Analysis of inviscid shear instability of axisymmetric flows

TL;DR

This paper develops new analytical criteria to determine the stability of axisymmetric flows in annuli and pipes, improving classical results and extending recent theorems to predict flow stability effectively.

Contribution

It introduces novel sufficient conditions for stability and instability of axisymmetric flows, extending Kelvin-Arnol'd and hurdle theorems for inviscid flow analysis.

Findings

New stability criteria outperform classical results

Analytical conditions accurately predict neutral parameters

Criteria validated against eigenvalue computations

Abstract

Simple analytical criteria are derived to determine whether axisymmetric base flows in annuli and pipes are stable or unstable. Both axisymmetric and non-axisymmetric inviscid disturbances are considered. Our sufficient condition for stability improves upon the classical result of \cite{Batchelor_Gill_1962}, following the idea of the second Kelvin-Arnol'd stability theorem. A novel sufficient condition for instability is also derived by extending the recently proposed hurdle theorem for parallel flows \citep{Deguchi_Hirota_Dowling_2024}. These analytical criteria are applied to annular and pipe model flows and are shown to effectively predict the neutral parameters obtained from eigenvalue computations of the stability problem.