The interaction force may also be derived by application of Lagally's 

 theory ' for a body immersed in a steady potential flow. The solution 

 for the force on a body due to an isolated point source is developed in 

 Appendix A as 



F = -p/ a(^^) V^(r^) dS(r^) , (13) 



where V (r ) is the undisturbed onset velocity generated by the singularity. 

 This equation is also valid for a point doublet singularity. It will be 

 shown subsequently that the propeller disturbance arises from suitable 

 distributions of sources (blade thickness) and doublets or equivalently, 

 line vortices (blade loading) . Thus , the axial force arising from the 

 propeller-hull interaction may be written as 



y a(rg) V(|)p(rg) dS(rg) (14) 



F _ = - e_ • p 



•'s 



a 

 or in discrete form 



F, = - e^ • E a. (r^) V* (r^) AS. (r^) (15) 



X X 



If only the total force is required, this form is simpler for computation 

 and, in any case, provides a convenient check in the numerical evaluation 

 of equation (11). 



Once the interaction force is found, the thrust deduction fraction, t, 

 is given by 



F 

 t=Y^ (16) 



13 



Cummins, W.E. , "The Force and Moment on a Body in a Time-Varying 



Potential Flow," Journal of Ship Research, Vol. 1, No. 1 (Apr 1957). 



14 



Milne-Thomson, L.M. , "Theoretical Hydrodynamics," The Macmillan Company, 



New York, N.Y. , 2nd edition (1950). 



