Application of the Wiener-Hermite Expansion to the Diffusion of a Passive Scalar in a Homogeneous Turbulent Flow

The stochastic expansion of Cameron, Martin, and Wiener is used for the velocity and concentration fields in a homogeneous turbulent flow convecting a passive scalar. The expansion is truncated and analytical solutions are obtained for the concentration field with the velocity distribution given and molecular diffusion neglected. For diffusion from a point source, the effective diffusivity is found in terms of Eulerian velocity correlations. For a statistically homogeneous scalar distribution, the spectrum function is determined in terms of the energy spectrum function. When the Prandtl number is large, the results obtained do not agree with the predictions of the universal equilibrium hypothesis