Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-17T07:50:05.506Z Has data issue: false hasContentIssue false
  • English
  • Français

Low-dimensional bifurcation phenomena in Taylor–Couette flow with discrete azimuthal symmetry

Published online by Cambridge University Press:  26 April 2006

J. J. Kobine  and
T. Mullin
J. J. Kobine
Affiliation:
Atmospheric, Oceanic and Planetary Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK Present address: Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge, CB3 9EW, UK
T. Mullin
Affiliation:
Atmospheric, Oceanic and Planetary Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We report the results of an experimental study of flow in a Taylor–Couette system where the usual circular outer cylinder is replaced by one with a square cross-section. The objective is to determine the validity of low-dimensional dynamical systems as a descriptive framework for flows in a domain without the special continuous symmetry of the original problem. We focus on a restricted version of the flow, where the steady flow consists of a single cell, thereby minimizing the multiplicity of solutions. The steady-state bifurcation structure is found to be qualitatively unchanged from that of the standard system. A complex but self-consistent bifurcation structure is uncovered for time-dependent flows, culminating in observations of dynamics similar to those of the finite-dimensional Sil’nikov mechanism. Such behaviour has been observed in the standard system with continuous azimuthal symmetry. The present results extend the range of closed-flow problems where there is an apparent connection between the infinite-dimensional Navier-Stokes equations and finite-dimensional dynamical systems.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 275 , 25 September 1994 , pp. 379 - 405
Copyright
© 1994 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aitta, A., Ahlers, G. & Cannell, D. S. 1985 Tricritical phenomena in rotating Couette–Taylor flow. Phys. Rev. Lett. 54, 673–676.Google Scholar
Benjamin, T. B. 1978 Bifurcation phenomena in steady flows of a viscous liquid. 1. Theory 2. Experiments. Proc. R. Soc. Lond. A 359, 1–43.Google Scholar
Benjamin, T. B. & Mullin, T. 1981 Anomalous modes in the Taylor experiment. Proc. R. Soc. Lond. A 377, 221–249.Google Scholar
Broomhead, D. S. & King, G. P. 1986 Extracting qualitative dynamics from experimental data. Physica 20 D, 217–236.Google Scholar
Cliffe, K. A. 1983 Numerical calculations of two-cell and single-cell Taylor flows. J. Fluid Mech. 135, 219–233.Google Scholar
Cliffe, K. A., Kobine, J. J. & Mullin, T. 1992 The role of anomalous modes in Taylor–Couette flow. Proc. R. Soc. Lond. A 439, 341–357.Google Scholar
Crawford, J. D., Golubitsky, M., Gomes, M. G. M., Knobloch, E. & Stewart, I. N. 1991 Boundary conditions as symmetry constraints. In Singularity Theory and its Applications: Warwick 1989, Part II (ed. R. M. Roberts & I. N. Stewart). Lecture Notes in Mathematics, vol. 1463. Springer.
Diprima, R. C. & Swinney, H. L. 1981 Instabilities and transition in flow between concentric rotating cylinders. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub). Topics in Applied Physics, vol. 45. Springer.
Glendinning, P. & Sparrow, C. 1984 Local and global behaviour near homoclinic orbits. J. Statist. Phys. 43, 479–488.Google Scholar
Golubitsky, M. & Langford, W. F. 1981 Classification and unfoldings of degenerate Hopf bifurcations. J. Diff. Eqns 41, 375–415.Google Scholar
Guckenheimer, J. & Holmes, P. 1986 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, 2nd edition. Springer.
Lensch, B. 1988 Über die Dynamik der Einwirbelströmung im Taylor–Zylinder. Diplomarbeit, University of Kiel, Germany.
Mullin, T. 1993 Chaos in fluid dynamics. In The Nature of Chaos (ed. T. Mullin). Oxford University Press.
Mullin, T. & Lorenzen, A. 1985 Bifurcation phenomena in flows between a rotating circular cylinder and a stationary square outer cylinder. J. Fluid Mech. 157, 289–303.Google Scholar
Mullin, T., Lorenzen, A. & Pfister, G. 1983 Transition to turbulence in a non-standard rotating flow. Phys. Lett. A 96, 236–238.Google Scholar
Mullin, T. & Price, T. J. 1989 An experimental observation of chaos arising from the interaction of steady and time-dependent flows. Nature 340, 294–296.Google Scholar
Pfister, G., Schmidt, H., Cliffe, K. A. & Mullin, T. 1988 Bifurcation phenomena in Taylor–Couette flow in a very short annulus. J. Fluid Mech. 191, 1–18.Google Scholar
Pfister, G., Schulz, A. & Lensch, B. 1991 Bifurcations and a route to chaos of an one-vortex-state in Taylor–Couette flow. Euro. J. Mech. B/ Fluids 10, 247–252.Google Scholar
šil'nikov, L. P. 1965 A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl. 6, 163–166.Google Scholar
Smith, G. P. & Townsend, A. A. 1982 Turbulent Couette flow between concentric cylinders at large Taylor numbers. J. Fluid Mech. 123, 187–217.Google Scholar
Snyder, H. A. 1968 Experiments on rotating flows between noncircular cylinders. Phys. Fluids 11, 1606–1611.Google Scholar