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A non-hydrostatic depth-averaged model for dry granular flows is proposed,taking into account vertical acceleration. A variable friction coefficientbased on the $\mu(I)$ rheology is considered. The model is obtained from anasymptotic analysis in a local reference system, where the non-hydrostaticcontribution is supposed to be small compared to the hydrostatic one. Thenon-hydrostatic counterpart of the pressure may be written as the sum of twoterms: one corresponding to the stress tensor and the other to the verticalacceleration. The model introduced here is weakly non-hydrostatic, in the sensethat the non-hydrostatic contribution related to the stress tensor is not takeninto account due to its complex implementation. A simple and efficientnumerical scheme is proposed. It consists of a three-step splitting procedure,and it is based on a hydrostatic reconstruction. Two key points are: (i) thefriction force has to be taken into account before solving the non-hydrostaticpressure. Otherwise, the incompressibility condition is not ensured; (ii) boththe hydrostatic and the non-hydrostatic pressure are taken into account whendealing with the friction force. The model and numerical scheme are thenvalidated based on several numerical tests, including laboratory experiments ofgranular collapse. The influence of non-hydrostatic terms and of the choice ofthe coordinate system (Cartesian or local) is analyzed. We show thatnon-hydrostatic models are less sensitive to the choice of the coordinatesystem. In general, the non-hydrostatic model introduced here much betterreproduces granular collapse experiments compared to hydrostatic models. Animportant result is that the simulated mass profiles up to the deposit and thefront velocity are greatly improved. As expected, the influence of thenon-hydrostatic pressure is shown to be larger for small values of the slope.

A weakly non-hydrostatic shallow model for dry granular flows