Exact solutions for steady granular flow in vertical chutes and pipes
Exact solutions for steady granular flow in vertical chutes and pipes
复制标题
垂直溜槽和管道中稳定颗粒流的精确解决方案
DOI：
10.1017/jfm.2021.909
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发表时间：
2021-11-11
影响因子：
3.7
通讯作者：
Sun, J.
中科院分区：
文献类型：
--
作者：
Barker, T.;Zhu, C.;Sun, J.
摘要 垂直溜槽和管道是许多用于粉末和颗粒运输和加工的工业设备的常见组件。这里，首先考虑一种典型的布置，其中顶部的料斗向溜槽供料，底部的会聚出口控制质量通量。离散元法 (DEM) 模拟表明，仅在中等流速下观察到稳定均匀流，干扰波和不稳定波在慢速流中占主导地位，在快速流中则出现非均匀壁脱离。着眼于稳定的均匀状态，通过与周期单元中的等效 DEM 模拟相匹配来进行渐进的理想化。这些研究证明了问题的一维连续体建模的合理性并提供了关键的测试数据。这里使用“$\mu(I),\varPhi(I)$-流变学”的线性版本导出垂直流的新颖精确解，其中体摩擦力 $\mu$ 和稳定固体体积分数 $\varPhi$ 取决于惯性数 I。尽管没有捕获完整的非线性复杂性，但这些解与 DEM 流场的重要方面相匹配，并揭示了连接许多感兴趣量的简单缩放定律。特别是，这项研究清楚地证明了溜槽宽度与壁上剪切区尺寸之间的线性关系。这一发现与之前关于纯准静态流的研究形成鲜明对比，后者预测了大致恒定的剪切带宽度，这一差异意味着对于此处研究的惯性流来说，有限尺寸效应是最小的。
Abstract Vertical chutes and pipes are a common component of many industrial apparatus used in the transport and processing of powders and grains. Here, a typical arrangement is considered first in which a hopper at the top feeds the chute and a converging outlet at the bottom controls the mass flux. Discrete element method (DEM) simulations reveal that steady uniform flow is only observed for intermediate flow rates, with jamming and unsteady waves dominating slow flows and non-uniform wall detachment in fast flow. Focusing on the steady uniform regimes, a progressive idealisation is carried out by matching with equivalent DEM simulations in periodic cells. These investigations justify a one-dimensional continuum modelling of the problem and provide key test data. Novel exact solutions are derived here for vertical flow using a linear version of the ‘$\mu(I),\varPhi(I)$-rheology’, for which the bulk friction $\mu$ and steady solid volume fraction $\varPhi$ depend on the inertial number I. Despite not capturing the full nonlinear complexities, the solutions match important aspects of the DEM flow fields and reveal simple scaling laws linking many quantities of interest. In particular, this study clearly demonstrates a linear relation between the chute width and the size of the shear zones at the walls. This finding contrasts with previous works on purely quasi-static flow, which instead predict a roughly constant shear zone width, a difference which implies that finite-size effects are minimal for the inertial flows studied here.