On steady compressible flows with compact vorticity; the compressible Stuart vortex

Abstract

Numerical and analytical solutions to the steady compressible Euler equations corresponding to a compressible analogue of the linear Stuart vortex array are presented. These correspond to a homentropic continuation, to finite Mach number, of the Stuart solution describing a linear vortex array in an incompressible fluid. The appropriate partial differential equations describing the flow correspond to the compressible homentropic Euler equations in two dimensions, with a prescribed vorticity–density–streamfunction relationship. In order to construct a well-posed problem for this continuation, it was found, unexpectedly, to be necessary to introduce an eigenvalue into the vorticity–density–streamfunction equation. In the Rayleigh–Janzen expansion of solutions in even powers of the free-stream Mach number M_∞, this eigenvalue is determined by a solvability condition. Accurate numerical solution by both finite-difference and spectral methods are presented for the compressible Stuart vortex, over a range of M_∞, and of a parameter corresponding to a confined mass-flow rate. These also confirm the nonlinear eigenvalue character of the governing equations. All solution branches followed numerically were found to terminate when the maximum local Mach number just exceeded unity. For one such branch we present evidence for the existence of a very small range of M_∞ over which smooth transonic shock-free flow can occur.