On new erosion models of Savage–Hutter type for avalanches

Abstract

In this work we study the modeling of one-dimensional avalanche flows made of a moving layer over a static base, where the interface between the two can be time dependent. We propose a general model, obtained by looking for an approximate solution with constant velocity profile to the incompressible Euler equations. This model has an energy dissipation equation that is consistent with the depth integrated energy equation of the Euler system. It has physically relevant steady state solutions, and, for constant slope, it gives a particular exact solution to the incompressible hydrostatic Euler equations. Then, we propose a simplified model, for which the energy conservation holds only up to third-order terms. Its associated eigenvalues depend on the mass exchange velocity between the static and moving layers. We show that a simplification used in some previously proposed models gives a nonconsistent energy equation. Our models do not use, nor provide, any equation for the moving interface, thus other arguments have to be used in order to close the system. With special assumptions, and in particular small velocity, we can nevertheless obtain an equation for the evolution of the interface. Furthermore, the unknown parameters of the model proposed by Bouchaud, Cates, Ravi and Edwards (BCRE model [7]) can be derived. For the quasistationary case we compare and discuss the equation for the moving interface with Khakhar's model [13].

Modeling Shallow Over-Saturated Mixtures on Arbitrary Rigid Topography

“…In view of (56), the closure relation for M int gives a closure relation for s int . According to (21), an estimation of M int reduces to that of the solid volume fraction and of the relative velocity normal to the surface S 1 of the two mixture constituents, evaluated on the layer interface. Giving a law for M int is similar to giving a law for the erosion/deposition rate in processes with mass entrainment at the basal surface.…”

“…Moreover, noting that the trend in avalanche modeling is to account for curvature effects (see, e.g., Sivakumaran et al [14], Dewals et al [15], Iverson [16], Pudasaini et al [17,18], Bouchut and Westdickenberg [19], De Toni and Scotton [20], Bouchut et al [21], Tai and Kuo [22], Tai and Lin [23], Pelanti et al [24], Luca et al [25][26][27]5]), emphasis is on the description of the flow on arbitrary terrain, using the approach initiated by Bouchut and Westickenberg [19], and not on the formulation of constitutive laws, even if we indicate such laws as possible choices. In particular, the shallow saturated mixture in the lower layer is treated in Truesdell's sense, in the same manner as it has been done by Luca et al [27].…”

Modelling shallow debris flows of the Coulomb-mixture type over temporally varying topography

“…Taking into account the mass exchange between the moving layer and static base, Bouchut et al (2008) proposed a one-dimensional model for a general topography, in which the basal surface is time dependent. The formulism of the unified coordinate (UC) system (see e.g.…”

Dry granular flows with erosion/deposition process

“…Thus, they are able to describe granular flows over a non-uniformly curved bed of general type, and the meshes are automatically fit the moving topography. Second, in contrast to the traditional description of governing equations over a moving coordinate (e.g., [4]), the physical quantity hu computed in (5), is expressed in the Cartesian coordinates. This fact avoids the complicated calculation of Christoffel symbols and computations of changing coordinate orientation.…”