Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-18T20:42:57.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Similarities in the structure of swirling and buoyancy-driven flows

Published online by Cambridge University Press:  26 April 2006

P. A. Davidson
P. A. Davidson
Affiliation:
Department of Mechanical Engineering, Imperial College, Exhibition Road, London SW7 2BX, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We look at two classes of contained flow: swirling flow and buoyancy-driven flow. We note that the strong links between these arise from the way in which vorticity is generated and propagated within each. We take advantage of this shared behaviour to investigate the structure of steady-state solutions of the governing equations. First, we look at flows with a small but finite viscosity. Here we find that, Batchelor regions apart, the steady state for each type of flow must consist of a quiescent stratified core, bounded by high-speed wall jets. (In the case of swirling flow, this is a radial stratification of angular momentum.) We then give a general, if approximate, method for finding these steady-state flow fields. This employs a momentum-integral technique for handling the boundary layers. The resulting predictions compare favourably with numerical experiments. Finally, we address the problem of inviscid steady states, where there is a well-known class of steady solutions, but where the question of the stability of these solutions remains unresolved. Starting with swirling flow, we use an energy minimization technique to show that stable solutions of arbitrary net azimuthal vorticity do indeed exist. However, the analogy with buoyancy-driven flow suggests that these solutions are all of a degenerate, stratified form. If this is so, then the energy minimization technique, which conserves vortical invariants, may mimic the stratification of temperature or angular momentum in a turbulent flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 252 , July 1993 , pp. 357 - 382
Copyright
© 1993 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

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.
Davidson, P. A. 1989 The interaction between swirling and recirculating velocity components in unsteady, inviscid flow. J. Fluid Mech. 209, 35–55.Google Scholar
Davidson, P. A. 1992 Swirling flow in an axisymmetric cavity of arbitrary profile, driven by a rotating magnetic field. J. Fluid Mech. 245, 669–699.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Elder, J. W. 1965 Laminar free convection in a vertical slot. J. Fluid Mech. 23, 77–97.Google Scholar
Gill, A. E. 1966 The boundary layer regime for convection in a rectangular cavity. J. Fluid Mech. 26, 515–536.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.
Japan Society of Mechanical Engineers (1988 Visualised Flow. Pergamon.
Kármán, T. von (1921 Uber laminare und turbulente reibung. Z. Angew. Math. Mech. 1, 233–252.Google Scholar
Moffatt, H. K. 1969 The degree of knottedness in tangled vortex lines. J. Fluid Mech. 35, 117–129.Google Scholar
Moffatt, H. K. 1985 Magnetostatic equilibrium and analogous Euler flows of arbitrary complex topology. Part 1. Fundamentals. J. Fluid Mech. 159, 359–378.Google Scholar
Moffatt, H. K. 1986 Magnetostatic equilibrium and analogous Euler flows of arbitrary complex topology. Part 2. Stability considerations. J. Fluid Mech. 166, 359–378.Google Scholar
Moffatt, H. K. 1988 Generalised vortex rings with and without swirl. Fluid Dyn. Res 3, 22–30.Google Scholar
Prandtl, L. 1952 Essentials of Fluid Dynamics. Blackie.
Pumir, A. & Siggia, E. D. 1992 Development of singular solutions of the axisymmetric Euler equations. Phys. Fluids A 4, 1472–1491.Google Scholar
Schlichting, H. 1979 Boundary Layer Theory. McGraw Hill.
Townsend, A. A. 1976 The Structure of Turbulence Shear Flow. Cambridge University Press.
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Vallis, G. K., Carnevale, G. E. & Young, W. R. 1989 Extremal energy properties and constructions of stable solutions of the Euler equations. J. Fluid Mech. 207, 133–152.Google Scholar
van Dyke, M. 1982 An Album of Fluid Motion. Parabolic.
Vives, C. & Perry, C. 1988 Solidification of pure metal in the presence of rotating flows. In Liquid Metal Flows: Magnetohydrodynamics and Applications (ed. H. Branover & H. Mond). Progress in Astronautics and Aeronautics, pp. 515–535. AIAA.
Vladimirov, V. A. 1985 Example of the equivalence of density stratification and rotation. Sov. Phys. Dokl. 30, (9)1 748–750.Google Scholar