C,14 • CONSIDERATIONS OF SIMILARITY 



Heisenberg [41] indicated an equivalent solution for the spectral problem. 

 It is easily seen that these solutions are at variance with Eq. 14-9. In 

 other words, full similarity is only possible when we reject Loitsiansky's 

 theorem. In addition, experimental evidence clearly indicates that the 

 law of decay and the behavior of the characteristic length during decay 

 exclude the possibiUty of adopting full similarity as a generally valid 

 assumption for all decay processes. 



Let us now consider two opposite approaches. In the first approach, 

 we assume that Loitsiansky's invariant exists and that it plays a role in 

 the similarity of the spectrum. In the second approach, we assume that 

 similarity of the spectrum is occurring only in the eddies contributing 

 appreciably to the dissipation process, and that the largest eddies play 

 no role in determining the similarity of the spectrum. Clearly, the first 

 approach will not yield valid results unless Loitsiansky's invariant does 

 exist. This is definitely known only in the decay of isotropic turbulence 

 at very low Reynolds numbers (case (a) below). The second approach is 

 naturally independent of Loitsiansky's invariant. 



Let us consider now two opposite specific cases in the first approach : 

 (a) the transfer term is negligible for all frequencies, and (b) the influ- 

 ence of viscous dissipation is restricted to high frequencies whereas for 

 low frequencies the transfer term is the prevaihng factor. 



Case (a), w{^) = 0, leads to a solution of Eq. 8-7 which has full simi- 

 larity for all frequencies and also satisfies Loitsiansky's relation. One ob- 

 tains with ^ = d and I = \/Vt 



F = const VH^'e-^^' (14-10) 



or 



F = const VH'K'e-^'^''' (14-11) 



By using the definition of J in Eq. 14-9, we write 



F = Jk^c-^^'^'' (14-12) 



The corresponding correlation function can be easily shown to be 



/(r, t) = e-'-^/s^' (14-13) 



by using Eq. 8-1 and 10-3. This correlation function was noted by von 

 Kdrman and Howarth [17], and discussed by Milhonshchikov [4-2], 

 Loitsiansky [29], and Batchelor and Townsend [43]. Kd,rmd,n and Howarth 

 also obtained a more general self-preserving solution in terms of the 

 Whittaker function, with a spectral form F = CK^e"^'"'''. It can be easily 

 shown that the solution must specialize into Eq. 14-13 if the Loitsiansky 

 invariant is to be finite. The law of decay in this case is the five-fourths 

 power law : 



u^'^it - U)-^, X2 = 4:v{t - to) (14-14) 



This law of decay and the corresponding correlation function have been 



< 227 ) 



