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Taylor—Couette instability of travelling waves with a continuous spectrum

Published online by Cambridge University Press:  26 April 2006

S. Ghosh Moulic  and
L. S. Yao
S. Ghosh Moulic
Affiliation:
Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106, USA Present address, Department of Mechanical Engineering, Indian Institute of Technology, Bombay, India.
L. S. Yao
Affiliation:
Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, AZ 85287-6106, USA
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Abstract

The nonlinear evolution of a continuous spectrum of travelling waves resulting from the growth of unstable disturbances in circular Couette flow has been investigated. Numerical solution of the governing integro-differential equations for different initial conditions shows that the equilibrium states of Taylor-vortex, wavy-vortex or spiralvortex flows are not unique, but depend on the initial disturbance. The presence of multiple solutions at a fixed Reynolds number for a given Taylor–Couette geometry has been known since Coles’ seminal contribution in 1965. The current study indicates that the equilibrium state of flows on a stable bifurcation branch is a natural consequence of nonlinear wave resonance and is dependent on the initial conditions. The resulting wavenumber can take any value within an accessible finite band. Since such multiple solutions have also been found numerically for mixed-convection flows and experimentally for several other flows, there is evidence to support the conclusion that a non-uniqueness in the sense of Coles is a generic property for all fluid flows.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 324 , 10 October 1996 , pp. 181 - 198
Copyright
© 1996 Cambridge University Press

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References

Andereck, C. D., Liu, S. S. & Swinney, H. L. 1986 Flow regimes in a circular Couette system with independently rotating cylinders. J. Fluid Mech. 164, 155–183.Google Scholar
Benjamin, T. B. 1978 Bifurcation phenomena in steady flows of a viscous fluid. II. Experiments. Proc. R. Soc. Lond. A 359, 27–43.Google Scholar
Burkhalter, J. E. & Koschmieder, E. L. 1974 Steady supercritical Taylor vortices after sudden starts. Phys. Fluids 17, 1929–1935.Google Scholar
Canuto, C., Mussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.Google Scholar
Davey, A., DiPrima, R. C. & Stuart, J. T. 1968 On the instability of Taylor vortices. J. Fluid Mech. 31, 17–52.Google Scholar
DiPrima, R. C. 1961 Stability of nonrotationally symmetric disturbances for viscous flow between rotating cylinders. Phys. Fluids 4, 751–755.Google Scholar
DiPrima, R. C. & Mabetler, G. J. 1969 A completeness theorem for non-self-adjoint eigenvalue problems in hydrodynamic instability. Arch. Rat. Mech. Anal. 34, 218–227.Google Scholar
Eagles, P. M. 1974 On the torque of wavy vortices. J. Fluid Mech. 62, 1–9.Google Scholar
Fenstermacher, P. R., Swinney, H. L. & Gollub, J. P. 1979 Dynamical instabilities and the transition to chaotic Taylor-vortex flow. J. Fluid Mech. 94, 103–129.Google Scholar
Ghosh Moulic, S. 1993 Nonlinear instability. PhD thesis, Department of Mechanical and Aerospace Engineering, Arizona State University.
Kleiser, L. & Schumann, U. 1980 Treatment of incompressibility and boundary conditions in three-dimensional numerical spectral simulations of plane channel flows. Proc. 3rd GAMM Conf. on Numerical Methods in Fluid Mechanics (ed. E. H. Hirschel), pp. 165–173. Vieweg.
Krueger, E. R., Gross, A. & DiPrima, R. C. 1966 On the relative importance of Taylor-vortex and non-axisymmetric modes in flow between rotating cylinders. J. Fluid Mech. 24, 521–538.Google Scholar
Marcus, P. S. 1984 Simulation of Taylor-Couette flow. Part 2. Numerical results for wavy-vortex flow with one traveling wave. J. Fluid Mech. 146, 65–113.Google Scholar
Moler, C. B. & Stewart, G. W. 1973 An algorithum for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10, 241–256.Google Scholar
Snyder, H. A. 1969 Wavenumber selection at finite amplitude in rotating Couette flow. J. Fluid Mech. 35, 273–298.Google Scholar
Stuart, J. T. & DiPrima, R. C. 1978 The Eckhaus and Benjamin-Feir resonstance mechanisms. Proc. R. Soc. Lond. A 362, 27–41.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–343.Google Scholar
Yao, L. S. 1995 Non-uniqueness in convection. In Symp. on Thermal Science and Engineering in Honor of Chancellor Chang-Lin Ten, Berkeley, California, November 14 (ed. R. O. Buckins), pp. 75–82. University of Illinois at Urbana-Champaign.
Yao, L. S. & Ghosh Moulic, S. 1994 Uncertainty of convection. Intl J. Heat Mass Transfer 37, 1713–1721.Google Scholar
Yao, L. S. & Ghosh Moulic, S. 1995a Taylor-Couette instability with a continuous spectrum. Trans. ASME E: J. Appl. Mech. 62, 915–923.Google Scholar
Yao, L. S. & Ghosh Moulic, S. 1995b Nonlinear instability of traveling waves with a continuous spectrum. Intl J. Heat Mass Transfer 38, 1751–1772.Google Scholar