A two-layer shallow flow model with two axes of integration, well-balanced discretization and application to submarine avalanches

We propose a two-layer model with two different axes of integration and a well-balanced finite volume method. The purpose is to study submarine avalanches and generated tsunamis by a depth-averaged model with different averaged directions for the fluid and the granular layers. Two-layer shallow depth-averaged models usually consider either Cartesian or local coordinates for both layers. However, the motion characteristics of the granular layer and the water wave are different: the granular flow velocity is mainly oriented downslope while water motion related to tsunami wave propagation is mostly horizontal. As a result, the shallow approximation and depth-averaging have to be imposed (i) in the direction normal to the topography for the granular flow and (ii) in the vertical direction for the water layer. To deal with this problem, we define a reference plane related to topography variations and use the associated local coordinates to derive the granular layer equations whereas Cartesian coordinates are used for the fluid layer. Depthaveraging is done orthogonally to that reference plane for the granular layer equations and in the vertical direction for the fluid layer equations. Then, a finite volume method is defined based on an extension of the hydrostatic reconstruction. The proposed method is exactly well-balanced for two kind of stationary solutions: the classical one, when both water and granular masses are at rest; the second one, when only the granular mass is at rest. Several tests are presented to get insight into the sensitivity of the granular flow, deposit and generated water waves to the choice of the coordinate systems. Our results show that even for moderate slopes (up to 30◦), strong relative errors on the avalanche dynamics and deposit (up to 60%) and on the generated water waves (up to 120%) are made when using Cartesian coordinates for both layers instead of an appropriate local coordinate system as proposed here.Ministerio de Economía y Competitividad (MINECO). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Agence Nationale de la Recherche. FranceEuropean Research Council (ERC

A weakly non-hydrostatic shallow model for dry granular flows

A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the $\mu(I)$ rheology is considered. The model is obtained from an asymptotic analysis in a local reference system, where the non-hydrostatic contribution is supposed to be small compared to the hydrostatic one. The non-hydrostatic counterpart of the pressure may be written as the sum of two terms: one corresponding to the stress tensor and the other to the vertical acceleration. The model introduced here is weakly non-hydrostatic, in the sense that the non-hydrostatic contribution related to the stress tensor is not taken into account due to its complex implementation. A simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure, and it is based on a hydrostatic reconstruction. Two key points are: (i) the friction force has to be taken into account before solving the non-hydrostatic pressure. Otherwise, the incompressibility condition is not ensured; (ii) both the hydrostatic and the non-hydrostatic pressure are taken into account when dealing with the friction force. The model and numerical scheme are then validated based on several numerical tests, including laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models. An important result is that the simulated mass profiles up to the deposit and the front velocity are greatly improved. As expected, the influence of the non-hydrostatic pressure is shown to be larger for small values of the slope

A bed pressure correction of the friction term for depth-averaged granular flow models

Depth-averaged models, such as the Savage-Hutter model with Coulomb or Pouliquen fric tion laws, do not in some cases preserve the physical threshold of motion. In particular, the simulated granular mass can start to flow (or stay at rest) even if the slope angle of its free surface is lower (or higher) than the repose angle of the granular material involved. The problem is related to the hydrostatic pressure assumption, associated with the direction of integration, which is orthogonal to a reference plane or a reference bottom. We propose here an initial method to correct this misleading behavior. Firstly, we define a correction of the friction term that accounts for the Jacobian of a change of coordinates, making it possible to reproduce the physical threshold of motion and thus the solutions at rest. Sec ondly, we observe that the 3D model presented in [F. Bouchut, I. Ionescu, and A. Mangeney. An analytic approach for the evolution of the static-flowing interface in viscoplastic granular flows. Commun, Math. Sci., 14(8):2101–2126, 2016] verifies the physical thresholds of mo tion because it is based on a second order correction of the pressure valid for slow granu lar flows. The correction proposed here ensures that the model preserves, up to the second order, the physical threshold of motion defined by the repose angle of the material. Sev eral numerical tests are presented to illustrate certain problems related to classical depth averaged models and the remedial effect of the proposed correction, in particular through comparisons with experimental data. We finally show that this correction is not exact far from the starting and stopping phases of the granular avalanche and should be improved by adding other second order terms in the pressure approximationMinisterio de Ciencia, Innovación y Universidades RTI2018- 096064-B-C22European Research Council ERC-CG-2013-PE10-617472 SLIDEQUAKE

A robust model for rapidly varying flows over movable bottom with suspended and bedload transport: modelling and numerical approach

We propose a coupled model for suspended and bedload sediment transport in the shallow water framework. The model is deduced under hydrostatic pressure assumptions and will not assume any Bossinesq hypothesis. The numerical resolution is carried out in a segregated way. First the underlying system of conservation laws is solved by using a first order path-conservative Riemann solver. Then, the source terms corresponding with the erosion and depositions rates are approximated in a semi-implicit way. The final scheme preserves the positivity of the density. Several numerical experiments were carried out in order to validate the model and the numerical scheme. The results obtained are in good agreement with the experimental data

Simulation des \'ecoulements gravitaires avec les mod\`eles d'\'ecoulement en couche mince: \'etat de l'art et exemple d'application aux coul\'ees de d\'ebris de la Rivi\`ere du Pr\^echeur (Martinique, Petites Antilles)

Quantifying the propagation of landslides is a key step for analyzing gravitational risks. In this context, thin-layer models have met a growing success over the past years to simulate the dynamics of gravitational flows, such as debris flows. They are easier to use and require less computing ressources than 3D models, and provide more detailed estimations of flow thicknesses and velocities than purely empirical methods. In this literature review, we present the main rheologies used to model homogeneous gravitational flows, and describe a practical case study with debris flow modeling in the Pr{\^e}cheur River (Martinique, Lesser Antilles). Then, we discuss possible developments for operational hazard and risk assessment with thin-layer models.Comment: in French languag

Depth-averaged and 3D Finite Volume numerical models for viscous fluids, with application to the simulation of lava flows

This Ph.D. project was initially born from the motivation to contribute to the depth-averaged and 3D modeling of lava flows. Still, we can frame the work done in a broader and more generalist vision. We developed two models that may be used for generic viscous fluids, and we applied efficient numerical schemes for both cases, as explained in the following. The new solvers simulate free-surface viscous fluids whose temperature changes are due to radiative, convective, and conductive heat exchanges. A temperature-dependent viscoplastic model is used for the final application to lava flows. Both the models behind the solvers were derived from mass, momentum, and energy conservation laws. Still, one was obtained by following the depth-averaged model approach and the other by the 3D model approach. The numerical schemes adopted in both our models belong to the family of finite volume methods, based on the integral form of the conservation laws. This choice of methods family is fundamental because it allows the creation and propagation of discontinuities in the solutions and enforces the conservation properties of the equations. We propose a depth-averaged model for a viscous fluid in an incompressible and laminar regime with an additional transport equation for a scalar quantity varying horizontally and a variable density that depends on such transported quantity. Viscosity and non-constant vertical profiles for the velocity and the transported quantity are assumed, overtaking the classic shallow-water formulation. The classic formulation bases on several assumptions, such as the fact that the vertical pressure distribution is hydrostatic, that the vertical component of the velocity can be neglected, and that the horizontal velocity field can be considered constant with depth because the classic formulation accounts for non-viscous fluids. When the vertical shear is essential, the last assumption is too restrictive, so it must relax, producing a modified momentum equation in which a coefficient, known as the Boussinesq factor, appears in the advective term. The spatial discretization method we employed is a modified version of the central-upwind scheme introduced by Kurganov and Petrova in 2007 for the classical shallow water equations. This method is based on a semi-discretization of the computational domain, is stable, and, being a high-order method, has a low numerical diffusion. For the temporal discretization, we used an implicit-explicit Runge-Kutta technique discussed by Russo in 2005 that permits an implicit treatment of the stiff terms. The whole scheme is proved to preserve the positivity of flow thickness and the stationary steady-states. Several numerical experiments validate the proposed method, show the incidence on the numerical solutions of shape coefficients introduced in the model and present the effects of the viscosity-related parameters on the final emplacement of a lava flow. Our 3D model describes the dynamics of two incompressible, viscous, and immiscible fluids, possibly belonging to different phases. Being interested in the final application of lava flows, we also have an equation for energy that models the thermal exchanges between the fluid and the environment. We implemented this model in OpenFOAM, which employs a segregated strategy and the Finite Volume Methods to solve the equations. The Volume of Fluid (VoF) technique introduced by Hirt and Nichols in 1981 is used to deal with the multiphase dynamics (based on the Interphase Capturing strategy), and hence a new transport equation for the volume fraction of one phase is added. The challenging effort of maintaining an accurate description of the interphase between fluids is solved by using the Multidimensional Universal Limiter for Explicit Solution (MULES) method (described by Marquez Damian in 2013) that implements the Flux-Corrected Transport (FCT) technique introduced by Boris and Book in 1973, proposing a mix of high and low order schemes. The choice of the framework to use for any new numerical code is crucial. Our contribution consists of creating a new solver called interThermalRadConvFoam in the OpenFOAM framework by modifying the already existing solver interFoam (described by Deshpande et al. in 2012). Finally, we compared the results of our simulations with some benchmarks to evaluate the performances of our model

The Origin of the Apparent Hysteretic Friction in Granular Rheology

In the transition from a flowing to a stationary regime, granular material exhibits hysteretic behaviour, illustrated by a difference between the critical starting and stopping angle. The hysteretic phenomena have commonly been attributed to the rheological parameters of the medium, leading to the development of sophisticated, but often complex constitutive relations based on different physical explanations occurring at the particle scale. While the definition of one all-encompassing rheological law is still open for debate, one of the most widely accepted theories is the μ(I)-rheology. Nevertheless, the theory is sometimes considered to be incomplete, since it is not an inherently hysteretic model. Here, however, it is argued that we should not look for the origin of the hysteretic behaviour in the rheological parameters, but instead the observed phenomena can be attributed to the role of momentum. To support this argument, this study firstly presents the development of a one-dimensional model which is capable, to an extent, of describing the observed phenomena, without requiring any new constitutive assumptions apart from the μ(I)-rheology. The solution is initially presented in a depth-averaged framework, and subsequently, takes form as a depth-resolved erosion model. Secondly, the role of the momentum, reflected through the control parameters: the angular rate at which the inclination angle is changed and the threshold velocity, defined as the velocity of the granular layer at the onset of flow, is evaluated through a set of experiments. The experimental results show that the observed phenomena in the transition from static to dynamic granular rheology can be explained by the generation and transfer of momentum. This study, therefore, stresses that before developing new constitutive concepts, the contribution of momentum as part of the boundary value problem needs to be isolated and properly understood

Gravity currents from non-axisymmetric releases

Les courants de gravité, écoulements issus de la présence d’un contraste de densité dans un fluide ou de la présence de fluides de densités différentes, sont rencontrés dans de nombreuses situations naturelles ou industrielles. Quelques exemples de courants de gravité sont les avalanches, les marées noires et les courants de turbidité. Certains courants de gravité peuvent représenter un danger pour l’homme ou l’environnement, il est donc nécessaire de comprendre et de prédire leur dynamique. Cette thèse a pour objectif d’étudier l’évolution de courants de gravité de masse fixée, et notamment l’influence d’une forme initiale non-axisymétrique sur la dynamique, effet jusque-là peu abordé dans la littérature. Pour cela, une large gamme de paramètres est couverte, incluant le rapport de masse volumique entre le fluide ambiant et le fluide dans le courant, le rapport de forme initiale, la forme de la section horizontale de la colonne de fluide (circulaire, rectangulaire ou en forme de croix), le nombre de Reynolds (couvrant jusqu’à 4 ordres de grandeur) et la nature du fluide lourd (salin ou chargé en particules). Deux campagnes d’expériences ont été menées et complétées par des simulations numériques hautement résolues. Le résultat majeur est que la propagation du courant et le dépôt de particules (lorsque particules il y a) sont fortement influencés par la forme initiale de la colonne de fluide. Dans le cas de la colonne initialement rectangulaire le courant se propage plus vite et dépose plus de particules dans la direction initialement de plus courte dimension. Ce comportement non-axisymétrique est observé dans une large gamme des paramètres étudiés ici. Pourtant les modèles analytiques existants et notamment le modèle dit de boîte (box model) qui prédit avec succès le comportement des courants de gravité/turbidité dans les cas plan et axisymétrique ne sont pas capables de reproduire ce phénomène. C’est pourquoi une extension du box model a été développée ici, et est en mesure de décrire la dynamique de courants de gravité de masse fixée dont la forme initiale est arbitraire. Le cas plus général d'un courant de gravité évoluant sur un plan incliné a été abordé et une dynamique intéressante a été observée. ABSTRACT : Gravity currents are buoyancy driven flows that appear in a variety of situations in nature as well as industrial applications. Typical examples include avalanches, oil spills, and turbidity currents. Most naturally occurring gravity currents are catastrophic in nature, and therefore there is a need to understand how these currents advance, the speeds they can attain, and the range they might cover. This dissertation will focus on the short and long term evolution of gravity currents initiated from a finite release. In particular, we will focus attention to hitherto unaddressed effect of the initial shape on the dynamics of gravity currents. A range of parameters is considered, which include the density ratio between the current and the ambient (heavy, light, and Boussinesq currents), the initial height aspect ratio (height/radius), different initial cross-sectional geometries (circular, rectangular, plus-shaped), a wide range of Reynolds numbers covering 4 orders of magnitude, as well as conservative scalar and non-conservative (particle-driven) currents. A large number of experiments have been conducted with the abovementioned parameters, some of these experiments were complemented with highly-resolved direct numerical simulations. The major outcome is that the shape of the spreading current, the speed of propagation, and the final deposition profile (for particle-driven currents) are significantly influenced by the initial geometry, displaying substantial azimuthal variation. Especially for the rectangular cases, the current propagates farther and deposits more particles along the initial minor axis of the rectangular cross section. This behavior pertaining to non-axisymmetric release is robust, in the sense that it is observed for the aforementioned range of parameters, but nonetheless cannot be predicted by current theoretical models such as the box model, which has been proven to work in the context of planar and axisymmetric releases. To that end, we put forth a simple analytical model (an extension to the classical box model), well suited for accurately capturing the evolution of finite volume gravity current releases with arbitrary initial shapes. We further investigate the dynamics of a gravity current resulting from a finite volume release on a sloping boundary where we observe some surprising features

Modélisation d’écoulements multiphasiques environnementaux et géophysiques

Ce travail recense les activités de recherches menées à l’IMFT depuis mon recrutement en 2008 et qui portent sur la « modélisation d’écoulements multiphasiques environnementaux et géophysiques ». Derrière cet intitulé, trois systèmes sont considérés dont le point commun est qu’ils mettent en jeu des écoulements d’un ou plusieurs fluides avec la présence éventuelle d’une phase solide dispersée sous forme de grains ou particules. Ces systèmes sont (i) les milieux granulaires immergés, (ii) les courants de gravité / turbidité et (iii) les milieux diphasiques visqueux, rencontrés dans de nombreuses situations naturelles (transport sédimentaire, avalanches, éruptions volcaniques) couvrant une large gamme d’échelles allant typiquement du millimètre au kilomètre. Je présenterai via quelques exemples de travaux issus des deux premiers axes de recherche, la méthodologie que j’ai adoptée et qui consiste à utiliser principalement la simulation numérique afin d’analyser les processus importants du problème dans des configurations idéalisées et d’en tirer des modèles simplifiés permettant de reproduire dans des configurations plus complexes la dynamique de ces systèmes

Book of Abstracts

no abstrac