Theory of the Ducted Propeller — A Review 



In order to show clearly the distinct efforts, we put, according to BoU- 

 heimer (116,120,122,125), 



7 = V{g^ + g,+ gT[g* + g:+g*]} , (3.6) 



q = V{q, + gTq*} . (3.7) 



Here, g^, gt, and q^ give the distributions of the ring airfoil and are deter- 

 mined from the equations 



qt=2t'(^), Tog^=p;(^), T„g, . -vq^ . (3.8) 



Similarly, from the mean axial velocity u = Vg^u * we obtain 



s ^ s 



Finally, a vortex distribution is required to cancel Vq* and the component 

 arising from the interaction of strong vorticity with thickness as described by 

 (1.15). So, g* must satisfy r; . . • ,, 



' ^^'"^oit-v-^lf.) . ■ ;„ ; (3.10) 



The operator of the four integral equations is the basic operator Tg of thin 

 ring airfoil theory. Of course, one can determine the sums g^ + g^ and g* + g* 

 from one equation. Numerical methods have been discussed in Sec. 1.3. To 

 solve the last three ("starred") equations, one has to determine g* first and, 

 from this, Ug* by numerical integration. These two functions depend only on the 

 parameter \.^ All equations can be approximated by relations between matrices 

 and vectors composed of Birnbaum and Fourier coefficients. These have been 

 tabulated by Bollheimer (116, 120) for several values of ^. 



So far, the value of the propeller thrust coefficient has not been used. This 

 value is needed to determine gj. First, the mean axial velocity u-j- induced by 

 7 and q at the trailing edge can be computed from (3.6) and (3.7) as a linear 

 function of g^. Insertion of uj into (3.3) gives a quadratic equation for g^-. The 

 coefficients of this equation can be easily computed by means of the vectors t 

 tabulated by Bollheimer. Finally, the velocity distribution on the duct surface 

 is the sum of V and the axial velocity induced by y and q on the reference 

 cylinder, multiplied by the Riegels factor {l + [p^C^) ± t '(^)]2 }" ^^^. In our 

 notation the distributions having a subscript c or t are zero if the camber p^ 

 or the thickness are zero, respectively. 



Obviously, the static case v = o is contained in this theory. Putting 



7 = gT[g: + g: + gt] . q = gT^t • (3.11) 



g-j- is obtained by virtue of (3.1) from 



/^(u * + u * + u , + u *) g2 = Ap , (3.12) 



^s ^c "^t ^t T 



if Ap is prescribed. 



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