The purpose of this paper is to show exact expressions of hydrodynamic forces and moments acting on a moving body with separating thin vortex layers, in a form of velocity potential integral, and to show accuracy and validity of slender body approximate expressions by deducing them from exact expressions. Prediction of hydrodynamic forces and moments acting on a body is one of important research themes in hydrodynamics, and we want exact and simple expressions of hydrodynamic forces and moments, which are easy to calculate and convenient to understand flow field phenomena. In a hydrodynamic research on a ship manoeuvring motion, we often use a simple flow model in which separation flow is approximated by thin free vortex layers. In this kind of lifting potential flow problems, there are several cases in which calculation of hydrodynamic forces from velocity potential itself is more convenient than calculation them from pressure distribution. Under these considerations, I examined three dimensional lifting potential flow subjects, and got following results. First, exact and simple expressions of hydrodynamic forces and moments in a three dimensional lifting potential flow were obtained. These expressions are the same forms as ones in nonlifting potential flow, if separation vortex sheets are excluded from velocity potential integral area. Second, after a long time arbitrary motion of a body, it is practically impossible to calculate hydrodynamic forces and moments by use of above obtained exact and simple expressions. So, more practical expressions appropriate to such cases were obtained. Third, applying these exact and practical expressions to gentle or small motions around a constant advancing motion of a slender body, approximate expressions were obtained. This analyzing process shows that usually used approximate expressions of lift and moment in a slender body theory are correct to second order and are exact for some special cases. Finally, we often use up-and-down symmetric flow model between two parallel planes, in hydrodynamic research of shallow water effect to a ship manoeuvring motion. It was shown that the above obtained expressions of hydrodynamic forces and moments acting on a moving body can be applied to this shallow water problem in the same form as ones in an unbounded fluid region.
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