C • STATISTICAL THEORIES OF TURBULENCE 

 where 



kHi{k) = =^ I h{r) sin Krdr 



"" -^0 (8-9) 



h{r) = / kHi{k) sin ktcIk 



It is clear that the quantity WiK, t) in Eq. 8-7 represents the transfer 

 of energy among various frequencies. The above formula for W{k, t) also 

 shows that 



WcLk = (8-10) 



i: 



which means that no energy is generated or lost while it is redistributed 

 among various scales. The rate of dissipation is obtained from Eq. 8-7 by 

 integrating it with respect to k from k = to k = oo : 





I ^-^dK= -2v f liWdK (8-11) 



Jo ot Jo 



Exactly as in the case of the correlation theory, one cannot proceed 

 much further with the basic equation (Eq. 8-7) without a more specific 

 knowledge of W. However, with the physical interpretation that W{k, t) 

 represents the transfer of energy among various frequencies, it has been 

 found possible to obtain certain plausible formulas connecting WiK, t) 

 with F(k, t) and to make reasonable deductions. (Cf. Art. 17.) 



C,9. Spectral Analysis in One Dimension. We shall now develop 

 briefly the one-dimensional spectral analysis of a field of turbulence and 

 derive the Fourier transform relations (Eq. 8-1). 



In a homogeneous (not necessarily isotropic) field of turbulence, let 

 u(x) be the velocity at the point x in the direction of the x axis. It re- 

 mains finite as a; — > + oo . This makes its Fourier analysis more difficult 

 than that of a function which vanishes rapidly at infinity. For such a 

 function, <j){x), we have the pair of Fourier transform relations 



</)(.t) = /"_" a{K)e-'''='dK 



1 ['" 

 «(«) = -FT / <{>ix)e^'"'dx 

 Zir J -K 



where a{ — K) is equal to the complex conjugate a*(K) for real 0(.t), and 

 |a(K)|'^ is a measure of the energy content associated with the wave num- 

 ber or spatial frequency k. However, since the velocity fluctuation u{x) 

 in a homogeneous field of turbulence does not approach zero as x — > ± °o , 

 we cannot put u{x) in place of (}){x) in the above relation. Instead we 



< 214 ) 



