Model reduction for fluids using frequential snapshots

Abstract

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.

Feedback control of unstable flows: a direct modelling approach using the Eigensystem Realisation Algorithm

“…Aubry et al 1988;Podvin & Lumley 1998;Graham, Peraire & Tang 1999;Ravindran 2000aRavindran ,b, 2002Ravindran , 2006Prabhu, Collis & Chang 2001;Ma & Karniadakis 2002;Noack et al 2003;Gloerfelt 2008;Siegel et al 2008;Barbagallo et al 2009Barbagallo et al , 2012Tadmor et al 2010) can yield successful ROMs, but 'balanced modes' have been shown to lead to superior performance in terms of robustness and required model order in a number of studies (e.g. Willcox & Peraire 2002;Ilak & Rowley 2008;Bagheri et al 2009c;Barbagallo et al 2009;Dergham et al 2011). If the system is stable, such modes can be approximately identified 44…”

“…Extensions have therefore been designed to circumvent this issue, but these require either the evaluation of all the unstable global modes of the flow field (Barbagallo et al 2009;Ahuja & Rowley 2010) or the inversion of large matrices (Dergham et al 2011), making them often computationally intractable. These restrictions have recently been lifted, as it was shown by Flinois, Morgans & Schmid (2015) that the snapshot-based approach can in fact be used directly to obtain ROMs of unstable flows, as long as the snapshots are chosen appropriately.…”

“…It has been successful in modelling the dynamics of a large range of flows (e.g. Willcox & Peraire 2002;Rowley 2005;Bagheri et al 2009a,b;Ahuja & Rowley 2010;Dergham et al 2011;Semeraro et al 2011;Tu & Rowley 2012).…”

“…For this reason, a number of methods have been developed to circumvent this issue (e.g. Barbagallo et al 2009;Ahuja & Rowley 2010;Dergham et al 2011). They require additional computations that render the algorithm significantly more costly and often completely…”

POD-spectral decomposition for fluid flow analysis and model reduction

“…Obviously, those models are strictly dedicated to a given flow situation. This introduces an important drawback for flow control applications and has motivated different strategies to improve the expansion basis like balanced proper orthogonal decomposition [6] or frequential snapshots [11].…”