C • STATISTICAL THEORIES OF TURBULENCE 



property of turbulent motion should have a universal character described 

 by the following concepts. First, it is locally isotropic whether the large 

 scale motions are isotropic or not.^^ Second, the motion at the very small 

 scales is chiefly governed by the viscous forces and the amount of energy 

 which is handed down to them from the larger eddies. The large eddies 

 tend to break down into smaller eddies due to inertial forces. These in 

 turn break down into still smaller eddies, and so on. At the same time, 

 viscous forces dissipate these eddies at very small scales into heat. In the 

 long series of processes of reaching the smallest eddies, the turbulent mo- 

 tion adjusts itself to some definite state. The further down the scale, the 

 less is the motion dependent on the large eddies. 



Furthermore, in line with Taylor's experimental findings, Kolmogoroff 

 essentially postulates that practically all the dissipation of energy occurs 

 at the smallest scales when the Reynolds number of turbulent motion is 

 sufficiently high. 



To formulate these concepts mathematically, he introduced the corre- 

 lation functions of the type 



{u-u'Y = y?[\ -/(r)] 



which is the mean square value of the relative velocity of turbulent mo- 

 tion. The introduction of the relative velocity stresses the local nature. 

 The moments {u — u'Y would then be emphasized instead of the usual 

 correlations at two points. (In fact, the third moment {u — u'Y is pro- 

 portional to k{r).) 



The second step in the formulation of the theory is to introduce the 

 assumption that, for small values of r, these correlation functions depend 

 only on the kinematic viscosity v and the total rate of energy dissipation e. 

 This is in accordance with the previously discussed physical concepts. 

 One can then make some dimensional analysis and construct universal 

 characteristic velocity and length for motion at very small scales. Indeed, 

 from e and v, one can only construct the length scale 



and the velocity scale 

 We may then write 



■n = (jj (13-9) 



V = (!/e)i (13-10) 



{W - uY = {veY^aai^^j (13-11) 



{W -uY = iveY^d,a[^^J (13-12) 



where ^dd and ^ddd are universal functions for small values of r. 

 ^^ See Sec. B on shear flows for the experimental confirmation of this fact. 



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