where T is the propeller thrust. Up to this point, it has been assumed 

 that the propeller has been represented by an appropriate distribution of 

 singularities external to the body. Within the framework of the potential 

 flow formulation, the solution to the interaction problem is derived com- 

 pletely in terms of propeller disturbance velocities at points on the body 

 surface. It is now appropriate to set forth the theoretical basis and 

 numerical techniques for calculating these velocities. 



PROPELLER FIELD POINT VELOCITIES 



In the foregoing analysis, the modified body flow in the presence of a 

 propeller is derived in terms of induced velocities on the body boundary. 

 It is primarily in the treatment of the propeller that the present inter- 

 action analysis differs from earlier investigations. Owing largely to 

 advances in design theory and high-speed computing capabilities, it has 

 been possible to introduce a more realistic analytical representation of 

 the propeller. 



Previously, the propeller was approximated as a sink disk. In that 

 model, the diameter, axial location, and radial distribution of loading are 

 explicitly represented. In fact, as will be shown shortly, a sink disk of 

 strength 



'^ dr 1 



^^^> =yj ^' --^— " ^--^ ^""^ 



dr' r' tan 3. (r') 



generates the circumferential average velocity field of a moderately loaded 

 lifting-line representation of a propeller with bound circulation r, pitch 

 2ir r tan g. (r) , and Z symmetrically spaced blades. Beveridge has demon- 

 strated that this model satisfactorily predicts the thrust deduction for 

 conventional propeller geometries. However, it is evident that this sim- 

 plification breaks down for raked propellers in which the blade sections 

 are displaced axially. Moreover, the effects of blade thickness and finite 

 chordlength, while perhaps of small consequence to the thrust deduction, 

 may be important to the body pressure distribution and boundary layer 

 characteristics in the immediate vicinity of the propeller. In view of 



10 



