xv 



G( 



x R ) 7* 



(52) 



In Reference 9. the gradient of a scalar function is derived 

 for a general helical coordinate system. From this expression 

 one obtains: 



n: 



/c dG ( Xq . ,\ 



07 - e ' + 2 ("D^J o 7 d "c- aG T)?r 



(53) 



1/c dG (\ „ ,\ 1 ,^o X 

 ■■ — | I T dx - aG7 |e,+ — G7 



Ad dx R J o T c > * &.1 



In general, the trailing edge point will not be zero. 

 The trailing-edge offset can be converted into a pitch-angle 

 increment, 



tan (0 -0 p ) = - E c (1, x R )/(. 



0„ = P + tan 



E,. ( 1 , x „ ) 



D / \D, 



for which the pitch of the nose-tail line becomes 



= n x D tan 



7rx R tan0p + 7rx R (-E c ( I , x R )/c) 

 I - tan0 p (-E c (1, x R )/c) 



(61) 



(63) 



Hence from Equation (40): 



c_ _dG_ 

 D dx D 



7 dx - aG 7 



(54) 



(55) 



and meanline ordinates measured from the nose-tail line can 

 be determined: 



E c"-*K> 

 D 



(64) 



c COS' 0p 



D x„ 



(x -0.5) 



2 - dx K 



At the trailing edge and beyond 

 7* (D = 



+ sin 0p 



V dx„ 



(56) 



(57) 

 (58) 



These values of corrected pitch and meanline shapes relative 

 to the nose-tail line are the essential data produced by the 

 lifting-surface analysis. 



In addition to the meanline and pitch distributions, 

 several other quantities of interest can be computed. These 

 are pressure distribution, total forces, and streamline coordi- 

 nates (this coordinate system is described in the Appendix). 



From Reference 9, the pressure coefficient is 



P-Poo 



(65) 



V 2 



(66) 



Hence 



J_ c dG 

 T D~ dxT 



for the shed vortex wake. 



(59) 



Thus, the strength of the vortex distribution is explic- 

 itly known and the integration to determine the average in- 

 duced velocity at points on the blade surface can be undertaken. 



Once the induced velocity field on the blade is 

 computed, the meanline slope can be determined and the 

 meanline offset can be found by integrating the slope: 



E c (x c ' x R ) f Xc ° 3 



E„ (x.,x R ) 



(60) 



9x„ 



From Equation (22), the meanline offset for the term with 

 the radial inflow velocity component can be directly com- 

 puted; it consists of an angle of attack term due to gradient 

 of the rake and skew terms and a parabolic arc meanline 

 due to gradients of the pitch. 



An approximation for the non-linear speed on the surface 

 of the blade in the chordwise direction is ( 1 3) 



V<l-w x ) 2 + h^) + 



\ 3x 



E T /c)V 



(67) 



Hence 



$ = {^(- 



(68) 



where e is the unit vector in the Nt xej direction (nearly 

 the radial direction over much of the blade): 



n! X 



(69) 



|N+Xe,l Vl+NR 5 " 



