B • TURBULENT FLOW 



where Re = U^x/v and C/ is again the local friction coefficient defined by 

 Eq. 15-1. With the constants evaluated from Kempf's measurements on a 

 flat plate [80], Eq. 18-4 becomes 



-y= = 4.15 log (ReCf) + 1.7 (i8-5) 



Schoenherr [47] found the coefficient of mean friction over the dis- 

 tance X to be given by 



949 



^ = log {ReCf) (12-17) 



and the relation between the local and the mean friction coefficients to be 



0.558 + 2 VCf 



Eq. 12-17 is one of the most widely used formulas for incompressible flow 

 and, as previously mentioned in Art. 12 and 14, is sometimes called the 

 Kdrman-Schoenherr formula. 



As reported by Prandtl [48], SchHchting proposed an interpolation 

 formula of the form 



Cf = 0.455 (log i^e)-2-58 (12-18) 



The comparison between Eq. 12-17 and 12-18 is shown in Fig. B,12a. The 

 corresponding interpolation formula for c/, also given by Schlichting [80], 

 is 



Cf = (2 log Re - 0.65)-2-3 (18-7) 



Schultz-Grunow [74] adopted the Prandtl law with constants as follows: 



Cf = 0.370 (log i?e)-2-584 (18-8) 



While the foregoing formulas are expressed in the form usually de- 

 sired for engineering purposes, they suffer from the drawback that the 

 boundary layer is often laminar for a significant distance before transition 

 occurs. In such cases formulas based on Re cannot be applied without 

 assuming some fictitious origin for x. A formula like Eq. 18-3, based on 

 the local parameter Res, does not involve this difficulty. Because of the 

 indefiniteness of the outer limit of the boundary layer, the momentum 

 thickness 6 is commonly used in place of 8, and Ree = Ued/u takes the 

 place of Res. Squire and Young [81] obtained from Eq. 18-4 the approxi- 

 mate relation 



-^ = A log Ree + B (18-9) 



VCf 



with the constants A and B chosen to give the best agreement with Eq. 



< 128 > 



