Theory of the Ducted Propeller~A Review 



evaluation and again a system of linear equation for the unknown values g^ = 

 g(d{^) sin 0,^ is obtained. That the accuracy in this evaluation of Kg is not as 

 high as that for the singular integral Gg does not matter very much, because 

 usually, i.e., if c/D is not too large, Kg is small compared with Gg. Compared 

 with the other three methods, this method has the advantage that no Fourier ex- 

 pansions are needed for constructing the system of linear equations. Essen- 

 tially, the matrix is immediately equal to the matrix of kernel values G + K at 

 the points (^i , fk ) - -(cos d-^ , cos 0{^ ) . Therefore, the programming will be 

 easier. The disadvantage consists in the fact that the accuracy in the numerical 

 evaluation of Kg is coupled with the number N of unknowns. In our experience 

 this methid is superior to the first— and probably to the other ones— with respect 

 to computation time. 



(3) Let go be the solution of the Glauert equation 



Ggo = f . (1.22) 



-I 



The Birnbaum coefficients of g^ are essentially (i.e., apart from factors such 

 as -2) equal to the Fourier coefficients of f . Then (1.21) is equivalent to 



g = go + Kg, K = -G-iK , (1.23) 



where the kernel of K is known because G" ^ is known. Then the solution g is 

 given by the Neumann series 



g :. {I + K + K2 + ...} g^ (1.24) 



and can be found by the iterative scheme 



g..i = go + Kg^ , u = 0, h ... . (1.25) 



This operator equation can be approximated by a matrix equation if the vector 

 of the first N Birnbaum coefficients is substituted for g and, for K, a matrix p 

 which is connected in a simple way with the Fourier coefficients of K. 



(4) Equation (1.21) is written in the form 



g-Kg=T, K = -G-iK, ? = G-i f . ..,.,,, . (1.26) 



If ic is represented approximately by a truncated double Fourier series, one ob- 

 tains a Fredholm integral equation of the second kind with a degenerate kernel. 

 This equation can be transformed into a system of linear equations by classical 

 methods. . . ... . i. .., 



2. THE RING AIRFOIL 



We assume that the profile satisfies the conditions of linearized two- 

 dimensional theory and that the angle between profile chord and axis is small 

 enough so that the surface formed by the chords can be approximated by a cir- 

 cular cylinder. Then the boundary condition (zero normal velocity on the sur- 

 face) can be satisfied within the limits of linear approximation by putting 



1219 



