The relative diffusion of a cloud of passive contaminant in incompressible turbulent flow

Abstract

A problem of major practical interest is the variation with x and t of the statistical properties of Γ(x, t), the distribution of concentration of a contaminant in a cloud containing a finite quantity Q of contaminant, released in a specified way at t = 0 over a volume of order L³₀. Of particular relevance is the case of relative diffusion (when x is measured throughout each realization relative to the centre of mass of the cloud), when important properties are L(t), the linear dimension of the cloud, C(x, t), the ensemble mean concentration, $\overline{c^2}({\bf x}, t)$, the variance of the concentration, and p(y, t), the distance-neighbour function. Much fundamental work has led to a knowledge of the way L varies with t, but not of the way the other properties vary. Hitherto therefore, prediction of such variation has normally used unjustifiable empirical concepts such as eddy diffusivities, but this is ultimately unsatisfactory, practically as well as theoretically. Hence the exact equations have been used to obtain a quite new description of the structure of a dispersing cloud, which it is hoped will serve as a basis for future practical work.When κ = 0 (where κ is the molecular diffusivity) the magnitude of p(y, t) is of order Q/L³ for most y, but of order Q/L³₀ when |y| is very small. By a variety of arguments it is shown that these facts can be explained (for many, if not all, flows) only if the distributions of C and $\overline{c^2}$, as well as that of p, have a core-bulk structure. In the bulk of the cloud C and $\overline{c^2}$ have magnitudes of order Q/L³ and Q²/L³₀L³ respectively, but there is a core region of thickness decreasing to zero surrounding the centre of mass within which they have much greater magnitudes. In one case, examined in some detail, the magnitudes in the core are of order Q/L³₀ and Q²/L⁶₀.It is then shown that the core and bulk exist even in the real case when κ ≠ 0. In the real case the core thickness no longer tends to zero but to a constant of order λ_c, the conduction cut-off length. As a consequence almost entirely of molecular diffusion acting in the core region, the magnitudes of C and $\overline{c^2}$ in both the core and the bulk decay to zero in a way which depends on the details of the fine-scale structure of the velocity field. Several examples of the decay are discussed.