and arises solely from the trailing-vortex sheets. As shown in Appendix B, 

 this is also the velocity field due to a sink disk of strength 



•'r X 



The method of Kerwin, originally developed to compute all harmonics 

 of the induced velocity, is based on a direct numerical integration of 

 equations (21) and (22) . The continuous-vortex sheet is divided into a set 

 of discrete helical vortex lines of constant strength. For computational 

 efficiency, the integration interval is divided into segments of increasing 

 size with distance from the field point and evaluated using a five-point 

 Gauss integration formula. The velocity is evaluated at a selected number 

 of angular positions between two blades and resolved into blade frequency 

 harmonics by Fourier analysis. Only the zero harmonic is used in the 

 present application. As an example of the numerical accuracy of this method, 

 a comparison between the computed and exact solutions of the axial velocity 

 at the propeller disk plane for two selected loading distributions are shown 

 in Figure 3. The velocity computed at the lifting line is also shown for 

 comparison with the circumferential average. 



At distances greater than about one radius, the propeller disturbance 

 velocity is generated essentially by the lifting-line sink disk. In order 

 to more accurately derive the near-field influence of the propeller, the 

 lifting-surface representation must be applied. 



LIFTING-SURFACE CORRECTIONS 



Propeller lifting-surface analyses have been developed as a logical . 

 extension of thin planar hydrofoil theory. Thus, blade loading and thick- 

 ness are represented in terms of vorticity and source-sink distributions 

 located on a reference surface approximating the actual blade. This is 

 normally taken to be the pitch surface derived from lifting-line calcula- 

 tions. Within the blade outline a distribution of bound and free vorticity 

 is established to represent both the radial and chordwise load variation. 

 A free-vortex sheet, previously assumed to be shed from the lifting line, 

 now originates from the blade trailing edge. 



17 



