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The interaction between swirling and recirculating velocity components in unsteady, inviscid flow

Published online by Cambridge University Press:  26 April 2006

P. A. Davidson
P. A. Davidson
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Present address: Westinghouse Research and Development Center, 1310 Beulah Road, Pittsburgh, PA 15235, USA.
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Abstract

In this paper we consider the transient evolution of a swirling, recirculating flow in a truncated cylinder. In particular, we consider an initial time period during which the evolution of the flow is controlled by inertia. Such flows exhibit a mutual interaction between the swirl and the poloidal recirculation, whereby any axial gradient in swirl alters the recirculation, which, in turn, redistributes the swirl. This interaction may be visualized as a flexing of the poloidal vortex lines, the best known example of which is the inertial wave. Physical arguments and numerical experiments suggest that, typically, a strong, oscillatory recirculation will develop. We examine the exchange of energy between the swirl and recirculation, and show that the direction of transfer depends on the relative signs of ψ and ηuθ/ηz. In addition, there is a limit to the amount of energy that may be exchanged, since conservation of angular momentum imposes a lower bound on the kinetic energy of the swirl. The characteristic reversal time for the recirculation is estimated by considering the history of fluid particles on the endwalls. Its magnitude depends on the relative strengths of the swirl and recirculation. When the recirculation is large, the reversal time exceeds the turn-over time for a poloidal eddy and, consequently, the vortex lines accumulate at the stagnation points on the endwalls. This leads to accelerated local diffusion on the axis. An elementary one-parameter model is proposed for these nonlinear oscillations. In the limit of very weak recirculation, this model is consistent with the exact solution for inertial waves, while for strong recirculation, it confirms that the reversal time is greater than the turn-over time, and that the vortex lines accumulate on the axis.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 209 , December 1989 , pp. 35 - 55
Copyright
© 1989 Cambridge University Press

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References

Batchelor, G. K. 1951 Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow. Q. J. Mech. Appl. Maths IV, 29–41.Google Scholar
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 Mechanics. Cambridge University Press.
Bloor, M. I. G. & Ingham, D. B. 1987 The flow in industrial cyclones. J. Fluid Mech. 178, 507–519.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
Dijkstra, D. & van Heijst, G. J. F. 1983 The flow between two finite rotating discs enclosed by a cylinder. J. Fluid Mech. 128, 123–154.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodyanic Stability, pp. 77–80. Cambridge University Press.
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Gupta, A. K., Lilley, D. G. & Syred, N. 1984 Swirl Flows. Energy and Engineering Science Series, Abacus Press.
Hall, M. G. 1967 The structure of concentrated vortex cores. Prog. Aeronaut. Sci. 7, 53–110.Google Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476.Google Scholar
Kojima, S., Ohnishi, T., Mori, T., Shiwaku, K., Wakasugi, I. & Ohgami, M. 1983 Application of advanced mild stirring to a new bloom caster; the latest Kosmostir-Magnetogyr process technique. Proc. 66th Steelmaking Conf., Atlanta, USA, Iron and Steel Soc. of AIME.Google Scholar
Lewellen, W. S. 1962 A solution for three-dimensional vortex flows with strong circulation. J. Fluid Mech. 14, 420–432.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids, pp. 14–47. Cambridge University Press.
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
Truesdell, C. & Toupin, R. 1960 The classical field theories. In Handbuch der Physik, Band III/1 (ed. S. Flügge), pp. 226–793. Springer.