On stability of stratified compressible shear flows

In this paper we study the linear normal mode stability of inviscid, compressible shear flows with variable temperature distribution in the presence of gravity. First, for plane, parallel flows, we prove the equivalent of Squire's theorem, namely, that for every unstable three-dimensional disturbance there corresponds a more unstable two-dimensional disturbance. Consequently, we study the instability problem only to two-dimensional disturbances and we obtain a sufficient condition for stability and an estimate for the growth rate of an unstable mode. Then, we prove the semicircle theorem. Moreover, we obtain instability regions to subsonic and a class of supersonic disturbances which depend on the Mach number, the wave number and the depth of the fluid layer. Finally, with the help of a set of transformations, we reduce the problem of non-parallel flows to the problem of parallel flows and obtain many results for the stability problem for non-parallel flows.