Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-04T17:47:26.401Z Has data issue: false hasContentIssue false
  • English
  • Français

Taylor vortices between two concentric rotating spheres

Published online by Cambridge University Press:  20 April 2006

Fritz Bartels
Fritz Bartels
Affiliation:
Aerodynamisches Institut RWTH Aachen, Aachen, Germany Present address: Fraunhofer-Institut für Hydroakustik, Waldparkstr. 41, 8012 Ottobrunn, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

The laminar viscous flow in the gap between two concentric spheres is investigated for a rotating inner sphere. The solution is obtained by solving the Navier-Stokes equations by means of finite-difference techniques, where the equations are restricted to axially symmetric flows. The flow field is hydrodynamically unstable above a critical Reynolds number. This investigation indicates that the critical Reynolds number beyond which Taylor vortices appear is slightly higher in a spherical gap than for the flow between concentric cylinders. The formation of Taylor vortices could be observed only for small gap widths s ≤ 0·17. The final state of the flow field depends on the initial conditions and the acceleration of the inner sphere. Steady and unsteady flow modes are predicted for various Reynolds numbers and gap widths. The results are in agreement with experiment if certain accuracy conditions of the finite-difference methods are satisfied. It is seen that the equatorial symmetry is of great importance for the development of the Taylor vortices in the gap.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 119 , June 1982 , pp. 1 - 25
Copyright
© 1982 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

Astafeva, N. M., Vvedenskaya, N. D. & Yavorskaya, I. M. 1977 Nonlinear axisymmetric fluid flows in spherical layers. Rep. no. 385, Institute of Cosmic Research of Akad. Nauk S.S.R., Moscow.
Bartels, F. 1978 Rotationssymmetrische Strömungen im Spalt konzentrischer Kugeln. Thesis, RWTH Aachen, Germany.
Bonnet, J.-P. & Alziary de Roquefort, T 1976 Écoulement entre deux sphères concentriques en rotation. J. Méc. 15, 373.Google Scholar
Dennis, S. C. R. & Singh, S. N. 1978 Calculation of the flow between two rotating spheres by the method of series truncation. J. Comp. Phys. 28, 297.Google Scholar
Greenspan, D. 1975 Numerical studies of steady, viscous, incompressible flow between two rotating spheres. Comp. & Fluids 3, 62.Google Scholar
Herbert, T. H. 1978 Berechnung der Strömung zwischen konzentrischen rotierenden Kugelflächen. Z. angew. Math. Mech. 58, T 275.Google Scholar
Khlebutin, G. N. 1968 Stability of fluid motion between a rotating and stationary concentric sphere. Fluid Dyn. 3, 31.Google Scholar
Kreiss, H.-O. & Oliger, H. 1972 Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24, 199.Google Scholar
Kirchgässner, K. 1961 Die Instabilität der Strömung zwischen zwei rotierenden Zylindern gegenüber Taylor-Wirbeln für beliebige Spaltbreiten. Z. angew. Math. Phys. 12, 14.Google Scholar
Munson, B. R. & Joseph, D. D. 1971a Viscios incompressible flow between concentric rotating spheres. Part 1. Basic flow. J. Fluid Mech. 49, 289.Google Scholar
Munson, B. R. & Joseph, D. D. 1971b Viscous incompressible flow between concentric rotating spheres. Part 2. Hydrodynamic stability. J. Fluid Mech. 49, 305.Google Scholar
Munson, B. R. & Menguturk, M. 1975 Viscous incompressible flow between concentric rotating spheres. Part 3. Linear stability. J. Fluid Mech. 69, 705.Google Scholar
Nakabayashi, K. 1978 Frictional moment of flow between two concentric spheres, one of which rotates. Trans. A.S.M.E. I, J. Fluids Engng, 100, 97.
Ovseenko, J. G. 1963 Über die Bewegung einer Flüssigkeit zwischen zwei rotierenden Kugelflächen. Isv. VUZ Math. 4, 129.Google Scholar
Pearson, C. E. 1967 A numerical study of the time-dependent viscous flow between two rotating spheres. J. Fluid Mech. 28, 323.Google Scholar
Hitter, C. F. 1973 Berechnung der zähen inkompressiblen Strömung im Spalt zwischen zwei konzentrischen rotierenden Kugelflächen. Thesis, Universität Karlsruhe, Germany.
Roesner, K. G. 1977 Numerical calculation of hydrodynamic stability problems with timedependent boundary conditions. In Proc. 6th Int. Conf. on Numerical Methods in Fluid Dynamics, Tbilisi, 1977, vol. 3, p. 3.
Sawatzki, O. & Zierep, J. 1970 Das Stromfeld im Spalt zwischen zwei konzentrischen Kugeln. Acta Mechanica 9, 13.Google Scholar
Walton, I. C. 1978 The linear stability of the flow in a narrow spherical annulus. J. Fluid Mech. 86, 673.Google Scholar
Wimmer, M. 1976 Experiments on a viscous fluid flow between concentric rotating spheres. J. Fluid Mech. 78, 317.Google Scholar
Wimmer, M. 1980 Experiments on the stability of viscous flow between two concentric rotating spheres. J. Fluid Mech. 103, 117.Google Scholar
Yakushin, V. J. 1970 Stability of the motion of a liquid between two rotating concentric spheres. Fluid Dyn. 5, 660.Google Scholar