Hamiltonian variational formulation of three-dimensional, rotational free-surface flows, with a moving seabed, in the Eulerian description

Abstract

Hamiltonian variational principles provided, since 60s, the means of developing very successful wave theories for nonlinear free-surface flows, under the assumption of irrotationality. This success, in conjunction with the recognition that almost all flows in the sea are not irrotational, raises the question of extending Hamilton Principle to rotational free-surface flows. The equations governing the fluid motion within the fluid domain, in the Eulerian description, have been derived by means of Hamilton Principle since late 50s. Nevertheless, a complete variational formulation of the problem, including the derivation of boundary conditions, seems to be lacking up to now. Such a formulation is given in the present work. The differential equations governing the fluid motion are derived as usually, starting from the typical Lagrangian, constrained with the conservation of mass and the conservation of fluid parcels identity. To obtain the boundary conditions, generic differential-variational constraints are introduced in the boundary variational equation, leading to a reformulation which permits us to derive both kinematic and dynamic conditions on all boundaries of the fluid, including the free surface. An interesting feature, appearing in the present variational derivation of kinematic boundary conditions, is a dual possibility of obtaining either the usual kinematic condition (the same as in irrotational flow) or a condition of different type, corresponding to zero tangential velocity on the boundary. The deeper meaning and the significance of these findings seem to deserve further analysis.