Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-19T02:14:49.805Z Has data issue: false hasContentIssue false
  • English
  • Français

An iterative method for 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 St., Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

In steady, two-dimensional, inviscid flows it is well-known that, in the absence of rotational forcing, the vorticity is constant along streamlines. In a bounded domain the streamlines are necessarily closed. In some circumstances, investigated in this paper, this behaviour is exhbited also by forced viscous flows, when the variation of vorticity across the streamlines is determined by a balance between viscous diffusion and the forcing. Similar results hold in axisymmetry. For such flows, an iterative process for finding the vorticity as a function of the stream function is described. The method applies whenever the viscous boundary condition can be expressed in terms of the vorticity or tangential stress rather then the tangential velocity. When it is applicable, the iterative method is faster than direct solution of the Navier-Stokes equations at high Reynolds numbers. As an example, the method is used to calculate the flow in a model of the electromagnetic stirring process. In this model, a conducting fluid in an elliptical region is driven by a rotating magnetic field and resisted by a surface stress. The functional dependence of the vorticity on the stream function is found for various values of the magnetic skin depth, surface stress and eccentricity of the ellipse. The form of the flow is discussed with particular reference to whether it consists of a single circulatory region or separates into two or more such regions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 200 , March 1989 , pp. 1 - 18
Copyright
© 1989 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

Abramowitz, M. & Stegun, I. A., 1965 Handbook of Mathematical Functions. Dover.
Batchelor, G. K.: 1956 On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177–190.Google Scholar
Batchelor, G. K.: 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bayly, B. J.: 1987 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.Google Scholar
Busse, F. H. & Clever, R. M., 1981 An asymptotic model of two-dimensional convection in the limit of low Prandtl number. J. Fluid Mech. 102, 75–83.Google Scholar
Davidson, P. A. & Hunt, J. C. R. 1987 Swirling recirculating flow in a liquid metal column generated by a rotating magnetic field. J. Fluid Mech. 185, 67–106.Google Scholar
Etay, J.: 1980 Formage et guidage des metaux liquides sous l'action de champs magnetiques alternatifs. Report de D. E. A. de Mécaniques des Fluides, Inst. Nain. Polytechnique, Grenoble.Google Scholar
Fautrelle, Y. R.: 1981 Analytical and numerical aspects of the electromagnetic stirring induced by alternating magnetic fields. J. Fluid Mech. 102, 405–430.Google Scholar
Jones, C. A., Moore, D. R. & Weiss, N. O., 1976 Axisymmetric convection in a cylinder. J. Fluid Mech. 73, 353–388.Google Scholar
Mestel, A. J.: 1982 Magnetic levitation of liquid metals. J. Fluid Mech. 117, 27–43.Google Scholar
Mestel, A. J.: 1984 On the flow in a channel induction furnace. J. Fluid Mech. 147, 431–447.Google Scholar
Mestel, A. J.: 1988 More accurate skin-depth solutions. Liquid Metal Magnetohydrodynamics. Proc. IUTAM Symp. Riga 1988. Kluwer.
Mestel, A. J.: 1989 On the stability of high Reynolds number flows with closed streamlines. J. Fluid Mech. 200, 19–38.Google Scholar
Moffatt, H. K.: 1965 On fluid flow induced by a rotating magnetic field. J. Fluid Mech. 22, 521–528.Google Scholar
Moffatt, H. K.: 1984 High frequency excitation of liquid metal systems. Metallurgical Applications of Magnetohydrodynamics. Proc. IUTAM Symp. Cambridge 1982. The Metals Society.
Read, P. L., Rhines, P. B. & White, A. A., 1986 Geostrophic scatter diagrams and potential vorticity dynamics. J. Atmos. Sci. 43, 3226–3240.Google Scholar
Shercliff, J. A.: 1981 Magnetic shaping of molten metal columns. Proc. R. Soc. Lond. A 375, 455–473.Google Scholar
Sneyd, A. D.: 1979 Fluid flow induced by a rapidly alternating or rotating field. J. Fluid Mech. 92, 35–51.Google Scholar
Sneyd, A. D. & Moffatt, H. K., 1982 The fluid dynamics of the process of levitation melting. J. Fluid Mech. 117, 45–70.Google Scholar
Trefethen, L. N.: 1986 (Ed.) Special issue on numerical conformal mapping. J. Comp. Appl. Maths. 14, 1 & 2.Google Scholar
Tutty, O. & Pedley, T. J., 1989 Computations of oscillatory flow in a non-uniform channel. (In preparation for J. Fluid Mech.)