On the stability of plane Couette-Poiseuille flow with uniform cross-flow

TL;DR

This paper investigates the linear stability of plane Couette-Poiseuille flow with uniform cross-flow, identifying stability regimes, mechanisms of stabilization and destabilization, and the role of Tollmien-Schlichting waves through spectral and energy analyses.

Contribution

It provides a detailed analysis of the stability behavior of Couette-Poiseuille flow with cross-flow, highlighting the mechanisms behind stabilization and destabilization at different flow parameters.

Findings

Existence of two stability regimes depending on cross-flow Reynolds number.

Stabilization occurs via velocity profile skewing and energy production decrease.

Destabilization is linked to resonant Tollmien-Schlichting wave interactions.

Abstract

We present a detailed study of the linear stability of plane Couette-Poiseuille flow in the presence of a cross-flow. The base flow is characterised by the cross flow Reynolds number, $R_{inj}$ and the dimensionless wall velocity, $k$. Squire's transformation may be applied to the linear stability equations and we therefore consider 2D (spanwise-independent) perturbations. Corresponding to each dimensionless wall velocity, $k\in[0,1]$, two ranges of $R_{inj}$ exist where unconditional stability is observed. In the lower range of $R_{inj}$, for modest $k$ we have a stabilisation of long wavelengths leading to a cut-off $R_{inj}$. This lower cut-off results from skewing of the velocity profile away from a Poiseuille profile, shifting of the critical layers and the gradual decrease of energy production. Cross-flow stabilisation and Couette stabilisation appear to act via very similar mechanisms in this range, leading to the potential for robust compensatory design of flow stabilisation using either mechanism. As $R_{inj}$ is increased, we see first destabilisation and then stabilisation at very large $R_{inj}$. The instability is again a long wavelength mechanism. Analysis of the eigenspectrum suggests the cause of instability is due to resonant interactions of Tollmien-Schlichting waves. A linear energy analysis reveals that in this range the Reynolds stress becomes amplified, the critical layer is irrelevant and viscous dissipation is completely dominated by the energy production/negation, which approximately balances at criticality. The stabilisation at very large $R_{inj}$ appears to be due to decay in energy production, which diminishes like $R_{inj}^{-1}$. Our study is limited to two dimensional, spanwise independent perturbations.