Theory of Turbulent Flow Between Parallel Plates 



"relaxations," equation by equation. (Find a solution that relaxes each equation 

 at least one order in (i^^ , and then continue to find higher- order relaxations. It 

 .was realized that it was essen tial to expand the perturbations in half powers, 

 say of 1, 1/^, lA, ... ; 1, 4W^', ... ; 1, 41^, ... since otherwise compensat- 

 ing zeros might arise.) This mathematically was satisfactory. It avoids the 

 very difficult question of straining one's detailed knowledge of cp" other than its 

 boundary condition. 



It is certain that the rigorous iterative task is best left to mathematicians. 

 Thus, only crude but suggestive ideas have been thrown into our developments. 



The minimal requirement is to satisfy the cp' and qp" boundary conditions on 

 (p. The second is to avoid stirring up too much trouble over the cp" impulse. 

 One perhaps may achieve this by a selected sequence of open forms ordered by 

 a single parameter, i.e., the maximum value of cp". Let this be cpq", and assume 

 this to be large, located at x^, nearly 1. One might regard this as a fifth bound- 

 ary condition on cp (with the requirement that it be estimated self -consistently). 



Proceeding now to the task of building solutions, we may then take for the 

 form of i', the following 



= oj + S Rq •■ ■ " ' ■ 



2( e/R -1 ") 2g/R ■ ^ ■ 



= Mg - 8g/2 X ^^ °° V S(g-2R„„)/2 x 



= Mq - SRoo (P • 



(Note: Since g ^ 0.02 R^^^^ ^g tj^e required result, g/Ro^ = 0,02/R°-75 

 has a value of approximately 2 and 4 for Rq^ = 500 and 1000. Thus, it is not 

 sufficiently larger than 1 or 2 to permit disregarding I'R^^/'g in the last term.) 



The working equation set [from Eqs. (6)] becomes, letting w = x + §/« v, 



[D2 - \- j0]U + qD [DU + j\/a V+ jSX] - DP = , 

 [D2- \-30] V + jaq [DU + j\/a V + j8X] - j aP = , 

 [D2-\-j^] X = -R^^CP'U , ^ 



[DU+ j \/a V+ jbX] - '//32gX - 76/32g/a V + yfi^\i<V - B^ j0T = , 

 l/a[D2-\- ]a4j] T - 2ej&R^^q)'U - [2eR„„cp'D + (7 - 1 ) g] X - S/a [2eR„„(p 'D 

 + (7- l)g] V + (y- 1) j0P . , 



= M - 8R_(p ; M, = o, + 8R ^ ; cp = - x2(N-l) _ ^LlI ^2^ 



00^ 



N = g/R, 

 741 



