Reduced order models for the incompressible Navier‐Stokes equations on collocated grids using a 'discretize‐then‐project' approach. (30th May 2021) - Star, Sabrina Kelbij
Abstract: A novel reduced order model (ROM) for incompressible flows is developed by performing a Galerkin projection based on a fully (space and time) discrete full order model (FOM) formulation. This 'discretize‐then‐project' approach requires no pressure stabilization technique (even though the pressure term is present in the ROM) nor a boundary control technique (to impose the boundary conditions at the ROM level). These are two main advantages compared to existing approaches. The fully discrete FOM is obtained by a finite volume discretization of the incompressible Navier‐Stokes equations on a collocated grid, with a forward Euler time discretization. Two variants of the time discretization method, the inconsistent and consistent flux method, have been investigated. The latter leads to divergence‐free velocity fields, also on the ROM level, whereas the velocity fields are only approximately divergence‐free in the former method. For both methods, accurate results have been obtained for test cases with different types of boundary conditions: a lid‐driven cavity and an open‐cavity (with an inlet and outlet). The ROM obtained with the consistent flux method, having divergence‐free velocity fields, is slightly more accurate but also slightly more expensive to solve compared to the inconsistent flux method. The speedup ratio of the ROM and FOM computation times is the highest for the open cavity test case with the inconsistent flux method. Abstract : A novel reduced order model is developed by performing a Galerkin projection based on a fully discrete (space and time) full order model formulation, that is, a 'discretize‐then‐project' approach. No pressure stabilization method is needed, even though the pressure term is present in the reduced order model. Moreover, no boundary control method is needed as the boundary conditions at the reduced order level are imposed via the projection of the boundary vectors that are specified at the discrete full order level.