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On the stability of high-Reynolds-number flows with closed streamlines

Published online by Cambridge University Press:  26 April 2006

A. J. Mestel
A. J. Mestel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
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Abstract

In steady, two-dimensional, inertia-dominated flows it is well known that the vorticity is constant along the streamlines, which, in a bounded domain, are necessarily closed. For inviscid flows, the variation of vorticity across the streamlines is arbitrary, while for forced, weakly dissipitative flows, it is determined by the balance between viscous diffusion and the forcing. This paper discusses the linear stability of flows of this type to two-dimensional disturbances. Arnol'd's stability theorems are discussed. An alternative functional to Arnol'd's is found, which gives the same stability criteria and which permits a representation of the problem in terms of a Schrödinger equation. Conditions for stability are derived from this functional. In particular it is shown that total flow reversals are potentially unstable. The results are illustrated with respect to the geometrically simple case when the streamlines are circular and the forcing is due to a rotating magnetic field, for which case the stability regions are calculated as a function of two parameters. It is shown that the entire theory, including Arnol'd's theorems, applies also to poloidal axisymmetric flows.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 200 , March 1989 , pp. 19 - 38
Copyright
© 1989 Cambridge University Press

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References

Andrews, D. G.: 1983 A conservation law for small-amplitude quasi-geostrophic disturbances on a zonally asymmetric basic flow. J. Atmos. Sci. 40, 85–90.Google Scholar
Arnol'd, V. I.: 1965 Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk. SSSR 162, 975–978. (English transl. Soviet Math. 6, 331–334.)Google Scholar
Arnol'd, V. I.: 1966 On an apriori estimate in the theory of hydrodynamic stability. Izv. Vyssh. Uchebn. Zaved. Matematika 54, No. 5, 3–5. (English transl. Amer. Math. Soc. Trans. 2 79, 267–269 (1969).)Google Scholar
Bayly, B. J.: 1988 Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows. Phys. Fluids 31, 56–64.Google Scholar
Blumen, W.: 1968 On the stability of quasigeostrophic flow. J. Atmos. Sci. 25, 929–931.Google Scholar
Drazin, P. & Reid, W., 1981 Hydrodynamic Stability. Cambridge University Press.
Fjørtoft, R.: 1953 On the changes in the spectral distribution of kinetic energy for two-dimensional, nondivergent flow. Tellus 5, 225–230.Google Scholar
Garabedian, P. R.: 1964 Partial Differential Equations, p. 412. Wiley.
Jones, C. A., Moore, D. R. & Weiss, N. O., 1976 Axisymmetric convection in a cylinder. J. Fluid Mech. 73, 353.Google Scholar
McIntyre, M. E. & Shepherd, T. G., 1987 An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems. J. Fluid Mech. 181, 527–565.Google Scholar
Mestel, A. J.: 1982 Matnetic levitation of liquid metals. J. Fluid Mech. 117, 27–43.Google Scholar
Mestel, A. J.: 1989 An iterative method for high Reynolds number flows with closed streamlines. J. Fluid Mech. 200, 1–18.Google Scholar
Moffatt, H. K.: 1965 On fluid flow induced by a rotating magnetic field. J. Fluid Mech. 22, 521.Google Scholar
Moffatt, H. K.: 1986 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 2. Stability considerations. J. Fluid Mech. 166, 359–378.Google Scholar
Murzbacher, E.: 1970 Quantum Mechanics. Wiley.
Rayleigh, Lord: 1880 On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 10, 4–13.Google Scholar
Richardson, A. T.: 1974 On the stability of a magnetically driven rotating fluid flow. J. Fluid Mech. 63, 593–605.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