where the plus and minus signs represent the upper and lower surfaces, respec- 

 tively. 



For three-dimensional flow such as that on propeller blades, the direction of 

 velocity jump depends on both the spanwise and chordwise vortices. In this case, 

 both spanwise and chordwise singularities have to be properly accounted for when 

 computing the velocity jump. The following is the algorithm adopted in the pre- 

 sent study. 



Suppose we want to compute the velocity jump at the i pressure point on i 

 spanwise vortex element. The total distributed vortex at this point, y_. , is the 

 sum of the spanwise and chordwise distributed vortices: 



^i = %h ^ ^^c^i (6) 



The spanwise distributed vortex, (y ) . , is approximated by: 



(r ). 



where Ac. is the length of the chordwise segment represented by the discrete span- 

 wise vortex, (r ) . . This is analogous to the two-dimensional distributed bound 

 ~s 1 



vortex (see Equation (2)). The chordwise distributed vortex, (x ) > is approximated 



by the vector average of the four adjacent chordwise vortices, (F ),, (F )„, (F )„, 



— c 1 — c Z — c J 



and (F ), (see Figure 6): 

 — c 4 



1 JL (r ) 



i=l 



where Ar is the length of the radial segment represented by each discrete chord- 

 wise vortex, (F ) . The total distributed vortex, Y., is then converted to the 

 —en -^1 



velocity jump in the tangential direction by using Equation (3). 



The velocity jump due to the source sheet is identical to the two-dimensional 

 case (see Equation (5)) since the boundary condition for thickness effects results 

 in the same relation between source strength and slope of the chordwise thickness 

 distribution with radius as a parameter. These velocity jumps due to vortices and 

 sources are added to the velocity induced by all other singularities to obtain the 

 total velocity induced by the propeller. 



6 



