C • STATISTICAL THEORIES OF TURBULENCE 



3. On the other hand, the theory gives the definite prediction that there 

 is no permanence of the largest eddies. Thus, there is a definite contra- 

 diction between Eq. 16-1 and the hypothesis leading to Loitsiansky's 

 invariant. This led Batchelor and Proudman [31] to the extensive 

 investigation mentioned in Art. 12. 



Chandrasekhar [56] used a different approach. He limited his study 

 to two-point correlations, but considered a difference in time. Thus Eq. 

 16-1 is replaced by relations of the kind 



Ui{x;, t')uj{x', t')Uk{x", t")ui{x", t") 



= QikQji + QiiQjk + QiAO, 0)Qm(0, 0) (16-5) 



where 



Qij = Ui{x', t')uj{x", t") (1.6-6) 



In this approach, the number of space variables is reduced, but another 

 time variable is introduced. Chandrasekhar then introduced the concept 

 of "stationary homogeneous and isotropic turbulence," and assumed that 

 the double correlation (Eq. 16-6) (and similar third order correlations) 

 depend only on the space vector x" — x' and the time interval \t" — t'\. 

 In this way, the theory leads to a single partial differential equation in two 

 independent variables, and a number of deductions were made. In particu- 

 lar, a discussion is given to show its compatibility with Kolmogoroff's 

 theory. 



C,17. Hypotheses on Energy Transfer. Another method for arriving 

 at a theory capable of yielding definite deductions is to assume, on the 

 basis of physical arguments, a relation between the spectrum function 

 F{k, t) and the transfer function W{k, t). Various hypotheses of this type 

 were proposed by Obukhoff [37], Heisenberg [S3], von Karmdn [46], and 

 Kovdsznay [57]. It should be recognized that there is no a priori reason 

 that such relations should exist. However, the success of these hypotheses 

 seems to indicate that this type of theory does give a reasonably accurate 

 description of the physical process. 



Heisenberg argued that the transfer mechanism is essentially similar 

 to viscous dissipation with the smaller eddies corresponding to molecular 

 motions. This is reasonable provided the smaller eddies are very much 

 smaller than the eddies from which they take energy. If this point of 

 view is accepted, the rate at which energy is lost to the smaller eddies is 

 proportional to 



- 1^ r FiKJdK = v + C j^ .y^^ dK" 1^ 2FWyHK' (17-1) 



{ 238 ) 



