A Lagrangian stochastic diffusion method for inhomogeneous turbulence

Abstract

A Lagrangian stochastic method of solving the diffusion equation for inhomogeneous turbulence is presented in this paper. This numerical method uses (1) a first-order approximation for the spatial variation of the eddy diffusivity and (2) the first three particle position moments to define a skewed, non-Gaussian particle position probability density function. Two cases of the variation of the eddy diffusivity near a boundary are considered. In the first case, the eddy diffusivity varies linearly with height and is zero at the boundary. The method handles this case efficiently and accurately by using the first few terms of the series representation of the analytic solution to construct components of the non-Gaussian position probability density function that are important near the boundary. In the second case, the eddy diffusivity is constant near the boundary, and the well-known solution for this case – a Gaussian function reflected at the boundary – is used. Comparison of numerical simulation results to analytic solutions of the diffusion equation show that this method is accurate. For cases where the eddy diffusivity varies linearly with height to zero at the boundary, we demonstrate that this method can be significantly more efficient than the commonly used method that assumes a Gaussian particle position probability density function.