Linear stability, transient energy growth and the role of viscosity stratification in compressible plane Couette flow

Abstract

Linear stability and the non-modal transient energy growth in compressible plane Couette flow are investigated for two prototype mean flows: (a) the {\it uniform shear} flow with constant viscosity, and (b) the {\it non-uniform shear} flow with {\it stratified} viscosity. Both mean flows are linearly unstable for a range of supersonic Mach numbers ($M$). For a given $M$, the critical Reynolds number ($Re$) is significantly smaller for the uniform shear flow than its non-uniform shear counterpart. An analysis of perturbation energy reveals that the instability is primarily caused by an excess transfer of energy from mean-flow to perturbations. It is shown that the energy-transfer from mean-flow occurs close to the moving top-wall for ``mode I'' instability, whereas it occurs in the bulk of the flow domain for ``mode II''. For the non-modal analysis, it is shown that the maximum amplification of perturbation energy, $G_{\max }$, is significantly larger for the uniform shear case compared to its non-uniform counterpart. For $\alpha =0$, the linear stability operator can be partitioned into $\mathcal{L}\sim \bar{\mathcal{L}}+R{e}^2{\mathcal{L}}_p$, and the $Re$-dependent operator ${\mathcal{L}}_p$ is shown to have a negligibly small contribution to perturbation energy which is responsible for the validity of the well-known quadratic-scaling law in uniform shear flow: $G(t/Re)\sim {Re}^2$. A reduced inviscid model has been shown to capture all salient features of transient energy growth of full viscous problem. For both modal and non-modal instability, it is shown that the {\it viscosity-stratification} of the underlying mean flow would lead to a delayed transition in compressible Couette flow.

References in corpus (2)

• Fractal Stability Border in Plane Couette Flow
• Nonmodal energy growth and optimal perturbations in compressible plane Couette flow

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