Static and flowing regions in granular collapses down channels: Insights from a sedimenting shallow water model

A two layer model for the collapse and spreading of a granular column is presented. This model builds upon that of Larrieu et al. [J. Fluid Mech. 554, 669 (2006)] where the free fall collapse of the column and subsequent flow of material onto a plane is represented by a “raining” mass source term into a thin flowing layer of constant density. These modified shallow water equations with Coulomb friction capture the free surface of the flows and key scaling laws for initial sand columns of aspect ratios up to a<10. However, unrealistically high coefficients of friction of μ=0.9 are required to reproduce run-outs observed. Key scaling laws for high aspect ratio columns are also not captured. We thus extend the model of Larrieu (2006) to include an estimation for the interface between the static and flowing regions observed within granular collapses in the laboratory by Lube et al. [Phys. Fluids 19, 043301 (2007)]. An empirical sedimentation term Ls and the instantaneous removal of a static deposit wedge, seen in the laboratory, are incorporated into the “raining” shallow water model. The growing static deposit surface provides a basal topography for the flowing layer. For a constant empirical sedimentation rate of Ls=0.20m∕s, a coefficient of friction of μ=0.4 simulates comparable run-outs to laboratory observations. The correct run-out dependence of a2∕3 for columns of aspect ratio a>3 is also captured. Simulating this behavior for values of a above 10 has not been possible with previous continuum models. In addition, this model captures the correct dependence of final run-out time upon a0.5. The application of this extends beyond observed and simulated collapses, to sedimenting highly concentrated debris flows, useful in the development of large mechanistic numerical models utilized in hazard assessment.