Accurate low dimensional models for deterministic fluid systems driven by uncertain forcing

This paper addresses the model reduction of high-order linear systems within the framework of the incompressible Navier-Stokes equations. We look for reduced-order models that capture the response of some specific sensor whatever the initial flow condition and in the presence of any time-dependent external forcing. Namely, this work deals with the accurate modeling of the input-output dynamics of a fluid system when considering each degree of freedom of the system as an input, and the given measurement as the output. In the case of complex or realistic flows, the number of inputs is too large to apply the standard balanced truncation procedure. To alleviate this problem, we introduce a method called input projection. Input projection is shown to be analogous to the output projection procedure introduced by Rowley, Int. J. Bifurcation. Chaos Appl. Sci. Eng. 15, 997 (2005). To illustrate the model reduction, we consider the dynamics of a globally stable flow over a rounded backward-facing step. Reduced-order models are obtained by projecting the full original system onto: (i) the basis of the leading balanced modes computed from the input-projected systems and (ii) the most observable modes. The balanced models are observed to accurately capture the transient growths along the separated flow whatever the input while outperforming the models based on the most observable modes.