C,21 • STATISTICAL THEORY OF SHEAR FLOW 

 The statistical average rate of temperature transfer is therefore 



-7d = (wYo (20-4) 



The value v Yo can be evaluated in a manner very similar to that used 

 in Art. 18. We obtain in this manner 



Wo = r v{t)v{jt')dt' = ^ j^ R{r')dr' (20-5) 



It should be noted that although this formula is very similar to Eq. 18-3, 

 the physical interpretation is different. The rate of temperature transfer 

 may now be written in the form 



-7d = D'^, D' =7' I R{r')dT' (20-6) 



ay Jo 



where D' is a "diffusion coefficient" in that it gives the rate of increase 

 of the mean square deviation (Art. 18). It is proportional to x at first, 

 and approaches a constant value D after a sufficient distance downstream. 

 Similarly it can be shown the standard deviation is given by 



T^ = ^2y2 n (^ _ r)R{T)dr (20-7) 



and that it becomes infinite as the first power of t or the distance down- 

 stream. In reality, this will be limited by molecular diffusion. 



In contrast to the above problem of heat transfer, an analysis of tem- 

 perature fluctuations in a statistically homogeneous field can be carried 

 out in much the same way as for homogeneous velocity fields. This was 

 done by Corrsin [71], who found that in the final stage of the decay process 

 the mean square of the temperature fluctuation decreases as the inverse 

 ■| power. This is different from the case of velocity fluctuations, and the 

 reason for this difference is the absence of the equation of continuity in 

 the present case (cf. Art. 15). Such a law was first obtained by Reissner 

 [60] in his asymptotic solution of the heat equation. 



C,21. Statistical Theory of Shear Flow. Although studies of 

 turbulent flow with shear date back further than studies of isotropic tur- 

 bulence, a complete statistical theory has not yet been developed. The 

 classical ideas of Reynolds still stand out as the best description of the 

 basic mechanism of turbulent shear. The mixture length theories, ^^ while 

 useful for practical purposes, are obviously not adequate statistical theo- 

 ries. Deviations from the classical concepts are especially evident near 

 the edge of the turbulent boundary layer and in turbulent jets and wakes. 

 Intermittency of the turbulent motion appears to be quite predominant 

 in such phenomena. It appears from these intermittent phenomena that 



^^ For details, see Sec. B. 



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