A fundamental question in fluid stability is whether a laminar flow 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. Here, we present a methodology to computationally construct Lyapunov functions more general than the energy, which is a quadratic function of the magnitude of the perturbation velocity. 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 solving a convex optimization problem through semidefinite programming (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 is the first improvement in over 110 years.

This is joint work with David Goluskin and Sergei Chernyshenko.