Level crossing instabilities in inviscid isothermal compressible Couette flow

TL;DR

This paper analyzes the linear stability of inviscid, isothermal compressible Couette flow, revealing stability transitions, eigenmode behaviors, and the existence of a continuous spectrum, with implications for understanding flow instabilities.

Contribution

It introduces a detailed stability analysis of compressible Couette flow, identifying eigenmode transitions, stability boundaries, and the spectral structure, including continuous modes.

Findings

Eigenmodes are neutrally stable at small wavenumber and Mach number.

Stability transitions occur with increasing wavenumber or Mach number.

A continuous spectrum of neutrally stable eigenmodes exists.

Abstract

We study the linear stability of inviscid steady parallel flow of an ideal gas in a channel of finite width. Compressible isothermal two-dimensional monochromatic perturbations are considered. The eigenvalue problem governing density and velocity perturbations is a compressible version of Rayleigh's equation and involves two parameters: a flow Mach number $M$ and the perturbation wavenumber $k$. For an odd background velocity profile, there is a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry and growth rates $\gamma$ come in symmetrically placed 4-tuples in the complex eigenplane. Specializing to uniform background vorticity Couette flow, we find an infinite tower of noninflectional eigenmodes and derive stability theorems and bounds on growth rates. We show that eigenmodes are neutrally stable for small $k$ and small $M$ but that they otherwise display an infinite sequence of stability transitions with increasing $k$ or $M$. Using a search algorithm based on the Fredholm alternative, we find that the transitions are associated to level crossings between neighboring eigenmodes. Repeated level crossings result in windows of instability. For a given eigenmode, they are arranged in a zebra-like striped pattern on the $k$-$M$ plane. A canonical square-root power law form for $\gamma(k,M)$ in the vicinity of a stability transition is identified. In addition to the discrete spectrum, we find a continuous spectrum of eigenmodes that are always neutrally stable but fail to be smooth across critical layers.