Closure relations for shallow granular flows from particle simulations

Abstract

The discrete particle method (DPM) is used to model granular flows down an inclined chute with varying basal roughness, thickness and inclination. We observe three major regimes: arresting flows, steady uniform flows and accelerating flows. For flows over a smooth base, other (quasi-steady) regimes are observed: for small inclinations the flow can be highly energetic and strongly layered in depth; whereas, for large inclinations it can be non-uniform and oscillating. For steady uniform flows, depth profiles of density, velocity and stress are obtained using an improved coarse-graining method, which provides accurate statistics even at the base of the flow. A shallow-layer model for granular flows is completed with macro-scale closure relations obtained from micro-scale DPM simulations of steady flows. We obtain functional relations for effective basal friction, velocity shape factor, mean density, and the normal stress anisotropy as functions of layer thickness, flow velocity and basal roughness. Granular avalanche flows are common in both the natural environments and industry. They occur across many orders of magnitude. Examples range from rock slides, containing upwards of 1,000 m 3 of material; to the flow of sinter, pellets and coke into a blast furnace for iron-ore melting; down to the flow of sand in an hour-glass. The dynamics of these flows are influenced by many factors such as: polydispersity; variations in density; non-uniform shape; complex basal topography; surface contact properties; coexistence of static, steady and accelerating material; and, flow obstacles and constrictions. KeywordsDiscrete particle methods (DPMs) are an extremely powerful way to investigate the effects of these and other factors. With the rapid recent improvement in computational power the full simulation of the flow in a small hour glass of millions of particles is now feasible. However, complete DPM simulations of large-scale geophysical mass flow will, probably, never be possible.One of the main goals of the present research is to simulate large scale and complex industrial flows using granular shallow-layer equations. In this paper we will take the first step of using the DPM [9,34,42,43,46] to simulate small granular flows of mono-dispersed spherical particles in steady flow situations. We will use a refined and novel analysis to investigate three particular aspects of shallow chute flows: (i) how to obtain meaningful macro-scale fields from the DPM simulation, (ii) how to assess the flow dependence on the basal roughness, and (iii) how to validate the assumptions made in depth-averaged theory.The DPM simulations presented here will enable the construction of the mapping between the micro-scale and 123

Force analysis of clogging arches in a silo

“…As long as there is an overlap between the interacting particles, ξ increases as dξ /dt = v T rel [27]. Moreover, the tangential spring is truncated and renormalized as necessary to be orthogonal to the normal vector [29].…”

From discrete particles to continuum fields in mixtures

“…The approach has been carefully studied in several publications: In [3], it was shown how to define macroscopic fields that are independent of the coarse-graining width. This approach was successfully applied to flows near boundaries/discontinuities, as well as layered flows [4]. Further, an expression for the displacement gradient was derived in [5], which provides a smooth measure of the shear rate that is accurate even in strongly sheared flows.…”

Asymmetric breaking size-segregation waves in dense granular free-surface flows

“…All of the simulation parameters were non-dimensionalised so that g = 1. A frictional springdashpot model (Cundall & Strack 1979;Weinhart et al 2012) with linear elastic and linear dissipative contributions was used for both the normal and tangential forces. The tangential force models the effects of particle surface roughness, and its spring stiffness was taken to be 2/7 of the spring stiffness for the normal direction.…”

“…The contact time for all head on collisions was fixed at 0.0054, with the collision properties chosen to be different for small/small, small/large and large/large collisions so that both the contact time and the coefficient of restitution were the same even in the mixed case. Further details of the precise DPM implementation may be found in Thornton et al (2012b) and Weinhart et al (2012). The simulations were conducted in a box of length 300d s with fixed end walls and width 8.4d s .…”

“…The layer is thick enough to ensure that no flowing particles can fall through the rough base during the simulations. More details of this base creation process can be found in Weinhart et al (2012) whereas a detailed description of different bed creation methods and their effect on the Asymmetric breaking size-segregation waves 469 macroscopic friction experienced by the flow can be found in Thornton et al (2012a). Before each time step t = 10 −4 √ d s /g, the base was moved upstream by a distance u belt t = 1.5 × 10 −4 d s .…”