Dispersion of Particle Pairs in Homogeneous Turbulence

The relative dispersion of particle pairs in homogeneous turbulence is treated by the Lagrangian-History Direct-Interaction closure approximation for a convected scalar field. Richardson's diffusion equation emerges, with the sole difference that the eddy diffusivity depends on the duration of the dispersion as well as on particle separation. The diffusivity tensor is evaluated in the inertial range, and the probability distribution for the separation distance of a particle pair is computed for longtime, inertial-range dispersion. From this solution, numerical values are obtained for the dimensionless constants in Richardson's law for mean-square separation distance and Oboukhov's scalar-field inertial-range spectrum law. An inequality relating these two constants is inferred from qualitative considerations, without use of the closure approximation.