Diffusion in a field of homogeneous turbulence

Abstract

In an earlier paper the author considered the problem of the turbulent diffusion, relative to a fixed origin, of a cloud of marked fluid whose initial position is given. This was found to be determined by the initial shape of the cloud and the statistical properties of the displacement of a single fluid particle. The present paper is concerned with the relative diffusion of the cloud, i.e. with the tendency to change its shape, or, more precisely, with that part of the relative diffusion which is described by the probability that a given vector y can lie with both its ends in marked fluid at time t. This aspect of the relative diffusion is found to be determined by the initial shape of the cloud and the statistical properties of the separation, at time t, of two fluid particles of given initial separation. The statistical functions introduced to describe the relative diffusion are found to be related to Richardson's distance-neighbour function.

The relative diffusion of two particles is a more complex problem than diffusion of a single particle about a fixed origin because the relative diffusion depends on the initial separation. The closer the particles are together, the smaller is the range of eddy sizes that contributes to their relative velocity; for the same reason, relative diffusion is an accelerating process, until the particles are very far apart and wander independently. The hypothesis is made that if the initial separation is small enough, the probability distribution of the separation will tend asymptotically to a form independent of the initial separation, before the particles move independently. This hypothesis permits various simple deductions, some of which make use of Kolmogoroff's similarity theory. The important question of the description of the relative diffusion by a differential equation is examined; Richardson has put forward one suggestion, and another, based on a normal distribution of the separation, is made herein.