C,18 ■ DIFFUSION BY CONTINUOUS MOVEMENTS 



process. Temperature measurements in the wake of a heated wire, such 

 as those made by Schubauer [61], and more recently by Uberoi and 

 Corrsin [62], give a more quantitative description of the phenomenon. 

 A complete theory of the phenomenon of turbulent diffusion is, however, 

 not available, even in the simplest case of homogeneous isotropic turbu- 

 lence, because of some inherent difficulties. In the first place, the usual 

 concept of a diffusion coefficient can in general be at best a crude first 

 approximation, because the variation of the statistical properties of inter- 

 est in turbulent diffusion (or transport) occurs over scales comparable to 

 that of the scale of the turbulent motion itself. The analogy in the molecu- 

 lar case would be variations at scales comparable to the mean free path. 

 Secondly, the mathematical difficulty encountered in trying to develop a 

 detailed theory is extremely heavy. Indeed, a theory of diffusion dealing 

 with the transport of material particles from one point to another sug- 

 gests the use of the Lagrangian description. This in itself makes the 

 theory difficult. On the other hand, the eventual diffusion of a certain 

 physical property, such as temperature, must be accomplished by the 

 molecular process, which is more conveniently described by the Eulerian 

 method. In the face of these difficulties, most of the existing theories are 

 far from being complete. In the present treatment, we shall therefore 

 limit ourselves to a brief account of some of the elementary concepts de- 

 veloped and some of the issues examined. For a more detailed treatment, 

 the reader is referred to the recent article by Batchelor and Townsend [63, 

 pp. 352-399]. 



A fundamental approach to turbulent diffusion was advanced by 

 Taylor [64] in 1921. While the theory deals with an idealized situation, 

 it does reveal some of the essential features of the process, and forms the 

 starting point of many later developments. In the simplest form of this 

 theory of diffusion by continuous movements, we restrict ourselves to the 

 idealized case of a homogeneous isotropic field of turbulence which is not 

 decaying. We consider diffusion from a plane x, z where all the diffusing 

 particles are concentrated at time f = 0. If F is the coordinate of a 

 particle at time T, then Y = jlvdt, and 



where the average is taken in the statistical sense or over planes parallel 

 to the X, z plane. We may now introduce the correlation coefficient R{t) by 



v{T)v{t) = v^Rir), T = T - t (18-2) 



Then Eq. 18-1 becomes 



T 



{ 241 > 



