Absolute and convective instabilities of an inviscid compressible mixing layer: Theory and applications

Abstract

This study aims to examine the effect of compressibility on unbounded and parallel shear flow linear instabilities. This analysis is of interest for industrial, geophysical, and astrophysical flows. We focus on the stability of a wavepacket as opposed to previous single-mode stability studies. We consider the notions of absolute and convective instabilities first used to describe plasma instabilities. The compressible-flow modal theory predicts instability whatever the Mach number. Spatial and temporal growth rates and Reynolds stresses nevertheless become strongly reduced at high Mach numbers. The evolution of disturbances is driven by Kelvin–Helmholtz instability that weakens in supersonic flows. We wish to examine the occurrence of absolute instability, necessary for the appearance of turbulent motions in an inviscid and compressible two-dimensional mixing layer at an arbitrary Mach number subject to a three-dimensional disturbance. The mixing layer is defined by a parametric family of mean-velocity and temperature profiles. The eigenvalue problem is solved with the help of a spectral method. We ascertain the effects of the distribution of temperature and velocity in the mixing layer on the transition between convective and absolute instabilities. It appears that, in most cases, absolute instability is always possible at high Mach numbers provided that the ratio of slow-stream temperature over fast-stream temperature may be less than a critical maximal value but the temporal growth rate present in the absolutely unstable zone remains small and tends to zero at high Mach numbers. The transition toward a supersonic turbulent regime is therefore unlikely to be possible in the linear theory. Absolute instability can be also present among low-Mach-number coflowing mixing layers provided that this same temperature ratio may be small, but nevertheless, higher than a critical minimal value. Temperature distribution within the mixing layer also has an effect on the growth rate, this diminishes when the slow stream is heated. These results are applied to the dynamics of mixing layers in the interstellar medium and to the dynamics of the heliopause, frontier between the interstellar medium, and the solar wind.