Variational Approach to Necessary and Sufficient Stability Conditions for Inviscid Shear Flow

TL;DR

This paper introduces a variational method to determine the linear stability of inviscid parallel shear flows, providing a necessary and sufficient condition that simplifies analysis compared to traditional approaches.

Contribution

It develops a novel variational framework based on Hamiltonian mechanics to establish stability criteria, applicable to a broad class of hydrodynamic problems.

Findings

The method characterizes instability via positive eigenvalues of a selfadjoint operator.

Stability conditions are derived by maximizing a quadratic form.

The approach is both theoretically sound and numerically efficient.

Abstract

A necessary and sufficient condition for linear stability of inviscid parallel shear flow is formulated by a novel variational method, where the velocity profile is assumed to be monotonic and analytic. Unstable eigenvalues of the Rayleigh equation are shown to be associated with positive eigenvalues of a certain selfadjoint operator. The stability is therefore simply determined by maximizing a quadratic form, which is theoretically and numerically more tractable than directly solving the Rayleigh equation. This variational approach is based on the Hamiltonian nature of the inviscid fluid and will be applicable to other hydrodynamic stability problems.