On the linear stability of a compressible inviscid parallel shear flow

Abstract

By applying an approach similar to that used in the Miles-Howard theory [1], [2] we derive simple constraints on the phase speedc r * of the neutral three-dimensional (3-D) monochromatic disturbances in an inviscid compressible parallel two-dimensional (2-D) shear flow. It is shown that for a boundary layer flow [a0*(y*)]2 —[U0*(y*)—cr*]2 must have a zero in [y1*,y2*) for the neutral 2-D modes whose phase speedcr* does not belong to the range ofU0*(y*). For the unstable waves the argument of Chimonas [3] applies leading to the Howard semi-circle theorem. HereU0*(y*) anda0*(y*) are the dimensional base velocity and local sonic speed respectively. It is suggested that hypersonic flows possess vertically highly undulated unstable normal modes.