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Reg-ROMs are stabilization strategies that leverage spatial filtering toalleviate the spurious numerical oscillations generally displayed by theclassical G-ROM in under-resolved numerical simulations of turbulent flows. Inthis paper, we propose a new Reg-ROM, the time-relaxation ROM (TR-ROM), whichfilters the marginally resolved scales. We compare the new TR-ROM with the twoother Reg-ROMs in current use, i.e., the L-ROM and the EFR-ROM, in thenumerical simulation of the turbulent channel flow at $Re_{\tau} = 180$ and$Re_{\tau} = 395$ in both the reproduction and the predictive regimes. For eachReg-ROM, we investigate two different filters: (i) the differential filter(DF), and (ii) a new higher-order algebraic filter (HOAF). In our numericalinvestigation, we monitor the Reg-ROM performance for the ROM dimension, $N$,and the filter order. We also perform sensitivity studies of the three Reg-ROMsfor the time interval, relaxation parameter, and filter radius. The numericalresults yield the following conclusions: (i) All three Reg-ROMs aresignificantly more accurate than the G-ROM and (ii) more accurate than the ROMprojection, representing the best theoretical approximation of the trainingdata in the given ROM space. (iii) With the optimal parameter values, theTR-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) Theoptimal parameters trained in the reproduction regime are also optimal for thepredictive regime for most $N$ values. (vi) All three Reg-ROMs are sensitive tothe filter radius and the filter order, and the EFR-ROM and the TR-ROM aresensitive to the relaxation parameter. (vii) The optimal range for the filterradius and the effect of relaxation parameter are similar for the two $\rmRe_\tau$ values.

A Time-Relaxation Reduced Order Model for the Turbulent Channel Flow