egbers c., pfister g. (eds.) physics of rotating fluids

Taylor, a powerful combination of theory andexperiment was brought to bear on the stability of flow between rotating cylin-ders, now referred to as Taylor–Couette flow.. The first part of t

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Christoph Egbers Gerd Pfister (Eds.)

Physics of Rotating Fluids

Selected Topics of the 11th International Couette–Taylor Workshop

Held at Bremen, Germany, 20-23 July 1999

1 3

Christoph Egbers

Lehrstuhl Aerodynamik und Str¨omungslehre

Fakult¨at Maschinenbau, Elektrotechnik

Cover picture: Plots of the velocity vectors of the spiral TG vortex flow, see K.Nakabayashi,

W Sha, Spiral and wavy vortices in th e sph erical Couette Flow, this issue.

Library of Congress Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Physics of rotating fluids : selected topics of the 11th International

Couette Taylor Workshop , held at Bremen, Germany, 20 - 23 July 1999 /

Christop h Egbers ; Gerd Pfister (ed.) - Berlin ; Heidelberg ; New

York ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singap ore ;

Tokyo : Springer, 2000

(Lecture notes in p hysics ; Vol 549)

(Physics and astronomy online library)

ISBN 3-540-67514-0

ISSN 0075-8450

ISBN 3-540-67514-0 Springer-Verlag Berlin Heidelberg New York

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“Lecture Notes in Physics”, having a strong publishing history in fundamentalphysics research, has devoted a special volume to recent developments in the field

of physics of rotating fluids and related topics The present volume will comprise

23 contributed papers on the different aspects of rotating fluids, i.e Taylor–Couette flow, spherical Couette flow, plane Couette flow, as well as rotatingannulus flow

In the seminal paper by G.I Taylor, a powerful combination of theory andexperiment was brought to bear on the stability of flow between rotating cylin-ders, now referred to as Taylor–Couette flow The significance of his work lies inthe fact that here, for the first time, an experiment in fluid dynamics and thetheory, using the Navier–Stokes equations, could be compared and led to excel-lent agreement Since that time ideas associated with rotating flows have beenextended and have resulted in classic texts such as Greenspan’s “The theory ofrotating fluids”

In this present book we report on modern developments in the field wherenew mathematical ideas have been applied to experimental observations on avariety of related flow fields

The aim of this volume is to provide the reader with a comprehensive overview

of the current state of the art and possible future directions of the Taylor–Couettecommunity and to include related topics and applications

The first part of this volume is devoted to several new results in the classicalTaylor–Couette problem covering diverse theoretical, experimental and numeri-cal works on bifurcation theory, the influence of boundary conditions, counter-rotating flows, spiral vortices, time-periodic flows, low dimensional dynamics, ax-ial effects, secondary bifurcations, spatiotemporal intermittency, Taylor–Couetteflows with axial and radial flow, Taylor vortices at different geometries and trans-port phenomena in magnetic fluids

The second part of this volume focuses on spherical Couette flows, includingisothermal flows, vortical structures, spiral and wavy vortices, the influence ofthroughflow, thermal convective motions, intermittency at the onset of convec-tion, as well as magneto-hydrodynamics in spherical shells

Further parts are devoted to Goertler vortices and flows along curved faces, rotating annulus flows, as well as superfluid Couette flows, tertiary andquarternary solutions for plane Couette flows with thermal stratification androtating disk flows

sur-We hope that the readers will find this volume useful, giving an overview ofthe latest experimental and theoretical studies on the physics of rotating fluids.

It is a pleasure for us to thank all those who contributed to the conference

“11th International Couette–Taylor Workshop” and, by the same token, to thisvolume We would like to thank the Dipl Phys Oliver Meincke, Markus Junk,Arne Schulz and Jan Abshagen for their invaluable and indispensable help inediting this book

Last, but not least, we are grateful to Dr Christian Caron for offering topublish this volume in the Springer Series “Lecture Notes in Physics” and forthe patient assistance of Mrs Brigitte Reichel-Mayer

Bremen, Kiel

Gerd Pfister

Dept of Mathematics and Statistics

The University of Newcastle Upon

Pascal Chossat

Universit´e de NiceSophia AntipolisI.N.L.N

1361, route des lucioles

06560 Sophia AntipoliFrance

USA

Antonio Delgado

TU M¨unchenLehrstuhl f¨ur Fluidmechanik undProzessautomation

Weihenstephaner Steig 23

85350 FreisingGermanydelgado@lfp.blm.tu-muenchen.de

Christoph Egbers

ZARMUniversit¨at Bremen

Am Fallturm

28359 BremenGermanyegbers@zarm.uni-bremen.de

Afshin Goharzadeh

Universit´e du Havre

Laboratoire de M´ecanique

Groupe d’Energ´etique et M´ecanique

25, rue Philippe Lebon, B.P 540

630090 NovosibirskRussia

laure@inln.cnrs.fr

Ming Liu

ZARMUniversit¨at Bremen

Am Fallturm

28359 BremenGermanyming.liu@promis.com

Manfred L¨ucke

Institut f¨ur Theoretische PhysikUniversit¨at des Saarlandes

66041 Saarbr¨uckenGermany

luecke@lusi.uni-sb.de

Richard M Lueptow

Northwestern UniversityDept of Mechanical Engineering

2145 Sheridan RoadEvanston, IL 60208-3111USA

r-lueptow@nwu.edu

Francesc Marqu`es

Universitat Polit`ecnica de CatalunyaDepartamenta de F´isica AplicadaJordi Girona Salgado s/nM`odul B4 Campus Nord

08034 Barcelona, SpainSpain

marques@chandra.upc.es

Numerical Analysis Group

Wolfson Building, Parks Road

Groupe d’Energ´etique et M´ecanique

25, rue Philippe Lebon, B.P 540

76058 Le Havre Cedex

France

mutabazi@univ-lehavre.fr

Tom Mullin

Department of Physics and Astronomy

The University of Manchester

Christiane Normand

C.E.A/Saclay,Service de Physique Th´eorique

91191 Gif-sur-Yvette CedexFrance

normand@spht.saclay.cea.fr

Stefan Odenbach

ZARMUniversit¨at Bremen

Am Fallturm

28359 BremenGermanyodenbach@zarm.uni-bremen.de

Gerd Pfister

Universit¨at KielInstitut f¨ur Experimentelle undAngewandte Physik

Olshausenstrasse 40

24098 KielGermanypfister@ang-physik.uni-kiel.de

Frank Pohl

MPI f¨ur PlasmaphysikEURATOM-Association

85748 GarchingGermany

Doug Satchwell

Department of Physics and AstronomyThe University of Manchester

Manchester M13 9PLUnited Kingdomsatch@reynolds.ph.man.ac.uk

Arne Schulz

Universit¨at KielInstitut f¨ur Experimentelleund Angewandte PhysikOlshausenstrasse 40

24098 Kiel

arne@ang-physik.uni-kiel.de

Nicoleta Dana Scurtu

Zentrum f¨ur Technomathematik

Yorinobu Toya

Nagano National College ofTechnology

Department of MechanicalEngineering

716 TokumaNagano, 381-8550Japan

toya@me.nagano-nct.ac.jp

Manfred Wimmer

Universit¨at KarlsruheFachgebiet Str¨omungsmaschinenKaiserstr 12

76128 KarlsruheGermanymanfred.wimmer@mach

uni-karlsruhe.de

Part I Taylor–Couette flow

Pitchfork bifurcations in small aspect ratio

Taylor–Couette flow

Tom Mullin, Doug Satchwell, Yorinobu Toya 3

1 Introduction 3

2A numerical bifurcation method 7

2.1 Governing equations 7

2.2 The finite element technique 9

2.3 Spatial discretisation and symmetry 11

2.4 Stability 13

2.5 Bifurcation points and extended systems 15

3 Results 16

3.1 Experimental apparatus 16

3.2Numerical and experimental bifurcation set 17

4 Discussion 18

References 19

Taylor–Couette system with asymmetric boundary conditions Oliver Meincke, Christoph Egbers, Nicoleta Scurtu, Eberhard B¨ansch 22

1 Introduction 2 2 2Experimental setup 2 3 3 Measurement techniques 2 3 3.1 PIV 2 3 3.2 LDV 25 4 Numerical method 2 6 5 Results 2 7 5.1 Symmetric system 2 7 5.2Asymmetric system 30

6 Conclusions 34

References 35

Bifurcation and structure of flow between counter-rotating cylinders Arne Schulz, Gerd Pfister 37

1 Introduction 37

2Experimental setup 37

3 Stability diagram 39

4 Primary instabilities 40

4.1 Transition to Taylor vortex flow (TVF) 40

4.2Transition to time-dependent flow states 42

5 Transition from Spirals to TVF 45

6 Wavy-vortex flow 46

7 Observation of propagating Taylor vortices 50

8 Comparison to theoretical investigations 51

9 Conclusion 53

References 53

Spiral vortices and Taylor vortices in the annulus between counter-rotating cylinders Christian Hoffmann, Manfred L¨ucke 55

1 Introduction 55

2System 56

3 Linear stability analysis of CCF 57

4 Bifurcation properties of Taylor vortex and spiral flow 58

5 Structure of Taylor vortex and spiral flow 64

6 Summary 64

References 66

Stability of time-periodic flows in a Taylor–Couette geometry Christiane Normand 67

1 Introduction 67

2Modulated base flow 71

2.1 Narrow gap approximation 73

3 Stability problem 74

3.1 Perturbative analysis 76

4 Nonlinear models 77

4.1 Amplitude equations 77

4.2Lorenz model 79

5 Conclusions 81

References 82

Low-dimensional dynamics of axisymmetric modes in wavy Taylor vortex flow Jan Abshagen, Gerd Pfister 84

1 Introduction 84

2Experimental setup 86

3 An intermittency route to chaos 86

3.1 Onset of ‘symmetric’ chaos 87

3.2Type of intermittency 90

3.3 Observation of Shil’nikov attractor 92

3.4 Transition to Hopf regime 94

4 A T3-torus in spatial inhomogeneous flow 96

4.1 Axially localised Large-jet mode 96

4.2Onset of VLF mode and transition to chaos 98

5 Discussion 100

References 100

Spatiotemporal intermittency in Taylor–Dean and Couette–Taylor systems Innocent Mutabazi, Afshin Goharzadeh and Patrice Laure 102

1 Introduction 102

2Pomeau model of spatiotemporal intermittency 103

2.1 Analogy with the directed percolation 104

2.2 Ginzburg–Landau amplitude equation 106

3 STI in the Taylor–Dean system 107

3.1 Main results on critical properties 107

3.2STI in other extended systems 108

4 STI in the Couette–Taylor system 109

4.1 Experimental setup 109

4.2Results 111

4.3 Physical origin of turbulent bursts 112

4.4 Kinematics of turbulent spiral 113

4.5 Hayot–Pomeau model for spiral turbulence 115

5 Conclusion 116

6 Acknowledgments 116

References 116

Axial effects in the Taylor–Couette problem: Spiral–Couette and Spiral–Poiseuille flows ´ Alvaro Meseguer, Francesc Marqu`es 118

1 Introduction 118

2Spiral–Couette flow 119

2.1 Linear stability of the SCF 12 1 2.2 Computation of the neutral stability curves 12 2 2.3 Stability analysis for η = 0.5 12 2 2.4 Comparison with experimental results (η = 0.8) 12 7 3 Spiral–Poiseuille flow 130

3.1 Linear stability results (η = 0.5) 131

4 Conclusions 133

References 135

Stability and experimental velocity field in Taylor–Couette flow with an axial and radial flow Richard M Lueptow 137

1 Introduction 137

2Cylindrical Couette flow with an imposed axial flow 139

2.1 Stability 139

2.2 Velocity field 143

3 Cylindrical Couette flow with an imposed radial flow 148

4 Combined radial and axial flow 150

5 Summary 153

References 154

Transport phenomena in magnetic fluids in cylindrical geometry Stefan Odenbach 156

1 Introduction 156

1.1 Magnetic fluids 157

1.2Magnetic properties of ferrofluids 158

1.3 Viscous properties of ferrofluids 160

2Taylor vortex flow in magnetic fluids 163

2.1 Taylor vortex flow as a tool for magnetic fluid characterization 163

2.2 Changes of the flow profile in magnetic fields 167

3 Taylor vortex flow in magnetic fluids with radial heat gradient 169

4 Conclusion and outlook 169

References 170

Secondary bifurcations of stationary flows Rita Meyer-Spasche, John H Bolstad, Frank Pohl 171

1 Stationary Taylor-vortex flows 171

2Convection rolls with stress-free boundaries 172

2.1 Critical curves of the primary solution 174

2.2 Pure-mode solutions 175

3 Secondary bifurcations on pure mode solutions 177

3.1 The 2-roll,4-roll interaction in a model problem 177

3.2The perturbation approach 179

3.3 A Hopf curve 180

3.4 The 2-roll, 6-roll interaction in a model problem 181

3.5 Other interactions 183

4 Numerical investigations 184

4.1 The Rayleigh–B´enard code used 184

4.2Convection rolls with rigid boundaries on top and bottom 187

4.3 Secondary bifurcations in the Taylor problem revisited 191

References 193

Taylor vortices at different geometries Manfred Wimmer 194

1 Introduction 194

2Flow between cones with a constant width of the gap 195

2.1 Experimental set-up 195

2.2 Flow field and Taylor vortices 195

2.3 Influence of initial and boundary conditions 198

3 Combinations of circular and conical cylinders 2 00

3.1 Rotating cylinder in a cone 2 013.2Rotating cone in a cylinder 2 01

4 Flow between cones with different apex angles 2 03

5 Flow between rotating ellipsoids 2 065.1 Oblate rotating ellipsoids 2 095.2Prolate rotating ellipsoids 2 10

6 Conclusions 2 11References 2 12

Part II Spherical Couette flow

Isothermal spherical Couette flow

Markus Junk, Christoph Egbers 215

1 Introduction 2 152Summary of previous investigations 2 18

3 Experimental methods 2 2 03.1 Spherical Couette flow apparatus 2 2 03.2LDV measuring system and visualisation methods 2 2 2

4 Transitions 2 2 44.1 Small and medium gap instabilities 2 2 44.2Bifurcation behaviour 2 2 74.3 Wide gap instabilities 2 2 8

3 Structure and formation of the spiral TG vortices 2 36

4 Motion of the azimuthally travelling waves 2 41

5 Spectral analysis of velocity fluctuations 2 44

6 Relaminarization 2 47

7 Concluding remarks 2 54References 2 54

Spherical Couette flow with superimposed throughflow

Karl B¨uhler 256

1 Introduction 2 562Numerical simulations 2 60

3 Experiments 2 60

4 Conclusion 2 67References 2 67

Three-dimensional natural convection in a narrow spherical shell

Ming Liu, Christoph Egbers 269

1 Introduction 2 69 2Mathematical formulation 2 70 3 Results and discussion 2 73 3.1 Axisymmetric basic flow 2 73 3.2Three-dimensional convective motions 2 74 3.3 Transient evolution 2 87 4 Concluding remarks 2 91 References 2 92 Magnetohydrodynamic flows in spherical shells Rainer Hollerbach 295

1 Introduction 2 95 2The induction equation 2 96 3 Kinematic dynamo action 301

4 The Lorentz force 304

5 Magnetic Couette flow 306

References 314

Intermittency at onset of convection in a slowly rotating, self-gravitating spherical shell Pascal Chossat 317

1 Introduction 317

2Heteroclinic cycles in systems with O(3) symmetry and the spherical B´enard problem 318

3 Perturbation induced by a slow rotation of the domain 32 2 References 32 4 Part III Goertler vortices and curved surfaces Control of secondary instability of the crossflow and G¨ortler-like vortices (Success and problems) Viktor V Kozlov, Genrich R Grek 327

Part I Active control over secondary instability in a swept wing boundary layer 32 7 Part II Transition and control experiments in a boundary layer with G¨ortler-like vortices 336

PART III Influence of riblets on a boundary layer with G¨ortler-like vortices 346

References 349

Part IV Rotating annulus

Higher order dynamics of baroclinic waves

Bernd Sitte, Christoph Egbers 355

1 Introduction 355

2The rotating annulus experiment 357

3 Stability 359

4 Nonlinear dynamics 362

4.1 Measurement technique 362

4.2Flow characterization 364

4.3 Bifurcation scenario 371

4.4 Comparison to Taylor–Couette flow 374

5 Conclusions 374

References 375

Part V Plane Couette flow Superfluid Couette flow Carlo F Barenghi 379

1 Liquid helium 379

2Helium II and Landau’s two-fluid model 379

3 Vortex lines and the breakdown of Landau’s model 381

4 The generalized Landau equations 383

5 The basic state 386

6 Rotations of the inner cylinder: absolute zero 389

7 Rotations of the inner cylinder: finite temperatures 390

8 Rotations of the inner cylinder: nonlinear effects 394

9 Rotations of the outer cylinder 394

10 Co-rotations and counter-rotations of the cylinders 396

11 Finite aspect ratios and end effects 396

12Discussion and outlook 397

References 398

Tertiary and quaternary solutions for plane Couette flow with thermal stratification R.M Clever, Friedrich H Busse 399

1 Introduction 399

2Mathematical formulation of the problem 401

3 Steady three-dimensional wavy roll solutions in an air layer 404

4 Wavy roll solutions in dependence on the Grashof number 408

5 Transition to quaternary states of fluid flow 413

6 Concluding remarks 414

References 416

On the rotationally symmetric laminar flow

of Newtonian fluids induced by rotating disks

Antonio Delgado 417

1 Introduction 417

2Isotherm, steady flow of a Newtonian fluid 419

2.1 Governing equations 419

2.2 Von K´arm´an’s solution for a single rotating disk 42 0 2.3 Flow between co-rotating disks 42 2 3 Conclusions and future investigations 437

References 438

Pitchfork bifurcations in small aspect ratio

Taylor–Couette flow

Tom Mullin1, Doug Satchwell1, and Yorinobu Toya2

1 Department of Physics and Astronomy,

The University of Manchester,

Manchester M159PL, UK

2 Department of Mechanical Engineering, Nagano National College of Technology,Nagano,

381–8550, Japan

Abstract We present a discussion of steady bifurcation phenomena in Taylor–Couette

flow The emphasis is on the role of pitchfork bifurcations in mathematical models andtheir relevance to the physical problem The general features of such bifurcations arereviewed before we discuss the numerical and experimental techniques used to ex-plore their properties New results are then presented for a wide-gap small aspect ratioversion of Taylor–Couette flow We find good agreement between numerical and exper-imental results and show that the qualitative features of the bifurcation sequence arethe same as those found with other radius ratios

in the subject of hydrodynamic stability theory Taylor used the powerful bination of theory and experiment to test the viscous formulation of Rayleigh’sstability criterion for circulating flows He established the principle of exchange

com-of stability between two fluid states and obtained remarkable agreement betweentheory and experiment for stability limits His success spawned a new subjectarea and to this date there have been over one thousand papers written onthe subject A comprehensive listing of references on the topic can be found in[20,21]

The onset of cells in the Taylor–Couette problem is widely believed to be

an example of a simple planar pitchfork bifurcation While evidence suggeststhat this is a good description, the connection between mathematical modelsand observations was shown by Benjamin to be very subtle In particular, thesymmetry of the abstract model is one of translation of the cellular pattern andthis is not readily achievable in the physical system As a result the onset of cellsremains sharp but the second branch of the pitchfork is removed to Reynoldsnumbers far in excess of those required for the first appearance of cells We will

C Egbers and G Pfister (Eds.): LNP 549, pp 3–21, 2000.

c

Springer-Verlag Berlin Heidelberg 2000

discuss these issues below and also review other important simple geometricalsymmetry breaking bifurcations in the problem.

A pitchfork bifurcation is so named because of its shape i.e it resembles athree pronged hayfork The handle and central prong correspond to the trivialsolution or zero state and the outer prongs relate to the bifurcating branches Afamiliar physical example of this mathematical entity is the Euler strut where

an initially straight elastic beam is buckled by the action of a compressive load.The straight configuration is the trivial solution which loses stability to a pair ofbuckled states as the load is increased This phenomenon can readily be demon-strated by applying an end loading to a plastic ruler using your hands Theruler will bend up or down (say) above a critical compression Here we have anexample of a simple symmetry breaking pitchfork bifurcation where the sym-metry of the originally straight ruler is destroyed above a critical load It willsoon become apparent to anyone who tries this demonstration that the rulerwill prefer to buckle in a particular direction This is because the ruler and theapplication of the load is not symmetric In fact it can never be so even in a lab-oratory where a high precision version of the plastic ruler experiment could bemade This important aspect of the physical system can be modelled by includ-ing an imperfection term in the model equations as discussed by Golubitsky andSchaeffer[12] The result is that there is a smoothly evolving state together with

a disconnected branch which is terminated at its lower end by a saddle–nodebifurcation

We will now consider the Taylor–Couette problem in the context of pitchforkbifurcations In the Taylor–Couette geometry the region between the surfaces

of two concentric cylinders is filled with fluid We consider the case where flow

is driven by the inner cylinder which rotates with a constant angular velocity,while the outer cylinder is held stationary In the configuration of interest herethe flow is terminated with fixed horizontal plates which span the gap betweenthe cylinders at the ends of the fluid annulus A sketch of the Taylor–Couettegeometry is presented in Fig 1 The coordinates system is cylindrical polar

(r, θ, z) with the origin located along the central axis and midway between the

end boundaries

The Reynolds number for this system is :

where ν is the kinematic viscosity of the fluid.

Two further independent dimensionless parameters may be defined for the Taylor–

Couette geometry These are the ratio of the length l of the fluid annulus to the gap width d, known as the aspect ratio :

and the radius ratio of the cylinders :

η = r i

Fig 1 The Taylor–Couette geometry Fluid contained between the surfaces of two

concentric cylinders is driven by a rotating inner cylinder

The Reynolds number and aspect ratio are continuously variable parameters.The radius ratio, on the other hand, is not easily adjusted and is therefore fixedthroughout an experiment At small Re the flow is observed to be mainly feature-less along most of the cylindrical gap except for some three dimensional motion

at the ends In practice, the laboratory flow appears to provide a reasonable proximation to rotary Couette flow where the principal action is shear betweenthe rotating cylinders It is this flow that is often considered to be related tothe trivial state of the mathematical model where the cylinders are taken to beinfinitely long The connection is appealing since it appears reasonable that thedistant ends in a long apparatus will act as small perturbations This suggeststhat the pitchfork in the model will be disconnected by a small amount Henceone branch would be continuously connected and show a sharp change in gradi-ent close to the bifurcation point of the perfect system while the other branchwould be disconnected and have a lower limit defined by a saddle–node How-ever, experimental evidence gathered over the last twenty years suggests thatthis view is misleading

ap-It is an experimental fact that when Re is increased above a certain welldefined value then cellular motion sets in rather quickly This is found to bethe case even when the aspect ratio of the system is as small as four If weconsider the onset of cells as a bifurcation then we must ask if it can be described

as a simple disconnected pitchfork One fact which would test this idea is anobservation of the second branch of the pitchfork and a measure of its lowerlimit of stability Surprisingly, Benjamin appears to be the first person to haveattempted this and in doing so he discovered that the second branch existsbut it is far away from the first onset of cells He termed these new solutions

‘anomalous modes’ and they have been the subject of a great deal of subsequentstudy [3,10] Included in these investigations is direct numerical evidence [7]

of the connection between the ‘periodic’ model and experimental observationsusing the Schaeffer [19] homotopy parameter This clearly elucidates the originand role of anomalous modes

The range of Re between the saddle-node and that for the onset of cells can

be as much as an order of magnitude and appears to be independent of aspectratio This suggests that ‘end effects’ are dominant in the Taylor–Couette prob-lem in practice no matter what the aspect ratio is One half of a simple planarpitchfork appears to provide a good model of the onset of cells However, thesymmetry involved is one of translation [1] and since the physical system doesnot easily permit this action the other half of the pitchfork is far removed fromthe mathematical idealisation of the model This finding has important conse-quences for the onset of low–dimensional chaos and in particular codimension–2organising centres [16,17]

Pitchfork bifurcations which give rise to pairs of solutions that break the

mirror plane Z2 symmetry are now known to be important in the organisation

of the dynamics found at higher Re [17] These are found on nontrivial symmetriccellular flows where one half of the pattern grows at the expenses of the other as

Re is varied As in any physical system the effects of imperfections are presentbut, unlike the onset of cells, the disconnection is small and is generally of theorder of a few percent of the range of the control parameter

A feature of the Taylor–Couette experiment which can be readily observed isthat there is a large multiplicity in the steady solution set [4] This feature washighlighted in the time–dependent regime by Coles [11] and also commented

on in [5] for steady flows Thus if one wishes to explore important details ofthe bifurcation structure it can be difficult if the aspect ratio is large Since

‘end effects’ are important for all aspect ratios it seems appropriate to carryout such investigations at small or modest aspect ratios where the solution set ismanageable This strategy has been adopted in several investigations which haveprovided an exacting challenge for comparison between the results of numericalcalculation and experiment [9]

The simplest example of a symmetry breaking pitchfork bifurcation in the

Taylor–Couette problem is found when the aspect ratio is O(1) In this case it was

shown [3] that a two-cell state can bifurcate into a pair of single-cell anomalousmodes This is the only known example of a continuously connected anomalous

mode and has been the subject of a great deal of subsequent numerical andexperimental study, as discussed in [18,23] We will use this flow as our example

to highlight a numerical bifurcation approach used to explore these and relatedproblems Then, we will present some new experimental results on a wide gapversion of the problem which shows the robustness of the basic mechanisms

Before proceeding to discuss the equations of motion and a numerical bifurcationmethod used in their study we will first discuss those symmetries which areimportant in the physical system The Taylor–Couette problem is invariant underreflections about the midplane or rotations through the azimuthal angle These

symmetries are embodied in the symmetry groups Z2 and SO2, respectively,that map :

and

where φ is an arbitrary phase.

The numerical methods used to calculate the Taylor–Couette flow make full

use of these symmetries The SO2 symmetry is used implicitly since all

calcu-lations are performed over the two-dimensional vertical cross-section The Z2

symmetry, on the other hand, is used to reduce the computational effort byapproximately half using a half-grid discretisation

2.1 Governing equations

The velocity components (u ∗ , u ∗

θ , u ∗), where ∗ denotes dimensional quantities,are made non-dimensional by scaling with the inner cylinder velocity :

With this notation, the dimensionless Navier–Stokes equations for an pressible Newtonian fluid are :

Thus u r and u z are zero on the entire boundary, and u θ = 1 at the inner

cylinder and u θ = 0 at the outer cylinder At the ends of the annulus u θ has

the dependence F (r) shown in Fig 2 This is the smooth function originally suggested [3] as a model for the corner singularity at (r, z) = (0, ±1

2) wherethe rotating inner cylinder meets a stationary end-boundary The dissipationrate in the fluid resulting from the singularity would otherwise be infinite, and

therefore physically unrealistic The particular form for F (r) used in this case

is a quadratic fitted to u θ from r = 0 to r =  Cliffe and Spence[8] report their numerical results to be insensitive to the precise value of , and conclude that

any sufficiently small value is adequate

0 1 0

1

uθ

Fig 2 The continuous function F (r) used as a model for the corner singularities.

2.2 The finite element technique

The discrete two-dimensional version of the Navier–Stokes equations for theTaylor–Couette flow were calculated on a Silicon Graphics Power Challenge us-ing the numerical bifurcation package ENTWIFE This is the same techniqueoriginally used by Cliffe [9] to calculate the 4/6 cell exchange mechanism for a

Newtonian fluid at radius ratios η = 0.6 and η = 0.507 he found good

agree-ment with experiagree-mental results and also showed the importance of symmetrybreaking bifurcations We now present an overview of the techniques used tocalculate these bifurcations and details of the basic numerical method may befound in Cliffe and Spence [8] and Jepson and Spence [13]

The steady version of equations (9a) - (9d) is solved using a primitive variable

Galerkin formulation The pressure terms p are required to lie in the space L2(D), the space of functions that are square integrable over the domain D Similarly the velocity components u r , u θ and u z are each required to lie in the space W 1,2 (D), the space of functions whose generalised first derivatives lie in L2(D) W 1,2 (D)3

is then the space of three-dimensional vector valued functions with components

existing in W 1,2 (D) This space is a natural setting for the problem, since the

total rate of viscous dissipation by the fluid is incorporated in the square of the

norm of the vector (u r , u θ , u z)

On the boundary of D the elements of W 1,2 (D)3 must vanish, and the

sub-space with this property is written W 1,20 (D)3 We therefore seek an axisymmetric

solution U + ˆ U where U = (u r , u z , u θ , p) ∈ H ≡ W 1,20 (D)3× L2(D) and the

function ˆU = (0, ˆuθ , 0, 0) ∈ W 1,2 (D)3×L2(D) matches the boundary conditions

on the azimuthal component of velocity

The domain D is covered with a finite-element mesh, the length of the longest

edge of which is denoted h W h and M h are two finite-dimensional subspaces

such that W h ∈ W 1,2 (D)3 and M h ∈ L2(D) The continuous solution U ∈

finite-dimensional Hilbert space H h = W h,0 × M h , W h,0 ⊂ W 1,20 (D)3

The steady finite-dimensional weak form of the Navier–Stokes equations are

expressed in D as a nonlinear operator f in finite-dimensional Hilbert space :

a prediction at each step s i, and Newton’s method is used to locate the solutionaccurately The same parametrisation applies equally to the continuation of so-

lutions in Re, Γ or η In this way paths of limit points in parameter space may

be computed The continuation procedure also extends naturally to the putation of bifurcations using the extended systems to be introduced in section2.5

com-2.3 Spatial discretisation and symmetry

The finite-dimensional space W h,0is generated using the nine-node ric quadrilateral elements shown in Fig 3 Each element has three components

isoparamet-of velocity at each node, totalling 27 velocity degrees isoparamet-of freedom in all, with each

component of velocity being approximated by biquadratic polynomials M h isgenerated by piecewise linear interpolation on the same elements In addition,

three pressure degrees of freedom p, p x and p y are associated with the centralnode and the interpolation is, in general, discontinuous across element bound-aries

In order to compensate for the rapid variation in velocity experienced near

(r, z) = (0, ±1

2), the corner elements are refined, as shown in Fig 4 The mainelement is successively subdivided into smaller elements as the corner is ap-proached Cliffe [6] shows that the numerical results are insensitive to any num-ber of subdivisions greater than four Here the number of subdivisions used forthe calculations is five

The full domain D was discretised using a 24×40 mesh The mesh is rical about the line z = 0 and, with the the exception of the corners, is uniform

symmet-over the domain As Cliffe and Spence [8] point out, this is an important sideration if the symmetry of the physical problem is to be correctly modelled

con-It has already been noted that the Taylor–Couette geometry is Z2 symmetric,

and therefore the Navier–Stokes equations in D are invariant under the following

Cliffe and Spence [8] calculate the form for the symmetry operator S ∗, as

the discretised analogue of the continuous symmetry operator S, and obtain the

operator S ∗ partitions the space X into symmetric and antisymmetric subspaces

X s and X a in the discrete case :

Fig 3 A nine-node quadrilateral element.

Fig 4 The five-fold corner refinement.

The symmetry of the problem may be utilised in order to reduce the

com-putational effort Flows that are symmetric about z = 0 can be calculated more

efficiently using the half domain :

cal-reflection of the half-grid solution about z = 0 The result is a reduction in the

number of degrees of freedom from 19124 for the full grid to just 9562 for thehalf grid

2.4 Stability

The weak form of the Navier–Stokes equations in D may be expressed as a set

of differential equations in t, with the pressure and velocity degrees of freedom

as dependent variables They are written in the form :

M dx dt + f(x(t); Re, Γ, η) = 0 (23)

where M is the mass matrix.

If a small axisymmetric perturbation ξ is introduced then the behaviour is

Thus the linear stability of solutions x to (13) is dependent upon solutions

of the generalised eigenproblem :

where γ is the generalised eigenvalue and  the corresponding eigenvector.

If σ > 0 the perturbation ξ decays with time, otherwise if σ < 0 the turbation ξ grows For the solution to be stable all generalised eigenvalues γ of

per-(27) must have positive real part However, the Jacobian matrix can in practice

be very large and it is not efficient to calculate all the generalised eigenvalues.The sign of the determinant of the Jacobian matrix, though, may be calculatedwith very little additional computational cost A necessary but not sufficientcondition for stability, then, is that the sign of the determinant of the Jacobian

matrix f xis positive On the other hand, a solution is necessarily unstable if the

determinant of f xis negative

For symmetric solutions x ∈ X s , the Jacobian f x maps X s → X s and

X a → X a Therefore the Jacobian matrix f x evaluated at x ∈ X s may bewritten in the block diagonal form :

For the symmetric solution to be stable, it must be stable to both symmetric

and antisymmetric disturbances Thus all the generalised eigenvalues γ s

corre-sponding to eigenvectors  s ∈ X s must have positive real part, and therefore

the sign of the determinant f x smust be positive In addition, all the generalised

eigenvalues γ a corresponding to eigenvectors  a ∈ X a must have positive real

part, and therefore the determinant of f x a must also be positive

Thus for symmetric solutions the determinants of f x s and f x a are lated on the half grid at each step of the continuation procedure The sign ofthe determinant indicates the stability of the solution with respect to symmetric

calcu-and antisymmetric disturbances respectively For asymmetric solutions x ∈ X a,

however, the Jacobian f x does not map X s → X s nor X a → X a, since

eigen-vectors are not exclusively elements of X s or X a , and f xcannot be partitionedinto block diagonal form Thus the stability of these solutions cannot be obtained

by finding the determinant of f x s and f x a, since these matrices do not exist.The asymmetric solutions must therefore be computed on the full grid

For asymmetric solutions the Jacobian f x may be expressed in the Jordanform :

axis This is a useful construct since the stability of the solution is dependent

upon the eigenvalues of B alone, and it is therefore used to indicate the stability

of asymmetric solution branches

2.5 Bifurcation points and extended systems

The linear stability analysis (24) fails whenever σ = 0 This occurs, for example,

at a simple singular point x0 where the Jacobian matrix f x has a single real

eigenvalue γ0= σ0+ iω0 such that σ0 = ω0 = 0 At such a point there exists a

unique null eigenvector φ0 If x0∈ X s then the null eigenvector φ0is either an

element of X s or an element of X a Thus if the determinant of f x sis zero then

the bifurcation is symmetry-preserving, otherwise if the determinant of f x a is

zero then the bifurcation is symmetry-breaking.

In order to compute such bifurcation points it is necessary to use extendedversions of the equations Moore and Spence [15] show that a limit point may

be characterised as an isolated solution of the following extended system :

where y = (x, φ, Re) ∈ X × X × R and l ∈ X  (the dual of X) In order to

calculate a path of limit points in the two-dimensional space (Re, Γ ) for instance

the Keller arc-length continuation method (15) is applied to (31) with u = y,

where y = (x, φ, Re) ∈ X s × X a × R This is similar to equation (31) but

there is an essential difference since now x must belong to X s and φ must belong

to X a Thus the basic solution is symmetric but the eigenvector, and thus thebifurcating branch, is asymmetric Other symmetry-breaking points include the

C+and C−coalescence points and quartic bifurcation points which can occur atcertain singularities of the system (32) Extended systems for these singularitiesmay be found in Cliffe and Spence [8]

The numerical problem is such that all of the symmetry-breaking tions are perfect, and therefore on perfectly symmetric boundary conditionsthere is no possibility of distinguishing between asymmetric solution branches.One practical point here is that it is possible to step on to an asymmetric solu-tion branch by perturbing the pitchfork bifurcation so that all solutions becomeslightly asymmetric It is then possible to step along the connected solutionbranch through the point at which the pitchfork occurs in the symmetric case.This solution is used as a first approximation to the perfectly symmetric prob-lem, which converges to the true value when the boundary conditions are reset

bifurca-to their original configuration In this way the paths of fold bifurcations alongasymmetric solution branches may be computed

Since both M and f x are real matrices it follows that the eigenvalues γ0are

either real or exist as complex conjugate pairs If at a simple singular point x0

the real part σ0 is zero and the imaginary part ω0 is non-zero, then a purelyimaginary pair of eigenvalues and corresponding complex conjugate eigenvectors

must exist In this case x0 is a Hopf bifurcation point

At a Hopf bifurcation point γ = ±iω and equation (27) becomes :

We will not calculate Hopf bifurcations here but refer the reader to [18] where

a discussion of Hopf bifurcations in small aspect ratio Taylor–Couette flows isgiven

3 Results

We will now discuss the application of the numerical techniques discussed above

to the study of pitchfork bifurcations in a wide-gap Taylor–Couette system Ourresults are concerned with two-cell and single-cell flows and the exchange ofstability between these flows as the aspect ratio is varied We will first present abrief description of the experimental apparatus before discussing the numericaland experimental results

3.1 Experimental apparatus

The fluid was contained in the annular gap between two concentric cylinders The

outer cylinder was a precision bore glass tube with inner diameter 74.6±0.02mm The inner cylinder was machined stainless steel with a diameter 25.3 ± 0.02mm

so that the radius ratio was 0.339 It was located in bearings and driven round

by a stepping motor via a gear box and belt drive system The motor speed wascontrolled by an oscillator and its speed was monitored The ends of the annular

gap were defined by two stationary PTFE collars which bridged the annulargap The upper collar was attached to a pair of posts so that it could be movedaccurately up and down The aspect ratio was measured using a cathetometerwhich was also used to measure the flow structure.

The fluid used was a water glycerol mixture whose viscosity was measured

to be 5.69 cSt The cylinders were surrounded by a water bath whose

tempera-ture was controlled to 0.02 o C by fluid pumped through commercial temperature

controller The flow was visualised using Mearlmaid AA pearlessence and mination was provided by a plane of light from a slide projector The cellularstructure was then clearly visible and the respective heights of cells were used todistinguish between flow states Estimates of the symmetry breaking bifurcationpoints were obtained by measuring the saddle–node points where the asymmetricstates collapsed to the symmetric ones by reduction in Re All other bifurcationpoints correspond to catastrophic changes in the flow structure with change in

illu-Re and so reliable estimates could be obtained

3.2 Numerical and experimental bifurcation set

We show in Fig 5 the bifurcation set in the (Re,Γ ) plane for the one-cell,

two-cell interaction.The solid lines have been calculated using the methods cussed above and the crosses are the measured points In general, there is verygood agreement between the numerical and experimental results Some ‘typical’streamline plots for these flows are shown in Fig 6 where we show both symmet-ric and asymmetric flows As discussed in [3] we call the asymmetric states singlecell flows since this the form they would have in a model problem where the endboundary conditions match Couette flow In the laboratory and in the numericalcalculations with stationary end–conditions there are always weak recirculationspresent in the corners Since these will be present for all cellular flows on finitedomains we choose to define the flow states in this way

dis-We next show in Fig 7 a sequence of schematic bifurcation diagrams which

we will now use in our discussion of Fig 5 In Fig 5 AB is the locus of symmetrybreaking bifurcations from the two-cell state to a pair of single cell flows Thecorresponding bifurcation diagram is given in Fig 7(a) where we see a simplepitchfork A ‘typical’ pair of streamline plots for such states are presented inFig 6 (a) and (c) respectively As the aspect ratio is increased towards B there is

an interaction with a second pitchfork which restabilises the two-cell branch Thissecond pitchfork is shown in Fig 7(b) where it can be seen that the bifurcation

to the pair of one single-cell states has become quartic The path of secondpitchforks is denoted by BC in Fig 5

As the aspect ratio is increased further the interaction increases such thatthere is hysteresis in the development of the single-cell states as shown schemat-ically in Fig 7(c) The hysteresis is very small and is hardly detectable on thenumerical results on the scale used in Fig 5 The influence of imperfections inthe experiment may be clearly seen in Fig 5 for the hysteresis is larger than

in the numerics and all points lie below the calculated ones Nevertheless, the

Fig 5 Comparison between experimental and numerical results for the bifurcation set

for one and two steady cell flows ABC is a locus of symmetry breaking bifurcationpoints and BD is a path of saddle–nodes

qualitative nature of the events is clear and we attribute the quantitative ence to the sensitivity of this feature to experimental imperfections Yet furtherincrease in aspect ratio causes the disconnection of the single-cell states through

differ-a codiffer-alescence of the pitchforks between Figs 7 (c) differ-and (d) Thus differ-along BD inFig 5 the pair of single-cell states are disconnected and this is the locus of limitpoints for these states

4 Discussion

The bifurcation sequence described above is consistent with those previously cussed by [3,6,18] for other values of the radius ratio These new results thereforeconfirm that these qualitative features are robust when the radius ratio is varied

dis-It is known [16] that pitchfork symmetry breaking bifurcations form organisingcentres for complicated dynamical motion in the Taylor–Couette problem andhence the robustness of the underpinning steady solution structure gives hopethat features such as Silnikov dynamics may also be relevant over a wide param-eter range

1 T B Benjamin 1978a Bifurcation phenomena in steady flows of a viscous liquid.

Part 1 Theory Proc R Soc Lond A 359, 1–26.

2 T B Benjamin 1978b Bifurcation phenomena in steady flows of a viscous liquid.

Part 2 Experiments Proc R Soc Lond A 359, 27–43.

3 T B Benjamin and T Mullin 1981 Anomalous modes in the Taylor experiment

Proc R Soc Lond A 377, 221–249.

4 T B Benjamin and T Mullin 1982 Notes on the multiplicity of flows in the Taylor–Couette experiment

5 J E Burkhalter and E L Koschmieder 1973 Steady supercritical Taylor vortex

flow J Fluid Mech 58, 547–560.

6 K A Cliffe 1983 Numerical calculations of two–cell and single–cell Taylor flows J.

Fluid Mech 135, 219–233.

7 K A Cliffe and T Mullin 1985A numerical and experimental study of anomalous

modes in the Taylor experiment J Fluid Mech 153, 243–258.

8 K A Cliffe, and Spence A 1986 Numerical calculations of bifurcations in the finite

Taylor problem In Numerical Methods for Bifurcation Problems (ed T Kupper,

H.D Mittleman and H Weber), pp 129–144 Birkhauser:ISNM

9 K A Cliffe 1988 Primary–flow exchange process in the Taylor problem J Fluid

Mech 197, 57–79.

10 K A Cliffe, J J Kobine and T Mullin 1992 The role of anomalous modes in

Taylor–Couette flow Proc R Soc Lond A 439, 341–357.

(a) (b)

Fig 7 A set of schematic bifurcation diagrams portraying the interaction between the

two-cell and pair of single-cell states Solid lines denote stable solutions and dashedlines unstable a)The pitchfork at small aspect ratios corresponding to AB in fig 5b)The pitchfork becomes quartic when there is interaction with a second pitchfork (BC

in fig 5) near B in fig 5 c) hysteresis in the development of the single cell pair near b

in fig 5d) the single-cell pair are disconnected along BD in Fig 5

11 D Coles 1965Transition in circular Couette flow J Fluid Mech 21, 385–425.

12 M Golubitsky and D G Schaeffer 1985 Singularities and Groups in Bifurcation

Theory Vol 1 Applied Mathematical Sciences 51 Springer.

13 A Jepson and A Spence 1985Folds in solutions of two parameter systems SIAM

J Numer Anal 22, 347–368.

14 H B Keller 1977 Numerical solutions of bifurcation and nonlinear eigenvalue

problems In Applications of Bifurcation Theory (ed P H Rabinowitz), pp 359–

384, Academic

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equa-tions SIAM J Numer Anal 17, 567–576.

16 T Mullin 1991 Finite-dimensional dynamics in Taylor–Couette flow IMA J App.

Math 46, 109-120.

17 T Mullin 1993 The Nature of Chaos Oxford University Press.

18 G Pfister, H Schmidt, K A Cliffe and T Mullin 1988 Bifurcation phenomena in

Taylor–Couette flow in a very short annulus J Fluid Mech 191, 1–18.

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Camb Phil Soc 87, 307–337.

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to Turbulence Topics in Applied Physics, Vol 45 Springer.

21 R Tagg 1994 The Couette–Taylor problem Nonlinear Science Today 4, 1–25.

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cylinders Phil Trans R Soc Lond A 223, 289–343.

23 Y Toya, L Nakamura, S Yamashita and Y Ueki 1994 An experiment on a Taylor

vortex flow in a gap with small aspect ratio Acta Mechanica 102, 137–148.

24 B Werner and A Spence 1984 The computation of symmetry–breaking bifurcation

points SIAM J Numer Anal 21, 388–399.

Taylor–Couette system with asymmetric

boundary conditions

Oliver Meincke1, Christoph Egbers1, Nicoleta Scurtu2, and Eberhard B¨ansch2

1 ZARM, Center of Applied Space Technology and Microgravity, University ofBremen, Am Fallturm, 28359 Bremen, Germany

2 Center of Technomathematics, University of Bremen, Postfach 33 04 40, 28334Bremen, Germany

Abstract We report on a study on stability, bifurcation scenarios and routes into

chaos in Taylor–Couette flow By increasing the Reynolds number with the angularvelocity of the driving inner cylinder, the flow bifurcates from laminar mid-plane-symmetric basic flow via a pitchfork bifurcation to mid-plane-symmetric Taylor vortexflow Both flow states are rotationally symmetric We now compare the dynamicalbehaviour in a system with symmetric boundary conditions with the effects in anasymmetric system We also could vary the gap widths The different flow states can

be detected by visualization with small aluminium flakes and also measured by LaserDoppler Velocimetry (LDV) and Particle Image Velocimetry (PIV) The dynamicalbehaviour of the rotating flow is discussed by time series analysis methods and velocitybifurcation diagrams and then compared with numerical calculations

1Introduction

The subject of hydrodynamic instabilities and the transition to turbulence isimportant for the understanding of nonlinear dynamic systems A classical sys-tem to investigate such instabilities is besides the Rayleigh–B´enard system theTaylor–Couette system It consists of two concentric cylinders where the soformed gap is filled with the working fluid The system was first examined the-oretically and experimentally by Taylor [16] Here, only rotation of the innercylinder is considered and the outer one is held at rest By increasing the speed

of the inner cylinder, the azimuthal Couette flow becomes unstable and is placed by a cellular pattern in which the fluid travels in helical paths around thecylinder in layers of vortices (Taylor vortex flow) By a further increase of therotation speed the system undergoes several bifurcations before the flow struc-ture becomes more complicated Different routes to chaos are possible by furtherincreasing the rotation rate One model was described by Ruelle & Takens [13].Benjamin [3], [4] showed the importance of the finite size of the cylinders and itseffects upon the bifurcation phenomena A summary of the current state of re-search was published by Ahlers [1], Chossat [6], Koschmieder [9], Meyer–Spasche[10] and Tagg [15]

re-In this work, short systems are investigated to reduce the multiplicity ofpossible solutions Some new aspects of the dynamical behaviour of the Taylor–Couette flow during the transition to turbulence for the case of symmetric and

C Egbers and G Pfister (Eds.): LNP 549, pp 22–36, 2000.

c

Springer-Verlag Berlin Heidelberg 2000

asymmetric boundary conditions and the small (η = 0.85) and the wide gap width (η = 0.5) are presented in this study.

2 Experimental setup

Most of our experiments were carried out by increasing the Reynolds number

of the inner cylinder in a quasistationary way from rest However, since theoccuring flow structures could depend on initial conditions, it is possible to varythe acceleration rate for the cylinder The temperature was precisely controlledand measured to allow the determination of a well defined Reynolds number ofthe flow The Taylor–Couette flow is characterized by the following three control

parameters: The aspect-ratio (Γ = H/d), the radius ratio (η = R i /R a) and the

Reynolds number Re = R i dΩ i

ν , where H, d, R i , R a , Ω i and ν are the height, the

gap width, the inner and outer radii, the angular velocity of the inner cylinderand the kinematic viscosity respectively The symmetric experimental setup isillustrated in Fig 1a It is only possible to obtain different aspect ratios byintegrating different inner cylinders due to the constant length of the system

The radius ratios we used during this work were (η = 0.5) to realize a wide cylindrical gap and (η = 0.85) a small one To realize asymmetric boundary

conditions, a new setup consisting of an inner cylinder with an attached bottom

plate is available (Fig 1b) In this system the radius ratio is (η = 0.5) and the

aspect ratio is variable

3 Measurement techniques

To observe the behaviour of the flow, two different techniques were used Usingthe PIV-technique one gets a 2–D vector map of the flow field whereas LDV leads

to time series with high resolution containing information about one component

of velocity at a special location in the working fluid depending on time

3.1 PIV

In our system a pulsed double cavity, frequency doubled Nd:YAG-Laser is used

for the Particle Image Velocimetry The second cavity is required to get a very

short time delay between the two pulses A single laser achieves only a repetitionfrequency of about 15Hz This time delay is too long for high flow rates and nocorrelation between the records would be achieved The emitted laser beam isfrequency doubled and then spread with a cylindrical lense to get a green lightsheet, because the original wavelength of a Nd:YAG-Laser is in the infrared

To get two images in a short time-interval, a fast CCD-camera is used In Fig.2a sketch of the Taylor–Couette system with the applied PIV-setup is shown.With the two recorded images one gets a light intensity distribution which showsthe particles suspended into the measuring fluid The recorded images are di-vided into smaller subareas, so called ‘interrogation areas’ The cross correlation

rotating bottom plate moving top plate

outer cylinder inner cylinder linear slider

tank to control temperature

working fluid

a) photograph of the

symmet-ric experimental setup b) principle sketch of the asymmetric setup with ro-tating bottom plate

Fig 1 The two different experimental setups which were used during this work

algorithm (see Eqn 1) calculates for every interrogation area a vector of themovement of the particles so that at least a 2–D vector map of the flow in the