H, (1975) Swir1ing flows between coaxial cylinders with
injection by radia!jets. AlAA Joumal13, 1435-1440,
McLEOD, A.R. (1918) On the action ofwind on flexib1e cab1es with application to cab1es
be10w aerop1anes and balloon cab1es. Aeronautica1 Research Counci1 (UK) (ARC)
R&M 554.
MEl, c., ABDEL-MOTAGALY, K. & CHEN, R. (1999) Review ofnonlinear panel flutter
at supersonic and hypersonic speeds, Applied Mechanics Reviews 52,321-332.
MEIROVITCH, L. (2001) Fundamentals oJ Vibrations. New York: McGraw-Hill.
MEIROVITCH, L. & BARUCH, H. (1983) On the inc1usion princip1e for the hierarchical
finite element method. lnternational Journal Jor Numerical Methods in Engineering
19,281-291.
MEKANIK, A. (1994) Genera1 solution for unsteady annu1ar flows between concent-
ric cylinders and annu1ar flow-induced instabi1ities, Ph,D. Thesis, Department oť
Mechanica! Engineering, McGill University, Montréa1, QC, Canada,
MENYAILOV, A. I. , PETERKA, F. & FORMÁNEK, P. (1986) Space-impact-flow-
induced oscillation of channe1-type nuc1ear reactor fuel casettes. The Institute of
Thermomechanics ČSAV, Report ZlOO3/86, Prague (in Russian).
MERCER, C.A. & HAMMOND, J.K. (1968) On the representation of continuous random
pressure fie1ds at a finite set of points. Journal oj Sound and Vibration 7, 49-61,
MEYERHOFF, W.K. (1970) Added masses of thin rectangu1ar p1ates calcu1ated from
potential theory, Joumal oj Ship Research 14, 100-111.
MIKKELSEN, L.P. & TVERGAARD, V. (1997) A nonloca! two-dimensional ana!ysis of
instabilities in tubes under interna! pressure. Danish Center for Applied Mathematics
and Mechanics, DCAMM Report No. 560, Lyngby.
MIKOLAJCZAK, A,A, (1957) The stresses in a 1iquid-filled flexib1e tube floating in
water. Dracone Deve10pment Report No, 5, Department of Engineering, University of
Cambridge, UK.
MILES, J.W. (1956) On the aerodynamic in stability of thin panels. Journal oj the
Aeronautical Sciences 23, 771-780.
MILES, J.W. (1957) Supersonic flutter of a cy1indrica1 shell. Joumal oJthe Aeronautical
Sciences 24, 107-118.
MILES, J.W, (1958) Supersonic panel flutter of a cylindrical shell- n, Joumal oj the
Aeronautical Sciences 25,312-316.
MILLER, D,R. (1960) Critical flow velocities for collapse of reactor parallel-plate fuel
assemblies. ASME Joumal oj Engineering Power 82, 83-95.

1546
SLENDER STRUCTURES AND AXIAL FLOW
MILLER, D,R. (1970) Generation of positive and negative damping with a flow restrictor
in axia! flow. In Proceedings oj the ConJerence on Flow-Induced Vibrations in Reactor
System Components (eds G,S, Rosenberg, M.W. Wambsganss & R.P. Carter) , Report
ANL-7685, pp. 304-311, Argonne National Laboratory, Argonne, IL, USA.
MILLER, D,R. &KENNISON, R.G. (1966) Theoreticalanalysisofflow-induced vibration
of a blade suspended in a flow channel. ASME paper 66-WNNE-1.
MISRA, A.K. & MODI, Y.J. (1982) Dynamics and control of tether connected two-body
systems - a brief review. 33rd Congress oJthe International Astronautical Federation,
Paris, France, paper 82-315,
MISRA, A.K., WONG, S.S.T. & PAIDOUSSIS, M,P. (2001) Dynamics and stability af
pinned-clamped and c1amped-pinned cylindrical shells conveying fluid, Journal oj
Fluids and Structures 15, 1153-1166,
MISTRY, J, & MENEZES, J.C, (1995) Vibration of cylinders partia!ly filled with liquid,
ASME Journal oJ Vibration and Acoustics 117, 87-93.
MIZOGUCHI, K. (1967) Vibration of a cylindrical shell containing a flowing fluid.
Bulletin oj JSME 10,59-67.
MIZOGUCHI, K. & KOMORI, S. (1978) Vibration and dynamic instability of a cylindrical
shell conveying a compressible fluid. Bulletin oj JSME 21, 628-636.
MODARRES-SADEGHI, Y., PAIDOUSSIS, M,P., SEMLER, C. & PICOT, P. (2003)
Nonlinear dynamics of slender cylinders supported at both ends and subjected to axia!
flow. Proceedings IUTAM Symposium on Integrated Modeling oJJully Coupled Fluid-
Structure Interactions (eds H. Benaroya & T, Wei), Rutgers University, USA.
MOHAMED, S.S. (1980) Prediction of velocity and wa1l pressure fluctuations of turbu-
lent dispersed two-phase flow. Ph,D, Thesis, Département de Génie mécanique, Ecole
Polytechnique de Montréa!, Montréal, Québec, Canada.
MONKEWITZ, P.A. (1989) The role of absolute and convective instabiIity in predicting
the behavior of fluid systems. Forum on Chaotic Dynamics in Fluid Mechanics, La
Jolla, CA, USA. ASME: New York.
MONKEWlTZ, P.A. & NGUYEN, L.N. (1987) Absolute instability in the near-wake of
two-dimensional bluffbodies. Journal oj Fluids and Structures 1,165-184.
MOODIE, T,B., TAIT, R.J. & HADDOW, J.B. (1982) Waves in compliant tubes. In Wave
Propagation in Viscoelastic Media (ed. F. Mainardi), pp. 124-168. Research Notes in
Mathematics, No. 52, Pitman.
MOODY, F.J. (1990) Introduction to Unsteady Thermofluid Mechanics. New York: John
Wiley & Sons.
MOON, F.C. (1992) Chaotic and Fractal Dynamics. New York: John Wiley & Sons.
MO ON, F.C. (ed.) (1999) New Applications oj Nonlinear and Chaotic Dynamics in
Mechanics. Dordrecht: Kluwer Academic Publishers.
MOORE, E,T, (1991) Avoiding vibration in penstocks. Waterpower '91, pp, 1585-1594.
MORAND, H. J.-P. & OHAYON, R. (1989) Fíníte element method applíed to the predic-
tíon of víbrations of liquid-propelled launch vehic1es. In Computational Experiments
(eds W.K. Liu, P. Smolinski, R. Ohayon, J, Navickas & J. Gvildys), pp. 75-84. New
York: ASME.
MORAND, H.J.-P. & OHAYON, R. (1995) Fluid Structure Interaction: Applied Numer-
ical Methods. Chichester: John Wiley & Sons.
MOREIRA, M., ANTUNES, J. & PINA, H. (2000) A theoretica! model for nonlinear
orbital motions of rotors under fluid confinement. Journal oj Fluids and Structures 14,
635-668,

REFERENCES
1547
MOREIRA, M., ANTUNES, J, & PINA, H, (2003) An improved 1inear model for rotors
subject to dissipative annu1ar flows. Journal oj Fluids and Structures 17, 813-832.
MORETTI, P.M. (2001) Private communications (15 Dec 2001, 22 Feb 2002),
MORETTI, P.M. & LOWERY, R.L. (1976) Hydrodynamic inertia coefficients for a tube
surrounded by rigid tubes. ASME Journal oj Pressure Vessel Technology 98,190-194.
MORGAN, G.W. & KlELY, J.P. (1954) Wave propagation in a viscous liquid contained
in a flexib1e tube. Journal oJthe Acoustical Society oj America 26,323-328.
MORKOVIN, M.V. (1988) Recent insights into stability and transition to turbulence in
open-flow systems. ICASE Report 88-44, NASA Lang1ey Research Center. Hampton,
VA: NASA.
MORLEY, L.S,D, (1959) An improvement on DonneU's approximation for thin-wal1ed
circular cylinders. Quarterly Journal oj Mechanics and Applied Mathematics 12,
89-99.
MORSE, P.M. (1948) Vibration and Sound, 2nd edition. New York: McGraw Hill; College
Park, MD: American Institute ofPhysics (1976),
MOTE, C.D. Jr., (1968) Dynamic stability of an axiaUy moving band. Journal oj the
Franklin lnstitute 285, 329-346.
MOTE, C.D. Jr., (1972) Dynamic stability ofaxially moving material, Shock and Vibration
Digest 14 (4), 2-11.
MOTE, C,D. Jr., (2002) Private communication (15 Mar 2002).
MŮFTŮ, S. & COLE, K.A. (1999) The fluid-structure interaction in supporting a thin
flexib1e cylindrical web with an air cushion. Journal oj Fluids and Structures 13,
681-708.
MUGGERlDGE, D.B. & BUCKLEY, T.J. (1979a) Dynamics of fluid conveying fiber-
reinforced shell. AlAA Journal17, 663-665.
MUGGERIDGE, D,B. & BUCKLEY, TJ. (1979b) Flexura! vibration of orthotropic
cylindrica! sheUs in a fluid medium. AlAA Journal17, 1019-1022.
MUJUMDAR, A.S, & DOUGLAS, W.J.M, (1976) Analytica! modeUing of sheet flutter.
Svensk Papperstidning 79,187-192.
MULCAHY, T.M. (1980) Fluid forces on rods vibrating in finite 1ength annu1ar regions.
Journal oj Applied Mechanics 47, 234-240.
MULCAHY, T,M. (1983a) A review of 1eakage-flow-induced vibrations of reactor
components. Report ANL-83-43, Argonne Nationa1 Laboratory, Argonne, IL, USA.
MULCAHY, T.M. (1983b) Leakage-flow-induced vibrations of reactor components. The
Shock and Vibration Digest 15(9), 11-18,
MULCAHY, T.M, (1984) Avoiding leakage flow-induced vibration by a tube-in-tube
slip joint. Report ANL-84-82, Argonne Nationa! Laboratory, Argonne, IL, USA.
A1so in Fluid-Structure lnteraction and Aerodynamic Damping (eds E.H. DoweU &
M.K. Au-Yang), (1985), pp. 159-169. ASME: New York.
MULCAHY, T.M. (1988) One-dimensiona11eakage flow vibration instabilities. Journal
oj Fluids and Structures 2, 383-403. See a!so Argonne Nationa! Laboratory Report
ANL-87.43 (1987),
MULCAHY, T.M., WAMBSGANSS, M.W., UN, W.H., YEH, T.T. & LAWRENCE,
W.P. (1981) Measurements of waU pressure fluctuations on a cylinder in annu1ar water
flow with upstream disturbances. Part 1: no flow sponers. General Electric Report
GEAP-24310, San José, CA, USA.
MULCAHY, TM., YEH, T.T & MISKEVICS, A.J. (1980) Turbulence androd vibrations
in an annular region with upstream disturbances. Journal oj Sound and Vibration 69,
59-69.

1548
SLENDER STRUCTURES AND AXIAL FLOW
MULCAHY, T,M., LAWRENCE, W. & WAMBSGANSS, M,W. (1982) Dynamic suďace-
pressure instrumentation for rods in para!le1 flow. Experimental Mechanics 22, 31-36.
MUNK, M.M. (1924) The aerodynamic forces on airship hulls. NACA Report No. 184.
MURDOCK, J.M. (1976) Fluid Mechanics and lts Applications. Boston: Houghton
Mifflin.
MUREITHI, N,W., PAIDOUSSIS, M.P. & PRICE, S.J. (1995) Intermittency transition to
chaos in the response of a 100se1y supported cylinder in an array, in cross-flow. Chaos,
Solitons & Fractals 5,847-867.
NAGAKURA, H. & KANEKO, S, (1993) Stability of a cantilever beam subjected to
one-dimensiona!leakage-flow. Proceedings Asia-Pacific ConJerence '93, Kitakyushu,
Japan, Vol. 1, pp. 248-253.
NAGIB, H.M" LA VAN, Z., FEJER, AA & WOLF, Z. (1971) Stability of pipe flow with
superposed solid body rotation. The Physics oj Fluids 14, 766-769.
NAKAO, T. (1983) Measurement of 1eakage flow induced vibrations of a cy linder. Joumal
oj Nuc/ear Science and Technology 20, 433-435.
NAMIAS, V, (1985) Load-supporting fluid-filled cyHndrica1 membranes. Joumal oj
Applied Mechanics 52,913-918.
NAUDASCHER, E. & ROCKWELL, D. (eds) (1980a) Practical Experiences with Flow-
lnduced Vibrations. Berlin: Springer-Verlag.
NAUDASCHER, E. & ROCKWELL, D. (1980b) Oscil1ator-mode1 approach to the iden-
tification and assessment of flow-induced vibrations of a system. Joumal oj Hydraulic
Research 18, 59-82.
NAUDASCHER, E. & ROCKWELL, D, (1994) Flow-lnduced Vibrations: An Engineering
Guide, Rotterdam: AA Ba1kema.
NAYFEH, A.H. (1981) lntroduction to Perturbation Techniques. New York: John Wiley
& Sons.
NAYFEH, A.H. & RAOUF, R.A. (1987) Non-linear osci1lation of circu1ar cyHndrica1
shells, lntemational Joumal oj Solids and Structures 23, 1625-1638.
NAYFEH, A.H., RAOUF, R.A. & NAYFEH, J,F. (1991) NonHnear response of infin-
ite1y 10ng circu1ar cylindrica! shells to subharmonic radia!loads. Joumal oj Applied
Mechanics 58, 1033-1041.
NEMAT-NASSER, S., PRASAD, S,N. & HERRMANN, G. (1966) Destabi1izing effect
of ve1ocity-dependent forces in non-conservative continuous systems. AlAA Joumal4,
1276-1280.
NEWMAN, J.N. (1977) Marine Hydrodynamics. Cambridge, MA: MIT Press,
NEWMAN,J,N. & WU, T,Y, (1973) Agenera1ized slender-bodytheoryforfish-like fonns.
Joumal oj Fluid Mechanics 57, 673-693.
NGUYEN, V.B. (1992) Theoretica! and experimenta1 study of the stability of c1amped-
free coaxia! cylindrica! shells subjected to interna! and annu1ar flows of viscous fluid.
Ph.D. Thesis, McGi1l University, Montréa1, QC, Canada.
NGUYEN, V.B., PAlDOUSSIS, M.P. & MISRA, AK. (1993a) A new outflow model for
cylindrica! shells conveying fluid, Joumal oj Fluids and Structures 7, 417-419.
NGUYEN, V,B., PAIDOUSSIS, M.P. & MISRA, AK. (1993b) An experimenta1 study of
the stability of canti1evered coaxia! cyHndrica! shells conveying fluid, Joumal oj Fluids
and Structures 7,913-930.
NGUYEN, V,B., PAlDOUSSIS, M.P, & MISRA, A.K. (1994c) A CFD-based model for
the study of the stability of canti1evered coaxial cylindrica! shells conveying viscous
fluid. Joumal oJSound and Vibration 176,105-125.

REFERENCES
1549
NICHOLS, W.w. & O' ROURKE, M.F. (1998) McDonald's Blood Flow in Arteries:
Theoretic, Experimental and Clinical Principles. London: Edward Arno1d.
NI, C.C. & HANSEN, R.J, (1978) An experimenta1 study ofthe flow-induced motions of
a flexible cy1inder in axia1 flow. ASME Journal oj Fluids Engineering 100, 389-394.
[Discussion 1979, 101,292-293; 1980,102, 119.]
NIEMI, J. & PRAMILA, A (1987) FEM-ana1ysis of transverse vibrations of an axi-
ally moving membrane immersed in idea1 fluid, lnternational Journal Jor Numerical
Methods in Engineering 24,2301-2313.
NIORDSON, F.!. (1953) Vibrations of a cy1indrica1 tube containing flowing fluid. Kung-
liga Tekniska Hogskolans Handlingar (Stockholm) No. 73. A1so Acta Polytechnica,
Mechanical Engineering Series, Vol. 3, No. 2, 1954, The Roya1 Swedish Academy of
Engineering Sciences.
NIORDSON, F.!. (1985) Shell Theory. Amsterdam: North-Holland.
NIORDSON, F.!. (2000) An asymptotic theory for circu1ar cy1indrica1 shells. lnternational
Journal oj Solids and Structures, 37, 1817-1839.
NOVOZHlLOV, V.V. (1964) The Theory oJ Thin Shells. Groningen: P. Noordhoff.
OHBA, K, YONEYAMA, N., SHIMANAKA, Y. & MAEDA, H. (1984) Self-excited
osci11ation of flow in a collapsib1e tube. Technology Report, Kansai University No. 25,
pp. 1-13.
OHBA, K, YONEYAMA, N., AONO, M. & SAKURAI, A (1985) Self-excited osci11ation
of flow in collapsible tube. II. Ana1ysis and experiment on initiation of osci11ation.
Technology Report, Kansai University No. 26, pp. 1-6.
OHBA, K, AONO, M" YONEYAMA, N. & SAKURAI, A (1989) F10w osci11ation and
wave propagation in collapsible tube. In Progress and New Directions oj Biomechanics
(eds Y.c. Fung et al.), pp. 213-227. Tokyo: Mita Press.
OHLMER, E. (1973) An experimenta1 investigation of an ana1ytica1 model for paralle1
flow induced vibrations of rod structures. In UKAEAlNPL Symposium on Vibration
Problems in lndustry, Session 5. Keswick, U.K
OHLMER, E., RUSSO, S. & SCHWEMMLE, R. (1972) Investigation of an ana1ytica1
model for paralle1 flow induced rod vibrations. Nuc/ear Engineering and Design 22,
272-289.
O'KEEFE, D.A (1965) Genera1 static aeroelastic ana1ysis for a body ofrevo1ution. AlAA
Journal3,2256-2261.
OLlVElRA, E.R. de A, GAGO, J. & REBELO, J. (1986) Adaptive macroelements using
sinusoida1 shape functions. In Accuracy Estimates and Adaptive Refinements in Finite
Element Computations (eds !. Babuska, O.c. Zienkiewicz, J. Gago & E.R. de A
Oliveira). New York: John Wi1ey & Sons.
OLSEN, J.G. & SHAPIRO, AH. (1967) Large-amp1itude unsteady flow in 1iquid-filled
elastic tubes. Journal oj Fluid Mechanics 29,513-538.
OLSON, M.D, (1965) Some experimenta1 observations on the non1inear vibration of
cy1indrica1 shells. AlAA Journal3, 1775-1777,
OLSON, M.D, & FUNG, Y.C. (1966) Supersonic flutter of circu1ar cyHndrica1 shells
subjected to interna1 pressure and axia1 compression. AlAA Journal 4, 858-864.
OLSON, M.D. & FUNG, Y.C. (1967) Comparing theory and experiments for the
supersonic flutter of circu1ar cylindrica1 shells. AlAA Journal5, 1849-1856.
OLSON, M.D. & LlNDBERG, G.M. (1971) Dynamic analysis of shallow shells with a
doub1y-curved triangu1ar finite element. Journal oj Sound and Vibration 19, 299-318.

1550
SLENDER STRUCTURES AND AXIAL FLOW
OLSSON, U. (1978) Supersonic flutter of heated circular cylindrical shells with
temperature-dependent materia! propertries. AlAA Journal16, 360-362.
OOMS, G., SEGAL, A. & VAN DER WEES, A.J. (1984) A theoretica! model for core-
annular flow of a very viscous oil core and a water annulus through a horizontal pipe.
lnternational Journal oj Multiphase Flow 10, 41-60,
ORTLOFF, C.R. & IVES, J. (1969) On the dynamic motion of a thin flexible cylinder in
a viscous stream. Journal oj Fluid Mechanics 38, 713-720.
OZ, H.R., PAKDEMIRLI, M. & BOYACI, H. (2001) Non-linear vibrations and stability
ofaxially moving beam with time-dependent velocity. lnternational Journal oj Non-
Linear Mechanics 36, 107-115.
PAIDOUSSIS, M.P. (1965a) Pressure waves in horizontal liquid-filled flexible tubes.
IMechE. Journal oj Mechanical Engineering Science 7,380-390.
PAIDOUSSIS, M.P. (1965b) The amplitude of fluid-induced vibration of cylinders in axial
flow. Atomic Energy of Canada Report AECL-2225, Chalk River, Ontario, Canada.
PAIDOUSSIS, M.P. (l966a) Dynamics of flexible slender cylinders in axial flow, Part 1.
theory. Journal oj Fluid Mechanics 26, 717-736.
PAlDOUSSIS, M.P. (1966b) Dynamics of flexible slender cylinders in axial flow. Part 2.
Experiments, Journal oj Fluid Mechanics 26,737-751.
PAlDOUSSIS, M.P. (1966c) Vibration of cylinders with supported ends, induced by axial
flow. Proceedings lnstitution oj Mechanical Engineers 180, 268-278.
PAIDOUSSIS, M.P. (1968) Stability of towed, totally submerged flexible cylinders.
Journal oj Fluid Mechanics 34, 273-297.
PAIDOUSSIS, M,P. (1969) An experimental study of vibration of flexible cylinders
induced by nominally axial flow. Nuclear Science and Engineering 35, 127-138.
PAIDOUSSIS, M.P. (1970) Dynamics of submerged towed cylinders. Eighth Symposium
on Naval Hydrodynamics: Hydrodynamics in the Ocean Environment. United States
ONR, ARC-179, pp. 981-1016.
PAIDOUSSIS, M.P. (1973a) Dynamics of cylindrical structures subjected to axia! flow.
Journal oj Sound and Vibration 29, 365-385.
PAIDOUSSIS, M.P. (1973b) Stability of conservative systems of tubular beams and
shells. MERL Technical Note 73-2, Department of Mechanical Engineering, McGill
University, Montreal, QC, Canada.
PAIDOUSSIS, M.P. (1974a) Stability of tubular cylinders conveying fluid. In Flow-
lnduced Structural Vibrations (ed. E. Naudascher), pp. 676-683. Berlin: Springer-
Verlag.
PAIDOUSSIS, M.P. (1974b) The dynamical behaviour of cylindrical structures in axial
flow. Annals oj Nuclear Science and Engineering 1, 83-106.
PAIDOUSSIS, M,P. (1974c) Vibrations of cylindrical structures subjected to axial flow.
ASME Journal oj EngineeringJor lndustry 96,547-552.
PAIDOUSSIS, M,P. (1975a) Flutter of conservative systems of pipes conveying incom-
pressible fluid. IMechE Journal oJMechanical Engineering Science 17, 19-25.
PAIDOUSSIS, M.P. (1975b) Stability of flexible slender cylinders in pulsatile axial flow.
Journal oJSound and Vibration 42, 1-11.
PAIDOUSSIS, M.P. (1975c) Mathematical model for the dynamics of an articulated string
of fuel bundles in axial flow. Paper D/5*, Transactions oj 3rd SMiRT (lnternational
Conf on Structural Mechanics in Reactor Technology), London, UK.
PAIDOUSSIS, M,P, (1976a) Hydroelastic ichthyoid propulsion. AlAA Journal oj Hydro-
nautics 10, 30-32.

REFERENCES
1551
PAIDOUSSIS, M.P, (1976b) Dynamics of fuel strings in axial flow, Annals oj Nuclear
Energy 3, 19-30,
PAIDOUSSIS, M.P, (1979) The dynamics of clusters of flexible cylinders in axial flow:
theory and experiments. Journal oj Sound and Vzbration 65,391-417.
PAIDOUSSIS, M,P, (1980) Flow-induced vibrations in nuclear reactors and heat
exchangers: Practical experiences and state of the knowledge, In Practical Experi-
ences with Flow-Induced Vibrations (eds E, Naudascher & D. Rockwell), pp. 1-81.
Berlin: Springer- Verlag.
PAIDOUSSIS, M,P. (1981) Fluidelastic vibration of cylinder arrays in axial and cross
flow: state of the art. Journal oj Sound and Vzbration 76, 329-360.
PAIDOUSSIS, M,P, (1982) Mter-dinner address. Nuclear Energy 21,302,
PAIDOUSSIS, M,P. (1983) A review of flow-induced vibrations in reactors and reactor
components. Nuclear Engineering and Design 74, 31-60,
PAlDOUSSIS, M,P. (1986) Stability of a chain of cylinders travelling underwater.
Proceedings 5th lnternational OMAE Symposium, Vol. 1, pp. 483-490. New York:
ASME.
PAIDOUSSIS, M,P, (1987) Flow-induced instabilities of cylindrical structures, Applied
Mechanics Reviews 40, 163-175.
PAIDOUSSIS, M,P. (1993) Calvin Rice Lecture: Some curiosity-driven research in fluid-
structure interactions and its current applications. ASME Journal oj Pressure Vessel
Technology 115, 2-14.
PAIDOUSSIS, M.P, (1997) Fluid-structure interactions between axial flows and slender
structures. In Theoretical and Applied Mechanics 1996 (eds T. Tatsumi, E, Watanabe
& T. Kambe), pp. 427-442. Amsterdam: Elsevier,
PAlDOUSSIS, M.P, (1998) Fluid-Structure lnterations: Slender Structures and Axial
Flow. Volume 1. London: Academic Press.
PAIDOUSSIS, M,P, & BESAN<;ON, P. (1981) Dynamics of arrays of cylinders with
internal and external axial flow, Journal oj Sound and Vzbration 76, 361-380.
PAIDOUSSIS, M.P, & BOTEZ, RM, (1993) Chaotic dynamics of articulated cylinders in
confined axial flow. Journal oj Fluids and Structures 7, 719-750.
PAlDOUSSIS, M.P. & BOTEZ, RM, (1995) Three routes to chaos for a three-degree-
of-freedom articulated cylinder system subjected to annular flow and impacting on the
outer pipe. Nonlinear Dynamics 7, 429-450.
PAIDOUSSIS, M.P, & CURLING, LI. R. (1985) An analytical model for vibration of
clusters of flexible cylinders in turbulent axial flow. Journal oj Sound and Vzbration 98,
493-517.
PAIDOUSSIS, M.P. & CURLING, LI.R, (1993) Unpublished report,
PAIDOUSSIS, M.P. & DEKSNIS, E.B. (1970) Articulated model s of cantilevers convey-
ing fluid: the study of a paradox. IMechE Journal oj Mechanical Engineering Science
12, 288-300.
PAlDOUSSIS, M,P, & DENISE, J,-P. (1970) Instabilities of cylindrical shells contain-
ing flow. Department of Mechanical Engineering, MERL Report No. 70-10, McGill
University, Montréal, QC, Canada.
PAIDOUSSIS, M.P. & DENISE, J.-P. (1971) Flutter of cylindrical shells conveying fluid.
Journal oj Sound and Vzbration 16, 459-461.
PAIDOUSSIS, M,P. & DENISE, J,-P, (1972) Flutter Df thin cylindrical shells conveying
fluid, Journal oj Sound and Vzbration 20, 9-26.

1552
SLENDER STRUCTURES AND AXIAL FLOW
PAIDOUSSIS, M.P. & GAGNON, J,O. (1984) Experiments on the vibration of cylinders
in axial flow: moda1 and spectra1 characteristics, Joumal oj Sound and Vibration 96,
341-352.
PAIDOUSSIS, M.P, & ISSm, N.T, (1974) Dynamic stability of pipes conveying fluid.
Joumal oj Sound and Vibration 33, 267-294,
PAIDOUSSIS, M.P. & LAKIS, AA (1972) Vibration characteristics of cylindrical shells
with severa1 axially equispaced constraints. Joumal oj Sound and Vibration 24, 51-62.
PAIDOUSSIS, M.P. & LI, G.x. (1992) Cross-flow-induced chaotic vibrations of heat-
exchanger tubes impacting on loose supports. Joumal oj Sound and Vibration 152,
305-326,
PAIDOUSSIS, M.P. & MATEESCU, AD. (1987) Dynamics of cylindrica1 shells
containing fluid flows with a developing boundary layer, AlAA Jouma125, 857-863,
PAIDOUSSIS, M.P. & OSTOJA-STARZEWSKI, M, (1981) Dynamics of a flexible
cylinder in subsonic axia1 flow. AlAA Joumall9, 1467-1475.
PAIDOUSSIS, M.P. & PETTIGREW, MJ. (1979) Dynamics of flexible cylinders in
axisymmetrica11y confined axial flow. Joumal oj Applied Mechanics 46,37-44,
PAIDOUSSIS, M.P. & SEMLER, C, (1993a) Nonlinear and chaotic oscillations of a
constrained cantilevered pipe conveying fluid: a full nonlinear ana1ysis. Nonlinear
Dynamics 4, 655-670.
PAIDOUSSIS, M.P. & SEMLER, C. (1993b) Nonlinear dynamics of a fluid-conveying
cantilevered pipe with an intennediate support. Joumal oj Fluids and Structures 7,
269-298.
PAIDOUSSIS, M.P. & SUSS, S, (1977a) Stability of a c1uster of flexible cylinders in
bounded flow. Journal oj Applied Mechanics 44,401-408. [Discussion 1978,45,455
and 703,]
PAIDOUSSIS, M.P. & SUSS, S. (1977b) Stability criteria of cylinders in axia1 flow.
Proceedings 6th Canadian Congress oj Applied Mechanics, University of British
Columbia, Vancouver, RC., Canada, pp. 705-706.
PAIDOUSSIS, M,P, & YU, RK. (1976) Elastohydrodynamics and stability of towed
slender bodies of revolution. AlAA Joumal oj Hydronautics 10, 127-134,
PAIDOUSSIS, M,P" SUSS, S. & PUSTEJOVSKY, M. (1977) Free vibration of c1usters
of cylinders in liquid-filled channels. Joumal oj Sound and Vibration 95, 443-459.
PAIDOUSSIS, M,P., ISSm, N,T, & TSUI, M, (1980a) Parametric resonance osci11a-
tions of flexible slender cylinders in hannonica11y perturbed axia1 flow. Part 1, Theory.
Joumal oj Applied Mechanics 47,709-714.
PAIDOUSSIS, M.P., ISSm, N.T, & TSUI, M. (1980b) Parametric resonance oscmations
of flexible slender cylinders in hannonical1y perturbed axial flow, Part 2. Experiments,
Joumal oj Applied M echanics 47, 715-719.
PAIDOUSSIS, M.P., ISSm, N.T, & TSUI, M. (1980c) Parametric resonance oscillations of
flexible slender cylinders in hannonically perturbed axia1 flow, MERL Report No, 80-2,
Department of Mechanical Engineering, McGill University, Montréa1, QC, Canada.
PAIDOUSSIS, M,P., CURLING, LI.R. & GAGNON, J.O. (1982) Experiments on
fluidelastic instability of cylinder clusters in axia1 flow, ASME Joumal oj Fluids
Engineering 104, 342-347.
PAIDOUSSIS, M,P., EL BARBIR, K.N., GENADRY, M.R. & CHAUBERNARD, J.-P.
(1983a) Dynamics of a cluster of flexibly interconnected cylinders. Part 1. In vacuum.
Joumal oj Applied Mechanics 50,421-428,

REFERENCES
1553
PAIDOUSSIS, M.P., CHAUBERNARD, J.-P., GENADRY, M.R. & EL BARBIR, K.N,
(1983b) Dynamics of a c1uster of flexibly interconnected cylinders. Part 2, ln axial flow.
loumal oj Applied Mechanics 50, 429-435,
PAIDOUSSIS, M.P., CHAN, S,P. & MISRA, A.K. (1984a) Dynamics and stability of
coaxia1 cylindrica1 shells containing flowing fluid. loumal oj Sound and Vibration 97,
201-235.
PAIDOUSSIS, M.P., MAVRIPLlS, D, & PRICE, S.J. (1984b) A potentia1 flow theory
for the dynamics of cylinder arrays in cross-flow. loumal oj Fluid Mechanics 146,
227-252.
PAlDOUSSIS, M.P" MISRA, A.K. & CHAN, S.P. (1985) Dynamics and stability of
coaxia1 cylindrica1 shells conveying viscous fluid. lournal oj Applied Mechanics 52,
389-396,
PAIDOUSSIS, M,P., LVU, T.P. & LAITHlER, RE. (1986) Dynamics of finite-Iength
tubular beams conveying fluid. loumal oj Sound and Vibration 106, 311-331.
PAIDOUSSIS, M.P., MATEESCU, D. & SIM, W.-G. (1990) Dynamics and stability of a
flexible cylinder in a narrow coaxia1 cylindrica1 duct subjected to annular flow. loumal
oj Applied Mechanics 57,232-240.
PAIDOUSSIS, M.P., NGUYEN, V.B. & MISRA, A.K. (1991) A theoretica1 study of the
stability of cantilevered coaxial cylindrica1 shells conveying fluid. loumal oj Fluids
and Structures 5, 127-164,
PAIDOUSSIS, M.P., MISRA, A.K. & NGUYEN, V.R (1992a) Intema1- and annular-
flow-induced instabi1ities of a clamped-clamped or cantilevered cylindrica1 shell in a
coaxia! conduit: the effects of system parameters. loumal oj Sound and Vibration 159,
193-205,
PAIDOUSSIS, M.P., MATEESCU, D. & BÉLANGER, F. (1992b) A computationa!
method for the dynamics of cylindrical structures subjected to annular flows by sim-
ultaneous integration of the Navier Stokes and structura1 equations, /ntemational
Symposium on Flow-/nduced Vibration & Noise, Vol. 5, PVP-Vol. 244, pp. 17-32.
New York: ASME,
PAIDOUSSIS, M.P., PRICE, S.J. & MUREITHI, W.N. (1993a) Nonlinear and chaotic
dynamics of a two-degree-of- freedom ana!ytica! model of a rotated triangular array in
cross-flow, loumal oj Fluids and Structures 7,497-520.
PAIDOUSSIS, M.P., TlAN, B. & MISRA, A.K. (1993b) The dynamics and stability of
pinned-clamped coaxia1 cylindrica1 shells conveying viscous flow, ln Proceedings 14th
Canadian Congress oj Applied Mechanics, Kingston, ON, Canada, pp. 259-260.
PAIDOUSSIS, M.P., LOPES, J.-L. & SEMLER, C. (1999) Nonlinear equations and
dynamica1 behaviour of a cantilevered cylinder in axial flow. Engineering Mechanics
6, 285-300,
PAIDOUSSIS, M.P., GRINEVICR, E., ADAMOVIC, D. & SEMLER, C, (2002) Lin-
ear and nonlinear dynamics of cantilevered cylinders in axial flow. Part I, Physical
dynamics. loumal oj Fluids and Structures 16, 691-713.
PAN, L.S, & CHEW, Y.T. (2002) A genera! fonnula for calculating forces acting on a 2-D
arbitrary body in any incompressible flow. loumal oj Fluids and Structures 16, 71-82.
PAO, R.P. (1970) Dynamical stability of a towed thin flexible cylinder. A/AA loumal oj
Hydronautics 4, 144-150.
PAO, S,K. & SIEKMANN, J. (1964) Note on the Smith-Stone theory of fish propulsion,
Proceedings oJthe Royal Society (London) A280, 398-408.

1554
SLENDER STRUCTURES AND AXIAL FLOW
PAO, H.P. & TRAN, Q. (1973) Response of a towed thin flexible cylinder in a viscous
fluid. Journal oJthe Acoustical Society oj America 53, 1441-1444.
PAPANASTASIOU, T.C., GEORGlOU, G.c. & ALEXANDROU, A.N. (2000) Viscous
Fluid Flow, Boca Raton: CRC Press.
PARKER, R (1966) Resonance effects in wake shedding from parallel plates: some
experimenta1 observations, Journal oj Sound and Vibration 4, 62-72.
PARKER, R (1967) Resonance effects in wake shedding from parallel plates: ca1culation
of resonance frequencies, Journal oj Sound and Vibration 5, 330-343.
PARKIN, M.M. (1980) Flow-Induced vibration problems in gas cooled reactors. In Prac-
tical Experiences with Flow-/nduced Vibrations (eds E. Naudascher & D. Rockwell),
pp. 126-136. Berlin: Springer- Verlag.
PARKIN, M.W. & WATSON, p,c. (1984) Reduction of vibration caused by flow in an
annular diffuser, Proceedings Symposium on Flow-/nduced Vibrations, Vol. 4 (eds
M.P. Pai"doussis & M,K Au-Yang), pp, 1-14, New York: ASME,
PARKlN, M,W" FOX, MJ.H., FRANCE, E.R & HOBSON, D.E. (1987) Annular
flow-induced vibration associated with on-load refueling of Advanced Gas cooled
Reactors. Proceedings BHRA /nternational ConJerence on Flow-/nduced Vibrations
(ed. R King), pp. 531-541. Cranfield: BHRA.
PARTHAN, S. & JOHNS, D.J. (1969) Flutter of circular cylindrical shells - a review
(90 pages, 143 references). Department of Transport Technology Report TT6917,
Loughborough University of Technology, U.K
PATEL, M.H. & SAYED, F.B. (1987) Analytica1 and design considerations fortowed pipe
lines. Proceedings Offshore and Arctic Pipelines - /987 (eds J .S. Chung & K Karal),
pp. 17-28. New York: ASME.
PATTANI, P,G. & OLSON, M.D, (1987) Periodic solutions of rigid body-viscous flow
interaction. /nternational JournalJor Numerical Methods in Fluids 7,653-696.
PATTANI, P.G. & OLSON, M.D. (1988) Forces on oscillating bodies in viscous fluid.
/nternational JournalJor Numerical Methods in Fluids 8, 519-536,
PATTON, KT, & SCHRAM, J.W. (1966) Equations of motion of a towed body moving in a
vertical plane. Report No. 736, U.S. Navy Underwater Sound Laboratory, Fort Turnbull,
Connecticut, USA.
PAVLlCA, RT. & MARSHALL, RC. (1966) An experimental study of fuel assembly
vibrations induced by coolant flow. Nuclear Engineering and Design 4, 54-60.
PEAKE, N, (1997) On the behaviour of a fluid-loaded cylindrical shell with mean flow.
Journal oj Fluids Mechanics 338,387-410.
PEAKE, N, (2001) Nonlinear stability of a fluid-loaded elastic plate with mean flow.
Journal oj Fluid Mechanics 434, 101-118.
PEDLEY, TJ. (1968) On the instability ofrapidly rotating shear flows to non-axisymmetric
disturbances. Journal oj Fluid Mechanics 31,603-607.
PEDLEY, T.J. (1969) On the instability ofviscous flow in arapidly rotating pipe. Journal
oj Fluid Mechanics 35,97-115.
PEDLEY, T.J. (1980) The FluidMechanicsoJLargeBloodVessels. Cambridge: Cambridge
University Press.
PEDLEY, T.J. & KESSLER, J.O. (1992) Hydrodynamic phenomena in suspensions of
swimming microorganisms. Annual Review oj Fluid Mechanics 24,313-358.
PEDLEY, T.J. & LUO, X,Y, (1998) Modelling flow and oscillations in collapsible tubes.
Theoretical and Computational Fluid Dynamics 10, 277-294.

REFERENCES
1555
PEDLEY, T,J" CARO, C,G, & NEREM, R,M, (eds) (1992) Specia! issue on blood flow
in large vessels, ASME Journal oj Biomechanical Engineering 114(1), 1-154,
PELLlCANO, F. & VESTRONI, F. (2000) Nonlinear dynallŮcs and bifurcations of an
axially moving beam. ASME Journal oJ Vibration and Acoustics 122, 21-30.
PELLlCANO, F. & VESTRONI, F. (2002) Complex dynamics of high-speed axia!ly
moving systems. Journal oj Sound and Vibration 258, 31-44.
PELLlCANO, F. & ZIRILLI, F. (1998) Boundary layers and non-linear vibrations in an
axiaI1y moving beam. International Journal oj Non-Linear Mechanics 33,691-711.
PELLlCANO, F., AMABILI, M. & VAKAKlS, A.F. (2000) Nonlinear vibrations and
multiple resonances of fluid-filled circular shells. Part 2. Perturbation analysis, ASME
Journal oJ Vibration and Acoustics 122, 355-364.
PELLlCANO, F., AMABILI, M, & PAIDOUSSIS, M.P, (2002) Effect of geometry on the
nonIinear vibration of circular cylindrica! shells. International Journal oj Non-Linear
Mechanics 37, 1181-1198.
PERCY, J.H., PIAN, T.H., KLEIN, S. & NAVARATNA, D.R. (1965) Application ofmatrix
displacement method to linear elastic analysis of shells of revolution. AlAA Journal 3,
2138-2145,
PERKlNS, N,C. & MOTE, C.D., Jr, (1987) Three-dimensiona! vibration of traveIing
elastic cables. Journal oj Sound and Vibration 114, 325-340.
PEROTlN, L. & GRANGER, S. (1997) A linearized unsteady model for computing
dynamics of cylindrical structures subjected to nonuniform annular flows at high
Reynolds numbers. Journal oj Fluids and Structures 11, 183-205.
PETERKA, F. (1991) F1ow-induced impact oscillation of a spherical pendulum, Joumal
oj Fluids and Structures 5, 627-650.
PETERKA, F. (1992) Dynamics of impact motions of a beam between stops and of
spherica! pendulum in a channel with a flow of liquid. Advances in Mechanics 15,
93-121.
PETERKA, F. (1997) DynallŮcs of the impact oscillator. In New Applications oj Nonlinear
and Chaotic Dynamics in Mechanics (ed, F.c. Moon). Dordrecht: Kluwer Academic
PubIishers,
PETERKA, F. & VACÍC, J. (1992) Transition to chaotic motion in mechanical systems
with impacts. Journal oj Sound and Vibration 154, 95-115.
PETTIGREW, M.J. (1977) F1ow-induced vibration of nuc1ear power station components.
AtOllŮC Energy of Canada Ltd. Report AECL-5852, Chalk River, Ontario, Canada.
PEITIGREW, M.J. (1981) F1ow-induced vibration phenomena in nuc1ear power station
components, Power Industry Research 1, 97-133.
PETTIGREW, M.J. (1993) The vibration behavior of nuc1ear fuel under reactor conditions,
Nuclear Science and Engineering 114, 179-189.
PETTIGREW, M,J. (1994) Two-phase flow-induced vibration: an overview. ASME
Journal oj Pressure Vessel Technology 116, 233-253.
PETTIGREW, M.J, (ed.) (2001) Flow-Induced Vibration - 2001, Vol. 2, PVP-Vol. 420-2,
New York: ASME.
PETTlGREW, M.J. & KNOWLES, G.D. (1997) Some aspects of heat-exchanger tube
damping in two-phase llŮxtures. Journal oj Fluids and Structures 11, 929-945.
PETTIGREW, M.J. & TAYLOR, C.E. (1993) Two-phase flow-induced vibration. In
Technology oJthe '90s (ed. M.K. Au-Yang), pp. 811-864. New York: ASME.
PETTIGREW, M.J., SYLVESTRE, Y. & CAMPAGNA, A.O. (1978) Vibration ana!ysis
of heat exchanger and steam generator designs. Nuclear Engineering and Design 48,
97-115,

1556
SLENDER STRUCTURES AND AXIAL FLOW
PETTIGREW, M.J., GOYDER, H.G.D., QIAO, Z.L. & AXISA, E (1986a) Damping in
mu1tispan heat exchanger tubes. Part 1. in gases. In Flow-/nduced Vibration-1986
(eds S,S. Chen, J.c. Simonis & Y.S. Shin), pp. 81-87. New York: ASME,
PETTlGREW, M.J., ROGERS, R.J. & AXISA, E (1986b) Damping in mu1tispan heat
exchanger tubes. Part 2. in liquids. In Flow-/nduced Vibration-1986 (eds S,S. Chen,
J,C, Simonis & Y.S. Shin), PVP-Vol. 104, pp, 89-98. New York: ASME.
PETTlGREW, MJ., TAYLOR, C.E. & SMITH, B.A.W, (1995) Vibration of cylindrica1
structures in two-phase axia1 flow: an overview. In Flow-/nduced Vibration (ed.
P.W. Bearman), pp. 85-96. Rotterdam: A.A. Balkema.
PETTlGREW, M.J., TAYLOR, C.E., FlSHER, N.J., YETlSIR, M. & SMITH, B.A.W.
(1998) F1ow-induced vibration: recent findings and open questions. Nuclear Engineer-
ing and Design 185, 249-276.
PHAM, T.T, & MISRA, A.K. (1981) DynallŮc stability of cylindrical sheUs subjected to
axial flow and a 1inear1y varying axia11oad, Proceedings 8th Canadian Congress oj
Applied Mechanics, Moncton, N.B., pp. 143-144.
PICOT, p, (2001) Nonlinear dynamics of a pinned-pinned cylinder subjected to an axia1
extemal flow. Rapport de DEA. Département de Mécanique, Eco1e Centra1e de Nantes
and Department of Mechanica1 Engineering, McGi11 University.
PLANCHARD, J. & IBNOU-ZAHIR, M, (1983) Natura1 frequencies of a tube bund1e
in an incompressib1e fluid. Computer Methods in Applied Mechanics and Engineering
41,47-68.
PLATUS, D,M, (1989) Aeroe1astic stability of slender, spinning llŮssi1es. AIAA Paper
89-3393-C.P.
PODDUBNY, V.I. & SALTANOV, N.V. (1991) Three-dimensional parametric vibrations
of fi1aments or wires in coaxia1 flows, Fluid Mechanics Soviet Research 20, 33--41.
PODDUBNY, v.I., SHAMARIN, J.E., CHERNENKO, D.A. & ASTAKHOV, L.S. (1995)
Dynamics oj Undersea Towing Systems. St. Petersburg: Sudostroenye.
POIREL, D,C. & PRICE, S.J. (2001) Structura1 nonlinear fluttering airfoi1 in turbu1ent
flow, A/AA Journal39, 1960-1968.
PONG, A. (2000) DynallŮc stability of e1astic sheUs containing rotating or swir1ing flows.
B.Eng, Honours Thesis, Department of Mechanica1 Engineering, McGiU University,
Montrea1, QC, Canada.
POPOV, A.A" THOMPSON, J,M.T. & McROBIE, EA. (1998) Low dimensional mode1s
of sheU vibrations. ParametricaUy excited vibations of cylindrica1 sheUs. Journal oj
Sound and Vibration 209,163-186.
PORCHER, G. (1994) Contribution a ľ étude des instabilités fluid-é1astiques de structures
tubu1aires, sous écou1ement axia1 confiné. Ph.D. Thesis, Université Pierre et Marie
Curie (Paris 6), France.
PORCHER, G. (2000) Influence of the boundary conditions in a friction based model
of fluid-e1astic in stability in axia1 flow, ln Flow /nduced Vibration (eds S. Ziada &
T. Staubli), pp. 371-378, Rotterdam: Balkema.
PORCHER, G. & DE LANGRE, E. (1997) A friction-based model for fluide1astic forces
induced by axia1 flow. Proceedings 4th /nternational Symposium on Fluid-Structure
/nteractions, Aeroelasticity, Flow-/nduced Vibration and Noise (eds M.P, Pai:doussis
et al,), Vol. n, AD-Vol. 53-2, pp. 67-74, New York: ASME.
POWELL, A. (1958) On the fatigue failure of structures due to vibrations excited by
random pressure fields, Journal oJthe Acoustical Society oj America 30,1130-1135.

REFERENCES
1557
PRAMILA, A. (1985) Sheet flutter - an analytical and experimental study. Report 30,
Department of Mechanical Engineering and Applied Mechanics, Tampere University
of Technology, Finland.
PRAMILA, A (1986) Sheet flutter and the interaction between sheet and air, TAPPI
Journa169,70-74,
PRAMILA, A. (1987) Natural frequencies of a submerged axially moving band. Journal
oj Sound and Vibration 113, 198-203.
PRAMILA, A, & ÁKESSON, B. (eds) (1987) Vibration and stability ofaxially moving
materials. Proceedings EUROMECH 223, Tampere, Finland.
PRATHAP, G. (1978) Comments on the large amplitude asymmetric vibrations of some
thin shells of revolution, Journal oj Sound and Vibration 56, 303-305.
PRICE, S.J. (1995) A review of theoretical models for fluidelastic instability of cylinder
arrays in cross-flow. Journal oj Fluids and Structures 9, 463-518.
PRINCE, A.S" GROTBERG, J.B, & REISS, E.L. (1993) Primary and secondary flutter
of leakage flow channels. Journal oj Sound and Vibration 161, 89-107,
PROKHOROVICH, V.A., PROKHOROVICH, PA & SMIRNOV, L.v. (1982) Dynamic
stability of a flexible rod in a longitudinal incompressible fluid flow. Prikladnaya
Mekhanika 18, 108-112 (translated from Russian).
PUSCH, C. & GRUNW ALD, G. (1991) Vibration model of a pressurized water reactor con-
sidering the fluid effect in the gap between core barrel and pressure vessel (in German).
Kernenergie 34(5), 198-203.
QUINN, KP. (1962) Vibration of fuel rods in parallel flow. U.S. Atomic Energy
Commission Report GEAP-4059,
QUINN, E.P. (1965) Vibration of SEFOR fuel rods in parallel flow. U,S. Atomic Energy
Commission Report GEAP-4966.
RAJU, K.K. & RAO, G.v. (1976) Large amplitude asymmetric vibrations of some thin
shells of revolution. Journal oj Sound and Vibration 44, 327-333.
RAMACHANDRAN, J. (1979) Non-linear vibrations of cylindrical shells of varying
thickness in an incompressible fluid. Journal oj Sound and Vibration 64, 97-106,
RAMAN, A. & MOTE, C.D. Jr (2001) Experimental studies on the nonlinear oscillations
of imperfect circular disks spinning near critical speed. lnternational Journal oj Non-
Linear Mechanics 36,291-305.
RAOUF, R.A. & NAYFEH, AH. (1990) One-to-one autoparametric resonances in
infinitely 10ng cyHndrical shells, Computers and Structures 35, 163-173,
RAPOPORT, LM, (1968) Dynamics oj Elastic Containers Partially Filled with Liquids,
Berlin: Springer- Verlag,
RAYLEIGH, Lord (1911) On the motion ofsolid bodies through viscous fluid. Philosoph-
ical Magazine 21,697-711.
RAYLEIGH, Lord (1916) On the dynamics ofrevolving fluids. Proceedings oJthe Royal
Society A 93, 148-154.
REAVIS, J,R. (1967) WVI - Westinghouse vibration corre1ation for maximum fuel-
element displacement in parallel turbulent flow. Transactions American Nuclear Society
10,369-371.
REAVIS, J.R. (1969) Vibration correlation for maximum fuel element displacement in
parallel turbu1ent flow. Nuclear Science and Engineering 38, 63-69,
REECE, J.W. & SIEKMANN, J. (1964) Swimming of a flexib1e hydrofoil near a free
surface. Zeitschriftfiir Angewandte Mathematik und Mechanik 44,559-571.

1558
SLENDER STRUCTURES AND AXIAL FLOW
REED, G.A. (1965) Major operating and refueling problems at the Yankee Atomic Electric
Company Rowe power plant. American Nuclear Society Transactions 8, (supp1.), 28.
REIMER, E., KATES, L. & PAIDOUSSIS, M.P, (1975) Computer modemng of the
dynamic behaviour of a string of fuel bundles in axial flow. Part A: Response to steady
hydrodynamic forces. Atomic Energy ofCanada Ltd. ReportCRNL-1340, ChalkRiver,
Ontario, Canada,
REISSNER, E. (1955) Nonlinear effects in vibrations of cylindrical shells. Ramo-
Wooldridge Corporation Report AM5-6.
RELF, E.F. & POWELL, C.H. (1917) Tests on smooth and stranded wires inc1ined to the
wind direction, and a comparison of results on stranded wires in air and water. Advisory
Committee for Aeronautics, London, R&M 307.
RENSHAW, AA (1998) Critica1 speed for floppy disks. Journal oj Applied Mechanics
65, 116-120.
RENSHAW, AA, D' ANGELO, C. & MmE, C.D, Jr (1994) Aerodynamically excited
vibration of a rotating disk. Journal oj Sound and Vibration 177, 577-590.
REUTOV, Y.P. & RIBUSHKINA, G,V. (1998) Hydroelastic in stability threshold in a
turbulent boundary-layer over a compliant coating, Physics oj Fluids 10,417-425,
REYNOLDS, AJ, (1965) The swimming of minute organisms. Journal oj Fluid
Mechanics 23, 241-260.
REYNOLDS, O. (1886) On the theory of lubrication and its application to Mr Beauchamp
Tower's experiments, inc1uding an experimental determination of the viscosity of olive
oB, Philosophical Transactions oJthe Royal Society (London) 177, Part 1, 157-234.
RICHARDSON, J.R, (1965) The dynamics of towed underwater systems. Engineering
Research Associates Report No. 56-1, Toronto, Canada.
RILEY, J.J., GAD- EL- HAK, M. & METCALFE, R. W. (1988) Compliant coatings. Annual
Review oj Fluid Mechanics 20, 393-420.
ROGERS, R.J. & PICK, R.J, (1976) On the dynamic spatial response of a heat exchanger
tube with intermittent baffle contacts. Nuclear Engineering and Design 36, 81-90.
ROPCHAN, C.B. & SWATERS, G.E, (1993) The role ofnegative energy waves in linear
and nonlinear shear-flow instability in hyperelastic fluid-filled tubes. Quarterly Journal
oj Mechanics and Applied Mathematics 46, 657-681.
ROSENBERG, G.S. & YOUNGDAHL, C.K (1962) A simplified dynamic model for the
vibration frequencies and critical coolant flow velocities for reactor para11el plate fuel
assemblies. Nuclear Science and Engineering 13, 91-102.
ROSTRŮM, K.G. (1964) Seven-rod fuel element - Vibration test. Ab. Atomenergi
(Stockholm), Arbetsrapport RRL-726.
ROSTRŮM, KG, & ANDERSSON, N. (1964a) Boiler element for Marviken. Vibration
tests with one rod, Ab. Atomenergi (Stockholm), Arbetsrapport RRL-724.
ROSTRŮM, KG. & ANDERSSON, N. (1964b) Superheater element for Marviken.
Vibration tests with one rod. Ab, Atomenergi (Stockholm), Arbetsrapport RRL-725.
ROUSSEAU, G. (1994) Prise en compte ďun fluide parfait incompressible au repos
comme masse ajoutée sur une structure. Electricité de France, Division des Etudes
et Recherches, Report 94NB00121.
RUBINOW, S.I. & KELLER, J.B. (1971) Wave propagation in a fluid.filled tube. Journal
oJthe Acoustical Society oj America 50, 198-223.
RUELLE, D. & TAKENS, F. (1971) On the nature of turbulence. Communications in
Mathematical Physics 20, 167-192; 23, 343-344,

REFERENCES
1559
RUUD, EO, (1991) Vibration of penstocks in hydroelectric insta1lations. Waterpower '91,
pp,2214-2222.
SAFFMAN, P,G. (1967) The self-propulsion of a deformable body in perfect fluid. Journal
o! Fluid Mechanics 28, 385-389.
SAKURAI, A. & OHBA, K. (1986) Self-excited osci11ation of flow in collapsible tube.
III. A resonant oscillation of air flow. Technology Report, Kansai University No. 28,
pp. 41-48.
SAN ANDRES, L.A. & VANCE, J.M. (1987a) Effects of fluid inertia on squeeze-film
damper forces for sma1l-amplitude circular-centered motions. ASLE Transactions 30,
63-68.
SAN ANDRES, L.A. & V ANCE, J .M. (1987b) Force coefflcients for open-ended squeeze-
film dampers executing small-amplitude motions about an off-center equilibrium
position, ASLE Transactions 30,69-76,
SANDERS, J.L. Jr. (1959) An improved first-approximation theory for thin shells. NASA
Report TR-R24.
SANDERS, J,V. (1982) A three-dimensional dynamic ana1ysis of a towed system. Ocean
Engineering 9, 483-499.
SAROUDlS, J. (1974) Propulsion by oscillating plates, B.Eng. Honours Thesis, Depart-
ment of Mechanica1 Engineering, McGill University, Montrea1, Canada.
SARPKAYA, T. (1961) Propagation of surges in highly deformable systems, Develop-
ments in Mechanics 1, 451-459.
SARPKAYA, T, (2001) On the force decomposition of Lighthi11 and Morison, Journal o!
Fluids and Structures 15, 227-233.
SARV, H. & JOHN, J. (2000) Technica1 and economical aspects of large-sca1e C02
storage in deep oceans. In Proceedings 25th lnternational Technical ConJerence on
Coal Utilization and Fuel Systems, Vol. I, Paper MTI 00-03, Clearwater, FL, USA.
SAUNDERS, H., WISNIEWSKI, E.J. & PASLAY, P.R (1960) Vibrations of conical shells.
Journal oJthe Acoustical Society oj America 32, 765-772.
SCANLAN, R,H. & ROSENBAUM, R (1951) lntroduction to the Study oj Aircraft
Vibration and Flutter. New York: The Macmil1an Co.
SCARTON, H.A. (1998) Private communications (Dec 1998 and Jan 1999).
SCARTON, H.A. & ROULEAU, W.T. (1973) Axisymmetric waves in compressible
Newtonian liquids contained in rigid tubes: steady-periodic mode shapes and dispersion
by the method of eigenval1eys. Journal oj Fluid Mechanics 58, 595-621,
SCAVUZZO, RJ. (1965) Post-buckling behaviorofhydraulica1ly collapsed assembly fuel
plates, Transactions oj American Nuclear Society 8,591-592,
SCHLlCHTING, H. (1960) Boundary Layer Theory, 4th edition. New York:
McGraw-Hill.
SCHLlCHTlNG, H. (1968) Boundary Layer Theory, 6th edition. New York:
McGraw-Hill.
SCHRAM, J,W, & REYLE, S.P. (1968) A three-dimensional dynamic ana1ysis ofa towed
system. Journal oj Hydronautics 2,213-220,
SCHUMANN, U. (1979) Dynamics of nuclear-reactor shell structure in an incompressible
fluid. In Recent Developments in Theoretical and Experimental Fluid Mechanics (eds
U. Miiller, K.G. Roesner & B. Schmidt), pp, 577-586. Berlin: Springer-Verlag.
SCHUMANN, U. (1980) Core barrel oscillations during blowdown of a pressurized water
reactor, ln Practical Experiences with Flow-lnduced Vibrations (eds E, Naudascher &
D. Rockwell), pp. 181-187. Berlin: Springer-Verlag,

1560
SLENDER STRUCTURES AND AXIAL FLOW
SCHUMANN, U. (1981) Homogenized equations oťmotion ťorrod bundles in fluid with
periodie structure, lngenieur-Archiv 50,203-216.
SCHWIRIAN, R.E, (1986) Hydroelastie stability and vibration oť inverted pendulum
structures in parallel flow. In Forum on Unsteady Flow (ed. P,H, Rothe), FED- Vol. 39,
pp. 28-30. New York: ASME,
SELMANE, A & LAKIS, AA. (1997a) Dynamic analysis oť anisotropie open cylindrical
shells. lnternational Journal oJComputers and Structures 62, 1-12.
SELMANE, A & LAKIS, A.A (l997b) Vibration analysis oť anisotropic open cyl-
indrical shells containing flowing fluid. Journal oj Fluids and Structures 11, 111-134;
corrigenda, 12,799.
SELMANE, A & LAKIS, AA (l997c) Influence oť geometrie non-linearities on ťree
vibrations oť orthotropie open cylindrical shells, lnternational Journal Jor Numerical
Methods in Engineering 40, 1115-1137.
SELMANE, A & LAKIS, AA (1997d) Non-linear dynamic analysis oť orthotropie open
cylindrical shells subjected to a flowing fluid. Journal oj Sound and Vibration 202,
67-93.
SEMLER, c., LI, G,X, & PAIDOUSSIS, M.P., (1994) The nonlinear equations oťmotion
oť pipes conveying fluid. Journal oj Sound and Vibration 169, 577-599.
SEMLER, C., GENTLEMAN, W,C. & PAIDOUSSIS, M,P. (1996) Numerical solutions
oť second order implicit non-linear ordinary differential equations. Journal oj Sound
and Vibration 195, 553-586.
SEMLER, C., LOPES, J.-L., AUGU, N, & PAIDOUSSIS, M.P. (2002) Linear and non-
linear dynamics oť cantilevered cylinders in axial flow. Part 3: Nonlinear dynamics.
Journal oj Fluids and Structures 16, 739-759.
SEW ALL, J.L. & NAUMANN, KC, (1968) An experimental and analytical vibration study
oť thin cylindrical shells with and without longitudinal stiffness. NASA Technieal Note
D-4705,
SHAPIRO, AH. (1977a) Physiologie and medieal aspects oť flow in collapsible tubes.
Proceedings 6th Canadian Congress oj Applied Mechanics, Vancouver, RC., Canada,
pp. 883-906.
SHAPIRO, AH. (1977b) Steady flow in collapsible tubes. ASME Journal oJBiomechan-
ical Engineering 99,126-147,
SHARMA, C.B. (1974) Ca1culation oť natural ťrequencies oť fixed- ťree circular cylindrical
shells. Journal oj Sound and Vibration 35, 55-76.
SHARMA, C.R (1979) Vibration characteristies oť thin circular cylinders. JournaloJ
Sound and Vibration 63, 581-592.
SHAYO, L.K. (1977) The stability oť elastic cylindrical pipes containing inviscid fluid in
uniťorm motion. Ph.D. Thesis, Mathematies Department, Imperial College oť Science
and Technology, London, U.K.
SHAYO, L.K. (1980) The stability oť cantilever panel s in uniťorm incompressible flow.
Journal oJSound and Vibration 68, 341-350,
SHAYO, L.K. & ELLEN, C.H. (1974) The stability oť finite length circular cross-section
pipes conveying inviscid fluid. Journal oJSound and Vibration 37,535-545.
SHAYO, L.K. & ELLEN, C.H. (1978) Theoretical studies oť internal flow-induced
instabilities oť cantilevered pipes. Journal oj Sound and Vibration 56, 463-474.
SHE, J. & BERTRAM, C.D. (1996) Numerical simulation oť collapsible-tube flows with
sinusoida! ťorced oscillations. Bulletin oj Mathematical Biology 58, 1023-1046.
SHIELDS, C.M. (1960) N.S. Savannah fuel design and development program - fuel rod
vibration. United States Atomic Energy Commission Report GEAP-3583.

REFERENCES
1561
SHIH, L. Y. (1975) Motion of elliptic ba1100ning of a trave1ing string, lnternational Journal
oj Non-Linear Mechanics 10, 183-191.
SHIN, Y.S. & WAMBSGANSS, M,W. (1977) F10w-induced vibration in LMFBR steam
generators: a state-of-the-art review. Nuc/ear Engineering and Design 40, 235-284.
SHIN, Y.S., JENDRZEJCZYK, J.A. & WAMBSGANSS, M.W. (1977) Effect of
tube/support interaction on the vibration of a tube on mu1tip1e supports. Argonne
Nationa1 Laboratory Report ANL-CT-77-5.
SHlN,Y.S" SASS, D,E. & JENDRZEJCZYK, J.A. (1978) Vibro-impact responses
of a tube with tube-baffte interaction. Argonne Nationa1 Laboratory Report ANL-
CT-78-11.
SHIN, Y.W, (1978) Two-phase flow-induced vibrations of rods in paralle1 flow: a state-
of-the-art review. Genera1 E1ectric Report GEAP-24148.
SHIMIZU, M. (1985) Characteristics of pressure-wave propagation in a comp1iant tube
with a fully collapsed segment. Journal oj Fluid Mechanics 158, 113-135.
SHlRAKI, K., NAGATA, o. & HONMA, T. (1975) Penstock vibration characteristics-
on the vibrations of simp1y supported and ring-stiffened cylindrica1 shells filled with
pressurized water. ASME paper 75-DET-114.
SHTERN, V, & HUSSAlN, F. (1999) Collapse, symmetry breaking, and hysteresis in
swirling flows. Annual Review oj Fluid Mechanics 31, 537-566.
SIEKMANN, J. (1962) Theoretica1 studies of sea animal10comotion. Part 1. /ngenieur-
Archiv 31, 214-228,
SIEKMANN, J. (1963) Theoretica1 studies of sea anima110comotion. Part 2. lngenieur-
Archiv 32, 40-50.
SILVESTRE, A.L. (2002) On the s10w motion of a self-propelled rigid body in a vis-
cous incompressib1e fluid. Journal oj Mathematical Analysis and Applications 274,
203-227.
SIM, W,-G. (1991) Study of unsteady annu1ar flows between eccentric cylinders. Ph.D,
Thesis, Department of Mechanica1 Engineering, McGi1l University, Montrea1, QC,
Canada,
SIVAK, V.F. & TELALOV, A.I. (1991) Experimenta1 investigation of vibrations of a
cylindrical shell in contact with a liquid. Soviet Applied Mechanics 27, 484-488.
SKALAK, R (1956) An extension of the theory of waterhammer. Transactions oj ASME
78, 105-116.
SKALAK, P., OZKAYA, N. & SKALAK, T.C. (1989) Biofluid mechanics. Annual Review
oj Fluid Mechanics 21, 167-204.
SKOP, RA. & CHOO, Y. (1971) The configuration of a cab1e towed in a circular path,
Journal oj Aircraft 8, 856--862.
SMISSAERT, G.E. (1968) Static and dynamic hydroe1astic instabHities in MTR-type fue1
e1ements. Part I. Introduction and experimenta1 investigation. Nuc/ear Engineering and
Design 7, 535-546.
SMISSAERT, G.E. (1969) Static and dynamic hydroe1astic instabi1ities in MTR-type fuel
elements, Part II. Theoretica1 investigation and discussion. Nuc/ear Engineering and
Design 9, 105-122.
SMITH, B.A.W, & DERKSEN, D.D. (1998) Measurement of steady and unsteady
hydrodynamic loads on a nuc1ear fue1 bund1e. Journal oj Fluids and Structures 12,
475-489.
SMITH, M,W. & LAMBERT, RF. (1960) Propagation of band limited noise. Journal oj
the Acoustical Society oj America 32,512-514.

1562
SLENDER STRUCTURES AND AXIAL FLOW
SOEDEL, W. (1993) Vibrations oj Shells and Plates, 2nd edition. New York: Marcel
Dekker.
SOGREAH (1962) Study of vibrations and 10ad 10sses in tubu1ar clusters. Société Gren-
ob10ise d'Etudes et Applications Hydrauliques, Special Report No.3, EURAEC-288.
SOH, W,Y. (1987) Time-marching solution for incompressib1e Navier-Stokes equations
for internal flows. Joumal oJComputational Physics 94,102-137.
SOH, w.y. & GOODRICH, J.w. (1988) Unsteady solution of incompressib1e Navier-
Stokes equations. Joumal oJComputational Physics 79,113-134.
SOROKlN, S.v. (2000) Nonlinear oscillations of a baffled e1astic p1ate in heavy fluid
10ading conditions. Joumal oj Sound and Vibration 232,619-643.
SPALDING, D,B. (1972) A novel finite difference fonnulation for differentia1 expres-
sions involving both first and second derivatives, /ntemational Joumal Jor Numerical
Methods in Engineering 4,551-559.
SPARENBERG, J.A. (1962) On the waving motion of a flag. Koninklijke Nederlandse
Akademie van Wetenschappen, Series B, 65, 378-392.
SPURR, A. & HOBSON, D.E. (1984) Forces on the vibrating centrebody of an annu1ar dif-
fuser. Proceedings Symposium on Flow-/nduced Vibrations, Vol. 4 (eds M.P, Pa.doussis
& M.K. Au- Yang) , pp. 41-52. New York: ASME.
SRlNIVASAN, A.v. (1971) Flutter ana1ysis ofrotating cylindrica1 shells immersed in a
circu1ar helica1 flowfie1d of air. A/AA Joumal9, 394-400.
STEARMAN, R.O., LOCK, M.H. & FUNG, Y.c. (1962) Ames tests on the flutter of
cybndrical shells. Graduate Aeronautical Labs, Cabfornia Institute of Technology,
GALCIT Structural Dynamic Report SM 62-37.
STEFAN, J, (1874), Versuche uber die scheinbare Adhasion. Sitzungsberichte Kaiser-
Uchew Akademie der Wžssenschaften (Wien) - Mathematisch-Naturwissenschaftiche
Klasse 69, 713-734.
STOCKMAN, M.G. & BEJAN, A. (1982) The nonaxisymmetric (buckling) flow regime
offast capil1ary jets. Physics oj Fluids 25, 1506-1511.
STOKES, G.G. (1843) On some cases of fluid motion. Proceedings oj the Cambridge
Philosophical Society 18, 105-137.
STOKES, G.G. (1851) On the effect of the internal friction of fluids on the motion of
pendulums. Transactions Cambridge Philosophical Society 9,8-106.
STRANDHAGEN, A.G. & THOMAS, C.P. (1963) Dynamics of towed underwater
vehicles. Report No. 219, United States Navy Mine Defense Laboratory, Panama City,
FL, USA.
STRANDHAGEN, A.G., SCHOENHERR, K.E. & KOBAYASHI, P.M. (1950) The
dynamic stability on course of towed ships. Transactions oj the Society oj Naval
Architects and Marine Engineers 58, 32-66.
STREETER. V.L. & WYLIE, E.B. (1974) Waterhammer and surge control. Annual Review
oj Fluid M echanics 6, 57-73.
STREITLlEN, K. & TRIANTAFYLLOU, G.S, (1998) On thrust estimates for flapping
foi1s. Joumal oj Fluids and Structures 12, 47-55.
STUDZINSKI, W. (1999) Private communication, Nova Chemicals, Calgary, Canada
(29 July 1999).
SUDARSAN, K., BHATTACHARYYA, S.K. & VENDHAN, C.P. (1997) Anexperimenta1
study of hydroelastic in stability of flexible towed underwater cylindrical structures.
Proceedings 16th OMAE ConJerence, Yokohama, Japan, Vol. A, pp. 73-80. New York:
ASME,

REFERENCES
1563
SUDARSAN, K., BHATTACHARYYA, S,K. & VENDHAN, c.P. (1999) Hydroe1astic
instabilities of ship and submarine towed underwater cylindrica1 structures. Proceedings
18th OMAE Conference, St. John's, Canada, Paper OMAE 99/0FT-4181. New York:
ASME.
SUESS, S,T. (1971) Viscous flow in a deforrnab1e rotating container. Journal oj Fluid
Mechanics 45, 189-201.
SUGIMOTO, N. (1996) Aeroe1astic stability of a beam trave1ing in a tunne1lined with
resonators, A/AA Journal34, 2005-2013.
SUGlMOTO, N. & KUGO, K. (2001) Wave propagation on an e1astic beam trave1ing in
a tube: linear theory of aerodynamic loading. Journal oj Computational Acoustics 9,
1227-1236.
SUGIYAMA, Y. & OSHIMA, T. (2002) Effect of aerodynamic loads on dynamic stability
of slender launch vehicIe subjected to an end rocket thrust. Paper IMECE2002-33034,
FS/, AE & FIV+N Symposium (ed. M,P. Paidoussis), New Orleans, LA, USA. New
York: ASME.
SUSS, S., PUSTEJOVSKY, M. & PAlDOUSSIS, M.P. (1976) The virtua1 mass matrix of
cylinders in 1iquid contained by a rigid outer cylinder. MERL Report 76-1, Department
of Mechanica1 Engineering, McGi11 University, Montrea1, Québec, Canada,
SVETLlTSKlI, v.A. (1982) Mechanika Truboprovodov i Shlangov (in Russian), Moscow:
Mashinostronye.
SVETLlSKII, v.A. & GRIGORIEV, Yu, V, (1988) Hydrodynamic forces on rod in
cylindrical channe1 (in Russian), Rascheti na Prochnost (Stress Ana1ysis), Vol. 28,
pp. 194-204, Moscow: Mashinostronye,
SWINSON, W.F., BATTISTE, R.L., LUTTRELL, c.R. & YAHR, G,T. (1992) Fue1-p1ate
stability experiments and analysis for the Advanced Neutron Source. ln Axial andAnnu-
lar Flow-/nduced Vibration and /nstabilities (eds M.P. Paidoussis & M.K. Au- Yang),
PVP-Vol. 244, pp. 133-143, New York: ASME.
SZERI, A,Z, (1998) Fluid Film Lubrication: Theory and Design. Cambridge: Cambridge
University Press.
SZYSZKOWSKI, W. & GLOCKNER, P,G, (1987a) On the statics of 1arge-sca1e cy1-
indrical floating membrane containers. /nternational Journal oJNon-Linear Mechanics
22,275-282.
SZYSZKOWSKI, W. & GLOCKNER, P,G, (1987b) On the statics ofaxisymmetric float-
ing membrane storage vesse1s. /nternational Journal oj Non-Linear Mechanics 22,
339-348,
TABARROK, B. & BEHDINAN, K. (2000) Dynamics of spacecraft antenna during
dep1oyment. In Proceedings /UTAM/1ASS Symposium on Deployable Structures:
Theory and Applications (eds S. Pellegrino & S.D. Guest), pp, 415-424, Dordrecht:
Kluwer,
TABARROK, B., LEECH, C.M. & KIM, Y.I. (1974) On the dynamics of an axially moving
beam. Journal oJthe Franklin /nstitute 297,201-220.
TACK, D,H. & LAMBERT, R.F. (1962) Response of bars and p1ates to boundary-1ayer
turbulence, Journal oJthe Aerospace Sciences 29,311-322.
TAIT, RJ. & MOODlE, T.B. (1984) Waves in non1inear fluid filled tubes, Wave Motion
6, 197-203,
TALEB, I.A. & MISRA, A.K. (1981) Dynamics ofaxia1ly moving beam submerged in a
fluid. A/AA Journal oj Hydronautics 15, 62-66.

1564
SLENDER STRUCTURES AND AXIAL FLOW
TANAKA, S., KANEKO, S, & WATANABE, T. (2000) Self-excited oscillations of dual
cyHndrical flexible weir shells due to overflow fluid. Joumal oj Vibration and Control
6,351-373,
TANAKA, S., HlRATA, T., KANEKO, S. & WATANABE, T. (2001) Aerodynamic vibra-
tion of combined rigid bodies due to leakage- flow. In Flow-lnduced Vibration-200 I,
Vol. 2 (ed. M.J. Pettigrew), PVP-Vol. 420-2, pp. 45-52.
TANEDA, S. (1968) Waving motions of flags. Joumal oj the Physical Society oj Japan
24, 392-401.
TANG, D.M., HERRY, J.K. & DOWELL, E.H. (1999) Limit cyc1e oscillations of delta
wing models in low subsonic flow. AlAA Joumal37, 1355-1362.
TANG, D.M., YAMAMOTO, H. & DOWELL, E.H. (2003) Flutter and limit cyc1e oscil-
lations of two-dimensional panels in three-dimensional axial flow. Joumal oj Fluids
and Structures 17, 225-242,
TANI, J. & DOKI, M. (1979) Effects of shearing loads and in-plane boundary conditions on
the stability of thin tubes conveying fluid. Joumal oj Applied Mechanics 46, 779-783.
TANIDA, y, (2001) Stability of a soft plate in channel flow (aerodynamic aspects ofpalatal
flutter). JSME lntemational Joumal 44, 8-13.
TAYLOR, G.I. (1923) Stability of a viscous liquid contained between two rotating
cylinders. Philosophical Transactions oJthe Royal Society (London) A 157, 289-343.
TAYLOR, G.I. (1951) Analysis of the swimming of microscopic organisms. Proceedings
oJthe Royal Society (London) A209, 447-461.
TAYLOR, G.I. (1952a) Analysis of the swimming of long and narrow animals.
Proceedings oJthe Royal Society (London) A 214,158-183.
TAYLOR, G.I. (1952b) The action of waving cylindrical tails in propelling microscopic
organisms. Proceedings oJthe Royal Society (London) A211, 225-239.
TAYMANS, E., DE GOEYSE, A., JAUMOTTE, A., MATHIEU, F. & PLANQUART, J.
(1975) Resonance frequencies of a tubular shel1 surrounded internally and externally
by fluid annuli of variable thickness - preliminary experimental results. Paper F2/12,
Transactions oj 3rd SMiRT, London, UK.
TELFORD, W.M., GELDART, L.P., SHERIFF, R.E. & KEYS, D.A. (1976) Applied
Geophysics, Chapter 4. Cambridge: Cambridge University Press.
THEODORACATOS, V.E. & CALKlNS, D,E. (1986) An experimental study of elasto-
hydrodynamics of towed flexible cylinders aided by video image processing. Ocean
Engineering 2,587-619,
THOMPSON, J.M.T. & BISHOP, SR (eds) (1994) Nonlinearity and Chaos in Engineer-
ing Dynamics. Chichester: John Wiley & Sons,
THOMPSON, J,M.T. & STEWART, H.B. (1986) Nonlinear Dynamics and Chaos.
Chichester: John Wiley & Sons.
THOMSON, W.T. (1951) Transmission of pressure waves in liquid filled tubes.
Proceedings I st US National Congress oj Applied Mechanics, pp. 927-933.
THOMSON, W.T. (1962) Continuous structures excited by corrrelated random forces.
lntemational Joumal oj Mechanical Science 4, 109-114.
TlJSSELlNG, A.G. (1996) Fluid-structure interaction in liquid-fil1ed pipe systems: a
review. Joumal oj Fluids and Structures 10, 109-146.
TlMOSHENKO, S. (1961) Theory oj Elastic Stability. New York: McGraw-Hill.
TIMOSHENKO, S. & WOINOWSKY-KRIEGER, S. (1959) Theory oj Plates and Shells,
2nd ed. New York: McGraw-Hill.

REFERENCE S
1565
TOEBES, G.H, & EAGLESON, P.S, (1961) Hydroe1astic vibrations of flat p1ates re1ated
to trailing edge geometry. ASME Journal oj Basic Engineering 83, 671-678.
TOMITA, H. (1984) Ana1ysis of vibration of a group of rods in a fluid by approximation
of flow-path network (Report II. Continuous body model for a group of rods). Bulletin
oj JSME 27, 965-973.
TOMITA, H. & SASAKI, y, (1984) Ana1ysis of vibration of a group of rods in a fluid by
approximation of flow-path network (Report I. SimpHfied vibration model for a group
of rods). Bulletin oj JSME 27, 956-964.
TOMSKI, L. & KUKLA, S. (1984) Dynamic response of bar-fluid-shell system simu-
1ating hydrau1ic cy1inder subjected to arbitrary axia1 excitation. Journal oj Sound and
Vibration 92, 273-284.
TOORANI, M.H. & LAKIS, AA (2001) Shear deformation in dynamic ana1ysis of anis-
tropic 1aminated open cy1indrica1 shells filled with or subjected to a flowing fluid.
Computer Methods in Applied Mechanics and Engineering 190, 4929-4966.
TORRES, M.R. (1980) F10w induced vibration testing of BWR feedwater spargers. In
Flow lnduced Vibration oj Power Plant Components (ed. M.K. Au- Yang), PVP-Vol. 41,
pp. 159-176. New York: ASME.
TRIANTAFYLLOU, G.S. (1992) Physica1 condition for abso1ute instabi1ity in inviscid
hydroe1astic coup1ing. Physics oj Fluids A 4, 544-552.
TRIANTAFYLLOU, G.S. & CHRYSSOSTOMIDIS, C, (1984) Ana1ytic determination
of the buckling speed of towed slender cy1indrica1 beams. ASME Journal oj Energy
Resources Technology 106, 246-249.
TRIANTAFYLLOU, G.S, & CHRYSSOSTOMIDIS, C. (1985) Stability of a string in
axia1 flow. ASME Journal oj Energy Resources Technology 107, 421-425.
TRIANTAFYLLOU, G.S. & CHRYSSOSTOMIDIS, C. (1989) The dynamics of towed
arrays. ASME Journal oj Offshore Mechanics and Arctic Engineering 111, 208-213.
TRIANTAFYLLOU, G,S., TRIANTAFYLLOU, M.S. & GROSENBAUGH, M.A (1993)
Optima1 thrust deve10pment in osci11ating foi1s with app1ication to fish propu1sion.
Journal oj Fluids and Structures 7,205-224.
TRIANTAFYLLOU, M,S. (1998) Private communication (3 Nov 1998).
TRIANTAFYLLOU, M.S. & TRIANTAFYLLOU, G.S, (1995) An efficient swimming
machine. Scientijic American, March.
TRIANTAFYLLOU, M.S" TRIANTAFYLLOU, G.S. & YUE, D.K.P. (2000) Hydro-
dynamics of fish swimming, Annual Review oj Fluid Mechanics 32, 33-53,
TRIITON, D.J. (1988) Physical Fluid Dynamics, 2nd edition. Oxford: C1arendon Press.
TRUFYN, N. (1970) A Guide to FORMAC. Institute of Computer Science, University of
Toronto, Toronto, Ontario, Canada.
TSUCHIYA, K. (1983) Dynamics of spacecraft during extension of flexible appendages.
AlAA Journal oJGuidance 6, 100-103,
TUOVILA, W.J. & LAND, N.S. (1968) Experimenta1 study of aeroe1astic instabmty of
overexpanded rocket nozzle extensions. NASA TN D-4471.
UGURAL, AC. & CHENG, S. (1968) Buckling of composite cy1indrica1 shells under
pure bending. A/AA Journal 6, 349-354.
ULDRICK, J.P. & SIEKMANN, J. (1964) On the swimming ofaflexib1e p1ate ofarbitrary
finite thickness. Journal oj Fluid Mechanics 20, 1-33.
U,S. AEC (1975) Power Reactor Docket Information, Dockets 50259, 50298-465,
50231-417 .

1566
SLENDER STRUCTURES AND AXIAL FLOW
VAIL, E.G. (1973) Pulmonary insufficiency with airway flutter, c10sure and collapse.
Aerospace Medicine 44,649-662.
VAN DER HOOGT, P.J.M., WEPPELlNK, H. & VAN CAMPEN, D.H. (1979) Three-
dimensional ca1culation of the coup1ed vibrations of a group of circular tubes in
unconfined liquid. Transactions 5th SMiRT, Paper B5/2, Berlin, Germany,
VARADAN, T.K., PRATHAP, G. & RAMANI, H,V. (1989) Nonlinear free flexural
vibration of thin circular cylindrical shells, AlAA Journal27, 1303-1304.
VENDHAN, c.P., BHATIACHARYYA, S,K. & SUDARSAN, K. (1997) Stability charac-
teristics of s1ender flexible cylinders in axial flow by the finite element method. Journal
oj Sound and Vibration 208, 587-601.
VENTRES, C,S. (1975) Shear flow aerodynaDŮcs lifting surface theory. AlAA Journall3,
1183-1189,
VENTRES, C,S. & DOWELL, E,H. (1970) Comparison of theory and experiment for
nonlinear flutter of loaded plates. AlAA Journal 8, 2022-2030.
VENTSEL, E. & KRAUTHAMMER, T. (2001) Thin Plates and Shells. New York: Marcel
Dekker.
VERNA, B. (ed.) (1976) Instrument vibration - channel box damage. In Nuc1ear Power
Experience, Section BWR-2, II, items 26, 35, 47.
VIDELER, J,J. (1993) Fish Swimming. London: Chapman & Hall,
VOLFSON, D, & VINALS, J. (2001) F10w induced by a randomly vibrating boundary.
Journal oj Fluid Mechanics 432, 387-408.
WAGNER, C,D. & PERSSON, P.B. (1995) Nonlinear chaotic dynaDŮcs of arterial blood
pressure and renal blood flow, American Journal oj Physiology 268, H621-H627.
WALD, M.L. (1983) Cracks extend nuc1ear shutdown. The New York Times, 24 July.
WALLlS, G.B. (1969) One Dimensional Two-Phase Flow. New York: McGraw Hi11.
WALSH, C. (1995) Flutter in one-dimensional collapsib1e tubes. Journal oj Fluids and
Structures 9, 393-408.
WALSH, C., SULLIVAN, P.A. & HANSEN, J,S. (1991) Subcritical flutter in collapsib1e
tube flow - a model of expiratory flow in the trachea. ASME Journal oj Biomechanical
Engineering 113, 21-26.
WALTER, J.L., McDANIEL, O.H. & REETHOF, G. (1977) The coincidence of higher-
order acoustic modes in pipes with the pipe vibrational modes. Journal oJthe Acoustical
Society oj America 62, S84.
W AMBSGANSS, M, W. (1967a) Vibration of reactor core components, Reactor and Fuel-
Processing Technology 10, 208-219.
WAMBSGANSS, M,W. Jr (1967b) Second-order effects as related to critical coolant flow
velocities and reactor parallel plate fuel assemblies. Nuclear Engineering and Design
5, 268-276,
WAMBSGANSS, M.W. & CHEN, S.S. (1971) Tentative design guide for calculating the
vibration response of flexible cylindrical elements in axial flow. Argonne Nationa!
Laboratory Report ANL-ETD-7I-07.
WAMBSGANSS, M.W. & JENDRZEJCZYK, J.A. (1979) The effect oftrailing end geo-
metry on the vibration of a circular cantilevered rod in nominally axial flow. Journal
oj Sound and Vibration 65, 251-258.
WAMBSGANSS, M.W. & ZALESKl, P.L. (1970) Measurement, interpretation and char-
acterization of nearfield flow noise. In Proceedings ConJerence on Flow.lnduced
Vibrations in Reactor System Components, Argonne National Laboratory Report
ANL-7685, pp. 112-140.

REFERENCES
1567
WANG, S. & RUSAK, Z. (1996) On the stability of an axisymmetric rotating flow in a
pipe, Physics oj Fluids 8, 1007-1016.
WANG, x. (1997) Finite element ana!ysis of air-sheet interactions and flutter suppression
devices. Computers & Structures 64, 983-994.
WANG, X. & BLOOM, F. (1999) Dynamics of a submerged and inclined concentric
pipe system with interna! and externa! flows. Journal oj Fluids and Structures 13,
443-460.
WARBURTON, G.B. (1961) Vibration of a cylindrica! shell in an acoustic medium.
IMechE. Journal oj Mechanical Engineering Science 3, 69-79.
WARBURTON, G.B. (1973) Dynamics of shells. Technica! Report UTME-TP-7307,
Department of Mechanica! Engineering, University of Toronto, Toronto, Canada,
WARBURTON, G.B. & AL-NAJAFI, A.M.J. (1969) Free-vibration of tmn cy1indrica!
shells with discontinuity in the tmckness. Journal oj Sound and Vibration 9, 373-382.
WARBURTON, G.B. & HIGGS, J, (1970) Natura! frequencies of thin canti1ever
cylindrica1 shells. Journal oj Sound and Vibration 11, 335-338.
WATANABE, Y., SUZUKI, S., SUGIHARA, M. & SUEOKA, Y. (2002a) An experimenta!
study of paper flutter. Journal oj Fluids and Structures 16, 529-542.
WATANABE, Y., ISOGAI, K., SUZUKI, S. & SUGIHARA, M. (2002b) A theoretica1
study of paper flutter. Journal oj Fluids and Structures 16, 543-560.
WATAWALA, L. & NASH, w.A. (1983) Influence of initia1 geometric imperfections on
vibrations of tmn circu1ar cylindrica1 shells. Computers and Structures 16, 125-130.
WAUER, J. & PLAUT, R.H. (1991) Vibrations of an extensible, air-inflated cylindrica!
membrane. Zeitschriftfiir angewandte Mathematik und Mechanik 71, 191-192.
WEAVER, D.S. (1972) Letter to the Editor on M.P. Pai'doussis & Denise's (1972) paper,
with authors' response. Journal oj Sound and Vibration 22, 247-248.
WEAVER, D.S. (1974) On the nonconservative nature of "gyroscopic conservative"
systems. Journal oj Sound and Vibration 36, 435-437.
WEAVER, D.S. (1980) F10w induced vibrations in va!ves operating at sma1l openings.
In Practical Experiences with Flow-Induced Vibrations (eds E. Naudascher & D.
Rockwell), pp. 305-319. Berlin: Springer-Verlag.
WEAVER, D.S. (1998) Private communication (17 Jun 1998).
WEAVER, D.S. & MYKLATUN, B. (1973) On the stability of tmn pipes with an interna1
flow. Journal oj Sound and Vibration 31, 399-410.
WEAVER, D.S. & PAIDOUSSIS, M.P. (1977) On collapse and flutter phenomena of tmn
tubes conveying fluid. Journal oj Sound and Vibration 50, 117-132.
WEAVER, D.S. & UNNY, T,E. (1970) Hydroe1astic stability of a flat p1ate. Journal oj
Applied Mechanics 37,823-827.
WEAVER, D.S. & UNNY, T.E. (1972) The influence of a free surface on the hydroe1astic
stability of a flat panel. Journal oj Applied Mechanics 39, 53-58.
WEAVER, D.S. & UNNY, T.E. (1973) On the dynamic stability of fluid-conveying pipes.
Journal oj Applied Mechanics 40, 48-52.
WEAVER, D.S. & ZIADA, S. (1980) A theoretica! model of self-excited vibrations of
hydraulic gates, va!ves, and seals. ASME Journal oj Pressure Vessel Technology 102,
146-151.
WEAVER, D.S., ZIADA, S., AU-YANG, M.K., CHEN, S.S., PAIDOUSSIS, M.P. &
PETTIGREW, M,J, (2000) F1ow-induced vibrations in power and process plant
components. Mi1lenium Issue, ASME Joumal oj Pressure Vessel Technology 122,
339-348.

1568
SLENDER STRUCTURES AND AXIAL FLOW
WEBER, W. (1866) Theorie der durch Wasser oder andere incompressibile Fliissigkeiten
in elastischen Rohren fortgepflanzten Wellen. (Theory of waves propagating in water or
other incompressible liquids contained in elastic pipes.) Berichte uber die Verhandlun-
gen der Koniglichen Sachsischen Gesellschaft der WissenschaJten zu Leipzig, Leipzig,
Germany, Mathematical-Physical Section 18, 353-357 (in German).
WEBSTER, P.M" SAWATZKY, RP., HOFFSTEIN, V" LEBLANC, R, HINCHEY, M.J.
& SULLlV AN, P.A. (1985) Wall motion in expiratory flow limitation: choke and flutter.
Journal oj Applied Physiology 59, 1304-1312.
WElNGARTEN, v.I. (1965) Free vibration of ring-stiffened conical shells. AlAA Journal
3, 1475-1481.
WELT, E & MODI, v.J. (1992) Vibration damping through liquid sloshing, Part I. A
nonlinear analysis. ASME Journal oj Vibration and Acoustics 114, 10-16,
WENDT, E (1933) Turbulente Stromungen zwischen zwei rotierenden koaxialen Zylin-
dem. lngenieur Archiv 4, 577-595,
WEPPELlNK, H. (1979) Free vibrations of finite circular cylindrical shells and tubes
with and without a surrounding fluid. M.Sc. Thesis, Technische Hogeschool Twente,
Enschede, The Netherlands.
WEYERS, P.ER (1960) Vibration and near-field sound of thin-walled cylinders caused
by internal turbu1ent flow. NASA Report TN-D-430.
WHlTE, EM. (1974) Viscous Fluid Flow. New York: McGraw Hill.
WICKERT, ],A. (1992) Non-linear vibration of a traveling tensioned beam, lnternational
Journal oj Non-Linear Mechanics 27, 503-517.
WlCKERT, J ,A, & MOTE, C.D. Jr. (1989) Current research on the vibration and stability
ofaxially-moving materials, The Shock and Vibration Digest 20,3-13,
WICKERT, J.A. & MOTE, C.D" Jr (1990) Classical vibration analysis oí axially moving
continua. Journal oj Applied Mechanics 57, 738-744.
WIDNALL, S,E. & DOWELL, E.H. (1967) Aerodynamic forces on an oscillating
cylindrical duct with an internal flow. Journal oj Sound and Vibration 6, 71-85.
WIGGERT, D.C. (1996) Fluid transients in flexible piping systems. In Proceedings VIII
IAHR Symposium on Hydraulic Machinery and Cavitation, Valencia, Spain, pp, 58-67.
Delft: IAHR
WILKlNSON, D,H. (1978) Acoustic and mechanical vibrations ofliquid-filled pipework
systems. proceedings BNES lnternational ConJerence on Vibration in Nuclear Plant,
Keswick, UK, Paper 8.5,
WILLlAMS, T,M. (1999) The evolution oí cost efficient swimming in marine mam-
mals: limits to energetic optimization, Philosophical Transactions oj the Royal Society
(London) B354, 193-201.
WILLMARTH, W.W. (1958) Wall pressure fluctuations in a turbulent boundary layer.
NACA Report TN-4139.
WlLLMARTH, W,W. (1975) Pressure fluctuations beneath turbulent boundary layers.
Annual Review oj Fluid Mechanics 7,13-38,
WlLLMARTH, W. W. & ROOS, F. W. (1965) Resolution and structure ofthe wall pressure
field beneath a turbulent boundary layer. Journal oj Fluid Mechanics 22,81-95.
WILSON, B, (1963) A geometrically nonlinear theory of shells. Report TR32-584, Jet
Propulsion Laboratory, Pasadena, CA, USA.
WlLSON, RJ, & JONES, B.G. (1983) Turbulent pressure-velocity measurements in a
fully developed concentric annular air flow, ASME Journal oj Vibration, Acoustics,
Stress, and Reliability in Design 105, 345-354,

REFERENCES
1569
WILSON, T.A., RODARTE, J.R. & BUTLER, J.P. (1986) Wave-speed and viscous flow
1imitation. In Handbook oj Physiology (eds P.T. Macklem & J. Mead), pp. 55-61.
Bethesda, MD: American Physio1ogica1 Society.
WOLFGANG, M.J., ANDERSON, J.M., GROSENGAUGH, M.A., YUE, D.K.P. &
TRIANTAFYLLOU, M.S. (1999) Near-body flow dynamics in swimming fish. Journal
oj Experimental Biology 202,2303-2327.
WOLFRAM, S. (1996) The Mathematica Book, 3rd edition. Cambridge: Cambridge
University Press.
WOMERSLEY, J.R. (1955) Osci11atory motion of a viscous liquid in a thin-walled e1astic
tube. I: The 1inear approximation for long waves. Philosophical Magazine 46, 199-221.
WONG, S.T.-M. (2000) Dynamics of fluid-conveying shells with symmetric and asym-
metric end-supports. M.Eng. Thesis, Department of Mechanica1 Engineering, McGill
University, Montrea1, Québec, Canada.
WRINCH, D. (1922) On the 1atera1 vibrations ofbars of conica1 type. Proceedings oJthe
Royal Society oj London AlOI, 493-508.
WU, M.-S., GARCIA-FOGEDA, P. & LIU, D.D. (1989) Potentia1 flow over bodies of
revo1ution in unsteady motion. AlAA Journal27, 725-733.
WU, S.F. & ZHU, J. (1997) Stability ana1ysis of a non1inear p1ate in mean flow. Journal
oj Computational Acoustics 5, 137-155.
WU, T.Y.-T. (1961) Swimming ofa wavingp1ate. JournaloJFluidMechanics 10,321-344.
WU, T.Y. (1971a) Hydromechanics ofswimming propu1sion. Part 1. Swimming of a two-
dimensional flexib1e p1ate at variab1e forward speeds in an inviscid fluid. Journal oj
Fluid Mechanics 46, 337-355.
Wu, T. Y. (1971 b) H ydromechanics of swimming propu1sion. Part 2. Some optimum shape
prob1ems. Journal oj Fluid Mechanics 46,521-544.
Wu, T.Y. (1971c) Hydromechanics of swimming propu1sion. Part 3. Swimming and
optimum movement of slender fish with side fins. Journal oj Fluid Mechanics 46,
545-568.
WU, T. Y.- T. (1971d) Hydromechanics of swimming fishes and cetaceans. In Advances in
Applied Mechanics 11, 1-63.
WUNSCH, A.D. (1994) Complex Variables with Applications, 2nd edition. Reading:
Addison- Wes1ey.
WYLlE, E.B. & STREETER, V.L. (1978) Fluid Transients. New York: McGraw-Hill;
a1so 1983, Ann Arbor: FEB Press.
YADYKIN, J.V. (1982) The energy stability ana1ysis of flexib1e cab1es in axia1 flow.
Dopovdi Seriia A: Fizyko-Matematychni ta Tekhnichni Nauky, No. 12, pp. 39-44
(in Ukrainian). Akademii Nauk URSR.
YADYKIN, Y. & LEVIN, D. (1998) The dynamic stability of a flexib1e surface with
varying tension in a paralle1 flow. AIAA Paper 98-2654.
YADYKlN, J.v., KALlUKH, Y.I. & GORBAN, V.O. (1987) Stability offlexib1e wire in
flow at small ang1e of attack. Dopovdi Seriia A: Fizyko-Matematychni ta Tekhnichni
Nauky, No. 1, pp. 31-34 (in Ukrainian). Akademii Nauk URSR.
YADYKlN, Y., TENETOV, V. & LEVIN, D. (2001) The flow-induced vibration of a
flexib1e strip hanging vertically in a paralle1 flow. Part 1. Tempora1 aeroe1astic instabi1ity.
Journal oj Fluids and Structures 15, 1167-1185.
YAMAKI, N. (1984) Elastic Stability oj Circular Cylindrical Shells. Amsterdam:
North-Holland.

1570
SLENDER STRUCTURES AND AXIAL FLOW
YAMAKI. N.. TANI. J. & YAMAJI. T. (1984) Free vibration of a clamped-damped
circular cylindrical shell partially filled with an intemalliquid. Journal oj Sound and
Vibration 94, 531-550.
YAMAMOTO, T. (1976) Hydrodynamic forces on multiple circular cylinders. ASCE
Journal oJthe Hydraulics Division 102, 1193-1210.
YAMAMOTO. Y. (1989) Aeroelastic divergence offront-free-aft-fixedelastic strip parallel
to uniform flow. Transactions oJthe Japan Society Jor Aeronautical and Space Sciences
31. 188-194.
YAMAMOTO. Y. (2001) Flutter of fixed-free elastic strip. Transactions oj the Japan
Society Jor Aeronautical and Space Sciences 44. No. 143, 8-12.
YAMANE. T. & ORITA, T. (1992) Self-excited oscillations with and without supercrit-
ical flow in collapsible tubes. Proceedings oj the 7th lnternational ConJerence on
Biomedical Engineering, Singapore, pp. 502-504.
YANG, C.!. & MORAN, T.J. (1979) Finite element solution of added mass and damping
of oscillating rods in viscous fluids. Journal oj Applied Mechanics 46, 519-523.
YASUDA, K & SHlMIZU, T. (1992) Self-excited osci1lations of a circular disk rotating
in air. JSME lnternational Journal35, 347-352.
YASUO, A. & PAIDOUSSIS, M.P. (1989) F1ow-induced instabi1ity of heat-exchanger
tubes in a diffuser-shaped, loose intermediate support. ASME Journal oj Pressure Vessel
Technology 111, 428-434.
YE, W. & DOWELL, E.H. (1991) Limit cycle oscillation of a fluttering cantilever plate.
AlAA Journa129, 1929-1936.
YEH, T.T. & CHEN, S.S. (1977) Dynamics of a cylindrica1 shell system coupled by a
viscous fluid. Journal oJthe Acoustical Society oj America 62, 262-270.
YEH, T,T. & CHEN, S.S. (1978) The effect of fluid viscosity on coupled tube/fluid
vibrations. Journal oj Sound and Vibration 59, 1-24.
YEH. Y.S. & CHEN. S.S. (1990) F1ow-induced vibration of component cooling water
heat exchangers. In Flow-lnduced Vibration - 1990 (eds S.S. Chen, K Fujita &
M.K Au-Yang), PVP-Vol. 189, pp. 153-163, New York: ASME.
YEO, KS. & DOWLING, A.P. (1987) The stability of inviscid flows over passive
compliant walls. Journal oj Fluid Mechanics 183, 265-292.
YODER. P.1. (1973) Equations goveming the smalllateral motions of a slender axisymmet-
ric body towed through an incompressible fluid. B.Eng. Honours Thesis, Department
of Mechanical Engineering, McGi1l University, Montréa1, QC, Canada.
YOSHlKAWA, S.. WILLIAMS, E.G. & WASHBURN, KB. (1994) Vibration of two
concentric submerged cylindrical shells coupled by the entrained fluid, Journal oj
Acoustical Society oj America 95, 3273-3286.
YOUNG, T. (1808) HydrauHc investigations, subservient to an intended Croonian Lecture
on the motion of the blood. Philosophical Transactions oj the Royal Society oj London
98, 164-186.
YOUSRI, S.N. & FAHY, F.J. (1976) Acoustically induced vibration of, and sound radiation
from, beams inside an enclosure. Journal oj Sound and Vibration 45, 583-594.
ZDRAVKOVICH, M.M. (1997) Flow around Circular Cylinders, Volume 1. Oxford:
Oxford University Press.
ZDRA VKOVICH, M.M. (2003) Flow around Circular Cylinders, Applications, Volume 2.
Oxford: Oxford University Press.
ZEDAN. M.F. & DALTON. C. (1980) Higher-order axial singularity distributions for
potential flow about bodies of revolution. Computer Methods in Applied Mechanics
and Engineering 21. 295-314.

REFERENCES
1571
ZHANG, J., ELLIS, J. & ROBERTS, J.B. (1993) Observations on the nonlinear fluid forces
in short cylindrical squeeze film dampers. ASME Journal oJTribology 15, 692-698.
ZHANG, l, ROBERTS, J.B. & ELLIS, l (1994) Experimental behavior of a short cyl-
indrical squeeze film damper executing circular centered orbits. ASME Journal oj
Tribology 116, 528-534.
ZHANG, Y.L., GORMAN, D.G. & REESE, lM. (1999) Eigenmode analysis of a cyl-
indrical shell conveying viscous fluid. Proceedings 3rd lnternational ConJerence on
Engineering Aero-Hydroelasticity, Prague, Czech Republic, pp. 398-402.
ZHANG, Y.L., REESE, lM, & GORMAN, D.G. (2001) Vibration ofprestressed ortho-
tropic cylindrical shells conveying fluid. In Flow-Induced Vibration - 2001, Vol. 2
(ed. M.J. Pettigrew), PVP-Vol. 420-2, pp. 191-198. NewYork: ASME.AIso, in Journal
Jor Numerical Methods in Engineering.
ZHANG, Y.L., REESE, J.M. & GORMAN, D.G. (2002) Initially tensioned orthotropic
cylindrical shells conveying fluid: a vibration analysis. Journal oj Fluids and Structures
16,53-70.
ZHU, F. (1994) Rayleigh quotients for coupled free vibrations. Journal oj Sound and
Vibration 171, 641-649.
ZHU, F. (1995) Vibration and stability analysis oftoroidal shells conveying fluid. Journal
oj Sound and Vibration 183, 197-208.
ZHU, F., HALL, K. & RAHN, C.D. (1998) Steady state response and stability ofballooning
strings in air. International Journal oj Non-Linear Mechanics 33, 33-46.
ZHU, Q., WOLFGANG, M.J., YUE, D.K.P. & TRIANTAFYLLOU, M.S. (2002) Three-
dimensional flow structures and vorticity control in fish-like swimming, Journal oj
Fluid Mechanics 468, 1-28.
ZHU, W.D. & NI, J. (2000) Energetics and stability oftranslating media with an arbitrarily
varying length. ASME Journal oJVibration and Acoustics 122, 295-304.
ZHU, W.D" GUO, B.Z. & MOm C.D. Jr, (2000) Stability of a translating tensioned beam
through a pointwise control force. ASME Journal oj Dynamic Systems, Measurement,
and Control122, 322-331.
ZHU, W.D., NI, J. & HUANG, l (2001) Active control oftranslating media with arbitarily
varying length. ASME Journal oj Vibration and Acoustics 123, 347-358.
ZIADA, S., BŮHLMANN, E.T, & BOLLETER, U, (1988) Model tests on shell flutter
due to flow on both sides. Journal oj Fluids and Structures 2, 177-196.
ZIENKIEWICZ, O.c. & CHEUNG, Y.K. (1968) The Finite Element Method in Structural
and Continuum Mechanics. New York: McGraw Hi11.
ZIENKIEWICZ, O.C. & TAYLOR, R.L. (1989) The Finite Element Method, 4th edition,
Vols. I and 2. New York: McGraw Hi11.
ZIENKIEWICZ, O,C., GAGO, J. & KELLY, D.W. (1983) The hierarchical concept of
finite element analysis. Computers and Structures 16, 53-65.
ZOLOTAREV,1. (1987a) Optimization of aerodynamic damping of a cylindrica1 shell with
flowing fluid with the aid of boundary conditions. Mashinovedenie (Soviet Machine
Science), No. 1,44-49.
ZOLOTAREV, 1. (1987b) N atural vibrations of cylindrical shells conveying fluid with
spring supported ends. (In Czech: Vlastní kmitání pružně uložené válcové skořepiny
protékané tekutinou). Strojnícky Časopis 38, 533-550.
ZOLOTAREV,1. & KRUNTCHEVA, M. (1995) Vibration of cylindrical shells invol-
ving coupling with flowing acoustica1 medium, In Flow-Induced Vibration (ed.
P.W. Bearman), pp. 535-544. Rotterdam: Balkema.
ZWILLINGER, D. (1989) Handbook oj Differential Equations. San Diego: Academic
Press.

Index


Absolute Instabi1ities; see Plates in axia! flow; absolute
and convective instabilities
Acoustic streamers 787,928,953-7, 1031
Added mass
cylinders in tluidltlow 790, 795-6, 844-5, 1034,
1045, 1234, 1239-42, 1244-5, 1272, 1280,
1320, 1343-4, 1361
plates in fluidlflow 1155, 1183, 1207-9, 1218, 1252,
1259
shells with fluidlflow 583,589-90, 1371, 1377,
1396, 1412
Added mass mattix 631, 1034, 1045, 1059, 1092,
1239-42, 1259, 1362
Aeroelasticity 573, 649, 846, 926, 1137
Anisotropic shells 646-9, 1428-9
Anisotropic shells conveying fluid 646-9
companson isotropiclanisotropic 647-9
Annular and leakage flow
annular flow-induced instabilities 1269-321
annular tlow, numerical methods 1321-55
applications, miscellaneous 1419-20
engineering problem cases 1225-30
aircraft engine heat shield 1417-19
BWR leakage-flow instabi1ity 1230
feedwater sparger problems 1231-3
other Superphénix-related case 1417
PWR core barrel 1226-7
PWR thenna! shield 1229-30
Superphénix shell-weir 1414-16
hydraulic gates 1222
instabilities in slip-joint geometry 1221
example 1222-3
leakage-flow instabilities 1249-69
mechanisms for instability 1223-5
AGR gag-bomb 1225
c1assica! mechanism 1224
control-rod blades 1223, 1225
representative problem cases 1223-5
plug, check valves 1222
vibrations of coaxia! shells in annuli, quiescent fluid
1246-9
blow-down conditions 1248
effect of liquid level 1248
forced vibration 1248-9
natura! frequencies 1247
vibrations of cylinders in narrow annuli, quiescent
fluid 1233-49
added mass coefficients 1239-43
eccentric configurations 1238


experiments 1237
finite element method, moving boundaries
1243-5
finite-Iength coaxia! cylinders 1234-7
inertia! and viscous coupling, coaxia! cylinders
1234
spectra! collocation method 1237-43
squeeze-film effects 1245-6
viscous damping coefficients 1240-3
Annular flow, dynamics, numerica! methods
1338-55
Bélanger et a/. model 1340-5
divergence 1343-4
large added-mass, modifications 1343-4
Runge-Kutta predictor-corrector scheme 1342,
1344
the model 1340-2
typical results 1343-4
Li, Kaneko & Hayama model 1345-6
MOCCA code 1345-55
comparison with Hobson model 1353-4
comparison with Porcher & de Langre model
1354-5
comparison with TAXI experiments 1353-4
flutter 1354
method of integral relations 1347
the model for narrow annuli 1347-53
other, semi-empirica! models 1345-6
flutter 1345
time.dependent coordinate transfonnation 1344
Annular flow, numerica! methods, prescribed motions
1321-38
CFD method of Mateescu, Bélanger & Pa'idoussis
1321-8
a!temating direction implicit (AD!) scheme 1323,
1326
artificia! compressibility 1322, 1326
boundary conditions 1325-6
Crank-Nicolson scheme 1326
decoupled sca!ar tridiagona! system 1325
pseudo-time relaxation 1322
pseudo-time 1322
typical results 1326-9
CFD method of Mekanik et a/, 1328-32
moving-boundary boundary conditions 1328
time-dependent coordinate transfonnation
1328
typical results 1331-2


1573




1574
Annular flow, numerical methods, prescribed
motions (continued)
comparison with experiments 1332-4
spectral collocation method 1334-8
finite difference solution 1336-7
semi-analytical model 1338
the model 1334-6
typical results 1338-9
Annular-flow-induced instabilities, flexible cylinders;
see a/so: Annular flow, dynamics, numerical
methods
experiments in nonuniform annuli 1317-18
effect of annular shape 1318
instrumentation tubes in PWRs 1317
experiments/theoretical prediction in uniform annuli
1303-10
cantilevered cylinders 1304-5, 1307-9
comparison with theory 1307-10
cylinders with supported ends 1305-6, 1309-10
divergence, flutter 1303
model for uniform narrow annuli, heat exchanger
supports 1318-21
divergence,flutter 1320
effectofannularshape 1320
model for uniform narrow annuli 1310-16
analytical model 1310-12
annular width effect 1313-16
effect of sliding at supports 1313-16
steady and unsteady viscous effects 1312-16
Annular flow-induced instabilities, rigid-body motions;
see a/so: Annular flow, dynamics, numerical
methods 1269-303
Arai & Tajima study 1300
experiments
Hobson et a/, experiments 1275
Mateescu et al. experiments 1290-3,1297
Sim and Mekanik experiments 1332-4
TAXI experiments 1298-300
Hobson et a/, studies 1269-76
analytical model 1269-73
comparison of theory to experiment 1275-6
diffuser efficiency effect 1273
experiments 1275
negative damping in stability (flutter) 1273,
1276
upstreamldownstream constriction effects 1273
Mateescu et a/. viscous models 1284-90
annular width effect 1288-9
comparison of laminar and turbulent flow
predictions 1292-3
comparison with experiments 1291-3, 1297
flutter, divergence 1287-8
mean-velocity vector oscillation 1286
model for laminar flow 1284-7
model for turbulent flow 1289-90
viscous effects on stability 1287-8
Mateescu & Pa'idoussis inviscid model 1277-84
analytical model 1277-81
comparison with experiment 1291-3,1297
convergent/divergent channel, effect on stability
1282-3
flutter, divergence 1281, 1283
INDEX
hinge location, effect on stability 1281-3
mechanism of in stability 1283-4
sensitivity to boundary conditions 1292-8
comparison with experiment 1297
divergence, flutter 1298
effect of specific sets of boundary conditions
1297-8
nonpenetration boundary condition refinement
1295
specification of up- and downstream perturbation
pressure 1294
train of interconnected cylinders 1301-3
Applications; see Engineering applications and
Physiological applications
Articulated cylinders; see Self-propelledlarticulated
cyHnders; a/so Articulated cylinder in confined
flow
Articulated cylinders in confined flow, dynamics
1003-5, 1010-21
chaotic dynamics 1005, 1010-21
dynamics of 2-dof system 1010-12
Feigenbaum number 1012
period-doubling bifurcations 1012
Poincaré map 1012
intenIŮttency route to chaos 1016-21
"Iaminar", "turbulence" phases 1017
return maps 1020
type 1ll intermittency 10 19
linear dynamics 1003-5
three routes to chaos, 3-dof system 1212-21
intenIŮttency route to chaos 1016-21
period-doubling route to chaos 10 12-14
quasipenodic route to chaos 1014-16
Asymmetric support conditions for shells with flow
639-46
energy considerations 641
new doubts 646
paradoxical dynamics, clues, resolution 640-3
Blue sky catastrophe 745, 764
Bodies of revolution, flow over 789
Boundary layer excitation of shells 717-21
Buckling flows 856
Cable-body configurations 927
CFD analysis 969, 1294, 1321-32, 1334, 1340-55,
1401-10, 1494-6
Chaos 600,726,729,745,763,773,779-81,925,989,
993-4,998,1002-3,1005-21,1199,1205-7,
1220
Clamped.pinned shells; see Asymmetric support
conditions
Clustered cylinders in axial flow 1033-135
design considerations 1127,1134-5
hydrodynamic coupling 1033-54
inertial coupling 1033-48
constrained resonance 1038-40
cross-sectional pattems of motion 1036-7
distinct/repeated eigenfrequencies 1036--7
effective added mass coefficient 1048

experimental results 1040
groups of natura) frequencies 1034
inviscid quiescent fluid 1034-48
nth mode frequency band 1036
PSDs of motion 1036, 1038
Stokes number effects 1040
upperllower frequency bounds 1048
intercylinder gap, dimensionless 1036
inviscid coupling, analytica) methods 1040-8
3-D potentia) flow ana)ysis 1046--7
Chen and Chen & Chung method 1046
finite element methods 1047, 1474-6
flow-path network methods 1047
homogenization schemes, periodic geometrie s
1047
large amplitude, nonlinear methods 1048
Lin & Chen method, compressible fluids 1046
malched-asymplotic-expansion method 1047
pardoussis & Suss melhod 1040-5
structura) coupling; see Clustered cylinders, flexibly
interconnected
viscous coupling 1049-54
Lin's analytica) method 1049-53, 1477-9
pardoussis & Suss approximate melhod 1053-4
Clustered cylinders in axial flow, dynamics 1054-81
dynamica) behaviour, comparison of theory and
experiments 1073-81
four-, two- and three-cylinder clusters 1074-5
Iransition 10 divergence 1075-81
dynamica) behaviour, experimenta) 1068-81
apparalus 1068-71
cross-sectiona) modal patterns 1071
divergence onsel 1075-81
divergence, flutter 1071
general observations 1071-3
dynamica) behaviour, theoretical 1060-8
Argand diagrams 1061, 1064-5
cross-sectiona) modal patterns 1062, 1066--8
divergence, coupled-mode flutter 1061
effect of contiguity 1063
effecl of syslem parameters 1063
flexibly interconnected cylinders 1081-91
flexural-mode groups 1064-7
stability boundaries 1064-5
equations of motion 1055-9
dimensionless equations 1059
solution of Ihe equalions 1060
Clustered cylinders, flexibly interconnected 1081-91
in flow 1087-91
in vacuum 1081-7
effecl of spring sliffnesses 1085-7, 1480-1
Cluslered cylinders, flow-induced vibralion 1097-127
analYlical models 1105-12,1482-3
approximate model 1108-12
far-field, acoustic componenls in excitation 1110
more elaborate models 1112
response cross-correlation 1107
response cross-spectral density 1110
shortesllateral fluid distance concepl 1109
spati a) pressure correlalion funclions 1109
turbulence wa)l-pressure field 1108-9
wa)l-pressure PSDs 1110-12
INDEX
1575
comparison of theory and experiment 1113-26
28-cylinder clusler 1122-4
coherence, CSD phase, adjacent cylinder-pair
1116
coherence, CSD phase, diagona) cylinder pair
1119, 1124
effect of G c 1119
effect of number of cylinders 1121
effecl of some cylinders being rigid 1122
empirica) versus analytica) vibration amplitude
1124
four-cylinder cluster 1113-22
PSDsofvibralion 1113-16,1119-22,1123-4
vibration amplilude 1124-6
experiments; see a)so comparison of theory and
experiment 1097-104
apparatus 1097-8
characteristic directions, modal planes 11 ()()
CSDs of vibration 1102-4
moda! characteristics 1100-5
PSDs of vibration 1098-9, 1101-2
radial, tangenlia) vibration 11 ()()
vibration characteristics 1098-105
Clustered cylinders, wilh both interna) and exlerna) flow
1091-7
Argand diagrams 1094-5
equations of molion 1092-3
Coaxial cantilevered shells, with flow 1394-410
approximate viscous model 1394-8
ana)ytica) model 1491-3
critical flow velocities, annular flow 1394-6
divergence, flutter 1395
effects of SYSlem parameters 1396--8
interna) and annular flow concurrendy 1396,
1398
inlerna) dissipation, effect of 1395-6
sleady viscous effects 1394-8, 1493-4
CFD model by Nguyen et al. 1401-10
approximation to deforming grid 1408
comparison oftheory 10 experimenl 1409-10
divergence, flutter 1409
fixed-grid results 1408
method of solution 1404-7, 1494-6
model fonnulation 1401-4
moving boundary effects 1408
Iypica) results 1408-10
experimenls 1398-401
annular gap, effect on stability 1399-401
flutter, divergence 1399
Fujita et al. experimenls 1401
moda1lspectral characlerislics 1399-400
Coaxia) shells with flow 1371-419
canlilevered shells 1394-410
finite element codes 1371
shel1s with supported ends 13 71-9
simplified theory 1410-14
wave propagation 1387-8, 1410-13
Coaxia) shells with supported ends, with flow 1371-94
expenmenls 1390-3
comparison ofvarious Iheories 1393,1412-13
comparison with theory 1391-3
crilica) flow velocily 1391-3

1576
Coaxial shells with supported ends, with ftow
( continued)
experiments (continued)
frequency-ftow characteristics 1390-2
modallspectral characteristics 1390-2
Horáček's simplified theory 1410-13
analytical model 1410-12
divergence, ftutter 1412
typical results 1412-13
inviscid model 1372-83
"y-gap" system, detinition 1375
analytical model 1372-5, 1488-91
divergence, ftutter 1376
dynamics, coaxial shells, annular ftow 1381-3
dynamics, coaxial shells, internal ftow 1377-81
dynamics, compressible ftows 1383
dynamics, outer shell rigid 1376-7
simplitied model 1383,1410-14
nonlinear effects 1388-90
ftutter, nonoccurrence of 1390
subcritical static bifurcations 1389
typical results 1389-90
steady viscous effects 1384-7
additional pressurization 1386
analytical model 1384-5, 1490-1
divergence, ftutter 1385
typical results 1385-7
unsteady viscous effects 1387-8
Collapsible physiological tubes 750--81
bibliography 1437-43
chaotic oscillations 779-81
intermittency 781
collapse 752
ftow limitation 751, 752
ftow-induced oscil1ation 760-81
blue sky catastrophe 764
choking 765
control-space diagrams 763
different types of oscillations 763
flapping 767
flutter 766
limit-cycle oscillations 769
long-wavelength oscillations 765
milking oscillations 764
physical characteristics 760-7
models for flow-induced oscillation 768-79,1443-7
Bertram & Pedley model 768
Cancelli & Pedley l-D model 770, 1445-6
Conrad model 768
Grotberg et a/, 2-D models 774-7, 1446-7
Huang model 778
Jensen's l-D weak1y nonlinear analysis 771
Jensen's equations, Cancelli & pedley model 770
lumped-parameter models 768-9, 1444-5
Luo & Pedley's 2-D model 778, 1447
one-dimensional models 769-73
two-dimensional models 773-9
Walsh et a/. I-D model 773
pulmonary passages, trachea 752, 773
self-excited osci1lation 752,760-81
Starling resistor 752-3
steady flow 752-60
INDEX
elastic jump 755
flow limitation, experiments 760
flow limitation 755-60
pressure-area relations 753-4
pressure-drop limitation 755
Shapiro number, "speed index" 756
tube law 754
viscous flow limitation mechanism 757-8
wave-speed flow limitation mechanism 756
transmural pressure 751,753
urinary conduits 752
veins, arteries 752
Col1apsible tubes 743-81, 1437-7
pressure waves horizontal systems 746-51
analogy to gravity waves 749
fractional filling, a 748
Moens-Korteweg equation 750
wave velocity, extensible tubes 750
wave velocity, long waves 748
Colostomy tubes 595, 754
Companion mode in shell motions 723
Compliant coatings 653, 1144, 1152, 1176
Compressible subsonic flow
cylinders in flow 836-44,848-50, 1034, 1046
plates in flow 1137,1167-72, 1150-1, 1174-6,
1190,1199-203
shells with flow 582,605,608,612,613,631,
649-51,652-4,674,697,731,748,1372-5.
1383, 1388-90
Continement effects, shells with ftow; see
viscous/confinement effects
Control-space diagrams 763
Convective Instabilities; see Plates in axial flow;
absolute and convective instabilities
Coriolis shell-type flow meters 783-6
Coupled-mode flutter; see various physica/ systems
Coupling
acoustic 598,652-4,660,664, 1027; see a/so: Wave
propagation
hydrodynanůc 667,755, 1033-54, 1057-81.
1087-91,1092-7,1097-105,1113-27,1134,
1164-5,1183,1234-45,1246,1342,1376-83,
1416
sttucnffal 723-4,732,740,773,875-6,924,
1081-91
Cross-ftow-induced vibration 787, 926
Cylinders in axial flow 787-1032
annular/leakage flow oscillations 788, 1221-33,
1269-371
arti culated cy linders 1003-23
boundary layer effects 887,891-2,894,897
buckling (divergence) 788,802,850,892,904,992,
1010
complex support conditions 922-6
loose supports 924-6
multispan cylinders 923-4
concurrent with internal flow 898-908
dynamics, linear; see Cylinders in axial flow, linear
dynanůcs
equations oi motion; see Equations of motion,
cylinders in axial ftow, linear or nonlinear
harmonica11y perturbed ftow 908-18

low-amplitude vibration 788, 858-84
mechanisms of in stability
analogy to low-aspect ratio wings 830, 798
divergence, effect of base drag 830
divergence, free downstream end 830
divergence, supported ends 829-30
flutter, effect of f 833-4
flutter, free downstream end 833-4
flutter, general 831-2
flutter, supported ends 832-3
nonlinear dynamics, see Cylinders in axial flow,
nonlinear dynamics
osci1latory in stability (flutter) 788, 803, 850, 892,
904,992,1010
parametric resonances 908-22
pendular systems 1003-23
practical applications 1024-32
subcritical vibrations 788, 858-84
discussion of mechanisms 860-8, 870-2, 880-2
semi-empirical prediction 788,868-70
tapered cylinders 885-97
truncated cones 885, 87
wave propagation 848-50
Cylinders in axial flow, complex support conditions
922-6
loose supports 924-6
chaotic osci1lations 925
cross-flow-induced vibrations 926
inactive intennediate supports 925
mode localizationlenergy trapping modes 924-5
squeeze- film effects 924
stopping bands 925
vibro-impact 924
multispan cylinders 923-4
Cylinders in axial flow, flow-induced vibration 858-84
acoustically excited vibration 884
cantilevered cylinders 882-3
free-end shape effect 883
eccentricity effects 880
empirical expressions 868-70
PaYdoussis semi-empirical model 869-70
excitation pressure field 876-7, 878-80
inhomogeneity in pressure fie1d 880
non-axisymmetry in pressure field 880
upstream disturbances 870, 878
forced vibration model s 864-8
a more rigorous model 1453-6
Chen & Wambsganss model 866, 872-8
cross-acceptance, joint acceptance 875
far-fie1d effects 872,883
Gonnan's model 865
main assumptions 864
Reavis' model 865
"system" disturbances 866
two-phase flow effects 865,866,883
mechanisms of excitation 860-8, 880-2
comparlson and discussion 870-2, 883-4
forced vibration models 864-8, 878-82
further work and refinements 878-82
parametric excitation model 861-3
self-excited model 860-1
multispan cylinders 882, 923-4
INDEX
1577
vibro-impact 882, 884, 924
Cylinders in axial flow, linear dynarnics 802-15
boundary layer effects 834-6, 887, 894, 897
effect of C N , CT and boundary layer thickness
836,891-2
modified inviscid fluid force 836, 887
canti1evered cylinders 803-7, 808-11
coupled-mode flutter 805
cylinder with f = I 805
divergence 803
effect of varying f 805, 808
single-mode flutter 803
comparlson with theory, cantnevered cylinders
821-8
effect of f3 826-7
effect of free end 824-6
effect of slendemess 828
effect of surface roughness 828
comparlson with theory, pinned-free cylinders
828-9
comparlson with theory, pinned-pinned cylinders
821
compressibi1ity effects 838-44
3-D potential flow model 840-1
compressibility/s1endemess effect 842-4
effect of compressibility 842-4
integral-transfonn solution 841-4
slender-body model 840
wave propagation 848-50
concurrent intema1 and extemal flow 898-908
cylinders with supported ends, confined flow 803
cylinders with supported ends. unconfined flow
802-3,807-8
coupled-mode flutter 803
divergence 802
mode veeringlmode exchange 803
typical Argand diagrams 802-7
effect of system parameters, canti1evered cylinders
808-11,824-8
effect of f3 809, 826-7
effect of eCN and eCT 828
effect of f and Cb 808,824-7
effect of f together with eCN and eCT 811-13
effect of system parameters, cylinders with supported
ends 807-8
approximate expression for Ucd 808
experiments
flow-induced damping 815-16, 1450
low-u experiments 815, 1450
more recent experiments, apparatus 820
more recent experiments, observations 820-1
PaYdoussis (1966) experiments, observations
817-20
Pai'doussis experiments (1966) apparatus 815-17
with holes in end-piece 821
pinned-free cylinders 811-15
divergence, critical velocity 814
tapered cantilevers 891-7
boundarylayereffects 887,891-2,894-7
comparlson oftheory and experiments 895-7
experiments 894-5
theoretical results 891-4

1578
Cylinders in axial flow, linear dynamics (continued)
two-phase flow effects 836-8, 844-8
I-dof model for random added-mass variations
846-8
added mass versus void fraction 844--5
effect ofrandom added-mass variations 847-8
homogeneous flow model 837
nonoccurrence of fluidelastic instabilities 837
parametric resonances 918-22
random added-mass variations 844-8
two-phase damping versus void fraction 844-5
wave propagation 848-50
Cylinders in axia! flow, nonlinear dynamics 988-1003
cantilevered cylinders, comparison with experiments
996-8
amplitude of divergence and flutter 998
divergence, 2 nd and 3 rd mode flutter 998
frequency of oscillation 998
quasiperiodic motions 998
cantilevered cylinders, theory 989-96
aperiodic solution 994
bifurcation diagrams 989
coexisting attractors 994
divergence, flutter 992
linear stability reassessed 994-6
penod bubbling 993
period-doubling bifurcations 992
pitchfork, Hopf, toros, saddle-node bifurcations
990-1
quasiperiodic solutions 994
transition, 2 nd to 3 rd mode flutter 992
cylinders with supported ends, theory 999-1003
chaotic motions 1002
pitchfork bifurcation 999
post -divergence flutter 999
quasiperiodic solutions 1002
Cylinders in concurrent interna! and externa! flow
898-908
design considerations 908
drill-strings, heat exchanger tubes 898
dynamics; see Cylinders in concurrent internal and
externa! flow, dynamics
equations of motion; see Equations of motion,
cylinders in concurrent internal and externa! flow
Galerkin-like Fourier-series solution 900
Cylinders in concurrent interna! and external flow,
dynamics
dynamics of cantilevered cylinders 905-7,908
dynamics of cylinders with both ends supported 904,
905,907
equiva!ent internallexternal flow 901-2
experiments 903-7
comparison with theory 904-7
general observations 904
theoretical results 901-3
Cylinders in harmonically perturbed flow, parametric
resonances 908-22
comparison between theory and experiment
916-18
equations 908-9
experiments 912-16
apparatus 912
INDEX
genera! observations 915-16
imperfect clamped support 914
primary, principa!, secondary resonances 910
theoretical results 910-12
cantilevered cylinders 910-12
clamped-pinned cylinders 912
cylinders with pinned ends 910
effect of dissipation, confinement and eCf 912
two-phase flows 918-22
"buried" parametnc resonances 921
experimental response 921
Hara's model for slug flow 918-21
Cylindrical shells
axial wavenumber 578-9
boundary conditions 577-8
Kirchhoff's stress resultants 577
circumferential wavenumber 578-9,586
equations ofmotion 575-81
Donnell's shell equations, linear 580
Donnell's shell equations, nonlinear 731-2
FlťJgge's shell equations 576-8,584
FlťJgge-Kempner shell equations 581,601
Leissa's operator form 581
Morley-Koiter shell equations 580
Sanders' shell equations 581, 628
Cylindrica! shells with swirling flow; see
Rotofluidelasticity of shells
Deploying cylinders in dense fluid 856
dynamics 857
equations of motion 856-7
Deployment of cylinders, strings, appendages 856-8,
988-9
Divergence; see various physical systems
Disk drives; see Spinning disks
Donnell's shallow shell theory 580,731
Dracone barges 787,927, 1031
Driven mode in shell motions 723
Dynamic divergence, shells with flow 611
arguments in favour 611-12,737-40
Ekman number 709
Energy trapping modes 924, 925
Engineering applications 781-6,856,859,882, 898,
908,924--5,926-9,957,970,1003,1024--32,
1081, 1091, 1097, 1127-8, 1137, 1182, 1195,
1197, 1207, 1213, 1221-3, 1226-33, 1269, 1301,
1317,1319,1355-6,1372,1414,1417,1419-20
Equations of motion, articulated cylinders in confined
axia! flow
dimensionless equations 1009
linear equations 1006-7, 1459
nonlinear equations 1006-8
impacting model 1008
Equations of motion, clustered cylinders in axia! flow
dynamics 1055-9
strings of clusters 1129-32
turbulence-induced vibration 1105-12
Equations of motion, cylinders in axia! flow, linear
789-800

added mass 790, 795. 844
analogy to internal flow 800
boundary conditions for free downstream
end 797-8
free-end slenderness parameter J 797
low-aspect ratio wing analogy 798
boundary conditions, generalized 796
dimensionless boundary conditions 799-800
dimensionless equation of motion
cluster or confined flow 798
solitary cylinder 799
dimensionless parameters 798-9
equations for cluster or confined flow 794-6
added-mass coefficient X 795-6
hydraulic diameter 796
linearized equation of motion 796
pressure/axial-force relationship 795
equations for solitary cylinder 792, 794
Euler-Bernoulli beam theory 791
general fonnulation 789-98
inviscid force 790
Kelvin-Voigt dissipation 791
methods of solution 800-2
pressure forces 791-2,794
tapered cantilevers 885-90
equations 889,890
viscous frictional forces 793, 795-6
base drag 794
range of values 1448-52
Taylor's expressions 793
Equations of motion, cylinders in axial flow, nonlinear
971-88, 1460-72
analysis methods 983-6
expanded domain method 983
linear and nonlinear coefficients 984-6
boundary conditions 982-3, 1465-8
equations for extensible centerline 986-8, 1468-72
equations for inextensible centerline 980-2
dimensionless equations 981
frictional forces 979-80
hydrostatic forces 977-8
earlier, erroneous fonn 986
inviscid fluid dynamic forces 974-7
lateral force (lift) 977, 1460-5
slender-body, Lighthill fonnulation 974
kinetic and potential energie s of cylinders 973-4
Lagrangian and Eulerian coordinates 971-2
Equations of motion, cylinders in concurrent internal
and external flow
dimensionless equations 899-900
equations 898-9
Galerkin-like Fourier-series solution 900-1
Equations of motion, cylinders in harmonically
perturbed flow
equations and boundary conditions 908-9
Hara's model for two-phase flows 918-21
modeling of rotationally constrained end 909,914
Equations of motion, plates in axial flow
l-D equations (2-D plates) 1138, 1139, 1145, 1152,
1162, 1176, 1186
2-D equations (3-D plates) 1138, 1167, 1183
edge flutter equations for travelling web 1211
INDEX
1579
nonlinear equations 1196-7
spinning disks 1215-16
travelling web equations 1207-9
Equations of motion, shells containing/immersed in flow
conical shells 1424-7
cylindrical shells
anisotropic shells 1428-9
nonlinear equations 731-2
shells in quiescent fluid 582-4
sheUs in vacuo 575-81
shells with internal or external flow 584-6, 601,
613,622,628,671,677-8,683-4,700,711
shells with swirling flow 697, 700, 708
pliable shells
horizontal systems 746-8
oscillatory behaviour 768-70,774-5, 1444-7
steady flow 756-8
Equations of motion, spherical pendulum in axial flow
1022-3
Equations of motion, towed flexible cylinders
analytical prediction of JI and h 933
boundary conditions 931-2
dimensionless equations 932
Dowling's model for acoustic streamers 953-7
equations of motion 930
Hawthorne-Pa'idoussis model 930-3
Paldoussis & Yu model 933
Equations of motion, towed rigid cylinders 949-50
dimensionless equations 950
Equations, l-D leakage-flow instabilities
Inada & Hayama 2-d,oJ, analysis for tapered
channels 1256-9
Mulcahy's l-d,oJ, analysis for channels 1249-53
Porcher & de Langre multi-d,oJ, model 1264-6
Equations, annular-flow-induced instabilities
Bélanger et a/, CFD model 1321-5, 1340-2
modifications for large added mass 1344
Hobson et a/, equations 1269-73
Mateescu & Pa'idoussis inviscid model 1277-81
Mateescu et a/, viscous models 1284-7, 1289-90
Mekanik et a/, CFD model 1328-31, 1344
MOCCA CFD code 1347-53
Pa'idoussis, Mateescu & Sim model for flexible
cylinders 1310-2
Equations, coaxial shells with flow
cantilevered shells 1491-4
CFD model, cantilevered shells 1402-7, 1494-6
inviscid model, shells with supported ends 1372-5,
1488
simplified theory, shells with simply supported ends
1411-2
steady viscous effects, shells with supported ends
1384-5, 1490-1
Equations, cylinders in annuli
CFD method with time-dependent coordinate
transfonnation 1328-31
CFD method, flowing fluid 1321-5
coaxial cylinders, quiescent fluid 1236-7
eccentric cylinders, quiescent fluid 1238-40
finite element method, quiescent fluid 1243-4
spectral collocation method, flowing fluid 1334-8
spectral collocation method, quiescent fluid 1238-40

1580
Equations, rotors in fluid-filled annu1i
Antunes et a/, 1inear model 1360-2
more genera11inear model 1367
Extrusion of cy1inders 856, 988-9
Finite element methods
cylinders in fluidlflow 815,924,933, 1047, 1049,
1241, 1243-5, 1345-6
p1ates in fluidlflow 1161, 1205
shells with fluidlflow 627-39,649,681,711-23,
725,728,731, 1246, 1371, 1372, 1383, 1474-6
Finite element methods for shells 627-39
anisotropic shells 649, 1429
conica! shells 633, 1424-7
empty shel1 627,630,634
fluid-filled shells 631-2, 635-6
free-surface effects 632
Lakis' hybrid method 627-31, 1423-7
open cy1indrica1 shells 632,639,649
shells with flowing fluid 631-2,636-9, 1424
sloshing modes 632
Fish propu1sion; see Swimming of slender fish
Flow 1imitation 755-60
Flow-induced oscil1ation; see a/so Flutter
collapsib1e tubes 760-81
mode1s for collapsib1e tubes 768-79, 1443-7
Flow-induced vibration
c1ustered cy1inders 1097-127
cy1inders 858-84
shells 71 0-23
Flutter in aircraft engines 782
F1utter in rocket engine exhaust 782-3
Flutter; see various physica/ systems
Fretting/wear 787,859,884,924, 1027, 1127, 1135,
1224-5, 1227-9
Globa! instabi1ities; see P1ates in axia! flow; 10ca1 and
globa! instabi1ities
Group ve10city 656,662, 1159, 1179, 1181
Hawthome-Pa'idoussis model, towed cy1inders 930-3
Heatexchangertubes 787,859,882,898,923-4,
1245-6, 1318-21
Heat exchangers 670,859,882, 1024, 1027, 1033,
1081,1134, 1372, 1419
He1ica! flow in shells 696
High-cyc1e fatigue 723, 787, 1232
High-speed trains 856, 1301-3
Hopf bifurcation
cy1inders in flow 803,806,811,990-1, 994-6,1002,
1010, 1012-7, 1273, 1289, 1305, 1354
p1ates in flow 1203. 1253, 1259
shells with flow 594,597,651,763-4,772-3,776,
781, 1394, 1409
Hysteresis; see Subcritica! bifurcation
Impacting 924, 1003, 1008, 1021, 1023; see a/so
Vibro-impact
INDEX
Incompressib1e flow
cy1inders in flow 789,842-4,851,856,887,898,
908,930,953,957,974,1003,1042,1055,
1278, 1321, 1334, 1340, 1346, 1360
p1ates in flow 1137, 1139-56, 1167, 1174,
1199-203, 1250
shells with flow 573-5,584,601, 622, 651, 674, 677,
683, 700, 734, 741, 1375, 1401
and wherever not otherwise specified
Intense Neutron Generator (ING) 706-8, 781
Interrrrittency 773,781, 1016-21
Intema1lextema1 flow inlaround shells, p1ates 649-52
Inverse spaghetti prob1em 856
Kite balloons 926
Korotkoff sounds 574,752
Leakage flow; see Annu1ar and 1eakage flow
Leakage-flow instabi1ities, 1-0 ana!yses 1249-69
feedback contro1 1269
flexib1e-p1ate ana!yses 1268
Inada & Hayama's 2-d,oJ, tapered channe1 ana!ysis
1256-64
ana!ytica! model 1256-9, 1485-7
comparison of theory to experiment 1262-4
convergent, divergent channe1 effects 1259
coup1ed-mode flutter 1261
effect of system parameters 1264
experiments 1261-2
flutter, divergence 1259
in1et and exit 10ss effects 1268
Mu1cahy's 1-d,oJ, channe1 ana1ysis 1249-56
ana!ytica! model 1249-53, 1484-5
convergent, divergent shape effects 1256
divergence, flutter instabilities 1253
effectofconsu1ction 1254
effect of entrance, exit, wall-friction 10sses 1253-4
Porcher & de Langre mu1ti-d,oJ, model 1264-7
ana!ytica! model 1264-6
divergence, flutter 1267
stability conditions 1266-7
Loca1 instabilities; see P1ates in axia! flow; Iaea! and
globa! instabi1ities
Loosesupports 882,914,924-6
LPRM (loca1 power range monitor) 1029
LPRM practica! prob1em case 1027-31
Mathieu-Hill equation 862,921
Medica! imp1ants 787
Missi1es, missi1e skin 573,699,729,782,857, 1137,
1197,1205
Mode 10ca!ization; see Energy trapping modes
Moored airships 926
NAVMlfactors 1162
Non1inear dynamics
cy1inders in flow 925,988-1003,1005-23
p1ates in flow 1195-207, 1220
shells with flow 652,723-43,769,773,776,781,
786

Nuclear reactor applications 1024-7
BWR reactors 1027
damages due to PIV 1027
PWR, AGR, PHWR reactors 1025
sample practica! problem case 1027-31
statistics of PSI-type problems 1027
Nuclear reactor components 787-8, 859-61, 882, 884,
923,1003,1021,1033,1081,1097,1127-9,1134,
1182,1221-3,1225,1226-31,1317-8,1355-6,
1414-9
Nuclear reactor fuel bundles 787, 1024-5
Nuclear reactor fuel rods, elements 859, 1026
Nuclear reactor monitoring tubes, control rods 787,
1003, 1027-31
Nuclear reactors 653,670,727,782,787-8, 1024-31,
1081,1127-8,1134,1182,1248,1372
Numerical methods for annular flow
dynanrics 1338-55
prescribed motions 1321-38
Orthotropic shells with fluid 646
Outflow model s for canti1evered shells 621-7
collector-pipe, free-flow, 10ng pipe model s 623-4
Ilgamov et a/, model 627
improved Nguyen et a!. model 626
Pa'idoussis et a/, models 624
Panel, panel flutter 573,649-51,850, 1137, 1165,
1169,1220
Parametric resonances 858,861-3,908-22, 1135
Pendular systems 787, 1003-23
dynamics; see Articulated cylinders in confined axia!
flow dynamics; Spherica! pendulum in axial
flow, dynamics
equations of motion; see Equations of motion,
articulated cylinders in confined axial flow;
Equations of motion, spherica! pendulum in
axial flow
Penetration depth 1050, 1052, 1236, 1239
Penrose tubing 595, 754
Period bubbling 993,996, 1022
Period doubling 992-3, 1011-3, 1021
Phase velocity 654-9,665,667,669,833, 849, 884,
957,964,966,1139-40,1143,1158-9,1212
Physiologica! applications 746,750-81, 1178, 1188,
1419
pulmonary passages, trachea 752, 773
snoring 1188
veins, urinary conduits 752
Pinned-clamped shells; see Asymmetric support
conditions
Pitchfork bifurcation 737,802,990-2,999, 1010,
1012, 1016-7
Plates in axial flow, dynamics 1137-1220
absolute and convective instabmties 1157-61
absolute frequency 1159
definition 1159
varicose and sinuous motions of plate channel
1178-82
zero-group-velocity wavenumber 1159
baff!ed plates 1162
INDEX
1581
cantilevered plates in flow 1186-95
circulatory theory 1187
experiments 1187-8,1194-5
Guo & PaYdoussis analysis 1192-3
Huang's analysis 1188-90
jumps in stability curve 1193
Komecki, Dowell & O'Brien ana!ysis 1186-7
mass-ratio effects 1191-3
noncirculatory theory 1186
Shayo's ana!ysis 1191
snoring 1188
tapes, foi1s, strips, flags 1194-5
vortex shedding at free end 1186
convective instabmties; see absolute and conveclive
instabilities
dissipative effects, plates in flow 1172-7
boundary layer effecls 1176
compressible flow 1174-6
incompressible flow 1174
duct noise control 1220
equations of motion; see Equations of motion, plates
in axia! flow
finile width and length plates in flow 1169-72
Ellen's standing wave ana!ysis 1169-72
high aspecl ralio asymptote 1171-2
10w aspect ratio asymptole 1169-70
finite width plales in flow 1166-9
divergence, flutter boundaries 1168-9
Dowelľs analysis 1166-8
effective flutter threshold 1169
in confined flow, with free surface 1169
subsonic/supersonic flow 1168-9
foi1s, tapes, flags in flow 1194-5
Datta & Gottenberg experimenls 1195
piezoelectric eels 1195
sheel-paper flutter 1195
Taneda's experimenls 1194
Watanabe et a/. work 1195
globa! instabmties; see loca! and global inslabi1ilies
infinile plate, incompressible fluid 1139-44
critica! wavenumber 1142
damping effecl 1143-4
divergence, flutter 1141
fonnulalion, equations, ana!ysis 1139-40
mass-ratio effect 1141-2
stabilily 1141-4
10cal and globa! instabmlies 1161
10ng hanging-pipe, string-cylinder examples
1161
nonHnear dynamics; see Plates in flow, nonlinear
dynamics
parallel plate assemblies 1181-6
aspect-ratio, thickness effects 1184-5
dissipation effecls 1186
Guo & Pa'idoussis analysis 1183-6
Mi11er's analysis 1182-3
plate channels in flow 1178-82
absolute and conveclive instabilities 1179-82
physiological applications 1178
shear flow effects 1178
sinuous motions 1181-2
varicose molions 1178-81

1582
Plates in axial flow, dynamics (continued)
plates in quiescent fluid 1161-6
circular plates 1162-4
dry, wet mode s 1162
Lamb's assumption 1162
NAVMI factors 1162
rectangular plates 1164-6
simply-supported plate, incompressible fluid
1145-52
compressibilityeffects 1151
confinement effects 1152-7
divergence, flutter 1147
Dugundji eta/, Galerkin-type solution 1145-7
Ellen solution 1151
EllenIMatsuzaki & Ueda criticisms 1150
experiments 1151-2
fonnulation, equations, analysis 1145-50
Ishii "exact" solution 1150
Kornecki et a/, solution 1147-9
Matsuzaki & Ueda solution 1151
spatial memory 1149
Weaver & Unny Fourier integra! approach 1150
spinning disks; see Spinning disks
trave\ling web, sheets, tapes; see Travelling web,
sheets, tapes
various boundary conditions, confined
incompressible flow 1152-6
added mass effects on stability 1155
Cauchy-type singular integral equation 1154
clamped-clamped, clamped-pinned, free-free and
pinned-clamped plates 1155-6
fonnulation. equations, analysis 1152-5
instability at U::: O 1155
stability for same/different end supports 1157
Plates in flow, nonlinear dynamics 1195-207
buckled plates in flow 1205-7
chaotic vibrations 1205
Dowell's analysis 1205-6
Poincaré maps 1206
nonlinear dynamics, cantilevered plates 1203-5
nonlinear dynamics, subsonic/incompressible flow
1199-203
divergence waves in subsonic flow 1201
Lucey et al. work 1200
negative energy waves (NEWS) 1201
nonlinear tension effects 1201
post-divergence regime in subsonic flow 1199
solitons 1203
wave focusing 1203
nonlinear dynamics, supersonic flow 1197-203
hard instabilities 1197
Holmes' work 1197-9
soft divergence, flutter 1197
Pliable shells; see Collapsible tubes
Post-divergence flutter, shells with flow 610-2
energy considerations 611
Potential flow
cylinders in fluidlflow 790,835,840-1,933,951,
964,974,1033,1040-7,1053,1057,1241,
1278,1285,1289.1292,1294-7,1313-15,
1460-5
INDEX
plates in fluidlflow 1140, 1145, 1152, 1162, 1165,
1187, 1195. 1199, 1203, 1208, 1211
shells with fluidlflow 584-6,601,613,631,683-5,
731,734,778,1372-3,1388, 1401
Power-generating system applications 1024-31
Practical experiences 706,782-3. 1024-31, 1127-8,
1221. 1225-33. 1414-19
Pressure waves 746-50
Quasiperiodicity 729,742,745,772-3,821,971,990,
993-6,998-9,1002,1012,1014-16,1021-2
Random pressure field excitation of shells 713-17
Rockets, V-2 573
Rossby number 706
Rotofluidelasticity of shells 696-710
boundary-Iayer effects 708, 1430-2
forward, retrograde wave propagation 697
gas centrifuge project 699
Intense Neutron Generator 706-8
Lai & Chow's theory 699-702
numerical difficulties 702-4
rotating sheU with co-rotating fluid 699
Srinivasan, Dowell et a/, work 697-9
stationary shell with rotating fluid 704-6
whirling vibrations 707
Rotors in fluid-filled annuli 1355-71
Antunes et a/, work 1359-71
analyticallinear model 1359-62
comparison of experiment to theory 1364-6
experiments 1364-6
flutter, divergence 1363, 1364, 1366, 1367
more generallinear model 1366-8
nonlinear effects 1368-71
typical theoretical results 1362-4
Campbell diagrams 1357-8
coolant pump of Superphénix 1356
fluid thickness effects 1355
forward, backward whirl 1359
mechanical analogy 1355-9
retrograde whirl 1359
whirling shafts, lubricated bearings,
labyrinth seals 1355
Seismic arrays; see Acoustic streamers
Self-propelledlarticulated cylinders 957-63,
969-71
dynamics of self-propelled system 960-2
dynamics of towed system 962-3
equations ofmotion 960,1457-9
hydrodynamic forces 958-9
Sea-Chain system 957
transition from discrete to continuous 960
undulatory propulsion 969-71
Shapiro number 756
Shells in vacuo, simply supported 578
Shells, nonlinear dynamics 723-43
axisymmetric mode in mode expansion 724, 732,
740
circumferential contraction 724
driven, companion mode s 723

fluid-conveying shells with forcing 741-3
blue sky catastrophe 743
chaotic motion 743
literature review 724-31
effect ofaxisymmetric modes 725
experiments, fluid-fi1led shells 728, 729
shells as rings 724
shells in contact with fluid 727-9
shells in vacuo 724-7
shells with flow 729-31
shells conveying fluid 731-41
boundary conditions 733
Donnelľs nonlinear equations 731-2
experiments 740
tinite-dimensiona! model 735-6,1433-6
fluid-dynamic model 734-5
in-plane constraints 734
mode expansion 732
nonlinear dynamics 736-41
subcritica! pitchfork bifurcation 737
travel1ing wave behaviour 739
shells with externa! flow 741
softeninglhardening response 723, 725, 728, 737
Shells with flow
applications 781-6
Coriolis shell-type flow meters 783-6
coupled-mode flutter 591,602
divergence 591,602
expenments 594-600
colostomy, Penrose, silicone rubber
tubes 594-5
comparison with theory 596-8
disturbance-induced flutter 598
nonlinear effects, hysteresis 600
qua!itative observations 595-6
retined experiments 598-600
self-excited flutter 600
flow-induced vibration 720-3
flutter 594
Galerkin-type solutions 601
comparison with experiments 603-5
standing wave solution 601
Weaver & Dnny analysis 601
generalized-forceIFourier -transform solutions
cantHevered shel1s 621-3
extended Galerkin method 622
shells with supported ends 613-6
Hamiltonian coupled-mode flutter 602
impermeability condition 586
moda! shapes 592-3,784
nonuniform support conditions 652
Pai'doussis & Denise ana!ysis 584-90
cantilevered shells 594
c1amped-clamped shells 590-3
dimensionless quantities 588
equations of motion 584
intemallextemal flow similarity 589
numerica! solution 588-9
travelling wave solution 586
Pai'doussis-type coupled-mode flutter 591
toroida! shells conveying fluid 786
INDEX
1583
wave propagation; see Wave propagation in
shells
Shel1s with quiescent fluid 582-4
added density 583
wave propagation 654-60
wet, dry modes 582
Shells. engineenng applications 781-6
air-springs, vibroacoustic suppression 781, 782
fluid transport 781
gyroscope 782
pollution barriers, flood control, dams 781
Shells/plates with supported ends with flow 649-52
post-divergence stability 651-2
proof that Ucf > Ucd for subsonic flow 649-51
stability in supersonic flow 651
Slender-body theory 606,697-8,790,797-8,840-2,
889,931,963-4,974,1057,1130,1281,1287,
1315
Sloshing 632,727,728, 1231, 1415-16
Softening/hardening nonlinear response of shells;
see Nonlinear dynamics of shells
Solitons 1203, 1220
Spaghetti problem 856
Spectral collocation methods 1238-43, 1334-8
Spherical pendulum in axial flow, dynamics
1021-3
Spinning disks 1213-20
centrifugal stiffening 1218
computer disk drives 1213
critical rotation, flutter speed, damping speed
1216
effect of adjacent rigid plate 1220
experiments 1218-9
formulation, equations, analysis 1215-8
wave coordinates 1218
Standing wave solution 601,612-21,622,644-5,767,
1169-72
Starling resistor 752
Streaming 1220
Stress-corrosion cracking 787
Strings in axial flow 850-6
Hansen & Ni experiments 854--6
both ends supported, cantilevered 855-6
"rigiditied" upstream length 855
literature review 850-1
locallglobal instabilities 1161
nonlinear dynamics 988-9
Triantafyl10u & Chryssostornidis analysis 852-4
stability 854
Strings of clustered cylinders in axial flow 1127-34
dynamics 1129-34
dynamical behaviour 1132-4
equations 1129-32
nuc1ear reactor appHcations 1127-8
Subcritica! bifurcation
dynamic 595,597,600,731,763-4,772,781,
998-9, 1203, 1233, 1302
static 737-8,740,990-2,1199,1389-90,1393
Superphénix shell-weir instability 1414-16
ana!ytica! model 1415
experiments 1416
flutter 1416

1584
Supersonic flow
cylinders in flow 858
plates in flow 1137, 1151, 1168-9, 1174, 1176,
1195,1197-9,1205-7
shells with flow 573-4,612,649,651,729,
741
Swimming of slender fish 963-9
anguilIiform motion 963, 964
brief literature review 969
Swimming of slender fish (continued)
carangiform swimming 963, 967
dynamics of auguiIliform propulsion 964-7
Lighthill's model 964-7
propulsive efficiency 966
dynamics of carangiform swimming 967-8
lunate tail 968
vortex street, Strouhal number 968, 970
other types of aquatic locomotion 968-9
flapping foil RoboTuna 970, 1032
flappinglundulating foils 969-71
Gray's paradox 968
small Reynolds number swimming 969
undulatory propulsion and self-excited oscillations
969-70
Swirling flow 696-710,781,786,1216,1230,1355,
1364
Thin/thick shells conveying fluid 605-10
beam-Iike divergence 608-10
approximate divergence condition 608-9
effect of bluntnesslslendemess 608
transition to beam-type pipes 606
Towed acoustic arrays 787,928,1031
Towed cylinders 926-63, 1031-2, 1458-9
Towed flexible cylinders, dynamics 933-49
applications 1031-2
comparison between theory and experiments 947-9
Dowling's work for acoustic streamers 953-7
forced vibration 957
negatively buoyant towed bodies 957
stability 956
experiments 937-47
apparatus 937-40
cylinder with different nose and tail shapes
941-4
cylinder with ring stabilizer 944
effectoftoWTopelength 941
Sudarsan et a/, experiments 945
flexural, "snaking" instabilities 935
mechanisms of instability 951-3
oscillatory instability 934
summary of dynamical behaviour 947
typical dynamics, Hawthome-Pa'idoussis model
933-7
typical dynamics, Pai'doussis & Yu model 937
yawing 934
Towed pipelines 928
Towed rigid cylinders, dynamics 951-2
Towed ships 926
Towing cables 926
TransmuraI pressure 751,753
TravelIing beams, strings 988-9
INDEX
Travelling versus standing wave solutions for shells
with flow 612-21
comparison between traveling and standing wave
divergence prediction 619-21
divergence of slender shelIs 617-19
generalized-forceIFourier-transform solution 614
Pa'idoussis, Chan & Misra analysis 613
Shayo & Ellen's analysis 613
short waves on long shells 619
simplified prediction of divergence 616-17
simplified, Fliigge- Kempner eigenfrequencies 616
TraveIling wave solution; see also Wave propagation
cylinders in flow 849-50, 966
platesinflow 1139,1174-5,1212,1216
shelIs with flow 586,612-21,644-5,686-7,689,
697,700,729,739,767,774,778, 1388, 1411,
1446
Travelling web, sheets, tapes 1207-13
Chang & Moretti work 1208-10
edge flutter 1210-13
equations of motion, analysis 1211-13
experiments, comparison to theory 1213
Pramila's work 1207-9
revised travelling threadline model 1207, 1209
travelIing threadline model (TIM) 1207, 1209
Tube law 754
Turbulence-induced vibration in sheIls 710-23
general formulation 711-13
high-cycIe fatigue 723
hybrid analyticaIlfinite-e1ement method 710
Iimitations of the analysis 722
far-fie1d noise 723
response to boundary-Iayer excitation 717-21
pressure PSD 718-19
r,m,s. vibration amplitude 720
sample results 720-1
spatial correlation functions 717, 718
response to random pressure field 713-17
spatial cross-correlation function 713
statistical energy analysis 715
Two-phase flow 723, 766, 836-8, 844-8, 860, 864-6,
869,871,882-3,908.918-22,1024,1045,1131,
1303-4
Vibration-induced flow 1220
Vibro-impact 882,884,924, 1008
Viscous coupling matrix 1051,1053-4, 1059, 1092,
1239, 1477-9
Viscouslconfinement effects, shells with flow 670-96
steady viscous effects 670,671-6
basic loads 671
equations with steady stress resultants 671
steady viscous effects, cantilevered shelIs 687-96
comparison inviscidlviscous 691-2
confinement effects 689
expenments 692-6
steady viscous effects, shells with supported ends
671
comparison inviscidlviscous 674-6
time-mean Navier-Stokes equations 672
unsteady viscous effects, developing boundary layer
676-81

comparison to inviscid flow 681
viscous flow in boundary layer 678
unsteady viscous effects, general analysis 681-7
comparison to inviscid flow 686-7
fluid-shell boundary conditions 685-6
general formulation 681-4
Volume 1 addenda 1497-501
Volume 1 corrigenda 1501
Waterhammer 653
Wave propagation
cylinders in flow 848-50
plates in flow 1143-4, 1159-60, 1268
shells with fluidlflow 646,649,652-69,687-8,
697-8, 700-6, 746-50, 765, 770, 775, 1387-8,
1410-11
INDEX
1585
Wave propagation in shells 652-69
coincidence 660
complex wavenumbers 665
cut-offfrequency 657,665
dispersion relation 655
effect of compressibility 653
gravity-related 653
in anisotropic shells 667-9
in empty shells 655
in fluid-conveying shells 660
structural, acoustical modes 660
in fluid-filled shells 654-60
effect of mass ratio 658
mode veering, mode exchange 656, 662
viscous effects 665
waterhammer 653
Wavenumber 578-9,654,700,775