Theory of the Ducted Propeller—A Reveiw 



Only the terms of order mN are left in the general Fourier series because the 

 flow must be periodic with a period 27r/iM. Evidently, Vg^ is identical with 7o 

 in the foregoing steady-part theory. 



It turns out that the two-dimensional integral equation, which equates the 

 radial downwash of the duct vortex system and the radial velocity induced by the 

 propeller at the duct, is equivalent to an infinite system (m=i, 2, • • • ) of coupled 

 integral equations for g^ and h^. According to (142), these equations can be 

 written in the form 



\ g„K'dx' -— r h^K dx' = Bp, 



(5.11) 



r h^K'dx'+— r g^K dx' = Ap^ + Ap, . ,• (5.12) 



I mm T I "mm i m i m ^ i -,.■'• - 



*'-x Jm -'-a ,, ■ ; 



Here, VAp<^ and VBp-^ are the coefficients of the mN-th sine and cosine har- 

 monics, respectively, in the Fourier expansion of the radial velocity induced on 

 the duct by the helical vortices shed from the blades. Similarly, VApm is the 

 sine coefficient of the contribution due to the propeller blades. For the rather 

 complex formulas see (139, 142). 



The kernel K^ is defined by 



K„ = ^{S^n(^') - G^(Ax')} 0)= 1 + (Ax')V2, Ax' := (x - x' )/R , (5.13) 

 G^(Ax') = n I {2j2/i2Q^_^^^(~) + s^(S)} sin^r dr , 







.2 



(5.15) 



o) = 1 + (Ax' - Jfj.Ty/2 



K^ denotes the derivative of K„ with respect to Ax' and has the well-known 

 Cauchy singularity. 



The pair of integral equations (5.11) and (5.12) can be solved simultane- 

 ously by methods similar to those used in the preceding sections. In (139, 142), 

 a method of decoupling the equations has been described. There, it is also indi- 

 cated that there may exist nontrivial solutions of the homogeneous equations. 

 The physical interpretation of these distributions is not clear. Evidence for the 

 existence of such periodic eigensolutions can be deduced also from theories of 

 Ludwieg (50, 51) and Rautmann (118, 120). 



As J -» 00, the helical trailing vortices of the duct become straight lines, 

 i.e., one obtains the vortex model of the thin-ring airfoil with deviations in 

 shape from axisymmetry such that y depends on also. So, one gets the un- 

 coupled equations of ring-airfoil theory with the integral operators T^,. The de- 

 coupling for i/J = can be observed immediately in (5.11) and (5.12), On the 



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