Abstract Permeability is a very important multiscale parameter providing information of dendritic structures in macroscale simulations of solidification with liquid flow during casting. In our previous study, it was demonstrated that the permeability of liquid flow normal to a columnar dendritic structure can be well approximated by the Kozeny–Carman (KC) equation with a coefficient kc = 9. In this study, the permeability of liquid flow parallel to a columnar dendritic structure is investigated using a method that combines both phase–field and lattice Boltzmann simulations. It is shown that the permeability of parallel flow can also be approximated by the KC equation with kc = 3. A permeability tensor is then proposed for a columnar dendritic structure. Validity of the permeability tensor is confirmed by simulations of liquid flow in arbitrary directions of a columnar dendritic structure.

中文翻译：

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柱状树枝状结构的渗透率张量：相场和晶格玻尔兹曼研究

摘要 渗透率是一个非常重要的多尺度参数，在铸造过程中液态流动凝固的宏观模拟中提供树枝状结构的信息。在我们之前的研究中，已经证明垂直于柱状树枝状结构的液体流动的渗透率可以通过系数 kc = 9 的 Kozeny-Carman (KC) 方程很好地近似。在这项研究中，液体流动的渗透率平行使用结合相场和晶格 Boltzmann 模拟的方法研究柱状树枝状结构。结果表明平行流的渗透率也可以用kc=3的KC方程近似。然后提出了柱状树枝状结构的渗透率张量。