A new model for shallow viscoelastic fluids

International audienceWe propose a new reduced model for gravity-driven free-surface flows of shallow elastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for elastic fluids. The viscosity is assumed small (of order epsilon, the aspect ratio of the thin layer of fluid), but the relaxation time is kept finite. Additionally to the classical layer depth and velocity in shallow models, our system describes also the evolution of two scalar stresses. It has an intrinsic energy equation. The mathematical properties of the model are established, an important feature being the non-convexity of the physically relevant energy with respect to conservative variables, but the convexity with respect to the physically relevant pseudo-conservative variables. Numerical illustrations are given, based on a suitable well-balanced finite-volume discretization involving an approximate Riemann solver

A robust well-balanced scheme for multi-layer shallow water equations

International audienceThe numerical resolution of the multi-layer shallow water system encounters two additional difficulties with respect to the one-layer system. The first is that the system involves nonconservative terms, and the second is that it is not always hyperbolic. A splitting scheme has been proposed by Bouchut and Morales, that enables to ensure a discrete entropy inequality and the well-balanced property, without any theoretical difficulty related to the loss of hyperbolicity. However, this scheme has been shown to often give wrong solutions. We introduce here a variant of the splitting scheme, that has the overall property of being conservative in the total momentum. It is based on a source-centered hydrostatic scheme for the one-layer shallow water system, a variant of the hydrostatic scheme. The final method enables to treat an arbitrary number of layers, with arbitrary densities and arbitrary topography. It has no restriction concerning complex eigenvalues, it is well-balanced and it is able to treat vacuum, it satisfies a semi-discrete entropy inequality. The scheme is fast to execute, as is the one-layer hydrostatic method

A Subsonic-Well-Balanced Reconstruction Scheme for Shallow Water Flows

International audienceWe consider the Saint-Venant system for shallow water flows with nonflat bottom. In past years, efficient well-balanced methods have been proposed in order to well resolve solutions close to steady states at rest. Here we describe a strategy based on a local subsonic steady state reconstruction that allows one to derive a subsonic-well-balanced scheme, preserving exactly all the subsonic steady states. It generalizes the now well-known hydrostatic solver, and like the latter it preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality. An application to the Euler-Poisson system is proposed

A low cost semi-implicit low-Mach relaxation scheme for the full Euler equations

We introduce a semi-implicit two-speed relaxation scheme to solve the compressible Euler equations in the low Mach regime. The scheme involves a relaxation system with two speeds, already introduced by Bouchut, Chalons, Guisset (2019) in the barotropic case. It is entropy satisfying and has a numerical viscosity well-adapted to low Mach flows. This relaxation system is solved via a dynamical Mach number dependent splitting, similar to the one proposed by Iampietro et al. (2018). Stability conditions are derived, they limit the range of admissible relaxation and splitting parameters. We resolve separately the advection part of the splitting by an explicit method, and the acoustic part by an implicit method. The relaxation speeds are chosen so that the implicit system fully linearizes the acoustics and requires just to invert an elliptic operator with constant coefficients. The scheme is shown to well capture with low cost the incompressible slow scale dynamics with a timestep adapted to the velocity field scale, and rather well the fast acoustic waves

An analytic approach for the evolution of the static/flowing interface in viscoplastic granular flows

International audienceObserved avalanche flows of dense granular material have the property to present two possible behaviours: static (solid) or flowing (fluid). In such situation, an important challenge is to describe mathematically the evolution of the physical interface between the two phases. In this work we derive analytically a set of equations that is able to manage the dynamics of such interface, in the thin-layer regime where the flow is supposed to be thin compared to its downslope extension. It is obtained via an asymptotics starting from an incompressible viscoplastic model with Drucker-Prager yield stress, in which we have to make several assumptions. Additionally to the classical ones that are that the curvature of the topography, the width of the layer, and the viscosity are small, we assume that the internal friction angle is close to the slope angle (meaning that the friction and gravity forces compensate at leading order), the velocity is small (which is possible because of the previous assumption), and the pressure is convex with respect to the normal variable. This last assumption is for the stability of the double layer static/flowing configuration. A new higher-order non-hydrostatic nonlinear coupling term in the pressure allows us to close the asymptotic system. The resulting model takes the form of a formally overdetermined initial-boundary problem in the variable normal to the topography, set in the flowing region only. The extra boundary condition gives the information on how to evolve the static/flowing interface, and comes out from the continuity of the velocity and shear stress across it. The model handles arbitrary velocity profiles, and is therefore more general than depth-averaged models

Viscoplastic modeling of granular column collapse with pressure-dependent rheology

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

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

Influence of the scar geometry on landslide dynamics and deposits: Application to Martian landslides

International audienceLandslides dynamics prediction remains difficult in spite of a considerable number of studies. The runout distance is widely used in analysis of landslide dynamics and in the calibration of the rheological parameters involved in numerical modeling. However, the unknown impact of the significant uncertainty in the shape of the initial released mass on the runout distance and on the overall shape of the deposit raises questions about the relevance of these approaches. The impact of the initial scar geometry on flow and distribution of the deposits is studied here using satellite data and numerical modeling of theoretical landslides, and Martian landslides informed by geomorphological analysis, by varying the initial scar geometry from spoon‐shaped to steep wall geometry. Our results show that the runout distance is a very robust parameter that is only slightly affected by the change in the geometry of the initial scar. On the contrary, the lateral extent of the deposit is shown to be controlled by the scar geometry, providing unique insights into the initial landsliding conditions on Mars and makes it possible to accurately recover the volume initially involved, an essential ingredient for volume balance calculation. A feedback analysis of Valles Marineris landslides can be drawn, showing good agreement between numerical results and geomorphological analysis; the geometry of the initial scar inferred from numerical modeling is strongly correlated with the regional tectonic history in Valles Marineris area