The Aerodynamics of Sails 



true wind velocity whose magnitude 

 increases with height, and the neg- 

 ative of the velocity of the vessel. 

 This small effect is neglected in 

 this analytical development, but is 

 taken into account in the numerical 

 calculations of the next section. 



The lifting line problem isfor- 

 mally posed as follows. Determine 

 the flow field and associated lift, 

 and the induced drag and heeling 

 moment for a lifting line of length 

 b perpendicular to an incident, x - 

 directed, shear flow and inclined to 

 the vertical by an angle . An in- 

 finite horizontal plane is located at 

 a distance h below the lifting line, 

 and the variation of free stream 

 velocity depends only on height in 

 a linear fashion such that its speed 

 is given by 



Uq + Ky 



(1) 



The origin of the coordinates 

 is at the midspan of the lifting line 

 (Fig. 2). Calling the disturbance 

 velocity potential by and the dis- 

 turbance velocity components by u ' , 

 v', and w', the boundary conditions 

 are: 



Fig. Z - The geometry for a heeled 

 lifting line in the presence of an 

 image plane 



y=-b/2-h 



(2) 

 (3) 

 (4) 

 (5) 

 (6) 



The method of solution is similar to that used by Glauert (1948) for an un- 

 bounded airfoil in a uniform stream. The circulation must vanish at the ends of 

 the lifting line to satisfy Eq.(6). This is because a nonzero value of circulation at 

 an end of the lifting line would necessitate a trailing vortex of nonzero strength, 

 since the vortex field is solenoidal. The circulation strength on the lifting line 

 is expanded in a Fourier series, each term of which vanishes at the ends of the 

 span. First the angular variable ./. is defined by the relation 



1399 



