Constitutive relations for compressible granular flow in the inertial regime
Constitutive relations for compressible granular flow in the inertial regime
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DOI：
10.1017/jfm.2019.476
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发表时间：
2019-07
影响因子：
3.7
通讯作者：
D. Schaeffer;T. Barker;D. Tsuji;P. Gremaud;M. Shearer;J. Gray
中科院分区：
文献类型：
--
作者：
D. Schaeffer;T. Barker;D. Tsuji;P. Gremaud;M. Shearer;J. Gray
颗粒流在工业、自然环境和我们的日常生活中广泛存在，具有实际意义。本文关注的是所谓惯性状态下的颗粒流动，当流变性与非常大的颗粒刚度无关时。这种流动已经用$\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -流变学建模，它假设体积摩擦系数$\unicode[STIX]{x1D707}$（即剪切应力与压力的比率）和固体体积分数$\unicode[STIX]{x1D719}$仅是惯性数$I$的函数。虽然$\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -流变学已经在稳定状态下通过几种不同几何形状的实验和离散粒子模拟得到验证，但最近表明该理论在数学上不适用于时间相关问题。直接结果是，使用这种流变学的计算可能会呈指数增长，随着离散化长度趋于零，增长率趋于无穷大，这在本文中首次得到了明确的证明。在开发新的数学模型时，由于病态造成的灾难性不稳定性是一个常见的问题，这意味着缺少一些重要的物理特性，或者模型没有得到适当的表述。在本文中，提出了一种替代$\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -流变学，它不会遭受这些缺陷。在可压缩流变学（CIDR）框架下，引入了惯性区新的本构律；它们与稳态极限下已建立的$\unicode[STIX]{x1D707}(I)$和$\unicode[STIX]{x1D6F7}(I)$关系相匹配，同时对于所有变形和所有填充密度都是适定的。对结果方程进行了随时间变化的数值解，证明了新的惯性CIDR模型可以使数值收敛于离散元法模拟支持的物理真实解。
Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology that does not suffer from such defects is proposed. In the framework of compressible $I$ -dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.