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On steady compressible flows with compact vorticity; the compressible Hill's spherical vortex

Moore, D. W. and Pullin, D. I. (1998) On steady compressible flows with compact vorticity; the compressible Hill's spherical vortex. Journal of Fluid Mechanics, 374 . pp. 285-303. ISSN 0022-1120. doi:10.1017/S0022112098002675.

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We consider steady compressible Euler flow corresponding to the compressible analogue of the well-known incompressible Hill's spherical vortex (HSV). We first derive appropriate compressible Euler equations for steady homentropic flow and show how these may be used to define a continuation of the HSV to finite Mach number M_∞ =U_∞/C_∞, where U_∞, C_∞ are the fluid velocity and speed of sound at infinity respectively. This is referred to as the compressible Hill's spherical vortex (CHSV). It corresponds to axisymmetric compressible Euler flow in which, within a vortical bubble, the azimuthal vorticity divided by the product of the density and the distance to the axis remains constant along streamlines, with irrotational flow outside the bubble. The equations are first solved numerically using a fourth-order finite-difference method, and then using a Rayleigh–Janzen expansion in powers of M^2_∞ to order M^4_∞. When M_∞ > 0, the vortical bubble is no longer spherical and its detailed shape must be determined by matching conditions consisting of continuity of the fluid velocity at the bubble boundary. For subsonic compressible flow the bubble boundary takes an approximately prolate spheroidal shape with major axis aligned along the flow direction. There is good agreement between the perturbation solution and Richardson extrapolation of the finite difference solutions for the bubble boundary shape up to M_∞ equal to 0.5. The numerical solutions indicate that the flow first becomes locally sonic near or at the bubble centre when M_∞ ≈ 0.598 and a singularity appears to form at the sonic point. We were unable to find shock-free steady CHSVs containing regions of locally supersonic flow and their existence for the present continuation of the HSV remains an open question.

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Additional Information:© 1998 Cambridge University Press. Received 5 December 1997 and in revised form 18 June 1998. Published online: 08 September 2000. D.I.P. was partly supported by a grant from the EPSERC of the UK. We acknowledge a referee who pointed out the relevance of the Munk-Prim transformation.
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Official Citation:On steady compressible flows with compact vorticity; the compressible Hill's spherical vortex D. W. MOORE and D. I. PULLIN Journal of Fluid Mechanics / Volume 374 / pp 285 - 303
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28837
Deposited By: Ruth Sustaita
Deposited On:18 Jan 2012 22:36
Last Modified:09 Nov 2021 17:01

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