A General Theory for Marine Propellers 



radius, or the p-^ angle obtained from a lifting line computation, can be taken as 

 the blade pitch angle. After the first computation the nose-tail line obtained at 

 each radius gives a new blade contour orientation from which pressure dipole 

 distribution is obtained and a second computation can be carried out. This itera- 

 tive procedure is continued until a convergence is obtained. 



It is appropriate to consider the design of a propeller working in the behind 

 condition, since it constitutes one of the problems which has motivated the de- 

 velopment of the present theory. However, for the sake of clarity the open- water 

 design problem is discussed first. The design conditions are as follows: 



'- Thrust .. ; -T ..: • , .:.;;,. 



Diameter ,. ., D; 



-.; Propeller velocity V 



Angular velocity fi 



Radial thrust distribution 

 Chordwise load distribution 

 Blade contour ,= , 



Number of blades " 



Our objective is to obtain the cambered surface which will produce the 

 specified load distribution over the blade in the design condition. We picture the 

 propeller starting from far behind with the blades carrying a specified lift dis- 

 tribution Apj (/0,i9). This lift distribution is derived from the desired thrust dis- 

 tribution and the blade contour and its orientation; these are supposedly known 

 when the computation in each iteration is started. Far ahead of the propeller we 

 choose a number of points along a line corresponding to the leading edge of the 

 blade at time t equal to zero when the blade reaches there. These are starting 

 points for a streamline tracing which defines the blade orientation and chamber. 

 Equations (19) can be used to compute the absolute velocity of the tracing at any 

 time for any point relative to the moving propeller reference axes. Thus the 

 streamline tracing in the propeller reference axes is equivalent to solving the 

 following first- order differential equations: 



dx T de dr ■ .^g. 



u(x,r,0) - V v(x,r,0) - rfi w(x,r,c/)) 



The streamlines so traced define the cambered surface of the propeller blade. 



Before discussing the general case of a propeller behind a surface ship, we 

 will mention briefly the case of a propeller working behind a body of revolution 

 where the flow to the space yet to be occupied by the propeller has a symmetry 

 with respect to the propeller axis of rotation. However, the flow has radial as 

 well as axial variations. Again we picture the propeller moving to a fixed space 

 from far behind. The only difference between this and the open-water case is 

 that the fluid velocity field already exists even when the propeller is still in- 

 finitely far behind. Hence, in addition to the induced velocity components cal- 

 culated by using Equations (19), the velocity components existing in this space 

 must be accounted for to obtain the absolute fluid velocity in that space at any 

 time. After this is done, the streamline tracing is exactly the same as the open- 

 water case (Eq. (79)). K the radial velocity component in the space induced by 



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