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

TL;DR

This paper develops a Lyapunov-based method to determine upper bounds on transient energy growth in accelerating and decelerating wall-driven flows, revealing larger growth in decelerating flows and providing stability certificates.

Contribution

It introduces a Lyapunov approach for linear time-varying systems to bound transient energy growth in wall-driven flows, matching results from singular value decomposition.

Findings

Decelerating flows exhibit larger transient growth than accelerating flows.

The Lyapunov method closely matches transient growth computed via SVD.

Provides certificates of stability and invariant sets for flow trajectories.

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 an upper bound on transient energy growth by solving linear matrix inequalities. This 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 Lyapunov method offers the advantages of providing a certificate of uniform stability and an invariant set to bound the solution trajectory.