The field due to a pair of line vortices in a compressible fluid

Abstract

The problem considered is that of a self-propagating pair of line vortices, of equal and opposite strengths, in a compressible inviscid fluid. An asymptotic solution is sought in the limit of small propagation speed compared with the sound speed in the medium. In most of the fluid region, the elementary incompressible flow solution limit is enhanced by a Rayleigh–Janzen approximation. This approximation fails at points that are either far from, or very close to, a vortex. For distant points, rescaled outer variables lead to an approximation that corresponds to the field induced by a moving dipole in compressible fluid. The main approximation also fails near each vortex line, where the analysis of Barsony-Nagy, Er-El & Yungster (J. Fluid Mech. vol. 178, 1987, p. 367) is used to express the local flow field in terms of a hypergeometric function. A particular feature of the problem is the propagation parameter $P$, which is proportional to $U' h'/K$, where $U'$ and $2h'$ denote the propagation speed and separation of the vortices and $K$ is the circulation. The parameter $P$ is a function of the Mach number $M$ and has the asymptotic value unity in the limit of incompressible flow. The analysis leads to the conclusion that $P=1+o(M^2)$ for small values of $M$; that is, the propagation number is unchanged to order $O(M^2)$. This differs from earlier work, which predicted the asymptotic development $P\sim 1-M^2/4$.