for subcavitating propellers, the pitch angle $ . of the blade-reference surface is 

 taken as the hydrodynamic advance angle given by the preliminary design calculations. 

 Then, were the source distributions known, all the induced velocities could be 

 computed, and the blade-cavity shape and the final pitch could be obtained in a 

 manner similar to that used for a subcavitating propeller. 



Two problems, strength and the location, are related to the source distribu- 

 tions. The location may be considered to be on the blade reference surface. But, 

 how far downstream should the sources be distributed? A two-dimensional supercavi- 

 tating cascade section could be considered to contribute to the source distribution. 

 When the Riabouchinsky cavity model is used, the cavity source will be confined 



inside a finite cavity domain. Thus the cavity domain supplied by two-dimensional 



12-14 

 theory could be used in an attempt to solve the source strength distribution. 



If the linear double spiral vortex model is used, the source distribution does not 

 terminate at the cavity end, but the wake source continues from the end of the 

 cavity. Although there exists a logarithmic singularity in the normal velocity at 

 the cavity end, the streamline represented by the integration of the normal velocity 

 continues smoothly at the cavity end. Any cavity mode is known to give reasonable 

 predictions of lift and drag. Therefore, it may be better to free ourselves from the 

 concept of a finite cavity platform and to consider the cavity wake-source distribu- 

 tion as beginning behind the leading edge. The major question is where to truncate 

 and how to reduce the truncation error. 



Even if the problem of the source domain were solved, the method of solution 

 would be far from definite because there is no established method to solve the 

 Fredholm singular integral equation of the first kind; there are suggestions, how- 

 ever, that a series of eigenf unctions could be used. * Thus solutions to the three- 

 dimensional problem of a supercavitating foil are complicated and time consuming. 



To arrive at a reasonable method to solve the present problem, we considered a 

 combination of three-dimensional corrections to a two-dimensional solution and the 

 series of eigenf unctions. That is, the two-dimensional solution for the supercavi- 

 tating cascade is multiplied by a double polynomial function with unknown coeffi- 

 cients 



J J J 



f (p,4>) = y a ±± p 1-1 + 5J y^ a... p 1_1 x j_1 in the blade plane (31) 

 i=l 1-1 j = 2 



12 



