Theory of Turbulent Flow Between Parallel Plates 



However, we also know that the coefficient of the x^ term is small com- 

 pared to unity in turbulent flow. (As may be found in Laufer's data, it is an ap- 

 preciable fraction, such as 0.2, 0.3. Thus for more precise perturbations, it 

 may not be neglected. However, at the present stage of theory, we may disre- 

 gard it.) Thus as an approximation, we may let the coefficient vanish 



N - 2 ^ M - 1 , ,^ -..:,,-'.. 



SO that a simpler two- parameter result may be obtained. " ' 



,2N ■.■.'•=.■ -^ - 



At the present, this is the simplest relation that has been found that satis- 

 fies the boundary conditions and yet is capable of fairly reasonable representa- 

 tion of the mean flow. Thus it is a good starting perturbation for a self- 

 consistent field study. 



Note that this relation contains only two constants, the "given" Reynolds 

 number R^o of the field, and an as-yet-undetermined parameter g, which is to 

 be determined finally as a function of Rq^. As such, this relation has the very 

 minimum number of constants. More complicated self- consistent expressions 

 will have to determine additional constants. What is specifically involved at 

 this point, is how much detail a priori can be involved in describing the bound- 

 ary layer. 



For example, the form is "wrong" when compared with experimental data. 

 It suggests that the "end" of the boundary layer, as marked by q?" ', occurs at 



1 



1 - X = — • 



2N 



Laufer's three points suggest, more nearly, 



1 



1 - X = • 



4N 



Also, it suggests that the maximum value of cp" is about 0,8 n^; Laufer's data 

 suggest 2.5 n2. However, what seems quite satisfactory in this form is the 

 proper dependence on the power of N (= g/R^o) and the emergence of a correct 

 order- or- magnitude estimate of boundary- layer parameters. 



Obtaining results from such self-consistent field theories (i.e., assume 

 an n -parameter open form for the time- independent solutions, then use them in 

 the inhomogeneous time-dependent equations to get the co indexed fluctuating 

 components, and then compute the undetermined parameters of the time- 

 independent solutions) is sensitive to the form of the time- independent solution 

 assumed — here, the mean velocity distribution. Even the slightly extended form 



739 



