Raining into shallow water as a description of the collapse of a column of grains

A modified shallow-water model is presented for the collapse of tall columns of grains. The flow is divided in two parts. Depth-averaged shallow-water equations are applied to a thin horizontally spreading layer which is subjected to Coulombic friction. The failing mass of grains is gradually added to the zone of the initial column during the free-fall time of the column. This 'rain' is assumed to have no horizontal momentum. The results obtained here are in agreement with both planar and axisymmetric experiments over a range of aspect ratio a. In particular, the runout distance is found to vary as a(0.65) (planar) and a(0.52) (axisymmetric). The flow dynamics compares well with discrete simulations which have been successfully compared with experiments.