Stability Criteria of 3D Inviscid Shears

TL;DR

This paper investigates the stability of 3D inviscid shear flows, deriving new criteria that extend classical 2D theories, and explores the limiting behavior of steady coherent structures in high Reynolds number flows.

Contribution

It introduces novel stability criteria for 3D inviscid shears, expanding understanding beyond classical 2D Rayleigh theory and analyzing their limiting shear states.

Findings

Derived general stability criteria for 3D inviscid shears.

Showed that 2D limiting shear must be the classical laminar shear.

Connected steady 3D structures to Euler equations stability analysis.

Abstract

The classical plane Couette flow, plane Poiseuille flow, and pipe Poiseuille flow share some universal 3D steady coherent structure in the form of "streak-roll-critical layer". As the Reynolds number approaches infinity, the steady coherent structure approaches a 3D limiting shear of the form ($U(y,z), 0, 0$) in velocity variables. All such 3D shears are steady states of the 3D Euler equations. This raises the importance of investigating the stability of such inviscid 3D shears in contrast to the classical Rayleigh theory of inviscid 2D shears. Several general criteria of stability for such inviscid 3D shears are derived. In the Appendix, an argument is given to show that a 2D limiting shear can only be the classical laminar shear.