Newman 



by our experiments quoted on page 216 of the paper. The form of the tail is of 

 minor importance in this case, and the addition of a rudder does not improve 

 the trailing edge condition but means the addition of an isolated lift force, which 

 in turn may give an unstable moment on the low side (Cf. Fig. 8). 



As for the total normal force results displayed in Fig. 7, it must be realized 

 that the experimental accuracy was less in case of the ship model — because of 

 the relatively shorter length of the balance beam — and that the dip at the highest 

 speed may partly be due to a local effect of the stern wave on the force on the 

 rudder. 



Finally I will make a brief comment on the formal higher order approxima- 

 tions. From physical considerations the cross -flow -drag lift suggests a square 

 term, which we have found to agree with experimental results. Due to the finite 

 time required for the development of the cross-flow boundary layer, however, 

 the contribution from sections with small fullness might be of a higher order, 

 and this contribution might in turn be reflected in the small nonlinear moment 

 term. 



REPLY TO DISCUSSION 



J. N. Newman 



In reply to Mr. Gertler I must first emphasize that my "conclusion" regard- 

 ing the relative importance of viscous and free -surface effects should be re- 

 garded for the present time as a hypothesis. Mr. Gertler's summary of the ex- 

 tensive experiments carried out on submarines at the David Taylor Model Basin 

 is a valuable supplement to the data in Table 2. Taken in conjunction with the 

 revised data which Admiral Brard reports in his discussion, it suggests that 

 submerged bodies of good hydrodynamic form will have a limiting reduced fre- 

 quency, below which their motion can be regarded essentially as pseudo-steady- 

 state, of between four and ten. This range of reduced frequency is substantially 

 higher than the corresponding values for surface ship models as reported in 

 Table 2, thus suggesting that frequency effects come into play sooner for surface 

 ships than for submarines. At first glance this conclusion implies that, contrary 

 to our hypothesis, the free surface-effects are dominant, but a likely alternative 

 is that the hull form of surface ships is inferior to that of submarines. This 

 situation could be clarified by performing planar -motion -type experiments with 

 a submerged double-body, analogous to the steady-state experiments of Norrbin 

 (1965a). 



I am very grateful to Admiral Brard for amplifying upon the results of 

 Casal's thesis and for bringing the results of his own research up to date. The 

 observation by Casal of a horseshoe vortex system generated near the bow dur- 

 ing a turn, as opposed to a steady yawed condition, is another indication of un- 

 steady viscous effects which in this case is associated with the vertical stem of 



236 



