Energy Theory of Nonlinear Stability of Plane Shear Flows of Thermally Nonequilibrium Gas

Abstract

The energy stability theory extended by the authors to the case of compressible flows of a vibrationally excited molecular gas is used to study stability of a subsonic Couette flow. Universal approach is developed for derivation of equations of the energy balance of disturbances for energy functionals. Based on these equations variational problems are posed for determining the critical Reynolds number of the possible beginning of the laminar-turbulent transition. Their asymptotic solutions are obtained in the limit of long-wave disturbances and yield an explicit dependence of the critical Reynolds number on the bulk viscosity coefficient, Mach number, and vibrational relaxation time. Neutral stability curves are calculated for arbitrary wavenumbers on the basis of the numerical solution of eigenvalue problems. It is shown that the minimum critical Reynolds numbers in realistic (for diatomic gases) ranges of flow parameters increase with increasing bulk viscosity coefficient, Mach number, vibrational relaxation time, and degree of excitation of vibrational modes. The results obtained in the study qualitatively confirm the asymptotic estimates for critical Reynolds number.