q = V ( 1 - w x (x R )) i_ + 27rn r e^ 

 + Vw R (x R ) e r + v 

 = q,,, + v 

 where V = the constant reference speed 



(12) 

 (13) 



1-w = the wake-fraction multiple to obtain the 

 local axisymmetric speed 1 



w R = the radial component of inflow, fraction of 

 the reference speed 



r. = the rotational speed, revolutions per unit 

 time, and 



v = the velocity component due to the presence 

 of the blades 



If 0p is the pitch angle and (3 is the advance angle 



(0 = tan _1 [ V(l -w x )/(2irnr)]) 

 then 



V(l-w x ) 2 + ^~j 



(14) 

 • {cos (0 p - 0) e i + sin (0 p - 0) e 2 } + w R e f 



where the advance coefficient, J v , is given by 



J v = V/(nD) (15) 



The boundary condition on the blade is that there be no flow 

 through the surface: 



q • N = for r_ on S (16) 



This condition applies to both the upper and lower surfaces: 



q + • N + = 

 q~ • N~ = 



The sum of Equations ( 17) and (18) is: 



q • (N + + N~) + y_ + • N + + v" • 



Now 



3E T /D 

 q= • (N + + N _ )=-D 2 — - (q 



(17) 

 (18) 



3x„ 



+ (E, w R ) 

 thus to first order in w R or E: 



(19) 



N + + v" 



N" =D 2 V\(l-w x ) 2 +f y^j 

 • cos (<t>p - 0) — 



»N+ • (v + - 



(20) 



1 As described in Reference 9, the inclusion of a 

 radially variable inflow introduces vorticity into the mathe- 

 matical model. No specific consideration is undertaken to 

 account for this vorticity. 



The difference of Equations ( 1 7) and ( 1 8) gives: 



v + • N + - vf • N" = -q«. • (N + - N") 



= -qoo-[2N + -D 2 _p_e, + 0(E)e r ] 



N + -D 2 y- -q, 



+ CHE, w p ) 



■ D 2 V 



K 1 - w % )' + 



^x R 



J. 



BE /D 



-0) 



3x . 

 + 0(E,w R ) 



* K ■ (v + - v") 



cos (0 p - (3 



+ D W R N R 



(21) 



3x„ 



tan(0 p -0) (22) 



N* • (v + + v-)/(D 2 Vc/D) + w R N Ro ~ 



COS (0p 



-/3)\(i-w x ) 2 +(y-^) 



This is the fundamental equation to be evaluated to determine 

 the pitch and meanline shape. The slope of the meanline is 

 given as a function of known geometry and inflow quantities 

 plus the normal component of the average induced velocity 

 on the blade surface. Hence to determine the meanline slope, 

 the mean perturbation velocity must be determined on the 

 blade surface. 



The velocity component due to the disturbance of the 

 blades, v, can be shown to be potential in nature (9). Hence, 

 it can be represented by an equation (9, 1 2) 



v(r) 



z 



v. = l -*4, 



b = l 



+ (n X v) X' 



lr - s h |- 



(23) 



where n is a unit normal pointing into the fluid from a point 

 s^ on the surface S (including both the blades and shed vortex 

 sheet), and do. is the area element. 



In several texts (11, 1 2), it is shown that 

 n do s = N dx c dx R 



(24) 



In Equation (23), it is convenient to let the region of integra- 

 tion be the blade reference surface £ 2 = 0, < x c < 1 , 



x h < X R " ' 



Xu < x D < 1, for which 



