全球笛卡尔坐标系下非静力浅水颗粒流模型的改进
Refinement on non-hydrostatic shallow granular flow model in a global Cartesian coordinate system
摘要：
目前适用于任意地形的浅水颗粒流模型可分为两大类，一类是在床面拟合曲线坐标下建立的模型，另一类是在整体笛卡尔坐标下建立的模型。Denlinger和Iverson的浅颗粒流模型\cite{Denlinger2004}和Castro-Orgaz的Boussinesq型浅颗粒流理论\emph{et al}。\cite{Castro2014}是在笛卡尔坐标系（与$z$垂直），并考虑到非零垂直加速度对深度平均动量通量和应力状态的影响。在本文中，我们首先重新制定的垂直法向应力Castro-Orgaz \emph{et al}。相对高程$η$中的二次多项式中的\cite{Castro2014}。该形式允许对垂直法向应力进行分析深度积分。然后根据基底摩擦定律和比例分析计算基底法向应力。这些计算，加上一定的本构关系，导致一个完善的全非静力浅层颗粒流模型，这是进一步改写的形式Boussinesq型水波动方程式为未来的数值研究。在目前的数值研究中，我们采用开放源代码TITAN 2D的数值解的低阶版本的完整模型只涉及一个平均垂直加速度校正项。为了解决重力增强离散化过程中的数值不稳定性问题，利用床面法向静水压力假设，提出了重力增强的近似计算公式。数值计算进行了几个测试情况下，涉及陡坡。与床拟合模型的比较表明，即使是简化的非静力笛卡尔模型也可以用来模拟任意地形上的浅颗粒流。
英文摘要：
Current shallow granular flow models suited to arbitrary topography can be divided into two types, those formulated in bed-fitted curvilinear coordinates, and those formulated in global Cartesian coordinates. The shallow granular flow model of Denlinger and Iverson \cite{Denlinger2004} and the Boussinesq-type shallow granular flow theory of Castro-Orgaz \emph{et al}. \cite{Castro2014} are formulated in a Cartesian coordinate system (with $z$ vertical), and both account for the effect of nonzero vertical acceleration on depth-averaged momentum fluxes and stress states. In this paper, we first reformulate the vertical normal stress of Castro-Orgaz \emph{et al}. \cite{Castro2014} in a quadratic polynomial in the relative elevation $η$. This form allows for analytical depth integration of the vertical normal stress. We then calculate the basal normal stress based on the basal friction law and scaling analysis. These calculations, plus certain constitutive relations, lead to a refined full non-hydrostatic shallow granular flow model, which is further rewritten in a form of Boussinesq-type water wave equations for future numerical studies. In the present numerical study, we apply the open-source code TITAN2D to numerical solution of a low-order version of the full model involving only a mean vertical acceleration correction term. To cure the numerical instability associated with discretization of the enhanced gravity, we propose an approximate formula for the enhanced gravity by utilizing the hydrostatic pressure assumption in the bed normal direction. Numerical calculations are conducted for several test cases involving steep slopes. Comparison with a bed-fitted model shows that even the simplified non-hydrostatic Cartesian model can be used to simulate shallow granular flows over arbitrary topography.
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