G,4 • TRANSPIRATION-COOLED BOUNDARY LAYER 



The bars representing time-average quantities have been dropped since 

 all the terms in Eq. 4-39 are mean values. The two boundary conditions 

 are: (1) the shearing stress t = Tw when ?/ -^ 0, and (2) the velocity dis- 

 tribution from the solution of Eq. 4-39 must reduce to von Karman's 

 logarithmic velocity distribution law for T^/T^ = 1, M^ = 0, and fw = 0. 

 The equation of state for zero pressure gradient in y direction leads to 



T_ 



(4-40) 



Upon the substitution of Eq. 4-40 and Eq. 4-38 into Eq. 4-39, the solu- 

 tion which represents the velocity distribution of a compressible turbu- 

 lent boundary layer with fluid injection at the wall is obtained: 



where 



/i = 



■u/U 



1. 



5u 



q(.Ii-DK) 



E = K 



2r, 



.U' 



and 



A2 = 



7- 1 



Ml 



5 = U^- 1 4- 



T^fy-l 



(4-41) 

 (4-42) 



(4-43) 

 (4-44) 



Mi (4-45) 



The empirical constants K and D can only be determined from the proper 

 experimental velocity distribution. 



The skin friction coefficient can be determined from the momentum 

 integral given in Eq. 4-1 as follows: 



= 9 — 

 ^ " dx 



8u 



Pt 



pu 



^'vyi 



- 2 



p^^v^, 



M 



(4-46) 



Eq. 4-46 can be simplified by substituting the expression for d(y/du) from 

 Eq. 4-41, after its differentiating with respect to u/U, along with the 

 relation between p, T, and u given in Eq. 4-38 and 4-40. The final ex- 

 pression for local and average skin friction coefficient is 



In 



Cf + 



Pw^^v 

 X Jo p^U 



dx 1 Rx 



+ ln 



K 



— CO In 



+ DK = \n h 



(4-47) 



<457 ) 



