Supercavitating Propeller Theory 



where 



N = r(x*-i9X.) + \jr* sin ^ + xd\-^ (r- r* cos (p) 

 p(e*\t- e,r*,6'*) = - - , ^lCO ^ ^* ^ ^t(0 ' q = 1 



p(0*X=J+ e,r*,0*) = - - , e^(r) < e* < ^^Cr) , q = 1 



and consideration of e is similar to that already discussed. Considerable 

 care has to be exercised when Eqs. (2), (3), and (7) are evaluated numerically 

 for the reference blade, since the integrands become singular for r = r*, d = 

 6* [9,10,11J. 



In order to use the lifting- surface equations (2), (3), and (7) for the purpose 

 of design, i.e., to determine the blade face shape, it is assumed that A.j(r) and 

 y{v,e) are known. After solving Eq. (7) for s(r,6' ), the axial and tangential in- 

 duced velocity components can be computed at desired positions (r*, d*) on the 

 blade, using Eqs. (2) and (3). Hence, the normal induced velocity components 

 u^(r*, 0*) can be obtained using 



r*u (r*,e*) - X.(r*)u.(r*,^*) 



Integration of the normal induced velocity components in a chordwise direction 

 will give the blade face shape at any desired radial position r* relative to the 

 helical line pitched at angle /3i(r*) = tan"^ {k\ /r*). 



In the solution of Eq. (7), s(r, 0) should be represented by suitable modes 

 possessing unknown coefficients, so that the problem reduces to the determina- 

 tion of these unknown coefficients. Attention is directed to the work of Widnall 

 which includes a useful discussion of the problem [9], including the influence 

 of various cavity closure conditions. As regards the prediction of lift force on 

 a supercavitating hydrofoil, Widnall concludes that an approximate representa- 

 tion is adequate for s (r, 6^) and that the cavity closure condition and closure 

 location is not important, provided that the cavity is sufficiently long. Parkin's 

 work [13] on linearized two-dimensional supercavitating hydrofoils operating at 

 nonzero cavitation number indicates exact chordwise modes for s(r,6') and 

 y{r,e), where the designer has the choice of either designing for an angle of 

 attack or shock-free entry with a prescribed cavity thickness at the trailing 

 edge. 



935 



