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

Abstract

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 to 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 behaviour of the avalanche, new schemes need to be designed, involving the classical notion of wellbalancing. 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. We derived such schemes and numerical experiments are presented to show their performances.

Laminar shallow viscoplastic fluid flowing through an array of vertical obstacles

“…Combining the previous relation with the transport equation (15) at the zeroth order, and replacing in (42) at z = f + h, leads to:…”

“…The same applies to nonvolcanic debris flows in forested mountainous or urban areas (see e.g. [15,32]). In terms of material sciences, flow of a viscoplastic fluid through arrays of solid cylinders needs to be considered in industrial processes, such as in the case of fresh concrete spreading through networks of steel bars (see e.g.…”