C • STATISTICAL THEORIES OF TURBULENCE 



range of wave number* corresponds to about one third of the total energy. 

 However, the contribution to the rate of dissipation from this range is 

 neghgible. This agrees with the concept of similarity of the vorticity 

 spectrum k}F{k, t), and lack of similarity for the energy F{k, t) for low 

 wave numbers (case (c) of the similarity hypothesis spectrum). 



Fig. C,17. The decay spectra for various values of R. The curves marked 1, 2, 3, 

 4, 5, 6, and 7 are for R'^ = 1.65, 1.34, 0.98, 0.55, 0.22, 10-^, and respectively. The 

 curve ior R = » is the decay spectrum for infinite Reynolds number and becomes 

 asymptotic to the Kolmogoroff spectrum f{x) = ix~i for a; ^ <» . 



It should be pointed out that this case has a special bearing with 

 Heisenberg's formula. If we differentiate Eq. 17-1, we obtain 



dF 

 dt 



= -2 



V + C 



F{k") 



dK' 



kW{k) + 2C 



F(/c) 



K'WiK')dK' 



(17-7) 



Thus, the behavior of the spectrum at frequency k depends on the lower 

 frequencies only to the extent of jlK'W{K')dii'. If this integral can be accu- 

 rately calculated by the similarity spectrum for high values of k, Eq. 17-7 

 is consistent with the hypothesis of similarity. Otherwise, there is some 

 doubt that similarity is possible at all, even at high frequencies, until the 

 last term in Eq. 17-7 becomes negligible. 



CHAPTER 4. TURBULENT DIFFUSION 

 AND TRANSFER 



C,18. Diffusion by Continuous Movements. One of the most 

 striking properties of turbulent motion is its diffusive property. Obser- 

 vation of smoke from a chimney stack gives a general idea of such a 



( 240 ) 



