Theory of Turbulent Flow Between Parallel Plates 

 but 



thus 



or 



Q OO 



10-^ R^o 



This is not g = 0.02 ^l^"^ , but it shows that the deviation from g proportional 

 to Re, the laminar flow branch (1) can come from a limiting frequency, and 

 (2) can come from an elastic parameter /3 ^ 0. 



One may note that the estimate of critical Reynolds number 



did not explicitly depend on compressibility, whereas now the turbulent branch 

 does. This is consistent with the concept that it is not impossible for a critical 

 transition to take place over a range rather than at one critical Reynolds number. 



Finally, with regard to inter mittency: The essential matter is the fourth 

 mode. It has two independent forms — sinh (c^x) or cosh (c^x). The two inde- 

 pendent secular determinants of differing symmetries must be made to vanish for 

 a continuous range of uj. This was accomplished by letting C4 approach zero, 

 leading to the specific secular result used. Another possibility has suggested 

 itself. 



The inquiry in this paper dealt with the question: Does there exist a set 

 for the fluctuating components of the field that represents stationary limit cy- 

 cles? These were probed at by a describing function technique in the form 



gj(ay+sz + a.T)^ a, §, o) real. An affirmative answer was supplied, and a set 

 was demonstrated. The equation set cannot support any other stationary fluc- 

 tuating systems. 



But turbulence is not made up of stationary fluctuations — it is locally epi- 

 sodic. It is only in the mean that the field is a stationary stochastic one. How 

 are the episodic fluctuations to be found? (So far we have a family whose phase 

 are stochastic, not the frequencies or amplitudes.) We propose a new answer. 

 It is related to the discarded solution set of opposite symmetry. 



It is true that the second set cannot exist as a stationary set. It cannot 

 satisfy the boundary conditions except by null amplitudes, basically because 

 cosh c^x ^ 0. 



c 2 zz K - /32cj2 ^ j-gn imaginary part] . 



745 



