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In the alloy solidification process, the movement behavior of free dendrites in the melt is one of the key issues in studying the formation process of the alloy solidification structure. The cellular automata-lattice Boltzmann coupling model has become a main numerical model for numerical simulation of solidification microstructure in recent years. In this paper, cellular automata and lattice Boltzmann model for simulating dendrite growth are improved to simulate the movement of equiaxed grains in undercooled melt. In the improved model, the Galilean invariant momentum exchange method is used to calculate the fluid force, the motion equation of the center of mass is solved to calculate the motion displacement of the dendrite, the motion of the dendrite is realized through the dynamic mesh technology, and the rigid sphere model is used to deal with the collision of the dendrite. The settlement of a single dendrite in the undercooled melt of Al-4.7% Cu alloy, the settlement of two round particles in the Newtonian fluid, and the elastic collision of two dendrites are all simulated by this model. The simulation results show that this model can maintain the morphology of dendrites well in the process of calculating dendrite growth motion, and this model can calculate the collision process of irregular dendrites. The movement of dendrite disturbs the melt around it, resulting in a significant change in the concentration of melt around it, which affects the growth of dendrite and aggravates the asymmetry of dendrite growth.
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Keywords:
- cellular automata /
- lattice Boltzmann /
- dendrite collision /
- dynamic grid
[1] Wang Y B, Peng L M, Ji Y Z, Chen X X, Wang C L, Wu Y J, Fu Y N, Chen L Q 2018 J. Mater. Sci. Technol. 34 1142
[2] Lesoult G 2005 Mater. Sci. Eng., A 413 19
[3] Zhu M F, Dai T, Lee S Y, Hong C P 2008 Comput. Math. Appl. 55 1620
Google Scholar
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Google Scholar
Pan S Y, Zhu M F 2012 Acta Phys. Sin. 61 228102
Google Scholar
[6] 潘诗琰, 朱鸣芳 2009 物理学报 58 278
Google Scholar
Pan S Y, Zhu M F 2009 Acta Phys. Sin. 58 278
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Google Scholar
[8] Liu S Y, Hong K M, Shin Y C 2021 Comput. Mater. Sci. 192 110405
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Google Scholar
[10] Meng S X, Zhang A, Guo Z P, Wang Q G 2020 Comput. Mater. Sci. 184 109784
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[11] Takaki T, Sato R, Rojas R, Ohno M, Shibuta Y 2018 Comput. Mater. Sci. 147 124
Google Scholar
[12] Ratkai L, Pusztai T, Granasy L 2019 npj Comput. Mater. 5 113
Google Scholar
[13] Sakane S, Takaki T, Ohno M, Shibuta Y, Aoki T 2020 Comput. Mater. Sci. 178 109639
Google Scholar
[14] 吴伟, 孙东科, 戴挺, 朱鸣芳 2012 物理学报 61 150501
Google Scholar
Wu W, Sun D K, Dai T, Zhu M F 2012 Acta Phys. Sin. 61 150501
Google Scholar
[15] Fang H, Tang Q Y, Zhang Q Y, Gu T F, Zhu M F 2019 Int. J. Heat Mass Transfer 133 371
Google Scholar
[16] Lee W, Jeong Y, Lee J W, Lee H, Kang S H, Kim Y M, Yoon J 2020 J. Mater. Sci. Technol. 49 15
Google Scholar
[17] Cu C, Ridgeway C D, Moodispaw M P, Luo A A 2020 J. Mater. Process. Technol. 286 116829
Google Scholar
[18] Liu L, Pian S, Zhang Z, Bao Y, Li R, Chen H 2018 Comput. Mater. Sci. 146 9
Google Scholar
[19] Wu J Y, Sun D K, Wang J C, Zhu M F 2020 Eur. Phys. J. E 43 30
Google Scholar
[20] Zhang Q Y, Sun D K, Pan S Y, Zhu M F 2020 Int. J. Heat Mass Transfer 146 118838
Google Scholar
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Google Scholar
[22] Rappaz M, Thévoz P H 1987 Acta Metall. 35 2929
Google Scholar
[23] Zhu M F, Stefanescu D 2007 Acta Mater. 55 1741
Google Scholar
[24] Wen B H, Zhang C Y, Tu Y S, Wang C L, Fang H P 2014 J. Comput. Phys. 266 161
Google Scholar
[25] Mei R, Yu D, Shyy W, Luo L S 2002 Phys. Rev. E: Stat. Nonliner Soft Matter Phys. 65 041203
Google Scholar
[26] Wu M, Ludwig A, Fjeld A 2010 Comput. Mater. Sci. 50 43
Google Scholar
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Google Scholar
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图 1 动网格方案示意图
Figure 1. Schematic diagram of dynamic grid scheme.
图 2 算法流程图
Figure 2. Algorithm flowchart.
图 3 自然对流下单枝晶沉降过程中溶质浓度和流体速度的时间演化 (a) t = 0.005 s; (b) t = 0.015 s; (c) t = 0.025 s
Figure 3. Evolution of solute concentration and fluid velocity during precipitation of single dendrite under natural convection: (a) t = 0.005 s; (b) t = 0.015 s; (c) t = 0.025 s.
图 4 t = 0.015 s时, 模拟域O点处竖直方向上的溶质浓度变化
Figure 4. When t = 0.015 s, the solute concentration change in the vertical direction at point O in the simulation domain.
图 5 枝晶尖端生长速度随时间的变化
Figure 5. The change of dendrite tip growth rate with time.
图 6 两个球形粒子沉降过程中, 四个时间的瞬时涡量轮廓 (a) 1 .0 s; (b) 1.5 s; (c) 2.5 s; (d) 4.0 s
Figure 6. The instantaneous vorticity profiles of two spherical particles in the process of settling at four times: (a) 1.0 s; (b) 1.5 s; (c) 2.5 s; (d) 4.0 s.
图 7 粒子质心坐标 (a) 横坐标; (b) 纵坐标
Figure 7. Coordinates of the particles centroid: (a) Transverse coordinates; (b) longitudinal coordinates.
图 8 枝晶偏心碰撞形貌图 (a) 0.008 s; (b) 0.0089 s; (c) 0.009 s; (d) 0.00915 s. 箭头表示速度v的大小和方向, 灰度表示流体溶质浓度C
Figure 8. The morphology of dendrite eccentric collision: (a) 0.008 s; (b) 0.0089 s; (c) 0.009 s; (d) 0.00915 s . The arrow indicates the size and direction of velocity v, and the gray scale indicates the concentration of solute C.
图 9 枝晶2附近熔体平均溶质浓度
Figure 9. Average solute concentration of melt near dendrite 2
图 10 熔体平均溶质浓度计算规则
Figure 10. Calculation rules of melt average solute concentration.
表 1 Al-4.7%Cu(质量含量)合金的热物性参数[26]
Table 1. Physical properties of Al-4.7%Cu (weight percent) alloy[26].
Physical parameter Value Melting temperature, Tm/K 933.3 Solidification temperature, T0/K 920.1 Diffusivity in liquid, D/(10–9 m2·s–1) 3.0 Partition coefficient, k 0.145 Gibbs-Tomson coefficient, Γ/(10–7 m·K) 2.4 Specific heat capacity, Cp/(J·kg–1·K–1) 1179 Latent heat, L/(103 J·kg–1) 397 Density, ρ/(kg·m–3) 2606 
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[1] Wang Y B, Peng L M, Ji Y Z, Chen X X, Wang C L, Wu Y J, Fu Y N, Chen L Q 2018 J. Mater. Sci. Technol. 34 1142
[2] Lesoult G 2005 Mater. Sci. Eng., A 413 19
[3] Zhu M F, Dai T, Lee S Y, Hong C P 2008 Comput. Math. Appl. 55 1620
Google Scholar
[4] Sun D K, Zhu M F, Pan S Y, Raabe D 2009 Acta Mater. 57 1755
Google Scholar
[5] 潘诗琰, 朱鸣芳 2012 物理学报 61 228102
Google Scholar
Pan S Y, Zhu M F 2012 Acta Phys. Sin. 61 228102
Google Scholar
[6] 潘诗琰, 朱鸣芳 2009 物理学报 58 278
Google Scholar
Pan S Y, Zhu M F 2009 Acta Phys. Sin. 58 278
Google Scholar
[7] Chen R, Xu Q Y, Liu B C 2015 Comput. Mater. Sci. 105 90
Google Scholar
[8] Liu S Y, Hong K M, Shin Y C 2021 Comput. Mater. Sci. 192 110405
Google Scholar
[9] Qi X B, Chen Y, Kang X H, Li D Z, Gong T Z 2017 Sci. Rep. 7 45770
Google Scholar
[10] Meng S X, Zhang A, Guo Z P, Wang Q G 2020 Comput. Mater. Sci. 184 109784
Google Scholar
[11] Takaki T, Sato R, Rojas R, Ohno M, Shibuta Y 2018 Comput. Mater. Sci. 147 124
Google Scholar
[12] Ratkai L, Pusztai T, Granasy L 2019 npj Comput. Mater. 5 113
Google Scholar
[13] Sakane S, Takaki T, Ohno M, Shibuta Y, Aoki T 2020 Comput. Mater. Sci. 178 109639
Google Scholar
[14] 吴伟, 孙东科, 戴挺, 朱鸣芳 2012 物理学报 61 150501
Google Scholar
Wu W, Sun D K, Dai T, Zhu M F 2012 Acta Phys. Sin. 61 150501
Google Scholar
[15] Fang H, Tang Q Y, Zhang Q Y, Gu T F, Zhu M F 2019 Int. J. Heat Mass Transfer 133 371
Google Scholar
[16] Lee W, Jeong Y, Lee J W, Lee H, Kang S H, Kim Y M, Yoon J 2020 J. Mater. Sci. Technol. 49 15
Google Scholar
[17] Cu C, Ridgeway C D, Moodispaw M P, Luo A A 2020 J. Mater. Process. Technol. 286 116829
Google Scholar
[18] Liu L, Pian S, Zhang Z, Bao Y, Li R, Chen H 2018 Comput. Mater. Sci. 146 9
Google Scholar
[19] Wu J Y, Sun D K, Wang J C, Zhu M F 2020 Eur. Phys. J. E 43 30
Google Scholar
[20] Zhang Q Y, Sun D K, Pan S Y, Zhu M F 2020 Int. J. Heat Mass Transfer 146 118838
Google Scholar
[21] Sun D K, Zhu M F, Pan S Y, Yang C R, Raabe D 2011 Comput. Math. Appl. 61 3585
Google Scholar
[22] Rappaz M, Thévoz P H 1987 Acta Metall. 35 2929
Google Scholar
[23] Zhu M F, Stefanescu D 2007 Acta Mater. 55 1741
Google Scholar
[24] Wen B H, Zhang C Y, Tu Y S, Wang C L, Fang H P 2014 J. Comput. Phys. 266 161
Google Scholar
[25] Mei R, Yu D, Shyy W, Luo L S 2002 Phys. Rev. E: Stat. Nonliner Soft Matter Phys. 65 041203
Google Scholar
[26] Wu M, Ludwig A, Fjeld A 2010 Comput. Mater. Sci. 50 43
Google Scholar
[27] Feng Z G, Michaelides E E 2004 J. Comput. Phys. 195 602
Google Scholar
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