Weissinger and Maass 



most simple model of a ducted propeller and shows many of the relevant fea- 

 tures. Next we must consider the propeller field. The blade circulation being 

 given, the main problem is the determination of the shape of the vortex sheets 

 shed from the blades and of the field induced by these sheets. Then, the inter- 

 ference problem can be attacked. The most important result of Sec. 5 is that 

 the steady part of the flow through a finite -bladed ducted propeller is the same 

 as the flow for the same configuration but with an infinite blade number. This 

 problem, simplified to constant radial distribution of propeller circulation, is 

 treated in Sec. 4, 



So far, axial flow is assumed. In the case of nonzero incidence, which is 

 considered next (Sec. 6), there does not exist a system of coordinates in which 

 the field is time independent. Obviously, this problem is very complex and only 

 rough approximate solutions have been found. As a matter of fact, only the 

 steady part of the solution has been determined. 



Before starting this program we will have a brief look at the two-dimen- 

 sional theory in order to clear some of the basic ideas. 



1.3 Two -Dimensional Considerations 



(A) Infinitely thin airfoils (Fig. 2). Assuming small camber and small curva- 

 ture of the profile, the linearized boundary condition is fulfilled by putting on the 

 chord a vortex distribution y {^) = Vg(^) which satisfies the Kutta condition and 

 the integral equation 



1 



2tt J , 



g(^') 



-d^' 3 -a(f) , -1 < ^ < 1 , 7(1) = 



or, written with the "Glauert operator" G in operator form, 



Gg - a . 



(1.1) 



(1.2) 



Fig. 2 - Two-dimensional airfoil 

 theory. The infinitely thin cam- 

 bered airfoil. (Actually, for this 

 configuration the vortices rotate 

 in the opposite direction.) 



1214 



