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Symmetry-breaking bifurcations in spherical Couette flow

Published online by Cambridge University Press:  26 April 2006

Oleg Yu. Zikanov
Oleg Yu. Zikanov
Affiliation:
Institute for Problems in Mechanics, Russian Academy of Science, 101 Vernadskii Ave., Moscow 117526, Russia Present address: Institut für Strömungsmechanik, Technische Universität Dresden, 01062 Dresden, Germany.
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Abstract

The solutions of the nonlinear and linearized Navier-Stokes equations are computed to investigate the instabilities and the secondary two- and three-dimensional regimes in the flow of an incompressible viscous fluid in a thin gap between two concentric differentially rotating spheres. The numerical technique is finite difference in the radial direction, spectral in the azimuthal direction, and pseudo-spectral in the meridional direction. The study follows the experiments by Yavorskaya, Belyaev and co-workers in which a variety of steady axisymmetric and three-dimensional travelling wave secondary regimes was observed in the case of a thin layer and both boundary spheres rotating. In agreement with the experimental results three different types of symmetry-breaking primary bifurcations of the basic equilibrium are detected in the parameter range under consideration.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 310 , 10 March 1996 , pp. 293 - 324
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
Astaf'eva, N. M. 1985a Numerical simulation of spherical Couette flow asymmetric with respect to the equator. Fluid Dyn. 20, 3, 383–387. (Transl. from Izv. AN SSSR. Mekh. Zhid. i Gaza 1985, 3, 56–62)Google Scholar
Astaf'eva, N. M. 1985b Numerical solution of the problem on linear stability of the spherical Couette flow (in Russian). Preprint 1027. Space Research Institute, USSR Academy of Science, Moscow.
Astaf'eva, N. M., Vvedenskaya, N. D. & Yavorskaya, I. M. 1978 Numerical study of nonlinear axisymmetric flow of fluid between Two concentric rotating spheres. In Sixth Intl Conf. on Numerical Methods in Fluid Dynamics (ed. H. Cabannes, M. Holt & V. Rusanov). Lecture Notes in Physics, vol. 90, pp. 56–63. Springer.
Bar-Yoseph, P., Solan, A., Hilen, R. & Roesner, K. G. 1990 Taylor vortex flow between eccentric coaxial rotating spheres. Phys. Fluids A 2, 1564–1573.Google Scholar
Bartels, F. 1982 Taylor vortices between two concentric rotating spheres. J. Fluid Mech. 119, 1–25.Google Scholar
Belyaev, Yu. N., Monakhov, A. A., Khlebutin, G. N. & Yavorskaya, I. M. 1980 Study of stability and nonuniqueness of secondary flows in rotating spherical layers (in Russian). Preprint 567. Space Research Institute, USSR Academy of Science, Moscow.
Belyaev, Yu. N., Monakhov, A. A. & Yavorskaya, I. M. 1978 Stability of spherical Couette flow in thick layers when the inner sphere revolves. Fluid Dyn. 13, N2, 162–168. (Transl. from Izv. AN SSSR, Mekh. Zhid. i Gaza 1978, N2, 9–15)Google Scholar
Bonnet, J.-P. & Alziary de Roquefort, T. 1976 Ecoulement entre deux sphères concentriques en rotation. J. Méc. 15, 373–397.Google Scholar
Buuhler, K. 1990 Symmetric and asymmetric Taylor vortex flow in spherical gaps. Acta Mechanica 81, 3–38.Google Scholar
Buuhler, K. & Zierep, J. 1984 New secondary flow instabilities for high Re-number flow between two rotating spheres. In Laminar-Turbulent Transition (ed. V. V. Kozlov), pp. 677–685. Springer.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21, 385–425.Google Scholar
Dumas, G. 1994 The spherical Couette flow and its large-gap stability by spectral simulation. In CFD94, CFD Society of Canada, Toronto.
Dumas, G. & Leonard, A. 1994 A divergence-free expansions spectral method for three-dimensional flows in spherical-gap geometries. J. Comput. Phys. 111, 205–219.Google Scholar
Godunov, S. K. & Ryabenki, V. S. 1964 Theory of Differerence Schemes. North-Holland
Joseph, D. D. 1976 Stability of Fluid Motion I. Springer.
Khlebutin, G. N. 1968 Stability of fluid motion between a rotating and a stationary concentric sphere. Fluid Dyn. 3, N6, 31–32. (Transl. from Izv. AN SSSR. Mekh. Zhid. i Gaza 1968, N 6, 53–56)Google Scholar
Kotschin, N. E., Kibel, I. A. & Rose, N. V. 1955 Theoretische Hydromechanik II. Akademie, Berlin.
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
Langford, W. F., Tagg, R., Kostelich, E. J., Swinney, H. L. & Golubitsky, M. 1988 Primary instabilities and bifurcality in flow between counter-rotating cylinders. Phys. Fluids 31, 776–785.Google Scholar
Mamun, C. K. & Tuckerman, L. S. 1995 Asymmetry and Hopf bifurcation in spherical Couette flow. Phys. Fluids 7, 80–91.Google Scholar
Marcus, P. S. & Tuckerman, L. S. 1987 Simulation of flow between concentric rotating spheres. J. Fluid Mech. 185, 1–65.Google Scholar
Munson, B. R., & Joseph, D. D. 1971 Viscous incompressible flow between concentric rotating spheres. part 1. basic flow. J. fluid mech. 49, 289–303.Google Scholar
Akabayashi, K. 1983 Transition of Taylor-Görtler vortex flow in spherical Couette flow. J. Fluid Mech. 132, 209–230.Google Scholar
Nakabayashi, K., & Tsuchida, Yo. 1988a Spectral study of the laminar-turbulent transition in spherical Couette flow. J. Fluid Mech. 194, 101–132.Google Scholar
Nakabayashi, K., & Tsuchida, Yo. 1988b Modulated and unmodulated travelling azimuthal waves on the toroidal vortices in a spherical Couette system. J. Fluid Mech. 195, 495–522.Google Scholar
Orszag, S. A. 1974 Fourier series on spheres. Mon. Weather Rev. 102, 56–75.Google Scholar
Ovseenko, Yu. G. 1963 On the motion of a viscous liquid between two rotating spheres (in Russian). Izv. Vyssh. Ucheb. Zaved. Mat. N 4, 129–139.Google Scholar
Ruelle, D. 1973 Bifurcations in the presence of a symmetry group. Arch. Rat. Mech. Anal. 51, 136–152.Google Scholar
Schrauf, G. 1986 The first instability in spherical Taylor-Couette flow. J. Fluid Mech. 166, 287–303.Google Scholar
Wimmer, M. 1976 Experiments on a viscious fluid flow between concentric rotating spheres. J. Fluid Mech. 78, 317–335.Google Scholar
Wimmer, M. 1977 Experimentelle Untersuchung der Strömung zwischen konzentrischen, rotierenden Kugeln. Z. Angew. Math. Mech. 57, 218–219.Google Scholar
Wimmer, M. 1981 Experiments on the stability of viscous flow between two concentric rotating spheres. J. Fluid Mech. 103, 117–131.Google Scholar
Yavorskaya, I. M. & Belyaev, Yu. N. 1986 Hydrodynamic stability in rotating spherical layers: application to dynamics of planetary atmospheres. Acta Astrounautica 13, N6/7, 433–440.Google Scholar
Yavorskaya, I. M., Belyaev, Yu. N. & Monakhov, A. A. 1977 Stability investigations and secondary flows in rotating spherical layers at arbitrary Rossby number. Sov. Phys. Dokl. 22, 717–719. (Transl. from Dokl. Akad. Nauk SSSR, 1977, 237, 804–807.)Google Scholar
Yavorskaya, I. M., Belyaev, Yu. N., Monakhov, A. A., Astaf'eva, N. M., Scherbakov, S. A. & Vvedenskaya, N. D. 1980 Stability, non-uniqueness and transition to turbulence in the flow between two rotating spheres. In Theoretical and Applied Mechanics (ed. F. P. J. Rimrott & B. Tabarrok), pp. 431–443. North-Holland.
Yavorskaya, I. M., Vvedenskaya, N. D. & Astaf'eva, N. M. 1978 On stability and nonuniqueness of fluid flows in rotating spherical layers. Dokl. Akad. Nauk SSSR 241, 52–55.Google Scholar
Young, R. E. 1974 Finite-amplitude thermal convection in a spherical shell. J. Fluid Mech. 63, 695–721.Google Scholar
Zikanov, O. Yu. 1993a Numerical study of instability of viscous fluid flows in rotating volumes (in Russian). Kand. of Sci. Thesis, Moscow State University, Moscow.
Zikanov, O. Yu. 1993b Numerical simulation of the first instability in spherical Couette flow. Preprint 531. Institute for Problems in Mechanics, Russian Academy of Science, Moscow.
Zikanov, O. Yu. 1995 Numerical simulation of instability and secondary regimes in spherical Couette flow (in Russian). Izv. AN SSSR, Mech. Zhid. i Gaza N 1, 1–15.Google Scholar