Theory of the Ducted Propeller — A Review 



one has to determine the strength y^ of the ring vortices from the pressure 

 condition on the slipstream surface, the radial velocity v^^ induced on the duct 

 and a ring vortex distribution y on the duct from the integral equation 



To7= -v^^ . -^ - - :- (4.9) 



or 



^o>^=-%' ""'-W- ^^■■■- (4,10) 



p 



If the duct is not cylindrical, the influence of camber and thickness can be taken 

 into account by superposing the singularity distributions of ring-airfoil theory 

 (within the framework of strict linearization). 



By satisfying the pressure condition far away from the duct, where the 

 duct-induced velocities are zero, one obtains 



^ ^ ' ^ % V ^-■• 



J +VJ2 + c-r "Rp :... - . 



p 



Therefore, the pitch j of the helical vortices is given by 



. •': . j = -y^/y^ = -^ (j + VJ2 + CT^ ). .;^ '.;."■■■ : ' (4.12) 



and we can write 



The velocity induced by a semi-infinite cylinder of ring vortices with constant 

 strength can be expressed by means of the Legendre functions Qi/2' So> o^^^ 

 obtains 



% = " ^ ^^Qi/2(-) -- - ^ %Qi/2(-) ■ ... (4.14) 



(1-M)' + (x-Xp)^ _ _ 



a=l+ -, X - Xp = (x-Xp)/Rj , (4.15) 



and the integral equation 



'ro^ = ^^TpQi/2(-) • ■ ■ . ' (4.16) 



Putting 



Qi/zC-^) = ^0 " Z] 'In ^°sn0 , X = -\ cos^ , (4.17) 



n= 1 



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