THE SOCIETY OF NAVAL ARCHITECTS AND MARINE ENGINEERS 

 One World Trade Center, Suite 13S9, New York, N.Y. 10048 



Lifting-Surface Hydrodynamics for Design of 

 Rotating Blades 



Terry Brockett, David Taylor Naval Ship Research and Development Center, Bethesda, MD 



No. 20 



ABSTRACT 



Analysis and numerical results are presented for the 

 design of a system of wide-bladed thin lifting surfaces rotat- 

 ing at constant angular velocity in an axisymmetric onset 

 flow field. Blade sections may be located arbitrarily in space. 

 General chordwise and spanwise loading functions are avail- 

 able as well as a variety of thickness forms. In addition to 

 the final meanline and pitch distribution determined from 

 a chordwise integration of an appropriate combination of 

 geometric variables and induced velocities, additional infor- 

 mation not avai'able from other existing techniques includes 

 pressure distributions and surface metrics for an orthogonal 

 streamline coordinate system on the blade surface, as defined 

 in the Appendix. The induced velocity field on the blades 

 is derived from the principal value of a singular integral; 

 the evaluation of this integral is discussed. The predictions 

 are generally supported by experimental data. A new term 

 in the analysis arises from a radial onset flow component 

 and an example illustrates its importance in design. Suf- 

 ficient information for manufacture is obtained for com- 

 puter run times from 400 to l 200 seconds on the Burroughs 

 7700 high-speed computer. 



NOTATION 



A, B Point values in linear approxima- 



tion for distance 



Cq Blade-section drag coefficient 



*^p = 'P " Poo)/(P» /2) Pressure coefficient 



Cp = 2?rQn/(pV 3 irD'/8) Power loading coefficient based 

 on reference speed 



C, 



: T/(pV-7rD-/8) Thrust loading coefficient based 

 on reference speed 



c(x R ) 

 D 



Df 



E(x c , x R ) = E c 

 E c (x c , x R ) 

 E T (x c , x R ) 

 (e, , e,, e r ) 



F 

 G(x R ) 



i T (x„) 



Blade-section chord length 



Rotor diameter 



Friction drag on blade section 



Profile shape function 



Meanline shape function 



Thickness shape function 



Unit base vectors in a helical 

 reference system 



Induction factor 



Non-dimensional circulation 



Total rake: axial displacement of 

 blade-section midchord point from 

 x = plane 



Uj.k) 

 J v = V/nD 



N(x c , x R > 



N R (x c ,x R ) 

 N„(x„.x R ) 



P 



P(x R ) 



Q 



q = q„ + v 



R = D/2 



ib( x c' x R» 

 V x c x R> 



s Wb (77,x R ) 



T 



t 



V 

 v 



v = (u, v, w) 

 <v> = (7, ii, a) 



Unit base vectors in a cylindrical 

 polar reference system 



Unit base vectors in a Cartesian 

 reference frame 



Advance coefficient based on 

 reference speed 



Vector normal to blade surface, 

 pointing into fluid 



Radial component of N 



Normal to blade-reference surface 

 (E = surface) 



Unit vector normal to blade 

 surface, pointing into fluid 



Propeller rotational speed, revolu- 

 tions per unit time 



Pressure 



Pitch of blade section 



Torque absorbed by blades 



Velocity vector 



Velocity vector far upstream 



Rotor tip radius 



Radius of rotor hub 



Position vector of field point 



Position vector of field point on 

 blade reference surface 



Position vector of point on b 

 blade surface 



Position vector of point on blade 

 reference surface 6 b = 



Position vector of point on shed 

 vortex sheet 



Thrust produced by blades 



Thickness of blade section 



Reference speed 



Velocity component due to 

 presence of the blades 



Average perturbation velocity 

 along blade surface, due to 

 presence of the blades 



Velocity difference across blade 

 surface 



Even perturbation velocity com- 

 ponent due to blade loading and 

 shed vortex sheet 



