B • TURBULENT FLOW 

 Eq, 12-12, according to the formula of momentum thickness, 



the momentum thickness 0(c/J must be a function of x. Since the solution 

 (Eq. 12-12) is vaHd only in the turbulent boundary layer and not in the 

 laminar sublayer, some error will be introduced in the integration of Eq. 

 12-13 by using Eq. 12-12. However, since the laminar sublayer is thin, 

 the error must be very small. 



We recall that the coefficient of skin friction C/^ is defined by Eq. 

 12-9a in terms of density pw, and that the momentum equation in the 

 integral form, for a fiat plate with zero pressure gradient, is 



'^'--j^r^t <i2-9b) 



where C/^ is the skin friction coefficient referred to pe, which will be fre- 

 quently used later on. It is to be noted that Cfjcf^ = T^/T^. After inte- 

 gration with respect to x, Eq. 12-9b can be rewritten as follows: 



T^ h^dd{cfj 1 

 X = 2-pfr \ — i dCf^ 



= 2(l+^M^)-| ^^/^.. 02-14) 



The value of the integrand of Eq. 12-14 is given by the differentiation of 

 Eq. 12-13. In Eq. 12-14, the limits of integration are (co, c/^) for C/^, 

 corresponding to (0, x) for x^ because at a; = the boundary is so thin 

 that the velocity gradient and the skin friction become infinite. If we 

 write X in terms of the Reynolds number py,JJ^xl\x^^ the integration of 

 Eq. 12-14 gives a relation between the skin friction coefficient and the 

 Reynolds number of the following form : ^ 



C/w = C/w (^, ^^ 1^, ^e) (12-15) 



Further the heat transfer coefficient may be found on the basis of the 

 skin friction coefficient by means of the Reynolds analogy as examined in 

 Art. 11. 



Instead of using the Kd,rmdn similarity hypothesis (Eq. 12-3), which 

 serves as the foundation of the differential equation (Eq. 12-5), we can 

 use the Prandtl hypothesis (Eq. 12-4) so that Eq. 12-2a now becomes 



at/^2 E (12.16a) 



( 110) 



