%T 1 J dx * J d " 



b = I x h 



drj 



The first term in Equation (27), the strength of the sources, 

 is known from Equation (20). The strength of the vortex 

 component: 



N* X v 1 + N" X v" * Nj X (v + -v~) ( 32 ) 



has yet to be determined both on the blades and along the 

 shed vortex sheet. To find a value for this term, it is appro- 

 priate to look at the condition that the blade section develop 

 a force at a given radial station 



F(x R ) 



= -/pn 



dJ 



To first order the local lift is 



L = F ■ e 2 =D-M (p-- p + )d 



(33) 



(34) 



Bernoulli's equation can be constructed for a coordinate 

 system rotating with the blade (9) and the pressure dif- 

 ference determined by 



.2 2| 



p'-p =-y{<s ) -(a") 



■ (q + ) 2 -(q-) 2 J (35) 



= p|s~ •(v + -v-)+-(v t + v-)-(v + -v-)l 



Let 



*pq.' (v + - v") 



<v> = — [y_ + - v~] = ^-e , + • 



a K X e, 



— jtD 2 V 



2 ....... 2 



(36) 



Ni 



Nil 



I NX I 



Then to first order 



pV\/(l -w„) 2 + 



COS (0p - / 



(37) 



(38) 



(39) 



•fc^M 



and from Equation (20): 



cos (0 p - 0) 



. = D 2 V\/(1 -w x ) 2 +(-jt) cos(0 p -|3) — (40) 



AC p = (p--p + )/(pV 2 /2) 



(41) 



It remains to determine o. Now, in general 



No X(v + -v-) = -7rD 2 Vae, +7(N+ Xe,) (42) 



and the value of o can be determined by setting the diver- 

 gence of the right-hand-side equal to zero (9). However, this 

 results in a differential equation to be solved for o. A more 

 direct procedure is to express the perturbation velocity 

 vector as the gradient of a potential: 



v = V0 



( r) = lim 



r 



d£ 



(43) 



(44) 



i-*5 - A i 



A-~ 



— lim Ivj^v • dfil 

 r -» r" J-\i 



(45) 



N! X <v> 



n! 



2jtD 2 V 4jtD 2 V 



V f (v + -v-)-d« 



le 



dS = D— e, dx. 

 D ' c 



2ttD 2 V 

 It is convenient to define 



K r c r x 



XV D— 



D 2 V L °Jo 



7dx. 



7(x r , x R ) = — 



1 T(x R ) 7 (x c ) 



c/D 



(46) 



(47) 



(48) 



where r (x R ) is the bound circulation (0 + - 0~ at the trail- 

 ing edge and points beyond), and 7 has unit magnitude 

 when integrated across the chord. Let the nondimensional 

 circulation G be 



irDV 



Then 



G(x R )7 (xj 



7 = irV 



c/D 



n;x<v> n; 



• xv 



2ttD 2 V 4-itD- 1 V 



irVDG( 



x R ) 7 dx c 



•Jo 



(49) 



(50) 



(51) 



