A General Theory for Marine Propellers 



potential is used in developing a vortex propeller theory. After this has been 

 done, it will be easy to understand how the acceleration potential is used in de- 

 veloping our theory for marine propellers. 



For a velocity potential we may (22) write ' ^ ■..;.'.- c 



where 0p is the value of the velocity potential at a field point p, 4> is the 

 velocity potential value on one side of the boundary surface, <p' is that on the 

 other side, and 3/3n is equal to -3/3n'. 



The first term is due to a surface distribution of simple sources of density 



It gives the discontinuity of the normal velocities on the two sides of the bound- 

 ary. The second term is due to a surface distribution of dipoles of density 

 - 0' . It gives the discontinuity of the tangential velocities across the boundary. 



In applying Eq. (12) to a propeller problem the discontinuity of the normal 

 velocities is due to the blade thickness. Hence the simple source distribution at 

 the boundary can be derived from the thickness. In a propeller problem we are 

 concerned with the lift distribution produced by the discontinuity of the tangential 

 velocities. Hence for the sake of convenience we specify the circulation distribu- 

 tion on the boundary directly rather than specify the discontinuity of the velocity 

 potential across the boundary. The boundary surface, in the case of a propeller 

 blade, extends from the leading edge to infinity behind, since there is a discon- 

 tinuity of the tangential velocities across the trailing free- vortex sheet as well 

 as across the blade surface. r -. 



The discussion about the velocity potential can be repeated for the accelera- 

 tion potential, except for replacing the word velocity by the word acceleration. 

 We may write in analogy with Eq. (12), 



In applying Eq. (13) to a propeller problem, the first term is again due to the 

 blade thickness. However, the pressure source distribution at the boundary is 

 derived from the blade- section curvature on both sides of the blade rather than 

 from the thickness. The second term is also due to the blade-load distribution. 



For the sake of convenience in the following discussion let us temporarily 

 approximate $ by p/Pf. As shown by Eq. (7) they differ only by a second- order 

 quantity qV2. The pressure dipoles in the second term of Eq. (13) then corre- 

 spond to the pressure jump across the boundary or the blade-load distribution. 



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