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Speaker: Federico Fuentes, Catholic University of Chile
Title: Global stability of fluid flows despite transient growth of energy
Abstract: A fundamental question in fluid stability is whether a laminar flow governed by the incompressible Navier-Stokes equations is nonlinearly stable to all perturbations. The typical way to verify this type of stability, called the energy method, is to show that the energy of a perturbation must decay monotonically under a certain Reynolds number called the energy stability limit. The energy method is known to be overly conservative in many systems, such as in plane Couette flow, meaning the flow is suspected to be globally stable even past this limit. Here, we present a methodology to computationally construct Lyapunov functionals more general than the energy which allow proving global stability of a fluid flow, even for Reynolds numbers where transient energy growth is observed. These new Lyapunov functions are not restricted to being quadratic, but are instead high-order polynomials that depend explicitly on the spectrum of the velocity field in the eigenbasis of the energy stability operator. The methodology involves numerically computing energy eigenmodes and using them to solve a convex optimization problem through a semidefinite program (SDP) constrained by sums-of-squares polynomial ansatzes. We then apply this methodology to 2D plane Couette flow and under certain conditions we find a global stability limit higher than the energy stability limit. For this specific flow, this was the first improvement in over a century.