C • STATISTICAL THEORIES OF TURBULENCE 



can be calculated from a hypothesis of self-preservation. This provides 

 an experimental basis for Lin's earlier hypothesis [4.8] which was obtained 

 from general considerations influenced by the theory of Kolmogoroff. 



It may be noted here that, at least in these experiments, there is as 

 yet no need for generalizing the hypothesis further in the line indicated 

 by Goldstein [Jf.9], although the need for such generalization is not ex- 

 cluded. (See also Art. 17.) Goldstein also proposed the measurement of 

 the law of decay of turbulence behind a grid when another grid of larger 

 mesh is placed upstream. In this case, the large eddies from the first grid 

 tend to cause the turbulent motion behind the second grid to depart 

 greatly from similarity. Such experiments were made by Tsuji and Hama 

 [62], showing strong departure from the law of decay y? '^ t~^ (Fig- C,15e, 

 upper). On the other hand, the more general law of decay (Eq. 14-17) is 

 verified with the additive term h 9^ (Fig. C,15e, lower). More recently, 

 Tsuji [53] examined the spectral distribution of the turbulent motion 

 behind the second grid and obtained results in agreement with the above 

 concepts. When the second grid is 70 mesh widths behind the first grid, 

 the similarity of the vorticity spectrum is not found to be accurate, as 

 one may also expect from the fact that a well-developed turbulent mo- 

 tion is not yet formed behind the first grid. 



C,16. The Quasi-Gaussian Approximation. As pointed out in 

 Art. 7, it is possible to obtain an infinite system of differential equations 

 for determining the correlation functions of all orders. In order to obtain 

 a "deductive theory," a closed system of a finite number of partial differ- 

 ential equations is needed. For this purpose, some approximation has to 

 be made. Now it is known that the probability distribution of the ve- 

 locity components at a given point is approximately the normal distribu- 

 tion. If this were true for the joint probability distribution at several 

 points, the triple correlation function would vanish, while the correlation 

 functions of the fourth order would be related to those of the second order 

 by the relation 



UiU'jUk'u'i" = UiUj u'kU'i" + UiU'^ u'jU'i" + UiU'i" u'jU'^ (16-1) 



It is known that triple correlations do not vanish in homogeneous turbu- 

 lence, but it is still natural to speculate whether Eq. 16-1 may still be true 

 or remains a good approximation without the vanishing of the triple corre- 

 lations." There is some support of such a step from experimental obser- 

 vations (cf. Fig. C,3a and C,3b). 



To give an idea of the application of the hypothesis (Eq. 16-1), let us 

 indicate how the pressure correlation at two points may be derived. One 



" A hypothesis of this type was first introduced into the theory of turbulence by 

 Millionshchikov U2\. 



{ 236 ) 



