A Vortex Theory for the Maneuvering Ship 



However, if we deal with integral equations, the conventional analog com- 

 puters are no more sufficient. Bigger computers, analog, digital, or hybrid, 

 are in fact necessary and the work to be undertaken to study all the possible 

 interesting cases becomes really huge . . . 



22. General Conclusions 



22.1. As already stated in the Introduction, this paper is devoted to the 

 effects of the circulation around a ship on the set of hydrodynamic forces 

 exerted on her. 



As a matter of fact, the subject is restricted to the case of a submerged 

 body in an infinite fluid. But the case of a surface ship is similar, apart of the 

 fact that the free surface effects have to be taken into account. 



22.2. Our mathematical model is defined in Section I, pars. 3 and 4. 



We start from the possibility to substitute for a submerged body moving in 

 a perfect fluid an equivalent distribution of bound vortices on its hull (and inside 

 the volume interior to the body when the angular velocity is not identically null). 

 Consequently, a motion is defined in the whole space; the fluid interior to the 

 body is at rest with respect to the latter. Then we introduce a new family of 

 bound and free vortices in order to get a wake. This new family has to be added 

 to the first. 



We consider firstly the case of a small motion with one degree of freedom 

 around an uniform motion of velocity u parallel to the x-axis. This small mo- 

 tion is assumed to be parallel to the (z,x) -plane of symmetry. Neglecting the 

 deflection of the fluid due to the reaction of the body on the fluid or, which is 

 equivalent, to the velocities induced by the vortices on themselves, we admit 

 that the free vortices are at rest with respect to the fixed axis. They are lying 

 on U-shaped arcs which are nearly located on planes parallel to the (x, y) - 

 plane; because the angle of attack (or the reduced angular velocity) is small, 

 these arcs are approximately located on a wake surface attached to the body 

 along a line which is assumed to be known (given by experience for each body), 

 and which acts as the trailing edge of a lifting surface. The bound vortices as- 

 sociated to these free vortices are distributed on the hull. The total distribution 

 fulfils the condition that the circulation around a closed fluid arc is equal to zero. 

 The total potential equivalent to the free and bound vortices of the second family 

 induces a velocity which is null inside the body and which, outside the body, is 

 tangent to the external face of the hull and to the surface of the wake. It is shown 

 that these conditions lead, when the motion is unsteady, to a formula which gives 

 the circulation in term of the circulation in the quasi -steady motion. This ex- 

 pression is a convolution function 



r(t') = r^(r') —J (t'-T')d' 



where t ' is the reduced time t ' = Ut L, L being the length of the body, and 

 t ' = the time at the beginning of the unsteady motion. 



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