Brard 



REPLY TO THE DISCUSSION BY NORRBIN 



Roger Brard 



Bassin d'Essais des Carenes de la Marine 



Paris, France 



Dr. Norrbin has drawn the attention to the mathematical model which I 

 choose as a starting point. I agree that the choice is a difficult one and, indeed, 

 I have hesitated for a fairly long time before deciding. That is why I wrote that 

 the NACA model only "suggests" the physical picture of a surface wake limited 

 by lines more or less parallel to the x-axis. As a matter of fact, the length and 

 the shape of the arc along which the separation occurs depends strongly upon the 

 hull form. For instance, for a thin surface ship, this arc is practically the keel 

 line and the maximum of the density of the free and bound vortices is located 

 near the bow. For a body of revolution, the arc depends also upon the angle of 

 attack. But it does not seem to me that the final structure of the formulae giving 

 the expression of the forces exerted on the ship or on the submerged body in an 

 unsteady motion strongly depends upon these circumstances. I hope that, in the 

 field of linearity, at least when the body is moving in its centerplane, the forces 

 are always given by convolution functions. The behaviour of these functions may 

 differ when the form of the hull changes, and their calculation should be very 

 intricate. My purpose was only to give means in order to get these functions 

 starting from experimental results and not through mathematical calculations. 



When the motion is not parallel to the centerplane of the body, the form of 

 the hull is of still greater importance. It is my intention to insist on this ques- 

 tion in the final text of the paper. For instance, in the case of a nonsymmetric 

 body with respect to the (x,y)-plane, strong forces along the z-direction and 

 moment about the x -axis may appear. The intensity of these force and moment 

 strongly depends on the form. 



Presently, the theory does not permit to predict which hydrodynamic forces 

 are exerted on the body whatever its form may be. But it leads to a method for 

 deducing these forces from these measured in particular motions, the steady and 

 harmonic forced motions. The theory also indicated that the wake has a great 

 influence on the forces acting on the stern planes and fins. 



Dr. Norrbin said that the surface of the body on which the wake acts is of 

 small area when the body is of revolution and when the angle of attack is small. 

 He expressed the opinion that it might explain why ordinary differential equations 

 with constant coefficients are seemingly sufficient. I have indicated in the paper 

 that, from my point of view, this question is presently not solved. It is quite 

 evident that the coefficients which depend upon the added masses or which con- 

 tain terms due to the rotation of the axis are much less sensitive to the history 

 of the motion than the others. For this reason the history of the motion should 

 act mainly on the lift coefficient due to the angle of drift. 



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