Weissinger and Maass 



0.12 



0.09 



0.06 



0.03 



As described in the first paragraph of 

 Sec. 4, the propeller is represented by n ra- 

 dial vortices with equal distribution r(r) of 

 circulation. The helical vortices with strength 

 -r' ( r ) shed from a blade form a quasi-helical 

 surface with pitch j ( r). Assuming a moder- 

 ately loaded propeller, the axial pitch depend- 

 ence can be neglected. From theory and ex- 

 periment it is known that a hub of usual shape 

 does not affect the duct distributions very 

 much. Therefore, it will at first be consid- 

 ered as nonexistent. r(r) is considered as a 

 known function with r(0) = and r(Rp) = 0. 

 Only if the distance between propeller tip and 

 duct surface is practically zero do we have 

 r(R ) ^ 0. The meaning of "practically zero" 

 is a question of boundary -layer theory. 



As in the theories of Sees. 3 and 4, the 

 duct is represented by a distribution y of ring 

 vortices and for nonzero thickness, by a dis- 

 tribution q of sources, both lying on a refer- 

 ence cylinder of radius R^. The main feature 

 now is that, obviously, y depends on 4> as well 

 as on X. Therefore, from each ring element 

 ydxfree vortices of strength By/R^B^ dx are shed. These are not straight lines, 

 as in the case of the ring airfoil at angle of attack but rather are of helical shape. 

 They are assumed to be regular helices with constant pitch j^ = v/nR^. This 

 assumption excludes the static case. It yields a reasonable approximation for 

 the case of moderately loaded propellers in cruise condition. It will easily be 

 perceived that the theory of this section and the computational labour involved 

 are not changed essentially by the choice of another constant value for j^ . 



Fig. 20 - (See (153)) 

 Duct pressure distri- 

 bution for two pro- 

 locations and 

 parameters as 

 18 



peller 

 other 

 in Fig. 



inner 

 outer 



surface 

 surface 



If, within the framework of strict linearization— contrary to the more gen- 

 eral Bollheimer theory described in Sec. 3.1, the interaction between the pro- 

 peller slipstream and both profile camber and thickness are neglected, then the 

 duct distributions due to camber and thickness can be calculated separately by 

 ring-airfoil theory. These have to be added to the distributions induced by the 

 propeller and its slipstream on a cylindrical duct, which shape will be assumed 

 for the rest of this section. As a practical choice for the duct radius R^, the 

 radius of the shroud camber line at the propeller plane is proposed in (147). 



Because of the helical shape of the vortices shed from the duct, the radial 

 velocity induced by the duct vortex system at the duct cannot be expressed by 

 the integral operators !„ used in ring-airfoil theory. But the main interest is 

 in the circumferential average 



7n(x) = - / 7(x,0) d0 



97T ^ 



(5.1) 



27T 



1238 



