C,12 • LARGE SCALE STRUCTURE OF TURBULENCE 

 and ^ij can be expressed in the form 



% = ^M{k^ - KiKj) + XiMxfilCm) (11-5) 



where i/'(0 is a scalar function of the vector k^, Xiium) is a vector perpen- 

 dicular to Km, and X* is its complex conjugate. When the turbulence is 

 isotropic, Xi = 0, and ^(/c™) is a function of the magnitude k only. The 

 form (Eq. 11-5) is due to Kampe de Feriet [^4]- 



The dynamical equations for anisotropic turbulence are more compli- 

 cated than those for isotropic turbulence, among other things, by the 

 presence of the pressure terms in the equations of the change of double 

 correlations. In the correlation form, the equations are 



dt 

 where 



^^'' = Pik + Tik + 2v^Rik (11-6) 



=U^-^"^.^'^*) ^^^-^^ 



Pa = -[^ 



Tik = Y~ (uiUpUk — UiUpU'k) (11-8) 



In the spectral form, we have 



-^ = Uik + Qik - 2uKWik (11-9) 



where 11^ and Qik are respectively the Fourier transforms of Pik and Tik. 

 Obviously Pa = 0, so that Ilji = 0. Thus the pressure fluctuations have 

 no effect on the total energy density Fkk', their influence produces a re- 

 distribution of energy among the various directions. It is not immediately 

 evident whether the net effect is to make the turbulent field more or less 

 isotropic, but general evidence seems to indicate that the former is the 

 case. 



The above developments are mostly due to Batchelor [25]. Other 

 detailed studies of anisotropic turbulence have been by Batchelor [26], 

 Chandrasekhar [27], and others. The reader is referred to the original 

 papers. 



CHAPTER 3, PHYSICAL ASPECTS OF THE 

 THEORY OF HOMOGENEOUS TURBULENCE 



C,12. Large Scale Structure of Turbulence. In the following 

 articles, we shall make use of the methods developed above — the corre- 

 lation and spectral theories — to study the nature of turbulent motion. 

 As pointed out above, the theory by itself allows us to reach only partial 



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