A pressure-free long-time stable reduced-order model for two-dimensional Rayleigh-Bénard convection

TL;DR

This work develops a pressure-free, long-time stable reduced-order model (ROM) for two-dimensional Rayleigh-Bénard convection by employing a POD-Galerkin projection of a structure-preserving, staggered-grid full-order model (FOM). The FOM preserves kinetic-energy and buoyancy-driven dynamics through skew-symmetric convective operators and a divergence-free discretization, which carries over to the ROM and yields stable integration without additional stabilization. Across RBC regimes ($Ra=10^4$, $3\times10^5$, $6\times10^6$), the ROM accurately captures global heat transport and temperature statistics, with convergence in steady and periodic cases achievable at modest modes, while chaotic turbulence demands many modes to resolve energy-dissipation balance. The framework offers a robust baseline for long-time RBC studies and provides a natural testbed for energy-consistent closure models to address unresolved small-scale dissipation in turbulent convection.

Abstract

The present work presents a stable POD-Galerkin based reduced-order model (ROM) for two-dimensional Rayleigh-Bénard 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.

Figures (15)

• Figure 1: Schematic of a $2D$ RBC case with boundary conditions.
• Figure 2: Schematic of a $2D$ finite volume cell $p_{i,j}$ showing the location of velocity, pressure, and temperature.
• Figure 3: Steady state RBC in a closed cavity: comparison of $\mathrm{Nu}$ obtained with our FOM to results of Cai_2019.
• Figure 4: Decay of singular values $(\sigma /\sigma_m)$ with increasing modes $(M)$ for (a) $\mathrm{Ra}=10^4$, (b) $3\times 10^5$, and (c) $6\times 10^6$. Here, $\sigma_m$ refers to the maximum singular value; the shaded region indicates the maximum number of modes ($M=16,32$ and $1000$ in (a), (b) and (c), respectively) used in the three $\mathrm{Ra}$ cases.
• Figure 5: Effect of initial condition and $M$ on the onset of convection at $\mathrm{Ra}=10^{4}$. Here FOM$_\text{BIC}$ refers to the FOM case whose initial condition is the best approximation computed by considering all the modes $(M^*=4560)$.