Morgan and Caster 



In the following sections of the paper, the theory of annular airfoils and 

 ducted propellers will be discussed first, then the experimental data available 

 for use in making comparisons will be presented. The review of the theory will 

 be very brief, as Weissinger's review (2) gives the details of the various theo- 

 ries. Theoretical-experimental comparisons will be made for pressure distri- 

 butions and forces, and from these comparisons conclusions will be drawn as to 

 the adequacy or inadequacy of the theory. 



DISCUSSION OF THE THEORY 



Theoretical investigation of the ducted propeller has concentrated to a large 

 extent on linearized theory. Some investigations have been made, however, 

 where the annular airfoil, or duct, is treated in a less restrictive manner. In 

 general, the following assumptions are made regarding the mathematical model 

 of the annular airfoil and the duct of a ducted propeller: 



(a) The fluid is inviscid and incompressible, and no separation occurs 

 on the duct. 



(b) Body forces such as gravity are neglected. 



(c) The free-stream flow is, in general, axisymmetric but may have a 

 small cross-flow component. The free-stream velocity is, of 

 course, zero for the static case. 



(d) The annular airfoil is axisymmetric and of finite length. Although, 

 Siekmann (3) has considered annular airfoils of elliptic cross- 

 section. 



For the linearized theory, the following two additional assumptions are made: 



(e) The annular airfoil can be represented mathematically by a distri- 

 bution of ring vortices and ring sources along a cylinder of con- 

 stant diameter. This implies that the boundary conditions are lin- 

 earized, i.e., the perturbation velocities are small in relation to 

 the free-stream velocity (in calculating the pressure distribution, 

 the Bernoulli equation without linearization is often used) and the 

 boundary condition (normal velocity zero) is satisfied on the cylin- 

 der rather than on the foil surface. 



(f ) The trailing vortex system of the annular airfoil, if one exists, has 

 the constant diameter of the annular airfoil and extends from the 

 annular airfoil to infinity. 



These foregoing assumptions apply only to the duct. Assumptions for the 

 propeller are usually more restrictive, and the propeller is often considered as 

 a crude approximation. Various mathematical models of the propeller have 

 been used, some of which are: [1] momentum theory (4), [2] variable -load actu- 

 ator disk (5), and [3] lifting -line theory (6). In the design process, the propel- 

 ler and duct are treated separately, and a process of iteration is used to obtain 



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