B,12 • BASIS OF SKIN FRICTION THEORIES 



Again by introducing the dimensionless quantities (Eq. 12-7), we can re- 

 write Eq. 12-16a in tiie following dimensionless form: 



r 



dy* 



= — p 



(12-16b) 



In Eq. 12-16 the assumption is again made that t is constant across the 

 boundary layer with the value Tw. For p*, the expression given by Eq. 

 12-10 is again used, and all further steps to compute U* and the skin 

 friction coefficient are similar to the treatment given above for the 

 Kdrman hypothesis. It is remarked that the differential equation (Eq. 



Fig. B,12a. Comparison of the mixing length and similarity hypotheses of skin 

 friction. Curve 1 illustrates the Falkner law according to Eq. 13-15, curve 2 illus- 

 trates the K^rman law (Eq. 12-17), based on the similarity hypothesis, and curve 3 

 is the Prandtl law (Eq. 12-18) drawn with a coefl&cient 0.472, based on the mixing 

 length hypothesis. 



12-16b) from the Prandtl hypothesis corresponds to the differential equa- 

 tion (Eq. 12-8) from the Karman hypothesis. However, Eq. 12-16b is of 

 the first order and needs only one boundary condition, namely the inter- 

 face condition 



U* = 11.5 at ?/* = 11.5 



It is interesting to compare the effect of the two hypotheses (Eq. 12-3 and 

 12-4) on the skin friction. For the sake of simplicity and in order to avoid 

 as much as possible other assumptions which may obscure the issue, the 

 comparison is made for skin friction coefficients of incompressible flow. 

 Fig. B,12a shows that the Kdrmd,n and Prandtl hypotheses do not lead 

 to an appreciable difference in skin friction coefficients. The curves are 

 drawn according to the following formulas: 



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