Flow Instability: Richtmyer-Meshkov and Pulsating Flows

Acceleration and deceleration significantly influence the transition to turbulence during the most dangerous phases of aviation: takeoff and landing. This work analyzes accelerating and decelerating wall-driven channel flows by quantifying the maximum transient energy growth of perturbations using a Lyapunov-type approach. By formulating the linearized Navier-Stokes equations as a linear time-varying system and constructing time-dependent Lyapunov functions, we obtain rigorous upper bounds on transient energy growth by solving linear matrix inequalities (LMI). Our analysis reveals that decelerating base flows exhibit significantly larger transient growth compared to accelerating or steady flows, which is consistent with existing analysis. The LMI approach can obtain worst-case transient energy growths that closely match transient growth computed via the singular value decomposition of the fundamental solution operator of linear time-varying systems. However, our approach offers advantages for a rigorous certificate of stability and provides an invariant ellipsoid to bound the solution trajectory. Our Lyapunov-based analysis also has the potential to be extended to input-output analysis or include nonlinear effects.