Brard 



DISCUSSION 



Nils H. Norrbin 



Swedish State Shipbuilding Experimental Tank 



Goteborg, Sweden 



Admiral Brard has presented a fine piece of work on the mathematics of a 

 changing system of lifting vortices on bodies in transient and periodic motions, 

 accepting the physical picture of a flow separating along lines more or less 

 parallel to the body axis and producing a downwash over almost the full length 

 and width of the after body. 



When the body changes its attitude the interference between the vortex wake 

 and after body, and fins, does also change, and this interference must be depend- 

 ent on the time history of the motion. The physical picture also brings with it 

 the concept of a certain time required before a change of boundary flow condi- 

 tions develops into a change of separation and vortex wake, thus complicating 

 the dependence of the history of a transient motion, or of the frequency of a 

 periodic one. The present speaker fails to see to which extent the results of the 

 oscillator experiments quoted by the author are in any quantitative support of 

 this theory. 



To the speaker again, the flow separation parallel to the axis as mentioned 

 is more associated either with surface ship forms with spontaneous separation, 

 or with bodies of revolution at angles of attack no longer small. For the body of 

 revolution at a small angle of attack, on the contrary, the Nonweiler theory sug- 

 gests separation to occur much further aft and along the contour of a plane 

 almost at right angles to the axis. The vortex wake then covers a narrow region 

 of the after body only, and the circumferential flow will also be more rapidly 

 adjusted to the boundary conditions, thus extending the domain of practically 

 frequency independent derivatives. This might explain why ordinary differential 

 equations with constant coefficients are seemingly sufficient to predict the nor- 

 mal motion of a fair shaped submarine, but it would be interesting to hear of the 

 authors experience of such predictions. 



DISCUSSION 



A. J. Vosper 



Admiralty Experimental Works 



Gasport, England 



The mathematical presentation in Admiral Brard' s paper is welcomed as 

 a laudable attempt to calculate the forces on a submerged body. This is a 



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