C • STATISTICAL THEORIES OF TURBULENCE 



In the case of a turbulent flow with steady mean motion, the time 

 average is taken at every point, and the above physical interpretation 

 of the Reynolds stress is clear. In the case of variable mean motion, such 

 as the case of the infinite plate mentioned above, where mean values 

 are defined as the averages over parallel planes, the interpretation of 

 — pUiUj as a local stress is not as direct. In the case of general variable 

 motion, where the averaging process is the arithmetical mean taken over a 

 large (infinite) number of unrelated identical systems (ensemble average), 

 the physical interpretation of an average quantity as an apparent stress 

 requires even more careful examination, since the average momentum 

 transfer is not directly associated with any one particular system. More- 

 over, the time average is usually measured in the case of steady mean 

 flow. Thus, if the general theory is developed on the basis of statistical 

 averages, an ergodic hypothesis must be introduced to identify these two 

 in that case. In this section, the statistical average shall be adopted, and 

 the vahdity of such a hypothesis shall be implied. Further investigations 

 of such basic problems are beyond the scope of the present treatment. 



C,3. Frequency Distributions and Statistical Averages. One 



basic concept in the discussion of statistical averages is the frequency of 

 occurrence, or the distribution function. For example, in the classical 

 kinetic theory of gases, one considers a distribution function /(w, y, w, x, 

 y, z, t) such that 



f{u, V, w, X, y, z, t)dudvdwdxdydz 



gives the fraction of molecules at time t, having velocities in the range 

 u, u -\- du] V, V -\- dv; w, w -]- dw and lying in the element of volume 

 X, X -{- dx; y, y -{- dy; z, z -\- dz. The kinetic theory of gases may then be 

 based on the law governing the change of this function /(w, v, w, x, y, z, t). 

 For a homogeneous gas at rest, it is the well-known Maxwellian function. 

 In the case of turbulent motion, a similar (but different) function 

 F{u, V, w, X, y, z, t) can be introduced giving, for each point {x, y, z) and 

 each instant t, the probabihty that the turbulent velocity shall lie in the 

 range u, u -\- du; v, v -\- dv; w, w -{- dw, or for shortness, Ui, Ui + dui. If 

 this function is known, then the Reynolds shear is given by formulas of 

 the kind 



— puv = —p III F{u, V, w, X, y, z, t)uvdudvdw (3-1) 



— 00 



for each point x, y, z at each instant t. 



To analyze the structure of turbulence one also needs to know the 

 joint probability distribution for quantities observed at several points. 

 For example, if we are interested in the correlation of velocities at two 

 points P' and P", then we must know a distribution function of the form 



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