C,14 • CONSIDERATIONS OF SIMILARITY 



he derived the law of decay 



w2 = a{t - U)-^ + h (14-17) 



where a and h are constants, with a > 0. This law can be easily obtained 

 from the general relations (Eq. 14-3, 14-4, 14-5, and 14-8), which are 

 valid for any similarity hypothesis. One obtains the positive and nega- 

 tive half-power laws for the change of the characteristic length and the 

 characteristic velocity V, and the inverse square law for the rate of dissi- 

 pation €. To be more specific, one finds that I and V may be identified 

 with Kolmogoroff's characteristic quantities (cf. Eq. 13-9 and 13-10) 



(IT 



77 = 1-1 and V = (ve)i (14-18) 



It can easily be seen by introducing these relations into Eq. 14-3, 14-4, 

 and 14-5 that the law of decay is of the form of Eq. 14-17. 



It is convenient to rewrite the results as follows, with definite physical 

 interpretations attached to the constants. The law of decay is given by 



(14-19) 



where u% is the additive constant giving the departure of the energy con- 

 tent from that in the case of similarity, and Do is the initial diffusion 

 coefficient 



Do = Km ^^ (14-20) 



defined according to a formula of the kind suggested earlier by von 

 Karman [13]. The changes with time of the characteristic velocity and 

 scale, and of the Reynolds number of turbulence are given by 



(-^^') 



y2 = (io)-|^^^,^-i^ ^2 = (loy.R^.'d, R^ = Rxo 11 - ^yP t ) (14-21) 

 where R\o is the initial Reynolds number of turbulence 



Rxo = lim ^ (14-22) 



It is evident from Eq. 14-19 and 14-21 that the solutions obtained 

 can only be applied to an early stage of the decay process, in which 

 10^1)^/2) remains small. 



Case {d). The above assumption is based on the idea that the low 

 frequency components do not have the time to adjust themselves to an 

 equilibrium state. (An investigation of such a concept was made by Lin, 

 and will be briefly presented in Art. 17.) It is specifically assumed that 

 e may be calculated by a similarity spectrum. Goldstein [49] further 



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