Carine Lucas

SWASHES: a compilation of shallow water analytic solutions for hydraulic and environmental studies

Numerous codes are being developed to solve Shallow Water equations. Because there are used in hydraulic and environmental studies, their capability to simulate properly flow dynamics is critical to guarantee infrastructure and human safety. While validating these codes is an important issue, code validations are currently restricted because analytic solutions to the Shallow Water equations are rare and have been published on an individual basis over a period of more than five decades. This article aims at making analytic solutions to the Shallow Water equations easily available to code developers and users. It compiles a significant number of analytic solutions to the Shallow Water equations that are currently scattered through the literature of various scientific disciplines. The analytic solutions are described in a unified formalism to make a consistent set of test cases. These analytic solutions encompass a wide variety of flow conditions (supercritical, subcritical, shock, etc.), in 1 or 2 space dimensions, with or without rain and soil friction, for transitory flow or steady state. The corresponding source codes are made available to the community (http://www.univ-orleans.fr/mapmo/soft/SWASHES), so that users of Shallow Water-based models can easily find an adaptable benchmark library to validate their numerical methods.

FullSWOF: A software for overland flow simulation / FullSWOF : un logiciel pour la simulation du ruissellement

Overland flow on agricultural fields may have some undesirable effects such as soil erosion, flood, and pollutant transport. To better understand this phenomenon and limit its consequences, we developed a code using state-of-the-art numerical methods: Full Shallow Water equations for Overland Flow (FullSWOF ), an object-oriented code written in C++. It has been made open-source and can be downloaded from http://www.univ-orleans.fr/mapmo/soft/FullSWOF/. The model is based on the classical system of shallow water (SW) (or Saint–Venant system). Numerical difficulties come from the numerous dry/wet transitions and the highly variable topography encountered inside a field. The code includes run-on and rainfall inputs, infiltration (modified Green-Ampt equation), and friction (Darcy-Weisbach and Manning formulas). First, we present the numerical method for the resolution of the SW equations integrated in FullSWOF_2D (the two-dimensional version). This method is based on hydrostatic reconstruction scheme, coupled with a semi-implicit friction term treatment. FullSWOF_2D has been previously validated using analytical solutions from the Shallow Water Analytic Solutions for Hydraulic and Environmental Studies library (SWASHES). FullSWOF_2D is run on a real topography measured on a runoff plot located in Thies (Senegal). Simulation results are compared with measured data. This experimental benchmark demonstrates the capabilities of FullSWOF to simulate adequately overland flow. FullSWOF could also be used for other environmental issues, such as river floods and dam breaks.

A faster numerical scheme for a coupled system modeling soil erosion and sediment transport

Overland flow and soil erosion play an essential role in water quality and soil degradation. Such processes, involving the interactions between water flow and the bed sediment, are classically described by a well-established system coupling the shallow water equations and the Hairsine-Rose model. Numerical approximation of this coupled system requires advanced methods to preserve some important physical and mathematical properties; in particular the steady states and the positivity of both water depth and sediment concentration. Recently, finite volume schemes based on Roe’s solver have been proposed by Heng et al. (2009) and Kim et al. (2013) for one and twodimensional problems. In their approach, an additional and artificial restriction on the time step is required to guarantee the positivity of sediment concentration. This artificial condition can lead the computation to be costly when dealing with very shallow flow and wet/dry fronts. The main result of this paper is to propose a new and faster scheme for which only the CFL condition of the shallow water equations is sufficient to preserve the positivity of sediment concentration. In addition, the numerical procedure of the erosion part can be used with any well-balanced and positivity preserving scheme of the shallow water equations. The proposed method is tested on classical benchmarks and also on a realistic configuration.

AN ENERGETICALLY CONSISTENT VISCOUS SEDIMENTATION MODEL

In this paper we consider a two-dimensional viscous sedimentation model which is a viscous Shallow–Water system coupled with a diffusive equation that describes the evolution of the bottom. For this model, we prove the stability of weak solutions for periodic domains and give some numerical experiments. We also discuss around various discharge quantity choices.

New Developments and Cosine Effect in the Viscous Shallow Water and Quasi-Geostrophic Equations

The viscous Shallow Water Equations and Quasi-Geostrophic Equations are considered in this paper. Some new terms, related to the Coriolis force, are revealed thanks to a rigorous asymptotic analysis. After providing well-posedness arguments for the new models, the authors perform some numerical computations that confirm the role played by the cosine effect in various physical configurations.

Multiscale Analyses for the Shallow Water Equations

This paper explores several asymptotic limit regimes for shallow water flows over multiscale topography. Depending on the length and time scales considered and on the characteristic water depth and height of topography, a variety of mathematically quite different asymptotic limit systems emerges. Specifically, we recover the classical “lake equations” for balanced flow without gravity waves in the single time, single space scale limit (Greenspan, Cambridge Univ. Press, (1968)), discuss a weakly nonlinear and a strongly nonlinear multi-scale version of these wave-free equations involving short-range topography, and we re-derive the equations for long-wave shallow water waves passing over short-range topography by Le Maitre et al., JCP (2001).

Cosine Effect on Shallow Water Equations and Mathematical Properties

This paper presents a viscous Shallow Water type model with new Coriolis terms, and some limits according to the values of the Rossby and Froude numbers. We prove that the extension to the bidimensional case of the unidimensional results given by [J.―F. GERBEAU, B. PERTHAME. Discrete Continuous Dynamical Systems, (2001)] including the Coriolis force has to add new terms, omitted up to now, depending on the latitude cosine, when the viscosity is assumed to be of the order of the aspect ratio. We show that the expressions for the waves are modified, particularly at the equator, as well as the Quasi-Geostrophic and the Lake equations. To conclude, we also study the mathematical properties of these equations.

SWASHES: A Library for Benchmarking in Hydraulics

Numerous codes are being developed to solve shallow water equations. Because they are used in hydraulic and environmental studies, their capability to simulate properly flow dynamics is essential to guarantee infrastructure and human safety. Hence, validating these codes and the associated numerical methods is an important issue. Analytic solutions would be excellent benchmarks for these issues. However, analytic solutions to shallow water equations are rare. Moreover, they have been published on an individual basis over a period of more than five decades, making them scattered through the literature. In this chapter, a significant number of analytic solutions to the shallow water equations are described in a unified formalism. They encompass a wide variety of flow conditions (supercritical, subcritical, shock …), in one or two space dimensions, with or without rain and soil friction, for transitory flow or steady state. An original feature is that the corresponding source codes are made freely available to the community (http://www.univ-orleans.fr/mapmo/soft/SWASHES), so that users of shallow water–based models can easily find an adaptable benchmark library to validate their numerical methods.

FullSWOF: Full Shallow-Water equations for Overland Flow

Numerical simulations of shallow flows are required in numerous applications and are typically performed by solving shallow-water equations. FullSWOF solves these equations by using up-to-date finite volume methods and well-balanced schemes. Several features make FullSWOF particularly suitable for surface water hydrologists: small water depths and wet-dry transitions are robustly addressed, rainfall and infiltration are incorporated, and grid-based digital topographies can be used directly. The modular structure of FullSWOF is also useful to numerical modelers willing to test new schemes or boundary conditions.