Weissinger and Maass 



straight axial vortices, each having constant strength y^ and y^, respectively. 

 Since the circulation outside the slipstream is zero, the total strength of the 

 axial vortices on the cylinder must be -r^ . Therefore, the axial vorticity is 



(4.2, 



VTQ ^-] .. . /x - 27rR, 



K the velocity components corresponding to cylindrical coordinates (x, r,0) are 

 denoted by u, v, w, then the vortex system without the ring vortices induces 



outside the slipstream 



inside the slipstream 



277r 



(4.3) 



If w is considered small in comparison with the angular rotational propel- 

 ler speed fi, we have constant pressure jump 



^P=^^^o (4.4) 



at the propeller disk. That gives the propeller thrust 



(4.5) 



(4.6) 



(4.7) 



Now, if the propeller is ducted, we have almost the same model as that 

 treated in Sec. 3.1. The only difference is that there is a contribution p^'^/l to 

 the pressure jump at the slipstream surface. But this can be neglected for the 

 same reasons as was done at the propeller disk. So, given the propeller thrust 

 or pressure jump, this problem can be solved by the methods discussed in Sec. 

 3.1. The important fact that the results do not depend on the location of the pro- 

 peller in the duct remains true. 



If there is a "tip clearance" (with respect to the reference cylinder), meas- 

 ured by the parameter 



/x= R/R^ (/x<l) (4.8) 



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