Global stability of fluid flows despite transient growth of energy

TL;DR

This paper introduces a novel polynomial optimization method to verify the global stability of fluid flows, overcoming limitations of traditional energy-based approaches, and successfully applies it to 2D plane Couette flow at high Reynolds numbers.

Contribution

It develops a broadly applicable polynomial optimization approach to construct non-quadratic Lyapunov functions for stability analysis, surpassing classical energy methods.

Findings

Verified global stability of 2D plane Couette flow at high Reynolds numbers

First to demonstrate stability beyond classical energy thresholds

Introduces a new computational method for nonlinear stability verification

Abstract

Verifying nonlinear stability of a laminar fluid flow against all perturbations is a central challenge in fluid dynamics. Past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. None show stability for the many flows that seem to be stable despite these energies growing transiently. Here a broadly applicable method to verify global stability of such flows is presented. It uses polynomial optimization computations to construct non-quadratic Lyapunov functions that decrease monotonically. The method is used to verify global stability of 2D plane Couette flow at Reynolds numbers above the energy stability threshold found by Orr in 1907. This is the first global stability result for any flow that surpasses the energy method.