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

TL;DR

This paper introduces a new reduced order modeling approach for incompressible Navier-Stokes equations that simplifies implementation by avoiding pressure stabilization and boundary control, while maintaining high accuracy and efficiency.

Contribution

The paper presents a 'discretize-then-project' ROM approach based on a fully discrete FOM, eliminating the need for pressure stabilization and boundary control techniques.

Findings

The consistent flux method produces divergence-free velocity fields at the ROM level.

The ROM achieves significant speedups over the FOM, especially in open cavity cases.

The consistent flux method is more accurate but slightly more computationally expensive.

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.