Nonlinear Stability Analysis of Transitional Flows using Quadratic Constraints

Aniketh Kalur, Peter Seiler, Maziar S. Hemati

Published 2020-04-11Version 1

The dynamics of transitional flows are governed by an interplay between the non-normal linear dynamics and quadratic nonlinearity in the Navier-Stokes equations. In this work, we propose a framework for nonlinear stability analysis that exploits the fact that nonlinear flow interactions are constrained by the physics encoded in the nonlinearity. In particular, we show that nonlinear stability analysis problems can be posed as convex optimization problems based on Lyapunov matrix inequalities and a set of quadratic constraints that represent the nonlinear flow physics. The proposed framework can be used to conduct global and local stability analysis as well as transient energy growth analysis. The approach is demonstrated on the low-dimensional Waleffe-Kim-Hamilton model of transition and sustained turbulence. Our analysis correctly determines the critical Reynolds number for global instability. We further show that the lossless (energy conservation) property of the nonlinearity is destabilizing and serves to increase transient energy growth. Finally, we show that careful analysis of the multipliers used to enforce the quadratic constraints can be used to extract dominant nonlinear flow interactions that drive the dynamics and associated instabilities.