A General Theory for Marine Propellers 



Once the pressure dipoles and pressure- source distributions at the blade 

 surface are determined, Eq. (13) is used to obtain the acceleration field from 

 which the velocity field is obtained. If a higher order solution is required, an 

 iterative procedure may be tried. 



It is not our intention here to show how to design a super cavitating propel- 

 ler. Our discussion is merely to indicate how the acceleration potential may be 

 conveniently used in such problems. « ;■ •■■' ■ •'- 



Let us return to the general propeller problem. For a thin blade with a 

 zero thickness, only the second term of Eq. (13) exists on the boundary. We 

 want to find the relationship between the jump in Pf $ and the jump in pressure 

 p across the blade surface. From Eqs. (7) and (9) we have 



Ap. = Ap + i PfCqL^ - q^') , - ■ . (18) 



where Ap. and Ap denote the jump across the blade of Pf and p respectively, 

 and ql and qy are the absolute velocities of the fluid at the lower and the upper 

 blade surfaces respectively. The velocity squared is equal to the sum of the 

 square of the normal velocity and the square of the tangential velocity. Since the 

 normal velocity is continuous across the blade due to zero thickness, the con- 

 tribution to qL^ - q^j^ must be from the difference in the tangential velocities 

 across the blade. The magnitude of the tangential velocities on both sides of the 

 boundary are nearly equal but with opposite signs. Hence, the difference between 

 the square of the tangential velocities is negligible, and the approximation of Ap 

 by Apj is correct to the second order of the induced velocity. If the blade thick- 

 ness is not zero, the thickness distribution produces continuous tangential and 

 discontinuous normal velocity components across the blade. Hence there may be 

 some differences of qL^ - qj, depending on the relative magnitudes of the induced 

 velocities due to blade loading and blade thickness respectively. In a perform- 

 ance prediction, Ap. instead of Ap is determined from the boundary condition, and 

 Ap is then computed from Eq. (18). In a design problem the Ap. distribution can 

 be taken equal to the specified Ap distribution as a first approximation. If their 

 differences are found to be appreciable, Ap. can be corrected accordingly and the 

 computation repeated. 



The feature of any propeller theory is to obtain the changes in fluid velocity 

 in the vicinity of the blade due to the direct action of the propeller. This can be 

 done quite conveniently by using the acceleration potential. Let us assume that 

 the time history of singularity distributions of pressure sources and dipoles are 

 specified on the blade surface and that the blade position in the past relative to 

 the present position is also known. By taking the negative gradient of Eq. (13) 

 the acceleration at any point relative to the present blade position is known from 

 t equal to -» to the present time. The time integration of the acceleration plus 

 the initial fluid velocity at the point under consideration, in the absence of the 

 propeller, gives the fluid velocity at the present time. This procedure is much 

 simpler than that involved when the velocity potential is used. 



In a vortex theory based on the velocity potential a trailing vortex sheet ex- 

 tends to infinity, and its geometry is not only a function of the blade loading but 

 also a function of the fluid flow when the propeller is absent. For an unsteady 



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