Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics

One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 4096 3 grid points. The DNS consist of two series of runs; one is with k(max)eta similar to 1 (Series 1) and the other is with k(max)eta similar to 2 (Series 2), where k(max) is the maximum wavenumber and eta the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number R-lambda in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on R-lambda of the skewness and flatness factors (S and F). The DNS data suggest that the R-lambda dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S similar to -0.32R(lambda)(0.11) and F similar to 1.14(lambda)(0.34), in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with R-lambda similarly to each other at R-lambda > 100, in contrast to R-lambda < 100. Regarding the statistics of time derivatives, the sccond-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at R-lambda approximate to 430. The flatness factors of the Eulerian and Lagrangian accelerations increase with R-lambda approximately proportional to R-lambda(alpha E) and R-lambda(alpha L), respectively, where alpha(E) approximate to 0.5 and alpha(L) approximate to 1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to R-lambda(beta E) and R-lambda(beta L), respectively, where beta(E) approximate to 1.5 and beta(L) approximate to 3.0.