Vortical mode instability of shear layers with temperature and density gradients

Abstract

The linear inviscid instability of a low Mach number shear layer with imposed temperature and density gradients is analyzed in terms of its vortical, acoustic, and entropy modes. The conditions under which the acoustic mode is decoupled from both the vortical and entropy modes in the linear instability region of the shear layer are first discussed. It is then shown that for a two-dimensiona l parallel shear layer, satisfying these conditions, the vortical mode also decouples from the entropy mode. The instability of such a shear layer is then identified with the instability of the vortical mode. Two examples that illustrate the strong influence of the steady-state temperature profile on the growth rates of the unstable vortical mode are presented and discussed. Finally, it is demonstrated that the initial value of the unstable vortical mode is determined by the acoustic mode. Nomenclature A = eigenfunction in Eq. (12) a — local speed of sound a, = reference speed of sound B ' = Bernoulli enthalpy fluctuation B = Fourier component of B ' C — constant in the solution of the unstable vortical wave Cf, = specific heat at constant pressure cpl} = phase velocity of vortical wave F —Fourier