14 



This equation is applicable to two-dimensional flows and to axl- 

 symmetric flows (<$ « r ). For purposes of calculation it is seen that rela- 

 tions for the variation of the shearing stresses in a pressure gradient are 

 still required. The main objective of this paper is the conversion of the 

 above equation into a usable auxiliary equation by supplying relations for the 

 variation in a pressure gradient of the local-skin-friction coefficient r A>U 2 

 and the integral of the shearing-stress distribution 1-^dHH. 



For comparison the various auxiliary equations for the variation of 

 the shape parameter in a pressure gradient are listed in Table 1 . It is seen 

 that Buri's method does not provide for an auxiliary equation since no allow- 

 ance is made for the effect of the previous development of the flow. On the 

 other hand, the Gruschwitz method is correct in utilizing the rate of change 

 of a shape parameter in its analytic formulation. It is seen that Kehl re- 

 moved one of the main objections to the Gruschwitz method by incorporating the 



TABLE 1 



Summary of Methods for Calculating the Turbulent Boundary Layer 



in a Pressure Gradient 



Investigator 



Criterion 



for 

 Separation 



Auxiliary Equation 

 (for use with von Karma'n Momentum Equation) 



Remarks 



Burl, 1951 

 Reference 9 



F= -0.06 



None explicitly; von Karman momentum equation modified to 



1 

 ^ (Rj 9 ) = a - \>T [19] 



r Is not proper param- 

 eter for shape of ve- 

 locity profile; no 

 allowance in method 

 for effect of previous 

 development of flow in 

 boundary layer 



Gruschwitz, 

 1931 

 Reference 5 



rj £0.8 



- e A ^ri) - 0.00894rj - 0.00461 [20] 

 u 2 ° x 



No Reynolds-number ef- 

 fect Included; based 

 on test data, 

 1 x 1 3 < R, < 4 x 1 3 



Kehl, 1943 

 Reference 6 





-• a 'f"' = 0.00894„ 0.016U + ^|5 [21] 

 [j2 dx 7 IoSkA (R # -300) l j 



Gruschwitz method ex- 

 tended to include ef- 

 fects of Reynolds num- 

 ber; based on test data 

 1 x 10 3 <R <3.5 x 10 4 



von Doenhoff 

 and Tetervin, 

 19"3, 

 Reference 7 



1 .8<H<2.6 



»§• [-2-°35(H - 1.286) --2- (||)] e «-»^'«» [ 22] 



W 



Empirical; based on 



test data, 



1 X 10 3 <R 8 < 7 x 10* 



^Tji - L 5-890 log 10 (4.075 R„ ) J 



Garner, 1944 

 Reference 12 



H = 2.6 



. dH _ f 0.0135(H-1.4) 9 dU"| M//-1.4) ., , 

 * dx [ R m 'u dxj e KHJ 



Based mostly on same 

 data as von Doenhoff 

 and Tetervin 



Tetervin and 

 Lin, 1950, 

 Reference 15 





„ dH H(H+1)(H 2 -1 ) 6 dU 

 dx " 2 U dx 



[271 

 +H(H 2 -1 ) — -(H+1 )(H 2 -1 )(-I-<ii£\ 

 PU 2 I PU 2 V * ' 



Theoretically 'derived 

 but lacking in shear- 

 ing-stress Information 

 required for numerical 

 solution 



