Theory of the Ducted Propeller — A Review 



With cos tto - 1> this is a consequent linearization if a^, (,f ), t (j) (and 

 the derivatives) are considered small. But it might be advisable in certain 

 cases not to neglect the product of quantities proportional to 7 and t, respec- 

 tively. U one looks at the symmetrical profile in Fig. 3b, where the vortex- and 

 source -distribution of the strict linear theory is placed on the chord, one sees 

 that there remains a normal component y^ t'/2 on the surface. Another compo- 

 nent comes from the first-order term in the Taylor expansion of the vertical 

 component induced by yg 



v,„(^.±t)= v,^(^,0) ±2-v,^(^,0)t .,: (1^14) 



The sum of both normal components is dd^" (y^ t )/2. Since it is continuous 

 through the chord, it must be annihilated by a vortex distribution y^ satisfying 



%(^-0) + ^"ro(^'O) t = v^^(e,0) + y; t/2 





(7ot)/2 . 



(1.15) 



Putting y = yo -^ Vt) the surface velocity is then given by (1.13). In this manner 

 one obtains a theory which is linearized in t, but not in qlq. For elliptic pro- 

 files the surface velocity is exact. Riegels (76) first introduced the thickness- 

 influences vortex distribution based on ideas of conformal mapping theory. But 

 the idea is more general and can be applied to rotational flows (123), too. A 

 theory in which y^ is neglected will be called a "strictly" linearized theory. 



For thick airfoils with small camber, the camber-induced velocity is usually 

 superposed linearly. 



(C) The biplane (Fig. 4). Aside from profile geometry, the configuration is 

 characterized by the "chord/diameter" ratio c/D and the inclination of the chord 

 relative to the axis. In order to avoid the second parameter in the kernels of the 

 integral equations, the singularity distributions are put on a "reference chord," 

 e.g., the projection of the profile on a line parallel to the axis through the lead- 

 ing edge. Obviously, the vertical velocity v^ induced at one profile by the vor- 

 tex distribution 7 located on both (reference) chords has the form 



v^ = G7 + Ky , (1.16) 



where G is the Glauert operator and K an integral operator with a continuous • 

 kernel depending on c/D. The term K7 represents the velocity v^ induced by 

 the vorticity of the opposite profile. Similarly, the source-induced velocity is 



\ = ±q/2 + v^ , (1.17) 



where v^ is the source-induced velocity from the opposite profile. The 

 (strictly) linearized boundary condition 



v^ + Vq = -Va(^) (1.18) 



1217 



