bound circulation, r(r). Conservation of vorticity requires that a free- 

 vortex "sheet of strength - — , ■ is shed i 



dr 



sheets are assumed to lie on the surfaces 



vortex "sheet of strength - , ■ is shed from each lifting line. These 



x(r,<(0) = r tan 3^(r) [v - <^, ] Rj, 1 r £ R 



k = 1,2, ...Z 



where (x,r,v) is a cylindrical coordinate system fixed to the propeller as 

 shown in Figure 1. ''V. is the angular position of the k blade, 



^ = 2TT(kill k = 1 2 Z 



and g.(r) is the flow angle defined in terms of the relative velocity at 

 the blade section as shown in Figure 2. 



V (r) + u'(0,r,<^ ) 



tan e . (r) = -^ ~.^ --^^^— (20) 



x^ ' Qr - u^(0,r,<A ) 



Expressions for the induced velocity u'(x,r,v') = (u'',u',u') can be 

 derived by applying the Biot-Savart law to the distributions of vortici- 

 ty. ' The induced velocity due to the bound vortex lines, u', is 



Z , rR e X D 

 %(^,r,^) = E y r(p) % ' dp (21) 



k=i ^ •'r,, d 



\ 



p 



where 



e = (0, cos f, , sin '^, ) 

 p k k 



D = (x, r cos V - p cos -P, , r sin <^ - p sin <P. ) 

 p k k 



20 



Wu, T.Y., "Some Recent Developments in Propeller Theory,' Schiffs- 



technik. Vol. 9, No. 47 (1962). 



12 



