Ordered and Turbulent Patterns in Taylor-Couette Flow NA TO ASI Series Advanced Science Instltutes Serles ro A series presenting the results of aetivities sponsored by the NA Seienee Committee, whieh aims at the dissemination of advaneed seientifie and teehnologieal knowledge, with a view to strengthening links between seientifie eommunitles. The series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences Plenum Publishing Corporation B Physlcs New York and London C Mathematlcal and Physlcal Sclences Kluwer Academic Publishers O Behavioral and Social Sclences Dordrecht, Boston, and London E Applled Sclences F Computer and Systems Sclences Springer-Verlag G Ecological Sciences Berlin, Heidelberg, New York, London, H Cell Blology Paris, Tokyo, Hong Kong, and Barcelona I Global Envlronmental Change Recent Volumes In thls Ser/es Volume 290-Phase Transitions in Liquid Crystals edited by S. Martellucci and A. N. 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Hayot The Ohio State University Columbus, Ohio Springer Science+Business Media, LLC Proceedings of a NATO Advanced Research Workshop on Ordered and Turbulent Patterns in Taylor-Couette Flow, held May 22-24, 1991, in Columbus, Ohio NATO-PCO-DATA BASE The electronic index to the NATO ASI Series provides lull bibliographical relerences (with key words and/or abstracts) to more than 30,000 contributions Irom international scientists published in aII sections 01 the NATO ASI Series. Access to the NATO-PCO-DATA SASE is possible in two ways: -via online FILE 128 (NATO-PCO-DATA SASE) hosted by ESRIN, Via Galileo Galilei, 1-00044 Frascati, Italy. Llbrary of Congress Cataloging-ln-Publication Data Ordered and turbulent patterns in Taylor-Couette flow I edited by C. David Andereck and F. Hayot. p. cm. -- (NATD ASI series. Series B. PhYS1CS ; v. 297) "Proceedings of a NATO Advanced Research Workshop an Ordered and Turbulent Patterns in Taylor-Couette Flow, held May 22-24. 1991, in Columbus, Ohio"--Verso t.p. "Published In cooperat ion with NATO Scientific Affairs Oivision." Includes bibliographical references and index. ISBN 978-1-4613-6521-1 ISBN 978-1-4615-3438-9 (eBook) DOI 10.1007/978-1-4615-3438-9 1. Vortex-motion--Congresses. 2. Fluid dynamics--Congresses. 1. Andereck, C. David. II. Hayot, F. III. North Atlantic Treaty Organization. Scientific Affairs Division. IV. NATO Advanced Resear~h Horkshap an Ordered and Turbulent Patterns in Taylor -Couette Flow (1991 Columbus, Ohio) V. Title, Taylor-Couette flow. VI. Series. OA925.073 1992 532' .0595--dc20 92-18440 CIP ISBN 978-1-4613-6521-1 © 1992 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1992 Softcover reprint of the hardcover lst edition 1992 AII rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher PREFACE Seldom does a physical system, particularly one as apparently simple as the flow of a Newtonian fluid between concentric rotating cylinders, retain the interest of scientists, applied mathematicians and engineers for very long. Yet, as this volume goes to press it has been nearly 70 years since G. I. Taylor's outstanding experimental and theoretical study of the linear stability of this flow was published, and a century since the first experiments were performed on rotating cylinder viscometers. Since then, the study of this system has progressed enormously, but new features of the flow patterns are still being uncovered. Interesting variations on the basic system abound. Connections with open flows are being made. More complex fluids are used in some experiments. The vigor of the research going on in this particular example of nonequilibrium systems was very apparent at the NATO Advanced Research Workshop on "Ordered and Turbulent Patterns in Taylor Couette Flow," held in Columbus, Ohio, USA May 22-24, 1991. A primary goal of this ARW was to bring together those interested in pattern formation in the classic Taylor Couette problem with those looking at variations on the basic system and with those interested in related systems, in order to better define the interesting areas for the future, the open questions, and the features common (and not common) to closed and open systems. This volume contains many of the contributions presented during the workshop. The organization of the contents is similar to that of the workshop itself. It begins with a detailed historical survey of the experimental aspects of the study of the Taylor-Couette system. This is followed by several contributions devoted to the classical Taylor-Couette problem, including numerical and experimental studies of Taylor vortices, wavy vortices, and spirals, and the phase dynamics approach to flows both near and far from onset. These papers highlight the fact that even the simplest form of the problem continues to provide a challenge to theory and experiment. The next section concerns Taylor-Couette flow, but with broken rotational symmetry. It has long been recognized that the basic system is very special in that it combines rotational symmetry with translational invariance. Recently several experiments have been undertaken to understand the effects on the flow patterns when the rotational symmetry is broken, either by adding a Coriolis force or by running the system horizontally with only a partially filled gap. In either case wholly new flow patterns have emerged and new routes to turbulence are seen. Turbulence in Taylor-Couette flow and in plane Couette flow are covered in the next section, the first instance in this volume in which direct connections between flows in open and closed systems are discussed. In the following section, connections with more general theoretical modeling are made in papers on the Eckhaus and Benjamin-Feir instabilities, and on turbulence in the one-dimensional complex Ginzburg-Landau equation. Several papers discussing variations on the basic system were presented. Throughflows have been imposed, both radially and axially, thus producing an effectively open system. v In contrast to most open systems, however, the modified Taylor-Couette system offers the experimentalist a very highly controlled environment for study. Other interesting variations described here include different geometries, such as rotating cones and cylinders, and magnetohydrodynamic instabilities with radial temperature gradients. Another paper in this group discusses the flow of a radically different fluid, superfluid helium, in an otherwise simple Taylor apparatus. The Taylor-Couette system is then actually serving as a testing ground not just for the solution of the governing equations, but for the very form of these equations. The final group of papers deals with related open flow systems. Of particular importance is the interest in vortex formation and transition in flows through curved channels or over curved surfaces. The centrifugal instabilities in these cases are naturally related to the Taylor-Couette problem, although the spatial development is typically quite different in the open flows. During a summary panel discussion a number of points were brought out regarding the status and future of work on the Taylor-Couette system and related pattern-forming systems. Motivation for further work remains strong: -The Taylor-Couette system is a paradigm of pattern-forming systems -It serves as a test bed for mathematical modeling of such systems, including symmetries, the identification of the correct normal form, the importance of nonlocal terms, the treatment of fluxes of various sorts, and the amplitude equations for various circumstances -It is a problem in which detailed quantitative comparisons between theory and experiment are possible, partly because remarkably good laboratory control is possible -It has long been a test bed for novel experimental techniques -It is an appropriate test bed for computational fluid dynamics modeling -It provides a suitable environment for new mathematical models of physical behavior such as that exhibited by complex fluids or superfluids. There were several broadly defined open questions: -Is fully developed turbulence universal, or is it special in this system? -What can we learn from the Taylor-Couette system about the evolution of complexity in general systems? -What can we learn about convective and absolute instabilities using variations of the basic system? What is the general relationship of the Taylor-Couette system to open flows? -Are there new flow visualization or other diagnostic methods deserving of use? It was clear that even though great progress has been made in understanding some of the basic instabilities and resulting flow patterns in the Taylor-Couette system, there remains a great deal of work to be done. We hope that this volume will serve to stimulate this line of research. To assist in this process we have included as an appendix a bibliography, prepared by Randall Tagg, that contains a very complete listing of references on the Taylor-Couette system and related problems. We wish to thank the other members of the organizing committee. Russell J. Donnelly, Yves Pomeau and Daniel Walgraef, for their ideas and suggestions in making the workshop a success. We much appreciate the key speakers, including Donnelly, Phil Hall, Lorenz Kramer, Pomeau and Harry Swinney, who gave overview talks that broadened considerably the scope of the workshop and provided a context for our discussions. We also thank the North Atlantic Treaty Organization Scientific Affairs Division, the Ohio State University vi Office of Research and Graduate Studies, the College of Mathematical and Physical Sciences, the Department of Physics and the Ohio State Fluids Research Institute for their generous financial support. Finally we thank Debra Dunson, Lynn McGraner and Natalie Novak for their dedication to the details of organizing the workshop and producing this volume. Columbus, December 1991 C. David Andereck F. Hayot vii CONTENTS TA YLOR-COUETTE FLOW, EXPERIMENT AND THEORY Evolution of Instrumentation for Taylor-Couette Flow R.J. Donnelly Mode Competition and Coexistence in Taylor-Couette Flow 29 J. Brindley and F.R. Mobbs Low-Dimensional Spectral Truncations for Taylor-Couette Flow 43 K.T. Coughlin and P.S. Marcus The Couette-Taylor Problem in the Small Gap Approximation 51 Y. Demay, G. Iooss, and P. Laure Structure of Taylor Vortex Flow and the Influence of Spatial Amplitude Variations on Phase Dynamics ........................................ 59 D. Roth, M. Lucke, M. Kamps, and R. Schmitz Chaotic Phase Diffusion Through the Interaction of Phase Slip Processes 67 H. Riecke and H.-G. Paap Phase Dynamics in the Taylor-Couette System 75 M. Wu and C.D. Andereck Spiral Vortices in Finite Cylinders 83 E. Knobloch and R. Pierce TAYLOR-COUETTE FLOW SYSTEMS WITH BROKEN ROTATIONAL SYMMETRY A Model of the Disappearance of Time-Dependence in the Flow Pattern in the Taylor-Dean System .................................... 91 L. Fourtune, I. Mutabazi, and C.D. Andereck End Circulation in Non-Axisymmetrical Flows 99 C. Normand, I. Mutabazi, and J.E. Wesfreid Bifurcation Phenomena in Taylor-Couette Flow Subject to a Coriolis Force ........ 107 P.W. Hammer, R.I. Wiener, and R.I. Donnelly ix Instability of Taylor-Couette Flow Subjected to a Coriolis Force 121 R.J. Wiener, P.W. Hammer, and R. Tagg Bifurcations to Dynamic States in Taylor-Couette Flow with External Rotation ...... 131 L. Ning, G. Ahlers, and D.S. CanneIl On the Stability of Taylor-Couette Flow Subjected to External Rotation ........... 141 M. Tveitereid, L. Ning, G. Ahlers, and D.S. CanneIl TURBULENCE IN T AY LOR-COUETTE AND PLANE COUETTE FLOW Numerical Simulation of Turbulent Taylor-Couette Flow 149 S. Hirschberg Intermittent Turbulence in Plane and Circular Couette Flow 159 J. Hegseth, F. Daviaud, and P. BergtS INSTABILITIES, PATTERN FORMATION AND TURBULENCE IN MODEL EQUATIONS On the Eckhaus and the Benjamin-Feir Instability in the Vicinity of a Tricritical Point ........................................ 167 H.R. Brand Phase vs. Defect Turbulence in the I-D Complex Ginzburg-Landau Equation ....... 173 A. Pumir, B.I. Shraiman, W. van Saarloos, P.C. Hohenberg, H. ChattS, and M. Holen Double Eigenvalues and the Formation of Flow Patterns 179 R. Meyer-Spasche EXTENSIONS OF TAYLOR-COUETTE FLOW The Effect of Throughflow on Rayleigh Benard Convective RoIls 187 H.W. MuIler, M. Lucke, and M. Kamps Taylor Vortex Flow with Superimposed Radial Mass Flux 197 K. Buhler Vortex Patterns Between Cones and Cylinders ........................... 205 M. Wimmer Instability of Taylor-Couette Flow of Helium II .......................... 213 C.J. Swanson and R. J. DonneIly Effects of Radial Temperature Gradient on MHD Stability of Couette Flow Between Conducting Cylinders - A Wide Gap Problem .................. 221 H.S. Takhar, M.A. Ali, and V.M. Soundalgekar OPEN FLOWS Structure and Perturbation in Glirtler Vortex Flow 245 P. Petitj eans and J.E. Wesfreid x Transition to Turbulence in Gortler Flow .............................. 253 W. Liu and J.A. Domaradzki Effect of Curvature Plane Orientation on Vortex Distortion in Curved Channel Flow ............................................ 263 H. Peerhossaini and Y. Le Guer Splitting, Merging and Wavelength Selection of Vortex Pairs in Curved and/or Rotating Channels ......................................... 273 W.H. Finlay and Y. Guo Transient, Oscillatory and Steady Characteristics of Dean Vortex Pairs in a Curved Rectangular Channel ................................... 281 P.M. Ligrani On the Subharmonic Instability of Finite-Amplitude Longitudinal Vortex Rolls in Inclined Free Convection Boundary Layers ........................ 289 C.C. Chen, A. Labhabi, H.-C. Chang, and R.E. KeJly Centrifugal Instabilities in Rotating Frame about Flow Axis .................. 297 I. Mutabazi, C. Normand, M. Martin, and J.E. Wesfreid APPENDIX A Guide to Literature Related to the Taylor-Couette Problem ................. 303 R. Tagg INDEX ................................................... 355 xi