Linear stability of a compressible flow in a channel

Abstract

Summary Modal instabilities in a compressible flow through a channel at high Reynolds numbers are studied for three-dimensional (3D) perturbations. In addition to the Tollmien–Schlichting (TS) mode, there exist compressible modes in a channel flow that do not have a counterpart in the incompressible limit. The stability characteristics of these compressible modes, obtained through numerical calculations, are compared with boundary layer and Couette flows that have been previously studied. The dominant compressible instabilities in a channel flow are shown to be viscous in nature, in contrast to compressible boundary layer modes. For general compressible bounded-domain flows, a necessary condition for the existence of neutral modes in the inviscid limit is obtained, and this criterion is used to determine critical Mach numbers below which the compressible modes remain stable. This criterion also delineates a range of wave-angles which could go unstable at a specified Mach number. Asymptotic analysis is carried out for the lower and upper branches of the stability curve in the limit of high Reynolds number for both the T-S and the compressible modes. A common set of relations are identified for the scaling exponents, and the leading order eigenvalues for the unstable modes are obtained through an adjoint-based procedure. The asymptotic analysis shows that the stability boundaries for 3D perturbations at high Reynolds numbers can be calculated from the strain rate and the temperature of the base flow at the wall.