C,17 • HYPOTHESES ON ENERGY TRANSFER 

 where C is a constant. The term 



e. = C j^ ^^-^dK" (17-2) 



represents an apparent kinematic viscosity coefficient associated with the 

 motions of wave number above k. 

 Von Karman proposed the form 



j^ WdK = C 11^" [F{k')Yk'Hk''^ ' {|^''F(K')«-V^-^d/c'} (17-3) 



for the transfer function. It reduces to Heisenberg's form for a = ^, 

 i8 = — f . It also reduces to a modified Obukhoff form for a = I, 13 = 0. 

 In the present discussion, we shall restrict ourselves to the Heisenberg 

 formula and use Eq. 17-1 as the basis for determining the spectrum 

 F{k, t) at any future time from its present knowledge. 



For large values of k, the rate of loss of energy to still larger wave 

 numbers is expected to be very small. Consequently, the left side of Eq. 

 17-1 is neghgible. It then follows (see Chandrasekhar [58]) from Eq. 17-1 

 that 



f(M) = const Q'j^^^ (17-4) 



where ko is a constant and Ks is inversely proportional to Kolmogorofi's 

 scale 17. For large values of k, F{k) ^^ kt"^. However, theoretical and ex- 

 perimental considerations indicate that F(k) probably decreases faster 

 than kt'' at high wave numbers. 



If we now introduce the hypothesis of similarity, we can determine 

 the spectrum function completely. In fact, if the hypothesis of complete 

 similarity is used, we have F{k, t) in the form 



where ko and ^0 are certain constants. According to Eq. 17-1, f{x) satisfies 

 j^J{x)dx - \ xKx) = ^-^ j^ yj^ dx" 1^ 2f{x')x'Hx' (17-6) 



where R = 1/vkIU. This equation has been solved numerically by Chan- 

 drasekhar [58] for a series of values of R\ (Fig. C,17). Proudman [59] 

 computed the correlation function from these spectral functions and 

 found quite good agreement with experiments. 



The spectrum defined by Eq. 17-6 has a relatively long range, at low 

 values of /c, for which F{k) '^ k. However, an examination of the stability 

 of the spectrum [60] indicates that this part of the spectrum is not stable. 

 At the Reynolds numbers obtained in usual experiments, this unstable 



< 239 ) 



