Model reduction for fluids using frequential snapshots

This paper deals with model reduction of high-order linear systems. An alternative method to approximate proper orthogonal decomposition (POD) and balanced truncation is exposed in this paper within the framework of the incompressible Navier-Stokes equations. The method of snapshots used to obtain low-rank approximations of the system controllability and observability Gramians is carried out in the frequency domain. Model reduction is thus performed using flow states that are long-time harmonic responses of the flow to given forcings, we call them frequential snapshots. In contrast with the recent works using time-stepping approach, restricted to stable systems, this one can always be computed for systems without marginal modes while it reduces to the same procedure for stable systems. We show that this method is efficient to perform POD and balanced proper orthogonal decomposition reduced-order models in both globally stable and unstable flows through two numerical examples: the flow over a backward-facing step and the flow over a square cavity. The first one is a globally stable flow exhibiting strong transient growths as a typical noise amplifier system while the second is a globally unstable flow representative of an oscillator system. In both cases, it is shown that the frequency-based snapshot method yields reduced-order models that efficiently capture the input-output behavior of the system. In particular, regarding the unstable cavity flow, our resulting unstable reduced-order models possess the same unstable global modes and stable transfer functions as those of the full system.