Upper bound of transient growth in accelerating and decelerating wall-driven flows using the Lyapunov method

Abstract: This work analyzes accelerating and decelerating wall-driven flows by quantifying the upper bound of transient energy growth using a Lyapunov-type approach. By formulating the linearized Navier-Stokes equations as a linear time-varying system and constructing a time-dependent Lyapunov function, we obtain a rigorous upper bound on transient energy growth by solving linear matrix inequalities (LMI). The LMI approach 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.