In Equations (67) and (68). the velocity components v • e j 

 and v • e are 



(70) 



v < v >\ y 



Vl + N? 



(71) 



Each of the components v and < y > has terms due to 

 thickness and terms due to loading. 



To first order, the pressure coefficient is 



-^-v-^iv-) 



Hence the pressure difference to first order is 



■■'Soffit 



j /irx R \ 2 7*(x c ) 



(72) 



(73) 



(74) 



which is independent of rake, skew, or pitch variations (to 

 first order). 



Once the pressure distribution on the blade surface 

 is known, the thrust and torque in inviscid flow may be cal- 

 culated (9). In terms of a thrust and power coefficient, these 

 inviscid overall performance values are: 



1 v2 V 



-jpv 2 *— 



(75) 



2ZT 1 f 1 



dx. AC„( — cos (4 D + 



' p 3x, 



P\D 



sin P 



3x„ 



27rnQ: 



1 \/3 ® 



(76) 



(77) 



2 zf' r 1 r a 



— x D dx p dx„ AC — ; 



E, 



' p 9x„ 



(78) 



where AC p = C" - C* 



and C p = C p + C p 



The effects of pitch, skew and rake enter into the determina- 

 tion of pressure and are not explicitly in the integrands for 

 thrust and torque. 



A correction for viscous drag may be made by assum- 

 ing the drag force acts along the pitch line. For a given drag 

 coefficient Cp, the force will be approximately 



D f = T pv2 [ n " w - )2+ ("T^)]d DCd 



(79) 



When the effects of this force are included in the thrust and 

 power coefficients, one obtains 



C Th" C Th. + AC Th 



C D = C D + AC D 



(80) 

 (81) 



=-f[c D r(i-w x ) 2 + (-^)]§si„0pd XR 



(82) 

 "J" I x R C D[ n - w x» 2+ (— )J^ cos *P dx R 



(83) 



Hence the thrust will be reduced and the torque increased 

 relative to the inviscid values, as expected. 



In the Appendix, the formulation for a streamline 

 coordinate system on the blade surface is given. 



NUMERICAL ANALYSIS PROCEDURE 



The computation of the meanline slope relative to 

 the blade-reference surface (Equation 22) involves some 

 straightforward computations of geometry gradients, which 

 are handled by spline functions, to determine N^ plus the 

 evaluation of the even velocity component across the blade 

 surface (Equation 30). This even velocity component arises 

 from the odd velocity component (Equation 37) integrated 

 over the blade surfaces and trailing vortex sheets. The mag- 

 nitude of the odd velocity component is known from 

 Equations (40), (50) and (55). A singularity occurs on the 

 reference blade and special procedures must be employed on 

 that blade. Integration over the other blades can be based 

 on conventional integration procedures. Integration over the 

 infinite shed vortex sheet is truncated to a finite extent. 

 Once the slope is known, it can be integrated across the chord 

 to produce the meanline offset (Equation 60). 



For regular integration over the span or chord of the 

 blade, trigonometric polynomials are employed which pre- 

 cisely fit a set of tabulated values given at equal angular 

 increments in a or oj where 



1 

 — ( 1 - cos a) , < a < n 



(84) 



( 1 - X h ) COS CO 1 , < CO < 7T 



