Estimating stability margins for transitional flows using quadratic constraints

Laminar to turbulent transition can be detrimental to many engineering systems. Yet, predicting precisely when a flow will transition has been notoriously difficult. In this work, we propose a method for estimating stability margins of nonlinear fluid systems. The approach considers the stability of the linear dynamics subject to quadratic constraints that describe input-output properties of nonlinear terms. We demonstrate the approach on the Burgers equation and a reduced-order model of plane Couette flow. Our results indicate that the proposed method reduces conservatism in estimating regions of attraction and permissible perturbation amplitudes compared to related quadratic-constraint-based analysis techniques. Additionally, we demonstrate that the approach, while introducing some conservatism, has lesser computational complexity than commonly used stability analysis techniques, including sum-of-squares optimization and direct-adjoint looping methods.