High-order structure functions for passive scalar fed by a mean gradient

Abstract

Transport equations for even-order structure functions are written for a passive scalar mixing fed by a mean scalar gradient, with a Schmidt number $\mathit{Sc}=1$. Direct numerical simulations (DNS), in a range of Reynolds numbers $R_\lambda \in [88,529]$ are used to assess the validity of these equations, for the particular cases of second-and fourth-order moments. The involved terms pertain to molecular diffusion, transport, production, and dissipative-fluxes. The latter term, present at all scales, is equal to: i) the mean scalar variance dissipation rate, $\langle \chi \rangle$, for the second-order moments transport equation; ii) non-linear correlations between $\chi$ and second-order moments of the scalar increment, for the fourth-order moments transport equation. The equations are further analyzed to show that the similarity scales (i.e., variables which allow for perfect collapse of the normalised terms in the equations) are, for second-order moments, fully consistent with Kolmogorov-Obukhov-Corrsin (KOC) theory. However, for higher-order moments, adequate similarity scales are built from $\langle \chi^n \rangle$. The similarity is tenable for the dissipative range, and the beginning of the scaling range.