Theory of the Ducted Propeller — A Review 



cylindrical. In other words, the generat- 

 ing curve of B is approximated by a con- 

 tinuous, piecewise linear graph. The 

 approximate surface s is determined by 

 M-l radii r j . 



(2) Corresponding to this approxi- 

 mation, 7 is approximated by a continu- 

 ous, piecewise linear function, which is 

 constant on the cylindrical part of s. For 

 numerical reasons y is written as the sum 

 of triangular distributions (see Fig. 12). 

 The first segment is loaded with a distri- 

 bution which has a square -root singu- 

 larity at the leading edge. The function 

 7 is determined by N + M parameters 



3.0r 



2.5 



2.0 



15 



1.0 



0.5 



y-y 



1, 



N+M. 



(3) With these approximations, the 

 boundary condition / = ^q is satisfied at 

 the leading edge and at the midpoints of 

 the segments, with the exception of the 

 last cylindrical one. The pressure con- 

 dition is satisfied at the trailing edge and 

 the following M - 1 midpoints. For a speci- 

 fied value 4jq this affords N+ 2M- i equa- 



-1.0 



-0.5 



\ 



0.5 



tions for the N + 2M 



7- , r . . 

 '1 1 



1 unknown values 



Fig. 11 - (See (1Z2)) Non- 

 dimensional vortex distri- 

 butions gj and g* due to 

 interaction of thickness 

 with the free- stream and 

 the slipstream vorticity, 

 respectively, for a ducted 

 actuator disc (x = 0.5, p,^ = 

 0) with synnmetrical Jou- 

 kowski profile 



Since ijj and the induced axial and ra- 

 dial velocity can be expressed as linear .. '^ . 

 combinations of the 7i, the unknowns 7i 



appear in a fairly simple way in the equations. But the "influence coefficients" 

 of the 7i contain the unknown radii in a very complex manner, so that the sys- 

 tem of equations is highly nonlinear and has to be solved by iteration. 



The iteration starts by putting the values r^, 7^ on s as constants (equal 

 to the trailing-edge value) into the equations. Then, in the first cycle the equa- 

 tions are solved for the unknowns 7i on D. This first approximation must coin- 

 cide exactly with the results obtainable by the linear theory for cylindrical 

 shrouds or by the somewhat more general theory developed by BoUheimer (116) 

 for the static case. Then, the resulting error in the boundary conditions on s is 

 evaluated and from this by certain rules, improved estimates of the 7-, rj on 

 S are obtained. From the improved r-, a better approximation of the boundary 

 conditions can be calculated by means of improved influence coefficients. Then 

 the second cycle starts by solving the improved equations for the y^ on d. 



Contrary to most other theories, the basic aerodynamic input parameter is 

 not the pressure jump (or some other thrust parameter), but the value i//q, i.e., 

 essentially the mass flow. Without loss of generality, by appropriate choice of 

 units one can take /^o as an arbitrary fixed value {4>q = 0.5 in (130)). The cyl- 

 inder approximation of S begins at a distance 4Rx from the leading edge. It 



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