32 



The insertion of the relations for local skin friction and for the integral 

 of the shearing-stress distribution, [40] and [8l], into the moment-of- 

 momentum equation [27], converts it into the following equation, which is ap- 

 plicable to the calculation of both two-dimensional and axisymmetric flows 

 where <5«r ) 



- dH H(H+1)(H 2 -1) 9 dU , ,„ a , 

 6 dx" = 2 U dx" + (H _1 



"^r-'"- 1 '^ 1 .]^ «» 



where y, y and I are defined in Equations [42], [43], and [78] respectively, 

 and the subscript refers to flat-plate values at the same Reynolds number R . 

 This equation is the desired auxiliary equation to be solved simultaneously 

 with one of the modified von Ka'rman momentum equations, [82a] or [82b], for 

 calculating the growth of boundary layers in pressure gradients. 



The modified moment -of -momentum equation of this report represents 

 an auxiliary equation whose form has a theoretical basis and whose coeffi- 

 cients have been evaluated indirectly from flat-plate data. On the other hand 

 the various auxiliary equations described in this report are almost wholly 

 empirical in origin. A comparison of the auxiliary equations may be made by 

 writing them in the following generalized form 



•S--*(tS)-» [84] 



The expression for the variation of 77 in the Gruschwitz or Kehl method is con- 

 verted into the above form by using the power-law formula, [15] » f° r the rela- 

 tion between 77 and H. The expressions for the coefficients, A and B, are 

 listed in Table 3 for the various auxiliary equations. It is seen that the 

 coefficients for the Kehl, Garner and the moment -of -momentum equations are 

 functionally similar with A = f(H) and B = f(H, R„ ) . The von Doenhoff and 

 Tetervin equation, nowever, has the Reynolds -number effect in coefficient A 

 instead of coefficient B. As expected, none of the coefficients for the 

 Gruschwitz method exhibits a Reynolds -number effect. 



The coefficients A and B are plotted against H in Figures 15, 16, 

 and 17 for two different values of R ff . It is observed that although all the 

 curves have the same general shape and trend, there are marked differences in 

 the magnitude of the ordinates. The exponential form of the coefficients of 

 the Garner and of the von Doenhoff and Tetervin equations is responsible for 

 the very large increase in value of the coefficients at the higher values of 

 H. The curves for the coefficients of the modified moment -of -momentum equa- 

 tion of this report maintain average positions among the other curves. 



