A VORTEX THEORY FOR THE 

 MANEUVERING SHIP 



Roger Brard 



Bassin d'Essais des Carenes de la Marine 



Paris, France 



FOREWORD 



The present text differs on many points from the draft which was prepared 

 for the 5th Symposium on Naval Hydrodynamics held at Bergen (10-12 September 

 1964). Firstly it appeared necessary to correct many misprints and also omis- 

 sions which made the reading difficult. Moreover it was useful to explain with 

 more details the theoretical views which lead to the introduction of a delayed 

 circulation around' a maneuvering submerged body. 



The line of thought is unchanged, but some results are presented with a 

 greater precision. 



The new paragraph on some experimental results (par, 16) shows that some 

 "apparent coefficients" may be found increasing and not decreasing when the re- 

 duced frequency increases. That seems to mean that the effects of the terms in 

 3/3t in the equations of the motion may be higher than in the case of a wing of 

 infinite aspect ratio. The effects of the wake on the stern planes are confirmed 

 to be very high. 



INTRODUCTION 



The work to be done in the naval hydrodynamic field in order to solve the 

 problems related to the unsteady motion of the ship is often a very difficult one. 

 A mathematical model of the physical phenomena has to be found. That requires 

 various compromises. For, if the equations, which the mathematical model 

 leads to, were too complicated in regard to the possibilities of an effective treat- 

 ment, no real improvement would have been obtained. 



That is undoubtedly why the equations of the classical ship hydrodynamics 

 are differential and of the second order. Nevertheless, in some cases, such 

 equations are not suitable at all, and the modern ship hydrodynamics must often 

 consider other classes of equations. It is, for instance, admitted that the equa- 

 tions which govern the rolling, heaving and pitching motions of a surface ship on 



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