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In a crystal growth system, the crystal quality is greatly affected by the coupling properties between unsteady melt flow and thermal transfer. In this paper, an improved lattice Bolzmann method is proposed. This incompressible axisymmetric model based method transforms the fluid equations of cylindrical coordinate into those of the two-dimensional Cartesian coordinate and constructs the evolutionary relationship of the external force terms, such as rotational inertia force and the thermal buoyancy. In the unsteady melt, the temperature distribution and the rotational angular velocity are determined based on the D2Q4 model and the velocity of axisymmetric swirling fluid is calculated based on the D2Q9 model. The mirror bounce format is adopted as the boundary conditions of the free surface and the axis symmetry. For the remaining boundary conditions, the non-equilibrium extrapolation format is used. In the simulation, 12 sets of flow function results are obtained by choosing different sets of Grashof number and Reynolds number. By comparing with the finite crystal growth results, the effectiveness of the proposed method can be shown. Furthermore, by studying the convection shape and the temperature distribution of the melt under coupling between high Grashof number and high Reynolds number, it can be concluded that the thermal coupling properties and flow in the unsteady melt relate to Grashof number and Reynolds number. By adjusting the high Reynolds number generated by the crystal and crucible rotation, the strength of the forced convection in the melt can be changed. Therefore, the natural convection in the melt can be suppressed effectively and the temperature distribution results can be improved significantly. In addition, it is worth mentioning that the findings in this paper can be straightforwardly extended to the silicon single crystal growth experiment by turning the dimensionless crystal rotation Reynolds number and crucible rotation Reynolds number into the actual rotation speed.
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Keywords:
- lattice Boltzmann /
- crystal growth /
- axisymmetric swirl /
- thermal-fluid coupling
[1] Gu X, Li R, Tian Y 2014 J. Cryst. Growth 390 109
[2] Sabanskis A, Bergfelds K, Muiznieks A, Schröck Th, Krauze A 2013 J. Cryst. Growth 377 9
[3] Niemietz K, Galindo V, Pätzold O, Gerbeth G, Stelter M 2011 J. Cryst. Growth 318 150
[4] Xing H, Chen C L, Jin K X, Tan X Y, Fan F 2010 Acta Phys. Sin. 59 8218 (in Chinese) [邢辉, 陈长乐, 金克新, 谭兴毅, 范飞 2010 59 8218]
[5] Liu Q Z, Kou Z M, Han Z N, Gao G J 2013 Acta Phys. Sin. 62 234701 (in Chinese) [刘邱祖, 寇子明, 韩振南, 高贵军 2013 62 234701]
[6] Shi D Y, Wang Z K, Zhang A M 2014 Acta Phys. Sin. 63 074703 (in Chinese) [史冬岩, 王志凯, 张阿漫 2014 63 074703]
[7] Xie J F, Zhong C W, Zhang Y, Yin D C 2009 Chin. J. Theor. Appl. Mech. 41 635 (in Chinese) [解建飞, 钟诚文, 张勇, 尹大川 2009 力学学报 41 635]
[8] Halliday I, Hammond L A, Care C M, Good K, Stevens A 2001 Phys. Rev. E 64 011208
[9] Peng Y, Shu C, Chew Y T, Qiu J 2003 J. Comput. Phys. 186 295
[10] Weinstein O, Miller W 2010 J. Cryst. Growth 312 989
[11] Huang H B, Lu X Y, Krafczyk M 2014 Int. J. Heat Mass Tran. 74 156
[12] Guo Z L, Shi B C, Wang N C 2000 J. Comput. Phys. 165 288
[13] He X, Luo L 1997 J. Stat. Phys. 88 3
[14] Guo Z, Shi B, Zheng C 2002 Int. J. Numer. Meth. Fl. 39 325
[15] Zou Q, He X 1997 Phys. Fluids 9 1591
[16] Bansch E, Davis D, Langmach H, Reinhardt G, Uhle M 2006 Comput. Fluids 35 1400
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[1] Gu X, Li R, Tian Y 2014 J. Cryst. Growth 390 109
[2] Sabanskis A, Bergfelds K, Muiznieks A, Schröck Th, Krauze A 2013 J. Cryst. Growth 377 9
[3] Niemietz K, Galindo V, Pätzold O, Gerbeth G, Stelter M 2011 J. Cryst. Growth 318 150
[4] Xing H, Chen C L, Jin K X, Tan X Y, Fan F 2010 Acta Phys. Sin. 59 8218 (in Chinese) [邢辉, 陈长乐, 金克新, 谭兴毅, 范飞 2010 59 8218]
[5] Liu Q Z, Kou Z M, Han Z N, Gao G J 2013 Acta Phys. Sin. 62 234701 (in Chinese) [刘邱祖, 寇子明, 韩振南, 高贵军 2013 62 234701]
[6] Shi D Y, Wang Z K, Zhang A M 2014 Acta Phys. Sin. 63 074703 (in Chinese) [史冬岩, 王志凯, 张阿漫 2014 63 074703]
[7] Xie J F, Zhong C W, Zhang Y, Yin D C 2009 Chin. J. Theor. Appl. Mech. 41 635 (in Chinese) [解建飞, 钟诚文, 张勇, 尹大川 2009 力学学报 41 635]
[8] Halliday I, Hammond L A, Care C M, Good K, Stevens A 2001 Phys. Rev. E 64 011208
[9] Peng Y, Shu C, Chew Y T, Qiu J 2003 J. Comput. Phys. 186 295
[10] Weinstein O, Miller W 2010 J. Cryst. Growth 312 989
[11] Huang H B, Lu X Y, Krafczyk M 2014 Int. J. Heat Mass Tran. 74 156
[12] Guo Z L, Shi B C, Wang N C 2000 J. Comput. Phys. 165 288
[13] He X, Luo L 1997 J. Stat. Phys. 88 3
[14] Guo Z, Shi B, Zheng C 2002 Int. J. Numer. Meth. Fl. 39 325
[15] Zou Q, He X 1997 Phys. Fluids 9 1591
[16] Bansch E, Davis D, Langmach H, Reinhardt G, Uhle M 2006 Comput. Fluids 35 1400
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