Reg-ROMs are stabilization strategies that leverage spatial filtering to alleviate the spurious numerical oscillations generally displayed by the classical G-ROM in under-resolved numerical simulations of turbulent flows. In this paper, we propose a new Reg-ROM, the time-relaxation ROM (TR-ROM), which filters the marginally resolved scales. We compare the new TR-ROM with the two other Reg-ROMs in current use, i.e., the L-ROM and the EFR-ROM, in the numerical simulation of the turbulent channel flow at $Re_{\tau} = 180$ and $Re_{\tau} = 395$ in both the reproduction and the predictive regimes. For each Reg-ROM, we investigate two different filters: (i) the differential filter (DF), and (ii) a new higher-order algebraic filter (HOAF). In our numerical investigation, we monitor the Reg-ROM performance for the ROM dimension, $N$, and the filter order. We also perform sensitivity studies of the three Reg-ROMs for the time interval, relaxation parameter, and filter radius. The numerical results yield the following conclusions: (i) All three Reg-ROMs are significantly more accurate than the G-ROM and (ii) more accurate than the ROM projection, representing the best theoretical approximation of the training data in the given ROM space. (iii) With the optimal parameter values, the TR-ROM is more accurate than the other two Reg-ROMs in all tests. (iv) For most $N$ values, DF yields the most accurate results for all three Reg-ROMs. (v) The optimal parameters trained in the reproduction regime are also optimal for the predictive regime for most $N$ values. (vi) All three Reg-ROMs are sensitive to the filter radius and the filter order, and the EFR-ROM and the TR-ROM are sensitive to the relaxation parameter. (vii) The optimal range for the filter radius and the effect of relaxation parameter are similar for the two $\rm Re_\tau$ values.

翻译：Reg-ROMs是一种利用空间滤波来缓解经典G-ROM在湍流欠解析数值模拟中通常出现的虚假数值振荡的稳定化策略。本文提出了一种新的Reg-ROM——时间松弛ROM（TR-ROM），它对临界分辨尺度进行滤波。我们将新提出的TR-ROM与当前正在使用的另外两种Reg-ROM（即L-ROM和EFR-ROM）在湍流通道流（$Re_{\tau} = 180$ 和 $Re_{\tau} = 395$）的复现预测和预测性数值模拟中进行了比较。对于每种Reg-ROM，我们研究了两种不同的滤波器：（i）微分滤波器（DF）和（ii）一种新的高阶代数滤波器（HOAF）。在数值研究中，我们监测了Reg-ROM在不同ROM维度、$N$ 和滤波器阶数下的性能。我们还对三种Reg-ROM在时间区间、松弛参数和滤波器半径方面进行了敏感性研究。数值结果得出以下结论：（i）所有三种Reg-ROM的精度均显著高于G-ROM；（ii）它们的精度高于ROM投影，而ROM投影代表了给定ROM空间中训练数据的最佳理论近似；（iii）在最优参数值下，TR-ROM在所有测试中均比其他两种Reg-ROM更精确；（iv）对于大多数 $N$ 值，DF能为所有三种Reg-ROM提供最精确的结果；（v）在大多数 $N$ 值下，于复现预测中训练得到的最优参数对预测性模拟同样最优；（vi）所有三种Reg-ROM对滤波器半径和滤波器阶数敏感，且EFR-ROM和TR-ROM对松弛参数敏感；（vii）两种 $\rm Re_\tau$ 值的滤波器半径最优范围和松弛参数影响相似。