Weissinger and Maass 



distributions of ring vortices and ring sources on a reference cylinder of ra- 

 dius R. If the vortex strength varies with the azimuthal angle, straight vortices 

 must also be placed on the cylinder. 



We introduce cylinder coordinates (x, r,0) and define the nondimensional 

 coordinates 



^--~-, P--^ or 7 = ^ (2.1) 



and the chord/diameter ratio 



W (2-2) 



The induced axial, radial, and azimuthal velocities are denoted by u, v, and 

 w, respectively. If the arithmetic mean of the values of a function at the outer 

 and inner point of the cylinder is denoted by a bar, we can write for the veloci- 

 ties induced at the cylinder by a vortex distribution 7 or a source distribution q 



■••'•• . V ■ ■ y _ _ 



' U = + + U , V = V , (2.3) 



y 2 y y y 



— Q _ 



u„=u„, v=+-+v, (2.4) 



q q • q 2 1 \ • / 



where the upper (lower) sign refers to the outer (inner) surface of the cylinder. 

 Then the (strictly) linearized boundary condition can be written in the form 



vy ± ^ + Vq + v„ . , (2.5) 



where v„ denotes the normal component of the given flow on the airfoil surface. 

 For zero thickness this reduces to 



v^ + v^ = . (2.6) 



For an axisymmetric ring airfoil the surface can be described by 



d(x) ,_ _, 



r(x) = R+ rjx) ± -^ (2.7) 



or, in nondimensional form, 



P(^) = 7+ Pm(^) ± t(^) , (2.8) 



P„<f)=^'. .(0 = ^ . (2.9) 



1220 



