D f["5" + D (x c" O5)sin0p ] 



e,«U 



(25) 





tlT? 



(27) 

 (cont.) 



|N + X(v + -y")| X (r-s ) 



Hence, Equation (23) can be reduced to an integral over only 

 one side of the blade surfaces and shed vortex sheets: 





(N + • v + + N" • v - )- 



.+ (N + X v + + N" X v~) X 



dx„ 



4 vEK/ d ^^x(v + -v 



where the symbol X means symmetry restrictions occur for 



the limiting region which excludes the singularity. For 



example (9), the region may be square, circular, or rectangular 



centered at r <) . In the present application, the rectangular 



region x r - e < x r < x,. + e, x h < x„ < 1 will be the 

 c — c — c h — K — 



shape of the excluded region. Then this principal value 



integral is defined 



1 1 |~ " c -e 1 



Jr dx c / dx R K = lim I dx c J dx R K 



X_+6 J X 



dx„ K 



The assumption 



lim [v(r)] = v + (r ) 



(26) 



(281 



(29) 



+ Tj sin P 1 i 



+ -7- e r (0 te + 2t) cos p /x R + b ) 



V =-(x c -1.0) 



As the field point r approaches a point r,, (x. , x R ) on 

 _ -u l r. 



the surface of the blade, Equation (23) or (26) becomes 



singular. If a small region about this point is excluded from 



the surface S and the limit of the integral taken for L^Io 



with the excluded area tending to zero, there results 



(see Reference 9): 



lim 

 r -► r 



Z 1 1 



"h 

 r„ -s„ 



(N + • v + + N" • V _ ); 



r„ - s„ 



+ (N + X v + + N" X v") X- 



u-i^J 



+ y [v + (ro>-»"feo)l 



(27) 



(i.e., that the velocity defined in the field does approach the 

 value on the boundary) leads to the following expression for 

 the average velocity component on the blade surface 



(vx)"! ^( x vx) + - v "( x v x r ) 



(30) 



ue | + ve-, + we_ 



Z 1 1 



£ / dx c/ dx R K (^c/R/r x R) + *w (311 



b= 1 J J x 



where the singular kernel is 



-( X V Xr o' Xc,Xr )" 4rr 



(N + ■ v + + N" • v") 



+ (N + X v + + N" X v") 



and the velocity induced by the shed vortex sheet is 



