15 



effects of Reynolds number. Mention is made of an experimental study of the 

 Gruschwitz method by Peters 16 who found that it gave fair results for 77 but 

 that it seemed useless for predicting separation. The form of the Garner 

 equation is superior to that of the von Doenhoff and Tetervin equations be- 

 cause it separates the pressure-gradient parameter and the Reynolds-number 

 effect. A critical comparison of the two equations just mentioned, made by 

 Fage and Raymer, 1 ' claimed serious numerical discrepancies between them even 

 though they had both been derived for the most part from the same test data. 

 The moment -of -momentum equation developed in the present report will be found 

 similar in form to that of the Garner equation. A graphical comparison of all 

 the methods with the exception of the Buri method is shown at the end of this 

 report . 



LOCAL SKIN FRICTION AS A FUNCTION OF PRESSURE GRADIENT 



The manner of the variation of the local skin friction or shearing 

 stress at the wall t with pressure gradient as well as with Reynolds number 

 is required in both the von Karman momentum equation and the moment-of- 

 momentum equation. Until recently the experimental study of wall shearing 

 stresses was hindered by the poor precision of the existing experimental tech- 

 niques, based on measuring the rate of change of the momentum and the pressure 

 of the flow through the boundary layer. In 1 9^+9 Ludwieg and Tillmann 18 intro- 

 duced a new method for accurately measuring the wall shearing stress in a pres- 

 sure gradient. The improved technique, described in Reference 19, consists in 

 measuring the rate of heat transferred to the fluid from a calibrated instru- 

 ment imbedded in the wall. From their measurements Ludwieg and Tillmann 

 proved that the wall shearing stress diminishes in an adverse pressure gradi- 

 ent, even close to the separation point. In addition, they demonstrated that 

 the so-called "law of the wall" can be used for a quantitative analysis of the 

 variation of the wall shearing stress in a pressure gradient. 



The "law of the wall" 20 states that within the boundary layer in a 

 pressure gradient there is a region of flow next to the wall which has veloc- 

 ity and skin-friction characteristics similar to that of a flat plate in a 

 zero pressure gradient. As seen in Figure 7 this region of flow or inner flow 

 has subregions of a laminar sublayer and a transition zone. The Prandtl 

 friction-velocity relationship for a flat plate 2 can then be applied to the 

 whole inner flow or 



u. . t £±) [28) 



