B,15 • POWER LAWS 



the constants also determined from pipe tests, Eq. 15-3, 15-4, 15-5, and 

 15-6 become 



m' 



Cf = 0.0466 ( ^^ ) (15-3a) 



-^ = 8.74 (^Y (15-4a) 



^^ = S.7,{^J (15-5a) 



The foregoing formulas for pipes would not be expected to apply to 

 other cases. However, they do apply to two-dimensional channels and 

 flat plates over a limited range of Reynolds number for particular coef- 

 ficients and exponents. Power-law velocity distributions fit the observed 

 distributions in an over-all way but not in all detail. 



When applied to the flat plate, Eq. 15-3a and 15-6a may be used to 

 calculate 5 and C/ as functions of x and of a Reynolds number based on x, 

 provided the boundary layer begins as a turbulent layer at the leading 

 edge. The loss of momentum flux through any section of the boundary 

 layer is given by 



f^%U{U.- U)dy 



The momentum thickness d, which when multiplied by pUl gives this 

 quantity, is accordingly defined by 



' = ;^ r ^^(^- - ^^"^ ^ r w. (i - w) "' '■''-'^ 



Since r^ alone accounts for the loss of momentum, it follows that 



dd T„ I , . 



With the velocity distribution given by Eq. 15-6a, Q = 75/72. By substi- 

 tuting this and Eq. 15-3a into Eq. 15-8, and integrating with the bound- 

 ary condition 5 = when a; = 0, the result is 



(^T 



= 0.381a; 1^1 (15-9) 



where x is the distance from the leading edge and U^x/v is a Reynolds 

 number based on x and the velocity of the free stream. From Eq. 15-3a 

 and 15-9 it follows that 



Cf = 0.0592 (^ j (15-10) 



( 121 > 



