B,12 • BASIS OF SKIN FRICTION THEORIES 



the elementary and fundamental transport phenomena by turbulence, as 

 was done in Eq. 10-12, one could consider a more general shear stress and 

 heat transfer, by defining them by the right-hand side of the boundary 

 layer equations, Eq. 8-1 and 8-4 respectively. Then a new coefficient of 

 skin friction and a new Stanton number are obtained with a rather com- 

 phcated ratio between them. Such an analogy between shear stress and 

 heat transfer is called in the Uterature "Modified Reynolds Analogy" [8]. 

 While it is already hard to express a satisfactory analogy between two 

 elementary transfers, one wonders sometimes whether an analogy between 

 a complex transfer of different nature (turbulent and laminar combined) 

 could be expected in a reasonably simple form. The results of some of 

 those analogy theories and the effects of the Prandtl number are shown in 

 Fig. B,lla, after Chapman and Kester [29]. The data mentioned in Fig. 

 B,lla are found in [16,30,31,32]. Fig. B,llb, after Pappas [33] shows how 

 the skin friction coefficient and the heat transfer coefficient vary with the 

 Mach number. The data of Fig. B,llb refer to [29,32,33,34,35,36,37,38, 

 39,40,41,42,43]. 



B,12. Basis of Skin Friction Theories. For the analysis of the 

 stresses acting on a body moving at high speeds, a study of the skin 

 friction in a compressible fluid becomes important. Not only is it neces- 

 sary for drag calculations, but it is useful for estimating heat transfer by 

 means of the Reynolds analogy discussed in Art. 11. 



Before discussing experimental results and their comparison with 

 theories, it appears desirable to review in a simple and general way the 

 main steps, concepts, and approximations underlying the theories which 

 have been proposed. 



According to Eq. 8-1, the turbulent shear stress for a compressible 

 flow is 



T = —puv — Up'v (12-1 a) 



The first term of the right-hand side of Eq. 12-1 a represents a momentum 

 transfer, and the second a mass transfer. The ratio of the second term to 

 the first is estimated to be proportional to the square of the local Mach 

 number. Now the theories of skin friction assume, as a first approxi- 

 mation, a value of r = Tw for Eq. 12-la, where Tw is the total shear stress 

 at the wall. Any variation of t through the boundary layer is taken into 

 account only in the higher order of approximations. Since the local Mach 

 number is small near the wall, the second term of the right-hand side of 

 Eq. 12-la can be neglected, and we obtain 



T = -puv (12-lb) 



The next step is to express the fluctuating quantities in terms of the 



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