Model reduction on 2-D incompressible laminar flows: Application on the flow over a backward-facing step

Abstract

Abstract. The computation of fluid mechanics problems usually leans on a discretization of the Navier-Stokes equations which has to be so fine that the dimensions of the linear systems to be solved are very high. As a direct consequence, the Central Processing Unit time needed to solve complex systems my become extremely large when accuracy is demanded. When coupling numerical modeling schemes to inversion or control problems, the size of linear systems to be solved has to be drastically reduced. Within this context, the identification method consists in identifying the components of a low-order matrix system. The identification process works as an inverse problem of parameter estimation. The test case shows the ability of the proposed method to reduced with accuracy a particular fluid mechanics problem. IntroductionThe numerical solution of fluid mechanics problems usually needs a fine spatial discretization of the considered domain. Consequently, the matrix systems may be large and the computational times may be elevated. This common approach gives the so-called detailed model. Though these models may be highly reliable and accurate, the computational cost can be cumbersome to treat applications such as in-time control or inverse problems for instance. The alternative is to build a model of much lower number of degrees of freedom that reproduces well enough the dynamics of the detailed model. Among the large variety of reduction methods, the modal identification method has shown to be efficient in our considered cases in transient heat transfer. The modal identification method was developed earlier on to identify reduced models for both linear [1] and nonlinear systems [2,3] in heat conduction problems.The identification method has recently been adapted for some fluid mechanics problems in stationary cases [4]. In [4], the stream function-vorticity (Ψ − ω) formulation was used for simplicity. However, the choice for the application of the boundary conditions is uneasy and impractical. This is the reason why the reduced model formulation has been revisited based on the primal variables that are the velocities and the pressure. The direct Navier-Stokes equations are discretized. The modal form is then derived leading to the structure for the reduced model. The components of the matrix system form the vector of parameters that is to be identified. The identification is based on the solution of an optimization problem that consists in minimizing a cost function. A gradient quasi-Newton method is used. The cost function gradient is computed through the solution of the adjoint state problem [5].