Nonlinear stability analysis of transitional flows using quadratic constraints

The dynamics of transitional flows are governed by an interplay between the non-normal linear and quadratic nonlinearity in incompressible Navier-Stokes equations. In this work, we propose a framework for nonlinear stability analysis that exploits fact flow interactions constrained physics encoded nonlinearity. particular, show problems can be posed as convex feasibility optimization based on Lyapunov matrix inequalities, set constraints represent physics. proposed used to conduct global stability, local transient energy growth analysis. approach is demonstrated low-dimensional Waleffe-Kim-Hamilton model transition sustained turbulence. Our correctly determines critical Reynolds number instability. For analysis, estimate size region attraction well amplitude largest permissible perturbation such all trajectories converge back equilibrium point. Additionally, predict bounds maximum growth. Finally, careful multipliers enforce extract dominant drive associated instabilities.