Hydrodynamic modeling of dilute and dense granular flow

We study numerically a continuum model for granular flow, which covers the regime of fast dilute flow as well as slow dense flow up to vanishing velocity. The constitutive relations at small and intermediate densities are equivalent to those derived from kinetic theory of granular flow. The existence of an inherent instability due to the vanishing kinetic or collisional pressure for small granular temperatures requires a cross over from a collisional pressure to an a thermal yield pressure at densities close to random close packing. Contrary to a kinetic viscosity, the viscosity turns into a function diverging for small temperatures analogous to the diverging viscosities of liquids close to the glass transition. In this respect the presented model is a simplified version of a model of Savage (J Fluid Mech 377:1–26, 1998), which nevertheless recovers many aspects of dense granular flow. As examples we show simulations of sandpiles with predictable slopes, hopper simulations with mass and core flow and angle dependent critical sand heights in flows down an inclined plane. We solve the system of the strongly nonlinear singular hydrodynamic equations with the help of a newly developed nonlinear time stepping algorithm together with a finite volume space discretization. The numerical algorithm is implemented using a finite volume solver framework developed by the authors which allows discretization on cell-centred bricks in arbitrary domains.