w 



w R (x R ) 



(x, y, z) 



Y T (x c ) 



Z 



a 



a(x,.,x p ) 



r<x D ) 



7(x r ,x R ) 



7*(xJ 



6 =tan-' -£- 



V*R> 



6 (x c ,x R ) 

 A = (o,7) 



Even perturbation velocity 

 component due to thickness 



Velocity induced by vortex 

 filament 



Local wake fraction 



Radial free stream velocity 

 component, fraction of V 



Cartesian coordinates 



Cartesian coordinate for field point 

 on blade surface 



Fraction of chord, measured from 

 leading edge 



Fraction of chord for field point 

 on blade surface 



Hub radius, fraction of tip radius 



Fraction of radius, measured from 

 axis of rotation 



Radial coordinate for field point 

 on blade surface 



Nondimensional thickness offset; 

 maximum Y-r- = 0.5 



Number of blades 



Angular variable in chordwise 

 direction 



Component of derivative of 

 surface coordinate 



_i Jd -wj 

 ! = tan — Advance angle of blade section 



Circulation distribution 



Chordwise component of disturb- 

 ance velocity difference across 

 blade section 



Chordwise velocity difference 

 scaled to give unit magnitude when 

 integrated across the chord 



Error bound; Increment to pitch 

 angle when radial inflow exists 



Integration variable along vortex 

 filament 



Angular coordinate in cylindrical 

 reference frame 



9 b = 27r(b - 1)/Z Angular coordinate of blade- 



reference line of b'* 1 blade 



Skew angle; circumferential dis- 

 placement of blade-section mid- 

 chord point from y = plane 



Angular coordinate of point on 

 blade-reference surface 



Vorticity vector 



M(x c ,x R ) 



P 



o(x c , x R ) 



</>p (x R ) = tan" 



« g (x R ) 

 Hx c ) 



CO 



INTRODUCTION 



(P/D) 



Normal component of disturbance 

 velocity difference across blade 

 section (source strength due to 

 thickness) 



Helical coordinates on pitch 

 reference surface 



Fluid density 



Component of disturbance velocity 

 difference across blade section 



Surface area 



Potential function for perturbation 

 velocity; polar coordinate for field 

 point 



Pitch angle of blade reference 

 surface; measured on cylinder of 

 radius r 



Geometric pitch angle 



Radius of streamline on blade 

 surface 



Angular variable in radial direction 



The design of an open marine propulsor is a complex 

 process, involving structural and hydrodynamic considera- 

 tions (1,2). For the hydrodynamic considerations during most 

 of the preliminary design process, approximate models of the 

 lifting surfaces are employed, e.g., the lifting-line model (3, 4) 

 for powering considerations, and two-dimensional flow over 

 equivalent blade sections for cavitation performance. More 

 sophisticated models of the lifting surfaces are used for pre- 

 dicting fluctuating loads (5) and some cavitation predic- 

 tions (6). These approximate models have been acceptable 

 during the preliminary design process and provide a basis for 

 choice of the maximum diameter, advance coefficient and 

 radial variations of chord, skew-angle, rake, thickness, and 

 circulation distribution. The chordwise variation in load has 

 usually been selected during this preliminary stage and is 

 often based on cavitation and propulsion considerations. 



For the final stage of the design, the meanline distribu- 

 tion and radial pitch variation are determined corresponding 

 to the selections for load and geometry already available. To 

 derive a geometry which accurately produces the specified 

 load distributions, a lifting-surface model of the blades is 

 required. 



Several procedures already exist for performing 

 lifting-surface calculations for wide-bladed open marine pro- 

 pulsors. In particular, two different approaches to the 

 analysis for blades with arbitrary locations in space have been 

 presented by Kerwin (7) and McMahon (8). Kerwin's numeri- 

 cal analysis procedure is based on three fundamental assump- 

 tions: (1) that the continuous loading distribution on the 

 nonplanar blade surface can be adequately approximated by 

 a multitude of discrete straight lines of constant-vortex 

 strength and that the source distribution arising from the 

 thickness distribution can be similarly approximated, (2) that 

 the minimum required spacing between lattice elements along 

 the chordline is Ad = 2 degrees, and (3) that the resulting 

 meanline shape for a given chordwise load is similar to the 

 two-dimensional shape for the same chordwise load. The 

 first two assumptions are not acceptable for very narrow 

 blades: for a blade with a 20 degree pitch angle at the 0.9 



