C • STATISTICAL THEORIES OF TURBULENCE 



Taylor made use of Eq. 8-1 to connect the observed time spectrum 

 with the spatial correlation function by way of his assumption. To do 

 this, the spatial distance r is replaced by Ut and the time frequency n is 

 related to k by 



kU = 2Trn 

 Then, 



u'^fif) = J F{n) cos -^^ dn 

 /o 'J 



(8-3) 



Pin) = -g- / fix) cos -jj- dr 



These relations were actually well verified, justifying his assumption ex- 

 perimentally. (See Fig. C,8; after Stewart and Townsend [22].) 



From Eq. 8-3 we can calculate the rate of energy dissipation in terms 

 of the spectrum. It is easy to show that 



u^f'iS)) = ^ / F(n)n'dn (8-4) 



fJ" Jo 



which is proportional to the rate of energy dissipation (cf. Eq. 7-5). This 

 formula shows that the high frequency components are more important 

 for the dissipation of energy. In fact, Taylor found from an analysis of 

 his measurements of spectrum that the dissipation of energy is practically 

 all associated with such high frequency components which contain a negligible 

 amount of energy. This has a very important bearing on later develop- 

 ments (see Art. 13 on Kolmogoroff's theory). 



The above spectral considerations do not give a proper representation 

 of the energy distribution among various scales. For theoretical purposes, 

 one should then consider spectral functions obtained by a three-dimen- 

 sional harmonic analysis. As it will be shown in Art. 10, the three-dimen- 

 sional spectrum F{k) is connected with the one-dimensional spectrum 

 Fi{k) by the relation [cf. Eq. 10-3], 



F{k) =U>^W['{k) -kF[{k)] (8-5) 



The kinetic energy per unit volume, for each component of the motion, 

 lying in the range of spacial frequencies (k, k + dK) is now given by 

 ^pF(K)dK, the functions F{k) and Fi{k) being both normalized to give 



w2 = fj F(iK)dK = r FliK)dK (8-6) 



It is now easy to obtain the equation for the change of spectrum. We 

 take the cosine transform of the Kd,rman-Howarth equation and then 

 apply the operation 



if 2'^ _ A 



3 y dK^ " dKj 



<212 ) 



