No pressure? Energy-consistent ROMs for the incompressible Navier-Stokes equations with time-dependent boundary conditions

This work presents a novel reduced-order model (ROM) for the incompressible Navier-Stokes equations with time-dependent boundary conditions. This ROM is velocity-only, i.e. the simulation of the velocity does not require the computation of the pressure, and preserves the structure of the kinetic energy evolution. The key ingredient of the novel ROM is a decomposition of the velocity into a field with homogeneous boundary conditions and a lifting function that satisfies the mass equation with the prescribed inhomogeneous boundary conditions. This decomposition is inspired by the Helmholtz-Hodge decomposition and exhibits orthogonality of the two components. This orthogonality is crucial to preserve the structure of the kinetic energy evolution. To make the evaluation of the lifting function efficient, we propose a novel method that involves an explicit approximation of the boundary conditions with POD modes, while preserving the orthogonality of the velocity decomposition and thus the structure of the kinetic energy evolution. We show that the proposed velocity-only ROM is equivalent to a velocity-pressure ROM, i.e., a ROM that simulates both velocity and pressure. This equivalence can be generalized to other existing velocity-pressure ROMs and reveals valuable insights in their behaviour. Numerical experiments on test cases with inflow-outflow boundary conditions confirm the correctness and efficiency of the new ROM, and the equivalence with the velocity-pressure formulation.