Reduced order models for the incompressible Navier-Stokes equations on\n collocated grids using a 'discretize-then-project' approach

Abstract

A novel reduced order model (ROM) for incompressible flows is developed by\nperforming a Galerkin projection based on a fully (space and time) discrete\nfull order model (FOM) formulation. This 'discretize-then-project' approach\nrequires no pressure stabilization technique (even though the pressure term is\npresent in the ROM) nor a boundary control technique (to impose the boundary\nconditions at the ROM level). These are two main advantages compared to\nexisting approaches. The fully discrete FOM is obtained by a finite volume\ndiscretization of the incompressible Navier-Stokes equations on a collocated\ngrid, with a forward Euler time discretization. Two variants of the time\ndiscretization method, the inconsistent and consistent flux method, have been\ninvestigated. The latter leads to divergence-free velocity fields, also on the\nROM level, whereas the velocity fields are only approximately divergence-free\nin the former method. For both methods, accurate results have been obtained for\ntest cases with different types of boundary conditions: a lid-driven cavity and\nan open-cavity (with an inlet and outlet). The ROM obtained with the consistent\nflux method, having divergence-free velocity fields, is slightly more accurate\nbut also slightly more expensive to solve compared to the inconsistent flux\nmethod. The speedup ratio of the ROM and FOM computation times is the highest\nfor the open cavity test case with the inconsistent flux method.\n