Theory of the Ducted Propeller — A Review 



Chaplin has been used by Hunt (35) for detern\ming the flow from a circular 

 orifice. 



4. THE UNIFORMLY LOADED DUCTED PROPELLER WITH 

 AN INFINITE NUMBER OF BLADES IN AXIAL FLOW 



For the rest of this paper each propeller blade will be represented by a ra- 

 dial vortex with a circulation distribution r( r). From each element (r, dr) a 

 helical vortex with strength -r'( r ) dr is shed. In an exact theory the radius and 

 the pitch of the helix will naturally depend on the axial coordinate x. In linear- 

 ized theory the radius is assumed constant and in most theories also the pitch, 

 i.e., the helix is assumed as a regular helix of radius r and pitch j ( r). The 

 pitch has to be determined. See Fig. 17. 



Fig. 17 - (See (142)) Vortex 

 system and geometry for ducted 

 propeller configuration 



In this section is is assumed that r(r) = r = const. Then the shed helical 

 vortices have strength r and lie on a cylinder of radius Rp = propeller radius. 

 On the axis lies a vortex, and "hub vortex," extending from x = Xp (= axial lo- 

 cation of the propeller) with strength NT, where N denotes the number of pro- 

 peller blades. It is assumed that n becomes infinite, such that 



Nr = r 



(4.1) 



is a finite constant. In practice, it can be expected that the following theory can 

 be applied for blade numbers as low as 3 or 4. 



In this model, the slipstream cylinder r = R 



> x„ is covered with a 



continuous and constant distribution of helical vortices. Taking orthogonal com- 

 ponents, this distribution can be split into a distribution of ring vortices and of 



1233 



