Enablers for robust POD models

Abstract

This paper focuses on improving the stability as well as the approximation properties of Reduced Order Models (ROM) based on Proper Orthogonal Decomposition (POD). The ROM is obtained by seeking a solution belonging to the POD subspace and that at the same time minimizes the Navier-Stokes residuals. We propose a modified ROM that directly incorporates the pressure term in the model. The ROM is then stabilized making use of a method based on the fine scale equations. An improvement of the POD solution subspace is performed thanks to an hybrid method that couples direct numerical simulations and reduced order model simulations. The methods proposed are tested on the two-dimensional confined square cylinder wake flow in laminar regime.

POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder

“…The time window used to calibrate the ROM model is indicated by the black arrow. Using only three modes the ROM model, for long time integration exhibits numerical instabilities as observed in [14]. Using more modes, including a percentage of energy up to 99%, instability phenomena vanish.…”

“…• Differently to what proposed in [13,14], where only the momentum equation is considered and where it is assumed that velocity and pressure share the same temporal coefficients, the Poisson equation for pressure reported in Equation (9) is projected onto the POD pressure modes in order to enforce the continuity equation constraint. Such approach, that is proposed in literature by several authors [12,24] for FEM approximations, is here adapted to a FVM framework.…”

“…This term is considered also in [13,14] but, there, it is assumed that velocity and pressure modes share the same temporal coefficients. More details about the treatment of this term are given in § 3.2.2.…”

“…In the first approach [13,14] it is assumed that velocity and pressure share the same temporal coefficients (a = b) and only the momentum equation is exploited at reduced order level.…”

“…In this method the governing equations, describing the phenomenon, are projected onto a low dimensional space called the reduced basis space [9,10] that is optimally constructed starting from high fidelity simulations. In particular in this work the ROM is constructed using a POD-Galerkin approach [11][12][13][14][15][16][17]. As previously mentioned, since the main purpose of this paper is the correct modelling of the vortex shedding phenomenon, particular attention is paid to the reconstruction in the reduced order model of the pressure field.…”