On the Rayleigh theorem for inflectional velocity instability of inviscid flows

TL;DR

This paper rigorously proves that the classical Rayleigh Theorem's necessary condition for inviscid flow instability is incorrect and reveals that 2D disturbances must evolve into 3D spiral waves, leading to streamwise vortex formation.

Contribution

It provides a rigorous proof challenging the classical Rayleigh Theorem and explains the 3D nature of disturbances in inviscid flows.

Findings

Rayleigh Theorem on inflectional velocity instability is incorrect.

Disturbances in 2D inviscid flows are necessarily 3D.

Streamwise vortices originate from spiral waves after T-S wave breakdown.

Abstract

It is exactly proved that the classical Rayleigh Theorem on inflectional velocity instability is wrong which states that the necessary condition for instability of inviscid flow is the existence of an inflection point on the velocity profile. It is shown that the disturbance amplified in 2D inviscid flows is necessarily 3D. After the break down of T-S wave in 2D parallel flows, the disturbance becomes a type of spiral waves which proceed along the streamwise direction. This is just the origin of formation of streamwise vortices.