Nonmodal energy growth and optimal perturbations in compressible plane Couette flow

Abstract

Nonmodal transient growth studies and estimation of optimal perturbations have been made for the compressible plane Couette flow with three-dimensional disturbances. The maximum amplification of perturbation energy over time, $G_{\max}$, is found to increase with increasing Reynolds number ${\it Re}$, but decreases with increasing Mach number $M$. More specifically, the optimal energy amplification $G_{\rm opt}$ (the supremum of $G_{\max}$ over both the streamwise and spanwise wavenumbers) is maximum in the incompressible limit and decreases monotonically as $M$ increases. The corresponding optimal streamwise wavenumber, $\alpha_{\rm opt}$, is non-zero at M=0, increases with increasing $M$, reaching a maximum for some value of $M$ and then decreases, eventually becoming zero at high Mach numbers. While the pure streamwise vortices are the optimal patterns at high Mach numbers, the modulated streamwise vortices are the optimal patterns for low-to-moderate values of the Mach number. Unlike in incompressible shear flows, the streamwise-independent modes in the present flow do not follow the scaling law $G(t/{\it Re}) \sim {\it Re}^2$, the reasons for which are shown to be tied to the dominance of some terms in the linear stability operator. Based on a detailed nonmodal energy analysis, we show that the transient energy growth occurs due to the transfer of energy from the mean flow to perturbations via an inviscid {\it algebraic} instability. The decrease of transient growth with increasing Mach number is also shown to be tied to the decrease in the energy transferred from the mean flow ($\dot{\mathcal E}_1$) in the same limit.