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A fundamental issue for describing gravity-driven flows over general topography is the search for an “optimal” coordinate. Bouchut and Westdickenberg [1] proposed an arbitrary coordinate system (BW) for general topography. The unified coordinate (UC) system (e.g., [2], [3]), which was developed for computational fluid dynamics, combines the advantages of both Eulerian and Lagrangian systems, so that the coordinates can instantaneously move with some singular surface within the flows. By utilizing the benefit of the BW coordinates and UC system, a new model of gravity-driven flows over general topography is derived, in which the erosion and deposition processes at the bed are considered. The depth-integrated mass and momentum equations are presented in the time-dependent and terrain-following coordinate system, which coincides with the interface distinguishing between the static and flowing layers. A shock-capturing numerical scheme is implemented to solve the derived equation system. Simulation results present the new features of this model and reveal a new physical insight of the erosion/deposition processes. [ABSTRACT FROM AUTHOR]

We use the non-Cartesian, topography-based equations of mass and momentum balance for gravity driven frictional flows of Luca et al. (Math. Mod. Meth. Appl. Sci. 19, 127–171 (2009)) to motivate a study on various approximations of avalanche models for single-phase granular materials. By introducing scaling approximations we develop a hierarchy of model equations which differ by degrees in shallowness, basal curvature, peculiarity of constitutive formulation (non-Newtonian viscous fluids, Savage–Hutter model) and velocity profile parametrization. An interesting result is that differences due to the constitutive behaviour are largely eliminated by scaling approximations. Emphasis is on avalanche flows; however, most equations presented here can be used in the dynamics of other thin films on arbitrary surfaces. [ABSTRACT FROM AUTHOR]