Global stability of fluid flows despite transient growth of energy

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.

6 pages + 4-page supplement, 3 figures

References in corpus (2)

• Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods
• On the existence of two-dimensional nonlinear steady states in plane Couette flow

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• A study of the double pendulum using polynomial optimization
• Stability of plane Couette and Poiseuille flows rotating about the streamwise axis
• A PIE Representation of Scalar Quadratic PDEs and Global Stability Analysis Using SDP