(4) = velocity due to a distribution of sources 



In the method of Kerwln (available as a FORTRAN computer program 

 FPV) , the numerical evaluation is again based on discrete singularities. 

 On the blade surface, a grid of radial and helical lines is constructed to 

 form a lattice of source and vortex elements as Illustrated in Figure 4. 

 The elemental singularity strengths are determined as follows: 



1. Radial vortex elements are required to produce the desired chord- 

 wise load distribution and the known bound circulation r(r) at each radius. 



2. Helical vortex elements must satisfy conservation of vorticity. 



3. Source and sink elements are required to generate the same chord- 

 wise velocity distributions as the section thickness form would produce in 

 two-dimensional flow. Also the sum of sources and sinks must equal zero. 



As in the lifting-line analysis, the velocity is computed at a set of angu- 

 lar positions for each field point (x,r) and resolved into blade frequency 

 harmonics. 



It may be noted that radial vortex lines (bound circulation) do not 

 contribute to the steady axial and radial velocities and need only be 

 computed to determine the strength of the helical (free) vortices on the 

 blade. It is also straightforward to show that the steady velocity is 



% + ^^ 

 L T 

 k k 

 independent of the skew angle, , as would be expected physically. 



On the other hand, blade rake, the axial displacement of blade sections aft 

 of the lifting line plane, is of marked importance. This is manifested in 

 the free vorticity term (2) in equation (34) which "corrects" for the 

 starting position of the trailing-vortex sheets. These and other features 

 of the propeller calculations are best illustrated by considering specific 

 examples. 



EXAMPLE CALCULATIONS AND COMPARISON WITH EXPERIMENTS 

 The equations derived in the foregoing theoretical analyses have been 

 programmed for computer-aided numerical solution. The calculation is per- 

 formed by interfacing three separate programs: (1) the hull potential flow 

 solution (PF), (2) propeller field point velocities (FPV), and (3) the 



22 



