Theory of the Ducted Propeller—A Review 



The solution is known to be :i =- > ii. :. : v 



The integral can be evaluated by means of the formula ' ■• '' ' ' 



f ^°""^' 60' -- rr liUJll n . 0, 1, 2, ... (1.4) 



•J ens. B' - cos 6 sin £? 



cos 6' - cos 6 sin 6 



u 



and the Fourier expansion 



= 1 

 The result is the Birnbaum series 



(^) = — + y a cos v0 , ^ - -cos^ . (1.5) 



2 ^ 



■^0 cot —+2^3^ siny( 

 ^ v= 1 



;i.6) 



Another method (114) of solving the integral equation 



1 f" g(i9') sini9' , /- „v 



— — dO' = -a{d) , < e < V (1.7) 



277 J^ cos 9' - cos d 



which is equivalent to (1.1), is a collocation method based on the quadrature 

 formula 



\ 'sn d.' = ^ E 



~n COS0' - cos 0. N i,= r 



-Q COS o - uus u ■ '"I k= <-US u. 



^^^^^ ■ 1 



1=1,..., N 



cos 6',' - cos 0. 



(1.8) 



, _ 21^ ^ _ fl for k = 1 , . . . , N - 1 



2N *" 2N ' '' I 1 , , ^ ^, 



- for k = , N . 



This formula is exact if i(0') is a cosine polynomial of degree <2N , i.e., the 

 accuracy has Gaussian character. Formally, it cna be interpreted as the appli- 

 cation of the rectangular rule on the singular integral (1.8). To obtain the high 

 accuracy, the collocation points di must be the midpoints of the subintervals of 

 length 7T/N. Applying (1.8) with f(e) = g(9) sin 9 and introducing the Kutta con- 

 dition, the integral equation (1.7) is transformed into a set of linear equations 



^T, h A =-«(^i). i=l,...- N (1.9) 



N r — '. cos 9, - cos 9. 



k - K 1 



for the N finite unknowns g^ = g(0k) sin 0^ . These values coincide with the 

 exact ones if a(e) is a cosine polynomial of degree ^2N - i. 



1215 



