radius and a chord to diameter ratio of 0.05, the 2 degree 

 spacing equals increments of about 1/3 chord length. The 

 last assumption permits calculations to be performed using 

 only a few points along the chord and the two-dimensional 

 shape is fitted to the data at these points. The resulting 

 computer code is relatively quick running and produces a 

 geometry which, in practice, has an overall speed and power- 

 ing performance generally within a few percent or so of the 

 predicted values, with a general tendency to produce a 

 greater thrust than predicted. The procedure of McMahon 

 employs continuous distributions for the loading and thick- 

 ness functions and calculates the meanline from the induced 

 velocity. Consequently, data at more chordwise points are 

 required to define the pitch and meanline distributions. 

 The resulting computer code is lengthy to run but has shown 

 remarkably different meanline shapes from the two-dimen- 

 sional one at the hub and tip region of the blade where the 

 meanlines can be s-shaped (8). Two models were constructed 

 and experimentally evaluated to provide data on the relative 

 cavitation and propulsion performance of designs having the 

 same input specifications but final geometry according to 

 the Kerwin and McMahon procedures. Some inconsistencies 

 occurred in the experimental measurements but the thrust 

 was closer to the predicted value and the operating point 

 centered in the cavitation bucket for the model designed by 

 the McMahon method. Hence, the determination of specific 

 meanline and pitch distributions, instead of fitting the two- 

 dimensional meanline, is considered to be a superior pro- 

 cedure when the design is based on a narrow range of permis- 

 sible operating conditions and the delay of cavitation is 

 critical. 



Because the numerical-analysis procedure employed 

 by McMahon results in lengthy computer runs and Kerwin's 

 procedure is not acceptable for narrow blades, alternative 

 numerical-analysis schemes are investigated in this paper. 

 In addition, a detailed description of the flow field across 

 the blade surface was desired as input into boundary-layer 

 calculations. Two different numerical analysis schemes are 

 described, each involving an expansion of the singular kernel 

 about the singular point. Both approaches employ integra- 

 tion of the specified thickness slope and load distribution 

 over the reference blade in the radial direction first and the 

 remaining chordwise integration then takes the form of the 

 velocity component corresponding to two-dimensional flow 

 modified by the presence of an induction factor in the 

 integral. Regular integration techniques are employed for 

 the other blades and the shed vortex sheet. The induced 

 velocity components are appropriately combined and inte- 

 grated to obtain meanline shapes. 



The present investigation describes the real-fluid 

 flow about a rotating system of lifting surfaces having both 

 loading and thickness. Several approximations are made. 

 The first of these is the mathematical model for which 

 potential flow equations are employed and the solution to 

 first-order in thickness-to-chord ratio, camber-to-chord ratio 

 and difference in pitch and flow angles derived. Compari- 

 sons with experimental results for other lifting-surface 

 configurations lead to confidence in this linearized approx- 

 imation. In addition to this mathematical model, further 

 approximations occur in the numerical analysis. Confidence 

 in the numerical analysis procedures is justified by compar- 

 ison with analytical solutions or experimental results. That 

 is, results are sought from some discretized numerical- 

 analysis procedures involving N by M approximations, which 

 have converged to within some specified tolerance, e, of the 

 real or analytical value of the quantity investigated. Math- 

 ematically this may be stated 



lf(x,y)-f NM (x,y)| < e 



J(x, y) on the surface S 

 N>N„ 



where f N M = the approximate calculation of a particular 

 quantity f 



S = a region of the surface of interest 



N M Q = minimum numbers of the discrete approx- 

 imations for which the computed results are 

 within e of the values for f 



For rotating lifting surfaces, neither measured nor analytical 

 solutions exist for details of the flow field on the blade. 

 Hence, comparisons will be made with other procedures. 

 It is assumed that numerical solutions which employ in- 

 creasingly greater pointwise definition of the input variable 

 without change in computed values have converged and that 

 the solution has converged when a smooth curve can be drawn 

 through point values in both the chordwise and radial direc- 

 tions. These assumptions are believed to be necessary but 

 not sufficient for convergence. 



In the following sections, the mathematical model 

 of the flow field on the blade surface is first reviewed and 

 numerical-analysis techniques for evaluating both regular and 

 singular integrals are described. A FORTRAN computer code 

 is discussed and sample calculations using this code are pre- 

 sented. From example calculations, it is found that greater 

 accuracy in the integral evaluations is required for the deter- 

 mination of smooth pressure distribution curves than for the 

 shape of the meanline and the pitch distributions. The choice 

 of a particular chordwise loading distribution is shown to 

 have an effect on the meanline shape and the pressure dis- 

 tribution. The effects of rake and skew are shown to be 

 important on both pressure distribution and meanline shape. 

 A particular thickness function has hardly any effect on pitch 

 or meanline but a significant effect on pressure distribution. 



MATHEMATICAL MODEL - THICK LIFTING BLADE 



The mathematical model of a system of rotating lifting 

 surfaces advancing in an unbounded irrotational flow field 

 with an inviscid fluid has been developed on a formal mathe- 

 matical basis by Brockett (9). A reformulation of that analysis 

 in terms of non-dimensional surface coordinates is presented 

 herein for completeness. The propulsor is assumed to be ade- 

 quately represented by the blades alone, i.e., neither the hub 

 nor fillet from the blades to the hub is included in the blade 

 specification. The onset flow is assumed to be directed along 

 the axis of rotation but a new feature included herein is that 

 it may have a small radial component. Overall geometry nota- 

 tion generally follows the definitions given in Reference 10. 



Coordinate systems are constructed with the same 

 orientation as in Reference 9, and in particular, the helical 

 coordinate system (£j , £2. f ) rotating with the blades is 

 shown in Figure 1 . Unit base vectors in a right-handed 

 Cartesian reference frame are the customary ( i, j, k) where i 

 is along the x axis and is positive pointing aft, j^ is along the 

 y axis and k is along the z axis which is generally along the 

 reference blade. Unit vectors along the helical coordinates are 



e j = sin <t>f i + cos 0p eg 

 e 2 = - cos 0p i_ + sin 4>p j>0 

 e r = - sin 6 j + cos 6 k 



e fl = - cos 6 ) - sin 6 k 



(1) 

 (2) 

 (3) 



(4) 



