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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:353

Article

Fernandez-Nieto, Enrique D.¹  Gallardo, Jose M.²  Vigneaux, Paul^3,4 

[1] Univ Seville, Dept Matemat Aplicada 1, ETS Arquitectura, Avda Reina Mercedes S-N, E-41012 Seville, Spain

[2] Univ Malaga, Dept Anal Matemat Estadist & IO & Matemat Aplicad, Ciencias F, Campus Teatinos S-N, E-29071 Malaga, Spain

[3] Ecole Normale Super Lyon, Unitee Math Pures & Appl, CNRS, UMR 5669, 46 Allee Italie, F-69364 Lyon 07, France

[4] Univ Lyon, Ecole Normale Super Lyon, INRIA, Team NUMED, 46 Allee Italie, F-69364 Lyon 07, France

关键词: Viscoplastic;    Shallow water;    Finite volume;    Well-balanced;    Variational inequality;    Bingham;    Taconnaz;   

DOI  :  10.1016/j.jcp.2017.09.054

来源: Elsevier

【 摘 要 】

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 Bermudez-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 twonon-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). (c) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free