Hydrodynamic forces in water exit problems

The three-dimensional steady problem of an elongated smooth body moving along the water free surface at a constant speed is considered within the 2D+T approximation. The corresponding unsteady two-dimensional problem describes both the water entry and the subsequent exit of a smooth contour from the water. The shape of the contour varies in time. The present paper is concerned with the exit stage. The draft of the body is small compared with the body length and beam. The hydrodynamic loads during the entry stage are evaluated by the original Wagner model of water impact. The linearized exit model (Korobkin, 2013) is generalized to account for time-dependent acceleration of the body and the body shape which also varies in time. The integral equation with respect to the size of the wetted area of the body is solved numerically. The theoretical predictions of the hydrodynamic forces acting on the body during its exit from the liquid are compared with the numerical results obtained by solving the Navier-Stokes equations. A simplified model of water exit with the body shape approximated by a parabolic contour with a time-dependent radius of curvature is proposed and validated. It is shown that the linearized water-exit model with non-linear correction terms predicts reasonably well the hydrodynamic loads