Weissinger and Maass 



In Fig. 10, the factor gx as a function of cj is shown for a duct with a 

 symmetrical Joukowski profile of relative thickness t = 0.15, chord/diameter 



ratio \ =0.5, and zero chord incidence 

 [p^(<^) = 0]. The two functions g^ and 

 g* are proportional to the thickness pa- 

 rameter T of the Joukowski profile. 

 Their values, divided by r, are plotted 

 over the chord in Fig. 11. Multiplication 

 of gt and gt by v and Vg^, respectively, 

 gives the two vortex distributions y^ and 

 7* due to thickness. One sees that y* 

 has at least the same order of magnitude 

 as 7t if cj is of order 1 or greater. 

 So, it does not seem consistent to neglect 

 7* and take into account 7^ as is done in 

 most theories. 



100 



Fig. 10 - (See (122)) Non- 

 dimensional vortex strength 

 g-j- at the trailing edge of a 

 ducted actuator disc {X = 

 0.5, f^ = 0) with a sym- 

 metric Joukowski profile 

 of relative thickness r = 

 0.15 plotted against the pro- 

 peller thrust coefficient C-j. 



3.2 Nonlinear Theory 



The only consistent nonlinear theory 

 for ducted propellers has been given by 

 Chaplin (130). The duct is assumed to 

 have zero thickness. The exaxt problem 

 is to find a harmonic stream function \p 

 having a constant value 4' - ^0 '^^ the 

 boundary B = D + s formed by the duct D and the slipstream surface s. On s, 

 the condition for continuous pressure v^ Av^ = const., where v^ denotes the 

 mean (tangential) velocity and ^v^. the jump of the velocity at s, must also be 

 satisfied. If the induced flow is produced by a distribution 7 of ring vortices on 

 B, the pressure condition on s can be written as v^y = const. The problem is 

 solved if 7 on B and the shape of s are determined. Then, the stream function, 

 the velocity field, and other characteristics can be found by numerical integra- 

 tion. Special emphasis is placed on the evaluation of the slipstream contraction 

 ratio 



(Rco'^Rt) 



(3.13) 



where Rx and R^ denote the radius of the duct at the trailing edge and of the 

 slipstream far away from the duct, respectively. 



At present a mathematical theory of existence and uniqueness does not 

 exist. No exact analytical solutions are known, not even for special cases. 

 Nevertheless, there are strong reasons, based on analogy and numerical expe- 

 rience, to believe that the results of the method developed by Chaplin for use on 

 high-speed computer are exact in a numerical sense. 



The boundary conditions are made discrete in the following manner. 



(1) The boundary surface B is approximated by N + M cone frustum seg- 

 ments such that the midpoint of the N-th segment coincides with the trailing 

 edge of the duct (see Fig. 12). The fN + M)-th segment is assumed to be 



1228 



