Stability and performance analysis of nonlinear and non-normal systems using quadratic constraints

We propose a system-theoretic approach for analyzing stability and transient energy growth performance of nonlinear fluid flow systems. The systems in consideration are composed of a non-normal linear element in feedback with a static and lossless nonlinearity—the NavierStokes equations being a special case. Specifically, we show that the input-output properties of the nonlinear element can be represented by a set of quadratic constraints. As a result, the nonlinear system can be analyzed by solving the Lyapunov inequalities of a linear system with a set of quadratic constraints that capture nonlinear behavior. Here, we investigate the proposed analysis framework on the Waleffe-Kim-Hamilton model—a low-dimensional mechanistic model of transitional and turbulent shear flows. Our proposed analysis framework can analyze global and local stability of a given equilibrium point of the nonlinear system. We show that nonlinear flow interactions have a destabilizing effect on the system response. The Lagrange multipliers in the proposed analysis provide additional information regarding the dominant nonlinear flow interaction terms.