Linear stability of viscous supersonic plane Couette flow

The linear stability of viscous compressible plane Couette flow is not well understood even though the stability of incompressible Couette flow has been studied extensively and has been shown to be stable to linear disturbances. In this paper, the viscous linear stability of supersonic Couette flow for a perfect gas governed by Sutherland viscosity law was studied using two global methods to solve the linear stability equations. The two methods are a fourth-order finite-difference method and a spectral collocation method. Two families of wave modes, modes I and II, were found to be unstable at finite Reynolds numbers, where mode II is the dominant instability among the unstable modes. These two families of wave modes are acoustic modes created by sustained acoustic reflections between a wall and a relative sonic line when the mean flow in the local region is supersonic with respect to the wave velocities. The effects of viscosity on the stability of the two families of acoustic modes were studied by comparing the viscous results at finite Reynolds numbers with the inviscid results published by Duck, Erlebacher, and Hussaini [J. Fluid Mech. 258 (1994)]. It was shown that viscosity plays a destabilizing role in both mode I and mode II stability for supersonic Couette flow in a range of Reynolds numbers and wavenumbers. The effects of compressibility, three-dimensionality, and wall cooling on the two wave families were also studied. The stability of supersonic Couette flow was found to be different from that of the unbounded boundary layers in many aspects because of the effects of additional boundary conditions at the upper wall.