C,6 • KINEMATICS OF TURBULENCE 



of parallel components taken. Now the velocity at a given point may be 

 expressed as a linear combination of three mutually perpendicular com- 

 ponents, taken along and perpendicular to PP'. The general velocity 

 correlation between P and P' can therefore be expressed in terms of the 

 nine correlations between Ur, Ut, Wp and u'^, u[, u'^, where u^ and u'^ are 

 components perpendicular to both Ur and Wf By isotropy, correlations hke 

 Uru[ and u^u'^ are zero, and we see that an arbitrary velocity correlation 

 can be expressed in terms of the two basic correlations /(r) and g{r). 

 In fact, if Ui{i = 1, 2, 3) are the components of velocity at P{xi), and 



f(r) 



U2 



U2 



g(r) 



h(r) 



■»- 



k(r) 



U2 



Ul 



q(r) 



Fig. C,6a. Diagram illustrating the definition of the principal 

 correlation functions in isotropic turbulence. 



u'j{j = 1, 2, 3) are those at P'{x'^, von K^rman [13] has shown by direct 

 calculation that 



■Lb ij U/'i ttj 



fir) - gix) 



TiTj + g{r)h 



(6-3) 



where r^ = x'i — Xi, and 8ij is the Kronecker delta (da = \\li = j, 8ij = 

 if ^ 5^ j) . The correlation coefficient UiUj/yr will be denoted by Rij, and 

 is equal to the expression in the brackets in Eq. 6-3. A derivation of Eq. 

 6-3, following the method of Robertson [14] will be given at the end of 

 this article. 



By using the correlation tensor (Eq. 6-3), correlations involving ve- 

 locity derivatives can be conveniently calculated. For example, if one 



< 203 ) 



