Hydrodynamics around cylindrical strucures

Advanced Series on Ocean Engineering — Volume 26

HYDRODYNAMICS AROUND CYLINDRICAL STRUCTURES Revised Edition

B. Mutlu S u m e r Jorgen Fredsoe

World Scientific

HYDRODYNAMICS AROUND CYLINDRICAL STRUCTURES Revised Edition

ADVANCED SERIES ON OCEAN ENGINEERING Series Editor-in-Chief Philip L- F Liu (Cornell University) Published Vol. 9 Offshore Structure Modeling by Subrata K. Chakrabarti (Chicago Bridge & Iron Technical Services Co., USA) Vol. 10 Water Waves Generated by Underwater Explosion by Bernard Le Mehaute and Shen Wang (Univ. Miami) Vol. 11 Ocean Surface Waves; Their Physics and Prediction by Stanislaw R Massel (Australian Inst, of Marine Sci) Vol. 12 Hydrodynamics Around Cylindrical Structures by B Mutlu Sumer and Jorgen Fredsoe (Tech. Univ. of Denmark) Vol. 13 Water Wave Propagation Over Uneven Bottoms Part I — Linear Wave Propagation by Maarten W Dingemans (Delft Hydraulics) Part II — Non-linear Wave Propagation by Maarten W Dingemans (Delft Hydraulics) Vol. 14 Coastal Stabilization by Richard Silvester and John R C Hsu (The Univ. of Western Australia) Vol. 15 Random Seas and Design of Maritime Structures (2nd Edition) by Yoshimi Goda (Yokohama National University) Vol. 16 Introduction to Coastal Engineering and Management by J William Kamphuis (Queen's Univ.) Vol. 17 The Mechanics of Scour in the Marine Environment by B Mutlu Sumer and Jorgen Fredsoe (Tech. Univ. of Denmark) Vol. 18 Beach Nourishment: Theory and Practice by Robert G. Dean (Univ. Florida) Vol. 19 Saving America's Beaches: The Causes of and Solutions to Beach Erosion by Scott L. Douglass (Univ. South Alabama) Vol. 20 The Theory and Practice of Hydrodynamics and Vibration by Subrata K. Chakrabarti (Offshore Structure Analysis, Inc., Illinois, USA) Vol. 21 Waves and Wave Forces on Coastal and Ocean Structures by Robert T. Hudspeth (Oregon State Univ., USA) Vol. 22 The Dynamics of Marine Craft: Maneuvering and Seakeeping by Edward M. Lewandowski (Computer Sciences Corporation, USA) Vol. 23 Theory and Applications of Ocean Surface Waves Part 1: Linear Aspects Part 2: Nonlinear Aspects by Chiang C. Mei (Massachusetts Inst, of Technology, USA), Michael Stiassnie (Technion-lsrael Inst, of Technology, Israel) and Dick K. P. Yue (Massachusetts Inst, of Technology, USA) Vol. 24 Introduction to Nearshore Hydrodynamics by lb A. Svendsen (Univ. of Delaware, USA)

Advanced Series on Ocean Engineering — Volume 26

HYDRODYNAMICS AROUND CYLINDRICAL STRUCTURES Revised Edition

B. Mutlu Sumer Jergen Fredsoe Technical University of Denmark, Denmark

\fc World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • H O N G K O N G • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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Cover: Flow around a marine pipeline placed over a trench during a half wave period, calculated by use of the discrete vortex method.

HYDRODYNAMICS AROUND CYLINDRICAL STRUCTURES (Revised Edition) Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Preface

Flow around a circular cylinder is a classical topic within hydrodynamics. Since the rapid expansion of the offshore industry in the sixties, the knowledge of this kind of flow has also attracted considerable attention from many mechanical and civil engineers working in the offshore field. T h e purpose of t h e present book is • To give a detailed, u p d a t e d description of t h e flow p a t t e r n around cylindrical structures (including pipelines) in the presence of waves a n d / o r current. • To describe t h e impact (lift and drag forces) of t h e flow on the structure. • And finally to describe the possible vibration p a t t e r n s for cylindrical structures. This part will also describe the flow around a vibrating cylinder and the resulting forces. T h e scope does not deviate very much from t h e book by Sarpkaya and Isaacson (1980) entitled "Mechanics of Wave Forces on Offshore Structures". However, while Sarpkaya and Isaacson devoted around 50% of the book to the drag-dominated regime and around 50% to diffraction, the present book concentrates mainly on the drag-dominated regime. A small chapter on diffraction is included for the sake of completeness. T h e reason for our concentration on the drag-dominated regime (large i f C - n u m b e r s ) is t h a t it is in this field the most progress and development have taken place during the last almost 20 years since Sarpkaya and Isaacson's book. In the drag-dominated regime, flow separation, vortex shedding, and turbulence have a large impact on the resulting forces. Good understanding of this impact has been gained by detailed experimental investigations, and much has been achieved, also in the way of the numerical modelling, especially during the last 5-10 years, when the computer capacity has exploded. In the book the theoretical and the experimental development is described. In order also to make the book usable as a text book, some classical flow solutions are included in the book, mainly as examples.

vi

Preface

Acknowledgement: T h e writers would like to express their appreciation of the very good scientific climate in t h e area offshore research in Denmark. In our country the hydrodynamic offshore research was introduced by professor Lundgren at our institute in the beginning of the seventies. In the late seventies and in the eighties the research was mainly concentrated in the Offshore Department at the Danish Hydraulic Institute. Significant contributions to the understanding of pipeline hydrodynamics were here obtained by Vagner Jacobsen and Mads B r y n d u m , two colleagues whose support has been of inestimable importance to us. In 1984 a special grant from the university m a d e it possible to ask one of the authors (Mutlu Sumer) to join the Danish group on offshore engineering so that he could convey his experience on fluid forces acting on small sediment particles to larger structures. This has been followed up by many grants from the Danish Technical Council ( S T V F ) , first through the F T U - p r o g r a m m e and next through the frame-programme "Marine Technique" (1991-97). T h e present book is an integrated o u t p u t from all these efforts and grants. T h e book has been typewritten by Hildur Juncker and the drawings have been prepared by Liselotte Norup, Eva Vermehren, Erling Poder, and Nega Beraki. Our librarian Kirsten Dj0rup has corrected and improved our written English.

Credits

T h e authors and World Scientific Publishing Co P t e Ltd gratefully acknowledge the courtesy of t h e organizations who granted permission to use illustrations and other information in this book. Fig. 3.4: Reprinted from H. Honji: "Streaked flow a r o u n d an oscillating circular cylinder". J. Fluid Mech., 107:509-520, 1982, with kind permission from Cambridge University Press, Publishing Division, T h e Edinburgh Building, Shaftesbury Road, Cambridge CB2 2RU, UK. Fig. 3.7: Reprinted from C.H.K. Williamson: "Sinusoidal flow relative to circular cylinders". J. Fluid Mech., 155:141-174, 1985, with kind permission from Cambridge University Press, Publishing Division, T h e Edinburgh Building, Shaftesbury Road, Cambridge CB2 2RU, UK. Figs. 4.51-4.53: Reprinted from E.-S. Chan, H.-F. Cheong and B.-C. Tan: "Laboratory study of plunging wave impacts on vertical cylinders". Coastal Engineering, 25:87-107, 1995, with kind permission from Elsevier Science, Sara Burgerhartstraat 25, 1055 KV Amsterdam, T h e Netherlands. Fig. 5.4b: Reprinted from J.E. Fromm and F.H. Harlow: "Numerical solution of the problem of vortex street development". T h e Physics of Fluids, 6(7):975-982, 1963, with kind permission from American Institute of Physics, Office of Rights and Permissions, 500 Sunnyside Blvd., Woodbury, NY 11797, USA.

viii

Credits

Fig. 5.9: Reprinted from P. Justesen: "A numerical study of oscillating flow around a circular cylinder". J. Fluid Mech., 222:157-196, 1991, with kind permission from Cambridge University Press, Publishing Division, T h e Edinburgh Building, Shaftesbury Road, Cambridge CB2 2RU, UK. Fig. 5.14: Reprinted from T. Sarpkaya, C. Putzig, D. Gordon, X. Wang and C. Dalton: "Vortex trajectories around a circular cylinder in oscillatory plus mean flow". J. Offshore Mech. and Arctic Engineering, 114:291-298, 1992, with kind permission from Production Coordinator, Technical Publishing Department, ASME International, 345 East 47th Street, New York, NY 10017-2392, USA. F i g . 5.26: Reprinted from P.K. Stansby and P.A. Smith: "Viscous forces on a circular cylinder in orbital flow at low Keulegan-Carpenter numbers". J. Fluid Mech., 229:159171, with kind permission from Cambridge University Press, Publishing Division, T h e Edinburgh Building, Shaftesbury Road, Cambridge CB2 2RU, UK. Fig. 8.50: Reprinted from R. King: "A review of vortex shedding research and its application". Ocean Engineering, 4:141-172, 1977, with kind permission from Elsevier Science Ltd., T h e Boulevard, Langford Lane, Kidlington 0 X 5 1GB, UK.

List of symbols

T h e main symbols used in the book are listed below. In some cases, the same symbol was used for more t h a n one quantity. This is t o maintain generally accepted conventions in different areas of fluid mechanics. In most cases, however, their use is restricted to a single chapter, as indicated in t h e following list.

Main symbols A A Ar,

a b C CD C'D CL C'L CLA CLT

Cld,

Cfn

CL max ^ Lrms

amplitude of vibrations cross-sectional area of b o d y (Chapter 4) m a x i m u m value of vibration amplitude amplitude of oscillatory flow, or amplitude of horizontal component of orbital motion acceleration (Chapter 4) distance between discrete vortices in an infinite row of vortices ( C h a p t e r 5) amplitude of surface elevation (Chapter 7) amplitude of vertical component of orbital motion concentration or passive quantity (or temperature) drag coefficient oscillating component of drag coefficient lift coefficient oscillating component of lift coefficient lift coefficient corresponding to FyA lift coefficient corresponding to FyT lift force coefficients (drag and inertia components, respectively) lift coefficient corresponding to FL m a x lift coefficient corresponding t o Firms force coefficient corresponding to Fxrms

X

CM Cm Cmc Cs c c cp D D(f,9) E E E Ex E& e F F Fp Fp FK FK,tot Fi F'L FL max Fhrms FN FT Frrms Fj, Ff Fp Fp,Fm FTms Fx,Fy Fx,tot Fy FyA FyT Fz FQ /

List of symbols

inertia coefficient hydrodynamic-mass coefficient hydrodynamic-mass coefficient in current force coefficient corresponding to force / viscous damping coefficient wave celerity (Chapter 4, Appendix III) pressure coefficient cylinder diameter (or pipeline diameter) directional spectrum ellipticity of orbital motion elasticity modulus (Chapter 11) mean wave energy total energy energy dissipated in one cycle of vibrations gap between cylinder and wall, or clearance between pipeline and seabed Morison force per unit length of structure external force drag force per unit length of structure oscillating component of drag force per unit length of structure Froude-Krylov force per unit height of vertical structure total Froude-Krylov force on vertical structure lift force per unit length of structure oscillating component of lift force per unit length of structure m a x i m u m value of lift force per unit length of structure root-mean-square value of lift force per unit length of structure force component normal to structure, per unit length of structure total (resultant) force per unit length of structure root-mean-square value of total (resultant) force per unit length of structure damping force friction drag per unit length of structure form drag per unit length of structure predicted and measured in-line forces, respectively (Chapter 4) root-mean-square value of in-line force per unit length of structure force components in Cartesian coordinate system total force on vertical cylinder lift force per unit length of structure m a x i m u m value of lift force away from wall per unit length of structure m a x i m u m value of lift force towards wall per unit length of structure lift force per unit length of structure force due to potential flow per unit length of cylinder frequency, frequency of vibrations

List of symbols

/ fl f„ fnc ft /„ /„, fx fy fQ g H Hm Hrms Hs H1/3 h h I Iu i Im K Ks KC KCr ks k k k fcr, k{ L L M M m m' mc m„ N N(z)

xi

impact force on vertical cylinder due to breaking waves (Chapter 4) fundamental lift frequency u n d a m p e d n a t u r a l frequency (or n a t u r a l frequency) n a t u r a l frequency in current frequency of transition waves vortex-shedding frequency frequency of oscillatory flow, frequency of waves frequency of in-line vibrations frequency of cross-flow vibrations in forced vibration experiments peak frequency acceleration due to gravity wave height m a x i m u m wave height root-mean-square value of wave height significant wave height significant wave height ( = Hs) water depth distance between two infinite rows of vortices (Chapter 5) inertia moment turbulence intensity imaginary unit imaginary part diffusion coefficient (or thermal conductivity) stability parameter Keulegan-Carpenter number Keulegan-Carpenter number for r a n d o m oscillatory flow Nikuradse's equivalent sand roughness cylinder roughness (Chapter 4) spring constant (Chapters 8-11) wave number real and imaginary parts of wave number k correlation length wave length (Chapter 6, Appendix III) mass ratio overturning moment (Chapter 6) mass of body, per unit length of structure unless otherwise is stated hydrodynamic mass, per unit length of structure unless otherwise is stated hydrodynamic mass in current, per unit length of structure unless otherwise is stated rjth moment of spectrum normalized vibration frequency in oscillatory flows or in waves f / fw ( = number of vibrations per flow cycle) tension ( C h a p t e r 11)

xii

List of symbols

NL n P Pr P P P' Po p+ 1 9o R R Re Rer

r,e r,6 ro ro St

S(f) Sa(f) SFAI)

Su(f) £,(/)

T

TR

Tc Ts Tv T Tz T T0 t U

uN

rms

uc Uf

um U rms

normalized lift frequency, /z,// T O (= number of oscillations in lift per flow cycle) normal direction pressure force probability of occurrence pressure probability density function (Chapter 7) fluctuating pressure hydrostatic pressure excess pressure spectral width parameter speed autocovariance function (Chapter 7) correlation Reynolds number Reynolds number for random oscillatory flow polar coordinates spherical coordinates (in axisymmetric flow) (Chapter 5) cylinder radius sphere radius (Chapter 5) Strouhal number spectrum function of surface elevation (wave spectrum) spectrum function of acceleration force spectrum spectrum function of velocity spectrum function of surface elevation (wave spectrum) period of oscillatory flow, period of waves return period mean crest period significant wave period vortex-shedding period period of oscillatory flow, period of waves mean zero-upcrossing period mean period peak period time outer flow velocity flow velocity component normal to cylinder root-mean-square value of resultant velocity current velocity wall shear stress velocity maximum value of oscillatory-flow velocity, maximum value of horizontal component of orbital velocity root-mean-square value of horizontal velocity

List of symbols

Uw u u,v,w u',v' u V Vm Vr Vrms v vr, v$ WQ,WI w x Xd Xf x, y y x, y z 2 z 3 T I\ 5 6 6 5 6* St e ep C C/ C» T] 6 6 K A fj,

wind speed flow velocity in boundary layer velocity components in Cartesian coordinates infinitesimal disturbances introduced in velocity components velocity vector volume of body m a x i m u m value of vertical component of orbital velocity reduced velocity root-mean-square value of vertical velocity speed velocity components in polar coordinates, or spherical coordinates (axisymmetric) complex potential complex potential streamwise distance, or horizontal distance " d y n a m i c " motion forced motion Cartesian coordinates distance from wall x- and ^-displacements of structure (Chapter 8-11) vertical coordinate measured from mean water level upwards (Chapter 6, Appendix III) spanwise separation distance, or spanwise distance complex coordinate, z = x + iy = re'e (Chapter 5) ratio of Reynolds number to Keulegan-Carpenter number circulation vortex strength, corresponding to zth vortex b o u n d a r y layer thickness goodness-of-fit parameter (Chapter 4) phase difference between incident wave and force (Chapter 6) logarithmic decrement (Chapter 8) displacement thickness of boundary layer time increment spectral width parameter 1 for p = 0; 2 for p > 1 total damping fluid damping structural damping surface elevation polar coordinate or spherical coordinate wave direction (Chapter 7) strength of individual vortices in an infinite row wave length of wavy trajectory of cylinder towed in still fluid dynamic viscosity

xiii

XIV

v p ffu av r T TO TW 4> 4> (j>i <j>, s if ij} %l>' u) u> Ud u>dv u>„ u)r,u>t UJV overbar overdot

List of symbols

fluid

kinematic viscosity density s t a n d a r d deviation of flow velocity s t a n d a r d deviation of quantity r/ shear stress normalized wave period (Chapter 7) wall shear stress wall shear stress (Chapter 4) angular coordinate phase difference between cylinder vibration and flow velocity ( C h a p t e r 3) potential function (Chapters 4, 6 and Appendix III) potential function for incident waves potential function for scattered (reflected plus diffracted) waves ( C h a p t e r 6) separation angle phase delay stream function infinitesimal disturbance in stream function angular frequency, also angular frequency of external force (for a vibrating system) vorticity defined by to = dv/dx — du/dy ( C h a p t e r 5) d a m p e d n a t u r a l angular frequency angular frequency of damped free vibrations u n d a m p e d natural angular frequency real and imaginary p a r t s of angular frequency 10 angular frequency of u n d a m p e d free vibrations time average differentiation with respect to time

Contents

PREFACE

v

CREDITS

vii

LIST O F SYMBOLS 1.

ix

F l o w a r o u n d a c y l i n d e r in s t e a d y c u r r e n t 1.1 Regimes of flow around a smooth, circular cylinder 1.2 Vortex shedding 1.2.1 Vortex-shedding frequency 1.2.2 Correlation length References

2.

1 6 10 28 33

F o r c e s o n a c y l i n d e r in s t e a d y c u r r e n t 2.1 Drag and lift 2.2 Mean drag 2.3 Oscillating drag and lift 2.4 Effect of cross-sectional shape on force coefficients 2.5 Effect of incoming turbulence on force coefficients 2.6 Effect of angle of attack on force coefficients 2.7 Forces on a cylinder near a wall References

3.

F l o w a r o u n d a c y l i n d e r in o s c i l l a t o r y

37 40 50 52 53 55 57 70

flows

3.1 Flow regimes as a function of Keulegan-Carpenter number 3.2 Vortex-shedding regimes 3.3 Effect of Reynolds number on flow regimes 3.4 Effect of wall proximity on flow regimes 3.5 Correlation length 3.6 Streaming References

..

74 78 89 92 104 116 120

xvi

4.

Forces o n a c y l i n d e r in r e g u l a r w a v e s 4.1 In-line force in oscillatory flow 4.1.1 Hydrodynamic mass 4.1.2 Froude-Krylov force 4.1.3 T h e Morison equation 4.1.4 In-line force coefficients 4.1.5 Goodness-of-fit of the Morison equation 4.2 Lift force in oscillatory flow 4.3 Effect of roughness 4.4 Effect of coexisting current 4.5 Effect of angle of attack 4.6 Effect of orbital motion 4.6.1 Vertical cylinder 4.6.2 Horizontal cylinder 4.7 Forces on a cylinder near a wall 4.8 Forces resulting from breaking-wave impact References

5.

M a t h e m a t i c a l a n d n u m e r i c a l t r e a t m e n t o f flow a r o u n d a c y l i n d e r 5.1 5.1.1 5.1.2 5.1.3 5.1.4 5.2 5.2.1 5.2.2

Direct solutions of Navier-Stokes equations Governing equations T h e Oseen (1910) and Lamb (1911) solution Numerical solutions Application to oscillatory flow Discrete vortex methods Numerical simulation of vorticity transport Procedure used in the implementation of discrete vortex method 5.2.3 Application areas 5.3 Hydrodynamic stability approach References

6.

123 124 129 130 133 147 149 153 157 161 163 163 169 180 187 201

210 211 211 219 227 233 234 237 242 248 266

D i f f r a c t i o n effect. F o r c e s o n large b o d i e s 6.1 Vertical circular cylinder 6.1.1 Analytical solution for potential flow around a vertical circular cylinder 6.1.2 Total force on unit-height of cylinder 6.1.3 Total force over the depth and the overturning moment .... 6.2 Horizontal circular cylinder near or on the seabottom. Pipelines References

276 276 282 287 289 295

xvii

7.

Forces o n a cylinder in irregular waves 7.1 Statistical t r e a t m e n t of irregular waves 7.1.1 Statistical properties of surface elevation 7.1.2 Statistical properties of wave height 7.1.3 Statistical properties of wave period 7.1.4 Long-term wave statistics 7.2 Forces on cylinders in irregular waves 7.2.1 Force coefficients 7.2.2 Force spectra 7.2.3 Forces on pipelines in irregular waves 7.2.4 Forces on vertical cylinders in directional irregular waves References

8.

F l o w - i n d u c e d v i b r a t i o n s o f a free c y l i n d e r in s t e a d y c u r r e n t s 8.1 A summary of solutions to vibration equation 8.1.1 Free vibrations without viscous damping 8.1.2 Free vibrations with viscous damping 8.1.3 Forced vibrations with viscous damping 8.2 Damping of structures 8.2.1 Structural damping 8.2.2 Fluid damping in still fluid 8.3 Cross-flow vortex-induced vibrations of a circular cylinder 8.3.1 Feng's experiment 8.3.2 Non-dimensional variables influencing cross-flow vibrations 8.4 In-line vibrations of a circular cylinder 8.5 Flow around and forces on a vibrating cylinder 8.5.1 Cylinder oscillating in the cross-flow direction 8.5.2 Cylinder oscillating in in-line direction 8.6 Galloping 8.7 Suppression of vibrations References

9.

..

297 298 312 315 318 319 319 325 328 330 330

.

335 336 336 338 342 342 346 353 354 364 376 383 383 396 397 407 413

F l o w - i n d u c e d v i b r a t i o n s o f a free c y l i n d e r in w a v e s 9.1 Introduction 9.2 Cross-flow vibrations 9.2.1 General features 9.2.2 Effect of mass ratio and stability parameter 9.2.3 Effect of Reynolds number and surface roughness 9.2.4 Cross-flow vibrations in irregular waves 9.3 In-line vibrations 9.4 In-line oscillatory motion 9.5 Flow around and forces on a vibrating cylinder References

418 421 423 432 432 436 441 443 445 450

xviii

10.

Vibrations of marine pipelines 10.1 Cross-flow vibrations of pipelines 10.1.1 Cross-flow vibrations of pipelines in steady current 10.1.2 Cross-flow vibrations of pipelines in waves 10.2 In-line vibrations and in-line motions of pipelines 10.3 Effect of Reynolds number 10.4 Effect of scoured trench 10.5 Vibrations of pipelines in irregular waves 10.6 Effect of angle of attack 10.7 Forces on a vibrating pipeline References

11.

M a t h e m a t i c a l modelling of

flow-induced

455 455 465 471 473 479 481 486 486 491

vibrations

11.1 T h e steady-current case 11.1.1 Simple models 11.1.2 Flow-field models 11.2 T h e wave case 11.3 Integrated models References A P P E N D I X I.

Force coefficients for various cross-sectional shapes

A P P E N D I X II.

Hydrodynamic-mass coefficients for two- and threedimensional bodies

497 497 499 503 506 510 ....

514 517

A P P E N D I X III. Small amplitude, linear waves

519

REFERENCES FOR APPENDICES

521

A U T H O R INDEX

522

SUBJECT INDEX

527

Chapter 1. Flow around a cylinder in steady current

1.1 Regimes of flow around a smooth, circular cylinder T h e non-dimensional quantities describing the flow around a smooth circular cylinder depend on the cylinder Reynolds number

Re=™

(1.1) v

in which D is the diameter of the cylinder, U is the flow velocity, and v is the kinematic viscosity. T h e flow undergoes tremendous changes as the Reynolds number is increased from zero. T h e flow regimes experienced with increasing Re. are summarized in Fig. 1.1. Fig. 1.2, on the other hand, gives the definition sketch regarding the two different flow regions referred to in Fig. 1.1, namely the wake and the boundary layer. While the wake extends over a distance which is comparable with the cylinder diameter, D, the boundary layer extends over a very small thickness, 6. which is normally small compared with D. T h e boundary layer thickness, in the case of laminar boundary layer, for example, is (Schlichting, 1979)

2

Chapter 1: Flow around a cylinder in steady current

b)

-c o^s>

e)

No separation. Creeping flow

Re h)

B: Turbulent boundary layer separation;the boundary layer partly laminar partly turbulent

C



Figure 1.14 (continued.) on the vortex shedding has been studied by various authors, for example by Cheung and Melbourne (1983), Kwok (1986) and Norberg and Sunden (1987) among others. Fig. 1.15 presents the Strouhal number data obtained by Cheung and Melbourne for various levels of turbulence in their experimental tunnel. Here, Iu is the turbulence intensity defined by

h=

(1-

in which V u ' 2 is the root-mean-square value of the velocity fluctuations and u is the mean value of the velocity. The variation of St with the Reynolds number changes considerably with the level of turbulence in the approach flow. The effect of turbulence is rather similar to that of cylinder roughness. The critical, the supercritical, and the upper transition flow regimes seem to merge into one transitional region.

18

Chapter 1: Flow around a cylinder in steady

n

0.4-

current

st

0.3"

o.i-

Mt. Isa stack full scale data for Iu = 7.8% 10 10

St = 0.20

at

Re = 4 x

St = 0.15

at

Re = 2 x 10 ? Re

10" Figure 1.15 Effect of turbulence in the approach flow on vortex-shedding frequency. Strouhal numbers as a function of Reynolds number for different turbulence intensities. Iu is the level of turbulence (Eq. 1.8). Cheung and Melbourne (1983). It appears from the figure t h a t the lower end of this transition range shifts towards the smaller a n d smaller Reynolds numbers with t h e increased level of turbulence. This is obviously due to the earlier transition to turbulence in the cylinder boundary layer with increasing incoming turbulence intensity. Effect o f s h e a r in t h e i n c o m i n g flow T h e shear in the approach flow is also an influencing factor in the vortex shedding process. T h e shear could be present in the approach flow in two ways: it could be present in the spanwise direction along the length of the cylinder (Fig. 1.16a), or in the cross-flow direction (Fig. 1.16b). T h e characteristics of shear flow around bluff bodies including t h e non-circular cross-sections have been reviewed by Griffin (1985a and b). In t h e case when the shear is present in the spanwise direction (Fig. 1.16a), the vortex shedding takes place in spanwise cells, with a

Vortex shedding

19

frequency constant over each cell. Fig. 1.17 clearly shows this; it is seen t h a t the shedding occurs in four cells, each with a different frequency. W h e n t h e Strouhal number is based on the local velocity (the dashed lines in the figure), t h e d a t a are grouped around t h e Strouhal number of about 0.25.

b)

a)

Figure 1.16 Two kinds of shear in the approach flow, a: Shear is in the spanwise direction, b: Shear is in the cross-flow direction.

Regarding the length of cellular structures, research shows that the length of cells is correlated with t h e degree of t h e shear. T h e general t r e n d is t h a t t h e cell length decreases with increasing shear (Griffin, 1985a). W h e n the shear takes place in the cross-stream direction (the conditions in the spanwise direction being uniform), the shedding is only slightly influenced for small and moderate values of the shear steepness s which is defined by D_du Uc dy

(1.9)

For large values of s, however, the shedding is influenced somewhat substantially (Kiya, T a m u r a and Arie, 1980). Fig. 1.18 shows the Strouhal number plotted against the Reynolds number for three different values of s. As is seen for s = 0.2, t h e Strouhal number is increased substantially relative t o t h e uniform-flow case

(s = 0).

20

Chapter 1: Flow around a cylinder in steady

current

ID

Stii

U,, 0.32

-

0.28

fD

0.240.20

i

0

2

10

I

I

14

I

i

-z/D

18

Figure 1.17 Effect of shear in the approach flow on vortex-shedding frequency. Shear in the spanwise direction. Circles: Strouhal number based on the centre-line velocity Uc. Dashed lines: Strouhal number based on the local velocity, U\ociLi. Re = 2.8 X 10 4 . The shear steepness: s = 0.025. Maull and Young (1973).

S

-g

Re

Figure 1.18 Effect of shear in the approach flow frequency. Shear in crossflow direction. The Strouhal number against the Reynolds number for three different values of the shear steepness s. Hatched band: Uniform-flow results. Circles: Shear-flow results. Kiya et al. (1980).

10

Vortex shedding

21

Effect of w a l l p r o x i m i t y This topic is of direct relevance with regard to pipelines. W h e n a pipeline is placed on an erodible sea bed, scour may occur below the pipe due to flow action. This may lead to suspended spans of the pipeline where the pipe is suspended above the bed with a small gap, usually in t h e range from 0 ( 0 . I D ) to 0 ( 1 D ) . Therefore it is important to know what kind of changes take place in the flow around and in the forces on such a pipe.

777777777777777777777

Figure 1.19 Flow around a) a free cylinder, b) a near-wall cylinder. S = separation points.

W h e n a cylinder is placed near a wall, a number of changes occur in the flow a r o u n d t h e cylinder. These changes are summarized as follows: 1) Vortex shedding is suppressed for the gap-ratio values smaller t h a n about e/D = 0.3, as will be seen later in the section. Here, e is the gap between the cylinder and the wall. 2) T h e stagnation point moves to a lower angular position as sketched in Fig. 1.19. This can be seen clearly from the pressure measurements of Fig. 2.20a and Fig. 2.20b where t h e mean pressure distributions around t h e cylinder are given for three different values of t h e gap ratio. While the stagnation point is located at about = 0° when e/D = 1, it moves to the angular position of about <j> = —40° when the gap ratio is reduced to e/D = 0.1. 3) Also, the angular position of the separation points changes. T h e separation point at t h e free-stream side of t h e cylinder moves u p s t r e a m and t h a t at the wall side moves downstream, as shown in t h e sketch given in Fig. 1.19. T h e

22

Chapter 1: Flow around a cylinder in steady

current

Free-stream side separation point

140

(a) 120 77777777777777-

100 8 0 *»>*60

(b) 140 120

7777^77777777'

Wall s i d e separation point

100 80 60 i

0

i

i

i

1

i

J_l

I

I

I

I

L e/D

Figure 1.20 Angle of separation as a function of the gap ratio, (a): At the free-stream side of the cylinder and (b): At the wall side of the cylinder. ije = 6 x 10 3 . Jensen and Sumer (1986). separation angle measured for a cylinder with Re = 6 x 10 3 is shown in Fig. 1.20; the figure indicates t h a t for example for e/D = 0.1 the separation angle at the free-stream side is (j> = 80°, while it is rf> = —110° at the wall side for the same gap ratio. 4) Finally, the suction is larger on the free-stream side of the cylinder t h a n on the wall-side of the cylinder, as is clearly seen in Fig. 2.20b and c. W h e n the cylinder is placed away from the wall, however (Fig. 2.20a) this effect disappears and the symmetry is restored.

Vortex shedding

At B

At A

-2.0 =2

-2.0

-®:

-3.0 -4.0 O

0.8 1.6 2.4 Log. frequency (Hz)

0.8 1.6 2.4 Log. frequency (Hz)

///*////

c b)

0.3

-2.0

-2.0

-3.0

-3.0

V

a (0

oio o

C

a;

-4.0

-4.0

0.8 1.6 2.4 Log. frequency (Hz)

0 0.8 1.6 2.4 Log. frequency (Hz)

0.8 1.6 2.4 Log. frequency (Hz)

0.8 1.6 2.4 Log. frequency (Hz)

I

XI

c)

0.2

-2.4

y

v -3.2 a. to

oio o

d)

i

-4.0

-2.4

-a

1 -3.2 s. -4.0 CD

•ats

23

_i

i

i

i_

0 0.8 1.6 2.4 Log. frequency (Hz)

2 Figure 1.21 Effect of wall proximity on vortex shedding. Power spectra of the hot-wire signal received from the wake. Bearman and Zdravkovich (1978).

24

Chapter 1: Flow around a cylinder in steady

current

Vortex shedding may be suppressed for a cylinder which is placed close to a wall. Fig. 1.21 presents power spectra of the hot-wire signals received from b o t h sides of the wake of a cylinder placed at different distances from a wall (Bearman and Zdravkovich, 1978). As is clearly seen, regular vortex shedding, identified by the sharply defined, dominant peaks in the power spectra, persists only for values of the gap-to-diameter ratio e/D down to about 0.3. This result, recognized first by B e a r m a n and Zdravkovich, was later confirmed by the measurements of Grass, Raven, Stuart and Bray (1984). T h e photographs shown in Fig. 1.22 demonstrate the supression of vortex shedding for gap ratios e/D below 0.3. T h e suppression of vortex shedding is linked with the asymmetry in the development of the vortices on the two sides of the cylinder. T h e free-stream-side vortex grows larger and stronger t h a n the wall-side vortex. Therefore the interaction of the two vortices is largely inhibited (or, for small e/D, totally inhibited), resulting in partial or complete suppression of the regular vortex shedding. Regarding the effect of wall proximity on the vortex-shedding frequency for the range of e/D where the vortex shedding exists, measurements show t h a t the shedding frequency tends to increase (yet slightly) with decreasing gap ratio. In Fig. 1.23 are plotted t h e results of two studies, namely Grass et al. (1984) and Raven, Stuart, Bray and Littlejohns (1985). Grass et al.'s experiments were done in a laboratory channel with b o t h smooth and rough beds. T h e surface of the test cylinder was smooth. Their results collapse onto a common curve when plotted in the normalized form presented in the figure where Stg is the Strouhal number for a wall-free cylinder. T h e d a t a points of Raven et al.'s study, on the other hand, were obtained in an experimental program conducted in the Severn Estuary (UK) where a full-scale pipeline (50.8 cm in diameter with a surface roughness of k/D = 8.5 x 1 0 - 3 ) was used. In both studies, St is defined by the velocity at the top of the cylinder. There are other d a t a available such as Bearman and Zdravkovich (1978) and Angrilli, Bergamaschi and Cossalter (1982). While Bearman and Zdravkovich's measurements indicate t h a t the shedding frequency practically does not change over t h e range 0.3 < e/D < 3, Angrilli et al.'s measurements show t h a t there is a systematic (yet, slight) increase in the shedding frequency with decreasing gap ratio in their measurement range 0.5 < e/D < 6 (they report a 10% increase in the shedding frequency at e/D = 0.5). It is apparent from the existing d a t a t h a t the vortex-shedding frequency is insensitive to the gap ratio, although there seems to be a tendency t h a t it increases slightly with decreasing gap ratio. This slight increase in the Strouhal frequency may be a t t r i b u t e d to the fact t h a t the presence of the wall causes the wall-side vortex to be formed closer to the free-stream-side vortex. As a result of this, the two vortices interact at a faster rate, leading to a higher St frequency.

Vortex shedding

a)^o0.4

b)

= 0.3

c)

= 0.2

d)

= 0.05

Figure 1.22 Effect of wall proximity on vortex shedding. Flow in the wake of a near-wall cylinder. Shedding is apparent for e/D = 0.4 and 0.3 but suppressed for e/D = 0.2and 0.05. Re = 7 x l 0 3 .

25

26

Chapter 1: Flow around a cylinder in steady

current

e/D Figure 1.23 Effect of wall proximity on vortex shedding frequency. Normalized Strouhal number as a function of gap ratio. St0 is the Strouhal number for wall-free cylinder. Circles: Raven et al. (1985). Solid curve: Grass et al. (1984).

Jensen, Sumer, Jensen and Freds0e (1990) investigated t h e flow around a pipeline (placed initially on a flat bed) at five characteristic stages of the scour process which take place underneath the pipeline. Each stage was characterized in the experiments by a special, frozen scoured bed profile, which was an exact copy of the measured bed profile of an actual scour test. T h e investigated scour profiles and the corresponding mean flow field are shown in Fig. 1.24. It was observed that no vortex shedding occurred for the first two stages, namely stages I and II, while vortex shedding did occur for stages III - V. Fig. 1.25 depicts t h e shedding frequency corresponding to the different stages. T h e variation of the Strouhal number, which goes from as high a value as 0.36 for Stage III to an equilibrium value of 0.17 in Stage V, can be explained by the geometry of the downstream scour profile as follows. For profiles III and IV, the steep slope of t h e u p s t r e a m p a r t of the dune behind the cylinder forces the shear layer originating from t h e lower edge of the cylinder to bend upwards, thus causing t h e associated lower vortex to interact with the upper one prematurely, leading to a p r e m a t u r e vortex shedding. T h e result of this is a higher vortex shedding frequency and a very narrow formation region. T h e flow visualization study carried out in the same experiments (Jensen et al., 1990) confirmed the existence of this narrow region.

Vortex shedding

y(cm) 6* 4-1 2

20 c m / s

I

///>//////////•/?//////?////////////////////;;;/;//;/

S )-

j

H j

J

II

1

ft?V s -

- * •

:

yx^-'/"/"X'//^[ \ J b > /

in

///////////////////////////

1

i

—"i

1

- 4 - 3 - 2 - 1 0 1 2 3 4 5 6 7 8

r-

x/D

Figure 1.24 Vector plot of the mean velocities, S = the approximate position of the stagnation point. Jensen et al. (1990).

A St 0.40.2

1

10

100

, time (mln)

Figure 1.25 Time development of Strouhal number during the scour process below a pipeline. Jensen et al. (1990).

27

28

Chapter 1: Flow around a cylinder in steady

1.2.2

Correlation length

current

As has been mentioned in Section 1.1, vortex shedding in the turbulent wake regime (i.e. iJe>200) occurs in cells along the length of the cylinder. These spanwise cell structures are visualized in Fig. 1.26 which shows the time evolution of the shedding process in plan view. T h e cells are quite clear from the photographs in Fig. 1.26. Shedding does not occur uniformly along the length of the cylinder, b u t rather in cells (designated by A, B and C in Fig. 1.26). It can also be recognized from the pictures in Fig. 1.26 that the cells along the length of the cylinder are out of phase. Consequently, the maximum resultant force acting on the cylinder over its total length may be smaller t h a n the force acting on the cylinder over the length of a single cell. T h e average length of the cells may be termed t h e correlation length. T h e precise determination of the correlation length requires experimental determination of the spanwise variation of t h e correlation coefficient of some unsteady quantity related to vortex shedding, such as fluctuating surface pressure, or a fluctuating velocity just outside the shear layer at separation. T h e correlation coefficient is defined by

R(z) =

X^-P'^

+ Z)

(1.10)

in which £ is the spanwise distance, z is the spanwise separation between two measurement points, and p' is the fluctuating part of the unsteady quantity in consideration. T h e overbar denotes the time averaging. T h e correlation length L, on the other hand, is defined by the integral /•oo

L=

/

R(z)dz

(1.11)

Jo Fig. 1.27 gives a typical example of the correlation coefficient obtained in a wind tunnel with a cylinder 7.6 cm in diameter and 91.4 cm in length with large streamlined end plates (Novak and Tanaka, 1977). T h e Reynolds number was 1.9 x 10 4 . T h e measured quantity was the surface pressure at an angle 60° to the main stream direction. T h e correlation length corresponding to the correlation coefficient, given in Fig. 1.27, on the other hand is found to be L/D = 3 from Eq. 1.11.

0.5 s

0.3 s

t = 0

0.9 s

U

Uu— a)

b)

c)

. d)

Figure 1.26 Photographs, illustrating the time evolution of spanwise cell structure. Cyli

SO

Chapter 1: Flow around a cylinder in steady

current

For a smooth cylinder, the correlation length changes with the Reynolds number. Table 1.1 presents the correlation-length d a t a compiled by King (1977).

Table 1.1 Correlation lengths and Reynolds numbers of smooth cylinders.

Reynolds number

Correlation length

Source

40 < Re < 150 150 < Re < 105 1.1 x 104 < Re < 4.5 x 104 > 105 2 x 105

(15-20)D (2-3)D (3-6)D 0.5D 1.56D

Gerlach and Dodge (1970) Gerlach and Dodge (1970) El-Baroudi (1960) Gerlach and Dodge (1970) Humphreys (1960)

T h e table shows t h a t the correlation length is (15-20)D for 40 < Re < 150 but experiences a sudden drop to (2-3)D at Re = 150. T h e latter Re number is quite close to the Reynolds number (see Fig. l . l d ) , at which t h e laminar vortex shedding regime disappears. Regarding the finite (although large) values of the correlation length in the range 40 < Re < 150, the correlation length in this flow regime should theoretically be infinite, since the vortex regime in this range is actually two-dimensional. However, purely two-dimensional shedding cannot be achieved in practice due to the existing end conditions. A slight divergence from the purely two-dimensional shedding, in the form of the so-called oblique shedding (see for example Williamson, 1989), may result in finite correlation lengths. Other factors also affect the correlation. T h e correlation increases considerably when the cylinder is oscillated in the cross-flow direction. Fig. 1.28 presents the correlation coefficient d a t a obtained by Novak and Tanaka (1977) for several values of t h e double-amplitude-diameter ratio 2A/D where A is the amplitude of cross-flow vibrations of t h e cylinder. T h e figure shows t h a t the correlation coefficient increases tremendously with t h e amplitude of oscillations. Similar results were obtained by Toebes (1969) who measured the correlation coefficient of fluctuating velocity in the wake region near the cylinder. Fig. 1.29 presents the variation of the correlation length as a function of the amplitude-to-diameter ratio (curve a in Fig. 1.29). Clearly, the correlation length increases extensively with increasing the amplitude of oscillations.

Vortex shedding

SI

z/D

Figure 1.27 Correlation coefficient of surface pressure fluctuations as function of the spanwise separation distance z. Cylinder smooth. Re = 1.9 X 10 4 . Pressure transducers are located at 60° to the main stream direction. Novak and Tanaka (1977).

R i

i

1.00.8-

2A/D == 0.20

0.6-

0.15

0.4-

«\V*. ^^>~~

o—

0 0H (3

1

2

0.10 0.05 ^0~^~

0.2•

0

" - •

i

1

4

6

A

1

*1

8

10

*

z/D

Figure 1.28 Effect of cross-flow vibration of cylinder on correlation coefficient of surface pressure fluctuations. Cylinder smooth. Re = 1.9 X 10 . Pressure transducers are located at 60° to the main stream direction. A is the amplitude of the cross-flow vibrations of cylinder. Novak and Tanaka (1977).

Turbulence in the approaching flow is also a significant factor for the correlation length, as is seen from Fig. 1.29. T h e turbulence in the tests presented in this figure was generated by a coarse grid in the experimental tunnel used in Novak and Tanaka's (1977) study. T h e figure indicates t h a t t h e presence of turbulence

S2

Chapter 1: Flow around a cylinder in steady

current

in the approaching flow generally reduces the correlation length. It is interesting to note t h a t with 2A/D = 0.2, while the correlation length increases from about 3 diameters to 43 diameters for a turbulence-free, smooth flow, the increase is not so dramatic when some turbulence is introduced into the flow; t h e correlation length increases to only about 10 diameters in this latter situation.

D ' 40-

Flow:

30-

a: S m o o t h

20-

b: Turbulent

100 -£ 0

1 0.1

1 *- 2 A / D 0.2

Figure 1.29 Correlation length. Cylinder smooth. Re = 1.9 X 10 4 . Pressure transducers are located at 60° to the main stream direction. A is the amplitude of cross-flow vibrations of the cylinder. Turbulence in the tunnel was generated by a coarse grid, and its intensity, Iu = 11%. Novak and Tanaka (1977).

T h e subject has been most recently studied by Szepessy and Bearman (1992). These authors studied the effect of the aspect ratio (namely the cylinder length-to-diameter ratio) on vortex shedding by using moveable end plates. They found t h a t the vortex-induced lift showed a m a x i m u m for an aspect ratio of 1, where the lift could be almost twice the value for very large aspect ratios. This increase of the lift amplitude was found to be accompanied by enhanced spanwise correlation of the flow. Finally, it may be noted t h a t Ribeiro (1992) gives a comprehensive review of the literature on oscillating lift on circular cylinders in cross-flow.

References

SS

REFERENCES

Achenbach, E. and Heinecke E. (1981): On vortex shedding from smooth and rough cylinders in the range of Reynolds numbers 6 x 10 3 to 5 x 10 6 . J. Fluid Mech., 109:239-251. Angrilli, F., Bergamaschi, S. and Cossalter, V. (1982): Investigation of wallinduced modifications to vortex shedding from a circular cylinder. Trans. of the ASME, J. Fluids Engrg., 104:518-522. ASCE Task Committee on W i n d Forces (1961): W i n d forces on structures. Trans. ASCE, 126:1124-1198. Batchelor, G.K. (1967): An Introduction to Fluid Dynamics. Cambridge University Press. Bearman, P.W. and Zdravkovich, M.M. (1978): Flow around a circular cylinder near a plane boundary. J. Fluid Mech., 89(l):33-48. Blevins, R.D. (1977): Flow-induced Vibrations. Van Nostrand. Bloor, M.S. (1964): T h e transition to turbulence in the wake of a circular cylinder. J. Fluid Mech., 19:290-304. Cheung, J.C.K. and Melbourne, W.H. (1983): Turbulence effects on some aerodynamic parameters of a circular cylinder at supercritical Reynolds numbers. J. of W i n d Engineering and Industrial Aerodynamics, 14:399-410. El-Baroudi, M.Y. (1960): Measurement of Two-Point Correlations of Velocity near a Circular Cylinder Shedding a K a r m a n Vortex Street. University of Toronto, UTIAS, T N 3 1 . Farell, C. (1981): Flow around fixed circular cylinders: Fluctuating loads. Proc. of ASCE, Engineering Mech. Division, 107:EM3:565-588. Also see the closure of t h e paper. Journal of Engineering Mechanics, ASCE, 109:1153-1156, 1983. Gerlach, C.R. and Dodge, F . T . (1970): An engineering approach to t u b e flowinduced vibrations. Proc. Conf. on Flow-Induced Vibrations in Reactor System Components, Argonne National Laboratory, pp. 205-225. Gerrard, J.H. (1966): T h e mechanics of the formation region of vortices behind bluff bodies. J. Fluid Mech., 25:401-413.

34

Chapter 1: Flow around a cylinder in steady

current

Gerrard, J.H. (1978): T h e wakes of cylindrical bluff bodies at low Reynolds number. Phil. Transactions of the Royal Soc. London, Series A, 288(A1354):351-382. Grass, A.J., Raven, P.W.J., Stuart, R.J. and Bray, J.A. (1984): T h e influence of b o u n d a r y layer velocity gradients and bed proximity on vortex shedding from free spanning pipelines. Trans. ASME, J. of Energy Res. Technology, 106:70-78. Griffin, O.M. (1985a): Vortex shedding from bluff bodies in a shear flow: A Review. Trans. ASME, J. Fluids Eng., 107:298-306. Griffin, O.M. (1985b): T h e effect of current shear on vortex shedding. Proc. Int. Symp. on Separated Flow Around Marine Structures. T h e Norwegian Inst. of Technology, Trondheim, Norway, J u n e 26-28, 1985, p p . 91-110. Homann, F . (1936): Einfluss grosser Zahigkeit bei Stromung u m Forschung auf dem Gebiete des Ingenieurwesen, 7(1):1-10.

Zylinder.

Humphreys, J.S. (1960): On a circular cylinder in a steady wind at transition Reynolds numbers. J. Fluid Mech., 9:603-612. Jensen, B.L. and Sumer, B.M. (1986): Boundary layer over a cylinder placed near a wall. Progress Report No. 64, Inst, of Hydrodynamics and Hydraulic Engineering, ISVA, Techn. Univ. Denmark, p p . 31-39. Jensen, B.L., Sumer, B.M., Jensen, H.R. and Freds0e, J. (1990): Flow around and forces on a pipeline near a scoured bed in steady current. Trans, of t h e ASME, J. of Offshore Mech. a n d Arctic Engrg., 112:206-213. King, R. (1977): A review of vortex shedding research and its application. Ocean Engineering, 4:141-171. Kiya, M., Tamura, H. and Arie, M. (1980): Vortex shedding from a circular cylinder in moderate-Reynolds-number shear flow. J. Fluid Mech., 141:721-735. Kwok, K.C.S. (1986): Turbulence effect on flow around circular cylinder. J. Engineering Mechanics, ASCE, 112(11):1181-1197. Maull, D.J. and Young, R.A. (1973): Vortex shedding from bluff bodies in a shear flow. J. Fluid Mech., 60:401-409. Modi, V.J., Wiland, E., Dikshit, A.K. and Yokomizo, T. (1992): On the fluid dynamics of elliptic cylinders. Proc. 2nd Int. Offshore and Polar Engrg. Conf., San Francisco, CA, 14-19 J u n e 1992, 111:595-614.

References

35

Nikuradse, J. (1933): Stromungsgesetze in rauhen Rohren. Forsch. Arb.Ing.-Wes. No. 361. Norberg, C. and Sunden, B. (1987): Turbulence and Reynolds number effects on the flow and fluid forces on a single cylinder in cross flow. Jour. Fluids and Structures, 1:337-357. Novak, M. and Tanaka, H. (1977): Pressure correlations on a vibrating cylinder. Proc. 4th Int. Conf. on W i n d Effects on Buildings and Structures, Heathrow, U.K., Ed. by K.J. Eaton. Cambridge Univ. Press, p p . 227-232. Raven, P.W.J., Stuart, R.J., Bray, J.A. and Littlejohns, P.S. (1985): Full-scale dynamic testing of submarine pipeline spans. 17th Annual Offshore Technology Conference, Houston, Texas, May 6-9., paper No. 5005, 3:395-404. Ribeiro, J.L.D. (1992): Fluctuating lift and its spanwise correlation on a circular cylinder in a smooth and in a turbulent flow: a critical review. Jour, of W i n d Engrg. and Indust. Aerodynamics, 40:179-198. Roshko, A. (1961): Experiments on the flow past a circular cylinder at very high Reynolds number. J. Fluid Mech., 10:345-356. Schewe, G. (1983): On the force fluctuations acting on a circular cylinder in crossflow from subcritical up to transcritical Reynolds numbers. J. Fluid Mech., 133:265-285. Schlichting, G. (1979): Boundary Layer Theory. 7.ed. McGraw-Hill Book Company. Szepessy, S. and Bearman, P.W. (1992): Aspect ratio and end plate effects on vortex shedding from a circular cylinder. J. Fluid Mech., 234:191-217. Toebes, G.H. (1969): T h e unsteady flow and wake near an oscillating cylinder. Trans. ASME J. Basic Eng., 91:493-502. Williamson, C.H.K. (1988): T h e existence of two stages in the transition to threedimensionality of a cylinder wake. Phys. Fluids, 31(11):3165-3168. Williamson, C.H.K. (1989): Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds number. J. Fluid Mech., 206:579-627.

Chapter 2. Forces on a cylinder in steady current

T h e flow around t h e cylinder described in Chapter 1 will exert a resultant force on the cylinder. There are two contributions to this force, one from the pressure and the other from the friction. T h e in-line component of the mean resultant force due to pressure (the in-line mean pressure force) per unit length of the cylinder is given by _

r2n

/•27T

Fv = / Jo

pcos((j>)r0d<j>,

(2.1)

(see Fig. 2.1 for t h e definition sketch), while t h a t due t o friction (the in-line mean friction force) is given by

I

•2TT

TO sin(^)rod

(2.2)

in which p is the pressure and To is the wall shear stress on the cylinder surface, and the overbar denotes time-averaging. T h e total in-line force, the so-called m e a n d r a g , is the sum of these two forces: FD = FP+Ff Fp is termed the f o r m d r a g and Ff the friction drag.

(2.3)

Drag and lift

SI

Figure 2.1 Definition sketch.

Regarding the cross-flow component of the mean resultant force, this force will be nil due to symmetry in the flow. However, t h e instantaneous cross-flow force on t h e cylinder, i.e., the instantaneous lift f o r c e , is non-zero and its value can be rather large, as will be seen in t h e next sections.

2.1 Drag and lift As has been discussed in Chapter 1, the regime of flow around a circular cylinder varies as t h e Reynolds number is changed (Fig. 1.1). Also, effects such as the surface roughness, the cross-sectional shape, the incoming turbulence, and the shear in the incoming flow influence the flow. However, except for very small Reynolds numbers (Re ~ 40), there is one feature of the flow which is common to all t h e flow regimes, namely t h e vortex shedding. As a consequence of the vortex-shedding phenomenon, t h e pressure distribution around the cylinder undergoes a periodic change as the shedding process progresses, resulting in a periodic variation in t h e force components on the cylinder. Fig. 2.2 shows a sequence of flow pictures of the wake together with the measured pressure distributions and t h e corresponding force components, which are calculated by integrating the pressure distributions over the cylinder surface (the time span covered in the figure is slightly larger t h a n one period of vortex shedding). Fig. 2.3, on the other hand, depicts the force traces corresponding to the same experiment as in t h e previous figure. T h e preceding figures show t h e following two important features: first, the force acting on t h e cylinder in the in-line direction (the drag force) does change periodically in time oscillating around mean drag, and secondly, although the incoming flow is completely symmetric with respect to the cylinder axis, there exists a non-zero force component (with a zero mean, however) on the cylinder

Pressure

t = 0.84s

U-

0.87s

0.90s

0.94s

0.97s

Figure 2.2 Time development of pressure distribution and the force components, as the Re = 1.1 X 10 5 , D = 8 cm and U = 1.53 m/s. cp = (p - p 0 ) / ( \ p U 2 ) . D

Drag and lift

C

D-

C

S9

L4

2

VortexShedding period

Figure 2.3 Drag and lift force traces obtained from the measured pressure distributions in the previous figure. Cp = Fo/^pDU2) and CL = FL/{\pDU2). Drescher (1956).

in the transverse direction (the lift force), and time. In the following paragraphs we will first mean drag, then we will focus on the oscillating the oscillating drag force and the oscillating lift

this, too, varies periodically with concentrate our attention on the components of the forces, namely force.

40

Chapter 2: Forces on a cylinder

in steady

current

2.2 Mean drag

Form drag and friction drag Fig. 2.4 shows the relative contribution to the total mean drag force from friction as function of the .Re-number. T h e figure clearly shows t h a t , for the range of Re numbers normally encountered in practice, namely Re ~ 10 4 , t h e contribution of the friction drag to the total drag force is less t h a n 2 - 3 % . So the friction drag can be omitted in most of the cases, a n d the total mean drag can be assumed to be composed of only one component, namely t h e form drag

FD

0.020 o±

0.010

Thorn ( 1 9 2 9 ) ^ -

A

A

A

u

io

°A

•

0.005

0.002 0.001

_l

10

10

10

Re

10

Figure 2.4 Relative contribution of the friction force to the total drag for circular cylinder. Achenbach (1968).

p cos((f>)rod

(2.4)

Jo Fig. 2.5a depicts several measured pressure distributions for different values of Re, while Fig. 2.5b presents the corresponding wall shear stress distributions. Fig. 2.5a contains also the pressure distribution obtained from t h e potential flow theory, which is given by P-l

-pU2(l-4:Sm2

(2.5)

Mean drag

R e = l x 10 2.6x 1 0 5 3.6x 10 6

300

3 6 0 t

^?f^\

\I

X"'

"^QL Separation point

1

1 1 1 •

(b)

1

1

• W^•*/

'

k'f

r"-'Mi ""»''•

1

_

Re=lx l O 5 " " ' ^ . T 2.6x l O 5 " " ' ^

-

3.6x 1 0 6 - ^ ^ ^

i

i

i

,

tV

1

tjf

* • N

// -'V

1

0

1

60

1

i

120

i

i

180

i

i

240

i

1

300

1

1

360

Figure 2.5 Pressure distribution and wall shear stress distribution at different Re numbers for a smooth cylinder. Achenbach (1968).

^1

42

Chapter 2: Forces on a cylinder in steady

Super critical

Subcritical

current

Upper transition

A

,

Transcritical

/

150 i

130

i i

140

1o

_

A o 0

0

Al o

u

o

= *

t^S

S

(f

0

120

110 100 90 _ _ 80 70

|°

Separation point

o

%

.A

i

i

*•

i i i ml

4

10

- A

10

2

i

i

| | nil 5 10

1

1

1 1 1 1 III

5

10

tab-

Re

Figure 2.6 Position of the separation point as a function of the Reynolds number for circular cylinder. Achenbach (1968).

in which po is t h e hydrostatic pressure. Fig. 2.6 gives t h e position of separation points as a function of Re. T h e main characteristic of the measured pressure distributions is t h a t the pressure at the rear side of the cylinder (i.e., in the wake region) is always negative (in contrast to what the potential-flow theory gives). This is due to separation. Fig. 2.5a further indicates t h a t the pressure on the cylinder remains practically constant across the cylinder wake. This is because the flow in t h e wake region is extremely weak as compared to the outer-flow region.

D r a g coefficient T h e general expression for t h e drag force is from Eqs. 2.1-2.3 given by /-27T

FD=

(pcos((f>) +T0sm((f>))rod

Jo This equation can be written in the following form

(2.6)

Mean drag

43

in which D = 2r 0 , the cylinder diameter. T h e right-hand-side of the equation is a function of t h e Re number, since b o t h the pressure term and the wall shear stress term are functions of the Re number for a smooth cylinder (Fig. 2.5). Therefore Eq. 2.7 may be written in t h e following simple form (2.8)

kpDU*

CD is called the mean drag coefficient, or in short, the drag coefficient, and is a function of Re.

0.1 I I ~

10

U-Ll

I

U-Ll

0

10

No separation

I 1

. 10

l_LLl

I

l_LLl

2

, 10 .

1 3

10

l l ll 4

10

I

' lit

|

1_LU | 6

5

10

..

10,

l_LLl

I 7

.10

• £T

10 Re

Transcritical •jy Subcritical Lam. Fixed ^ r t p „ Transition /Super pair of sS un e"d ^ to turbulence Cr Hr lHt i„c a 10 5 . The diagram minus Schewe's data was taken from Schlichting (1979).

Fig. 2.7 presents the experimental d a t a together with t h e result of the laminar theory, illustrating t h e variation of Co with respect to the Re number, while Fig. 2.8 depicts t h e close-up picture of this variation in the most interesting

44

Chapter 2: Forces on a cylinder in steady

*ooo^>

current

(a)

(

1.0

0.5 o 8

w 0 O OOO>° O o

4

I I I I 11

2 x 10

5

I

I

I

°°o *°°

I

Ill

10

10

Re

10 (b)

° „8

0.3

^4

0.2 0.1 f0

°|Q P P P ° i l ° E

0 2 j

X

10

10

10

0.4

Re

10

i

0.5

St

° ° ° o oooo I

1 i

3 0 OO

_

(c)

_

"""bo

o

'

0.3

-

0.2

-o Oo 0 o o o o o c o x o o

0.1

-

0°°°

o°o°

o oo ° 1

°o

oo o 1

1

1

1 11 l i

4

2 x 10

1

1

1 1 1 11

10

1

6

5

10

/ Super/ critical Critical

1

1 1 1 1 ll

Re

4

Subcritical

1

^

10

»

Upper transition

Transcritical

Figure 2.8 Drag coefficient, r.m.s. of the lift oscillations and Strouhal number as function of Re for a smooth circular cylinder. Schewe (1983).

Mean drag

J^5

range of Re numbers, namely Re ~ 10*. T h e latter figure also contains information about the oscillating lift force and the Strouhal number, which are maintained in the figure for t h e sake of completeness. T h e lift force d a t a will be discussed later in the section dealing with the oscillating forces. As seen from Fig. 2.7, Co decreases monotonously with Re until Re reaches the value of about 300. However, from this Re number onwards, Co assumes a practically constant value, namely 1.2, throughout the subcritical Re range (300 < Re < 3 x 10 5 ). W h e n Re attains the value of 3 X 10 5 , a dramatic change occurs in Co', the drag coefficient decreases abruptly and assumes a much lower value, about 0.25, in the neighbouring Re range, the supercritical Re range, 3.5 x 10 5 < Re < 1.5 x 10 6 (Fig. 2.8a). This phenomenon, namely the drastic fall in Co, is called the d r a g crisis. T h e drag crisis can best be explained by reference to the pressure diagrams given in Fig. 2.5. Note t h a t the friction drag can be disregarded in the analysis because it constitutes only a very small fraction of t h e total drag.

Re=lxlO (Subcritical)

Re = 8 . 5 x 1 0 (Supercritical)

Figure 2.9 Pressure distributions. cp = (p — p0)/(^pU2). separation points. Achenbach (1968).

S denotes the

Two of t h e diagrams, namely the one for Re = 1 x 10 5 (a representative Re number for subcritical flow regime) and t h a t for Re = 8.5x 10 5 (a representative Re number for supercritical flow regime) are reproduced in Fig. 2.9. From the figure,

1^6

Chapter 2: Forces on a cylinder in steady

current

it is evident t h a t the drag should be smaller in t h e supercritical flow regime t h a n in the subcritical flow regime. Clearly, the key point here is t h a t t h e separation point moves from <j>3 = 78° {Re = 1 x 10 5 , the laminar separation) to <j>3 = 140° (Re = 8.5 x 10 5 , the turbulent separation), when the flow regime is changed from subcritical to supercritical (Fig. 2.6), resulting in an extremely narrow wake with substantially smaller negative pressure, which would presumably lead to a considerable reduction in the drag. Returning to Figs. 2.7 and 2.8 it is seen t h a t the drag coefficient increases as the flow regime is changed from supercritical to upper-transition, and then Co attains a constant value of about 0.5, as Re is increased further to transcritical values, namely Re > 4.5 x 10 6 . Again, the change in Co for these higher flow regimes can be explained by reference to the pressure distributions given in Fig. 2.5 along with the information about the separation angle given in Fig. 2.6. Effect o f surface r o u g h n e s s In the case of rough cylinders, the mean drag, as in t h e case of smooth cylinders, can be assumed to be composed of only one component, namely the form drag; indeed, Achenbach's (1971) measurements demonstrate t h a t t h e contribution of the friction drag to the total drag does not exceed 2 - 3 % , thus can be omitted in most of t h e cases (Fig. 2.10).

0.03

0.002

Figure 2.10 Relative contribution of the friction force to the total drag. Effect of cylinder roughness. Achenbach (1971).

T h e drag coefficient, Co, now becomes not only a function of Re number but also a function of the roughness parameter ks/D

Co = Co (Re, ^ )

(2.9)

Mean drag

I

I I I

I

I

I I

4 4

10

I

I

I

10

I

1_|

5

6 10

^7

I

L*.

„ Re

Figure 2.11 Drag coefficient of a circular cylinder at various surface roughness parameters k3/D. Achenbach and Heinecke (1981).

in which ks is the Nikuradse equivalent sand roughness. Fig. 2.11 depicts Co plotted as a function of these parameters. T h e way in which Co varies with Re for a given ka/D is sketched in Fig. 2.12. As seen from t h e figures, the Reynolds-number ranges observed for the smooth-cylinder case still exist. However, two of the high Re n u m b e r ranges, namely t h e supercritical range and t h e upper transition range seem to merge into one single range as the roughness is increased. Furthermore, the following observations can be m a d e from the figure: 1) For small Re numbers (i.e., the subcritical Re numbers), Co takes the value obtained in the case of smooth cylinders, namely 1.4, irrespective of the cylinder roughness. 2) T h e CD~versus-Re curve shifts towards the lower end of the .Re-number range indicated in the figure, as the cylinder roughness is increased. Clearly, this behaviour is related to the early transition to turbulence in the b o u n d a r y layer with increasing roughness. 3) T h e drag crisis, which is characterized by a marked depression in the Co curve, is not as extensive as it is in the smooth-cylinder case: while Co falls from 1.4 to a value of about 0.5 in the case of smooth cylinder, it falls from 1.4 only to a value of about 1.1 in t h e case of rough cylinder with k3/D = 30 x 1 0 - 3 . This is directly linked with t h e angular location of t h e separation points. Fig. 2.13 compares the latter quantity for cylinders with different roughnesses. It is seen t h a t , in the supercritical range, while <j>s is equal to 140° in the case of a smooth

J8

Chapter 2: Forces on a cylinder in steady

Super critical Subcritical

Critical

current

Upper transition

/.

Transcritical

Re

Figure 2.12 General form of CD = Co(Re)

curve for a rough cylinder.

Smooth

Figure 2.13 Circular cylinder. Angular position of boundary-layer separa tion at various roughness parameters. Achenbach (1971).

Mean drag

1)9

cylinder, it is only 115° for the case of a rough cylinder with ks/D = 4.5 x 10~ 3 . (This is because of t h e relatively weaker m o m e n t u m exchange near the wall in the case of rough wall due to the larger boundary-layer thickness). Therefore, the picture given in Fig. 2.9b for t h e smooth-cylinder situation (where (j>s = 140°) will not be the same for the rough cylinder (<j>s = 115°). As a m a t t e r of fact, the pressure-distribution picture for the rough cylinder in consideration (<j>a = 115°) must lie somewhere between the picture given in Fig. 2.9a a n d t h a t given in Fig. 2.9b, which implies t h a t the fall in the mean drag due to the drag crisis in this case will not be as extensive as in the case of a smooth cylinder, as clearly indicated in Fig. 2.11. Regarding the transcritical Re numbers in Fig. 2.11, t h e transcritical range covers smaller and smaller Re numbers as the roughness is increased. Also, the CD coefficient in t h e transcritical range takes higher and higher values with increasing roughness, see Table 2.1. Clearly, this behaviour is closely linked with the behaviour of the cylinder boundary layer. Finally, Fig. 2.14 gives the drag coefficient as a function of cylinder roughness for t h e transcritical .Re-number range.

Table 2.1 Transcritical Re number range for various values of the relative roughness. Data from Fig. 2.11.

k3/D

Transcritical Reynolds number range

Cylinder roughness

0 0.75 x i r r 3

3 x icr- 33

9 x lO 30 x 10~3

Re Re Re Re Re

> > > > >

(3 - 4) x 106 9 x 105 5 x 105 3 x 105 (1 - 2) x 105

T h e reader is referred to t h e following work for further details of the effect of the cylinder roughness on the mean drag: Achenbach (1968, 1971) and Giiven, Patel and Farell (1975 and 1977), Giiven, Farell and Patel (1980), Shih, Wang, Coles and Roshko (1993) among others.

50

Chapter 2: Forces on a cylinder in steady

current

' c 1.5

• • _-^-~-" *

1.0

^

^

0.5 ks

-5-

A

0 0

'

1

xl

3

°

10

Figure 2.14 Drag coefficient for rough cylinders in the transcritical number range (Table 2.1). Data from Fig. 2.11.

Re-

2.3 Oscillating drag and lift A cylinder which is exposed to a steady flow experiences oscillating forces if Re > 40, where t h e wake flow becomes time-dependent (Section 1.1). T h e origin of the oscillating forces is t h e vortex shedding. As already discussed in Section 1.1, the key point is t h a t t h e pressure distribution around t h e cylinder undergoes a periodic change as the vortex shedding progresses, resulting in a periodic variation in the force (Figs. 2.2 and 2.3). A close inspection of Fig. 2.2 reveals t h a t the upward lift is associated with the growth of the vortex at the lower edge of the cylinder (t = 0.87 - 0.94 s), while the downward lift is associated with t h a t at the upper edge of t h e cylinder (t = 1.03 - 1.10 s). Also, it is readily seen t h a t b o t h vortices give a temporary increase in the drag. As seen from Fig. 2.3, t h e lift force on the cylinder oscillates at the vortexshedding frequency, / „ ( = 1/T„), while the drag force oscillates at a frequency which is twice t h e vortex-shedding frequency. Fig. 2.3 further indicates t h a t the amplitude of the oscillations is not a constant set of value. As is seen, it varies from one period to t h e other. It may even happen t h a t some periods are missed. Nevertheless, t h e magnitude of the oscillations can be characterized by their statistical properties such as the root-mean-square (r.m.s.) value of t h e oscillations. Fig. 2.15 gives the oscillating-force d a t a compiled by Hallam, Heaf and Wootton

Oscillating drag and lift

51

CD,Ci

w^ cL

0.1

a)

2

(C D ) * 0.05

\ * * * * * * J

i

i M

I

i

i i i

I

I

I II

10

10

Re

b)

Range of r e s u l t s for stationary cylinders

i

i

i i i

i

i

i i i

10"

i

i

i i

1 0 7 Re

Figure 2.15 R.m.s.-values of drag and lift oscillations. C'D = F'Dj {\pDU2) and C'L = F[l(\pDU2). Hallam et al. (1977). (1977), regarding the magnitude of the oscillations in the force coefficients where C'D and C'L are defined by the following equations K = -2PCDBV2

(2.10) (2.11)

52

Chapter 2: Forces on a cylinder in steady

current

in which FD is the oscillating part of the drag force FD = FD-FD

,

(2.12)

and F[ is t h e oscillating lift force F[ = FL-FL

= FL-0

= FL

,

(2.13)

(CD2) and (C'L2) are the r.m.s. values of the oscillations CD and C'L, respectively. T h e magnitude of the oscillating forces is a function of Re, which can be seen very clearly from Fig. 2.8, where C'L d a t a from a single set of experiments are shown along with the Co and the Si-number variations obtained in the same work. It is evident t h a t the r.m.s.-value of C'L experiences a dramatic change in the same way as in the case of Co and St in the critical flow regime, and then it attains an extremely low value in the supercritical flow regime. This point has already been mentioned in Section 1.2.1 in connection with the frequency of vortex shedding with reference to the power spectra of t h e lift oscillations illustrated in Fig. 1.10 (cf. Fig. 1.10a and 1.10b, and note the difference in t h e scales of the vertical axes of the two figures). T h e main reason behind this large reduction in the r.m.s.-value of C'L is t h a t , in the supercritical flow regime, t h e interaction between the vortices in the wake is considerably weaker, partly because the b o u n d a r y layer separates at an extremely large angular position (Fig.2.6) meaning t h a t the vortices are much closer to each other in this flow regime, and partly because the boundarylayer separation is turbulent (Fig. 1.1).

2.4 Effect of cross-sectional shape on force coefficients T h e shape of the cross-section has a large influence on the resulting force. A detailed table giving the variation in the force coefficient with various shapes of cross-sections is given in Appendix I. There are two points which need to be elaborated here. One is the Reynolds number dependence in the case of cross-sectional shapes with sharp edges. In this case, practically no Reynolds number dependence should be expected since the separation point is fixed at t h e sharp corners of the cross section. So, no change in force coefficients is expected with Re number for these cross-sections in contrast to what occurs in t h e case of circular cross-sections. Secondly, non-circular cross-sections may be subject to steady lift at a certain angle of attack. This is due to the asymmetry of t h e flow with respect to the principle axis of the cross-sectional area. A similar kind of steady lift has been observed even for circular cylinders in t h e critical flow regime (Schewe, 1983) where the asymmetry occurs due to t h e one-sided transition to turbulence (Section 1.1). Fig. 2.16 presents the force coefficient regarding this steady lift for different cross-sections.

Effect of incoming

0

turbulence

on force coef

ficients

53

5

10 15 20 25 a(deg) Figure 2.16 Steady lift force coefficients, Re = 33,000 to 66,000. Parkinson and Brooks (1961).

2.5 Effect of incoming turbulence on force coefficients T h e turbulence in t h e approaching flow may affect the force coefficients, Cheung and Melbourne (1983), Kwok (1986), and Norberg and Sunden (1987). T h e effect is summarized in Fig. 2.17 based on the d a t a presented in Cheung and Melbourne (1983). T h e dashed lines in t h e figure correspond to t h e case where the turbulence level is very small, and therefore the flow in this case may be considered smooth. T h e figures clearly show t h a t the force coefficients are affected quite considerably by the incoming turbulence. Increasing the turbulence level from almost smooth flow (the dashed curves) to larger and larger values acts in t h e same way as increasing t h e cylinder roughness (cf. Fig. 2.17a and Fig. 2.11). As has been discussed in the context of t h e effect of roughness, the increased level of incoming turbulence will directly influence the cylinder boundary layer and hence its separation. This will obviously lead to changes in the force and therefore in the force coefficients.

54

Chapter 2: Forces on a cylinder in steady

current

Mt. Isa stack full scale data C D =0.6 for I u =6.5%,Re=10

Figure 2.17 Effect of turbulence on the force coefficients. Iu is defined in Eq. 1.8. Cheung and Melbourne (1983).

Effect of angle of attack on force coefficients

55

2.6 Effect of angle of attack on force coefficients W h e n a cylinder is placed at an angle to the flow (Fig. 2.18), forces on the cylinder may change. Experiments show, however, t h a t in most of the cases the so-called independence or cross-flow principle is applicable (Hoerner, 1965). Namely, t h e component of t h e force normal to t h e cylinder may be calculated from FN = \pCDD

U2N

(2.14)

in which Upi is t h e velocity component normal to the cylinder axis. T h e drag coefficient in the preceding equation can be taken as t h a t obtained for a cylinder normal t o t h e flow. So, Co is independent of the angle of attack, 8.

Figure 2.18 Definition sketch. Angle of attack of flow, 6, is different from 90°.

It may be argued t h a t t h e flow sees an elliptical cross-section in t h e case of an oblique attack, and therefore separation may be delayed, resulting in a value of Co different from t h a t obtained for a cylinder normal t o t h e flow. Observations show, however, t h a t , although t h e approaching flow is at an angle, the streamlines in the neighbourhood of t h e cylinder are bent in such a way t h a t the actual flow past t h e cylinder is at an angle of about 8 = 90° (Fig. 2.19). Therefore, the position of t h e separation point practically does not change, meaning t h a t Co should be independent of 8. Kozakiewicz, Freds0e and Sumer (1995), based on their flow-visualization experiments, give t h e critical value of 8 approximately 35°. For 8 ~ 35°, t h e streamlines do not bend, implying t h a t , for such small values of 8, Co is no longer independent of 6, a n d therefore the independence principle will be violated.

56

Chapter 2: Forces on a cylinder in steady current

Figure 2.19 Visualization of flow past a circular cylinder in the case of oblique attack {6 being different from 90°). Kozakiewicz et al. (1995). Regarding the lift, Kozakiewicz et al. (1995) report that the independence principle is valid also for the lift force for the tested range of 6 for their force measurements, namely 45° < 6 < 90°. They further report that the vortex shedding frequency (obtained from the lift-force spectra) is close to the value calculated from the Strouhal relationship. The lift force power spectrum becomes broader, however, as 6 is decreased. Kozakiewicz et al.'s (1995) study covers also the case of a near-bottom cylinder (the pipeline problem) with the gap between the cylinder and the bottom being 0.1 D in one case and nil in the other. Apparently, the independence principle is valid also for the near-bottom-cylinder situation for the tested range of 6(45° < 0 < 90°). Finally, it may be noted that, although, theoretically, the independence principle is justified only in the subcritical range of Re, it has been proved to hold true also in the postcritical flows (Norton, Heideman and Mallard, 1981). However, there is evidence (Bursnall and Loftin, 1951) that for the transcritical values of Re the independence principle may not be applied.

Forces on a cylinder near a wall

57

2.7 Forces on a cylinder near a wall T h e changes in t h e flow caused by the wall proximity is discussed in Section 1.2.1; these changes will obviously influence the forces acting on the cylinder. This section will describe the effect of wall proximity on t h e forces on a cylinder placed near (or on) a wall. T h e following aspects of t h e problem will be examined: t h e drag force, t h e lift force, t h e oscillating components of t h e drag and the lift, and finally the forces on a pipeline placed in/over a scour trench.

D r a g force o n a c y l i n d e r n e a r a p l a n e wall Fig. 2.20 depicts t h e pressure distributions around a cylinder placed at three different distances from a plane wall (Bearman and Zdravkovich, 1978). Fig. 2.21, on the other h a n d , presents the experimental d a t a on the drag coefficient from the works by Kiya (1968), Roshko, Steinolffron and Chattoorgoon (1975), Zdravkovich (1985) and Jensen, Sumer, Jensen and Freds0e (1990). T h e drag coefficient is defined in t h e same way as in Eq. 2.8. T h e general trend is t h a t the drag coefficient decreases with decreasing gap ratio near the wall. This result is consistent with the pressure distributions given in Fig. 2.20. T h e differences between the various experiments in t h e figure may be attributed t o t h e change in t h e Reynolds number. One characteristic point in the variation of CQ with respect to e/D is t h a t , as seen from t h e figure, Cp increases in a monotonous manner with increasing e/D up to a certain value of e/D, and then it remains reasonably constant for further increase in e/D (Fig. 2.22). This behaviour has been linked by Zdravkovich (1985) t o t h e thickness of t h e boundary layer of t h e approaching flow: t h e flat portion of the curve occurs for such large gap ratios t h a t the cylinder is embedded fully in the potential flow region. At lower gap ratios the cylinder is embedded partly in the potential flow region and partly in t h e boundary layer of the incoming flow. T h e curves belonging to Zdravkovich's (1985) d a t a in Fig. 2.21 with two different values of S/D, namely S/D = 0.5 a n d S/D = 1 where S = t h e thickness of the boundary layer in the approaching flow, demonstrates this characteristic behaviour.

Lift force o n a c y l i n d e r n e a r a p l a n e wall T h e m e a n flow around a near-wall cylinder is not symmetric, therefore a non-zero mean lift must exist (in contrast to the case of a free cylinder). Fig. 2.20 shows t h a t , while the mean pressure distribution around t h e cylinder is almost symmetric when e/D = 1, meaning t h a t practically no lift exists, this symmetry

58

Chapter 2: Forces on a cylinder in steady

current

a)i-l

Stagnation

777777 b)^=0.1

TTT777TTJ Stagnation

/////////

\\7 V_V

_^s

Figure 2.20 Pressure distributions on a cylinder near a wall as a function of gap ratio e/D. cp = (p — Po)l(\pU2) where po ' s the hydrostatic pressure. Bearman and Zdravkovich (1978).

clearly disappears for t h e gap ratios e/D = 0.1 and 0, resulting in a non-zero m e a n lift on the cylinder. This lift, as seen from t h e figure, is directed away from the wall. T h e variation of the lift force with respect to the gap ratio can best be described by reference to the simple case, the shear-free flow situation, depicted in Fig. 2.23. In the figure are plotted Freds0e, Sumer, Andersen and Hansen's (1985) experimental d a t a , Freds0e a n d Hansen's (1987) modified potential-flow solution and also the potential-flow solution for a wall-mounted cylinder (see, for example, Yamamoto, Nath and Slotta (1974) for the latter). T h e shear-free flow in Freds0e et al.'s study was achieved by towing t h e cylinder in still water. T h e CL coefficient

Forces on a cylinder near a wall

tl^l

y 3) depicted in Fig. 3.16, t h e presently available d a t a are not very extensive. It is evident that no detailed account of various upper Reynolds-number regimes, known from the steady-current research (such as the lower transition, the supercritical, the upper transition a n d t h e transcritical regimes), is existent. Nevertheless, Sarpkaya's

92

Chapter 3: Flow around a cylinder in oscillatory

flows

(1976a) extensive d a t a covering a wide range of KC for lower Re regimes along with Williamson's (1985) and Justesen's (1989) d a t a may indicate what happens with increasing the Reynolds number. Regarding the vortex-shedding regimes, it is evident from t h e figure that the curves begin to bend down, as Re approaches to t h e value 10 5 , meaning that in this region t h e normalized lift frequency Ni increases with increasing Re. This is consistent with the corresponding result in steady currents, namely t h a t the shedding frequency increases with increasing Re at 3.5 x 10 5 when the flow is switched from subcritical to supercritical through the critical (lower transition) flow regime (Fig. 1.9). Finally, it may be mentioned t h a t Tatsumo and B e a r m a n (1990) presented the results of a detailed flow visualization study of flow at low KC numbers and low /?(= Re/KC) numbers.

3.4 Effect of wall proximity on flow regimes T h e influence of wall proximity on the flow around and forces on a cylinder has already been discussed in t h e context of steady currents (Sections 1.2.1 and 2.7). As has been seen, several changes occur in the flow around t h e cylinder when the cylinder is placed near a wall, such as the break-up of symmetry in t h e flow, the suppression of vortex shedding, etc.. T h e purpose of the present section is to examine the effect of wall proximity on the regimes of flow around a cylinder exposed to an oscillatory flow. T h e analysis is mainly based on the work of Sumer, Jensen and Freds0e (1991) where a flow visualization study of vortex motions around a smooth cylinder was carried out along with force measurements. T h e .Re-range of the flow-visualization experiments was 10 3 —10 4 , while t h a t of the force measurements was 0.4 x 10 5 —1.1 x 10 5 .

Flow regimes

4 < KC < 7 Fig. 3.17 illustrates how the vortices evolve during t h e course of the oscillatory motion for KC = 4 for three different values of the gap-to-diameter ratio e/D, e being the gap between the cylinder and the wall. T h e symmetry observed in the formation and also in the motion of the vortices (Fig. 3.17a) is no longer present when e/D = 0.1 (Fig. 3.17b). This is also clear from t h e lift-force traces given in Fig. 3.18 where almost no lift force is exerted on the cylinder for e/D = 2, while a non-zero lift exists for e/D = 0 . 1 . Here Ci is the lift coefficient defined by

Effect of wall proximity

a)

— ^ cot =90°

—

- ~

158°

on flow regimes

135°

- ^ 180°

eg b)

135"

GL3

77777777777777777

158°

180" K

&

2)

77777777777777777

c)

77777777777777777

cot = 0

60" K

>5> 77777^7777/

120"

77777777777777777

7777777)l777777777

180°

7777777^,'77777777

Figure 3.17 Evolution of vortex motions. KC = 4. Gap-to-diameter-ratio values: (a) e/D = 2, (b) e/D = 0.1, (c) e / D = 0. Sumer et al. (1991).

93

94

Chapter S: Flow around a cylinder in oscillatory

Fy =

flows

-pCLDUl

(3.14)

T h e vortex regime is quite simple for the wall-mounted cylinder (Fig. 3.17c): a vortex grows behind the cylinder each half-period, and is washed over the cylinder as the next half-period progresses. Jacobsen, B r y n d u m and Freds0e (1984) give a detailed account of the latter where the motion of the lee-wake vortex over the cylinder is linked to the maximum pressure gradient in the outer flow. T h e liftforce trace is presented in Fig. 3.18c. T h e peaks in the lift force are associated with the occurrences where the vortices (Vortex K, Vortex £,... in Fig. 3.17c) are washed over the cylinder.

Velocity, U(t)

0

360

Figure 3.18 Lift-force traces. KC = 4. Sumer et al. (1991).

Effect of wall proximity on flow regimes

95

96

Chapter 3: Flow around a cylinder in oscillatory

oot=0°

M

////////WW

90°

;//;/////////;

////////////// 150°

s/////////////

////////////// 75°

a

CO /S///7/7777777

75°

/s/ssssss/s/s; 135°

40°

t = 10

KG,

Q

Jy^

•77777777777777

77777777777777

- ^

M 1V1

150

? L

7^777777777777

?s7/S/////,

^

/MW////////

120

90 M

75

>r 77777777777777

Figure 3.20 Evolution of vortex motions in the range 7 < i f C < 15. In the tests presented here KC = 10. Gap-to-diameter-ratio values: (a) e/D = 1, (b) e/D = 0.1, (c) e/D = 0. Sumer et al. (1991).

Effect of wall proximity

on flow regimes

97

7 < K C < 15 One of the interesting features of this KC regime for a wall-free cylinder is the formation of t h e transverse vortex street where the shed vortices form a vortex street perpendicular to the flow direction (Figs. 3.6a and 3.7). Sumer et al.'s (1991) work shows t h a t the transverse street regime disappears when the gap between the cylinder and the wall becomes less than tibout 1.7-1.8 times the cylinder diameter. Figs. 3.19a and 3.19b illustrate two different vortex flow regimes, one with a gap ratio above this critical value (the transverse street regime) and the other below it, where the transverse vortex street is replaced by a wake region which lies parallel to t h e flow oscillation direction. (a) e / D = 1. Fig. 3.20a illustrates the time development of vortex motions during one half-period of the motion, while Fig. 3.21b presents the corresponding lift-force trace. Fig. 3.20a indicates t h a t there is only one vortex shed (Vortex L) during one half-period of the motion. Fig. 3.21b shows how the lift force evolves during the course of the motion. T h e negative peak (B in Fig. 3.21b) is caused by the development of Vortex K (Fig. 3.20a, cot = 0° - 45°) (see Maull a n d Milliner (1978) for the relation between the vortex motion and the forces). As Vortex K is washed over t h e cylinder, the cylinder experiences a positive lift force, and the development of Vortex L also exerts a positive lift force (C in Fig. 3.21b). As Vortex L moves away from t h e cylinder (wi = 135° — 150°), t h e positive lift exerted on the cylinder by Vortex L is diminished. (b) e / D = 0 . 1 . T h e main difference between this case and the previous one is that here the wall-side vortex (Vortex TV) grows quite substantially. It is this latter vortex which is washed over the cylinder, whereas in the former case it was the free-stream-side vortex (Vortex M). T h e positive peak in the lift force (D in Fig. 3.21c) is caused by the development of Vortex L. T h e negative peak in t h e lift force (E in Fig. 3.21c) is caused by the development of Vortex TV combined with t h e high velocities in the gap induced by t h e flow reversal. (c) e / D = 0. In this case, the vortex which develops behind the cylinder in the previous half-period (Vortex K in Fig. 3.20c) a n d t h e vortex which is newly created (Vortex L in Fig. 3.20c) form a vortex pair. This pair is then set into motion owing to its self-induced velocity field, and t h u s steadily moves away from the cylinder in t h e downstream direction (see Fig. 3.20c, urt = 40° — 120°). Following t h e removal of Vortex L, a new vortex (Vortex M) begins t o develop behind the cylinder. T h e visualization results show t h a t the way in which the vortex flow regime develops for the wall-mounted cylinder ( e / D = 0) remains the same, irrespective of the range of KC. It should be noted, however, t h a t the individual events such as the formation of t h e vortex pair etc. may occur at different phase (tvt) values for different KC ranges. T h e peaks in t h e lift-force trace are caused by the passage of Vortex K over the cylinder.

98

Chapter S: Flow around a cylinder in oscillatory

flows

Velocity, U(t)

7I7Z /////*////// 7 < KC < 13

\

a)

%

13 < KC < 15

b) £ - l

c) H =0.1

d)

0

Figure 3.21 Lift-force traces in the range 7 < KC < 15. Positive lift is directed away from the wall. The wall-free' cylinder traces (a), e/D = oo, are taken from Williamson (1985). For the tests presented here KC = 10. Sumer et al. (1991).

Effect of wall •proximity on flow regimes

99

15 < K C < 2 4 a n d f u r t h e r K C r e g i m e s First, the KC regime 15 < KC < 24 will be considered. (a) e / D = 1. In this KC regime for wall-free cylinders there is no symmetry between the half-periods, as far as the vortex motions are concerned (Figs. 3.8 and 3.12), and this also applies to t h e present case where e/D = 1, as seen from Fig. 3.22a; t h e vortex which is washed over the cylinder alternates between the wall side and the free-stream side each half-period. T h e lift-force variation (Fig. 3.23b) supports this asymmetric flow picture. (b) e / D = 0 . 1 . Here, the flow is asymmetry; it is always the wall-side vortex (Vortex P , Fig. 3.22b) which is washed over the cylinder before the flow reverses to start a new half-period. T h e lift force is directed away from t h e wall most of the time (Fig. 3.23c). Furthermore, it contains distinct, short-duration peaks in its variation with time (F, G in Fig. 3.23c). T h e flow-visualization tests show t h a t these peaks are associated with t h e vortex shedding at the wall side of the cylinder: such peaks occur whenever there is a growing vortex on t h a t side of the cylinder (Fig. 3.22b: wi = 50° - 60° and ut = 80° - 93°). Fig. 3.24 represents t h e lift-force traces separately for t h e interval 0.05 < e/D < 0.4. For values of the gap ratio smaller t h a n approximately 0.3, the lift force becomes asymmetric, being directed away from the wall for most of t h e time, containing t h e previously mentioned distinct short-duration peaks. These peaks are present even for the gap ratio e/D = 0.05. These short-duration peaks indicate t h a t t h e vortex shedding is maintained even for very small gap ratios such as e/D = 0.1, in contrast to what occurs in steady currents where the vortex shedding is maintained for values of gap ratio down to only about e/D = 0.3 (Section 1.2.1, Fig. 1.21). This aspect of the problem will be discussed in greater detail later in this section. (c) e / D = 0. It is apparent from Fig. 3.22c t h a t t h e m a n n e r in which t h e vortex flow regime develops is exactly the same as in t h e range 7 < KC < 15 (cf. Figs. 3.20c and 3.22c). However, the streamwise distance t h a t the vortex pair travels is now relatively larger. T h e lift force (Fig. 3.23d) varies with respect to time in the same way as in Fig. 3.21d where 7 < KC < 15. However, the peaks in t h e present case occur relatively earlier t h a n those in Fig. 3.21d. T h e visualization tests of Sumer et al. (1991) indicate t h a t , as in Williamson (1985), t h e flow p a t t e r n s for the KC regimes beyond KC = 24 differ only in t h e number of vortices shed with no basic changes in the actual flow p a t t e r n s . Vortex shedding W h e t h e r vortex shedding will be suppressed for small values of the gap ratio can be detected from t h e flow-visualization films as well as from t h e lift-force traces. T h e results of such an analysis are plotted in Fig. 3.25. From t h e figure, the following observations can be made.

100

Chapter 3: Flow around a cylinder in oscillatory

flows

O

cot = -10

.

120

60

o

K'

V;

M

K

(jj/

7CD

"

^

7777777777777777-

w/«w/«wr 205° M

Jf/77/J?J?/7???} o

350

3 i*cp

&

0

275

cot= 10°

Gj' 9 o

50°

60

— K

7777777777777777

7/JJM77M77/77 o

o

o

93

80

7777777777777777

•7777777777777777

cot= 10

140

7777/77777777777

40 M L

7777777777777777

7777777777777777

75

go

90 M

K

77777777777777777777777777777?

L-l

'7777777777777 77777777777

150 M

7777777777777777

Figure 3.22 Evolution of vortex motions in the range 15 < A'C < 24. In the tests presented here KC = 20. Gap-to-diameter-ratio values: (a) e/D = 1, (b) e/D = 0.1, (c) e/D = 0. Sumer et al. (1991).

Effect of wall proximity

on flow regimes

Velocity, U(t) 0/

\

360/

\

tot

V77777

0

n =01

Figure 3.23 Lift-force traces in the range 15 < KC < 24. Positive lift is directed away from the wall. The wall-free cylinder (e/D = oo) trace (a) is taken from Williamson (1985), see Fig. 3.12. In the tests presented here KC = 20. Sumer et al. (1991).

101

102

Chapter S: Flow around a cylinder in oscillatory

flows

Velocity, U(t) 0 / ~ \

360". /////X/S//// 7Z77.

a)

Figure 3.24 Lift-force traces for the KC < 24. Positive lift tests presented here KC (c) e/D = 0.1, (d) e/D

§ = 0.4

b)

D

0

% = 0.1

d)

% = 0.05

ranges 0.05 < e/D < 0.4 and 15 < is directed away from the wall. In the = 20. (a) e/D = 0.4, (b) e/D = 0.2, = 0.05. Sumer et al. (1991).

Effect of wall proximity

onflow

regimes

103

1) For large values of KC, it appears t h a t the gap ratio below which the vortex shedding is suppressed approaches the critical value e/D « 0.25 deduced from the work by B e a r m a n and Zdravkovich (1978) and by Grass et al. (1984) for steady currents, (Section 1.2.1). 2) Although t h e borderline between the two regions in the figure, namely t h e vortex-shedding region and the region where the vortex shedding is suppressed, is not expected to be a clean-cut curve, there is a clear tendency t h a t the vortex shedding is maintained for smaller and smaller values of the gap ratio as KC is decreased. Vortex shedding is maintained even for very small gap ratios such as e/D = 0.1 for KC = 10 — 20, as shown in the photograph in Fig. 3.19c. Likewise, Fig. 3.24c implies t h a t shedding occurs for t h a t value of t h e gap ratio, as t h e shortduration peaks in the lift-force time series are associated with vortex shedding. T h e reason why vortex shedding is maintained for such small gap ratios is because the water discharge at the wall side of the cylinder is much larger in oscillatory flow at small KC t h a n in steady currents due to the large pressure gradient from the wave.

\

D

•

0.4

•

Vortex shedding

•

l

0.2

I

,—— o 20

O A O O O A

OA O OA O —I

_L_

40 KC

O O O O

A A

Vortex shedding suppressed

_L_

oo (steady current)

Figure 3.25 Diagram showing where the vortex shedding is suppressed in the (e/D, iirC)-plane. Open symbols: vortex shedding is suppressed. Filled symbols: vortex shedding exists, o, A , experiments of Sumer et al. (1991). (o from flow visualization, A from lift-force traces); a, Bearman and Zdravkovich (1978); \/, Grass et al. (1984).

104

Chapter S: Flow around a cylinder in oscillatory

flows

T h e frequency of vortex shedding can be defined by an average frequency based on the number of the short-duration peaks in the lift force over a certain period, as sketched in Fig. 3.26. T h e figure depicts t h e Strouhal number, based on this frequency and the maximum flow velocity S* = JT

(3-15)

as a function of the gap ratio. T h e shedding frequency actually varies over the cycle. T h e /„-value used in the definition of St in the preceding equation is averaged over a sufficiently long period of time. Fig. 3.27 presents the same d a t a in the normalized form St/Stg where Sto is the value of St attained for large values of e/D. Also plotted in Fig. 3.27 are the results of two studies conducted in steady currents, namely Grass et al. (1984) and Raven et al. (1985). T h e details regarding these two latter studies have already been mentioned in the previous chapter (see Fig. 1.23 and t h e related text). From Figs. 3.26 and 3.27 t h e following conclusions can be drawn. 1) For a given e/D, St increases (albeit slightly) with decreasing KC (Fig. 3.26). 2) T h e measurements collapse remarkably well on a single curve when plotted in the normalized form S i / S i o v e r s u s e/D (Fig. 3.27), where the influence of the close proximity of the wall on St can be seen even more clearly. 3) It is apparent t h a t St increases as t h e gap ratio decreases. T h e increase in St frequency can be considerable (by as much as 50%) when the cylinder is placed very near the wall (e/D = 0.1 — 0.2). This is because t h e presence of the wall causes the wall-side vortex to be formed closer to t h e free-stream-side vortex. As a result of this, the two vortices interact at a faster rate, leading to a higher St frequency. Finally, Sumer et al.'s (1991) work indicates t h a t there is almost no noticeable difference between the shedding frequency obtained in their smooth-cylinder experiments and t h a t obtained in their supplementary experiments with a rough cylinder (the cylinder roughness in the latter experiments is about k3/D = 10~ 2 ).

3.5 Correlation length It has been seen that vortex shedding around a cylinder occurs in cells along the length of the cylinder (Section 1.2.2), and t h a t the spanwise correlation coefficient is one quantity which gives information about t h e length of these cells. T h e studies concerning the effect of Re number, the effect of cylinder vibration, and the effect of turbulence in the incoming flow on correlation in steady currents have been reviewed in Section 1.2.2. In the present section, we will focus on the correlation measurements m a d e for cylinders exposed to oscillatory flows.

Correlation

0.4

105

bo •Da

o

2 9

A

9

0.2 4 Vortex i shedding

0

J

0

1

L

2

e/D Figure 3.26 Strouhal number versus gap ratio, o, KC = 20; A , KC = 30; a, KC = 55; V . A'C = 65. Sumer et al. (1991).

These measurements have been m a d e by Obasaju, Bearman and G r a h a m (1988), Kozakiewicz, Sumer and Freds0e (1992) and Sumer, Freds0e and Jensen (1994). Obasaju et al.'s (1988) study has clearly demonstrated t h a t the correlation is strongly dependent on the Keulegan-Carpenter number. Fig. 3.28 depicts their results, 2 being the spanwise separation (see Eq. 1.10). In t h e study of Obasaju et al., the correlation measurements were m a d e by measuring the pressure differential, i.e. the difference between the pressures on the diametrically opposite points at the top and b o t t o m of the cylinder. Fig. 3.28 indicates t h a t the correlation coefficient takes very large values when KC is small, while it takes t h e lowest value when KC is at about 22. Obasaju et al. (1988) give a detailed accoi'nt of the behaviour of the correlation coefficient as a function of the KC number. They link the low correlation measured at KC = 22 to the fact t h a t KC = 22 lies at the boundary between t h e two A'C-regimes, 15 < KC < 24 and 24 < KC < 30, while they argue t h a t the correlation is measured to be high at KC = 10 because KC = 10 lies in the center of the A'C-regime 7 < KC < 15 (see also Bearman, 1985). Fig. 3.29 illustrates t h e time evolution of the correlation coefficient for a given value of the spanwise separation distance, namely z/D = 1.8, as the flow

106

Chapter S: Flow around a cylinder in oscillatory

flows

St St,

Vortex shedding

e/D Figure 3.27 Normalized Strouhal number as function of gap ratio, o, KC = 20; A , KC = 30; •, KC = 55; V , KC = 65; x , steady current (Raven et al., 1985); - -, steady current (Grass et al., 1984). Sumer et al. (1991).

progresses. Here KC = 65, and the figure is taken from Kozakiewicz et al.'s (1992) study where the cylinder was placed at a distance from a plane wall with the gapratio e/D = 1.5, sufficiently away from t h e wall so t h a t the wall effects could be considered insignificant. T h e correlation coefficient is calculated from the signals received from the pressure transducers mounted along the length of the cylinder using t h e following equations, Eqs. 3.16 and 3.20):

R(z,

ujt)

P'(C, o r t M C + z, wQ 2

b' (C, ^)] 1 / 2 b' 2 (C + ^ ^ ) ] 1 / 2

(3.16)

in which ( is t h e spanwise distance, z is t h e spanwise separation between two pressure transducers, and p' is t h e fluctuation in pressure defined by p'=p-p the pressure transducers being at the free-stream-side of the cylinder. T h e overbar in t h e preceding equations denotes ensemble averaging:

(3.17)

Correlation

107

I

R

\

^ K C = 10

1.0

N^ - * ^ C 2 T ~

"--«--•

•

°

0

0.8

*

**""

. 18

0.6 0.4 -~->

•

~~~~-t**~^~

• +

26 -34

a - ~ ——JVZ^ " 4 2

0.2 i

l

1

— 1 — e ^ H S -22 8 z/D

Figure 3.28 Average values of correlation coefficients versus spanwise separation. (a) V , KC = 10; o, 18; *, 18; D, 22; A , 26; + , 34; ., 42. Note /?(= Re/KC) = 683 except for the case denoted by * where fi = 1597. Obasaju et al. (1988).

M

1

P=I?£PK>

w

(* + 0'-i):r)]

(3.18)

J'=l

M

p'2 = i E M c ^ ( ' + (j-i)T)]} 2

(3.19)

i=i M

p'(C, U*)P'(C + *, ci) = — J^p'lC,

u{t+(j-l)T\p'[C+z,

u,(t+(;-l)T] (3.20)

in which T is the period of the oscillatory flow, and M is t h e total number of flow cycles sampled. Fig. 3.29 shows t h a t the correlation coefficient increases towards the end of every half period, and attains its m a x i m u m at the phase tot = 165°, about 15° before the outer flow reverses. This phase value corresponds to the instant where the flow at the measurement points comes to a standstill, as can be traced from the pressure traces given in Kozakiewicz et al. (1992). As the flow progresses from this point onwards, however, the correlation gradually decreases and assumes its

108

Chapter S: Flow around a cylinder in oscillatory

R(»t)z

=

flows

!. 8 D

Figure 3.29 Correlation coefficient as a function of phase ujt. KC = 65, Re — 6.8 X 10 4 , e/D = 1.5 (sufficiently large for the wall effects to be considered insignificant), z/D = 1.8. Kozakiewicz et al. (1992).

minimum value for some period of time. T h e n it increases again towards the end of t h e next half period. Fig. 3.30 shows three video sequences at the phase values u>t = 113°, 165° and 180°. T h e flow picture in Fig. 3.30b shows t h a t the shear layer marked by the hydrogen bubble has rolled up into its vortex (A in Fig. 3.30b) and is standing motionless. As time progresses from this point onwards, however, this vortex begins to move in the reverse direction and is washed over the cylinder as a coherent entity along t h e length of the cylinder (Fig. 3.30c). Now, comparison of Fig. 3.30a with Fig. 3.30b indicates t h a t while spanwise cell structures can easily be identified in the former (ut = 113°), no such structure is apparent in Fig. 3.30b (u>t = 165°), meaning t h a t t h e spanwise correlation should be distinctly larger in the latter t h a n in the former case. T h e same is also t r u e for cot = 180° where, again, large correlations should be expected. This is indeed the case found in the preceding in relation to Fig. 3.29.

Effect of w a l l p r o x i m i t y o n c o r r e l a t i o n Kozakiewicz et al.'s (1992) study covers also the near-wall cylinder case. Fig. 3.31 shows t h e correlation coefficients for four different test d a t a with e/D = 2.3, 1.5, 0.1 and 0 where e is the gap between the wall and t h e cylinder.

Figure 3.30 Hydrogen-bubble flow visualization sequence of pictures showing the time d cell structures for a stationary cylinder. D = 2 cm, KC = 40, Re = 2 (1992). The cylinder is located well away from a wall, namely the gap-t therefore, the effect of wall proximity could be considered insignificant.

110

Chapter 8: Flow around a cylinder in oscillatory

flows

1

(c)

KC = 65

aDw

R

0.2

e WW/////// \+ svg x+ "*"*" 1

-*-*-

f £l

1 8

z/D

Figure 3.31 Period-averaged correlation coefficient. Wall proximity effect regarding the pressure fluctuations. See Fig. 3.32 for the wall proximity effect regarding the correlation of the lift force. Kozakiewicz et al. (1992).

Correlation

111

T h e correlation coefficients presented in Fig. 3.31 are the period-averaged correlation coefficient, which is defined by 1 /27r R(z) = — / R(z, 2TT J0

ut) d{ojt)

(3.21)

T h e general trend in Fig. 3.31 is t h a t the correlation coefficient decreases with decreasing gap ratio. However, caution must be exercised in interpreting the results in the figure. While R for e/D = 2.3 and 1.5 can be regarded as the correlation coefficient also for t h e lift force on the cylinder (since the fluctuations p' for which R is calculated are caused by the vortex shedding), this is not the case for e/D = 0.1 and 0. First of all, for e/D = 0, t h e vortex shedding is totally absent (Fig. 1.21), and t h e fluctuations in the measured pressure, p', in this case degenerate from those induced by the highly organized vortex-shedding phenomenon (e/D = 2.3 and 1.5) to those due to disorganized turbulence. So, the correlation, R, for this case, namely e/D = 0, only give information about the length scale in the spanwise direction of this turbulence. For e/D = 0.1, on t h e other hand, the vortex shedding may be maintained particularly for small KC numbers (see Fig. 3.25). However, the lift in this case consists of two p a r t s , a low frequency portion which is caused by the close proximity of the wall and the superimposed high-frequency fluctuations which are caused by vortex shedding (Fig. 3.23c). As such, t h e correlation, R, calculated on the basis of fluctuations, p', which are associated with the vortex shedding only, cannot be regarded as the correlation coefficient also for the lift force for the case of e/D = 0.1. Regarding the correlation of the lift force itself, Kozakiewicz et al. (1992) did some indicative experiments for t h e wall-mounted cylinder situation with the pressure transducers positioned on the flow side of the cylinder. Clearly, with this arrangement t h e pressure time-series can be substituted in place of t h e lift force ones, as far as t h e correlation calculations are concerned. Regarding the lift force itself, t h e lift in this case (e/D = 0) is not caused by the pressure fluctuations (as opposed to what occurs in the case of a wall-free cylinder, Fig. 3.23a), but rather by t h e contraction of t h e streamlines near t h e flow side of t h e cylinder as well as by the movement of t h e lee-wake vortex over the cylinder, which results in the observed peak in the lift force prior to the flow reversal in each half-cycle of the motion (Fig. 3.23d). Hence, the correlation in connection with t h e lift force in this case cannot be calculated by Eq. 3.16 (which is based on the pressure fluctuations rather t h a n on t h e pressure itself); instead, t h e usual time-averaging should be employed, i.e. the correlation is calculated by Eq. 1.10. Fig. 3.32 presents the spanwise correlation coefficients obtained for the wall-mounted cylinder, where the results for e/D = 2.3 of Fig. 3.31 are replotted to facilitate comparison. T h e correlations in these diagrams are now all associated with the lift force; therefore comparison can be m a d e on t h e same basis. T h e figure indicates t h a t , as expected, t h e correlation increases tremendously as the gap ratio changes from 2.3 (the wall-free cylinder) to nil (the wall-mounted cylinder).

112

Chapter 3: Flow around a cylinder in oscillatory

R,RT A

D

K C = 6 /RT ^D: r-o-o-o-