Some Progress in Turbulence Theory 



Q kL(0) 



12 



Fig. 3 - Evolution of the energy spectrum 

 from the initial values of Eq. (19) (initial 

 R\ = 35) 



E(k,0) 



exp 



k. 



(20) 



In contrast to the previous case, tne energy spectrum here is nearly self- 

 preserving, particularly at the higher wave numbers, with very little change of 

 R^ during decay. Figure 7 shows the dissipation, energy transfer, and vorticity- 

 production spectra near the end of the time interval covered in Fig. 6. 



The normalizing parameter k^ used in Fig. 7 is a characteristic dissipation- 

 range wave number of the direct- interaction solutions [12,14]. It is related to K 

 by kj = (15rx)^/^A. The numerical integrations for a number of different 

 shapes of initial spectra have been found to lead to evolved dissipation-range 

 spectra which are nearly independent of initial conditions, and only weakly de- 

 pendent on R;^. This is true whether \ or k^ is used for normalization, in the 

 range r^ < 50 covered by the integrations. 



Figure 8 shows the evolution of the skewness factor [10] 



795 



