A pressure-free long-time stable reduced-order model for two-dimensional Rayleigh-B\'enard convection

TL;DR

This paper introduces a pressure-free, energy-conserving reduced-order model for 2D Rayleigh-Bénard convection that remains stable over long times without additional stabilization, effectively capturing various flow regimes.

Contribution

It develops a novel POD-Galerkin ROM with a staggered-grid discretization that ensures long-term stability and pressure independence across different Rayleigh numbers.

Findings

ROM accurately reproduces FOM results for steady, periodic, and chaotic flows.

Stability achieved without stabilization techniques, even outside training data.

High mode counts needed for chaotic regimes to capture low-energy structures.

Abstract

The present work presents a stable POD-Galerkin based reduced-order model (ROM) for two-dimensional Rayleigh-B\'enard convection in a square geometry for three Rayleigh numbers: $10^4$ (steady state), $3\times 10^5$ (periodic), and $6 \times 10^6$ (chaotic). Stability is obtained through a particular (staggered-grid) full-order model (FOM) discretization that leads to a ROM that is pressure-free and has skew-symmetric (energy-conserving) convective terms. This yields long-time stable solutions without requiring stabilizing mechanisms, even outside the training data range. The ROM's stability is validated for the different test cases by investigating the Nusselt and Reynolds number time series and the mean and variance of the vertical temperature profile. In general, these quantities converge to the FOM when increasing the number of modes, and turn out to be a good measure of accuracy. However, for the chaotic case, convergence with increasing numbers of modes is relatively difficult and a high number of modes is required to resolve the low-energy structures that are important for the global dynamics.