Morgan and Caster 



inverse problem would imply the physical impossibility that the duct shape could 

 change with the rotation of the propeller. 



For the design of a ducted propeller it may be necessary to consider the ef- 

 fect of the finite number of blades, but this effect hasn't been completely inves- 

 tigated. The principal interference with the duct is the average velocity induced 

 on the duct by the propeller. The form of the equation of this average component 

 is identical with the actuator disk solution with the same average radial load, but 

 the induced velocity differs somewhat because of the fact that the propeller pitch 

 changes with the number of blades and that the viscous drag of the blades changes 

 with the propeller blade area, i.e., if viscous effects are considered, the theo- 

 retical thrust of the propeller must be increased to account for this additional 

 drag. 



A few investigations of the nonlinear theory of ducted propellers have been 

 made. In these cases the nonlinear theory was applied to the duct only and not to 

 the propeller. A nonlinear approximation, based on second-order airfoil theory, 

 has been given by Morgan (10), but more exact theories have been given by 

 Chaplin (18) and Meyerhoff (22), Chaplin places a distribution of ring vortices 

 on the surface of the duct and downstream along the slipstream of the wake. The 

 slipstream boundary is not known a priori but is obtained by an iteration proce- 

 dure. The pressure distribution and duct forces are obtained once the slip- 

 stream location is known. Meyerhoff uses a finite -difference approach (itera- 

 tion and relaxation) and calculates the stream functions, and streamlines, for a 

 known duct shape. Both the Chaplin and Meyerhoff treatments of the theory are 

 mathematically correct, but certain numerical approximations must be made in 

 each. Also, both treat the propeller as a pressure jump, which means that the 

 mathematical model of the propeller is less sophisticated than that of Ordway 

 et al. (9) or Morgan (10). As with numerical approximations of the type used by 

 Chaplin and Meyerhoff, some difficulty is encountered in obtaining solutions for 

 arbitrary shapes. 



Besides the work of Chaplin and Meyerhoff on the nonlinear theory, the cal- 

 culation procedure developed by Douglas Aircraft Division for the solution of the 

 Neumann problem has been applied to the calculation of pressure distributions 

 on the forward part of a ducted propeller (23). In the use of the computer pro- 

 gram, the duct must be treated as semi-infinite, which means that only the pres- 

 sures calculated near the leading edge are meaningful and no duct forces can be 

 determined. Here again, the propeller is treated as a pressure jump. 



The theoretical approaches discussed have generally been for the case of 

 the ducted propeller moving at a constant velocity. However, there have been 

 some investigations of the static condition, i.e., zero forward velocity. Linear- 

 ized theories for this condition have been developed by Kriebel (24) and Green- 

 berg et al. (25). Also, the Chaplin development of the nonlinear theory was car- 

 ried out mainly to obtain a solution for the static case (18). In the linearized 

 theory of the static case, the axial perturbation velocity cannot be assumed 

 small, as in the free-running case. Both Kriebel and Greenberg neglect the 

 slipstream contraction behind the duct even though the ducted propeller is heav- 

 ily loaded. On the other hand, Chaplin, in his nonlinear treatment of the duct in 

 the static condition, concentrates mainly on calculating this contraction. 



1314 



