On the dispersion of fluid particles

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Abstract
The dispersion of fluid particles in a turbulent flow is described by transition-probability density functions. A renormalized expansion is made to establish the evolution equations of these functions. The resulting equations are nonlinear integrodifferential equations written in terms of Eulerian velocity-correlation functions. For the dispersion of a single particle, the equation at the zeroth order is the same as the one obtained by Roberts (1961). For the relative dispersion of a pair of particles, the equation is more convenient for applications than those of other theories. With this equation, Richardson's -power law for relative diffusion is recovered analytically based upon the Kolmogoroff spectrum and numerically based upon the von Kármán spectrum and a smoothed experimental spectrum.