Relative Diffusion of a Pair of Fluid Particles in Turbulence

Abstract

The effective Hamiltonian method is used to find the distribution function describing the positions as a function of time of a given pair of fluid particles subject to convection in steady, incompressible, statistically isotropic turbulent flow. By assuming a joint Gaussian distribution for the velocity field, a reasonable time dependence is obtained for the statistical average of the distance between the pair of fluid particles. The basis for Richardson’s four-thirds law is indicated and the proportionality constant C^* is shown to be related to the Kolmogorov constant C by C^*=3^1⁄3C^1⁄2⁄2^1⁄2.