A linear stability method based on generalized hydrodynamics: application to high-speed plane Couette flow beyond continuum regime

Linear stability analysis of compressible shear flows beyond continuum regime attracts increasing attention in recent years accounting for high-altitude applications. In this paper, the temporal stability of high-speed plane Couette flow in near-continuum regime is addressed. A novel linear stability method based on the generalized hydrodynamic equations (GHE) with slip boundary conditions is developed within the 13-moment framework. Meanwhile, the linear stability equations of the nonlinear coupled constitutive relations (NCCR) and the Navier--Stokes (NS) equations are also given. Based on the developed methods, the linear stability of plane Couette flows at $Ma=2.5$--10 and Kn=2.7× 10⁻⁵--0.07 are analyzed, and the rarefaction effects on the base flow, eigenspectrum, mode synchronization and least stable mode are investigated. The results show that the GHE with slip boundary conditions agrees well with direct simulation Monte Carlo (DSMC) results in predicting base flow profiles of velocity, temperature, all stress and heat flux components, while the NS equations show considerable differences, especially for capturing the nonlinear normal stress and tangential heat flux. The GHE-based linear stability theory (LST) recovers to the classical NS-based LST in the continuum limit, while they show considerable differences when $Kn$ reaches $O(0.01)$. As rarefaction degree increases, the synchronization between acoustic modes and viscous modes becomes weak, and all modes become stable in rarefied plane Couette flow. The surface slip effect and nonlinear transport increase the disturbance phase speeds, while they have different effects on the growth rate of different modes. Generally, rarefaction effects increase the growth rate of the least stable Mode II at low wavenumbers, and show more pronounced effects on higher modes and high-wavenumber disturbances.