Explicit solutions to a free interface model for the static/flowing transition in thin granular flows

Abstract

This work is devoted to an analytical description of the dynamics of the static/flowing interface in thin dry granular flows. Our starting point is the asymptotic model derived by Bouchut et al. (2016) from a free surface incompressible model with viscoplastic rheology including a Drucker--Prager yield stress. This asymptotic model is based on the thin-layer approximation (the flow is thin in the direction normal to the topography compared to its down-slope extension), but the equations are not depth-averaged. In addition to the velocity, the model includes a free surface at the top of the flow and a free time-dependent static/flowing interface at the bottom. In the present work, we simplify this asymptotic model by decoupling the space coordinates, and keeping only the dependence on time and on the normal space coordinate $Z$. We introduce a time- and $Z$-dependent source term, assumed here to be given, which represents the opposite of the net force acting on the flowing material, including gravity, pressure gradient, and internal friction. We prove several properties of the resulting simplified model that has a time- and $Z$-dependent velocity and a time-dependent static/flowing interface as unknowns. The crucial advantage of this simplified model is that it can provide explicit solutions in the inviscid case, for different shapes of the source term. These explicit inviscid solutions exhibit a rich behaviour and qualitatively reproduce some physical features observed in granular flows.

Basal boundary conditions for granular surface flows over fragile and brittle erodible beds

“…Possibly with minor variations, this description has been applied to a variety of parallel flows and found to closely match discrete element simulations (da Cruz et al 2005;Parez et al 2016) and laboratory experiments (Berzi & Jenkins 2008;Capart et al 2015). Regarding basal boundary conditions for flows over erodible substrates, however, there is less consensus (Iverson & Ouyang 2015;Lusso et al 2021;Pudasaini & Krautblatter 2021). What is needed is to specify boundary conditions for the flow velocity u, shear rate γ and/or shear stress τ at the base, so that the flow thickness h or its evolution over time h(t) can also be determined.…”

“…Such processes occur down natural inclines such as scree slopes and dune faces, and in laboratory configurations such as slowly rotating drums and grain piles driven by low inflow (Evesque 1991;Arran & Vriend 2018). Challenges are then to determine how rates of entrainment and detrainment are set, and how these affect flow acceleration and deceleration (Iverson & Ouyang 2015;Lusso et al 2021;Pudasaini & Krautblatter 2021). In narrow channels, wall friction is known to be an effective mechanism to limit erosion at the base of granular avalanches (Taberlet et al 2003;Jop, Forterre & Pouliquen 2005).…”

“…Depth-integrated and continuum models were first applied to granular avalanches over rigid boundaries (Savage & Hutter 1991;Silbert et al 2001). For erodible beds, the position of the basal interface becomes an additional degree of freedom (Capart, Hung & Stark 2015;Iverson & Ouyang 2015;Lusso et al 2021). To model the resulting dynamics, various approaches have been adopted.…”

“…Challenges are then to determine how rates of entrainment and detrainment are set, and how these affect flow acceleration and deceleration (Iverson & Ouyang 2015; Lusso et al. 2021; Pudasaini & Krautblatter 2021). In narrow channels, wall friction is known to be an effective mechanism to limit erosion at the base of granular avalanches (Taberlet et al.…”

“…For erodible beds, the position of the basal interface becomes an additional degree of freedom (Capart, Hung & Stark 2015; Iverson & Ouyang 2015; Lusso et al. 2021). To model the resulting dynamics, various approaches have been adopted.…”