The Aerodynamics of Sails 



craft, this effect is very small and will not be taken into account in the 

 following theory. 



It is shown in appendix A that the effect of heeling on sail aerodynamics is 

 negligible. Hence, the following developments will be carried out for zero heel. 

 The dominant effects of heeling can then be accounted for by proper resolution 

 of forces. The lift and heeling moment are determined by the circulation distri- 

 bution on the lifting line and the free stream velocity distribution. These inde- 

 pendent variables, along with the induced velocity component w', determine the 

 induced drag to the first order in the velocity ratio. 



We can now determine w'. From the law of Biot and Savart, 



w'(y) = ± f '' ZlZll dr, - -i- r ' ^Zm dT, . (14) 



^^477 j y-V 4^ J y + h + 2h + T] ' 



The negative sign occurs on the second term because the sign of the image vor- 

 tex system is the negative of the sign of the lifting line and trailing vortex sheet 

 system. 



Let 



7] = - — COS 'p on the lifting line 



(15) 



7] = - — cos (p on the image line , 'lt>) 



and, as before, 



b 

 y - - — cos lA 



2 



In terms of the variables 0, 0', and 4)', w'(y) is given by 



(17) 



(18) 



/_l_y~'_. ^ aJT cosn0' d4'' _ j cos n(p' d0' \ 



"" '•^' ~ 4^ L-J 0" "]] cos - cos -A' J„ 2+4h/b-(cos'/'+ cos(p') 



n- 1 [^0 ^ 



The first integral was evaluated by Glauert (1948) as 



r cos n0 dp _ sin n^ (19^ 



J cos ^' - cos 'p siniA 



The second integral is evaluated in appendix B as 



1401 



