B,18 ■ INCOMPRESSIBLE SKIN FRICTION LAWS 



pressure in the case of a channel, and a constant pressure in the case of a 

 boundary layer. 



The boundary layer bounded by a free stream of negligible turbulence 

 is known to have a sharp but very irregular outer limit. This is illustrated 

 schematically in Fig. B,17b. The phenomenon is common to all turbu- 

 lent shear flows which are limited only by the extent to which they have 

 diffused into nonturbulent fluid. There is no such limit for fully developed 

 turbulent flow in pipes and channels where turbulent motions may freely 

 cross the center. To a limited extent a similar condition can be produced 

 in boundary layer flows by introducing turbulence into the free stream 

 by means of a grid. It has been noted that the profiles then deviate toward 

 those for the pipe and channel. 



B,18. Smooth Wall Incompressible Skin Friction Laws. So far 



our laws have been so general that pipes and channels on the one hand 

 and boundary layers on the other could be treated as one subject. We 

 may continue in this vein in expressing the general form of the skin fric- 

 tion law, but shortly it will be necessary to make a distinction. 



Since skin friction depends on conditions near the wall, Eq. 17-3 and 

 17-4 are used to derive a formula for skin friction, as was first done by 

 von Kdrmdn [71]. If these equations are added, the result is 



Ur K 



By using Eq. 15-1 and 15-2 and introducing Res = U^b/v, Eq. 18-1 

 becomes 



J^ = ^ log {Rei VFf) + const (18-2) 



Eq. 18-2 has been verified by a number of rehable measurements in 

 pipes. With the constants for pipe flow as given by von Karmd,n [78], 

 Eq. 18-2 becomes 



\= = 4.15 log {ReB Vcf) + 3.60 (18-3) 



where Re^ is based on the velocity at the center f7e and the radius of the 

 pipe. The constant 4.15 corresponds io K = 0.39, this value having been 

 chosen to give the best all-around agreement. 



The Karman skin friction formula for flat plates [78,79] results from 

 conversion of Eq. 18-3 into terms involving x, where x is the distance 

 from the leading edge and the assumed beginning of the turbulent bound- 

 ary layer. It is expressed as 



2^ 2 3 



— = -^ log {Recf) + const (18-4) 



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