propeller. It is of interest to point out that the cascade corrections' 14 ' 15 ' were originally used in 

 subcavitating propeller design methods before the lifting-surface theory was fully developed. Both 

 theoretical and experimental data for supercavitating cascades which would be useful for the design 

 of supercavitating propellers are very scarce. The cascade section theory and its associated computer 

 program developed by Yim' 16 " 21 ' are used here to aid in designing blade shape. 



It must be recognized, however, that a supercavitating cascade is not a perfect model for blade- 

 section design principally because of the disparity of the cavitation numbers between the cascade and 

 the propeller' 4 '. That is, in a cascade, the cavitation number can never be smaller than the infinite 

 cavity cavitation number o^ which is always larger than zero, while a ventilating propeller has zero 

 cavitation number. Even for supercavitating propellers, the cavitation number a near the blade tip, 

 in general, is smaller than a M . Yet, the drag-lift ratio of a meaningful supercavitating foil, influenced 

 by neighboring cavities, is considered to be properly analyzed in supercavitating cascade theory, and 

 approaches the drag-lift ratio of a blade in the infinite medium near the blade tip. In addition, the 

 cascade effect is very sensitive to the shock-free entry angle and to the relation between the leading- 

 edge cavity thickness and the cavitation number' 21 '. These also seem to be important blade cavity 

 interferences. The effect of the cavitation number should be considered again in the final design' 22 ' 



SUPERCAVITATING CASCADE THEORY 



A two-dimensional supercavitating cascade theory is applied to each blade section. Each section 

 has a different hydrodynamic pitch angle P y and blade chord length c. The distance d between the 

 neighboring leading-edges of propeller blades at a blade section is 



-=(2*r)/(Zc) (1) 



c 



where r is the radial coordinate and Z is the number of blades. This ratio, d/c represents the solidity 

 of the cascade at the blade section. The stagger angle is 



7=f-0i (2) 



The parameters d/c and 7 are typical of cascades. The angle )3j is not known initially. Thus the 

 geometrical advance angle (3 is used first and then the approximate j3j is used for the second iteration. 



In using supercavitating cascade theory there are two ways to select blade shapes: one is from 

 a given mode of chordwise pressure distributions' 16 " 20 ' and the other is from a given mode of foil 

 shapes' 21 '. The present program can handle either of the two approaches although the pressure mode 

 is restricted l 16 - 18 ' to special mathematical forms. The details of cascade theory and the associated 

 program are explained in References 17, 18, and 20 for the pressure mode and in Reference 21 for 

 the foil mode. In practice, the foil mode is better for designing a supercavitating propeller. 



It is well known that the thrust and power coefficients of propellers are influenced by the 

 viscous drag/lift ratio e as shown by the following equations' 23 '. 



C T = 4zA G(r) 



I -1 

 X V 



(1-w)-- 



(l-etan0j)dr (3) 



(1 +e/tan/3j) dr (4) 



