F,ll • HEAT TRANSFER 



Finally, taking I = Ky and putting Eq. 11-24 into Eq. 11-20, with t = t, 

 yields the following velocity distribution result: 



1 . , We , 1 . , B 



A {B^ + 4:A^)i ' A (B2 + 4^2)^ 



- £ 



F + ^m 





(11-26) 



where F is a constant and I'w, the kinematic viscosity at the wall, is intro- 

 duced because of its influence in the laminar sublayer. 



Eq. 11-24 and 11-26 can now in turn be substituted into the von 

 K^rmdn momentum integral relation for a flat plate with zero pressure 

 gradient, viz. 



- A 



' W 7 



ax 



/ pw(we - u)dy (11-27) 



to yield a complicated integral which, however, can be expanded into a 

 series by means of integration by parts. Upon neglect of terms of higher 

 order, the resulting series can be approximated by a simple expression 

 which leads to an engineering formula for the local skin friction coef- 

 ficient in terms of Reynolds number, Mach number, and wall-to-free 

 stream temperature ratio when the final constant involving F is adjusted 

 to reduce the formula to the von Karman friction law for incompressible 

 flow. In this way van Driest [21] obtained the formula 



0.242 



(sin-i a -t- sin-i /3) = 0.41 + log (Re ■ Cf) 



-g + n)log@ (11-28) 

 where 



2^'-^ and ff ^ 



(52 -t- 4A2)J "'^" ^ (52 4- 4A2)i 



and n is the exponent in the viscosity law ju = const • jT". For air, the 

 exponent n ranges from 0.76 at ordinary room temperature to 0.5 at 

 higher temperatures. 



If, now, one assumes the similarity law for mixing length, viz. 

 I = —K{du/dy)/(d^u/dy^), instead of I = Ky, then, following the same 

 procedure given above, one obtains 



242 T 



AcHT /T)^ ^^^^~' " "^ ^^^~' ^^ ^ ^'^^ "^ ^^^ ^^^ -Cf) - n log Y 



(11-29) 



< 383 ) 



