A Vortex Theory for the Maneuvering Ship 



When ./ = 0, one has ^ = 6 t^ 4>/cos 6=0, ^ = constant. The motion is also 

 parallel to a vertical plane. 



Motions Parallel to the (x,y)- Plane — The z -component w + py - qx of the 

 absolute velocity of any point attached to the body is null. Consequently 



w=0, p=0, q=0. 



Therefore 



• • _ • cos 9 



0=0 sin 6 , 6 = - ip cos a tg, cfj , r = i/^ . /o\ 



cos (^ V"^/ 



This motion is parallel to the horizontal plane when 4> and 6 are simultaneously 

 equal to zero. 



Quasi -Rectilinear Motions Parallel to the x-Axis — In the case, we substi- 

 tute u + u for u . 



One considers that 



U V w 



U ' Tj ' u" ('^^ 



are small. B being the breadth of the body, 



Bp _ B Lp Lq Lr . , 



U ~ L ' U ' U ' U ^^' 



are small too. The square and products of ratios (4) and (5) are negligible. 



3. Vortices Attached to a Body on Steady Motion 

 in its (z,x)- Plane 



It is well known that a submerged body may be considered as equivalent to 

 a distribution of bound vortices when no wake exists and to a distribution of free 

 and bound vortices when a wake is shed. On the other hand, it is well known, 

 too, that a closed filament vortex is equivalent to a distribution of doublets. To 

 write the expressions of the forces generated by such a distribution of vortices 

 or doublets, it is helpful to bear in mind the main aspects of the theory. 



3.1. Bound Vortices are Equivalent to a Submerged 

 Body in a Perfect Fluid 



As a matter of fact, when the fluid is quite perfect, the absolute motion of 

 the fluid may be considered as generated by a distribution of vortices located 

 on the hull when the angular velocity Q is equal to zero, on the hull and inside 

 the body, when Q + . 



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