A NUMERICAL STUDY OF THE MIXING OF A PASSIVE SCALAR IN 3 DIMENSIONS IN THE PRESENCE OF A MEAN GRADIENT

The mixing of a passive scalar in the presence of a mean gradient is studied in three dimensions by direct numerical simulations. The driving velocity field is either a solution of the three-dimensional (3-D) Navier-Stokes equations, at a microscale Reynolds number in between 20 and 70, and with a Prandtl number varying between 1/8 and 1, or a solution of the Euler equation restricted to a shell of wave numbers, which formally corresponds to an infinite Prandtl number. The probability distribution function (PDF) of the scalar gradients parallel and perpendicular to the direction of the mean gradient are studied. The gradients parallel to the mean gradient have a skewness of order 1 in the range of Peclet number considered. The PDFs are sharply peaked and their maxima correspond to a perfect mixing of the scalar. The PDF of the scalar gradient perpendicular to the mean gradient are reasonably well fit by stretched exponentials. Similar properties are observed for the restricted Euler model. In physical space, the scalar is well mixed in large domains, separated by narrow regions, where very large gradients concentrate. These ''cliffs'' are found to sit in regions where the flow is hyperbolic, whereas the scalar gradients are much weaker where the flow is elliptic. The present results are generally in agreement with the conclusions reached in a comparable study in two dimensions by Holzer and Siggia (to appear in Phys. Fluids). The stretching acting on the scalar is studied by computing various correlations between scalar gradient and velocity derivatives, as well as the correlations between vorticity and scalar gradient.