Self-similar, slightly compressible, free vortices

Abstract

Exact and numerical similarity solutions for compressible perturbations to an incompressible, two-dimensional, axisymmetric vortex reference flow are presented. The reference flow consists of a set of two-dimensional, self-similar, incompressible vortices. Similarity variables, which give explicit expressions for the decay rates of the velocities and thermodynamic variables in the vortex flows, are used to reduce the governing partial differential equations to a set of ordinary differential equations. The ODEs are solved analytically and numerically for a Taylor vortex reference flow, and numerically for an Oseen vortex reference flow. The solutions are employed to study the dependences of the temperature, density, entropy, dissipation and radial velocity on the Prandtl number. Additionally, several integral relations, which allow one to trace the energy transfer in a slightly compressible vortex, are derived.