Effects of Numerical Forcing on the Two-Time Correlation of Fluid Velocity Differences in Stationary Isotropic Turbulence

Abstract

In direct numerical simulations (DNS) of homogeneous isotropic turbulence, numerical forcing is needed to achieve statistically stationary velocity fields. The Eulerian two-time correlation tensor of the fluid velocity difference field, Δu(r,t)=u(x+r,t)−u(x,t), characterizes the temporal evolution of turbulent eddies whose sizes scale with separation r=|r|. In this study, we investigate the effects of two spectral forcing schemes on the temporal decay of the Eulerian two-time correlation of fluid velocity differences ⟨Δu(r,t′)Δu(r,t)⟩. Accordingly, DNS of homogeneous isotropic turbulence were performed for two grid sizes, 1283 and 5123, corresponding to the Taylor micro-scale Reynolds numbers Reλ≈80 and 210, respectively. Statistical stationarity was achieved by employing deterministic and stochastic spectral forcing schemes. In the stochastic scheme, one needs to specify the time scale, Tf, of the Uhlenbeck–Ornstein (UO) processes that constitute the forcing. We considered four values of the UO time scale (Tf=TE/4,TE,2TE, and 4TE) for each Reλ, where TE is the large-eddy time scale obtained from the DNS run with deterministic forcing at the same Reλ. It is seen that the correlations ⟨Δu(r,t′)Δu(r,t)⟩ obtained from the deterministic-forcing DNS runs decay more slowly than those from stochastic-forcing DNS runs of all four Tf values. The slower decay of correlations in deterministic DNS runs is more pronounced at larger separations and for higher Reλ.