OPTIMUM DISTRIBUTION OF HYDRODYNAMIC PITCH 



The optimum distribution of hydrodynamic pitch for a supercavitating propeller was recently 

 found' 23 '. The optimum-pitch relation can be adopted as an option in the present design program. 

 Namely, instead of using 



(r/X) tan ft = c, (6) 



the corresponding new relation can be used' 23 ', 



(r/X) tanflj = {c, - (e + G ^ ^/{l + c, (e + G j^j X/rj^ (7) 



where e(G) is the drag-lift ratio derived from supercavitating cascade theory as a function of 

 circulation G, and c l is a constant which allows a specified thrust requirement to be met. 



INFLOW RETARDATION 



The performance of propellers is highly dependent on the advance coefficient J = V/(DN), where 

 D is the diameter and N is revolutions per unit time, or i/ir = X. It is well known that if J becomes 

 small, the angle of attack with respect to the inflow velocity grows, and accordingly, the cavity thick- 

 ness increases. Thus, investigations have found that the velocity retardation in front of the super- 

 cavitating propeller results in a considerable change in performance' 7 ' 25 ' 26 '. This can be a serious 

 problem, due to the local perturbation velocities caused by three-dimensional cavity sources in a 

 rotating helical surface of the propeller. This effect is qualitatively different from that of the two- 

 dimensional cascade approximation of blade sections' 25 '. The former is a near-field effect with 

 respect to the outer flow of a propeller and the latter is purely an inner flow effect without consider- 

 ation of the outer flow. An axisymmetric model was used' 25 ' 26 ' to evaluate the flow retardation. 

 However, this can be more properly handled by the lifting surface theory of supercavitating 

 propellers' 22 '. 



COMPUTER PROGRAM 



A flow chart for the computer program is shown in Figure 1 . Essentially the computer program 

 is a combination of the lifting-line design program for subcavitating propellers and the appropriate 

 supercavitating cascade theory. After reading the inputs, such as the propeller geometry, design speed, 

 revolutions per minute, and average wake fraction, flow field calculations for a supercavitating cascade 

 having infinite cavity length at each section are performed and saved for later use. Then the first 

 approximation of the hydrodynamic pitch angle j3j, is used to calculate the radial lift distribution and 

 the section lift coefficients C L by lifting line theory' 10 '. For this calculation there are two options; 

 (1) tan ^j(r) is proportional to any given set of tan 0j(r) and (2) the optimum pitch relation, 

 Equation (7) is satisfied. Then the computed value of C L is used to calculate a/C L which is used to 

 find the cavity-length. Then, the infinite cavity drag-lift ratio is corrected, and the thrust (or power) 

 is computed and compared with the design thrust (power). If the value of the thrust is not within a 

 specified error band, of the design thrust, a new 0j is obtained according to the Newton-Raphson rule 

 to compute a new thrust as previously described. This process is iterated until the design thrust is 

 obtained; usually two or three iterations are required. The final outputs are the power (thrust) 

 coefficient, the efficiency, the nose-tail line angle of attack with respect to 0j, the pitch distribution, 



