Supercavitating Propeller Theory 



The induction factor integral, summed over all blades, can be expressed as an 

 infinite series of products of modified Bessel functions of the first and second 

 kind. By the use of Nicholson's asymptotic formulas for the modified Bessel 

 functions, the numerical computation of the induction factor is rendered straight- 

 forward. A very full description of the process is given by Morgan and Wrench 

 who offer refinements to Nicholson's approximation [17]. 



By applying Eq. (26) to Eq. (Bl) and using the transformation (27) 



1 M r 1 x-(l-x)l/2i (x*^^) . 



u rr*)p = > A^^(m+1) dx 



^^ '^ 2(1- r,) ^ " r 1 



(x*-x) 



(B3) 



^ x"*l i (x*,x) 



dx 



2 I (1 - x)i/2 (x*- x) 



The integrals of (B3) can be made suitable for straightforward numerical com- 

 putation by subtracting out the singularity at x = x* , and eliminating the square- 

 root singularity at x = 1 by applying the transformation w = (1 - xY^'^. 



Hence , 



M r 1 



^ Y A^ \ -(m+ 1) 



"a(^*)r " r-T^^ : Y. A^^'C'^+l) I x™( l - x)^/^^^^ ,^*) ^^ 



(B4) 



+ - 1-— idx+ (l-w2)" 'G(w,x*) dw + J^(x*) cos /3.(r*)L 



2-!, (l-x)i/2 J^ "> ^ J 



where ii:,/- 1-,'.;^;; ..;;•_,•. .;,,,::,. .q> -^- 



i (x,x*) - i rx*,x*) 



G(x,x*) 



(X- x*) 

 i (x*,x*) = cos /3^(r*) , 



J (x*) = — - [(2m+3) x* - 2 (m+1)] in 



"A' [2^-1(5- 1)! 

 + (2m+3) 2^ 



1 + (1-x*)*/' 



-2(m+l) Y. 



ti (2s- 1)! 



[2^-i(s- 1)!]' 



; (2s- 1)! 



955 



