This work is devoted to the numerical simulation of Shallow Water Equations involving dry areas and a moving shoreline. The space and time discretization using the Runge-Kutta Discontinuous Galerkin approach is applied to nonlinear hyperbolic Shallow Water Equations. Problems with dry areas are challenging problems for such methods. To counter this issue, special treatment is applied around the shoreline. This work compares two treatments, one based on slope modification and one based on p-adaptation.

This paper deals with the numerical simulation of time-dependant flows in partially saturated porous media modelled by Richards equation. A Discontinuous Galerkin method together with an implicit scheme are used to approximate the solution. The mathematical framework and a procedure for solving this nonlinear equation are presented. In particular, treatment of seepage boundary condition and adaptive time step are underlined. An adaptive mesh refinement technique is also outlined in order to capture wetting front efficiently. Some numerical simulations are performed to illustrate the performance of the method.

This work is devoted to the numerical simulation of flows in partially saturated porous media. We describe the Richards equation governing the subsurface flow and discuss its range of applicability. A discontinuous Galerkin formulation is used to approximate the steady-state Richards equation. To this end, we present the mathematical framework and a procedure for solving the nonlinear equation. Numerical tests are carried out to highlight properties of the discontinuous Galerkin method and a test case is compared to experimental data to validate the model.

We consider the interaction between an air-water flow and a rigid structure using a fast accurate numerical model.Following the 3D low Mach compressible Euler model previously developed and validated, a compressible fluid-structure interaction model is proposed.In the context of a fictitious domain, a volumic penalization is applied inside the body to ensure a rigidity constraint through a penalized velocity in order to get the correct motion of the rigid body.The accuracy of this simple model is improved using block-based adaptive mesh refinement technique.The improvement and the validity of our fluid-structure interaction procedure is investigated through the numerical simulation of the water entry of a cylinder freely falling and impacting the water.This test case is compared to experimental data and other numerical simulations.

In the tsunami waves context, efficient numerical methods are necessary to simulate multi scales events. One way to reduce the computational cost is to use an adaptive mesh refinement method on unstructured meshes. This approach is used in this paper with a finite volume scheme solving the multi-dimensional Saint-Venant system. The adaptive mesh refinement method follows a block-based decomposition (called BB-AMR), which allows quick meshing and easy parallelization. One step of the AMR method is the so-called projection step where the new mesh values have to be defined from the old ones. For vertically integrated model the bathymetry variation during the projection step plays a crucial role on the mass conservation. To avoid large deficit or excess of mass a special attention is given to the projection step. Finally a 3D test case simulation is compared to experimental results to illustrated the quasi conservation of the mass with an appropriated projection method.

We propose to bring a better understanding of the interaction between an air-water flow and a floating structure by means of a fast accurate numerical model. Following the 3D low Mach compressible Euler model developed in [14] and previous works from [20], [11] and [13], we are now interested in simulating the motion of a floating structure in an air-water flow. In the context of a fictitious domain, a volumic penalization is applied inside the body to ensure a rigidity constraint through a penalized velocity in order to get the correct motion of the rigid body. The tracking of the solid is insured by the reconstruction of a Heaviside function thanks to a ray-casting algorithm. The validity of our fluid-structure interaction (FSI) procedure is investigated through some examples.

This study focuses on the numerical modeling of the surface erosion occurring at a fluid/soil interface undergoing a flow process. Following a previous work [3], the balance equations with jump relations are used and a penalization procedure is used to compute Navier-Stokes equations around obstacles, with a fictitious domain method, in order to avoid body-fitted unstructured meshes. The water/soil interface evolution is described with a Level Set function coupled to a threshold erosion law. In order to allow adaptive mesh refinement, we develop a Discrete Duality Finite Volume scheme (DDFV). The ability of the model to predict the interfacial erosion of soils is confirmed by presenting several simulations.

We propose an adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). Thus it is used as an a posteriori error which provides information on the need to refine the mesh in the regions where discontinuities occur and to coarsen the mesh in the regions where the solutions remain smooth. Nevertheless, due to the CFL stability condition the time step is restricted and leads to time consuming simulations. Therefore, we propose a local time stepping algorithm. This approach is validated in the case of classical 1D numerical tests. Then, we present a 3D application. Indeed, according to an eulerian bi-fluid formulation at low Mach, an hyperbolic system of conservation laws allows an easily parallelization and mesh refinement by "blocks".

This study focuses on the numerical modelling of the concentrated leak erosion of a cohesive soil by turbulent flow in axisymmetrical geometry, using the Hole Erosion Test (HET). The numerical model is based on the adaptive remeshing of the water/soil interface to ensure the accurate description of the mechanical phenomena occurring near the soil/water interface. The erosion law governing the interface motion is based on two erosion parameters: critical shear stress and the erosion coefficient. The model is first validated in the case of 2D piping erosion caused by laminar flow. Then, the numerical results are compared with the interpretation model of the HET. Three HETs performed on different soils are modelled with rather good accuracy. Lastly, a parametric analysis of the influence of the erosion parameters on erosion kinetics and the evolution of the channel diameter is performed. Finally, after this validation by comparison with both the experimental results and the interpretation of Bonelli et al. [2], our model is now able to accurately reproduce the erosion of a cohesive soil by a concentrated leak. It also provides a detailed description of all the averaged hydrodynamic flow quantities. This detailed description is essential in order to achieve better understanding of erosion processes.

In this work, we present a fast and parallel finite volume scheme on unstructured meshes applied to complex fluid flow. The mathematical model is based on a three-dimensional compressible low Mach two-phase flows model, combined with a linearised 'artificial pressure' law. This hyperbolic system of conservation laws allows an explicit scheme, improved by a block-based adaptive mesh refinement scheme. Following a previous one-dimensional work, the useful numerical density of entropy production is used as mesh refinement criterion. Moreover, the computational time is preserved using a local time-stepping method. Finally, we show through several test cases the efficiency of the present scheme on two- and three-dimensional dam-break problems over an obstacle.

Evaluating the erodibility of a soil, both in terms of erosion threshold (initiation) and erosion rate (progression), is critical for the evaluation of the safety of water retaining structures. Indeed different soils can erode at different rates. However, the relationship between the erosion parameters and the geotechnical and chemical properties of soils remains largely unknown. The jet erosion test appears to be an efficient and simple means for quantifying the two erosion parameters involved. The first parameter is the critical stress while the second parameter is the erosion coefficient. A simplified model of this test has been drawn up by G. Hanson et al. to interpret the experimental curves. Few attempts have been made so far to model the whole process, however. The aim of this study is to simulate the impinging jet and to take into account the erosion of the soil by means of computational fluid dynamics (CFD) numerical modelling. The key point was the time dependence of the problem, due to erosion processes, however the turbulent flow could be considered as steady because of the assumption of low kinetics erosion assumption. The results of the present modelling study are compared to the simplified model and to experimental data. This comparison is a first confirmation of the validity of the simplified model as a means of assessing the critical stress and the erosion coefficient with jet erosion tests.

The jet erosion test (JET) is an experimental device increasingly used to quantify the resistance of soils to erosion. This resistance is characterized by two geotechnical parameters: the critical shear stress and the erosion coefficient. A previously published JET interpretation model provides an estimation of these erosion parameters. But the existing model is simplified and semiempirical and several assumed hypotheses can be discussed. The aim of this study is to determine the relevance of the JET interpretation model. Therefore, a numerical model was developed that is able to predict the erosion of a cohesive soil by a turbulent flow. The numerical model was first validated on a benchmark: erosion of an erodible pipe by a laminar flow. The numerical results were satisfactorily compared with the theoretical solution. Then, three JETs were modeled numerically with values of erosion parameters obtained experimentally. A parametric study was also conducted to validate the accuracy of the numerical results and a good agreement was observed. The erosion parameters found experimentally permit the numerical prediction of the evolution of the erosion pattern within good accuracy. This result contributes to the validation of the JET's semiempirical model. The numerical model also gives a complete description of the flow, including vortices which can be observed in the cavity created by erosion. The entire erosion pattern evolution was given by the numerical results. This numerical model gives information that is not available otherwise.

This study focuses on 2D Computational Fluid Dynamics (CFD) numerical modelling of the erosion of a cohesive soil by a circular impinging turbulent jet. Initially, the model is validated in the case of a non erodible flat plate. Several turbulence models are compared to experimental results and to simplified formulas available in the literature. The results obtained show that the Reynolds Stress Model (RSM) is in good agreement with the semi-empirical results in the literature. Nonetheless, the RSM cannot be used with successive remeshings, due to its convergence issues. The shear stress at the wall is well-described by the k–ε model while the pressure is better-described by the k–ω model. The numerical model of erosion is based on adaptive remeshing of the water/soil interface to ensure the good precision of the mechanical values at the wall. The two erosion parameters are the critical shear stress and the erosion coefficient. The results obtained are compared with the semi-empirical model interpreting the Jet Erosion Test. The k–ε model underestimates the shear stress and does not allow simulation of the entire erosion process, whereas the results obtained with the k–ω model agree well with the semi-empirical model and experimental data. A study of the influence of erosion parameters on erosion kinetics and scouring depth shows that the shape and depth of scouring are influenced solely by the critical shear stress while the duration of scouring depends on both erosion parameters. Further research is nonetheless required to better understand the erosion mechanisms in the stagnation zone.

The aim of this study was to develop a numerical mo del for simulating the surface erosion occurring at a fluid/soil interface undergoing a flow process. The balance equations with jump relations are used. A penalization procedure is used to compute Stokes eq uations around obstacles, with a fictitious domain method, in order to avoid body-fitted unstructured meshes and to use fast and efficient finite volumes approximations on Cartesian meshes. The water/soil interface evolution is described with a Level Set function. The ability of the model to predict the i nterfacial erosion of soils is confirmed by present ing several 2-D simulations. Mots clefs : Erosion, Fictitious domains, Level Set

This paper is devoted to the numerical simulation of wave breaking. It presents the results of a numerical workshop that was held during the conference LOMA04. The objective is to compare several mathematical models (compressible or incompressible) and associated numerical methods to compute the flow field during a wave breaking over a reef. The methods will also be compared with experiments.

We study a problem of structural optimisation using the fictitious material approach. This is connected with the equilibrium of locking materials, which can be approximated by strongly non linear materials. A finite element simulation allows us to experiment with some conjectures about the topology of the optimal solutions.