C,6 • KINEMATICS OF TURBULENCE 



The idea of using the correlation function for isotropic turbulence 

 was first introduced by Taylor [15], who also gave the above proof that 

 they are even functions. The correlation tensor was first introduced by 

 von Karmdn [13], who also deduced Eq. 6-6. Detailed experimental veri- 

 fication of this relation (Eq. 6-6) seems to have first been made by 

 Macphail [16], and reconfirmed by later experimenters (see Fig. C,6b). 



Fig. C,6b. Experimental verification of von Kdrm^n's relation for isotropic turbu- 

 lence, after Macphail [16]. Ri = /(r), R2 = g(r). M denotes mesh width, y and z are 

 distances parallel to the grid. 



Triple velocity correlations. Continuing the study of velocity corre- 

 lation, one would naturally be led to correlations for velocity components 

 at three points P, P' , and P" : 



Tij,k = u'Tij,k = Ui{P)uj{P')ukiP") 



(6-8) 



This triple correlation tensor is a function of the two vectors PP' and 

 PP", say, and shows clearly that we are dealing with multiple-point 

 tensors. Often one needs only the correlation tensor Tij,k for two points 

 PandP': 



Tij.k = u'fij,k = Ui{P)uj{P)uk{P') (6-9) 



It then becomes a function of the vector PP'. Such a two-point triple 

 correlation tensor was first studied by von Karman and Howarth [17], 

 who showed that, because of isotropy, it can be expressed in terms of 



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