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

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.

submitted copy (33 pages, 9 figures)- accepted copy on first author's website

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Cited by in corpus (31)

• A Validated Nonlinear Kelvin-Helmholtz Benchmark for Numerical Hydrodynamics
• Optimal Control of Active Nematics
• An adjoint-based approach for finding invariant solutions of Navier-Stokes equations
• Nonlinear hydrodynamic instability and turbulence in pulsatile flow
• Edge states as mediators of bypass transition in boundary-layer flows
• Bounding extreme events in nonlinear dynamics using convex optimization
• Disruption of SSP/VWI states by a stable stratification
• Structured input-output analysis of transitional wall-bounded flows
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• Nonlinear Stability Analysis of Transitional Flows using Quadratic Constraints
• Experiments on transient growth of turbulent spots
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• Optimising energy growth as a tool for finding exact coherent structures
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• Computing Resilience Measures in Dynamical Systems
• Unraveling Self-Similar Energy Transfer Dynamics: a Case Study for 1D Burgers System
• On nonlinear transitions, minimal seeds and exact solutions for the geodynamo