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*10^5 (periodic), and 6*10^6 (chaotic). Stability is obtained through a particular (staggered-grid) 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 for the non-chaotic cases. However, for the chaotic case, convergence with increasing numbers of modes is not evident, and additional measures are required to represent the effect of the smallest (neglected) scales.

翻译：本文提出了一种基于POD-Galerkin的稳定降阶模型（ROM），用于方形几何中二维瑞利-贝纳德对流问题，涵盖三个瑞利数：10^4（稳态）、3*10^5（周期态）和6*10^6（混沌态）。该稳定性通过一种特定的（交错网格）全阶模型（FOM）离散化实现，从而得到无压力且具有斜对称（能量守恒）对流项的ROM。这使得即使在训练数据范围之外，也无需稳定机制即可获得长期稳定的解。通过研究努塞尔数和雷诺数时间序列，以及垂直温度剖面的均值和方差，验证了ROM在不同测试案例中的稳定性。总体而言，随着模态数量的增加，这些量收敛至FOM结果，并可作为非混沌案例中准确度的良好度量。然而，对于混沌案例，随着模态数量增加，收敛性并不明显，需要额外措施来表征最小（被忽略）尺度的影响。