Friction dependence of shallow granular flows from discrete par- ticle simulations

A shallow-layer model for granular flows is completed with a closure relation for the macroscopic bed friction or basal roughness obtained from micro-scale discrete particle simulations of steady flows. We systematically vary the bed friction by changing the contact friction coefficient between basal and flowing particles, while the base remains geometrically rough. By simulating steady uniform flow over a wide parameter range, we obtain a friction law that is a function of both flow and bed variables. Surprisingly, we find that the macroscopic bed friction is only weakly dependent on the contact friction of bed particles and predominantly determined by the properties of the flowing particles. INTRODUCTION. – Free-surface flows of granular material occur in many geophysical and engineering applications, such as rockslides, avalanches, or productionline transport. They have been studied extensively both experimentally and numerically. The most direct way to simulate granular flows is by methods such as the Discrete Particle Method (DPM), which computes the movement of individual particles based on a model of the contact forces between the particles [1, 2]. However, realistic flow situations often involve billions of particles, and can only be modeled on a coarser level by continuum solvers (or hybrid methods), in which the particulate flow is described by a small number of continuum fields governed by the conservation of mass, momentum, and often energy. For shallow flows, the mass and momentum conservation equations can be further simplified by averaging over the flow depth, yielding granular shallow-layer equations [3–5]. In order to obtain a closed system of equations, closure relations for first normal stress ratio, velocity shape factor, and macro basal friction, have to be developed in terms of the flow variables: height, h and the depth-averaged velocity, ū = (ū, v̄). While closure models are usually developed to retain the qualitative behaviour of the microscopic system, they often cannot describe the quantitative behaviour as the relations between the microand macroscopic quantities are not well known. Here, we focus on one closure relation: the effective macro-friction coefficient μ = μ(h, |ū|) and its dependence on the bed friction. It is informative to make a note about the nomenclature used in this paper. In the literature, the word friction gets used to mean both the macroscopic frictional forces felt by a large mass of material moving over a surface, as well as the contact frictional force between two individual flow particles, i.e., the contact friction used in the DPM contact model. In this paper we will refer to the macroscopic (shallow-layer) friction as μ, and use μ for the particle-particle contact friction between flowing particles. There is one final complication: we will take a different value for the contact friction for contacts between flowing and base particles; this will be named μ. The effective macro-friction coefficient, μ, determines the range of inclinations and heights at which the flow either arrests, reaches steady flow, or accelerates indefinitely. The rougher the base, the larger the range of inclinations at which steady flow is reached. Basal roughness can be modeled in various ways: in [6], a basal roughness was created by glueing particles onto a flat base. The roughness was changed by varying the diameter ratio between fixed basal and free flowing particles. They observed a peak in measured macro-friction coefficient at