C • STATISTICAL THEORIES OF TURBULENCE 



CHAPTER 2. MATHEMATICAL FORMULATION 



OF THE THEORY OF HOMOGENEOUS 



TURBULENCE 



C,6. Kinematics of Homogeneous Isotropic Turbulence. Cor- 

 relation Theory. In this chapter, we shall develop the theoretical 

 concepts used for the description of homogeneous turbulence. The main 

 body of the discussion will be limited to the isotropic case. The general 

 case of anisotropic turbulence will be taken up in Art. 11. 



The statistical correlation of velocity fluctuations at two points is the 

 most commonly used quantity for describing the structure of an isotropic 

 field of turbulence. Clearly, the larger the size of the eddies, the further 

 the correlation extends. Velocity correlations are used not only because 

 they are the easiest to measure, but also because correlations involving 

 pressure fluctuations are theoretically representable in terms of them. 

 Experimentally, it is as yet difficult to determine correlations involving 

 pressure fluctuations. In general, we shall be dealing with the correlation 

 of quantities at several points and at different instants of time. For ex- 

 ample, for three points P, P' , and P" in a field of turbulent motion, we 

 may wish to consider the correlation mi(P)w2(P')p(-P")j where U\{P) and 

 U2{P') are respectively the components of velocity in the direction of the 

 Xi axis at the point P and in the direction of the X2 axis at the point P', 

 and p{P") is the pressure at the point P". Statistical correlations of the 

 velocity components at one point are exemplified by the Reynolds stresses. 

 In the order of increasing complexity, we next consider correlations at two 

 points. As explained above, we now deal with the special case of isotropic 

 turbulence. 



Double velocity correlations. Since there is no preferred choice of the 

 coordinate system in the isotropic case, it is clear that the correlations 

 must be basically characterized by the directions of the velocity com- 

 ponents relative to the vector PP' joining the two points at which the 

 velocities are considered. It is therefore convenient to consider a longi- 

 tudinal correlation coefficient /(r) defined by (see Fig. C,6a)^ 



UrK = u^f(r) (6-1) 



Similarly, one may define a transverse correlation coefficient g{r) by 



Utu[ = u^gir) (6-2) 



for two parallel velocity components perpendicular to PP'. It is obvious 

 from isotropy that this correlation is independent of the particular pair 



^ Here the line PP' lies in the direction of the Xi axis, and the three mutually per- 

 pendicular components Ut, Ut, Up are Ui, 112, uz. 



{ 202 > 



