  The dynamics of transitional flows are governed by an interplay between the
non-normal linear dynamics and quadratic nonlinearity in the incompressible
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 feasibility
and 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 stability, local stability, and
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. For local stability analysis, we show that the framework
can estimate the size of the region of attraction as well as the amplitude of
the largest permissible perturbation such that all trajectories converge back
to the equilibrium point. Additionally, we show that the framework can predict
bounds on the maximum 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.
