On geophysical applications of two-layer and two-phase shallow models

The main objective of this talk is to present some geophysical applications that can be studied with two-layer or two-phase shallow models. Namely, applications on avalanches and sediment transport problems. Firstly, several models will be presented, by studying the relations between them. Actually, all the models considered in this talk include a source term corresponding to a Coulomb friction law. One of the difficulties of this term is that it is multi-evaluated for the case of a material at rest. The second part of the presentation focus on an unified formulation to approximate numerically these models, taking into account the difficulty introduced by multi-evaluated source terms.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Modelling and numerical simulation of submarine sediment shallow flows: transport and avalanches

This work presents two different types of models to study submarine sediment movements and its numerical discretization by finite volume methods. First, bedload sediment transport models are introduced. They are characterized by the slow movement of the sediment layer. The movement depends on the type of interaction between the fluid and the sediment layer, that generally is very weak. Secondly, a submarine avalanches model is presented. This model takes into account the coupling between the sediment and the fluid layer. The generated tsunamis by a submarine avalanche can be studied with this model. A family of two-dimensional finite volume methods is introduced. It is based on the fact that the models are invariant under rotations. Finally, two numerical tests are considered. In the first numerical test the spread angle for a sediment bump is studied. In the second one a submarine avalanche is simulated over a bottom with a rectangular bump

Efficient numerical schemes for viscoplastic avalanches. Part 2: the 2D case

This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez–Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. Here we derive such schemes in 2D as the follow up of the companion paper treating the 1D case. Numerical tests include in particular a generalized 2D benchmark for Bingham codes (the Bingham–Couette flow with two non-zero boundary conditions on the velocity) and a simulation of the avalanche path of Taconnaz in Chamonix—Mont-Blanc to show the usability of these schemes on real topographies from digital elevation models (DEM)

Exploring the Interplay between CAD and FreeFem++ as an Energy Decision-Making Tool for Architectural Design

The energy modelling software tools commonly used for architectural purposes do not allow a straightforward real-time implementation within the architectural design programs. In addition, the surrounding exterior spaces of the building, including the inner courtyards, hardly present a specific treatment distinguishing these spaces from the general external temperature in the thermal simulations. This is a clear disadvantage when it comes to streamlining the design process in relation to the whole-building energy optimization. In this context, the present study aims to demonstrate the advantages of the FreeFem++ open source program for performing simulations in architectural environments. These simulations include microclimate tests that describe the interactions between a building architecture and its local exterior. The great potential of this mathematical tool can be realized through its complete system integration within CAD (Computer-Aided Design) software such as SketchUp or AutoCAD. In order to establish the suitability of FreeFem++ for the performance of simulations, the most widely employed energy simulation tools able to consider a proposed architectural geometry in a specific environment are compared. On the basis of this analysis, it can be concluded that FreeFem++ is the only program displaying the best features for the thermal performance simulation of these specific outdoor spaces, excluding the currently unavailable easy interaction with architectural drawing programs. The main contribution of this research is, in fact, the enhancement of FreeFem++ usability by proposing a simple intuitive method for the creation of building geometries and their respective meshing (pre-processing). FreeFem++ is also considered a tool for data analysis (post-processing) able to help engineers and architects with building energy-efficiency-related tasks

A Class of Computationally Fast First Order Finite Volume Solvers: PVM Methods

In this work, we present a class of fast first order finite volume solvers, named as PVM (Polynomial Viscosity Matrix), for balance laws or, more generally, for nonconservative hyperbolic systems. They are defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a Roe matrix. These methods have the advantage that they only need some information about the eigenvalues of the system to be defined, and no spectral decomposition of Roe Matrix is needed. As consequence, they are faster than Roe method. These methods can be seen as a generalization of the schemes introduced by Degond et al. in [12] for balance laws and nonconservative systems. The first-order path conservative methods to be designed here are intended to be used as the basis for higher order methods for multi-dimensional problems. In this work, some well known solvers as Rusanov, Lax-Friedrichs, FORCE (see [30], [8]), GFORCE (see [31], [8]) or HLL (see [18]) are redefined under this form, and then some new solvers are proposed. Finally, some numerical tests are presented and the performance of the numerical schemes are compared among them and with Roe schem

A consistent intermediate wave speed for a well-balanced HLLC solver.

In this work, we present a HLLC scheme modification for application to nonhomogeneous shallow-water equations with pollutant transport. This new version is related to the definition of a consistent approximation of the intermediate wave speed. Numerical results are presented to illustrate the importance of such approximation to get appropriate pollutant concentration profiles

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 di usive 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

Formal derivation of a bilayer model coupling shallow water and Reynolds lubrication equations: evolution of a thin pollutant layer over water

In this paper a bilayer model is derived to simulate the evolution of a thin film flow over water. This model is derived from the incompressible Navier-Stokes equations together with suitable boundary conditions including friction and capillary effects. The derivation is based on the different properties of the fluids, thus, we perform a multiscale analysis in space and time, and a different asymptotic analysis to derive a system coupling two different models: the Reynolds lubrication equation for the upper layer and the shallow water model for the lower one. We prove that the model is provided of a dissipative entropy inequality, up to a second order term. Moreover, we propose a correction of the model by taking into account the second order extention for the pressur that admits an exact dissipative entropy inequality. Two numerical tests are presented. In the first one we compare the numerical results with the viscous bilayer shallow water model proposed in [G. Narbona-Reina, J.D.D. Zabsonré, E.D. Fernández-Nieto, D. Bresch, CMES Comput. Model. Eng. Sci., 2009]. In the second test the objective is to show some of the characteristic situations that can be studied with the proposed model. We simulate a problem of pollutant dispersion near the coast. For this test the influence of the friction coefficient on the coastal area affected by the pollutant is studied

Parametric Study of Thermodynamics in the Mediterranean Courtyard as a Tool for the Design of Eco-Efficient Buildings

Traditionally, people in the Mediterranean region knew that the temperatures in their courtyards were cooler in summer than outside temperature. This paper provides a quantitative study on the usefulness of Mediterranean courtyards as passive energy saving systems. This work is based on the creation of a Computational Fluid Dynamics (CFD) numerical model developed using the open source Freefem++ language. In this work, first the numerical model is tested using simplified-shape courtyards which have been previously studied both physically under controlled parameters, and mathematically through numerical simulations. We also study the most appropriate depth ratio for a courtyard, based on these simplified shapes, depending on the climate. Secondly, we apply the numerical model in a real geometry, the Monte Málaga hotel. We compare the numerical results with the monitored data of the temperature in the courtyard of the hotel. The numerical model takes into account precomputed solar radiation in the walls of the courtyard, the predominant wind and buoyancy effects