干颗粒流的弱非静力浅水模型
A weakly non-hydrostatic shallow model for dry granular flows
摘要：
提出了一个考虑垂向加速度的非静力学干颗粒流深度平均模型。考虑了基于$μ(I)$流变性的可变摩擦系数。该模型是从一个渐进的分析，在当地的参考系，其中的非流体静力学的贡献应该是小的相比，流体静力学的。压力的非流体静力学对应物可以写成两项之和：一项对应于应力张量，另一项对应于垂直加速度。这里介绍的模型是弱非流体静力学的，在这个意义上，非流体静力学的贡献有关的应力张量不考虑由于其复杂的实施。提出了一种简单有效的数值格式。它包括一个三步分裂过程，它是基于流体静力学重建。两个关键点是：（i）在求解非静水压力之前必须考虑摩擦力。否则，不可压缩性条件是不能保证的;（ii）流体静压和非流体静压时，考虑到处理摩擦力。然后，模型和数值方案进行了验证的基础上，几个数值试验，包括颗粒坍塌的实验室实验。分析了非静力项和坐标系（笛卡尔坐标系或局部坐标系）选择的影响。我们发现，非流体静力学模型是不太敏感的坐标系的选择。总的来说，与流体静力学模型相比，这里引入的非流体静力学模型更好地再现了颗粒崩塌实验。一个重要的结果是，模拟的质量分布的沉积物和前锋速度大大改善。正如预期的那样，非静水压力的影响被证明是更大的小值的斜率。
英文摘要：
A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the $μ(I)$ 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. A simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure, and it is based on a hydrostatic reconstruction. Two key points are: (i) the friction force has to be taken into account before solving the non-hydrostatic pressure. Otherwise, the incompressibility condition is not ensured; (ii) both the hydrostatic and the non-hydrostatic pressure are taken into account when dealing with the friction force. The model and numerical scheme are then validated based on several numerical tests, including laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models. An important result is that the simulated mass profiles up to the deposit and the front velocity are greatly improved. As expected, the influence of the non-hydrostatic pressure is shown to be larger for small values of the slope.
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