Linear instability of compressible plane Couette flows

The stability of compressible plane Couette flow, which is a simple case of hypersonic wall-bounded shear flows, is not well understood even though incompressible Couette flow has been studied extensively by linear stability analysis and shown to be stable to linear disturbances. As a first step in studying the stability of three-dimensional hypersonic boundary layers, this paper studies the temporal stability of compressible Couette flows with a perfect gas model. The full compressible linear stability equations are solved by both a high-order finite-difference global method and a Chebyshev spectral collocation global method. The accuracy of the linear stability codes are validated by comparing the solutions from the two approaches with known solutions for compressible boundary layer. Unstable first and second modes are found for compressible Couette flow at finite Reynolds numbers. The inviscid second modes are found to be the dominant instability. The results are consistent with the prediction by Duck et al. that unstable modes are possible for compressible Couette flow. The second modes are found to be two-dimensional and are stabilized by viscousity. The Mach number corresponding to the most unstable second modes increase with Reynolds number but has a finite limit. The second modes are destabilized first by wall cooling and then stabilized by further reduction of the lower wall temperature. This is different with what is known for boundary layer second mode instability. The first mode instability characteristics are also discussed in this paper.