B,24 • INTEGRAL METHODS 



effect of pressure gradient on skin friction, von Doenhoff and Tetervin 

 arrived at the following expression for QdH/dx: 



JTT 



zfL = g4.680(H-2.975) 



dx 



qdx T^ 



(24-4) 



Given dq/dx, the two equations (Eq. 24-4 and 24-3) were solved by 

 a step-by-step procedure for 6 and i? as a function of x. Starting with 

 some initial value, Qq and H^, dQ/dx and dH/dx were found. Each when 

 multiplied by an increment of x and added to the initial values gave the 

 next value of d and H to repeat the process. 



Garner [100] undertook to improve on the method of von Doenhoff 

 and Tetervin by using different auxiliary expressions for skin friction and 

 H, again disregarding the effect of pressure gradient on skin friction. The 

 method, however, remains basically the same. 



Tests of this general method have shown a closeness of agreement 

 with observations sufficient to make it worthy of consideration when con- 

 ditions are not out of the ordinary ; that is, when profiles can be expected 

 to have the form of Fig. B,19a. Since adverse pressure gradient domi- 

 nates the development of the layer, the use of an incorrect expression for 

 the skin friction apparently has minor consequences. 



Tetervin and Lin [101] initiated a fresh attack on the problem, again 

 built around the JY-parameter family. They set up integral expressions 

 for momentum, moment of momentum, and kinetic energy in a form suf- 

 ficiently general to include axially symmetric flow as well as two-dimen- 

 sional flow, subject to the restriction that 5 is small compared to the 

 radius of curvature about the axis of symmetry. Their principal objective 

 was to avoid an empirical expression for H if possible. The moment of 

 momentum equation was found to be best suited for this purpose, but it 

 required auxiliary expressions for velocity and shear stress distributions 

 across the layer. A power-law fitting of the i?-parameter profiles was 

 adopted as an approximate but reasonable procedure. More serious was 

 insufficient information about the value and distribution of shear stress. 

 While the work of Tetervin and Lin fell short of immediate success, it 

 pointed the way to future progress. 



It must be remembered that while r^ may be reduced to small values 

 by an adverse pressure gradient, r may rise to large values away from the 

 wall before falling to zero at the outer edge of the layer. Fediaevsky [102] 

 proposed a method for calculating the distribution of t/t^ with y/b 

 employing a polynomial expression that would satisfy boundary condi- 

 tions at the wall and the outer edge of the layer. Certain large discrepan- 

 cies were observed between shear stress distributions calculated by this 

 method and those directly measured by the hot wire method by Schu- 

 bauer and Klebanoff [87]. Ross and Robertson [103] modified the Fediaev- 

 sky method and obtained some improvement in accuracy. 



< 155 > 



