Weissinger and Maass 



source rings on the cylinder used in ring-airfoil theory. The free vortex rings 

 are placed on the same cylinder (extended to x = ») and their strength yj is as- 

 sumed to be constant. Since the velocity field must be continuous everywhere 

 with the exception of the jump at the slipstream boundary, the value of the free 

 vorticity 7f must be equal to that of the bound vorticity yj at the trailing edge. 

 Then the boundary condition (3.1) can be satisfied in one cross section only, 

 which will be taken here at the trailing edge. These simplifications are reason- 

 able because one is usually interested in the flow near the duct and this flow is 

 influenced mainly by the behaviour of the slipstream in the neighbourhood of the 

 duct. 



Using the propeller thrust coefficient 



' ■ Ct- = 2Ap/(pV2) , (3^2) 



the boundary condition (3.1) is simplified to 



""Tp = 2 (^1 + -f j gj ' gj = -yx^v , (3.3) 



where u^- is the mean induced axial velocity at the trailing edge. 



Now we consider the most simple case of an infinitely thin cylinder of 

 length c as a duct. The constant free vorticity y^ must be extended continu- 

 ously to a bounded vorticity on the duct such that the radial velocity v^^, in- 

 duced by the sum y^ of free and bounded vorticity, is zero. This can be 

 achieved in the following manner. 



The constant vorticity y{ of the slipstream is extended continuously on the 

 duct, e.g., by a linear distribution y^ that vanishes at the leading edge of the 

 duct. The radial velocity Vy^+y^ induced at the duct can be calculated explicitly 

 in terms of complete elliptic integrals of the first and second kind. On the duct, 

 a vortex distribution y^ satisfying the Kutta-Joukowski condition is then deter- 

 mined from the integral equation 



To>b = -vyj+y, • (3.4) 



Thus, 7s is the superposition of the two bounded distributions y^, y^, and the 

 free distribution yj. From now on we shall consider only the entire distribu- 

 tion y^. All further distributions will be confined to the duct and must satisfy 

 the Kutta-Joukowski condition. We put 



7s = VgT-g* . (3.5) 



In a theory that is linearized with respect to all singularity distributions, the 

 distributions of the strictly linearized airfoil theory can be added to y^ in order 

 to obtain the solution for the ducted disk. This is dome essentially in all the 

 theories developed at NSRDC and THERM. But, for small V and heavy propel- 

 ler loading, the axial velocity u^ induced by y^ can have the same order of 

 magnitude as V and should not be neglected in a consistent theory. 



1226 



