B,13 • EMPIRICAL LAWS OF SKIN FRICTION 



where 



re is the recovery factor defined by Eq. 9-9b, Me is the free stream Mach 

 number, and 7 is the ratio of specific heats. 



Cf 



' 0.4 



Fig. B,13. Compressibility effect on skin friction (empirical laws), c/e/c/i is the ratio 

 of average skin friction coefficients respectively at free stream Mach numbers M^, 9^ 

 and Mf. = 0. Curve 1 represents the theory of Frankl-Voishel [44,45]. Curves 2 and 3 

 represent Eq. 13-7b and 13-11, based respectively on the power law and the loga- 

 rithmic law of incompressible skin friction coefficient. The experimental results of 

 Coles [S4] are shown in circles for comparison. A viscosity-temperature exponent 

 a = 1 is used in plotting curves 1, 2, and 3. Curve 4 is plotted with a = 0.75, according 

 to Eq. 13-11. 



Instead of selecting the power law (Eq. 13-5) on which to base com- 

 pressibility effect, we may take as an alternative example a logarithmic 

 law of the form (decimal basis) : 



Cf^ir) = A{\ogRe)-- (13-10) 



Then by the procedures of Eq. 13-6, 13-7, and 13-8 we find the following 

 compressibihty effect: 



Cf,{Ree) 



1 + 



(1 +c.)log/3 ' 

 log Ree 



^ 



(13-11) 



where jS = Te/Tt is given by Eq. 13-9, when Tr assumes the value given 

 by Eq. 13-8. 



It is interesting to note that the compressibihty effect as given by 

 Eq. 13-7b, on the basis of the power law (Eq. 13-5), is separated from the 



< 115) 



