22 



Furthermore 



S 8 



r! / r , \ _ // r 



[68] 







a' ^ \ dr 



ady 1 — — is empirically of a lower order of magnitude. Hence the momentum 



8 

 since 



'0 



equation, [61], becomes for 8 « r 



dx 



^ ^ V dx -l .2 .2 dx ] n 



°'\^w^i /,, . n\ w- au . I • w . i u I _ ^.,1 r^m 



The sum of terms — ^ and — <^^y n^^Y be considered as an effective 



pV^ p V dx 



coefficient of local skin friction, i.e., 



\pU^I pU^ pU^ dx I 



A f f ^ 



^y [70] 







In their study of two-dimensional boundary layers in adverse pressure gradients, Wieghardt 

 and Tillmann found that the values of ( r^ / p 6'^) jj computed from the momentum changes of 

 the mean flow displayed unexpected increases near separation. ^^ Since tests by Ludwieg and 

 Tillmann^-^ have shown the values of r / p t/" to decrease in an adverse pressure gradient 

 faster than those for flat plates without pressure gradients, the explanation of the increase in 

 ( ''■^/p ^^)eit ^^^ ^ b® sought elsewhere. Wieghardt has attributed this apparent increase 

 wholly to the convergence of the flow caused by the thickening of the boundary layer on the 

 opposite sides of the wind tunnel. ^'^^ On the other hand, Newman^'* and various investigators 

 of the National Advisory Committee for Aeronautics^^ '^^'^^ have shown this increase to be 



partly accounted for by the increase in the value of the normal-stress term — j ady, 



P^ d^ Jo 

 especially close to separation. Hence for adverse pressure gradients, the decrease in the 



value of T^ / pU^ tends to be compensated by the increase in the value of 



8 



pressure gradient { t^ / p U^)„ a close approximation to the effective skin friction, at least 

 for moderate pressure gradients. Lyon's experimental results tend to substantiate this.-^' 

 Therefore with 



-i^ -4— I a dy. This makes the coefficient of local skin friction for flat plates without 

 ^ Jo 



(t^) =(^) 



Equation [69] becomes 



