Effect of Prandtl number on the linear stability of compressible Couette flow

Accurate prediction of laminar to turbulent transition in high-speed flows is a challenging task. Compressibility, and the resultant large variations in the transport properties can affect the transition process significantly. In this paper, we study the influence of Prandtl number, the ratio of momentum to heat diffusivity, in Couette flows at high Mach numbers. It is a part of an ongoing research programme to isolate and understand the transport property effects on the stability of high-speed flows. As a first step, we neglect the high-temperature effects and vary the Prandtl number in the range 0.9 to 0.2, by changing the relative magnitudes of viscosity and conductivity. A temporal linear stability analysis shows that the variation of phase speed with Prandtl number leads to synchronization between two acoustic modes, with peaks in growth rate at the synchronization points. Two types of branching patterns are observed depending on the Prandtl number, and the branch type determines which of the two modes is destabilized and which one is stabilized due to synchronization. Further, the mode shapes are either retained as earlier or interchanged between the two acoustic modes depending on the branching pattern. The stability diagrams for varying Mach and Reynolds numbers show a destabilizing role of decreasing the Prandtl number, both in terms of increased disturbance growth rates, and of larger regions of instability in the parameter space. It also results in a significant reduction in the critical Reynolds number of the flow, especially at high Mach numbers with wall cooling.