The compressible vortex pair

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Abstract
We consider the steady self-propagation with respect to the fluid at infinity of two equal symmetrically shaped vortices in a compressible fluid. Each vortex core is modelled by a region of stagnant constant-pressure fluid bounded by closed constant-pressure, constant-speed streamlines of unknown shape. The external flow is assumed to be irrotational inviscid isentropic flow of a perfect gas. The flow is therefore shock free but may be locally supersonic. The nonlinear free-boundary problem for the vortex-pair flow is formulated in the hodograph plane of compressible-flow theory, and a numerical solution method based on finite differences is described. Specific results are presented for a range of parameters which control the flow, namely the Mach number of the pair translational motion and the fluid speed on each vortex bounding streamline. Perturbation-theory predictions are developed, valid for vortices of small core radius when the pair Mach number is much less than unity. These are in good agreement with the hodograph-plane calculations. The numerical and the perturbation-theory results together confirm the recently discovered (Barsony-Nagy, Er-El & Yungster 1987) existence of continuous shock-free transonic compressible flows with embedded vortices. For the vortex-pair geometry studied, solution branches corresponding to physically acceptable flows that could be calculated using the present hodograph-plane numerical method were found to be terminated when either the flow on the streamline of symmetry separating the vortiqes tends to become superonic or when limiting lines appear in the hodograph plane giving a locally multivalued mapping to the physical plane.