Onset of nonlinear instabilities in monotonic viscous boundary layers

TL;DR

This paper investigates the nonlinear stability of shear layer profiles in Navier-Stokes equations near boundaries, focusing on how nonlinear interactions influence the growth of linear instabilities and proposing that perturbations saturate at small magnitudes.

Contribution

It analyzes the effect of cubic nonlinear interactions on linear instability growth in viscous boundary layers, providing conjectures on the saturation magnitude of perturbations.

Findings

Nonlinear interactions can tame linear instabilities in certain shear profiles.

Perturbations may grow only up to an order of (.25) in viscosity.

The study suggests small perturbations form small rolls near the boundary.

Abstract

In this paper we study the nonlinear stability of a shear layer profile for Navier Stokes equations near a boundary. This question plays a major role in the study of the inviscid limit of Navier Stokes equations in a bounded domain as the viscosity goes to $0$. The stability of a shear layer for Navier Stokes equations depends on its stability for Euler equations. If it is linearly unstable for Euler, then it is known that it is also nonlinearly unstable for Navier Stokes equations provided the viscosity is small enough: an initial perturbation grows until it reaches $O(1)$ in $L^\infty$ norm. If it is linearly stable for Euler, the situation is more complex, since the viscous instability is much slower, with growth rates of order $O(\nu^{-1/4})$ only (instead of $O(1)$ in the first case). It is not clear whether linear instabilities fully develop till they reach a magnitude of order $O(1)$ or whether they are damped by the nonlinearity and saturate at a much smaller magnitude, or order $O(\nu^{1/4})$ for instance. In this paper we study the effect of cubic interactions on the growth of the linear instability. In the case of the exponential profile and Blasius profile we obtain that the nonlinearity tame the linear instability. We thus conjecture that small perturbations grow until they reach a magnitude $O(\nu^{1/4})$ only, forming small rolls in the critical layer near the boundary. The mathematical proof of this conjecture is open.