Instability of shear layers and Prandtl's boundary layers

TL;DR

This paper investigates the nonlinear instability of shear and Prandtl boundary layers in incompressible Navier-Stokes flows, revealing conditions under which these layers become unstable and describing the resulting complex flow structures.

Contribution

It proves generic nonlinear instability of shear and boundary layers at high Reynolds numbers and shows the failure of Prandtl's boundary layer theory for Sobolev regular initial data.

Findings

Shear layers are nonlinearly unstable at high Reynolds numbers.

Prandtl's boundary layer analysis fails for Sobolev regular initial data.

Secondary instabilities lead to flow sublayers and complexity.

Abstract

This paper is devoted to the study of the nonlinear instability of shear layers and of Prandtl's boundary layers, for the incompressible Navier Stokes equations. We prove that generic shear layers are nonlinearly unstable provided the Reynolds number is large enough, or equivalently provided the viscosity is small enough. We also prove that, generically, Prandtl's boundary layer analysis fails for initial data with Sobolev regularity. In both cases we give an accurate description of the first instability which arises. In some cases a secondary instability appears, leading to several sublayers and to an unexpected complexity of the flow.