B,12 • BASIS OF SKIN FRICTION THEORIES 

 where C/^ is the skin friction coefficient at the wall, defined by 



The expression (Eq. 12-7c) is based on the equation of state of a perfect 

 gas and the constancy of pressure across a boundary layer (compare the 

 assumptions underlying Eq. 9-1). 



In order to formulate the differential equation for U*, it is necessary 

 to express p* in terms of U*. This is possible by using relations between 

 the temperature and velocity, such as those discussed in Art. 9. However, 

 it is more proper to regard the boundary layer as a composite layer, con- 

 sisting of a laminar sublayer very close to the wall with a superposed 

 fully developed turbulent layer. Obviously the computation for such a 

 condition becomes more elaborate, requiring matching of flow conditions 

 at the interface and consideration of the heat transfer through it. But 

 the final result of the temperature-velocity relation turns out to be rather 

 simple and is of the form 



P*-' ^-^ = Ao + AiU* + A2U*'' (12-10) 



■t w 



as could be expected from the elementary considerations of Art. 9, al- 

 though the coefficients Ao, Ai, and A 2 are more complicated functions 

 of Pr, Cf^, Tl/Ty,, and M^. For the details of the analysis by which these 

 are found the reader is referred to [44 A^]- 



When p* in Eq. 12-8 is replaced by Eq. 12-10 there is obtained an 

 ordinary nonlinear differential equation of second order for U*{y*), with 

 Pr, Cf^, T^/T^, Me as parameters. The integration gives two constants 

 to be determined by two boundary conditions taken at the interface be- 

 tween the laminar sublayer and the turbulent layer. According to experi- 

 mental data for incompressible flow [46], these are 



U* == y* = 11.5 



dU* 



j^ = 0.218 (12-11) 



In principle, Eq. 12-8 and 12-10 with the boundary conditions (Eq. 12-11) 

 can be solved, with the solutions of the following general form: 



U* = U* (y*; Pr, Cf^, ^, M^ (12-12) 



In practice the solution is very elaborate and various numerical and 

 approximate methods must be used. 



Now we assume that all the parameters in Eq. 12-12 are constant, 

 except Cf^ which depends on x. Thus after integration of U*{y*) given by 



< 109 ) 



