On the use of Saint Venant equations to simulate the spreading of a granular mass

摘要

Cliff collapse is an active geomorphological process acting at the surface of the Earth and telluric planets. Recent laboratory studies have investigated the collapse of an initially cylindrical granular mass along a rough horizontal plane for different initial aspect ratios a = H i /R i , where H i and R i are the initial height and radius, respectively. A numerical simulation of these experiments is performed using a minimal depth-integrated model based on a long-wave approximation. A dimensional analysis of the equations shows that such a model exhibits the scaling laws observed experimentally. Generic solutions are independent of gravity and depend only on the initial aspect ratio a and an effective friction angle. In terms of dynamics, the numerical simulations are consistent with the experiments for a ≤ 1. The experimentally observed saturation of the final height of the deposit, when normalized with respect to the initial radius of the cylinder, is accurately reproduced numerically. Analysis of the results sheds light on the correlation between the area overrun by the granular mass and its initial potential energy. The extent of the deposit, the final height, and the arrest time of the front can be directly estimated from the “generic solution” of the model for terrestrial and extraterrestrial avalanches. The effective friction, a parameter classically used to describe the mobility of gravitational flows, is shown to depend on the initial aspect ratio a. This dependence should be taken into account when interpreting the high mobility of large volume events.