Gravity effects in two-dimensional and axisymmetric water impact models

Abstract

The effect of gravity during the water entry of two-dimensional and axisymmetric bodies is investigated analytically and numerically. An extension to the Wagner model of water impact is proposed in order to take into account the effect of gravity. For this purpose, the free-surface condition is modified. The pressure is computed using the modified Logvinovich model of Korobkin (Eur. J. Appl. Maths, vol. 6, 2004, pp. 821–838). The model has been implemented and validated through comparisons with fully nonlinear potential flow simulations of different two-dimensional and axisymmetric water entry problems. Our investigation shows that it is equally important to account for gravity when computing the pressure distribution and to account for gravity when computing the size of the wetted surface in order to obtain accurate force results with the Wagner model. Simulations of wedges and cones with different values of deadrise angle ( $\beta$ ) entering water at constant speed ( $V$ ) demonstrate the accuracy of the semi-analytical model and show that the effect of gravity in such water impacts is governed by the effective Froude number defined as $Fr^*=V/(\sqrt {gh}\sqrt {\tan \beta })$ , with $g$ the acceleration due to gravity and $h$ the penetration depth. The accuracy of the semi-analytical model for decelerated water entries is also demonstrated by investigating the water entry of a wedge and a cone with a $15^\circ$ deadrise angle with deceleration until full stop. The semi-analytical model is able to accurately predict the effect of gravity during both two-dimensional and axisymmetric water entry problems with deceleration.