31 



When the values of ip Q , based on the same Reynolds number as those 

 appearing in the experiments of Schubauer and Klebanoff , are plotted in Figure 

 13 for comparison, they produce a curve of almost constant value which is in 

 excellent agreement with the experimental points of ip. Based on this evidence 

 it will be reasonable to conclude that at the same Reynolds number ip is a 

 close approximation to ip for other boundary layers in pressure gradients sim- 

 ilar to the boundary layer of Schubauer and Klebanoff. Further substantiation 

 depends on additional experimentation, especially at high Reynolds numbers. 



The integral of the shearing-stress distribution as required by the 

 moment -of -momentum equation, Equation [27], is expressed in terms of \p as 

 follows: 



f^dfy) = $1 [''''jLtin .'£> [8o] 



For power-law velocity profiles 



6l _ H - 1 

 6 ~ H + 1 



12 



and with the assumption rp = ip , where ip is defined in Equation [71]. the 

 integral of the shearing-stress distribution finally becomes 



[Mil- (fH)K)(%) .*. ™ 



pit 



This completes the shearing-stress relations required for transform- 

 ing the moment -of -momentum equation into an auxiliary equation for character- 

 izing the shape of the velocity profiles. 



MODIFIED MOMENTUM AND M0MENT-0F -MOMENTUM EQUATIONS 



The substitution of the relation for local skin friction in a pres- 

 sure gradient, Equation [40], into the von Karman momentum equations, [6a] and 

 [6b], modifies them as follows: For two-dimensional flows, 



I-<H + 2)lI-(if ^ 182.] 



and for axisymmetric flows (where i<<r ), 



d(r w 0) rj> 



<"♦*>-§-£♦(£] ' V-£ to*] 



dx -^ 



