Super cavitating Propeller Theory 



i Cr \ ^2 



:i7T r (24) 



".(r^) 



"tC-*) 



— - 1 + 



A. 



(r*2 + \.2) 



— - 1 - - ]r*\. 

 k 2 



(r*2 + \.2) 



(25) 



The problem for numerical solution is to solve Eqs. (12) or (13) and (14) 

 simultaneously for r(r), assuming a known k^. This involves suitable repre- 

 sentation of r(r) and s(r,5) in terms of unknown coefficients, so that the equa- 

 tions can be expressed as a set of linear algebraic equations to be solved for the 

 unknown coefficients. The terms associated with the unknown coefficients are 

 definite integrals, which have to be suitably arranged or simplified to enable 

 their evaluation, see Appendix B. According toWidnall's supercavitating hydro- 

 foil calculations [9], foil force prediction is not very sensitive to precise ^^(r) 

 values or a particular cavity closure condition. In addition, a reasonably sim- 

 ple representation of s(r,t^) appeared to suffice. Hence, 0^ can be assumed 

 independent of radius and determined by use of a convenient closure condition 

 at one representative radius. It may even suffice to use an estimate of cavity 

 length based on Parkin's two-dimensional supercavitating foil theory [13], or 

 some such similar reference. 



A suitable representation of radial circulation r(r) is 



M 



r(x) = (l-x)i/2 2 A^x-^1 , (26) 



m=l 



where 



^ = r^^ (27) 



The pressure source distribution s(r,(9) can be represented by 



M 

 ni= I 



(28) 



M N 



V 1 ' I ■ / . mn ^ ' m 1 m' ^ n- 1 n ' ' 



where 



941 



