Pien and Strom-Tejsen 



In front of the moving blade there is no vorticity, and the cross product in 

 Eq. (4) is zero. Our first conclusion was that in the wake, vortex lines are 

 parallel to the streamlines, and the cross product is again zero. Later we 

 realized this conclusion is not true, but for the reasons discussed in the appen- 

 dix we still feel justified in assuming the simplification that the cross product 

 is zero in the wake. With this simplification the following equation is valid 

 through the space except on the blade surface: 



|i=-,^vp-v(i<,^) (5) 



For incompressible fluid p^ is a constant, and we have 



^ „ / P 1 



5f=-'L-*^<.M (6) 



We introduce a function $ as follows: 



^-^+y^'- (7) 



Then 



3q 



Bt 



-Vd) . (8) 



For later convenience we also define an induced pressure p- as follows 



p. = PjO = p + ip^ q2 . ^9) 



From the continuity equation 



'(!?) = ^('"'> = °' (lo* 



and ^ consequently satisfies the Laplace equation 



V2 d. = . (11) 



The function ^ is defined as an acceleration potential, the negative gradient 

 of which according to Eq. (8) yields an acceleration field. Since it satisfies the 

 Laplace equation, it is an "exact" acceleration potential. It should be noted that 

 <I> is not the same as defined in Refs. 1 and 19 through 21, in which the accelera- 

 tion potential is based on the linearized equations of motion. The exact accel- 

 eration potential differs from the linearized one by a second-order term q^/2. 



The exact acceleration potential is the foundation for developing a general 

 theory for marine propellers. Since the acceleration potential is analogous to 

 the velocity potential, it is helpful to discuss very briefly how the velocity 



90 



