Weissinger and Maass 



right-hand side in (5.11), the Bp-^ become zero. Therefore, putting g^ = 0, the 

 first equation of each pair drops out from the computation. The expressions for 

 Apm + ^r'm can be simplified. 



So far, numerical results have been presented only for the steady part of 

 the solution. This is the most important part because it yields the time average 

 of all linear quantities in linearized theory such as the distribution of velocity 

 and pressure on the duct, the radial sectional force, etc. On the other hand, the 

 duct thrust is a nonlinear quantity and depends not only on the coefficient of 

 zero order, which is used in the usual computations, but on all Fourier coeffi- 

 cients. 



6. THE DUCTED PROPELLER AT ANGLE OF ATTACK 



The problem of this section is much more difficult than the previous ones, 

 because, for a propeller at angle of attack, a system of coordinates in which the 

 flow is time independent does not exist. Therefore, in its present state, the 

 theory is more rough than those presented in the previous sections. 



The first rough approximation is based on the so-called "superposition 

 model," in which the time dependence is eliminated. The flow is determined by 

 superposing the flow of the ducted propeller at zero incidence and the flow of a 

 cylindrical ring airfoil at angle of attack. Both flows can be determined sepa- 

 rately by the theory of Sees. 5 and 2, respectively. Essentially, this model was 

 first used by Kriebel (160). The gross forces resulting from this model are the 

 thrust due to the ducted propeller at zero incidence and a lift force due to the 

 ring airfoil at incidence. 



The most important result of the following theory developed at THERM 

 (157) is that it shows the existence of a mean net force due to the time -dependent 

 part of the flow which, in turn, produces a side force that can be as great as 20% 

 or more of the lift. Approximately one half results from the propeller and the 

 other half from the duct. A decrease in shroud lift of about 10% is also pre- 

 dicted by the interaction model, although it is approximately balanced by a lift 

 force on the propeller. Atypical example (N = 3, Xp/c = -0.219, d/Rp = 0.213, 

 \ = 0.5, /x = 0.956, J = 0.344) is shown in Table 1. The coefficients of lift and 

 side forces and the corresponding pitching and yawing moments are tabulated 

 separately for the duct and the propeller. The moments are referred to the 

 leading edge with the nose-up and nose -right directions taken as positive. The 

 positive direction of the side force is oriented to the left (Fig. 21). 



The problem to be solved may be stated as follows: Given the geometry of 

 the configuration and the mean propeller blade circulation r(r) (design circula- 

 tion), determine the duct source distribution q and the steady part of the bound 

 duct vortex distribution such that the tangent-flow condition is satisfied at the 

 reference cylinder r = R^ and the Kutta condition, at the trailing edge. 



A duct-fixed system of coordinates (x, r, 0) is used, as shown in Fig. 21. 

 If y and r denote the duct and propeller distributions of the superposition 

 model, then the corresponding distributions of the interaction model are taken 



to be r + 7 and r + r. The trailing vortices of the duct -3 (y + y)/'d4> are 



1242 



