A minimum residual projection to build coupled velocity-pressure POD-ROM for incompressible Navier-Stokes equations

The pressure term which appears in the ROM (reduced order model) associated to the incompressible Navier-Stokes equations, in particular for the shear flows, plays an important role on the velocity. The aim of this paper is to propose a Proper Orthogonal Decomposition based reduced order model (POD-ROM) to obtain both the velocity and pressure fields for incompressible flows. Two PODs are performed, one for the velocity and the other for the pressure. Contrary to existing projection methods available in the literature, the temporal velocity and pressure coefficients are sought by minimizing the residual of the momentum equation only, without the need of a Poisson equation. For the numerical test cases considered in this paper, the proposed minimum residual projection enables to obtain accurately the pressure field, and in turn to slightly improve the velocity one. The method is tested on two fluid flows: a transient mixed-convection flow and a periodic flow around a circular cylinder. In this last case, the drag, lift and pressure coefficients, as well as the Strouhal number are properly recovered compared to those of the full model.