Lyapunov stability analysis of time-varying shear flows

This work employs the Lyapunov method to analyze the linear stability of time-varying shear flows. The first part of this talk formulates the linear matrix inequalities to certify an upper bound of the growth rate. This method identifies instabilities of a linear time-varying system that describes cold fresh water on top of hot salty water with a periodically time-varying background shear flow. The Lyapunov method predicts the growth rate consistent with numerical simulations and the Floquet theory. We also use the Lyapunov method to analyze the instantaneous principal direction of instabilities and compare the computational resources required by the Lyapunov method, numerical simulations, and the Floquet theory. The second part of this talk uses the Lyapunov method to obtain an upper bound on the transient growth of accelerating and decelerating wall-driven flows. The Lyapunov method can obtain the upper bound of transient energy growth that closely matches transient growth computed via the singular value decomposition of the state-transition matrix of linear time-varying systems. Our analysis captures that decelerating base flows exhibit significantly larger transient growth compared with accelerating flows. Our approach offers the advantages of providing a rigorous certificate of uniform stability and an invariant ellipsoid to bound the solution trajectory. This Lyapunov-based analysis also has the potential to be extended to input-output analysis and nonlinear analysis.