Nonlinear stability results for plane Couette and Poiseuille flows

TL;DR

This paper proves nonlinear stability of plane Couette and Poiseuille flows for all Reynolds numbers with respect to streamwise perturbations, and establishes bounds for tilted perturbations, improving previous results and aligning with experimental and numerical findings.

Contribution

It introduces a weighted $L_2$-energy method to establish nonlinear stability for all Reynolds numbers and extends stability bounds to tilted perturbations, surpassing earlier work.

Findings

Nonlinear stability holds for any Reynolds number with streamwise perturbations.

Stability bounds are derived for tilted perturbations based on Orr's critical Reynolds number.

Results align well with experimental and numerical data.

Abstract

In this article we prove, choosing an appropriately weighted $L_2$-energy equivalent to the classical energy, that the plane Couette and Poiseuille flows are nonlinearly stable with respect to streamwise perturbations for any Reynolds number. In this case the coefficient of time-decay of the energy is $\pi^2/(2 {\rm Re})$, and it is a bound from above of the time-decay of streamwise perturbations of linearized equations. We also prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less then ${\rm Re}_{Orr}/\sin \varphi$ when the perturbation is a tilted perturbation, i.e. 2D perturbations with wave vector which forms an angle $\varphi \in [0, \pi/2]$ with the direction ${\bf i}$ of the motion. ${\rm Re}_{Orr}$ is the Orr (1907) critical Reynolds number for spanwise perturbations which, for the Couette flow is ${\rm Re}_{Orr}=177.22$ and for the Poiseuille flow is ${\rm Re}_{Orr}=175.31$. In particular these results improve those obtained by Joseph (1966), who found for streamwise perturbations a critical nonlinear value of $82.6$ in the Couette case, and those obtained by Joseph and Carmi (1969), who found the value $99.1$ for plane Poiseuille flow for streamwise perturbations. The results we obtain here are, for any angle, in a good agreement with the experiments (see Prigent et al. 2003) and the numerical simulations (see Barckley and Tuckerman 2005, 2007).