25 



INTEGRAL OF THE SHEARING -STRESS DISTRIBUTION 

 ACROSS THE BOUNDARY LAYER 



In order for the moment -of -momentum equation [27] to be used numer- 

 ically a relationship is required which describes the effect of pressure gradi- 

 ents on the integral of the shearing-stress distribution across the boundary 

 layer f -—gdf-j-). Although almost no study, either theoretical or empirical, 

 has been made of the integral of the shearing-stress distribution, some stud- 

 ies have been made of the shearing-stress distribution itself. 



The hot-wire measurements of Schubauer and Klebanoff , 2 shown in 

 Figure 12, demonstrate the large changes produced in the shape of the shearing - 

 stress profile by an adverse pressure gradient. The shearing-stress curves 

 have a positive slope at the wall determined by the positive pressure gradient; 

 they then come to a peak, and finally drop to zero at the outer edge of the 

 boundary layer. Several attempts have been made to derive analytical expres- 

 sions for the general shearing-stress distribution; these are described below. 



Fediaevsky, 26 following the Pohlhausen method for laminar flow, 

 tried to fit a polynomial expression for the shearing-stress distribution that 

 would satisfy the boundary conditions at the inner and outer edges of the 

 boundary layer. The coefficients A. of the following polynomial are to be 

 evaluated: 



T w j=0 2Kd ' 



The boundary conditions imposed by Fediaevsky are: 



a. At y = 0, -^-= 1 by definition, 



T w 



b. At y = 0, —^ = -J- from the boundary-layer 



°y ax equation, Equation [1], 



c. At y = 5, t=o from the definition of 



boundary-layer thickness, 



d. At y = 6 , -2^1= from the assumption that 



dy the derivative of the total 

 head of the outer streamline 

 is continuous, 



~2 



e. At y = 0, ^~ = from the derivative of the 



dy boundary-layer equation, 



Equation [1 ] and dp/dy = 0. 



