where e includes the cavity drag effects; r is a nondimensional radial coordinate; r n < r < 1 with r n 

 the nondimensional hub radius; and X = V/(J2R) with ship speed V, angular speed J2, and propeller 

 radius R. 



Since e is very sensitive to section cascade parameters' 16,21 ', it has to be obtained from super- 

 cavitating cascade theory. In addition, e is, in general, a function of the cavitation number and the 

 section lift coefficient. Thus, for a different lift coefficient, e has a different value. However, the 

 lifting-line propeller design program does not supply the lift coefficient distribution as an input but 

 rather as an output. The lift coefficient distribution is obtained by iteration, as for subcavitating 

 propellers. Thus, as usual, all the physical quantities such as pressure p, velocity u, drag-lift ratio e, 

 cavitation number o, etc, are normalized by lift coefficient C L . Then even if C L changes, the actual 

 values can be obtained by multiplication of C L . However, a not o/C L is given as an input. Therefore 

 a/C L changes with C L while all the physical quantities normalized by C L are functions of a/C L , not a 

 alone' 21 '. In addition, even if a/C L were known, the cavity problem with finite cavity length should 

 be solved by iteration because'the geometry of the problem associated with the cavity length is not 

 known a priori. Thus the design of a supercavitating propeller involves double iterations. 



To circumvent this difficulty, the supercavitating cascade problem is divided into two parts' 21 ': 

 one is the problem of infinite cavity length and the other is that of finite cavity correction. Then, if 

 the solution for the infinite cavity length is obtained once at each blade section it can be used 

 repeatedly for each iteration for a different value of C L . Fortunately the finite cavity correction is 

 simple' 17,21 ' in a linear design theory where the load distribution is fixed, and can be readily computed 

 during each iteration for a different cavity length. 



In the problem for the infinite-cavity cascade' 21 ' the basic camber shape, such as two-term 

 camber in an infinite medium, is given as an input. Then the shock-free angle is found at each section. 

 The shock-free angle is very sensitive to cascade parameters' 21 '. To this basic camber, which has its 

 own shock-free angle, an angle of attack and a point drag are combined to meet the desired lift and 

 leading-edge cavity thickness. There are three options at this stage: (1) the amount of camber is 

 given (2) the amount of angle of attack is given, and (3) no point drag is given. The method of 

 hydrofoil airfoil correspondence' 16 ' and the Fast Fourier Transform Technique' 21 ' are used to 

 compute the drag-lift ratio, the normal velocity on the foil and cavity, the foil cavity shape, and the 

 pressure distribution. 



For the finite cavity correction, a simple superposition method' 17 ' is used for many different 

 values of cavity lengths assuming that the load distribution on the foil is exactly the same as the case 

 of infinite cavity. Therefore, the cavity drag, the angle of attack, and the cavity thickness for a given 

 load distribution decrease when the cavity length decreases, although for a fixed foil shape the 

 opposite results are obtained. However, by a proper correction, the minimum leading-edge cavity 

 thickness is maintained. The cavity length corresponding to a/C L is obtained by interpolation. 



The local cavitation number is smallest at the blade tip of the propeller; much larger at the hub 

 and is the function of camber shape, load distribution and the given leading-edge cavity thickness. 

 As shown in Reference 21, a/C L is a linear function of the leading-edge cavity thickness. Also, near 

 the hub ct/C l varies very little even when the cavity length changes' 21 '. That is, near the hub where 

 the solidity is large, the lift coefficient is proportional to a and almost independent of cavity length. 



Since o/C L varies considerably from the blade tip to the propeller hub, sometimes a/C h may be 

 too large to have a stable cavity near the hub. If the supercavitating propeller must have a stable 

 cavity at every section, the foil shape should be designed accordingly. Although the leading-edge 

 cavity thickness is prescribed such that enough blade thickness can be accommodated inside the 

 cavity, the thickness may have to be larger than the strength analysis requires in order to have a 

 stable cavity at least as long as 1 .5 chord lengths. When this situation applies, the leading-edge cavity 

 thickness is computed for the 1 .5-chord-length cavity length (see Appendix A), and the leading-edge 

 point drag' 18 ' is superposed to supply the leading-edge thickness without changing the load distribution. 



