C,14 • CONSIDERATIONS OF SIMILARITY 



For very high Reynolds numbers, Kolmogoroff visuahzes that, at the 

 larger end of the universal range, there is a range of r for which the vis- 

 cosity coefficient does not play an expHcit role. This range may be con- 

 veniently referred to as an inertial subrange. The above relation then 

 implies that 



{u' - uY '^ (er)? (13-13) 



A definite form of the correlation function is thereby obtained. 



The concept of Kolmogoroff can also be introduced into the spectral 

 formulation. Thus, at high Reynolds numbers the spectrum F{k) at very 

 high frequencies can be expressed as 



F{k) = vSKKfi) (13-14) 



where the function fix) has a universal form for large values of x. 



For the inertial subrange, the spectral function can again be deter- 

 mined completely from dimensional arguments. This gives 



F{k) ^ e?K-=^ (13-15) 



This form was first given by Obukhoff [37]. It has received some experi- 

 mental support at high Reynolds numbers. ^^ With a spectrum of this 

 form, it can be expHcitly demonstrated that the dissipation of energy lies 

 essentially in the universal range of Kolmogoroff (cf. [39]). 



The actual form of the spectrum in the universal range is obviously 

 of basic theoretical interest. By following the general ideas discussed in 

 this section, Townsend [38] developed a more concrete model giving a 

 definite form for the spectrum of the small eddies. The results are in 

 general agreement with experimental observations. 



The scales -q and v defined above also occur in the study of the small 

 scale structure even when the Reynolds number is not high. This cannot 

 be interpreted on the basis of Kolmogoroff's theory, but follows from 

 considerations of self-preservation during the process of decay (see next 

 article) . 



C,14. Considerations of Similarity. As noted above, the general 

 theory of turbulent motion, as developed in Chap. 2, cannot lead to 

 specific predictions without auxihary considerations. For this reason, 

 von Karman and Howarth [17] introduced the idea of self-preservation 

 of correlation functions. ^^ In terms of the spectral language, this states 

 that the spectrum remains similar in the course of time. Since the energy 

 distribution among the various frequencies is changing through the trans- 

 fer mechanism, this may be reasonably expected provided that there is 

 enough time for the necessary adjustments. In this article, we shall con- 



12 Cf. [99] and [101] for detailed discussions. A different form of the spectrum has 

 been recently obtained by Kraichnan [100]. 



13 This article follows closely the treatment of von Kdrmdn and Lin [39, p. 1]. 



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