Pien and Strom-Tejsen 



Since no singularity exists except at the blade surface, the complexity of a pro- 

 peller problem is greatly reduced. 



It may be noted that the concept of the acceleration potential may be used 

 for super cavitating propeller problems. A cavity can be taken as the blade 

 thickness as far as viewed from the fluid, but the geometry of the cavity is un- 

 known before the problem is solved. Hence we cannot specify the simple source 

 distribution in Eq. (12) a priori if we base our analysis on the velocity potential. 

 This situation makes the problem extremely difficult. However, if the accelera- 

 tion potential is used in our analysis, the pressure- source distribution in the 

 first term of Eq. (13) can be determined from the blade loading and the cavita- 

 tion number a-. This is shown as follows, beginning with 



. ..-,^c. -,„.,. ..,,, .. . P" = Po - ^Ap , (14) 



where p" is the pressure on the upper or suction side of the blade, Pq is the 

 ambient pressure in the absence of the blade, and Ap is the pressure jump 

 across the boundary or the lift. The cavity pressure p^, is 



_,_-,_ ... ' Pc = Po - |pf -v^ , ^ ^ (15) 



where v is the speed used in defining the cavitation number a. Subtracting 

 Eq. (14) from Eq. (15) we have 



F = |-(Ap- PfCrV^) , (16) 



where F is the required additional pressure distribution to make the pressure 

 on the suction side equal to the cavity pressure. 



It has been observed that the curvature of the cavity wall may be very large 

 at the leading edge, depending upon the angle of attack. However, the curvatures 

 are small near the trailing edge and become large again near the end of the 

 cavity. Since there is no pressure discontinuity across the cavity wall, the 

 singularity distribution required to represent that portion of the cavity which is 

 trailing the blade is small to begin with and becomes appreciable at the end of 

 the cavity. It has been found that various models of the cavity closure condition 

 does not affect the loading significantly if the cavity is sufficiently long. There- 

 fore, at least as a first approximation, we may ignore the singularity of the ac- 

 celeration potential beyond the blade surface and consider the blade surface as 

 the only boundary where the pressure dipole Ap and the pressure source p^ are 

 distributed. The pressure dipole is derived directly from the specified blade- 

 load distribution. The pressure- source distribution is obtained by solving the 

 integral equation 



92 



