Abstract

Verifying nonlinear stability of a laminar fluid flow against all perturbations is a classic challenge in fluid dynamics. All past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. This “energy method” cannot show global stability of any flow in which perturbation energy may grow transiently. For the many flows that allow transient energy growth but seem to be globally stable (e.g. pipe flow and other parallel shear flows at certain Reynolds numbers) there has been no way to mathematically verify global stability. After explaining why the energy method was the only way to verify global stability of fluid flows for over 100 years, I will describe a different approach that is broadly applicable but more technical. This approach, proposed in 2012 by Goulart and Chernyshenko, uses sum-of-squares polynomials to computationally construct non-quadratic Lyapunov functions that decrease monotonically for all flow perturbations. I will present a computational implementation of this approach for the example of 2D plane Couette flow, where we have verified global stability at Reynolds numbers above the energy stability threshold. This energy stability result for 2D Couette flow had not been improved upon since being found by Orr in 1907. The results I will present are the first verification of global stability – for any fluid flow – that surpasses the energy method. This is joint work with Federico Fuentes (Universidad Católica de Chile) and Sergei Chernyshenko (Imperial College London).