This report explores some numerical alternatives that can be exploited to derive efficient low-order models of the Navier-Stokes equations. It is shown that an optimal solution sampling can be derived using appropriate norms of the Navier-Stokes residuals. Then the classical Galerkin approach is derived in the context of a residual minimization method that is similar to variational multiscale modeling. Finally, calibration techniques are reviewed and applied to the computation of unsteady aerodynamic forces. Examples pertaining to both non-actuated and actuated flows are shown.

We present a low-order modeling technique for actuated flows based on the regularization of an inverse problem. The inverse problem aims at minimizing the error between the model predictions and some reference simulations. The parameters to be identified are a subset of the coefficients of a polynomial expansion which models the temporal dynamics of a small number of global modes. These global modes are found by Proper Orthogonal Decomposition, which is a method to compute the most representative elements of an existing simulation database in terms of energy. It is shown that low-order control models based on a simple Galerkin projection and usual calibration techniques are not viable. They are either ill-posed or they give a poor approximation of the solution as soon as they are used to predict cases not belonging to the original solution database. In contrast, numerical evidence shows that the method we propose is robust with respect to variations of the control laws applied, thus allowing the actual use of such models for control