C • STATISTICAL THEORIES OF TURBULENCE 



a statistical theory of shear flow can be developed only after adequate 

 descriptions are obtained, both for motions on a small scale and for 

 motions on a scale comparable with that of the mean flow. The fact that 

 turbulent transfer is most effectively carried out by such large scale mo- 

 tions is in direct contrast to the phenomenon of molecular transfer. In 

 that case, the mean free path is much smaller than the scale of the mean 

 motion, and a definite coefficient of transfer is established. The difference 

 in scales of the random motion responsible for the transfer mechanism 

 in the two cases makes the analogy imperfect, and is at the root of the 

 difficulties in developing a theory of turbulent transfer. 



For steady flow through pipes and channels, the phenomenon is sim- 

 pler in the sense that intermittency is not apparent. Attempts to develop 

 a statistical theory, based on the use of correlations, have been made by 

 Keller and Friedmann [72], von Kdrmdn [IS], Chou [73], and Rotta [74] . 

 While the theory predicts the mean velocity and turbulence level in 

 reasonable agreement with experiments, the presence of arbitrary con- 

 stants shows their weakness. In the following, only an indication of the 

 approach is given; the reader is referred to the original papers for the 

 details. 



The equations of motion for the turbulent fluctuations can be ob- 

 tained by subtracting the Reynolds equations (Eq. 2-3) from the com- 

 plete equations of motion (Eq. 2-2), 



dUi ^j dUi dUi dUi 1 dp 1 dnj mi ^\ 



-^ -^ Uj- \- Uj- 1- wy -— = — ^ 5— + vAUi (21-1) 



ot dXj dXj oxj p dxi p oXj 



where p is the fluctuation of pressure, and Ty are the Reynolds stresses. 

 The velocity fluctuations w,- also satisfy the equation of continuity, 



^ = (21-2) 



dXk 



Instead of dealing with the velocity fluctuations themselves, one may 

 attempt to deal with the statistical correlation of the velocity fluctuations 

 in analogy with the study of homogeneous turbulence. To simplify mat- 

 ters, one may also eliminate the pressure fluctuation by using the Poisson 

 equation obtained by taking the divergence of Eq. 21-1 : 



and solving it under appropriate boundary conditions. However, the fun- 

 damental difficulty encountered in the homogeneous case — that higher 

 correlations are invariably brought into the picture when an equation is 

 constructed for dealing with correlations of a given order — also occurs 

 here. Approximations are therefore introduced by the various authors at 

 this stage, and the theory is not entirely free from arbitrariness. 



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