Cox 



Section 4 gives a brief outline of initial design procedure together with 

 the necessary equations. As already mentioned, it is considered justifiable to 

 account for section pressure drag along with viscous drag when determining 

 thrust loading and power coefficients. 



2. LIFTING SLfRFACE - INDUCED VELOCITIES 



Lifting surface equations for pressure and induced velocity components at 

 any position (x*, r*, 5*) relative to the propeller, are derived in Appendix A, 

 i.e., Eqs. (A9), (AlO), (All), and (A12), respectively. They are obtained by use 

 of the inviscid linearized equations of motion for which the acceleration poten- 

 tial (or pressure) is a solution of the Laplace equation. In addition, by use of 

 Green's theorem and linearized boundary conditions, the blade loading and 

 cavity are represented by pressure doublet and source distributions, 

 respectively. 



In propeller theory it is usual and convenient to replace the pressure 

 doublet strength Ap(r,6') by a bound vortex strength / (r, 9) per unit length, 

 where r(r,0) is nondimensionalized by 277U. By use of the Kutta-Joukowski 

 theorem it is possible to obtain the linearized relationship 



Ap(r,^) = y(T,e) ^ • (1) 



K 



It should also be noted that the radial distribution of advance ratios k (r) are 

 usually replaced by induced advance ratios /Vi(r) in the equations referred to 

 above. Strictly speaking this is a nonlinear refinement to the pitch of the lift- 

 ing surface which allows consideration to be extended from light to moderate 

 propeller loading. In addition, it is usual to neglect the effect of the radial 

 component of induced velocity Ur(x*, r*, 9*) for moderately loaded propellers. 

 Thus, it is only necessary to consider axial and tangential components, i.e., 

 Ug(x*, r*, d*) and u^ix*, r*, 0*), respectively. By the same token, it is then 

 sufficient to put 



M = r2 + \.2 + (r0\. )2 ^ r2 + \.2 . 



Hence, by applying the transformation 



(v-x*) 



k. 



1 



the axial and tangential velocity Eqs. (AlO) and (A12)^ become 



Note a change in definition: Nondimensional axial and tangential induced veloc- 

 ities, i.e., u and ug, respectively, in Appendix A, are henceforth defined as 

 Ug and u^.. 



932 



