B,13 • EMPIRICAL LAWS OF SKIN FRICTION 



trary. Therefore it is not surprising that they yield a great number of 

 different predictions of skin friction at high speeds. Fig. B,12b shows that 

 the various theories differ appreciably at large Me. They are based on the 

 recovery factor ^e = 1, with the exception of the theory of Wilson which 

 is based on re = 0.89. The viscosity-temperature exponent a covers the 

 range of a = 0.75 to 1, and the Reynolds number covers the range of 

 Re = 7 to 10 X 10^ The discrepancies between the values of skin friction 

 predicted by the various theories increase as the Mach number increases. 

 At a Mach number of 5, the theoretical values of the skin friction differ 

 by a factor greater than 3. Only a small portion of the discrepancies can 

 be attributed to the different values of Re, r^, and a used in the various 

 theories. Because of their uncertainties, we shall not enter into the detail 

 of the theories, and the readers who are interested in such details are 

 referred to [19,38,49,50,51,52,53,54,55,66,57,58,59,60]. In view of the diffi- 

 culties of such theories of the skin friction coefficient, some empirical 

 formula of skin friction coefficient may often be more useful in practice. 

 These will be treated in Art. 13. 



B,13. Empirical Laws of Skin Friction. Let us define again the 

 various skin friction coefficients used in the theories and experiments. 

 The local skin friction coefficient is defined in its general form by 



c/ = ^ (13-la) 



The wall and free stream values are obtained by writing in Eq.'^13-la, 

 respectively, p = Pw and p = p^: 



c/w = ;^ (13-lb) 



which have already been introduced in Eq. 12-13. The local skin friction 

 coefficient C/ is a function of x. Its average value is called the ''average 

 skin friction coefficient" Cj 



1 ['' 

 Cf = ~ Cfix)dx (13-2) 



^ Jo 

 or inversely 



Cf{x) = Cf{x) +x^ (13-3) 



Corresponding to C/^ and C/^, we can write their average values C/^ and 

 Cf^. In the dimensionless form x can be replaced by the Reynolds numbers 



Re = J /tCw = } or Ree = (13-4) 



M Mw Me 



in respective cases. 



< 113 ) 



