Theory of the Ducted Propeller — A Review 



For brevity we shall call p^{^) = dpjd^ the camber of the ring airfoil, al- 

 though—interpreted as local angle of attack— it also includes the incidence of the 

 profile chord. 



Then, for parallel flow with angle of attack ao> v„ is obtained by linear ap- 

 proximation as - ■ 



V^ = +Vt'(^) - Vp;(f") + Vfflo cose/. , (2.10) 



and putting 



q - Vq^ , 7 = V(g^ + g^ + ttgg^ COS0) (2.11) 



Eq. (2.5) splits into 



q, = 2t'(^), T,g, = /.;(a. T,g, = -Vq^, T^g, = -1, (2.12) 



where T is an integral operator that transforms a vortex distribution into the 

 radial velocity induced by it. It follows that the effects of thickness (q^, g^) 

 can be obtained by considering the ring airfoil without camber ip'^ - 0) in axial 

 flow, the effect of camber from the infinitely thin ring airfoil in axial flow and 

 the effect of angle of attack from the infinitely thin cylinder at angle of attack. 

 Addition of these separate effects gives the entire distribution. From these the 

 surface velocity is determined by 



(v + Uq + u^) . (2.13) 



Vl + [t'(<j)]2 



The continuous part of v^ can be considered as the first terms of the more 

 general Fourier series 



(CO ro^ "1 



^ ajf) cosm0 + 2] b^(^) sinm</)l . (2.14) 



m= m= 1 J 



This happens when the ring airfoil is not exactly axisymmetric or if there is an 

 interaction with a nonaxisymmetric velocity field. Then the distribution of ring 

 vortices has the form 



>' = ^ j E gm(^) cosm0 + £] hj^) sinm^l. (2.15) 



[m=0 m=l J 



The Fourier coefficients gn,(^) and h„(^) which satisfy the Kutta condition 

 are determined by the equations 



Tg=a,Th=b, (2.16) 



m "=m m m m m ' \ • / 



with the integral operator T^ defined by 



^^^ = 7-/7^^^'%-/ ^.(^')U.(^)d^'-v/ ^n,(^'')d^'- (2.17) 



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