Theory of the Ducted Propeller — A Review 



because this, evidently, is the steady part of the solution y as referred to a 

 duct-fixed system of coordinates. This can be determined by the ideas of the 

 foregoing section. 



Obviously, TqC'^) is that part of y that must cancel the average radial ve- 

 locity induced by the propeller vortex system, and the average velocity is the 

 same as the velocity induced by the average vortex system, which is equivalent 

 to the propeller with an infinite number of blades. With respect to the model 

 considered in Sec. 4 the only difference lies in the dependence of r on r. This 

 difficulty can be overcome by integrating the pertinent formulas of Sec. 4 with 

 respect to r in the following manner. 



The sought radial velocity Vp is written as -' •• "^ 



where dvp is the radial velocity induced by an annular part of the slipstream of 

 radius r < Rp and width dr. By (4.14) we have 



1 / (r- Rj)^ + (x- X )^ 



^^-^/^'■^'J./aMdr. -- '^ ,,,^ • (5.3) 



and 



'>'* = -^/j(r) , ''-■ '■■ ;\" ;;-.;V...„ . (^•'*) 



where j ( r) is the pitch of the helical vortices lying on the cylinder of radius r 

 and where y^ and y^ denote their azimuthal and axial components, respectively. 

 Since 



'>'x = ~ ^ — ':r~ ' (5.5) 



" 27Tr dr \ • / 



we obtain the integral equation i - 



T y = -V (5.6) 



with 



-N 

 v_ = 



r ' ^-=r4^Qi.2(-)dr . - (5.7) 



" 47r2 J j(r)VrRd d 



If the propeller -shed vortices are assumed to be convected with the free stream, 

 then 



JV_ ^ _v_ 



^^'^ %r ^ -^ I- ' -^ ' fiRp ' (5.8) 



and the integral equation (5.6) can be written in the form 



1239 



