  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.
