Weissinger and Maass 



20 



m 

 '"- 5 



Fig._6 - (See (106)) The func- 

 tion Fg(t) for ellipsoidal bodies 



Fig. 7 - (See (106)) The 

 kernel B(Ti) = -B(-Ti), rj = 



P^ to be small. Nevertheless, the value of 

 zero obtained for p^ - l is exact for the 

 cylindrical body because there is no lift 

 when the wing and the cylindrical body 

 coincide. Most of the results shown in 

 these figures have been obtained by the 

 three -quarter -point method. It can be seen 

 that they agree fairly well with the results 

 derived from the "exact" (linearized) the- 

 ory. No example of results for the moment 

 will be shown here. 



The numerical results of (110) may be 

 used immediately for determining the in- 

 fluence of a hub on the lift and moment of a 

 ducted propeller at angle of attack by means 

 of a "superposition model" as described in Sec. 6. The general framework of the 

 theory may also be used for a more thorough solution of this problem. These 

 remarks may be true also for the theory developed in (109) on the influence of 

 struts. 



3. THE DUCTED ACTUATOR DISC WITH 

 CONSTANT PRESSURE JUMP 



3.1 Linear Theory 



The simplest model of a ducted propeller is a ring airfoil with a constant 

 pressure jump Ap at a cross section x = Xp (the "disk"). Of course, this can 

 be used only for axisymmetric flows. It can be interpreted as the representa- 

 tion of a ducted propeller with many blades and with the swirl neglected or can- 

 celled either by a counterrotating propeller or by guide vanes. 



Since the disk does not produce vorticity, we have two regions of potential 

 (irrotational) flow: the propeller slipstream and the rest. The slipstream is 

 bounded by the disk, part of the duct and a free surface beginning at the trailing 



1224 



