G • STATISTICAL THEORIES OF TURBULENCE 



three scalar triple correlation coefficients h, k, and q in the following 

 manner :^ 



Tij,k = ^ ~ ^3~ ^^ r.Tjr, + 5,,n ^ + 8,,r,- 1 + Sy.fi | (6-10) 



The definitions of h, k, and g are shown in Fig. C,6a. Again, the equation 

 of continuity leads to 



^^ = (6-11) 



drk 



From this, the following two relations between the three quantities h, k, 

 and q may be deduced: 



k = -2h 



, rdh (6-12) 



expressing all triple correlation functions in terms of a single scalar 

 function. 



It can be shown, by the method of power-series expansion used above 

 for the study of the double correlations, that h, k, and q are odd functions 

 of r, and that their series expansions begin with the third powers of r. 



Triple correlations seem to have been first measured directly by 

 Townsend [10]. The more recent results of Kistler, O'Brien, and Corrsin 

 [18] are shown in Fig. C,6c. 



Higher velocity correlations. Correlation tensors involving one ve- 

 locity component each from n different points are multiple-point tensors 

 involving n — 1 positional vectors. 



Correlations involving pressure. Correlations involving pressure are 

 exemphfied by 



(1) piP)viP') and (2) vi.P)Ui{P') 



The first one is obviously a scalar quantity, which, from kinematical 

 considerations alone, is not connected with velocity correlations. How- 

 ever, by making use of dynamical relations, it can be connected with 

 velocity correlations of the fourth order. This will therefore be taken up 

 in later sections (Art. 7 and 16). On the other hand, von Kdrman and 

 Howarth [17] showed that 



^. = (6-13) 



from the requirements of isotropy and incompressibility alone. 



Robertson's invariant theory. As the need for developing more com- 

 pHcated correlations arises, one must employ some systematic methods 



• Derivation of Eq. 6-10 can be carried out by direct transformation of coordinates 

 and application of the condition of continuity. That is the original method used by 

 von Kdrmdn and Howarth. The method of Robertson described below yields the 

 result more readily. 



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