Abstract:
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.