A time-relaxation reduced order model for the turbulent channel flow

Regularized reduced order models (Reg-ROMs) are stabilization strategies that leverage spatial filtering to alleviate the spurious numerical oscillations generally displayed by the classical Galerkin ROM (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 Leray ROM (L-ROM) and the evolve-filter-relax ROM (EFR-ROM) and one eddy viscosity model, the mixing-length model, in the numerical simulation of the turbulent channel flow at $R{e}_{\tau }=180$ and $R{e}_{\tau }=395$ in both the reproduction and the predictive regimes. For each Reg-ROM, we investigate two different filters: (i) the differential filter (DF), and (ii) the higher-order algebraic filter (HOAF). In our numerical investigation, we monitor the Reg-ROM performance with respect to the ROM dimension, N, and the filter order. We also perform sensitivity studies of the three Reg-ROMs with respect to 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. (ii) All three Reg-ROMs are more accurate than the ROM projection in terms of Reynolds stresses. (iii) With the optimal parameter values, the new TR-ROM yields more accurate results than the L-ROM and the EFR-ROM in all tests. (iv) The new TR-ROM is more accurate than the mixing-length ROM. (v) For most N values, DF yields the most accurate results for all three Reg-ROMs. (vi) The optimal parameters trained in the reproduction regime are also optimal for the predictive regime for most N values, demonstrating the Reg-ROM predictive capabilities. (vii) All three Reg-ROMs are sensitive to the filter order and the filter radius, and the EFR-ROM and the TR-ROM are sensitive to the relaxation parameter. (viii) The optimal range for the filter radius and the effect of relaxation parameter are similar for the two $R{e}_{\tau }$ values.