Resolution effects and scaling in numerical simulations of passive scalar mixing in turbulence

The effects of finite grid resolution on the statistics of small scales in direct numerical simulations of turbulent mixing of passive scalars are addressed in this paper. Simulations at up to 2048³ grid points with grid spacing Δx varied from about 2 to 1/2 Batchelor scales (η_B) show that most conclusions on Schmidt number (Sc) dependence from prior work at less stringent resolution remain qualitatively correct, although simulations at resolution Δx≈η_B are preferred and will give adequate results for many important quantities including the scalar dissipation intermittency exponent and structure functions at moderately high orders. For Sc≥1, since η_B=ηSc^−1/2 (where η is the Kolmogorov scale), the requirement Δx≈η_B is more stringent than the corresponding criterion Δx≈η for the velocity field, which is thus well resolved in simulations aimed at high Schmidt number mixing. A simple argument is given to help interpret the effects of Schmidt and Reynolds numbers on trends towards local isotropy and saturation of intermittency at high Schmidt number. The present results also provide evidence for a trend to isotropy at high Reynolds number with fixed Sc=1.0. This is a new observation apparently not detected in less well resolved simulations in the past, and will require further investigation in the future.