Free Surface Effects tn Hull Propeller Interactton 
ristics as the connecting link between propeller geometry and forces 
after taking account of the velocity perturbation induced by the trailing 
vortices . Without going into details , which are given in Appendix C , 
Fig. 10 is presented as evidence for the close fit finally achieved bet- 
ween calculations and measurement . Note that the results of two dif- 
ferent calculations are displayed . The four sets of crosses mark the 
calculated performance of a series of hypothetical propellers indivi - 
dually designed at each respective advance coefficient so as to produce 
the known measured thrust with a minimum loss of energy (i.e. optimum 
distribution) . The exact agreement with the measured Ky values is 
therefore trivial , while the good agreement with the measured Ky 
values proves that hydrodynamic losses were reasonably estimated in 
the calculation and that the actual performance of the propeller is near- 
ly optimum over the range 0.6 <J<0.9 . On the other hand , the four 
sets of squares in Fig.10 mark at each respective advance coefficient 
the calculated performance of the given propeller with predetermined 
geometry . Hence , the perfect agreement with measured Ky and Kg 
values is trivial only at the design point , assumed to be at J = 0.8, 
whereas at the three other points it demonstrates the usefulness of the 
scheme devised to calculate the off-design performance with the aid of 
assumed (or empirically adjusted) foil characteristics . In particular , 
it may be anticipated from the trend visible in Fig. 10 that a more 
elaborate off-design analysis (as compared to the simpler design point 
analysis) would probably pay off at higher loadings (lower J values) by 
producing a more accurate simulation of actual propeller performance . 
The heart of the vortex model of the propeller used above is the 
calculated distribution of bound circulation along the blade . This is 
shown in a suitable nondimensional form in Fig. 11 for each of the four 
advance coefficients marked in Fig.10. It serves to illustrate the effect 
of loading and variation with radius , and is the basis of all further 
analysis . In passing we note that the two different calculations just 
discussed produced practically identical (within one percent) circulation 
distributions in the four cases considered here . 
The vortex model of the propeller was in turn transformed into 
a sink disk model by means of the Hough and Ordway (1965) relation , 
and linearized wavemaking theory was applied to calculate its self-indu- 
ced wake when operating near the free surface , see Appendix B , es- 
pecially Equations (B 13) and (B61) . The final results of four such 
calculations for a relatively shallow submergence of h/Rp = 1 .ore 
shown in Fig. 12. It is a rather remarkable coincidence that although 
the calculated self-induced wake varied strongly over the disk , its 
circumferential averages came out almost independent of the radius . 
1857 
