Consistency conditions for random-walk models of turbulent dispersion

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Abstract
Random‐walk models have long been used to calculate the dispersion of passive contaminants in turbulence. When applied to nonstationary and inhomogeneous turbulence, the model coefficients are functions of the Eulerian turbulence statistics. More recently the same random‐walk models have been used as turbulence closures in the evolution equation for the Eulerian joint probability density function (pdf) of velocity. There are, therefore, consistency conditions relating the coefficients specified in a random‐walk model of dispersion and the Eulerian pdf calculated using the same random‐walk model. It is shown that even if these conditions are not satisfied, the dispersion model does not violate the second law of thermodynamics: all that is required to avoid a second‐law violation is that the mean pressure gradient be properly incorporated. It is also shown that for homogeneous turbulence the consistency conditions are satisfied by a linear Gaussian model; and that for inhomogeneous turbulence they are satisfied by a nonlinear Gaussian model.