Efficient numerical schemes for viscoplastic avalanches. Part 2: the 2D case

This paper deals with the numerical resolution of a shallow water viscoplastic flow model. Viscoplastic materials are characterized by the existence of a yield stress: below a certain critical threshold in the imposed stress, there is no deformation and the material behaves like a rigid solid, but when that yield value is exceeded, the material flows like a fluid. In the context of avalanches, it means that after going down a slope, the material can stop and its free surface has a non-trivial shape, as opposed to the case of water (Newtonian fluid). The model involves variational inequalities associated with the yield threshold: finite volume schemes are used together with duality methods (namely Augmented Lagrangian and Bermúdez–Moreno) to discretize the problem. To be able to accurately simulate the stopping behavior of the avalanche, new schemes need to be designed, involving the classical notion of well-balancing. In the present context, it needs to be extended to take into account the viscoplastic nature of the material as well as general bottoms with wet/dry fronts which are encountered in geophysical geometries. Here we derive such schemes in 2D as the follow up of the companion paper treating the 1D case. Numerical tests include in particular a generalized 2D benchmark for Bingham codes (the Bingham–Couette flow with two non-zero boundary conditions on the velocity) and a simulation of the avalanche path of Taconnaz in Chamonix—Mont-Blanc to show the usability of these schemes on real topographies from digital elevation models (DEM)

Multilayer models for hydrostatic Herschel-Bulkley viscoplastic flows

This is an open access article under the CC BY-NC-ND licenseStarting from Navier-Stokes’ equation we derive two shallow water multilayer models for yield stress fluids, depending on the asymptotic analysis. One of them takes into account the normal stress contributions, making possible to recover a pseudoplug layer instead of a purely plug zone. A specific numerical scheme is designed to solve this model thanks to a finite volume discretization. It involves well-balancing techniques to be able to compute accurately the transitions between yielded and unyielded (or pseudoplug) zones, an important feature of the original partial differential equations’ model. We perform numerical simulations on various test cases relevant to these physics: analytical solution of a uniform flow, steady solutions for arrested state, and a viscoplastic dam break. Simulations agree well when we perform comparisons with physical experiments of the group of Christophe Ancey (EPFL) and we make a comparative study including shallow water models and lubrication models that they present in Ancey et al. (2012) [3]. Thanks to the multilayer structure of our model, we can go further on the description of the vertical structure associated to the (bottom) sheared layer and the top (pseudo-)plug layer

Simulation of an avalanche in the Taconnaz path, with a shallow-water Bingham model. Mont-Blanc massif, France

Simulation of a viscoplastic material (also called yield stress fluid) on a digital elevation model from ASTER GDEM v2. For more details, please see the associated article: Enrique D. Fernández-Nieto, José M. Gallardo, Paul Vigneaux, Efficient numerical schemes for viscoplastic avalanches. Part 2: The 2D case, Journal of Computational Physics, Volume 353, 2018, Pages 460-490

: Simulations numériques 2D pour des écoulements de type Bingham

International audienceIn this talk, we debrief two aspects done in the CNRS Tellus Insu-Insmi project (2016 Call) and which are now supported by the CNRS InFIniTi program (2017-2018). First, we present numerical simulations in expansion-contraction geometries and comparison with physical experiments of Chevalier et al EPL 2013 and Luu et al PRE 2015. It is shown that the Bingham law alone still allows to retrieve non trivial features of the flow. Second, we present 2D schemes for a shallow Bingham model (Bresch et al AMFM 2010) which allow to accurately compute arrested states at the end of the avalanche of a viscoplastic material. It blends Well-Balanced Finite Volumes and duality methods. These two aspects are described respectively in (i) A. Marly, P. Vigneaux : Augmented Lagrangian simulations study of yield-stress fluid flows in expansion-contraction and comparisons with physical experiments - Journal of Non Newtonian Fluid Mechanics 239 : 35-52 - 2017 and (ii) E. D. Fernandez-Nieto, J. M. Gallardo, P. Vigneaux. Efficient numerical schemes for viscoplastic avalanches. Part 2: the 2D case. Submitted, March 2017

Accurate arrested states for finite-volumes schemes on a shallow Bingham model

International audienceWe start here from a prototype of a 2D Shallow Water Bingham model as derived in Bresch et al (2010), see also Ionescu (2013). For such integrated models, Finite Volume methods are known to be very efficient. In this talk, we present a new 2D well-balanced scheme for such discretizations which is able to compute accurately the arrested states of avalanches of Bingham fluids on real topographies and in the presence of wet/dry fronts. This well-balanced approach needs to be coordinated with the duality method used to solve the viscoplastic nature of the model (leading to variational inequalities). The Bingham law is here solved unregularized via the Augmented Lagrangian or the Bermudez-Moreno (BM) methods: for both methods a rigorous study of the numerical cost is performed and an a priori estimation of the optimal duality parameter is given for the BM (extending the 1D case treated in Fernandez-Nieto et al (2014)). It is shown that this a priori estimate allows to be close to the shortest computation times.To illustrate the ability of these schemes to handle the numerical difficulties encountered in geophysical applications (complex DEM topographies, long space domains and long time scales), we present in particular a simulation of an avalanche in the Taconnaz path (Chamonix, Mont-Blanc). This is a joint work with E.D. Fernandez-Nieto and J. M. Gallardo. /-/ References:D. Bresch, E.D. Fernandez-Nieto, I. Ionescu, P. Vigneaux. Augmented Lagrangian Method and Compressible Visco-Plastic Flows : Applications to Shallow Dense Avalanches. Advances in Mathematical Fluid Mechanics, 2010, pp. 57-89.I. Ionescu. Augmented Lagrangian for shallow viscoplastic flow with topography. Journal of Computational Physics, 2013, Vol. 242, pp 544-560.E.D. Fernandez-Nieto, J. M. Gallardo, P. Vigneaux. Efficient numerical schemes for viscoplastic avalanches. Part 1: the 1D case. Journal of Computational Physics, 2014, Vol. 264, pp 55-90.E.D. Fernandez-Nieto, J. M. Gallardo, P. Vigneaux. Efficient numerical schemes for viscoplastic avalanches. Part 2: the 2D case. Journal of Computational Physics, accepted 2017. https://hal.archives-ouvertes.fr/hal-01593148/-/ Acknowledgements : This work has been partially supported by CNRS through the interdisciplinary program InFIniti 2017