Efficient numerical schemes for viscoplastic avalanches. Part 1: The 1D 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: finitevolume 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 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. We derived such schemes and numerical experiments are presented to show their performances.

Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

“…Note however that in these works nonconforming methods are used, and require further works for proving their convergence. Related works [5,17] use the augmented Lagrangian method.…”

Finite stopping times for freely oscillating drop of a yield stress fluid

“…in computations we set ρ = 1 and γ = 1 (density and surface tension parameters), while varying K, τ s , α, and n (consistency, yield stress, flow index parameters and the number n of spherical harmonic H n used to set the initial perturbation). The perturbation parameter in (18) is chosen to beε = 0.3. Further we use a sequence of descritizations with the following parameters: The maximal mesh size is h max = 16 ; the mesh is aggressively refined towards the free surface, where the mesh size equals h min .…”

“…Since the full problem poses a serious challenge for numerical simulations, it is common in the literature to consider simplified models of free-surface yield stress fluids. The shallow approximation is one of the most common reduced model for viscoplastic fluids flows over inclined planes and more complex 2D topographies, see [4,6,23] for recent reviews on this subject and [1,9,18,27] for more recent advances. The previous studies of free surface viscoplastic fluid flows also include axisymmetric squeezing flows, bubble Bingham type flows [2,29,46], the free interface lattice Boltzmann model [19], and the dam-break problem [35].…”

Numerical Simulation of Mudflow With a Two-Dimensional Depth-Averaged Model

“…Therefore, to simulate flow on a steep slope, the inclusion of the inertia term in the governing equation of motion is important. The inclusion of inertia term in the study of mudflow can be seen in the work of Fernandex-Nito et al [8] where the topography and inertia effects are included in the viscoplastic model. Ionescu [9] also included the inertia effect in the modelling of shallow avalanche onset.…”