G,16 • THE QUASI-GAUSSIAN APPROXIMATION 

 first notes that 



^P = a^("-^') (i«-2) 



SO that 



d 



^^^'P' = dndrjdndn '''''^'''''''^ ^^^"^^ 



We now break up the quadruple correlation by Eq. 16-1. In this manner, 

 one finds that 



W = 2u^ j^ [^ - ^) WmH^ (16-4) 



Eq. 16-4 was given by Batchelor [12] while an equivalent relation in the 

 spectral formulation was obtained earlier by Heisenberg [23]. 



An interesting observation may be made here. If the relation (Eq. 

 16-4) is accepted, it is possible to show that, at high Reynolds numbers 

 of turbulence, the magnitude of the pressure term in the Navier-Stokes 

 equations is much smaller in magnitude than either the local acceleration 

 or the convective acceleration taken individually [20]. 



With the help of Eq. 16-1, one can derive a system of partial differ- 

 ential equations with the same number of equations as unknowns. The 

 equation for the change of double correlation functions involves the triple 

 correlation functions. The equation for the change of the triple correlation 

 function involves the fourth-order correlations, which may be reduced to 

 the double correlations by Eq. 16-1. In this manner, a closed system of 

 equations is obtained. Such a theory was independently developed by 

 Proudman and Reid [54] and by Tatsumi [55] for decaying isotropic 

 turbulence in the spectral formulation. Without giving a detailed account, 

 a few of the outstanding features are discussed in the following paragraphs. 



1. The above reasoning for the establishment of a closed system was 

 obviously only true for multipoint correlations and not for two-point 

 correlations alone. While correlations of the fourth order are repre- 

 sented in terms of two-point correlations by Eq. 16-1, the triple corre- 

 lations must be kept in the general form of a three-point correlation. 

 In an isotropic case, there are then three independent space variables, 

 e.g. the three sides of the triangle with vertices at the points in ques- 

 tion. In the spectral formulation, the final equations contain three 

 independent wave numbers. 



2. According to the experimental results of Stewart (Fig. C,3b) the 

 hypothesis (Eq. 16-1) may become poor for small distances. Thus it 

 would be desirable to examine the behavior of the small eddies accord- 

 ing to this theory, and to compare it with the Kolmogoroff-Obukhoff 

 spectrum k~^ in the case of infinite Reynolds number. Such a com- 

 parison has not yet been carried out. 



< 237 ) 



