Some Remarks on Avalanches Modelling: An Introduction to Shallow Flows Models

Enrique D. Fernandez-Nieto (1) , Paul Vigneaux (2, 3)

1 IMUS - Departamento de Matemática Aplicada I
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
3 NUMED - Numerical Medicine

The main goal of these notes is to present several depth-averaged models with application in granular avalanches. We begin by recalling the classical Saint-Venant or Shallow Water equations and present some extensions like the Saint-Venant-Exner model for bedload sediment transport. The first part is devoted to the derivation of several avalanche models of Savage-Hutter type, using a depth-averaging procedure of the 3D momentum and mass equations. First, the Savage-Hutter model for aerial avalanches is presented. Two other models for partially fluidized avalanches are then described: one in which the velocities of both the fluid and the solid phases are assumed to be equal, and another one in which both velocities are unknowns of the system. Finally, a Savage-Hutter model for submarine avalanches is derived. The second part is devoted to non-newtonian models, namely viscoplastic fluids. Indeed, a one-phase viscoplastic model can also be used to simulate fluidized avalanches. A brief introduction to Rheology and plasticity is presented in order to explain the Herschel-Bulkley constitutive law. We finally present the derivation of a shallow Herschel-Bulkley model.

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]; Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]

Enrique D. Fernandez-Nieto, Paul Vigneaux. Some Remarks on Avalanches Modelling: An Introduction to Shallow Flows Models. Parés, Carlos; Vazquez Cendon, Carlos; Coquel, Frederic. Advances in Numerical Simulation in Physics and Engineering - Lecture Notes of the XV 'Jacques-Louis Lions' Spanish-French School, 3, Springer, pp.51-106, 2014, SEMA SIMAI Springer Series, 978-3-319-02838-5. ⟨10.1007/978-3-319-02839-2_2⟩. ⟨hal-01066445⟩