Numerical solution of granular hydrodynamics from dilute, fast to dense, quasi-static flow

Many contributions on the modelling of granular flow and the solving of such models numerically restrict themselves either to fast, dilute or to dense, quasi-static granular flow. In many engineering applications both regimes appear at the same time in different parts of the simulations or consecutively at different stages of flow. We numerically investigate a continuum model for granular flow, which covers the regime of fast dilute flow as well as slow dense flow down to vanishing velocity. Our model is at small and intermediate densities equivalent to the model used by Bocquet et. a]. [1). An inherent instability in this model for vanishing granular temperature and vanishing velocities is removed by a cross over from a kinetic pressure to an athermal yield pressure at densities close to random close packing. Also the model for the viscosity is modified such that it diverges for small granular temperatures analogous to the diverging viscosities of liquids close to the glass transition. The presented model is a simplified version of a model of Savage [2], which nevertheless recovers many aspects of dense granular flow. We solve the system of the strongly nonlinear singular hydrodynamic equations with the help of a newly developed nonlinear predictor corrector algorithm together and a finite volume space discretisation. To validate our numerical approach we solve a Couette problem to demonstrate the existence of shear bands and simulate the formation of sandpiles with angle of repose consistent with the analytically estimated internal friction angle of the model.