Supercavitating Propeller Theory 

 Q 



1^ 1 Su+S- 



x*-ex n 



r ±:l Ml/2 d^dr 



J R I 



J 



// 



+ Ap(r,^) 



' x*-5X N 

 J R.3 



d^dr 



(A8) 



where x, x*, r, r*, v are nondimensionalized by propeller radius R, by RU, 

 p by u2 and written p, Ap by 2n\j^ and written Ap, (Bp/Bn) by 2ttU^/r and 

 written s(r,0). 



Also, 



M = r2 + \2 + (r0\^)2 , 



N^ = TV + T* \. sin ip - — (v - X*) \^(t - T* cos i//) 



A. 



^= - -(l-X*) - 0* + Cq 



R = [1^2 + J.2 + r*2 - 2rr* cos i//] 



Likewise, Eq. (A6) becomes 



p(x*,r*,5*) = - y J ff S(r,,'^) ^^!? -^d^di 



2 f^i J.J. «$ 



+ [fAp(r,0) — deidr I 



(A9) 



where 



N= r (x* - 6k) + T* k sin<I)+ r0\j.(r-r* cos <P) , 

 R^ = [(X*- 5\)2 + r2 + r*2 - 2rr* cos (D] *^^ , 



^ = - e* + a . 



Now u^(x*, r*, 0*) = 30/dx*, u,(x*, r*, 0*) = B^/^r*, and u^(x*, r% 0*) = 



(l/r*)(B0/B0*), hence the induced velocities nondimensionalized by u are 



u,(x*,r*,0*)= -^ E i- /J S(r,^) 



r 



■t'x 



vdv 



Mi''2 d^dr 



(AlO) 



./jAp(..)r/ ''(^-^^)- 



d0dr 



953 



