环形双层液体内热对流过程的二维数值模拟
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摘要
在晶体生长过程中,由表面张力梯度引起的热毛细对流成为影响晶体材料品质的重要因素。液封Czochralski生长技术能有效地抑制熔体内的热毛细对流,但是,目前对该生长技术中环形双层液体系统内热对流过程基本特征的了解非常缺乏,因此,研究在水平温度梯度作用下环形双层液体内热对流过程的稳定性、发生转变的物理机制及其基本特性等,对于改进生产过程、提高产品质量具有重要的理论意义和实用价值。
本文建立了环形双层液体内存在水平温度梯度时热对流过程的物理模型和数学模型,采用有限容积法进行了二维数值模拟,基于双层流体内的流场和温度场分布,分析了Marangoni (Ma)数、浮力以及几何参数等对热对流的影响,得到了各种不同条件下的临界Ma数,探讨了流动失稳的物理机制。
结果表明:(1)当Ma数较小时,流动较弱,为稳态的二维流动。随着Ma数和深宽比的增大,流动强度增强,等温线发生强烈的非线性变形;浮力的引入强化了熔体层内的流动。(2)当Ma数超过临界值后,流动会失去稳定性,在冷壁附近出现附加的小流胞,流动会转化为非稳定的多胞流动。小的流胞在冷壁附近形成,并向热壁运动,最后在热壁附近消失。当Ma数较小时,多胞结构主要集中在冷壁附近,速度、温度振荡仅在冷壁附近较窄的区域内出现,随着Ma数和深宽比的增大,速度、温度振荡增强并向热壁方向扩展,多胞区域在更大的范围内出现。(3)临界Ma数随着深宽比的增大而减小,随着半径比的增大而增大。在有自由表面的环形液池内,浮力的引入会使临界Ma数增大,推迟流动的失稳,液池越浅,浮力的影响越明显。相同条件下,上部为固壁时临界Ma数更小,流动更易失稳。在上部为固壁的环形腔内,浮力的引入会使临界Ma数减小,流动更易失稳。
In the crystal growth process, the thermocapillary flow induced by the interface tension gradient becomes a prominent factor influencing the quality of crystal materials. The liquid encapsulation Czochralski (LEC) technology can effectively suppress thermocapillary convection in the melt. But there is a lack of the understanding on the basic characteristics of thermocapillary convection in the annular two-layer liquid system. It is of great theoretical significance and practical value to investigate the stability, physical mechanism and characteristics of thermocapillary convection in the two-layer system under the horizontal temperature gradient so as to improve the production processes and quality of materials.
The physical and mathematical models of thermal convection in the annular two-layer systems under the horizontal temperature gradient were established. A set of two-dimensional numerical simulations was carried out using the finite-volume method. The distributions of temperature and velocity in the two liquid layers were then obtained and effects of parameters, such as Marangoni (Ma) number, buoyant and geometric parameter et al, on the thermal convection were analyzed. Also, the critical Ma number at different conditions was determined and the physical mechanism of the unstable thermal convection was revealed.
It is found that: (1) for small Ma number, the flow is steady. With the increase of Ma number and the aspect ratio, the flow strength becomes much stronger and the deformation of the isotherms increases sharply. In considering the effect of buoyancy, the thermal convection in the melt is enhanced. (2) While Ma number exceeds the critical value, an unsteady multicultural structure is developed, and the additional cells appear near the cold lateral wall. During the oscillatory process, the flow cells move to the hot wall and then fade away in the vicinity of the hot wall. When the Ma number is small, the multicultural structure and the oscillations of velocity and temperature mainly appear near the cold wall. With the increase of the Ma number, the oscillations of velocity and temperature increase and propagate to the hot wall, and the multicultural structure expands to a wider region. (3) The critical Ma number decreases with the increase of the aspect ratio and the decrease of the radius ratio. In the annular pool with a free surface, the buoyancy can delay convective instability of the two-layer system and make the critical Ma number increase. The effect of buoyancy is much obviously in the shallow pool. The critical Ma number decreases and the instability of the system enhances when the top surface is bounded by the rigid wall, and in this condition the buoyancy makes the system apt to lose stability and the critical Ma number decrease.
本文建立了环形双层液体内存在水平温度梯度时热对流过程的物理模型和数学模型,采用有限容积法进行了二维数值模拟,基于双层流体内的流场和温度场分布,分析了Marangoni (Ma)数、浮力以及几何参数等对热对流的影响,得到了各种不同条件下的临界Ma数,探讨了流动失稳的物理机制。
结果表明:(1)当Ma数较小时,流动较弱,为稳态的二维流动。随着Ma数和深宽比的增大,流动强度增强,等温线发生强烈的非线性变形;浮力的引入强化了熔体层内的流动。(2)当Ma数超过临界值后,流动会失去稳定性,在冷壁附近出现附加的小流胞,流动会转化为非稳定的多胞流动。小的流胞在冷壁附近形成,并向热壁运动,最后在热壁附近消失。当Ma数较小时,多胞结构主要集中在冷壁附近,速度、温度振荡仅在冷壁附近较窄的区域内出现,随着Ma数和深宽比的增大,速度、温度振荡增强并向热壁方向扩展,多胞区域在更大的范围内出现。(3)临界Ma数随着深宽比的增大而减小,随着半径比的增大而增大。在有自由表面的环形液池内,浮力的引入会使临界Ma数增大,推迟流动的失稳,液池越浅,浮力的影响越明显。相同条件下,上部为固壁时临界Ma数更小,流动更易失稳。在上部为固壁的环形腔内,浮力的引入会使临界Ma数减小,流动更易失稳。
In the crystal growth process, the thermocapillary flow induced by the interface tension gradient becomes a prominent factor influencing the quality of crystal materials. The liquid encapsulation Czochralski (LEC) technology can effectively suppress thermocapillary convection in the melt. But there is a lack of the understanding on the basic characteristics of thermocapillary convection in the annular two-layer liquid system. It is of great theoretical significance and practical value to investigate the stability, physical mechanism and characteristics of thermocapillary convection in the two-layer system under the horizontal temperature gradient so as to improve the production processes and quality of materials.
The physical and mathematical models of thermal convection in the annular two-layer systems under the horizontal temperature gradient were established. A set of two-dimensional numerical simulations was carried out using the finite-volume method. The distributions of temperature and velocity in the two liquid layers were then obtained and effects of parameters, such as Marangoni (Ma) number, buoyant and geometric parameter et al, on the thermal convection were analyzed. Also, the critical Ma number at different conditions was determined and the physical mechanism of the unstable thermal convection was revealed.
It is found that: (1) for small Ma number, the flow is steady. With the increase of Ma number and the aspect ratio, the flow strength becomes much stronger and the deformation of the isotherms increases sharply. In considering the effect of buoyancy, the thermal convection in the melt is enhanced. (2) While Ma number exceeds the critical value, an unsteady multicultural structure is developed, and the additional cells appear near the cold lateral wall. During the oscillatory process, the flow cells move to the hot wall and then fade away in the vicinity of the hot wall. When the Ma number is small, the multicultural structure and the oscillations of velocity and temperature mainly appear near the cold wall. With the increase of the Ma number, the oscillations of velocity and temperature increase and propagate to the hot wall, and the multicultural structure expands to a wider region. (3) The critical Ma number decreases with the increase of the aspect ratio and the decrease of the radius ratio. In the annular pool with a free surface, the buoyancy can delay convective instability of the two-layer system and make the critical Ma number increase. The effect of buoyancy is much obviously in the shallow pool. The critical Ma number decreases and the instability of the system enhances when the top surface is bounded by the rigid wall, and in this condition the buoyancy makes the system apt to lose stability and the critical Ma number decrease.
引文
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[2] D. Villers, J. K. Platten. Thermal convection in superposed immiscible liquid layers [J]. Appl. Sci. Res., 1988, 45:145-152.
[3] A. Prakash, L. J. Peltier, D. Fujita. Convection in a two layer fluid system [J]. AIAA paper, 1991, 91-0313.
[4]胡良,刘秋生,刘方,张璞,姚永龙,胡文瑞.实践五号卫星两层流体Marangoni对流和热毛细对流研究[J].中国空间科学学会实践五号卫星空间探测与试验成果学术会议论文集,2002, 84-90.
[5] Q. S. Liu, B. H. Zhou, L. Hu, P. Zhang, Y.L. Yao, W.R. Hu. Space experiments of convection in a system of two immiscible liquid layers [J]. ESA SP, 2000, 117-124.
[6]周炳红,刘秋生,胡良.两层流体热毛细对流空间实验研究[J].中国科学,2002, 32(3):316-322.
[7] B. H. Zhou, Q. S. Liu, L. Hu, Y. L. Yao, W. R. Hu. Space experiments of thermocapillary convection in two-liquid layers [J]. Sci. China, Ser. E, 2002, 45:552-560.
[8] S. Someya, T. Munakata, M. Nishio. Flow observation in two immiscible liquid layers subject to a horizontal temperature gradient [J]. J. Cryst. Growth, 2002, 235:626-632.
[9]张嘉锋.两层不混溶液体中Benard-Marangoni的对流的实验研究[D].合肥:中国科学技术大学,2001.
[10] I. B. Simanovskii, P. Georis, A. A. Nepomnyashchy, J. C. Legros. Oscillatory instability in multilayer systems [J]. Phys. Fluids, 2003, 15:3867-3870.
[11]洪勇,金蔚青,潘秀红,依田真一.氧化物熔体自由表面变形的实验研究[J].中国科学,2005, 35(9):897-903.
[12] D. Villers, J. K. Platten. Influence of interfacial tension gradients on thermal convection in two superposed immiscible liquid layers [J]. Appl. Sci. Res., 1990, 47:177-191.
[13] T. Doi, J. N. Koster. Thermocapillary convection in two immiscible liquid layers with free surface [J]. Phys. Fluids A, 1993, 5:1914-1927.
[14] Q. S. Liu, G. Chen, B. Roux. Thermogravitational and thermocapillary convection in a cavity containing two superposed immiscible liquid layers [J]. Int. J. Heat Mass Transfer, 1993, 36 (1):101-117.
[15] Q. S. Liu, B. Roux. Thermocapillary convection in two-layer systems [J]. Int. J. Heat Mass Transfer, 1998, 41(11):1499-1511.
[16] P. Wang, R. Kahawita, D. L. Nguyen. Numerical simulation of Buoyancy-Marangoni convection in two superposed immiscible liquid layers with a free surface [J]. Int. J. Heat Mass Transfer, 1994, 1111-1112.
[17] L. M. Braverman, K. Eckert, A. A. Nepomnyashchy, I. B. Simanovskii, A. Thess. Convection in two-layer systems with an anomalous thermocapillary effect [J]. Phys. Rev. E, 2000, 62 (3):3619-3631.
[18] Th. Boeck, A. A. Nepomnyashchy, I. B. Simanovskii, A. Golovin, L. M. Braverman, Andre Thess. Three-dimensional convection in a two-layer system with anomalous thermocapillary effect [J]. Phys. Fluids, 2002, 14:3899-3911.
[19]刘秋生.多层流体的Marangoni对流[J].力学学报,2002, 481-490.
[20] A. A. Nepomnyashchy, I. B. Simanovskii. Combined action of anticonvection and thermocapillary mechanisms of instability [J]. Phys. Fluids, 2002, 14 (11):3855-3867.
[21] A. A. Nepomnyashchy, I. B. Simanovskii.Influence of buoyancy on thermocapillary oscillations in a two-layer system [J]. Phys. Rev. E, 2003, 68 (026301):1-8.
[22] A. A. Nepomnyashchy, I. B. Simanovskii. Influence of thermocapillary effect and interfacial heat release on convective oscillations in a two-layer system [J]. Phys. Fluids, 2004, 16: 1127-1139.
[23] S. J. Tavener, K. A. Cliffe. Two-fluid Marangoni–Benard convection with a deformable interface [J]. J. Comput. Physics, 2002, 182: 277–300.
[24] B. H. Zhou, Q.S. Liu, Z.M. Tang. Rayleigh-Marangoni-Benard instability in two-layer fluid system [J]. Acta Mechanica Sinica, 2004, 20:366-373.
[25] Q.S. Liu, B. H. Zhou, Z. M. Tang. Oscillatory instability of Rayleigh-Marangoni-Bénard convection in two-layer liquid system [J]. J. Non-Equilib. Thermodyn., 2005, 30:305–319.
[26] Q. S. Liu, B. H. Zhou, R. Liu, H. N. Thi, B. Billia. Oscillatory instabilities of two-layer Rayleigh-Marangoni-Benard convection [J]. Acta Astronautica, 2006, 59:40– 55.
[27] G. B. McFadden, S. R. Coriell, K. F. Gurski, D. L. Cotrell. Onset of convection in two liquid layers with phase change [J]. Phys. Fluids, 2007, 19, 104109:1-13.
[28] S. Madruga, C. Pérez–García, G. Lebon. Convective instabilities in two superposed horizontal liquid layers heated laterally [J]. Phys. Rev. E, 2003, 68 (041607):1-12.
[29] S. Madruga, C. Pérez–García, G. Lebon. Instabilities in two-liquid layers subject to a horizontal temperature gradient [J]. Theor. Comp. Fluid Dyn., 2004, 18:277-284.
[30] A. A. Nepomnyashchy, I. B. Simanovskii. Convective flows in a two-layer system with a temperature gradient along the interface [J]. Phys. Fluids, 2006, 18, 032105:1-7.
[31] I. B. Simanovskii, A. A. Nepomnyashchy. Nonlinear development of oscillatory instability in a two-layer system under the combined action of buoyancy and thermocapillary effect [J]. J. Fluid Mech., 2006, 555:177-202.
[32] N. R. Gupta, H. H. Hossein, A. Borhan. Thermocapillary flow in double-layer fluid structures: An effective single-layer model [J]. J. Colloid Interface Sci., 2006, 293: 158-171.
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