velocity induced by the source outside of the sphere is canceled by the image source 

 system; however, outside the sphere, the velocity field due to the image system 

 decays as flow caused by a doublet inside the sphere. 



The image source distribution is multiplied by the same double polynomial as 

 Equations (31) and (32), and the hub boundary conditions are considered together with 

 the cavity boundary conditions, Equations (28) and (29), to solve the simultaneous 

 Equations (33), (34), and (36) by the least-squares method. The solution is checked 

 to see if it actually satisfies the hub boundary condition in addition to the cavity 

 boundary conditions. 



BLADE SECTION SHAPE 

 When the unknown coefficients are obtained from Equation (38) , the source 

 distribution is computed from Equations (31) and (33). Thus, all the induced veloc- 

 ities can be calculated. It is possible to obtain the section shape of the propeller 

 blade by integrating the velocity field along the blade reference surface. However, 

 this may require a large amount of velocity information. Thus, the method used by 

 Kerwin is followed to correct the foil shape derived from supercavitating cascade 

 theory in the preliminary design process. At a field point of each section the 

 normal velocity component V„ is obtained on the pressure surface of the blade from 

 the lifting-surface theory, and V~ is matched with the corresponding normal velocity 

 obtained from supercavitating cascade theory 



z 



a. x + c v„ = v„ (39) 



o 2n 3n 



where a. and c = unknown coefficients 

 1 o 



v„ = normal velocity component on a foil of a 



supercavitating cascade 



x = distance from the leading edge 



The unknown coefficients are determined by the least-squares method. That is, the 



coefficient c and the terms a. x are the correction to the normal velocity compo- 

 o 1 



nent of the two-dimensional supercavitating cascade caused by the effects of the 

 lifting surface, three-dimensional cavity, flow retardation, etc. For a 



17 



