In this work, we implement the local $\mu (I)$ rheology law and its non-local modification in a PISO-VOF numerical scheme with a sharp-interface-capturing THINC method for two-dimensional dense granular flows in both steady and transient states. The non-local effect on the constitutive relation is addressed by the gradient expansion model to account for additional momentum transport in view of the spatial variation of flow inertial number and the stress model singularity near the static state, $I=0$, is handled by the Bercovier-Engelman regularization scheme. Our simulations provide the first numerical evidence for the crucial role of non-local momentum transport on degrading the velocity profiles predicted with a local rheology model as those reported from two-dimensional discrete element simulations. For example, we successfully reproduce the sub-Bagnold velocity profile for steady inclined flows of height comparable to the $H_{stop}$ and capture how the linear velocity profile in a simple shear creeping flow is degraded to a S-shape. More importantly, we exploit the current solver to study the layer-accumulated deviation, ε, between the local and the non-local velocity profile (Bagnold to its degradation) for inclined flows. For steady flows, ε decays monotonically with flow Froude number Fr and a critical $F{r}_c\approx$0.25 is detected below which ε rises rapidly necessitating a non-local constitutive relation. For a layer developing from rest to its steady state, the time evolution of $\epsilon (t)$ shows further dependence on flow height and can be collapsed into a monotonically growing trend with local instantaneous inertial number $I(t)$. A critical $I_{cr}$ is also determined and the local model prediction can be erroneous whenever $I(t)<I_{cr}$. Interestingly, the result for simple shear flows without gravity also confirms that non-local effect becomes non-negligible when the macroscopic inertial number drops to the order of $I_{cr}$. Finally, we simulate the two-dimensional column collapse flows with both local and non-local rheology and find a shorter run-out distance with the presence of non-local momentum transport.

在这项工作中，我们实施了本地 $\mu （\mathrm{一世}）$在稳态和瞬态状态下二维稠密颗粒流的锐化界面捕获THINC方法的PISO-VOF数值方案中，流变规律及其非局部修改。考虑到流动惯性数的空间变化和静态附近的应力模型奇异性，梯度扩展模型解决了对本构关系的非局部影响，以考虑额外的动量传输。$\mathrm{一世}=0$，由Bercovier-Engelman正则化方案处理。我们的模拟提供了第一个数值证据，证明了非局部动量传输在降低由局部流变模型预测的速度分布方面的关键作用，如二维离散元素模拟所报道的那样。例如，我们成功地复制了与高度相等的高度的稳定倾斜流的亚Bagnold速度分布图。$H_{sŤØp}$并捕获简单剪切蠕变流中的线速度曲线如何退化为S形。更重要的是，我们利用电流求解器来研究倾斜流的局部和非局部速度分布（Bagnold对其退化）之间的层累积偏差ε。对于稳定的流量，ε随着流量Froude数Fr和临界值而单调衰减。$F{\mathrm{[R}}_C\approx$当检测到0.25以下时，ε迅速上升，这需要非局部本构关系。对于从静止状态发展到稳态的层，其时间演化$\epsilon （Ť）$ 显示出对流动高度的进一步依赖性，并且可以被压缩为具有局部瞬时惯性数的单调增长趋势 $\mathrm{一世}（Ť）$。关键${\mathrm{一世}}_{C\mathrm{[R}}$ 也被确定，并且每当局部模型预测可能是错误的 $\mathrm{一世}（Ť）<{\mathrm{一世}}_{C\mathrm{[R}}$。有趣的是，在没有重力的情况下，简单剪切流的结果也证实了当宏观惯性数下降到1的数量级时，非局部效应变得不可忽略。${\mathrm{一世}}_{C\mathrm{[R}}$。最后，我们用局部和非局部流变学来模拟二维柱塌方流，并发现存在非局部动量传递时跳动距离更短。