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离散单元法(Discrete Element Method,简称DEM)是颗粒流数值模拟的重要数值方法,被用于各种各样的颗粒流定量研究中。颗粒流的粗粒化(Coarse Graining),便是将颗粒流流场的大量数据转化为连续场数据的数值方法。
在开始介绍之前,先简要列出主要的参考资料:
Breard, E. C. P., Dufek, J., Fullard, L., & Carrara, A. (2020). The Basal Friction Coefficient of Granular Flows With and Without Excess Pore Pressure: Implications for Pyroclastic Density Currents, Water‐Rich Debris Flows, and Rock and Submarine Avalanches. Journal of Geophysical Research: Solid Earth, 125(12), e2020JB020203. DOI:10.1029/2020JB020203.
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Weinhart, T., Labra, C., Luding, S., & Ooi, J. Y. (2016). Influence of coarse‐graining parameters on the analysis of DEM simulations of silo flow. Powder Technology, 293, 138–148. DOI:10.1016/j.powtec.2015.11.052
单种颗粒的粗粒化
为简单起见,先从仅存在单种球形颗粒的颗粒流开始说起,并且只考虑获取空间内一点 r 在 t 时刻的粗粒化数据。
粗粒化函数
在这里,引入高斯函数
W(r)={Vw1exp(−2w∣r∣2),0,∣r∣<cotherwise
其中: w 为粗粒化宽度; c=3w 为粗粒化截断长度; Vw 是确保其密度积分等于总质量的系数
Vw=22⋅w3π23erf(2wc2)−4cw2πexp(2w2−c2)
Weinhart 等建议 w 的取值范围应在 [0.75dmean,1.25dmean] 之间,其中 dmean 为平均颗粒直径。
密度和动量的计算
在 t 时刻,单种颗粒的集合为 q 。对于其中的第 i 个球,其位置矢量为 ri ,质量为 mi ,速度为 ui。
Chen, Z., Shu, C., Wang, Y., Yang, L. M., & Tan, D. (2017). A Simplified Lattice Boltzmann Method without Evolution of Distribution Function. Advances in Applied Mathematics and Mechanics, 9(1), 1–22. DOI:10.4208/aamm.OA-2016-0029 ↩︎
Z. Chen, C. Shu, D. Tan; Three-dimensional simplified and unconditionally stable lattice Boltzmann method for incompressible isothermal and thermal flows. Physics of Fluids. 1 May 2017; 29 (5): 053601. DOI:10.1063/1.4983339 ↩︎↩︎↩︎
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