Theory of Turbulent Flow Between Parallel Plates 



then 



■t: 



DW,i) %a2 



Dt 



Since the velocity 



and in the core this is given by 



a ^ 250 



W = W + W 



" " ' ( 1 ) 



W = R + W 



oo (1) 



a fluctuating amplitude of 250 compared to the Reynolds number of 20,800 is of the 

 proper order of magnitude for the rms fluctuation found in the core. Illustrated 

 in an average sense, such acoustic fluctuations, thus, can account for the momen- 

 tum discrepancy in turbulent channel flow. 



We can demonstrate further that this is no accident by the following: The 

 "complete" z momentum equation is 



[U^j^d/Bx + V^j^3/9y + W^ ^ ^B/Bz] W^ ^ ^ = g + d2R^/dx2 



= g - Koo<P" ■ 



Drop U(i) and v^^^ as before. Now consider this equation at the "end" of the 

 boundary layer, i.e., where cp" has its peak. In particular, apply this to Laufer's 

 Roo = 30,800 data. Whereas |g/Rool has the value of about 45.5, |(p"| has a 

 value of about 6000, at least a hundred times larger. Thus, 



or 



a = 250 yjR^y/g 

 = 2800 . 



This represents a peak amplitude of about 10% of the mean flow in the center. 

 This is the magnitude of the peak rms fluctuation that Laufer shows at about the 

 same location in his Fig. 11. This stresses the need for considerable attention 

 to a theory for d2Rydx2 (= -\y), or qp". 



733 



