We consider the Saint-Venant system for shallow water flows, with non-flat bottom. It is a hyperbolic system of conservation laws that approximately describes various geophysical flows, such as rivers, coastal areas, 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 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 semi-discrete entropy inequality.

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[1] When not laterally confined in valleys, pyroclastic flows create their own channel along the slope by selecting a given flowing width. Furthermore, the lobe-shaped deposits display a very specific morphology with high parallel lateral levees. A numerical model based on Saint Venant equations and the empirical variable friction coefficient proposed by Pouliquen and Forterre (2002) is used to simulate unconfined granular flow over an inclined plane with a constant supply. Numerical simulations successfully reproduce the self-channeling of the granular lobe and the levee-channel morphology in the deposits without having to take into account mixture concepts or polydispersity. Numerical simulations suggest that the quasi-static shoulders bordering the flow are created behind the front of the granular material by the rotation of the velocity field due to the balance between gravity, the two-dimensional pressure gradient, and friction. For a simplified hydrostatic model, competition between the decreasing friction coefficient and increasing surface gradient as the thickness decreases seems to play a key role in the dynamics of unconfined flows. The description of the other disregarded components of the stress tensor would be expected to change the balance of forces. The front's shape appears to be constant during propagation. The width of the flowing channel and the velocity of the material within it are almost steady and uniform. Numerical results suggest that measurement of the width and thickness of the central channel morphology in deposits in the field provides an estimate of the velocity and thickness during emplacement.

International audienceA mechanical and numerical model of dry granular flows is proposed that quantitatively reproduce laboratory experiments of granular column collapse over inclined planes. The rheological parameters are directly derived from the experiments.The so-called \mu(I) rheology is reformulated in the framework of Drucker-Prager plasticity with the yield stress and viscosity \eta(||D||,p) depending on both the pressure p and the norm of the strain rate tensor ||D||. The granular domain, velocities, stress deviator and pressure fields are calculated using a finite element method based on an iterative decomposition-coordination formulation coupled with the augmented Lagrangian method. 2-D simulations using this model well reproduce the dynamics and deposits of collapsing granular columns. The flow is essentially located in a surface layer behind the front, whereas it is distributed over the whole depth near the front where basal sliding occurs. The computed runout distances and slopes of the deposits agree very well with the values found in the experiments. Using an easily calculated order of magnitude approximation of the mean viscosity during the flow (\eta = 1 Pa s here), we show that a Drucker-Prager rheology with a constant viscosity gives results very similar to the \mu(I) rheology and agrees with experimental height profiles, while significantly reducing the computational cost. Within the range of viscosities 0.1 < \eta < 1 Pa s, the dynamics and deposits are very similar. The observed slumping behavior therefore appears to be mainly due to the flow/no-flow criterion and to the associated strain-independent part of the "flowing constitutive relation" (i.e. related to plastic effects). However, the results are very different when an unrealistically large value of viscosity (10 Pa s) is used

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Abstract. We derive new models for gravity driven shallow water flows in several space dimensions over a general topography. A first model is valid for small slope variation, i.e. small curvature, and a second model is valid for arbitrary topography. In both cases no particular assumption is made on the velocity profile in the material layer. The models are written for an arbitrary coordinate system, and several formulations are provided. A Coulomb friction term is derived within the same framework, relevant in particular for debris avalanches. All our models are invariant under rotation, admit a conservative energy equation, and preserve the steady state of a lake at rest.

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.

Abstract.We study a depth-averaged model of gravity-driven flows made of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. The model system consists of mass and momentum balance equations for the solid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the unknowns, and by interphase drag source terms. The system is hyperbolic at least when the difference between solid and fluid velocities is sufficiently small. We solve numerically the one-dimensional model equations by a high-resolution finite volume scheme based on a Roe-type Riemann solver. Wellbalancing of topography source terms is obtained via a technique that includes these contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problems of perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions. Mathematics Subject Classification. 65M99, 76T25.