Ordered and Turbulent Patterns in Taylor-Couette Flow

350 Pages·1992·11.15 MB·English

by  Russell J. Donnelly (auth.)

Ordered and Turbulent Patterns
in Taylor-Couette Flow
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Series B: Physics
Ordered and Turbulent Patterns
in Taylor-Couette Flow
Edited by
C. David Andereck
and
F. 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
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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
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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