Variational Models for Nonhydrostatic Free-Surface Flow: A Unified Outlook to Maritime and Open-Channel Hydraulics Developments

The computation of three-dimensional unsteady non-hydrostatic flows over large domains and/or for long simulation times is frequently conducted in research and practice using approximate methods to avoid the cost of a fully 3D solution. Among these methods are the shallow-water perturbation theories of St. Venant and Boussinesq, and the theory of directed fluid sheets by Green and Naghdi. The latter theory is not limited to shallow-water flows, and it is essentially a weighted-average residual method of Galerkin type, further developed by Shields and Webster for maritime hydraulics. The method is in essence equivalent to the vertically-averaged and moment (VAM) equations developed by Steffler and Jin for open channel flows, although this has not been recognized in the literature. A general framework for constructing VAM models of high-order, by linking maritime and open channel flow developments, is not available in the literature. In this work, a generalized framework for designing weighted-average residual equations for free-surface flow is presented based on the Kantorovich and Krylov method. The development of physically sound expansions for the hydrodynamic variables, and the construction of general systems of VAM equations to determine the unknowns in the expansions by selecting suitable weighting functions, is discussed in detail. The approach produces high-order models, thereby generalizing Steffler and Jin’s development. A hierarchy of high-order VAM models is demonstrated to progressively converge to the exact dispersion relation of periodic waves by increasing the vertical resolution. These models are not limited by any shallowness assumption and exhibit more accurate wave dispersion properties compared to the Serre-Green-Naghdi equations. Computational results show that the VAM equations produce an accurate prediction of dam-break waves and dispersive wave effects over submerged bars.