We consider the Saint-Venant system for shallow water flows, with nonflat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, and oceans when completed with a Coriolis term, or granular flows when completed with friction. Numerical approximate solutions to this system may be generated using conservative finite volume methods, which are known to properly handle shocks and contact discontinuities. However, in general these schemes are known to be quite inaccurate for near steady states, as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions. This difficulty can be overcome by using the so-called well-balanced schemes. We describe a general strategy, based on a local hydrostatic reconstruction, that allows us to derive a well-balanced scheme from any given numerical flux for the homogeneous problem. Whenever the initial solver satisfies some classical stability properties, it yields a simple and fast well-balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality.

A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows

We review flows of dense cohesionless granular materials, with a special focus on the question of constitutive equations. We first discuss the existence of a dense flow regime characterized by enduring contacts. We then emphasize that dimensional analysis strongly constrains the relation between stresses and shear rates, and show that results from experiments and simulations in different configurations support a description in terms of a frictional visco-plastic constitutive law. We then discuss the successes and limitations of this empirical rheology in light of recent alternative theoretical approaches. Finally, we briefly present depth-averaged methods developed for free surface granular flows.

Flows of Dense Granular Media

A review on the angle of repose of granular materials

Foreword 1. Introduction 2. Interactions at the grain level 3. The granular solid: statics and elasticity 4. The granular solid: plasticity 5. Granular gases 6. The granular liquid 7. Immersed granular media 8. Erosion and sediment transport 9. Geomorphology References Index.

Granular Media: Between Fluid and Solid

The article surveys the current state of our understanding of turbidity currents, with an emphasis on their fluid mechanics. It highlights the significant role these currents play within the global sediment cycle, and their importance in environmental processes and in the formation of hydrocarbon reservoirs. Events and mechanisms governing the initiation of turbidity currents are reviewed, along with experimental observations and findings from field studies regarding their internal velocity and density structure. As turbidity currents propagate over the seafloor, they can trigger the evolution of a host of topographical features through the processes of deposition and erosion, such as channels, levees, and sediment waves. Potential linear instability mechanisms are discussed that may determine the spatial scales of these features. Finally, the hierarchy of available theoretical models for analyzing the dynamics of turbidity currents is outlined, ranging from dimensional analysis and integral models to both depth-averaged and depth-resolving simulation approaches.

Turbidity Currents and Their Deposits

The transient surface flow occurring when a cylindrical pile of dry granular material is suddenly allowed to spread on a horizontal plane is investigated experimentally as a function of the released mass M, the initial aspect ratio a of the granular cylinder pile, the properties of the underlying substrate (smooth or rough, rigid or erodible) and the bead size. Two different flow regimes leading to three different deposit morphologies are observed as a function of the initial aspect ratio a, whatever the substrate properties and the bead size. For a≲3, the granular mass spreads through an avalanche on its flanks producing either truncated cone or conical deposits. For a≳3, the upper part of the column descends conserving its shape while the foot of the pile propagates radially outward. The obtained deposit looks like a “Mexican hat” and the slope angle at the foot of the deposit is observed to saturate at a value of the order of 5°. For a given ground and bead size, the flow dynamics and the deposit morph...

https://www.ipgp.jussieu.fr/~mangeney/etal_exp.pdf

Spreading of a granular mass on a horizontal plane

[1] Numerical modeling of debris avalanche is presented here. The model uses the long-wave approximation based on the small aspect ratio of debris avalanches as in classical Saint Venant model of shallow water. Depth-averaged equations using this approximation are derived in a reference frame linked to the topography. Debris avalanche is treated here as a single-phase, dry granular flow with Coulomb-type behavior. The numerical finite volume method uses a kinetic scheme based on the description of the microscopic behavior of the system to define numerical fluxes at the interfaces of a finite element mesh. The main advantage of this method is to preserve the height positivity. The originality of the presented scheme stands in the introduction of a Dirac distribution of particles at the microscopic scale in order to describe the stopping of a granular mass when the driving forces are under the Coulomb threshold. Comparisons with analytical solutions for dam break problems and experimental results show the efficiency of the model in dealing with significant discontinuities and reproducing the flowing and stopping phase of granular avalanches. The ability of the model to describe debris avalanche behavior is illustrated here by schematic numerical simulation of an avalanche over simplified topography. Coulomb-type behavior with constant and variable friction angle is compared in the framework of this simple example. Numerical tests show that such an approach not only provides insights into the flowing and stopping stage of the granular mass but allows observation of interesting behavior such as the existence of a fluid-like zone behind a stopped solid-like granular mass in specific situations, suggesting the presence of horizontal surfaces in the deposited mass.

/pdf/numerical-modeling-of-avalanches-based-on-saint-venant-51atsyuf5f.pdf

Numerical modeling of avalanches based on Saint-Venant equations using a kinetic scheme

[1] Cliff collapse is an active geomorphological process acting at the surface of the Earth and telluric planets. Recent laboratory studies have investigated the collapse of an initially cylindrical granular mass along a rough horizontal plane for different initial aspect ratios a = Hi/Ri, where Hi and Ri are the initial height and radius, respectively. A numerical simulation of these experiments is performed using a minimal depth-integrated model based on a long-wave approximation. A dimensional analysis of the equations shows that such a model exhibits the scaling laws observed experimentally. Generic solutions are independent of gravity and depend only on the initial aspect ratio a and an effective friction angle. In terms of dynamics, the numerical simulations are consistent with the experiments for a ≤ 1. The experimentally observed saturation of the final height of the deposit, when normalized with respect to the initial radius of the cylinder, is accurately reproduced numerically. Analysis of the results sheds light on the correlation between the area overrun by the granular mass and its initial potential energy. The extent of the deposit, the final height, and the arrest time of the front can be directly estimated from the “generic solution” of the model for terrestrial and extraterrestrial avalanches. The effective friction, a parameter classically used to describe the mobility of gravitational flows, is shown to depend on the initial aspect ratio a. This dependence should be taken into account when interpreting the high mobility of large volume events.

/pdf/on-the-use-of-saint-venant-equations-to-simulate-the-4go6xfmh0i.pdf

On the use of Saint Venant equations to simulate the spreading of a granular mass

Numerical modelling of debris avalanches is presented here. The model uses the long waves approximation, based on the small aspect ratio of debris avalanches, as in classical Saint-Venant models for shallow water. Depth-averaged equations using this approximation are derived in a reference frame linked to the topography. Debris avalanches are treated here as a dry granular flow with Coulomb-type behavior. The numerical finite volume method uses a kinetic scheme, based on the description of the microscopic behavior of the system, to define numerical fluxes at the interfaces of a finite element mesh. The main advantage of this method is to preserve the height positivity. The originality of the present scheme stands in the introduction of a Dirac distribution of particles at the microscopic scale, in order to describe the stopping of a granular mass when the driving forces are under the Coulomb threshold. Comparisons with analytical solutions for dam-break problems show the efficiency of the method to deal with significant discontinuities. The ability of the model to describe debris avalanche behavior is illustrated here by schematic 1D numerical simulations of an avalanche over simplified topography. Coulomb-type behavior with constant and variable friction angle are compared in the framework of this simple example. Numerical tests show that such approach does not only provide insights into the flowing and stopping stage of the granular mass but it also allows us to observe interesting behaviors, such as the existence of a fluidized zone behind a stopped granular mass in specific situations, suggesting the presence of horizontal surfaces in the deposited mass.

https://hal.inria.fr/inria-00071939/document

A new kinetic scheme for Saint-Venant equations applied to debris avalanches

L’analisi tridimensionale del run out delle valanghe di roccia viene affrontata nel presente articolo attraverso l’applicazione di un modello della meccanica del continuo. Dopo una prima parte di descrizione delle caratteristiche del modello base viene data una descrizione delle modifiche che sono state apportate e di quelle che si apporteranno nello sviluppo futuro del lavoro. Obiettivo finale è la messa a punto di uno strumento che sia, una volta testato, utilizzabile nello studio di fenomeni reali.

A. Mangeney-Castelnau

Papers

Spreading of a granular mass on a horizontal plane

Numerical modeling of avalanches based on Saint-Venant equations using a kinetic scheme

On the use of Saint Venant equations to simulate the spreading of a granular mass

A new kinetic scheme for Saint-Venant equations applied to debris avalanches

Sviluppo di un modello basato sulla meccanica del continuo per lo studio della propagazione di valanghe di roccia