Energy Method and Stability of Shear Flows: an Elementary Tutorial

TL;DR

This paper offers an accessible tutorial on the energy method for analyzing the nonlinear stability of shear flows, detailing the formulation of eigenvalue problems and numerical solutions using Chebyshev polynomials.

Contribution

It provides a clear, step-by-step pedagogical approach to stability analysis of shear flows, including formulation and numerical solution of eigenvalue problems with Mathematica code.

Findings

Eigenvalue problems formulated for transverse and longitudinal modes

Numerical solutions achieved using Chebyshev polynomial-based Galerkin method

Critical analysis of stability thresholds for plane Poiseuille and Couette flows

Abstract

This paper provides a pedagogical introduction to the classical nonlinear stability analysis of the plane Poiseuille and Couette flows. The whole procedure is kept as simple as possible by presenting all the logical steps involved in the application of the energy method and leading to the Euler-Lagrange equations. Then, the eigenvalue problems needed for the evaluation of the nonlinear energy threshold of the Reynolds number for stability are formulated for transverse modes and for longitudinal modes. Such formulations involve the streamfunction and, in the case of longitudinal modes, also the streamwise component of velocity. An accurate numerical solution of the eigenvalue problems, based on Galerkin's method of weighted residuals with the test functions expressed in terms of Chebyshev polynomials, is discussed in details. The numerical codes developed for the software Mathematica 14 (\copyright{} Wolfram Research, Inc.) are also presented. A critical analysis of the obtained results is finally proposed.