B • TURBULENT FLOW 



Reynolds number effect, while the compressibility effect (Eq. 13-11), on 

 the basis of logarithmic law (Eq. 13-10), includes a Reynolds number 

 effect. 



The two formulas (Eq. 13-7b and 13-11) are illustrated in Fig. B,13, 

 by taking a = 1, re = 1. The Falkner constants [61] 



A = 0.0262, n = i 



have been used in Eq. 13-5 and 13-7b, and the Prandtl constants [48] 



A = 0.455, n = 2.58 



in Eq. 13-10 and 13-11. It is seen that they are in quite good agreement. 

 Also plotted are the theoretical results of Frankl and Voishel [44,4^], 

 originally in tabulated form, and the experimental results of Coles and 

 Goddard [35]. It seems that the empitical formulas (Eq. 13-76 and 13-11) 

 agree rather well with the theory of Frankl- Voishel and with the experi- 

 ments of Coles. Although the experiments of Coles are run with a slightly 

 different Reynolds number {Re^ = 8 X 10®) than the Reynolds number 

 of the theoretical curves (Ree = 7 X 10®), the correction for such a dis- 

 crepancy is not significant. The viscosity-temperature coefficient a has 

 the value between 0.75 and 1. 



Eq. 13-11 is also plotted in Fig. B,14b with C/^ vs. Ree, to be com- 

 pared with experiments, by taking a = 0.75, n = 2.58. C/j is based on 

 Eq. 13-10 with A = 0.455. It is seen that the theoretical formula (Eq. 

 13-11) is in good agreement with experiments. 



B,14. Comparison between Experiments and Theories. There 

 exists an extensive history of experiments on skin friction. Because of the 

 importance of skin friction to naval architecture, experiments on skin 

 friction were started as early as 1793 by Beaufoy. Schoenherr [4-7] gives a 

 good review of experiments prior to 1932. 



In Fig. B,14a are plotted the experimental values of skin friction in 

 compressible flow. The ratio Cf^/Cf. or Cf^/c/^ is illustrated. Except for the 

 measurements of Chapman- Kester and Liepmann-Dhawan, wherein the 

 incompressible skin friction values are deduced experimentally, all data 

 points shown are based on the incompressible skin friction formula (Eq. 

 12-17) of Kdrmdn-Schoenherr. There are two methods of determining the 

 skin friction coefficient. Liepmann-Dhawan, Coles, and Chapman-Kester 

 determine the skin friction coefficient by direct force measurements. 

 Others determine it by surveying the boundary layer and then calcu- 

 lating the friction coefficient by the usual momentum method. At the 

 Mach number of 5, the two methods yield a discrepancy of about 5 per 

 cent. 



Since the empirical theories, as given in Art. 13, do not differ very 

 much according to Fig. B,13, we have plotted the theoretical formula 



< 116 > 



