In order to perform calculations, the form of 7* and 

 non-dimensional thickness must also be specified. A general 

 family of loading functions has been selected (18) with the 

 property that they have zero values at the leading and trail- 

 ing edges and resemble conventional NACA loading func- 

 tions ( 1 9). The zero values at the ends are necessary for accu- 

 rate fit with a sine-series trigonometric interpolation poly- 

 nomial. For loading distributions which approximate the 

 NACA a series riieanlines. the following chordwise form is 

 used 



(sin a) 



1/K 0<x c <0.5 



05 <x c <b, 0.5 <b< 1.0 



b < x„ < 1 .0 



(112) 



x = — ( 1 - cos a) 



The value of K can be taken sufficiently large to make the 

 load distribution in the leading-edge region nearly rectangu- 

 lar. A previous investigation ( 18) of this loading function for 

 K=8 and b = 0.7 demonstrated that it was an acceptable 

 approximation of the NACA a = 0.8 meanline, see Figure 2. 

 Symmetrical chordwise loading functions were selected to 

 be 



(sin a) 



1/K 



< x „ < 1 



(113) 



Each selected load distribution must be integrated across the 

 chord and scaled to produce a unit value for the integral. 



[ 1 



NACA 



1 1 



1 1 



1 1 



•0.8 



" 



/"- 







\v 





C L 



■ £81 









- 











- 



- 











- 



1 1 









1 



V 



X FRACTION OF CHORD 



Fig. 2 Load distribution 

 The thickness offset is assumed in the form 



L) 



— (x n ) • Y T (xJ 



(114) 



where the chordwise distribution, Y T (x c ), remains the same 

 from root to tip, only the maximum value changes with 

 radius. This is true for current propulsor designs. Specific- 

 examples of the thickness function included in the computer 

 code are the NACA 4 and 5 digit sections (19), the NACA 1 6 

 section (20), an elliptic nose quartic tail section similar to 

 that described in Reference 21, and an approximate NACA 

 06 (Mod) (22) section. All have been analytically defined. 



Computer Code Convergence and Run Time 



The complexity of the numerical analysis is such that 

 error estimates are difficult to establish. A few limiting cases 

 exist for which analytic integrations may be performed but 

 comparisons do not usually evaluate the general case. The 

 procedure selected to evaluate convergence was to vary the 

 number of intervals in the radial direction, NR, and the 

 number of intervals in the chordwise direction, NX. In addi- 

 tion, some radii and chordwise points were eliminated from 

 the calculations. Computed values of the pitch and camber 

 at selected radii are shown in Table 1, together with a similar 

 variation of data calculated according to the procedure 

 described in Reference 7 for the same propulsor which is 

 similar to NSRDC Model 4498. Computer central processing 

 time is for computations at 1 3 radii between the extreme 

 radii listed and is in seconds for the Burroughs 7700 High- 

 Speed Computer. Current charges are 3 cents per CPU second, 

 resulting in a maximum charge of about $40. All procedures 

 presented produce about equally satisfactory results with 

 about only one percent difference in pitch or camber values, 

 about the same as found for Kerwin's numerical analysis. 

 The computed pitch, however, is a few percent less than 

 computed by Kerwin's method. Since unpublished experi- 

 ence at DTNSRDC to date has been that Kerwin's procedure 

 produces designs that are generally slightly overpitched. 

 perhaps some improvement in performance may be expected 

 using the present method. 



Predictions of the pitch and camber by the two 

 procedures developed for computing the induced velocity 

 field on the blade surface which contains the field point, 

 "direct" and "approximate-plus-difference." are shown in 

 Table I to be nearly the same. However, it has been found 

 that overall, the "approximate-plus-difference" procedure is 

 preferable when dense chordwise spacing is chosen (e.g., 

 NX= 19) or narrow blades (maximum c/D * 0.05) are 

 involved. In these situations, the "direct" procedure produces 

 locally erratic values of the induced velocity because of the 

 decreased spacing between adjacent lines of integration with a 

 corresponding lack of accuracy in the numerical integrations 

 for the resulting near-singular integrals. This effect is illus- 

 trated in Figure 3 which shows values of one of the helical 

 components of the average induced velocity at the 0.946 

 radius of the reference blade. This velocity component is due 

 to only loading on the blade itself; the effects of thickness, 

 the other blades and the shed vortex sheet arc not included. 

 All data shown in subsequent figures have been computed by 

 the "approximate-plus-difference" procedure although only 

 the pressure distribution near the leading edge in the tip 

 region of the blade was significantly different between the 

 two procedures. 



Overall run time varies with number of points, number 

 of blades, and blade width. Since computer usage charges are 

 so low, the 181x19 array size is recommended. For a narrow 

 blade, the linear "approximation-plus-difference" procedure is 

 recommended, and the run time may increase by a few hun- 

 dred seconds because of special care taken with the shed 

 vortex sheet calculations. Computer execution time for 

 Kerwin's program is unknown for the Burroughs 7700 high- 

 speed computer but is estimated to be about 1 50 seconds, 

 for data calculations at 4 chordwise points at 8 radial stations. 

 For the results shown in Table I, data are computed at 1 3 

 radial stations with either 8. 1 1 or 17 chordwise points, 

 depending on input data specification. 



Further details of the geometry of this example are 

 given in Table II. Radial variables are titled according to the 

 symbols suggested in Reference 10. 



13 



