Vertically Averaged and Moment Equations for Dam-Break Wave Modeling: Shallow Water Hypotheses

Francisco N Cantero-Chinchilla,Oscar Castro-Orgaz,Rafael J Bergillos,Luis Cea,Pedro Gamero,Willi H Hager

doi:10.3390/w12113232

Francisco N Cantero-Chinchilla, Oscar Castro-Orgaz + Show 4 more

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https://doi.org/10.3390/w12113232

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Abstract

The dam-break wave modeling technology relies upon the so-called shallow water equations (SWE), i.e., mass and momentum vertically averaged equations by implementing the shallow water hypotheses, namely (i) horizontal velocity component independent of the vertical coordinate, (ii) vertical velocity component is null, (iii) pressure distribution is hydrostatic, (iv) turbulence is neglected. While this model often yields a satisfactory answer from an engineering standpoint, flows with vertical length scales not negligible cannot be modeled with accuracy, including the undular surge generated after a dam break for relatively high tailwater levels. These flows are modeled by the Serre–Green–Naghdi equations (SGNE), which fail to mimic wave breaking for low tailwater levels, however. Neither SWE nor SGNE produce a fully satisfactory answer for modeling dam break waves, therefore. A higher-order model using vertically averaged and moment equations (VAM) is used in this work to simulate dam break waves, thereby showing good results for arbitrary values of the tailwater level. The model contains four perturbation parameters implemented to overcome the shallow water hypotheses; two for the velocity components and two for fluid pressure. The role of each parameter in relaxing the limitations of the SWE is systematically investigated, depicting a complex and necessary interplay between the dynamic component of fluid pressure and the modeling of the velocity profile in producing accurate solutions for both non-hydrostatic and broken waves in dam break flows. The results highlight how the shallow water hypotheses can be relaxed in the vertically averaged modeling of dam break waves, producing an outcome of both theoretical and practical interest in the field. The results generated are tested with available experimental data, resulting in acceptable agreement.

Highlights

Dam-break waves are rapidly varied, unsteady free surface phenomena occurring frequently in hydraulic and environmental engineering
The shallow water hypotheses in dam-break flow modeling are analyzed within a vertically averaged framework using a vertically averaged and moment (VAM) equations model containing perturbation parameters introduced to overcome the limitations of the shallow water equations (SWE)
The results reveal that the predictions by the VAM model are more accurate than those obtained with the SWE model, concluding that the inclusion of non-hydrostatic pressure effects is important to model dam-break flows

Summary

Introduction

Dam-break waves are rapidly varied, unsteady free surface phenomena occurring frequently in hydraulic and environmental engineering. A feasible approximation widely used in practice is to simulate dam-break waves resorting to vertically integrated models. In those models, the basic idea is to remove the vertical coordinate from the model equations by undertaking a vertical integration and adopting suitable hypotheses on the field variables, e.g., the velocity and pressure fields, and, sometimes, on the turbulence. The most applied vertically integrated system of equations in this modeling technology appears to be the so-called de Saint Venant equations or shallow water equations (SWE) [3,6] These equations are a system of two hyperbolic conservation laws for mass and momentum (1D case) derived by implementing the so-called shallow water hypotheses in the vertical-integration of the RANS equations, namely (i)

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