Similarly, the velocity induced by the free-vortex sheets, uj, is given by 



e X D 

 s 



u^(x,r,v') = X, - TT / d^J\-(p)+P ^_, f da dp (22) 

 k=l ^R^dp M . J^^ |^^,3 



where 



>f^ 



(■'^.(p)> - P cos a, p sin a) 



(P) + P^ 



D = (x - A . (p) a, r cos <P - p cos a, r sin <P - p sin a) 

 X. (p) = p tan 3. (p) 



If these expressions are evaluated at the lifting line (0, r, f ) and 

 inserted into equation (20), an integral equation is obtained relating 

 r(r) and 3.(r). 



In design applications where a prescribed thrust, T, is to be developed, 

 a second relationship can be derived by applying the Kutta-Joukowski law 

 (with an empirical correction for profile drag - (see Figure 2) yielding 



J dr 



"■ (23) 



/.R 

 = pZ J r(r) [nr - u^(0,r,Vj^)] [l - e(r) tan e^(r)] dr 



The solution is normally found by an iterative procedure, starting with an 

 estimate of tan g.. r(r) is then computed from equations (20) and (22) and 

 used to calculate a thrust according to equation (23) . This procedure is 

 repeated until the desired value of thrust is obtained. Highly efficient 

 niomerical techniques based on asymptotic series expansions have been 

 developed to perform these computations. In the present work, it is assumed 



14 



