It is convenient to consider the induced velocities as the sum of the 



lifting-line result and separate lifting-surface corrections. The induced 



velocity due to the bound and free vorticity, e y (.r,<p) and e y {r,<p), may 



be written as 



¥> (p) 



Z /-R f- r f\ e D 



Ug(x,r,<p) = X; i:;, J V^i(p)+P J YgCP.ct) -^^-3^dp da (28) 



k 



and 



R /— K /-" e X D 



^ ^ (29) 



li-(x.r,<.) = E i^j y^4(p)+pj ^s^P'")-~:i ,3 



«H 



where <^ ,T (r) are the blade leading and trailing edge positions. Continu- 



^k " 



ity 



of 



vorticity 



requires 



that 









^s 



.(r,<?) = - 



■'^^i 



' (r) 

 ^R 











dr 





(30) 



and hence u;', can be rewritten as 



T 



e X D 



Z ^ /-R f-2 2" ( ^ ^s 



u'(x,r,v) = E 17 / J^Ap)+P J Ys^P''^>~;3,3 



^ k=i '^'^ V ^ ^ <e(p) D_r 



^^ •^«^, Id 



k 

 e X D 



(31) 



1^ 

 s ' 



20 



