C,19 • ANALYSIS INVOLVING MORE THAN ONE PARTICLE 



use of a diffusion coefficient is valid only for large distances downstream 

 (provided the idealization used is valid). Close to the source, there is no 

 basis for using a diffusion coefficient. Indeed, Taylor [15] pointed out 

 that the Gaussian distribution of temperature near the source is not 

 associated with a usual diffusion coefficient but should be accounted 

 for by Gaussian distribution of the velocity of fluctuation (cf. Eq. 18-9). 

 If the frequency distribution of velocities had obeyed some other law, 

 the distribution of temperature near the source would also have deviated 

 from an error curve. On the other hand, the temperature distribution 

 very far from the source must necessarily fit an error curve, whatever 

 the frequency distribution of velocities may be. In reality, however, the 

 analysis at very large distances downstream is complicated by the fact 

 that the turbulence dies away downstream so that the above analysis is 

 not accurate. 



For other approaches to the problem of turbulent diffusion, see 

 Frenkiel [66], where some semiempirical calculations are given. 



C,19. Analysis Involving More Than One Particle. In the above 

 analysis, we do not consider the joint configuration of a number of fluid 

 particles. This is of course necessary if we wish to get a more complete 

 description of turbulent diffusion. Indeed, one may consider a large num- 

 ber of particles, and their joint statistical behavior during the course of 

 time would give an almost complete statistical description of the turbu- 

 lent motion in the Lagrangian scheme. 



In practice, one is often limited to the consideration of the separation 

 of two particles. If Si is the separation between two particles, the rate of 

 variation of the statistical average s^ is clearly a measure of the rate of 

 diffusion. We shall now consider a special case where a concrete formula 

 can be obtained for the variation of s^ as a function of the time of sepa- 

 ration T = t — U between the initial instant U and the present time t. 

 If T is sufficiently large, then the influence of the initial separation must 

 be neghgible. If, furthermore, the separation between the particles lies in 

 the range of scales of turbulence for which the spectral law (Eq. 13-15) 

 holds, there is only one parameter — namely the rate of energy dissipation 

 e — characterizing the properties of the turbulent motion. Thus, dimen- 

 sional reasoning shows that s^ (which depends only on t and e) must be 

 of the form 



¥ ^ er^ (19-1) 



or 



^' ^ er2 -- 6H72)i (19-2) 



This means that the dispersive effect becomes larger and larger as the 

 particles separate further and further from each other, the diffusion coef- 



< 243 ) 



