A Vortex Theory for the Maneuvering Ship 



That is the case, because the lift is not null. For instance, when the lift 

 component on the z-axis is opposite to the direction of this axis, the mean 

 pressure on the upperside of the body is smaller than on the lowerside; on the 

 contrary, the mean velocity is greater on the upperside. And the circulation 

 around the body along closed circuits parallel to the (x,y)-plane is necessarily 

 non-null. 



The same reasoning holds in the case of a maneuvering surface ship, the 

 lift being now in a horizontal plane. In 1950, one of our assistants has calcu- 

 lated a distribution of free and bound vortices for a thin surface ship in a 

 steady turning motion and obtained by this way some results which help under- 

 standing several phenomena unexplained to this time (see [3] and also [4]). 



Some authors [5-7] probably have ideas quite similar to the one expressed 

 above. But they are principally interested in the configuration of the vortex 

 wake and in the mechanism of the transport into the wake of the vorticity which 

 originates in the boundary layer. Such a line of thought is the best from a sci- 

 entific point of view. Unfortunately, such a study is very difficult and will not 

 lead rapidly to results that the naval architects may easily use. That is why 

 we have chosen here another way. 



A mathematical model of the vortex shedding has to be defined. Preferably 

 it has to be flexible enough to be adaptable to the various hull forms we encoun- 

 tered in the practice. Consequently, this model is not made for giving all the 

 means necessary for a complete calculation, in each case, of the hydrodynamic 

 set of forces in steady and unsteady motions. In return, it has to yield the gen- 

 eral form of the expression of this set, and also, to supply a criterion which 

 permit to decide whether, according to the experimental results, the differences 

 between the quasi-steady forces and the real forces are negligible or not. 



The present paper gives a first answer to this problem. 



Section I defines a mathematical model of the wake vortex and leads to the 

 Volterra's integro-differential equations which govern, in an unsteady motion, 

 the circulation and the forces exerted on the body. Attention is drawn — as in 

 [8] — to the pressure distribution on the hull, and also to the effects on the stern 

 planes and rudders of the wake generated by the submerged body itself. 



Section 11 shows that in a harmonic forced motion, the forces differ from 

 those given by the quasi-steady motion theory. Some experimental results show 

 that there is a possibility to estimate the magnitude of the errors involved in the 

 quasi-steady theory. Some of them are small. Others are significant. 



Section III is devoted to possible further developments of our present views. 

 It is shown that tests in various steady and harmonic forced motions are able to 

 yield all the unknown coefficients and functions found in the so-called "true" 

 equations of the free quasi -rectilinear motions. Unfortiinately, other motions 

 are of great interest too, those which require non-linear equations. In these 

 cases, the technique of the steady and harmonic forced motions is unable, in its 

 present state, to supply all the necessary information. Moreover, the "true" 

 equations are more complicated than those of the quasi-steady theory and lead, 



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