Energy estimate of critical Reynolds numbers in the supersonic Couette flow of a vibrationally excited diatomic gas

The supersonic plane Couette flow of a vibrationally excited diatomic gas is investigated within the energy theory of hydrodynamic stability. The flow was described by a system of equations of two-temperature aerodynamics, which takes into account the dependence of the transport coefficients on the flow temperature. The corresponding spectral problem for the critical Reynolds number $\mathrm{Re_{cr}}$ determining the possible start of the laminar-turbulent transition was solved numerically using the method of collocations and $\mathrm{QZ}$-algorithm. The calculations showed that in the supersonic range, when $M > 1$, the calculated values $\mathrm{Re_{cr}}$ may exceed the corresponding values for subsonic Mach numbers $M > 1$ by about two orders of magnitude. Investigation of how $\mathrm{Re_{cr}}$ is affected by changes in the degree of vibrational energy excitation of gas molecules, vibrational relaxation time, bulk viscosity, and Mach number showed that the greatest impact on the increase in $\mathrm{Re_{cr}}$ at $M>1$ is exerted by the growth of the Mach number (compressibility). In the range of $M = 2\div5$, critical Reynolds numbers increase more than by an order of magnitude. However, the excitation of vibrational modes of gas molecules and the vibrational relaxation time which determine main effects at $M \leqslant 1$ have an effect at the same level with the transition to the supersonic regime.