Asymptotic solution of the equation for the probability distribution of a passive scalar in grid turbulence with a uniform mean scalar gradient

A fully explicit asymptotic solution of the (simplified) equation for the probability density function (pdf) of a passive scalar in grid turbulence with a uniform mean cross-stream scalar gradient is obtained (the equation considered is exact for homogeneous turbulence). The asymptotic pdf is a function of the conditional expectations of the scalar dissipation and the transverse velocity. The solution generalizes the well-known asymptotic solution of Sinai and Yakhot [Phys. Rev. Lett. 63, 1962 (1989)] for homogeneous turbulence in the absence of a mean scalar gradient. An exact expression that relates the conditional expectations of molecular diffusion and transverse velocity is derived. This expression generalizes the result of Valiño et al. [Phys. Rev. Lett. 72, 3518 (1994)] for the zero mean gradient case.