C • STATISTICAL THEORIES OF TURBULENCE 



sider the theoretical aspects. Comparison with experiments will be made 

 in the next article. 



Let us consider the equation (Eq. 8-7) for the change of spectrum 



dF 



^ + T7 = -2vk''F 

 at 



and try to find a similarity solution. If F is a characteristic velocity, 

 and I is a characteristic length, then, from dimensional arguments, 



F = VHrPik), W = VM^), ^ = d (14-1) 



Thus, the above equation becomes 



y I [^^'(^) + ^m ^Y^~ H^) + w{^) = -% e^k) (14-2) 



If the similarity solution is to be valid, one must have 



j| = a. (U-3) 



Y^^ = a. (14-4) 



Y = a, (14-5) 



where ai, a^, and as are all constants. Eq. 14-2 becomes 



ai?^'(^) + (ai + a^)rP{k) + 2a3eV(^) + w{^) = (14-6) 



Besides Eq. 14-3, 14-4, and 14-5, it is evident that the mean square 

 value w^ and the rate of energy dissipation have to satisfy the relations 

 (cf. Eq. 8-6 and 8-11) 



^2 _ 72 fj ^(^^)d^ (14-7) 



-f = 2^]^ mm (14-8) 



Finally, if the convergence criteria for Loitsiansky's relation (Eq. 12-6) 

 are assumed to be valid, we have 



iKj) 



This system of equations presumes that the transfer term in Eq. 14-2 

 is considered generally of equal importance with the term expressing the 

 viscous dissipation. It has been shown by Dryden [40] in the equivalent 

 problem of self-preserving correlation functions that such a solution is 

 connected with the statement that the square of the characteristic length 

 is proportional to the time t and the law of decay is expressed by u^ -^ t~'^. 



( 226 > 



y2^Mim!^=J (14-9) 



