FLUID VORTICES FLUID MECHANICS AND ITS APPLICAnONS Volume 30

Series Editor: R. MOREAU MADYLAM Ecole Nationale Superieure d'Hydraulique de Grenoble BOlte Postale 95 38402 Saint Martin d'Heres Cedex, France

Aims and Scope of the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modelling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advance• ment. Fluids have the ability to transport matter and its properties as well as transmit force, therefore fluid mechanics is a subject that is particulary open to cross fertilisation with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological ~nd ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

For a list of related mechanics titles, see final pages. Fluid Vortices

Edited by

SHELDON 1. GREEN Department ofMechanical Engineering, University ofBritish Colwnbia, Vancouver, Canada.

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. Library of Congress Cataloging-in-Publication Data

Fluid vart1ces I edited by Sheldan 1. Green. p. cm. -- (Fluid mechanics and its applicatians v. 30) Includes index. ISBN 978-94-010-4111-9 ISBN 978-94-011-0249-0 (eBook) DOI 10.1007/978-94-011-0249-0 1. Vartex-matian. 2. Turbulence. 3. Fluid mechanics. 1. Green. Sheldan 1. II. Series. aC159.F58 1995 ·S20.1'OS4--dc20 95-11

ISBN 978-94-010-4111-9

Printed on acid-free paper

AH Rights Reserved © 1995 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995 Softcover reprint ofthe hardcover lst edition 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means. electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Table of Contents

. Editor's Preface Xl . Colour Plates XVI

CHAPTER I INTRODUCTION TO VORTICITY . 1 S.1. GREEN 1.1 Vorticity and Vortices ...... 1 1.2 Vorticity Kinematics and Dynamics - Physical Principles . 9 1.3 The Vorticity Equation with Examples .... 18 1.4 Summary ...... 31 LA Vorticity in Orthogonal Curvilinear Coordinates 32

CHAPTER II MIXING LAYER VORTICES 35 M. R. LESIEUR 11.1 Introduction to Mixing Layers . . . 35 II.2 Methods for Large Eddy Simulations 40 11.3 Temporal Mixing-Layer Simulations 42 II.4 Spatially-Developing Simulations 50 II.5 Rotating Mixing Layers 53 11.6 Summary ...... 59

CHAPTER III VORTICITY IN JETS 65 F. F. GRINSTEIN, M. N. GLA USER, and W. K. GEORGE III.1 Introduction 65 III.2 Transitional Jet Flows ...... 67 IIL3 Entrainment and Three-Dimensionality in Free Jets . . 76 IlIA Statistical Methods Applied to Structure Determination 82 IlLS By Way of Conclusion - A Two-Ring Model for the Axisymmetric Jet . . . . . 88 vi TABLE OF CONTENTS CHAPTER IV VORTEX RINGS . 95 T. T. LIM and T. B. NICKELS IV.1 Introduction ...... 95 IV.2 Mathematical Background 97 IV.3 The Inviscid Vortex rung 100 IVA The Viscous Vortex rung 110 IV.5 Vortex rung Interactions 137 IV.6 Closing Remarks 147

CHAPTER V VORTEX DYNAMICS IN THE WAKE OF A CYLINDER . 155 c. H. K. WILLIAMSON V.1 Introduction ...... - 155 V.2 Overview of Vortex Shedding Regimes 160 V.3 Three-Dimensional Vortex Patterns in the Laminar Shedding Regime...... 169 V.4 Three-Dimensional Vortex Dynamics in the Wake Transition Regime ...... 183 V.5 Three-Dimensional Phenomena at Higher Reynolds Numbers 206 V.6 Wave Interactions in the Far Wake ...... 212 V.7 Three-Dimensional Effects in Other Wake Flows 218 V.8 Concluding Remarks ...... 221

CHAPTER VI TURBULENT WALL-LAYER VORTICES .. 235 C. R. SMITH and J. D. A. WALKER VI. 1 Introduction . . . . 235 VI.2 Background. . . . . 236 VI.3 Mean Profile Structure 240 VIA Coherent Structure 247 VI.5 Hairpin Vortices . . . 253 VI.6 Evolution of Hairpin Vortices in Shear 258 VI. 7 Low-Speed Streaks. 266 VI.8 Regeneration 270 VI.9 Summary ..... 283 FLUID VORTICES vii CHAPTER VII COMPRESSIBLE VORTICES 291 D. BERSHADER VII. 1 Further Remarks on the Basic Relations ...... 291 VII.2 Vortex Flows With Compressibility ...... 295 VII.3 Studies Of Two-Dimensional Compressible Transverse Vortices . 303

CHAPTER VIII VORTEX STABILITY . . 317 R. L. ASH and M. R. KHORRAMI VIlLI Basic Formulation · 319 VIII.2 Inviscid Analysis . . . . · 322 VIIL3 Viscous Analysis . . . . · 352 VIII A Experimental Verification · 365 VIIL5 Concluding Remarks · 366

CHAPTER IX BREAKDOWN OF SLENDER VORTICES . 373 W. ALTHAUS, CH. BRUCKER, and M. WEIMER IX.I Introduction · 373 IX.2 Basic Theoretical and Numerical Aspects . . . · 376 IX.3 Experimental Investigations...... · 385 IXA Comparison of Experimental and Numerical Results · 394 IX.5 Criteria For Vortex Breakdown · 417 IX.6 Summary. · 419

CHAPTER X WING TIP VORTICES 427 S.1. GREEN X.l The Origin of Tip Vortices · 427 X.2 The Technological Relevance of Tip Vortices · 430 X.3 Lifting Line Theory ...... · 435 X.4 Tip Vortices - Our Knowledge to Date · 444 X.5 Other Topics in Tip Vortices. . . . . · 460 viii 'I:ABLE OF CONTENTS CHAPTER XI VORTICES IN AERO-PROPULSION SYSTEMS . . 471 1. A. WAITZ, E. M. GREITZER, and C. S. TAN XI. 1 Introduction ...... 471 XI.2 Secondary Flow ...... · 472 XI.3 'Vortex Line Stretching, Intensification of Vorticity, . and Vortex Formation ...... · 477 XI.4 Tip Clearance Flows; Clearance Vortices · 490 XI.5 Vortex-Enhanced Mixing . . . . . · 502 XI.6 Shock-Enhanced Mixing . . . . . · 520 XI. 7 Summary and Concluding Remarks · 527

CHAPTER XII VORTEX-STRUCTURE INTERACTION . 533 R. D. BLEVINS XII. 1 Vorticity Generation; the Vortex Street ...... · 533 XII. 2 Periodic Vortex Shedding from Structures ...... · 536 XII.3 Influence of Structural Motion and Sound on Vortex Shedding · 542 XII. 4 Vortex-Induced Vibration ...... · 545 XII. 5 Wake Oscillator, Discrete Vortex, and Numerical Models . · 552 XII. 6 Vortex-Induced Sound . · 557 XII. 7 Bridges and Buildings · 563

CHAPTER XIII MATHEMATICAL MODELLING OF SOLITARY OCEANOGRAPHIC VORTICES 575 G. E. SWATERS XIII. 1 Introduction...... 575 XIII.2 Some Simple Models ...... 578 XIII.3 Variational Principles, Stability and Andrews' Theorem. . 592 XIII.4 Modulation Theory . 598 XIII.5 Concluding Remarks ...... 611 FLUID VORTICES ix CHAPTER XIV ATMOSPHERIC VORTICES . 617 R. A. PIELKE, J. L. EASTMAN, L. D. GRASSO, J. B. KNOWLES, M. E. NICHOLLS, R. L. WALKO, and X. ZENG XIV.1 Introduction...... 617 XIV.2 Derivation of Equations; Definitions . 617 XIV.3 Three-Dimensional Atmospheric Vortices - Atmospheric Turbulent Vortices . . 619 XIVA Solenoidally-Generated Vortices . . . . 620 XIV.5 Small-Scale Vortices Generated by Vertical Shear of the Horizontal Wind . 625 XIV.6 Synoptic-Scale Vortices ...... 625 XIV.7 Cumulonimbus-Generated Mesoscale Vortices . 628 XIV.8 Mesoscale Vortices Generated by Horizontal Shear of the Horizontal Wind . 629 XIV.9 Tornadoes. . . . . 629 XIV.I0 Tropical Cyclones . 636 XIV.ll Conclusions . . 643

CHAPTER XV THREE-DIMENSION AL VORTEX DYNAMICS SIMULATIONS . 651 E. MEIBURG XV.1 Introduction ...... · 651 XV.2 From Physical Principles to Computational Algorithms · 653 XV.3 Summary and Outlook ...... · 677 CHAPTER XVI VORTICITY INTERACTIONS WITH A FREE SURFACE . 687 E. P. ROOD XVL1 Introduction...... 687 XVL2 Stress Conditions . 693 XVL3 Free-Surface Deformation . 697 XVL4 Free-Surface Vorticity . 701 XVL5 Vorticity Flux ...... 715 XVL6 Chapter Summary . . . . 726 XVLA Derivation of Vw . n in Generalized Curvilinear Coordinates. . 727 XVLB Flux of Surface-Parallel Vorticity for Surfaces Normal to () and to z . . . 728 XVLC Flux of Surface-Normal Vorticity for Surfaces Normal to () and to z . . . 728 x TABLE OF CONTENTS CHAPTER XVII VORTEX CAVITATION. · 731 R. E. A. ARNDT XVII. 1 Introduction . . . . . · 731 XVII. 2 Occurrence of Cavitation 734 XVII.3 Basic Information . . . · 738 XVII. 4 Forms of Vortex Cavitation . · 749 XVII. 5 Cavitation in Trailing Vortices 750 XVII. 6 Hub Vortex Cavitation. 764 XVII. 7 Vortex Models 770 XVII. 8 Summary 773

CHAPTER XVIII BUBBLE INTERACTIONS WITH VORTICES · 783 G. L. CHAHINE XVIII. 1 Introduction . . . · 783 XVIII.2 Order of Magnitude Considerations · 786 XVIII. 3 Bubble Capture by a Vortex . . . · 788 XVIII A Numerical Study ...... · 794 XVIII.5 Numerical Results and Discussion · 800 XVIII.6 Validation Study: Bubble Interaction with a Vortex Ring · 813 XVIII. 7 Other Relevant Studies ...... · 820 XVIII.8 Full Viscous Interaction Between a Cylindrical Bubble and a Line Vortex · 820 XVIII.9 Conclusions ...... · 825

CHAPTER XIX PARTICLE INTERACTIONS WITH VORTICES · 829 C. T. CROWE, T. R. TROUTT, and J. N. CHUNG XIX.1 Fundamental Concepts . · 829 XIX.2 Quantitative Predictions · 834 XIX.3 Experimental Studies . · 844 XIXA Particle-Fluid Coupling · 848 XIX.5 Future Directions · 858

Index . · 863 Editor's Preface

Welcome to Fluid Vortices. I shall devote this preface to an explanation of why the concept of a book on Fluid Vortices is a worthy one, to elucidation of mechanisms that have been employed to ensure its readability, and to a much deserved "thank you" to those who contributed to this book. The vorticity, w, defined as the curl ofthe velocity, is a derived quantity. Unlike the flow velocity, which may be readily measured using hot wire anemometers, laser doppler velocimeters, particle image velocimeters, or by numerous other means, the vorticity field of a flow is extremely difficult to measure (because W is related to differences of velocity gradients, very accurate measurements of u at closely spaced locations must be made before w can be determined with acceptable accuracy). Why, then, do we study the vorticity, and more particularly, vortices? There is not one single reason why we study vorticity, but rather several. One excellent reason is that vortices are highly conspicuous structures in many flows. I certainly remember how, as a child, I was fascinated by the small vortices cre• ated by the stroke of a canoe paddle through the water. A cursory glance at most satellite weather maps reveals a series of clockwise and counterclockwise rotating weather systems, and on a smaller scale, tornadoes and waterspouts are intense at• mospheric vortices (Chapter XIV). The whirlwinds of dust and leaves that appear on windy days near building entrances, the beautifully-visualized vortices apparent in the diesel fuel films near marine refuelling stations, and the propagating smoke rings (Chapter IV) blown by an expert, are further examples of fluid vortices. Even Norman Maclean's book Young Men and Fire includes a fascinating account of forest fire "blow-ups" caused by vortices generated in the separated flow off a hillside. Fluid vortices are also central features of many industrial flows: the centrifugal (or 'cyclone') separator used in chemical engineering processes, the tip vortices gen• erated by wings (made visible as condensation trails), and the large unsteady side forces caused by wind action on chimneys (Chapter XII) are just a few examples. There is even evidence that the formation of vortical flows in the circulation system contributes to a harmful response (hyperplasia) by the arterial walls - an observation that lends new meaning to Kiichemann's description of vortices as 'the muscles and sinews' of fluid motion. A second salient reason for our interest in vorticity is it has certain simple properties - like the fact that vortex lines, the analogues of streamlines, do not begin or end in a fluid (a property discussed in Chapter I) - which make vorticity conceptualization easier. Furthermore, it is possible to derive from the momentum

xi XlI EDITOR'S PREFACE equation a vorticity equation (refer to Chapter I) that has virtually the same information content as the momentum equation. Any solution to the continuity and vorticity equations (subject to appropriate boundary conditions) is therefore also a valid solution to the N avier-Stokes equations. There are certain classes of flows (particularly two and three-dimensional incompressible flows) for which it may be easier to solve the vorticity equation than the momentum equation. If the fluid velocity and pressure fields are subsequently required, they may be found by integration and solution to a simple Poisson equation, respectively. In summary, knowledge of vorticity is relevant because, like the streamfunction, it can be an aid to flow analysis. The circulation (integrated vorticity, see Chapter I) around a two-dimensional lifting body is proportional to the lift (Kutta-Joukowski Law). The endpoint of many aerodynamics calculations, therefore, is the circulation distribution on a body; the velocity field is of only secondary importance. A fluid in rotational (vortical) motion must generate a pressure gradient to drive the centripetal acceleration of fluid towards the center of rotation. This pressure gradient results in regions of intense low pressure in a flow. The low pressure may cause cavitation in liquid flows (Chapter XVII), and the strong pressure gradient may produce centrifuging in particle-laden liquid or gas flows (Chapter XIX). Hence, vortical motions playa key role in the dynamics of multiphase flows. A final property of vortices that makes them tremendously important features of any flow is their role in mixing enhancement. Imagine two layers of water dyed different colours and separated by a planar interface (see, for example, the dis• cussion of vortex rings in Chapter IV and of mixing layers in Chapter II). If a fluid vortex is introduced into the flow at the interface, the two fluids are rotated around one another, forming a spiral in which one colour of fluid forms an inter• leaved layer with the other. The thickness of bands of fluid near the center of the vortex is reduced to virtually the Kofmogorov scale. Chemical diffusion of species (i.e. true mixing) can now occur rapidly owing to the large surface area and large concentration gradient of the interface. In summary, five reasons why vortices are studied have been enumerated:

1. Vortices form prominent features of most flows.

2. Vortices have fairly simple characteristics, and essentially all the information of the velocity field is prescribed by specifying the vorticity field.

3. Circulation, the integrated vorticity, is a fundamental variable in Aerodynamics.

4. Vortex cores are regions of intense low pressure.

5. Vortices playa crucial role in accelerated mixing.

There are without doubt other reasons to study vorticity, but these five should have whetted your interest sufficiently ... FLUID VORTICES xiii

Because of the near ubiquity of vortices in Fluid Mechanics, the history of vortex study is long and peppered with the names of illustrious researchers. Kelvin and Helmholtz, early mathematicians in the field, developed the first simple equations related to vorticity. The "Fathers of Fluid Mecha.nics," Prandtl (wing vorticity), von Karma.n (the vortex street), and Taylor (Taylor vortices and Taylor columns) all devoted substantial attention to fluid vortices. Vortices playa central role in Kolmogorov's turbulent energy cascade, and they are the primary feature of the coherent structures in turbulent flow, which have been studied intensively for the past two decades. Thus, it is clear from an historical perspective that vortices have captivated fluids researchers. There is no sign that their appeal has lessened. The extent to which fluid vortices pervade the topic of Fluid Mechanics justifies (in my unbiased opinion!) the production of this eponymous book. How then to make this book as effective as possible? Fluid vortices represent a very broad topic. I certainly don't have the knowledge to write intelligently about all the topics covered in this text. If a sole author were to attempt to write this book, the result would likely be either as broad a coverage as available here, but lacking depth in the different areas, or a deep discussion of just a few topics with no great breadth of coverage. By having many authors contribute to this book, I believe the result is adequate both in depth and breadth of topic coverage. Electing to have a multi-authored volume has attendant problems. Some of those problems are purely technical - coordinating contributions from all over the globe was non-trivial - but fortunately FAXes, e-mail, and long distance calls ame• liorated these problems. Other problems were of greater concern. In particular, the threat to the unity and readability of the text arising from the varied backgrounds and writing styles of the contributing authors was a major concern. I attempted to overcome this threat by ensuring the authors communicated with one another. The introductory chapter was given to all authors before writing of the other chap• ters commenced. This chapter served, to a degree, as a style template. It was also used by the authors as a baseline of vorticity knowledge common to all chapters. First drafts of chapters were circulated to other authors in an effort to maintain a reasonably consistent style. The authors were also given instructions about the level of presentation of the material. I also took significant pains to proofread fi• nal drafts from authors. Finally, and perhaps most trivially, all contributions are printed with a uniform font and heading/reference schemes. I cannot pretend that these measures have made Fluid Vortices a seamless whole; to the extent that the text is more than simply a compilation of separate, related review articles, I will accept some credit. This book has an overall structure. The first nine chapters comprise fundamen• tal studies in single phase flow vortices. Together they form the basis for a one term graduate course. The following six chapters form a group for which the underlying theme is applications - industrial, environmental, and computational - involving vortices. The whole is completed by four chapters on multiphase flow vortices. xiv EDITOR'S PREFACE

The entire book can be covered in a full year graduate course; demonstrations and simple experiments involving vortices would form a desirable complement to this text. Fluid Vortices represents the cumulative efforts of numerous people. All the authors should be recognized for their yeoman's work. This tome could not have been brought to you without their dedicated efforts. All the contributions to this book were externally reviewed by as many as six qualified people. Some of the chapters went through as many as three drafts prior to text setting: an original draft, a second draft after receipt ofthe referees' reports, and a third draft after re-review of the second draft. Many scholars gave generously of their time to the review process. Some wished to remain anonymous. The following people reviewed one book chapter and con• sented to have their good efforts acknowledged:

Prof. Susan Allen (UBC) Dr. Wolfgang Althaus (Aachen) Prof. Hassan Aref (Illinois) Prof. Bob Ash (Old Dominion) Prof. John Blake (Wollongong) Prof. Howard Bluestein (Oklahoma) Prof. Robert Breidenthal (Washington) Dr. Susan Brown (UC London) Dr. S. S. Chen (Argonne) Dr. Yan Kuhn de Chizelle (Schlumberger) Prof. John Cimbala (Penn State) Prof. A. Crespo (Madrid) Prof. Werner Dahm (Michigan) Prof. Charles Dalton (Houston) Dr. David Dritschel (Cambridge) Prof. Helmut Eckelmann (Max Planck) Prof. Ian Gartshore (UBC) Prof. Mory Gharib (Caltech) Prof. Mike Graham (Imperial College) Prof. Doug Hart (MIT) Dr. Mehdi Khorrami (High Tech. Corp.) Prof. Steven Kline (Stanford) Dr. Harri Kytomaa (Failure Analysis) Prof. Juan Lasheras (UCSD) Prof. Tony Leonard (Caltech) Mr. Justin McCarthy (David Taylor) Dr. Ian McLean (BC Research) Prof. Dan Meiron (Caltech) Prof. M. Melander (Southern Methodist) Prof. Peter Monkewitz (UCLA) Prof. John Moore (VPI) Prof. Ronald Panton (Texas at Austin) Prof. Bill Phillips (Clarkson) Prof. Andrea Prosperetti (Johns Hopkins) Dr. Edwin Rood (ONR) Prof. Steven Rogak (UBC) Prof. Barry Ruddick (Dalhousie) Prof. Robert Scanlon (Johns Hopkins) Dr. Karim Shariff (NASA Ames) Prof. Ronald Smith (Yale) Prof. George Triantafyllou (Levich) Prof. Gretar Tryggvasan (Michigan) Prof. E. A. Weitendorf (Hamburg) Dr. Davinder Virk (Houston) Prof. Harry Yeh (Washington) Dr. Yung Yu (NASA Ames) FLUID VORTICES XV

The select group of reviewers listed below graciously agreed to review more than one chapter.

Prof. Fazle Hussain (Houston) Dr. James McWilliams (NCAR) Prof. Eckart Meiburg (USC) Prof. Geoff Parkinson (UBC) Prof. Kenneth Powell (Michigan) Prof. Don Rockwell (Lehigh) Prof. Turgut Sarpkaya (Naval PGS)

Not only were the reviewers' comments helpful to the authors, they were often very encouraging for me. One reviewer wrote "I think the chapter represents an excellent contribution to the planned book." A second reviewer commenting on a different chapter wrote "overall an excellent article on an important topic in the area of vortex flows." A third chapter was described by a reviewer as making "an excellent contribution to the book." A fourth reviewer kindly wrote: "Overall, the manuscript looks very good to me. If the other chapters are also of this quality, then the book will be a fine one." Comments such as these sustain an editor. A number of people helped me in other ways, for example by suggesting suitable authors and topics:

Prof. Allan Acosta (Caltech) Prof. Holt Ashley (Stanford) Prof. Chris Brennen (Caltech) Prof. Fred Browand (USC) Prof. Frank Champagne (U. Arizona) Prof. Paul Dimotakis (Caltech) Prof. Ari G lezer (Georgia Tech) Prof. Joe Katz (Johns Hopkins) Prof. Tony Maxworthy (USC) Prof. Anatol Roshko (Caltech) Dr. Richard Rotunno (NCAR) Prof. Phillip Saffman (Caltech) Prof. Gary Schajer (UBC) Prof. Douw Steyn (UBC) Prof. Charles Williamson (Cornell)

A graduate student, Shizhong Duan, helped me with some of the text figures. Four undergraduate students, Curtis Robin, Adrian Post, Chloe Morrow, and Fran• cis Pun, helped with the difficult process of text setting. Professor Turgut Sarpkaya kindly organized the external reviewers for the two book chapters I contributed. Dr. Bernard Green assisted me with the editing process, and several people at Kluwer were instrumental in seeing the text through to completion. I am grateful to my colleagues, who were collegial and friendly with me. Finally, I would like to thank my family, who have been as supportive for me as the oil in a good journal bearing.

Sheldon Green Colour Plates FLUID VORTICES xvii

Figure 2.1.3 DNS of an incompressible two-dimensional temporal mixing layer, with a time evo• lution of the vorticity (left) and a passive scalar (right) (Comte 1989). xviii COLOUR PLATES

Figure 2.3.4 Top view of the low-pressure (top) and high-vorticity (bottom) fields in 3D LES of an incompressible temporal mixing layer subject to helical pairing. FLUID VORTICES xix

Figure 3.2.4 Instantaneous flow visualization for M = 0.57, D /00 = 140 in terms of isocontours of azimuthal vorticity and mixedness. xx COLOUR PLATES

Figure 3.3.1 Volume visualization of the vorticity magnitude for the square jet.

Figure 3.3.4 Axis switching of rectangular vortex rings. FLUID VORTICES xxi

Figure 4.5.2 Head-on collision of two vortex rings (made visible using colored dyes in w&• ter)(reproduced from Lim and Nickels (1992)). Note the instability and the formation of small rings. xxii COLOUR PLATES

(a) (b)

(c) (d)

Figure 4.5.5 A series of front-view photographs showing different stages of the collision of two vortex rings at an angle. Note that each resultant ring contains material from both the original rings. The viewing direction is shown in Figure 4.5.4 (reproduced with permission from Lim (1989». FLUID VORTICES xxiii

Figure 5.4.9 Colour stereographic view of streamwise vortex pairs. A hydrogen bubble technique, developed by Koenig (1993), has enabled stereoscopic flow visualization using a single photograph, with the use of standard red-green glasses. This picture, when viewed with such glasses, indicates three-dimensionally some of the features of turbulent vortex structure, including the streamwise vortex pairs of mode 'B' vortex shedding, as well as a 'vortex dislocation' forming in the right half of the picture. Flow is upwards, bubble wire is horizontal and placed at z/ D=2, Re=250. xxiv COLOUR PLATES

Figure 5.4.12 Colour visualization of a forced two-sided 'vortex dislocation.' In this photo, a two-sided 'vortex dislocation' has been induced at a local spanwise position in the wake by plac• ing a small 'ring' disturbance around the cylinder. The frequency of vortex shedding behind the ring is slightly lower than to either spanwise side, and thus a sequence of periodic large-scale dislo• cations ensue. In the picture, the flow is upwards past the horizontal cylinder (at the lower edge). &=140. The spanwise extent of the symmetric dislocation is far larger than the width of the ring disturbance wake itself (shown in yellow dye), in similarity with the growth of turbulent spots by 'spanwise contamination,' discussed by Gad-el-Hak et al. (1981). These three-dimensional struc• tures can grow to the order of 30 primary wavelengths in downstream extent, and are therefore very much larger than the small-scale wake instabilities (Williamson 1992a). FLUID VORTICES xxv

0 N ... + .., --0

0 +

-0

.. ~ '- -c? O~------O~------OL------OL------~- ... '" a/ 'rJ 1 N _ 0

Figure 5.4.13 A.-vortex signatures of two.!sided 'vortex dislocations.' (a) The A.-vortex structure is suggested by phase-locked velocity measurements in the wake centre-plane. The velocity data are taken at z/D=40, over a spanwise traverse z/D, and with normalized time representing downstream distance in the vertical direction. Levels of velocity u(t)/U are given by the colour code below (Williamson 1992a). (b) PlY velocity contour measurements of an induced two-sided dislocation from a cylinder with a local reduction of diameter, using a technique developed by Don Rockwell and co-workers. Regions of different colour for the velocity contours are used to highlight the symmetric two-sided nature of the structures. Horizontal distance is spanwise, vertical distance is downstream. (Towfighi and Rockwell 1993). xxvi COLOUR PLATES

Figure 5.4.15 Excellent experimental-computational agreement demonstrated using modern tech• niques. Wake vorticity is evaluated from experiment (using DPIV, or Digital Particle Image Velocimetry) and from computation (DNS, or Direct Numerical Simulation, in this case using an unstructured spectral element discretization). The comparison between these results yields an impressively close agreement between modern experimental and computational methods, for the two-dimensional wake of a cylinder at Re=100. These contour plots show the spanwise vorticity distribution at one time during the shedding cycle. Normalized vorticity levels are colour-coded from red (-2.5) to blue (+2.5), with a spacing of 0.1. In nondimensional units, the cylinder has a diameter of 1/2 and is centered at the origin. (DPIV: Gharib, Hammache, Maheo, and Dabri (Caltech); DNS: Henderson and Karniadakis (Princeton) - private communications). FLUID VORTICES xxvii

Figure 5.4.16 Period-doubling of three-dimensional structure, discovered from simulations. Spec• tral element computations of the transverse vorticity in the cylinder wake center-plane have indicated a structure that repeats itself every second period of Karman vortex shedding (a 1/2 subharmonic). The above contour plots of the vorticity Wy in the wake center-plane, each evaluated at one primary shedding period apart, demonstrate this period doubling of the three-dimensional vorticity structure. Re=250. The flow is from left to right (Tomboulides et al. 1992). xxviii COLOUR PLATES

1///////.-""".-___ •• ·· .b 1///////_____ " • • . d I lawIAW...... _ • • . I I/,@Z'~_, .• .. / / /

Figure 9.4.16 Contours of axial vorticity component in three axial sheets through bubble-type breakdown region. Top: numerical results from Visbal (1993a) of flow in a delta wing vortex breakdown. Bottom: experimental results from Briicker and Althaus (1992) of breakdown in a pipe. FLUID VORTICES xxix

a)

b)

c) Figure 9.4.17 a) Circumferential vorticity isosurface of a well developed bubble, b) Pressure iso• surfaces ofa well-developed bubble at the same time as (a) and c) Pressure isosurfaces ofa bubble at a later time, during transition to spiral breakdown (Althaus et al. 1994c). See text for details of colour coding. xxx COLOUR PLATES

Figure 11.4.10 Total pressure contours in the passage indicating association of tip leakage vortex with region oflowest total pressure :fluid (Crook et al. 1992). FLUID VORTICES xxxi

VI c:i

o c:i

Figure 11.6.3 Rayleigh scattering images of flow field development following M 1.146 shock passage over a cylinder of helium embedded in air (Budzinski 1992). xxxii COLOUR PLATES

Figure 13.1.1 Satellite views of the ocean near Vancouver Island, Canada. The photos are AVHRR images acquired by the NOAA9 satellite, and show the surface temperature of the ocean water in degrees Celsius. Lines of latitude and longitude appear on the images. The temperature corre• sponding to each colour is indicated at the upper right of each photograph. Note the well-defined oceanographic vortices visible at 50 0 N 129°W and 51°N 131°W. Photos courtesy of Gordon Sta• ples, Dept. of Oceanography, U.B.C.