C • STATISTICAL THEORIES OF TURBULENCE 



wants to calculate the correlation between Ui at P and the derivatives 

 du'j/dx'k at P' , one has only to use the following identity: 



The above transformations are made by using the general rule for aver- 

 aging a derivative and the definition of ru- Similarly 



dUi dUj _ d d 7 



dxi dx'k dn dfk ' ^ 



If we now make use of the equation of continuity bu'Jbx] = 0, we may 

 obtain from Eq. 6-4 



^-^ = (6-5) 



By using Eq. 6-3, this eventually gives rise to the single relation 



= f + ^% (6-6) 



connecting the two correlation functions / and g. Thus, in homogeneous 

 isotropic turbulence, all the correlation functions of the second order can be 

 expressed in terms of a single correlation function, say f{r) . 



We shall now show that the correlation tensor (Eq. 6-3) is an even 

 function of Vi. To do this, it is only necessary to show that f(r) is an even 

 function of r. It then follows from Eq. 5-6 that g{r) is also an even func- 

 tion, and the desired result becomes obvious from the formula (Eq. 6-3). 



Consider two points P and P' along the x axis at a distance r apart. 



Then 



u2/(r) = u{x)u{x + r) 



Expanding u{x -\- r) into a Taylor series, we obtain 



(•/ \ -(I UUx I J- UUxx n I 



The coefficients of this power series can be simplified as follows by using 

 the condition of homogeneity: 



UUx = ■i-(w^)x ^ 



UUxx = {uUx)x — ul = —III 



It can be easily seen that all coefficients of the odd powers of r are zero. 

 Thus, 



fir) = l-^^;r^+ • • • (6-7) 



^ u^ 



is an even function of r. 



< 204 ) 



