A General Theory for Marine Propellers 



1. Lifting surface where the general equation applies: , . ;■./.' -■. 



, : ,- V^* = V(q X f + F) . •:■,. •■-... ■'.■■... 



2. Wake: . -■ ■ ' - ' :• :■ :-^'' > ■'■.:•-■ 



V2$ = V(q. ^) - : : o,, ■: ■!,. (A8) 



since the force F does not exist in the wake, and 



3. Remaining field where both forces and vorticity are zero: . 



v^o) = . .. r (A9) 



In the text we have discussed the significance of the last equation and briefly 

 how a solution might be obtained. Also, we made a gross simplification, apply- 

 ing this equation to the wake region as well as neglecting the term q x ^ in the 

 wake. Now we shall proceed to discuss the implication of such a simplification. 



In developing a theory for wings with finite span, von Karman and Burgers 

 (Aerodynamic Theory edited by Durand, Vol, 2, Chapters III and V) presented a 

 thorough treatise on the solution of the general governing equation similar to 

 Equation (A7). They pointed out the difficulties encountered toward an exact 

 solution to the real problem; they also showed that an approximate solution to 

 the real problem might be obtained by neglecting in the wake the generalized 

 force term which they referred to as the induced "second order" forces com- 

 pared with either the k forces or the F forces and that the influence of the cor- 

 rections to be deduced from these second order forces is only of the third order 

 of magnitude. Hence, the resulting solution is correct to the second order of 

 magnitude, the reason being that "notwithstanding their smallness, they will have 

 a certain influence on all quantities considered; pressure, potential, and vortex 

 motion. As a force, however, can never produce vortex motion at a point up- 

 stream from it, the distribution of the vorticity within the region of lifting sur- 

 face is not affected." They also showed how a higher order solution could be 

 obtained by iteration based on the blade path rather than the slip- stream 

 geometry. 



Thus, the general theory for propellers developed in this paper, neglecting 

 the q X f term in the wake region, is considered to be a second order theory 

 which is consistent with the definition of the "exact acceleration potential" in the 

 text. 



On a lifting surface the vorticity ^ is always tangential to the surface since 

 the velocity discontinuity is in the tangential direction only. If we denote q^ and 

 q^ to be the tangential and normal components of q, respectively, we have on 

 the lifting surface the components of the generalized force: 



k, = q„ X ^ (AlO) 



k„ = q, X ^ + F . (All) 



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