Conditional statistics for a passive scalar with a mean gradient and intermittency

The conditional dissipation and diffusion for a passive scalar with an imposed mean gradient are studied here. The results are obtained for an elementary model consisting of a random shear flow with a simple time-periodic transverse sweep. As the Peclet number is increased, scalar intermittency is observed; the scalar probability density function departs strongly from a Gaussian law. As a result, the conditional dissipation undergoes a transition from a quadratic behavior for the near-Gaussian probability distribution case at low Peclet number to a more complex shape at large Peclet. The conditional diffusion also undergoes a transition, this time from a linear to a nonlinear dependence, for cases with sufficient intermittency as well as a significant contribution from multiple spatial modes. The present analysis sheds some light on similar behaviors observed recently in numerical simulations of more complex models. The statistics in the present study are obtained by exact processing of one-dimensional quadrature results so that all sampling errors are eliminated, including in the tails of the distribution. This allows for a quantification of typical sampling errors when the conditional statistics are processed from numerical databases. The robustness of models based on polynomial fits for the conditional statistics is also assessed.