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

by P. G. Saffman

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Application of a truncated Wiener-Hermite expansion of the velocity and concentration fields to analyze (1) the diffusion and mixing of a passive scalar from a point source in a turbulent fluid, and (2) the spectrum and intensity decay of a homogeneous distribution in an approximately homogeneous flow. The problem has important practical applications in reentry physics and air and water pollution. The truncated Wiener-Hermite expansion is treated as an approximation to the velocity field that must satisfy the Navier-Stokes equations; it is truncated at the first stage that gives nontrivial results. The mean concentration from a point source is shown to satisfy an integro-differential equation with diffusive and wavelike properties. The effective diffusivity is determined in terms of Eulerian velocity coordinates. For the homogeneous case, the concentration spectrum equals the energy spectrum times the mean-square concentration divided by the mean-square velocity component. A modification of Batchelor's theory is combined with this equation to predict the decay rate.

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