4D-Var identification of DMD Reduced-Order Models

A reduced-order modelling (ROM) strategy is crucial to achieve model-based control in a wide class of flow configurations. In turbulence, ROMs are mostly derived by Galerkin projection of first principles equations onto the proper orthogonal decomposition (POD) modes. POD is widely used since it extracts from a sequence of data an orthonormal basis which captures optimally the flow energy. Unfortunately, energy level is not necessarily the correct criterion in terms of dynamical modelling and deriving a dynamical system based on POD modes leads sometimes to irrelevant models. In this communication, the Dynamic Mode Decomposition (DMD) as recently proposed by Schmid (JFM 2010) is used to determine the DMD modes. A DMD ROM is then derived by Galerkin projection of the Navier-Stokes equations onto a selected set of optimized-DMD modes. Finally, a four-dimensional variational assimilation approach (4D-Var) is employed to identify the coefficients of the DMD ROM. Essentially, 4D-Var combines imperfect observations, a background solution and the underlying dynamical principles governing the system under observation to determine an optimal estimation of the true state of the system. The methodology is illustrated for a DNS cylinder wake flow at Re=200 and PIV measurements at Re=13000.