A dynamical equation for the distribution of a scalar advected by turbulence
Résumé
A phenomenological model for the dissipation of scalar fluctuations due to the straining by the fluid motion is proposed in this Brief Communication. An explicit equation is obtained for the time evolution of the probability distribution function of a coarse-grained scalar concentration. The model relies on a self-convolution process. We first present this model in the Batchelor regime and then extend empirically our result to the turbulent case. This approach is finally compared with other models.
The turbulent transport of tracers such as temperature or salinity is of great importance for many applications.1 Available models usually provide a set of closed equations for the mean quantities and their variance, and sometimes for the third- and fourth-order moments.2 For many practical problems, it would, however, be relevant to predict the dynamical evolution of the whole scalar probability distribution. This problem was first addressed in the context of reactive flows,3 but such a description can also be important to properly model the turbulent mixing of water masses in a stably stratified fluid, in which case the sedimentation under the influence of gravity has an opposite effect for fluid particles heavier or lighter than the surrounding fluid. The case of vorticity in two-dimensional turbulence is also of interest. Indeed, the statistical mechanics of two-dimensional turbulence gives predictions for the final flow organization depending on an initial distribution of vorticity.4 This theory can be used as a starting point for nonequilibrium transport models,5 expressed in terms of the local vorticity probability density function (PDF). In the presence of small viscosity, however, turbulent cascades modify this probability distribution by dissipation of fluctuations. For two-dimensional turbulence, this effect leads to modification of the equilibrium state resulting from turbulent mixing.6 In this Brief Communication, we introduce a simple model for the cascade effects, which can be combined with transport equations in the presence of spatial gradients.
