Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T22:52:05.219Z Has data issue: false hasContentIssue false
  • English
  • Français

Flow Bifurcation and Transition in a Gap Between Two-Rotating Spheres

Published online by Cambridge University Press:  05 May 2011

W.-J. Luo  and
R.-J. Yang
W.-J. Luo*
Affiliation:
Department of Electrical Engineering, Far East College, Tainan, Taiwan 74404, R.O.C.
R.-J. Yang*
Affiliation:
Department of Engineering Science, National Cheng Kung University, Tainan, Taiwan 70101, R.O.C.
*
*Assistant Professor
**Professor
Get access

Abstract

This study investigates the flow bifurcation and transition between stable states in the gap between two concentric rotating spheres. A continuation method is used together with linear stability analysis to investigate the bifurcation structure of the discretized governing equations and to determine the stability of the calculated states. The constructed bifurcation diagram is used to illustrate the restricted range of Reynolds number within which each equilibrium state exists. The diagram also identifies the permissible transitions between these states and indicates their terminative states. In the present study, it is shown how appropriate control of the angular velocity of the outer sphere results in the evolution of the flow state through a series of permitted stable states. The time-dependent transitions between these states are investigated by means of a backwards-Euler time stepping formulation. The terminate state of transition process can also be used to confirm the stability of the flow. The flow evolution between each transition is illustrated by means of temporal sequences of the meridional streamlines and transition curves. The present results indicate that all the flow transitions in a gap between two rotating spheres are produced symmetrically with respect to the equator.

Keywords

Continuation methodBifurcationTransition
Type
Articles
Information
Journal of Mechanics , Volume 20 , Issue 2 , June 2004 , pp. 95 - 105
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2004

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

1.Wimmer, M., “Experiments on a Viscous Fluid Flow between Concentric Rotating Spheres,” J. Fluid Mech., 79, pp. 317–335 (1976).CrossRefGoogle Scholar
2.Bartels, F., “Taylor Vortices between Two Concentric Rotating Spheres,” J. Fluid Mech., 119, pp. 1–25 (1982).CrossRefGoogle Scholar
3.Schrauf, G., “The First Instability in Spherical Taylor-Couette Flow,” J. Fluid Mech., 166, pp. 287–303 (1986).CrossRefGoogle Scholar
4.Buhler, K., “Symmetric and Asymmetric Taylor Vortex Flow in Spherical Gaps,” Acta Mechanica, 81, pp. 3–38 (1990).CrossRefGoogle Scholar
5.Yang, R.-J., “A Numerical Procedure for Predicting Multiple Solutions of a Spherical Taylor-Couette Flow,” International Journal for Numerical Methods in Fluids, 22, pp. 1135–1147 (1996).3.0.CO;2-N>CrossRefGoogle Scholar
6.Luo, W.-J. and Yang, R.-J., “Flow Bifurcation and Heat Transfer in a Spherical Gap,” International Journal of Heat and Mass Transfer., 43, pp. 885–899 (2000).CrossRefGoogle Scholar
7.Nakabayashi, K. and Tsuchida, Y., “Spectral Study of the Laminar-Turbulent Transition in the Spherical Couette Flow,” J. Fluid Mech., 194, pp. 101–132(1988).CrossRefGoogle Scholar
8.Sha, W., Nakabayashi, K. and Ueda, H., “An Accurate Second-Order Approximation Factorization method for Time-Dependent Incompressible Navier-Stokes Equations in Spherical Polar Coordinate,” J. Comput. Phys., 142, pp. 47–66 (1998).CrossRefGoogle Scholar
9.Zikanov, Oleg Yu, “Symmetry-Breaking Bifurcations in Spherical Couette Flow,” J. Fluid Mech., 310, pp. 293–324 (1996).CrossRefGoogle Scholar
10.Yavorskaya, I. M. and Belyaev, Yu. N., “Hydrodynamic Stability in Rotating Spherical Layers: Application to Dynamics of Planetary Atmospheres,” Acta Astrounautica, 13, N6/7, 433–440 (1986).CrossRefGoogle Scholar
11.Yavorskaya, I. M., Belyaev, Yu. N. and Monakhov, A. A., “Stability Investigations and Secondary Flows in Rotating Spherical Layers at Arbitrary Rossby Number,” Sov. Phys. Dokl. (Transl. from Dokl. Akad. Nauk SSSR), pp. 804–807 (1977).Google Scholar
12.Yavorskaya, I. M., Belyaev, Yu. N., Monakhov, A. A., Astafeva, N. M., Scherbakov, S. A. and Vvedenskaya, N. D., “Stability, Non-Uniqueness and Transition to Turbulence in the Flow between Two Rotating Spheres,” Theoretical and Applied Mechanics, Rimrott, F. P. J. and Tabarrok, B., eds., North-Holland, pp. 431–443 (1980).Google Scholar
13.Yang, R.-J., Luo, W.-J., “Flow Bifurcations in a Thin Gap between Two Rotating Spheres,” Theoret. Comput. Fluid Dynamics, 16, pp. 1–17 (2002).CrossRefGoogle Scholar
14.Dennis, S. C. R. and Quartapelle, L., “Finite Difference Solution to the Flow between Two Rotating Spheres,” Computers and Fluids, 12, pp. 77–92 (1984).CrossRefGoogle Scholar
15.Keller, H. B., “Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problem,” Applications of Bifurcation Theory, Rabinowitz, P., ed., Academic Press, New York, pp. 359–384 (1977).Google Scholar
16.Sorensen, D. C., “Implicit Application of Polynomial Filters in a k-Step Arnoldi Method,” SIAM (Soc. Ind. Appl. Math.) J. Matrix Anal. Appl., 13, pp. 357–367 (1992).CrossRefGoogle Scholar
17.Saad, Y., Numerical Methods for Large Eigenvalues Problems, Halsted (1992).Google Scholar