C • STATISTICAL THEORIES OF TURBULENCE 



and 



175 = u' f^ dt' // R(j)dT = v" /J {T - t)Rit)dt (18-4) 



In general, we may expect R{t) to decrease with increasing t. Suppose 

 that, for all times t greater than Ti, R(t) is practically zero. Then 



r R{T)dT = r R{T)dr, for t > Tx (18-5) 



and Eq. 18-4 gives 



The integral (Eq. 18-5) is a measure of the time scale of diffusion, and 



D = v-' jj R{T)dT (18-7) 



may be regarded as a diffusion coefficient, since for molecular diffusion, 

 T^ = 2DT. Thus the concept of a diffusion coefficient is justified for 

 large values of time of diffusion. On the other hand, for small values of 

 time t {t much less than the time scale defined by Eq. 18-5), R{t) '^ 1, 

 and Eq. 18-4 gives 



72 = y22^2 or VT^ = vT (18-8) 



It appears that when T is small, Y^ is proportional to T^ instead of T, 

 as in an ordinary diffusion process. This is clearly so, because over the 

 time interval in which R{t) is nearly equal to unity, the velocities of the 

 particles are nearly constant so that for each particle 



Y = vT (18-9) 



In this case, therefore, not only is Eq. 18-8 valid, but the frequency dis- 

 tribution of Y is the same as the frequency distribution of v. 



We shall now apply these ideas to the problem of the spread of heat 

 behind a heated wire. If heat is spread from a concentrated plane source, 

 after an interval of time t, the distribution of temperature according to the 

 usual process of molecular diffusion is proportional to t~^ exp [ — y^/4:kt], 

 where y is the normal distance from the plane source. The temperature 

 distribution behind a heated wire corresponds to such a problem, if there 

 is only molecular diffusion. The distribution is given by the above ex- 

 pression with t replaced by x/U, where x is the distance downstream 

 from the wire and U is the speed of the wind. In a turbulent stream, 

 diffusion due to turbulent motion must be superposed on molecular dif- 

 fusion. In fact, Schubauer has observed the error law for the distribution 

 of temperature, and it would appear that the phenomenon of turbulent 

 diffusion can be described by an adequate diffusion coefficient associated 

 with the turbulent motion. However, the above analysis shows that the 



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