An optimisation approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar

TL;DR

This paper presents an optimization method for analyzing nonlinear stability and transition to turbulence in fluid flows, bridging linear and nonlinear approaches and applicable to high-dimensional systems.

Contribution

It introduces a simple optimization technique for nonlinear stability analysis, applicable to both low-dimensional ODEs and high-dimensional fluid systems like Navier-Stokes.

Findings

Technique successfully applied to a bistable 2-ODE system

Extended to Navier-Stokes equations for turbulence transition

Bridges gap between linear and nonlinear stability analysis

Abstract

This article introduces, and reviews recent work using, a simple optimisation technique for analysing the nonlinear stability of a state in a dynamical system. The technique can be used to identify the most efficient way to disturb a system such that it transits from one stable state to another. The key idea is introduced within the framework of a finite-dimensional set of ordinary differential equations (ODEs) and then illustrated for a very simple system of 2 ODEs which possesses bistability. Then the transition to turbulence problem in fluid mechanics is used to show how the technique can be formulated for a spatially-extended system described by a partial differential equation (the well-known Navier-Stokes equation). Within that context, the optimisation technique bridges the gap between (linear) optimal perturbation theory and the (nonlinear) dynamical systems approach to fluid flows. The fact that the technique has now been recently shown to work in this very high dimensional setting augurs well for its utility in other physical systems.