B,24 • INTEGRAL METHODS 



not affected, the variation of H may be expressed as a function of C/ by 

 Eq. 20-5 for both smooth and rough walls. 



It is worth noting before we leave the subject that experimental de- 

 terminations of roughness effect in terms of tJJ [Vr vs. V^klv may be 

 made optionally in boundary layers, pipes, or channels. Application of 



0.012 



0.008 



Cf 



0.004 



Fig. B,23d. Local skin friction coefficient for smooth 

 plates with constant pressure, after Clauser \8S\. 



the results then merely requires the introduction of ^JJ/Uj into the 

 appropriate smooth wall formula. 



B,24. Integral Methods for Calculating Boundary Layer Develop- 

 ment. A number of methods have been proposed for calculating 

 boundary layer parameters and separation as functions of x for boundary 

 layers developed on a smooth wall in the presence of pressure gradients. 

 Most of the attention has been given to cases involving adverse pressure 

 gradients, and the methods are mostly restricted to two-dimensional flow, 

 although sometimes the problem is set up so as to include axially sym- 

 metric flow for the conditions where the boundary layer is thin compared 

 to the radius of the body about its axis. 



It is generally assumed that the boundary layer is so thin that pres- 

 sure changes across it may be neglected. Then the equations of motion 

 and continuity for two-dimensional flow reduce to Eq. 8-1, 8-2, and 8-3. 

 For incompressible flow, and by neglecting viscous stress and turbulent 

 normal stresses, these become 



1 dp 



dx dy^ 



dx dy 



1 ar 

 p dx p dy 



(24-1) 



(24-2) 



< 153 ) 



