Comparison of Theory and Experiment on Ducted Propellers 



the mutual interaction. This means that for practical purposes we can discuss 

 each separately. 



In general, the linearized theories of the duct are based on the so-called 

 Dickmann-Weissinger mathematical model, i.e., a distribution of ring vortices 

 and ring sources lie on a cylinder of a diameter representative of the duct di- 

 ameter and a length equal to the duct length. Some pertinent references are 

 (7 - 17). In the various references quoted, the theory of the annular airfoil is 

 essentially the same but with differences in numerical approach. Generally, two 

 problems have been considered: [1] the direct problem, and [2] the inverse 

 problem. The direct problem of the annular airfoil is: Given the annular air- 

 foil shape, determine the pressure distribution and forces (7 - 12). The inverse 

 problem is: Given the pressure distribution, determine the annular airfoil 

 shape (13). Both the direct and inverse problems require the solution of a sin- 

 gular integral equation, the first for the ring vortex distribution and the second 

 for the ring source distribution. Another version of the inverse problem is to 

 assume the ring vortex strength and, if the effect of thickness is considered, to 

 assume either the thickness or source distribution (14 - 17). The shape and 

 pressure distribution of the annular airfoil are then calculated. For the pur- 

 poses of this paper, any approach is pertinent as long as the theoretical and ex- 

 perimental data are for the same shape. 



Some of the previously mentioned references considered the annular airfoil 

 at an angle of incidence (8, 10 - 12). In the linearized theory, the effect of the 

 angle of incidence on the pressure distribution, forces and moments are inde- 

 pendent of the actual duct section shape, except for a small moment component, 

 and only dependent on the chord-diameter ratio. This means that the angle-of- 

 incidence effect is that of a circular cylinder at an angle of attack with a length- 

 diameter ratio equal to the chord -diameter ratio of the duct. 



As stated previously, various mathematical models of the propeller have 

 been added to the theory of the annular airfoil. In some cases, a constant pres- 

 sure jump with no clearance has been used to represent the propeller (17 - 19) 

 and in others, a variable-load actuator disk (16, 20, 21), or lifting-line theory 

 (9, 10). The use of a pressure jump at the propeller location would not appear 

 to represent a realistic flow, as this would imply that a pressure jump could 

 occur on the inner surface of the duct at the plane of the propeller (13). For the 

 normal number of blades used in ducted propellers and an adequate tip clear- 

 ance, it would not be possible to maintain a pressure jump on the duct surface. 

 A more realistic approach, it would seem, would be to use an actuator disk which 

 does not have a constant pressure jump or to assume a small tip clearance (5). 

 In any case, the propeller induces a radial velocity on the duct which, for the in- 

 verse problem, causes a change in the duct shape (13). 



When the lifting-line theory is used as the mathematical model for the pro- 

 peller, the finite blade effect causes the ring vortex strength to vary in the cir- 

 cumferential direction and free vortices are shed from the duct. The vortex 

 strength is steady with respect to the rotating propeller but unsteady with re- 

 spect to a coordinate system fixed in the duct. The finite blade effect can only 

 be considered for the direct problem as in this case, the pressure distribution 

 can fluctuate with the rotation of the propeller, but the finite blade effect in the 



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