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.
[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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