B,19 • EFFECT OF PRESSURE GRADIENT 



the adverse pressure gradient on the profile becomes more marked, i.e. as 

 H is greater. They also occur at lower values of UrV/v as the Reynolds 

 number decreases. Landweber [86] has shown that the logarithmic part 

 no longer exists if U^b*/v is less than 725. At the wall side the law begins 

 to merge into Eq. 16-2 somewhere around UrV/v = 50. The outer limit of 

 the laminar sublayer is usually taken as 11.5, representing the point where 

 the curve of Eq. 16-2 and the logarithmic law intersect. 



For a number of years there was considerable uncertainty about the 

 effect of the adverse pressure gradient on the skin friction, and most 

 methods of treating turbulent boundary layers assumed that the gradient 

 had no effect. Estimates by means of the momentum equation were un- 

 reliable and in some cases showed an apparent increase in the skin friction 



TOO 



U.y/v 



Fig. B,19c. Test of the law of the wall, after Clauser [83]. 



10,000 



coefficient in regions of strongly rising pressure. When data based on 

 more direct methods became available, such as those of Schubauer and 

 Klebanoff [87] and Newman [88], based on hot wire measurements of 

 shear stress, and those of Ludwieg and Tillmann [89], based on the heated- 

 element method, it became clearly evident that C/ was decreased by an 

 adverse pressure gradient, and was steadily reduced toward zero as sepa- 

 ration was approached, as logic dictates that it should be. The whole 

 question has been considerably clarified by the universal character of the 

 law of the wall which establishes a unique relation between velocity near 

 the wall and skin friction without explicitly involving the pressure gradi- 

 ent. The effect of the pressure gradient on the skin friction is thereby 

 seen to result from its reduction of velocity near the wall. 



The relation between the integral characteristics of a two-dimensional 

 boundary layer and the pressure gradient is obtained by integrating the 

 equation of motion from y = Q to y = b. The commonly used form, 



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