Variational Models for Nonhydrostatic Free-Surface Flow: A Unified Outlook to Maritime and Open-Channel Hydraulics Developments
Publication: Journal of Hydraulic Engineering
Volume 149, Issue 7
Abstract
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.
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Data Availability Statement
The MATLAB scripts developed for algebraic symbolic computations in this research are available from the corresponding author upon reasonable request.
Acknowledgments
The work of the first Author was supported by the Spanish project PID2020-114688RB-I00 and grant María de Maeztu for Centers and Units of Excellence in R&D (Ref. CEX2019-000968-M). He acknowledges Prof. James T. Kirby, University of Delaware, for his comments on the use of weighted-residual approaches in maritime hydraulics, and Prof. Michael Bestehorn, Brandenburg University of Technology Cottbus, for his advice on water wave modeling with sigma mapping. The Authors acknowledge the work of the Associate Editor and three referees of this work, who offered many comments to improve the paper.
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Published In
Journal of Hydraulic Engineering
Volume 149 • Issue 7 • July 2023
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© 2023 American Society of Civil Engineers.
History
Received: May 9, 2022
Accepted: Jan 19, 2023
Published online: Apr 19, 2023
Published in print: Jul 1, 2023
Discussion open until: Sep 19, 2023
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