Milgram 



■~" '" "^^ Appendix A 



THE EFFECT OF HEELING ON THE AERODYNAMICS OF SAILS 



The aerodynamic effect of heeling will be considered for lifting-line theory 

 inasmuch as almost identical results are obtained for lifting- surface theory. 

 Consider the lifting line and image system of vorticity shown in Fig. 2. The 

 problem is to determine w' on the lifting line. For a given circulation distri- 

 bution, the value of w' induced by the lifting line and its trailing vortex sheet is 

 independent of the heel angle 4> . Therefore, only the velocity induced by the 

 image system of vorticity will be considered. Since the line representing the 

 direction of w' and the image of the lifting line are coplanar, there is no w'- 

 directed induced velocity on the lifting line due to the image of the lifting line. 

 Hence, only the image-trailing vorticity need be considered. Calling the w'- 

 directed induced velocity at the lifting line by the image-trailing vorticity by w'. 



1 r' 



7(T)) COS (0+a)d7] 



^(y + T7+ 2b+ h)2 cos^^i (^-y)^ sin^^ 



(Al) 



where 



a ~ tan 



y + b + 2h + T7 



The integrand of Eq. (Al) can be written as 



tan 4> 



(A2) 



I (y,^) 



7(T1) 



y + b + 2h + 7? 



(A3) 



where the term outside the brackets is what the integrand would be if the heel 

 angle were zero. The bracketed term can be interpreted as a weighting function 

 due to heel. The difference between the weighting function and unity increases 

 with increasing heel angle and decreasing lower-end gap, h. Under normal cir- 

 cumstances, the largest heel angle of most boats does not exceed 30°. Values 

 of the weighting function for various values of y and 77 , with equal to 10, 20, 

 and 30° and the gap, h, equal to five percent of the span are shown in Table Al. 

 This table shows that the weighting function is positive and differs significantly 

 from unity only when the heel angle is large, and even then only for values of y 

 near the lower end of the lifting line and values of 77 near the lower end of the 

 image of the lifting line; i.e., for points on the lifting line near the image plane 

 and points on the image of the lifting line far from the image plane. The influ- 

 ence of the image vorticity is least for large values of 77, as shown in Table A2 

 which is a table of values of the integrand of the integral in Eq. (Al) divided 



by 7(77). 



1428 



