C • STATISTICAL THEORIES OF TURBULENCE 

 K2 and Kz while keeping ki constant. Thus, the one-dimensional spectrum is 



Fi(kx) = 11^(^1- fj HKj)dK2dK, 



Now, the three-dimensional spectrum is isotropic, so that 



F{k) = ^H>c) 



considering all k/s with the same magnitude k. We have finally 



00 



Fi(ki) = Fx(«0 + F^{-K^) = jj {^ - 72) ^ dK^dK^ 



— 00 



This is easily transformed into Eq. 10-2 by carrying out the integration 

 in a polar coordinate system in the plane of k2, ks. 



C,ll. General Theory of Homogeneous Anisotropic Turbulence. 



The above development of the theory of homogeneous isotropic turbu- 

 lence can be generaUzed to remove the restriction of isotropy. Such a 

 generalization is necessary because anisotropy of turbulence, particularly 

 in the largest eddies, does occur in practice. We shall outline here only 

 the main features of the developments and conclusions, pointing out 

 especially the difference between the isotropic and anisotropic cases. 



The concept of correlation functions requires very little modification, 

 although it is now obviously impossible to represent the double corre- 

 lation functions, for example, in terms of a single scalar function. The 

 spectral function must be replaced by a spectral tensor, which may be 

 defined as the three-dimensional Fourier transform of the double corre- 

 lation tensor. Thus, 



^ik{Kj) =^zjjj Rik{r^)e'^'"'"'^dr{r^) (11-1) 



and 



Rij = W fif $iy(K„)e-^('''"-")<iT(0 (11-2) 



It can be shown that $yy represents the energy density in the wave num- 

 ber space. In the case of isotropic turbulence, 



F = 47r/c2$,.y (11-3) 



Because of the condition of vanishing divergence of the correlation tensor, 

 we obtain 



^ijKi = (11-4) 



<218> 



