Hamiltonian Variational Formulation of Three-Dimensional, Rotational Free-Surface Flows, with a Moving Seabed, in the Eulerian Description

Abstract

Hamiltonian variational principles have provided, since the 1960s, 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’s principle to rotational free-surface flows. The Euler equations governing the bulk fluid motion have been derived by means of Hamilton’s principle since the late 1950s. Nevertheless, a complete variational formulation of the rotational water-wave problem, including the derivation of the free-surface boundary conditions, seems to be lacking until now. The purpose of the present work is to construct such a missing variational formulation. The appropriate functional is the usual Hamilton’s action, constrained by the conservation of mass and the conservation of fluid parcels’ identity. The differential equations governing the bulk fluid motion are derived as usually, applying standard methods of the calculus of variations. However, the standard methodology does not provide enough structure to obtain the free-surface boundary conditions. To overcome this difficulty, differential-variational forms of the aforementioned constraints are introduced and applied to the boundary variations of the Eulerian fields. Under this transformation, both kinematic and dynamic free-surface conditions are naturally derived, ensuring the Hamiltonian variational formulation of the complete problem. An interesting feature, appearing in the present variational derivation, is a dual possibility concerning the tangential velocity on the boundary; it may be either the same as in irrotational flow (no condition) or zero, corresponding to the small-viscosity limit. The deeper meaning and the significance of these findings seem to deserve further analysis.

A canonical Hamiltonian formulation of the Navier–Stokes problem

“…The application of analytical mechanics (Goldstein 1980;Arnold 1989;Fetter & Walecka 2003;Gelfand & Fomin 2012) to the field of fluid mechanics (Lanczos 1970) has recently seen a resurgence in interest (Salmon 1983(Salmon , 1988Brenier 2017;Giga, Kirshtein & Liu 2018;Mottaghi, Gabbai & Benaroya 2019;Taroco, Blanco & Feijoo 2020;Bedford 2021;Mavroeidis & Athanassoulis 2022) after a long history. In the absence of non-conservative forces, an inviscid fluid is a Hamiltonian system, and so the classical Hamiltonian theory applies.…”