Publication Date:
1961-09-01
Description:
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 and which approaches a constant value ≑ lv 0 for t t 〉 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 (〉 l). For smaller separations (r 〈 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 [formula omitted] is very sensitive to the spectrum. © 1961, Cambridge University Press. All rights reserved.
Print ISSN:
0022-1120
Electronic ISSN:
1469-7645
Topics:
Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
,
Physics
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