Energy estimate of the critical Reynolds numbers in a compressible Couette flow. Effect of bulk viscosity

A variational problem of determining the critical Reynolds number of the laminar-turbulent transition is numerically solved within the framework of the nonlinear energy theory of stability of compressible flows. Stability of various modes in the Couette flow of a compressible gas is estimated by the method of collocations. It is demonstrated that the minimum critical Reynolds numbers in the range of the ratio of the bulk viscosity $\eta_b$ to the shear viscosity $\eta$, which is realistic for diatomic gases, are reached for modes of streamwise disturbances. The critical Reynolds numbers increase as the bulk viscosity is increased in the interval $\eta_b=0-2\eta$, with the maximum increase in the limit being approximately 30%.