G • STATISTICAL THEORIES OF TURBULENCE 



verified experimentally by Batchelor and Townsend for the final stage 

 of decay (see Art. 15 for further details). 



Case (b) has also been treated in the theory of self-preserving corre- 

 lations by von Karman and Howarth [17] and later by Kolmogoroff [44]- 

 The former authors came to the conclusion that any power law for the 

 decay-time relation may prevail in the decay process. Kolmogoroff 

 pointed out that if one assumes the validity of Loitsiansky's theorem 

 the relations 



w2 = const t-'^' and X^ = 7vt (14-15) 



must apply. ^^ Von Karman [43, 4^] dealt with the corresponding spectral 

 problem in two communications assuming the specific decay law (Eq. 

 14-15). It should be reiterated, however, that this first approach, especi- 

 ally in case (b), can only be regarded as tentative because of the un- 

 certainty in the constancy of the Loitsiansky integral. 



Consider now the second approach. Clearly, the idea of complete 

 similarity, with the rejection of Loitsiansky's relation, belongs to this 

 case. However, there are physical and mathematical reasons for believing 

 that the large eddies do not play a significant role in the determination 

 of the similarity characteristics in the smaller eddies. We therefore con- 

 sider cases where the similarity requirement is relaxed for an increasing 

 range of frequencies at the end of largest eddies. 



Case (c). We first consider the assumption that similarity extends 

 over the whole frequency range, with the exception of the lowest. More 

 specifically, we assume that the deviation from similarity shall occur for 

 such small values of k that, whereas the contribution of the deviation is 

 neghgible for computation of e (Eq. 8-11), it enters in the calculation of 

 energy (Eq. 8-6). 



It is easy to see the corresponding assumption in the correlation for- 

 mulation by using Eq. 13-2 and 13-4 in the following form: 



n = l 



The above assumptions imply that all the higher moments of F{k) are not 

 appreciably influenced by the deviation from similarity. Hence, they are 

 all proportional to F^/"-". Similarity is therefore assumed for ^^[1 — /(r)]. 

 This form of the similarity hypothesis was introduced by Lin [48]. 

 Assuming the self-preservation of 



{u - u'y = u^[l - f(r)] 

 and 



{u - u'y = I2u%{r) 



1* See Frenkiel [47] for some discussion of the comparison of Eq. 14-15 with some 

 experiments. 



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