Theory of the Ducted Propeller — A Review 



1.2 Program of This Review ■.■..,■. .... 



This review will be concerned only with incompressible, nonviscous, and, 

 primarily, steady flow. Some hints referring to other problems can be found in 

 the list of references, e.g., cavitation (Tulin, Chen), boundary layer (Nickel), 

 compressibility (Zierep, Laschka). 



Since the development in the first two periods is well covered by the review 

 of Sacks and Burnell (78), where emphasis has been placed on theories of the 

 first kind, this review will be restricted to theories of the second kind developed 

 during the third period. Papers and reports published before 1955 are listed in 

 the references only insofar as they are referred to in the text. This is also true 

 for publications concerned mainly with experiments. 



The presentation is guided by a systematic and not by a historical point of 

 view. Emphasis is placed on the analysis of the phenomena and not on design 

 methods. Because of the complexity of the mathematical apparatus, we cannot 

 always give mathematical formulas in detail. Instead, we shall describe the 

 underlying models and the basic assumptions, characterize the numerical meth- 

 ods used for the solution, and state the main theoretical results. We will con- 

 centrate our attention on the theories developed at Karlsruhe, THERM, and 

 NSRDC. 



In aerodynamics and hydrodynamics two kinds of problems are distinguished. 

 In the "direct" problem the geometry is given and the flow field or special char- 

 acteristics such as lift or pressure distribution are sought, vice versa in the 

 "inverse" problem. Something inbetween is the case where a mathematical sin- 

 gularity distribution is given and the corresponding geometry and/or flow field 

 is to be determined. If the mathematical relations between the singularity dis- 

 tribution and the geometry as well as the induced flow are known, it is purely a 

 question of mathematical skill to solve the direct or inverse problem by elimi- 

 nation of the singularity distribution. Of course, this can usually be done by 

 numerical methods. 



The problem of the ducted propeller is a problem of interference, of inter- 

 action between two bodies, the duct and the propeller. From the four possible 

 direct/inverse combinations we shall choose the mixed one with given duct 

 geometry and propeller distribution. From the solution of this problem, the 

 propeller geometry can be determined by slight modifications (128, 134) of well- 

 known methods (48). 



Problems of interference can often be solved by an iteration scheme in the 

 following manner. First, one has to find a method for determining the flow field 

 of each separate body in a very general (nonuniform) flow. Then, starting with 

 an arbitrary field, this field is modified by the presence of the first body, the 

 new field is modified by the second body, this field again by the first body, and 

 so on. 



Following this line of thought, we deal first with the duct alone (Sec. 2), i.e., 

 we solve the direct problem for the ring airfoil in fairly general flows. These 

 contain also the case where a jet is emitted out of the ring airfoil by an actuator 

 disk of constant pressure jump. This case (Sec. 3) can be considered as the 



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