C • STATISTICAL THEORIES OF TURBULENCE 



ficient increasing as the |- power of the separation. ^^ The law (Eq. 19-2) 

 was obtained experimentally by Richardson [66] and deduced theoreti- 

 cally by Batchelor [67] in a somewhat different manner. However, the 

 agreement is partly fortuitous since the length scale involved in Richard- 

 son's data does not fulfill the requirements imposed in the theory. 



More detailed analyses of the magnitude of separation can be carried 

 out when the distance of separation is small. Such analyses can be used 

 to examine the deformation of a fluid element — material lines, surfaces, 

 and volumes. An interesting question is whether a material volume will 

 eventually be stretched into a needle-shaped line or a disk-shaped surface. 

 For the details of such investigations, the reader is referred to the original 

 articles of Batchelor [68] and Reid [69] or to the article of Batchelor and 

 Townsend [63]. 



C,20. Temperature Fluctuations in Homogeneous Turbulence. 



The above concepts can be extended to a continuous distribution of 

 sources. The results are particularly instructive when the distribution 

 has a uniform gradient. Following Corrsin [70], let us consider a dis- 

 tributed heat source in the plane x = with a hnear distribution in the 

 y direction: 



T = T^-i-ay (20-1) 



Consider the flow of fluid with nondecaying isotropic turbulent motion 

 past these sources with a uniform mean speed U which is much higher 

 than the velocities of the turbulent motion. If molecular conduction is 

 omitted, the instantaneous temperature at any point downstream is de- 

 termined by that of the fluid particle present at that instant. This in 

 turn depends on the location where the particle crossed the plane x = 0. 

 Since a point may be reached by a fluid particle from above and from 

 below with equal probability, it is clear that the mean temperature dis- 

 tribution (Eq. 20-1) persists downstream. 



Consider now a fixed point a: > in the plane y = (which is in fact 

 a typical plane), and the particle occupying that point at any instant. 

 If the level of turbulence is low, this particle passed by the plane x = 

 at the time t = x/U earher. The position of the crossing point is given 

 by — Fo, where 



Fo = /"' v{t')dt' (20-2) 



Jt-T 



t being the time under consideration. This introduces a temperature 

 deviation 



9 = -aYo (20-3) 



»» This result can also be obtained by formally using Heisenberg's formula (Eq. 

 17-2). 



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