A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the \(\mu (I)\) rheology is considered. The model is obtained from an asymptotic analysis in a local reference system, where the non-hydrostatic contribution is supposed to be small compared to the hydrostatic one. The non-hydrostatic counterpart of the pressure may be written as the sum of two terms: one corresponding to the stress tensor and the other to the vertical acceleration. The model introduced here is weakly non-hydrostatic, in the sense that the non-hydrostatic contribution related to the stress tensor is not taken into account due to its complex implementation. The motivation is to propose simple models including non-hydrostatic effects. In order to approximate the resulting model, a simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure and the resulting scheme is well-balanced for granular material at rest with slope smaller than the fixed repose angle. The model and numerical scheme are validated by means of several numerical tests, including a convergence test, a well-balanced test, and comparisons with laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is also analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In addition, the non-hydrostatic Cartesian model produces deposits similar to the hydrostatic local model as suggested by Denlinger and Iverson (J Geophys Res Earth Surf, 2004. https://doi.org/10.1029/2003jf000085), the flow dynamics being however different. Moreover, the proposed model, when written in Cartesian coordinates, can be seen as an improvement of their model, since the vertical velocity is computed and not estimated from the boundary conditions. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models, especially at the beginning of the flow.

考虑到垂直加速度，提出了一种用于干燥颗粒流的非静水深度平均模型。基于\（\ mu（I）\）的可变摩擦系数考虑流变学。该模型是通过局部参考系统中的渐近分析获得的，该模型中的非静水贡献与静水相比要小。压力的非静液压的对等可以写成两个项的总和：一个对应于应力张量，另一个对应于垂直加速度。在此引入的模型是弱非静液压的，在某种意义上，由于其复杂的实现方式，未考虑与应力张量相关的非静液压作用。目的是提出包括非静水效应的简单模型。为了近似结果模型，提出了一种简单有效的数值方案。它由三步拆分程序组成，并且对于静止状态下的颗粒材料（坡度小于固定的休止角）而言，生成的方案非常平衡。该模型和数值方案通过几种数值试验进行了验证，包括收敛试验，平衡试验和与颗粒坍塌实验室试验的比较。还分析了非静水力项和坐标系（笛卡尔坐标系或局部坐标系）选择的影响。我们证明了非静水模型对坐标系的选择不那么敏感。此外，非流体静力学笛卡尔模型产生的沉积物类似于Denlinger和Iverson（J Geophys Res Earth Surf，2004. https://doi.org/10.1029/2003jf000085）建议的流体静力学局部模型。不同。此外，提出的模型以笛卡尔坐标表示时，可以看作是模型的改进，因为垂直速度是通过计算得出的，而不是根据边界条件估算的。通常，与静水模型相比，此处介绍的非静水模型可以更好地重现颗粒崩溃实验，尤其是在流动开始时。