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The present work presents a stable POD-Galerkin based reduced-order model(ROM) for two-dimensional Rayleigh-B\'enard convection in a square geometry forthree Rayleigh numbers: 10^4 (steady state), 3*10^5 (periodic), and 6*10^6(chaotic). Stability is obtained through a particular (staggered-grid) FOMdiscretization that leads to a ROM that is pressure-free and has skew-symmetric(energy-conserving) convective terms. This yields long-time stable solutionswithout requiring stabilizing mechanisms, even outside the training data range.The ROM's stability is validated for the different test cases by investigatingthe Nusselt and Reynolds number time series and the mean and variance of thevertical temperature profile. In general, these quantities converge to the FOMwhen increasing the number of modes, and turn out to be a good measure ofaccuracy for the non-chaotic cases. However, for the chaotic case, convergencewith increasing numbers of modes is not evident, and additional measures arerequired to represent the effect of the smallest (neglected) scales.

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