Journal of Fluid Mechanics

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

D. W. MOORE a1 and D. I. PULLIN a2
a1 Department of Mathematics, Imperial College of Science, Technology and Medicine, Queens Gate, London SW7 2BZ, UK
a2 Graduate Aeronautical Laboratories 105-50, California Institute of Technology, Pasadena, CA 91125, USA


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[infty infinity]=U[infty infinity]/C[infty infinity], where U[infty infinity], C[infty infinity] 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 M2[infty infinity] to order M4[infty infinity]. When M[infty infinity]>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[infty infinity] equal to 0.5. The numerical solutions indicate that the flow first becomes locally sonic near or at the bubble centre when M[infty infinity][approximate]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.

(Received December 5 1997)
(Revised June 18 1998)