Analytical theory of turbulent diffusion

Recently Kraichnan (1959) has propounded a theory of homogeneous turbulence, based on a novel perturbation method, that leads to closed equations for the velocity covariance. In the present paper, this method is applied to the theory of turbulent diffusion and closed equations are derived for the probability distributions of the positions of marked fluid elements released in a turbulent flow. Two topics are discussed in detail. The first is the probability distribution, at time t, of the displacement of an element from its initial position. In homogeneous flows, this distribution is found to resemble that for classical diffusion but with a variable coefficient of diffusion which is proportional to v²₀ t for t l|v₀ and which approaches a constant value eDot lv 0 for t t Gt l / v 0 (l = macroscale, v 0 = r. m. s. turbulent velocity). The second topic treated is the joint probability distribution of the displacements of two fluid elements. Particular attention is focused upon the probability distribution of relative displacement, i. e. Richardson's distance-neighbour function. This is found to be Gaussian for separations r which are large (Gt l). For smaller separations (r Lt l), its behaviour at high Reynolds numbers is found to be quite well expressed in terms of a variable diffusion coefficient K (r, t), as suggested by Richardson (1926). For all but extremely short times, K (r, t) is found to depend only on r and on the form of the inertial range spectrum E (k). On assuming E (r) v²₀ l (kl) ^- {32} as results from Kraichnan's approximation (1959), one finds E (r) v₀ l (r|l) ^ {32}. On the basis of similarity arguments of the Kolmogorov type, which give E (r) v²₀ l (kl) ^- {52}, one finds E (r) v₀ l (r|l) ^ {43} as, in fact, Richardson originally proposed. The dispersion r 2 is proportional to l² (v₀ t|l) ⁴ on Kraichnan's theory; while r² l² (v₀ t|l) ³ on the similarity theory. This illustrates that the behaviour of r² is very sensitive to the spectrum.