Theory of Turbulent Flow Bet'>"i'een Parallel Plates 



Roo = 12,300 , . . _ _ ; 



u* = (y^) - 0.99 X 10"3(y+)2 - 0.91 X 10"3(y+)3 , 



< y^ < 4.8 ; 



R„„ = 30,800 , 

 u^ = (y*) - 4.2 X 10"" (y^) 2 - 7.7 x 10"''(y^)3 , ; : .-^ . 



< y* < 7.7 ; 



R„„ = 61,600 , 

 u+ = (y^) - 2.2 X 10""(y^)2 - 2.4 x 10"4(y+)3 ^ 



< y^ < 7.9 . 



It is impossible to detect the graphic difference from u* = y^ for this 

 "boundary- layer" region. Since experimental data (e.g., [8] or [9]) show no de- 

 viation from u^ = y^ over a range y* up to 7 or 8, Fig. 1 derived from Laufer 

 [8] is a more sensitive presentation of the boundary layer. However, what Fig. 

 1 succeeds in doing is to show quite explicitly the existence of a boundary- 

 layer region (namely, the region in which qp' is nearly constant). 



Intercomparison with [9], the logarithmic law, and the form of qp' and qp" 

 provides some measure of the so-called laminar sublayer. In agreement with 

 [9], a sublayer may be identified below y^ = 6, more probably below y^ = 4. 

 It is not surprising that a simple linear gradient is found only within the range 

 up to y^ = 1.6. 



It thus appears safe to infer that there is a region — typically x = 0.998 - 1 

 (or y* = 0- 1.5) — in which the variation in cp' is essentially small; and a 

 region — typically x = 0-0.8 (or y^ above 500) — in which the variation in qp' 

 is again small. In this report this transition zone will be considered very cur- 

 sorily. The complexity arises from the rapidly changing magnitude of cp". 



In a preliminary report of this work [lO], the form 



V^oo = 1 - ^O'^' - (1-%) ^'"^ ,.. 



had been used, a two-parameter form, consistent with the same proposal by Pai 

 [llj. The form is not satisfactory at the wall. It cannot satisfy all the cp bound- 

 ary conditions. A basic conclusion drawn from [10] was that trapping limit 

 cycle in parallel-plate flow was sensitive both to the mean flow in the core and 

 its form in the boundary layer. Thus, a more suitable form for the mean flow 

 must be selected. 



In order to avoid the mass of algebraic detail that arises when the problem 

 is not treated as an eigenvalue problem, the suggestion of a mathematical col- 

 league was accepted of breaking the field into two parts, a core and a layer near 

 the wall. The simplest form to take for <v is then 



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