In the present study, a two-dimensional phase field model coupled with Lattice Boltzmann method (PF-LBM) is proposed to predict the dendritic growth and motion in the melt of Fe-C binary alloy, where the phase field method (PF) is used to calculate the dendritic growth, including the phase field and the concentration field, and the lattice Boltzmann method (LBM) is used to calculate the flow field. The dendrite motion is determined by Newton’s Second Law and tracked by Lagrangian point in a Cartesian coordinate system. Later, the model validations were performed with the benchmark of a solid particle settlement in a stagnant fluid and particle motion in a shear flow, and the results show that the present model is capable of predicting the solid particle motion in the fluid flow. Finally, the model is adopted to investigate the dendritic growth and motion in a forced fluid flow (laminar flow or rotational flow), and the dendrite settlement in a terrestrial environment. The results show that when the forced fluid flow is a laminar flow, the free dendrite would be driven to translate, and the relative velocity between the dendrite and flow fluid decreases, resulting in weak influence of fluid flow on the dendritic growth. When the forced fluid flow is a rotational fluid flow, the dendrite would centrifugally rotate on the domain center with a counterclockwise self-spinning, and the rotation radius becomes larger and larger. For the case of dendrite settlement in a terrestrial environment, the relative movement between the dendrite and melt promotes the downward branch growth, but inhibits the upward branch growth, and two vortices form at the wake region of dendrite. Therefore, the settling dendrite shows a significant asymmetrical morphology.

中文翻译：

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Fe-C 二元合金过冷熔体中枝晶生长和运动的 PF-LBM 建模

在本研究中，提出了一种结合格子玻尔兹曼法 (PF-LBM) 的二维相场模型来预测 Fe-C 二元合金熔体中的枝晶生长和运动，其中相场法 (PF) 是用于计算枝晶生长，包括相场和浓度场，使用格子玻尔兹曼法（LBM）计算流场。枝晶运动由牛顿第二定律确定，并由笛卡尔坐标系中的拉格朗日点跟踪。随后，模型验证以静止流体中的固体颗粒沉降和剪切流中的颗粒运动为基准进行，结果表明本模型能够预测流体中固体颗粒的运动。最后，该模型用于研究强制流体流动（层流或旋转流）中的枝晶生长和运动，以及陆地环境中的枝晶沉降。结果表明，当强制流体流动为层流时，自由枝晶将被驱动平移，枝晶与流动流体之间的相对速度降低，导致流体流动对枝晶生长的影响较弱。当强制流体流动为旋转流体流动时，枝晶将在域中心以逆时针自旋的方式离心旋转，并且旋转半径越来越大。对于陆地环境下枝晶沉降的情况，枝晶与熔体之间的相对运动促进了向下的分支生长，但抑制了向上的分支生长，并且在枝晶尾流区域形成两个涡流。因此，沉降枝晶表现出明显的不对称形态。