高径比对GaAs熔体液桥热毛细对流失稳的影响

周游; 曾忠; 刘浩; 张良奇

doi:10.6052/0459-1879-21-227

高径比对GaAs熔体液桥热毛细对流失稳的影响

doi: 10.6052/0459-1879-21-227

周游^*,
曾忠^*,,
刘浩^†,
张良奇^*

*.
重庆大学航空航天学院, 重庆 400044
†.
重庆交通大学西南水运工程科学研究所, 重庆 400016

基金项目: 国家自然科学基金(12172070, 12102071)、中央高校基本科研业务费(2021 CDJQY-055)和重庆市教委科学技术研究(KJQN202100706)资助项目

详细信息

作者简介:
曾忠, 教授, 主要研究方向: 流体力学、热毛细对流、多相流. E-mail: zzeng@cqu.edu.cn

中图分类号: O35
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- 被引次数: 0
出版历程
- 收稿日期: 2021-05-26
- 录用日期: 2021-11-11
- 网络出版日期: 2021-11-12
- 刊出日期: 2022-02-17

EFFECT OF ASPECT RATIO ON THERMOCAPILLARY CONVECTION INSTABILITY OF GaAs MELT LIQUID BRIDGE

*.
College of Aerospace Engineering, Chongqing University, Chongqing 400044, China
†.
Chongqing Southwest Research Institute for Water Transport Engineering, Chongqing Jiaotong University, Chongqing 400016, China

摘要

摘要: 采用基于谱元法线性稳定性分析方法, 研究了高径比对GaAs熔体(Pr = 0.068)液桥热毛细对流失稳的影响, 同时结合能量分析揭示了热毛细对流的失稳机制. 研究结果表明: 与典型低普朗特数(例如Pr = 0.011)熔体静态失稳模式和典型高普朗特数(例如Pr>1)熔体振荡失稳模式不同, GaAs熔体热毛细对流失稳模式依赖于液桥高径比(As). 随高径比的变化, GaAs熔体热毛细对流存在两种失稳模式. 高径比As 在0.4≤As≤1.18范围内, 热毛细对流失稳是从二维轴对称定常对流转变为三维周期性振荡对流(振荡失稳); 高径比在1.20≤As≤2.5范围内, 热毛细对流失稳是从二维轴对称定常流动转变为三维定常流动(静态失稳). 典型的高普朗特数熔体液桥热毛细对流失稳机制是热毛细机制; 典型的低普朗特数液桥热毛细对流失稳机制是水动力学惯性机制. 本文基于扰动能量分析的结果表明: GaAs熔体热毛细对流失稳同时包括水动力学惯性失稳机制和热毛细失稳机制的贡献, 其中水动力学惯性失稳机制占主导作用, 两种机制对热毛细对流失稳能量贡献的占比随高径比的变化而变化.
- 谱元法 /
- 液桥 /
- 热毛细对流 /
- 线性稳定性分析
Abstract: In this paper, we explore the effect of aspect ratio on the instability of thermocapillary convection in GaAs melt (Pr = 0.068) liquid bridge by using the linear stability analysis in the context of spectral element method. Besides, we provide physical insight on the underlying instability mechanism via energy analysis. Differing from the cases of typical low Prandtl number (such as Pr = 0.011) and typical high Prandtl number (such as Pr > 1), which correspond to stationary instability and oscillatory instability respectively, the instability of the thermocapillary convection of GaAs melt (Pr = 0.068) is of note due to its noticeable dependence on the aspect ratio (As). In particular, we observe two instability modes for the flow considered here with the variation of the aspect ratio. When the aspect ratio As ranges from 0.4 to 1.18, thermocapillary flow transits from two-dimensional axisymmetric steady convection to three-dimensional periodic oscillatory convection (oscillatory instability). While for 1.20 ≤ As ≤ 2.5, the stationary instability appears and the two-dimensional axisymmetric steady flow transits to three-dimensional steady flow. As for the instability mechanism of the thermocapillary convection, the liquid bridge of high Prandtl number is characterized by thermocapillary mechanism, while the case of low Prandtl number features the hydrodynamic inertia mechanism. Based on disturbance energy analysis, it is shown that the instability of the present thermocapillary convection arises from the combined action of the hydrodynamic inertial instability and thermocapillary instability, in which the hydrodynamic inertial instability mechanism is dominant, and the specific proportion of these two contributions varies with the aspect ratio.
- spectral element method /
- liquid bridge /
- thermocapillary flow /
- linear stability analysis

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图 1 半浮区液桥模型

Figure 1. Floating half-zone model

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图 2 Ma_c随高径比(As)的变化(Pr = 0.068)

Figure 2. The variation of Ma_c versus the aspect ratio As for Pr = 0.068

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3 扰动能量随高径比As变化

3. The variation of the kinetic energy budgets with respect to the aspect ratio

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图 3 扰动能量随高径比As变化(续)

Figure 3. The variation of the kinetic energy budgets with respect to the aspect ratio (continued)

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图 4 高径比As = 0.48, Pr = 0.068的临界状态(Ma_c = 1936.3)下:(a)扰动能量平衡条状图, (b)局部扰动动能极值图

Figure 4. (a) The disturbance energy balance and (b) distribution of the local disturbance kinetic energy extreme value under critical condition (Ma_c = 1936.3) at As = 0.48, Pr = 0.068

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图 5 高径比As = 0.48, Pr = 0.068的临界状态 (Ma_c = 1936.3) 下:(a) z = 0.51截面扰动温度 (云图) 和扰动速度矢量 (箭头), (b)自由液面扰动温度 (云图) 和扰动速度矢量 (箭头)

Figure 5. The disturbance temperature (colored contour) and disturbance velocity vector (arrow): (a) the z = 0.51 cross section, (b) the free surface under the critical condition (Ma_c = 1936.3) at As = 0.48, Pr = 0.068

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图 6 高径比As = 0.77, Pr = 0.068的临界状态 (Ma_c = 1083.9) 下: (a)扰动能量平衡条状图, (b)局部扰动动能极值图

Figure 6. (a) The disturbance energy balance and (b) distribution of the local disturbance kinetic energy extreme value under critical condition (Ma_c = 1083.9) at As = 0.77, Pr = 0.068

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图 7 高径比As = 0.77, Pr = 0.068的临界状态 (Ma_c = 1083.9) 下: (a) z = 0.51截面扰动温度 (云图) 和扰动速度矢量 (箭头), (b)自由液面扰动温度 (云图) 和扰动速度矢量 (箭头)

Figure 7. The disturbance temperature (colored contour) and disturbance velocity vector (arrow): (a) the z = 0.51 cross section, (b) the free surface under the critical condition (Ma_c = 1083.9) at As = 0.77, Pr = 0.068

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图 8 高径比As = 1.11, Pr = 0.068的临界状态 (Ma_c = 1273.6) 下: (a)扰动能量平衡条状图, (b)局部扰动动能极值图

Figure 8. (a) The disturbance energy balance and (b) distribution of the local disturbance kinetic energy extreme value under critical condition (Ma_c = 1273.6) at As = 1.11, Pr = 0.068

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图 9 高径比 As = 1.11，Pr = 0.068的临界状态(Ma_c = 1273.6)下扰动温度(云图)和扰动速度矢量(箭头)

Figure 9. The disturbance temperature (colored contour) and disturbance velocity vector (arrow) under the critical condition (Ma_c = 1273.6) for the case of As = 1.11 and Pr = 0.068

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图 10 高径比As = 1.43, Pr = 0.068的临界状态 (Ma_c = 531.9) 下: (a)扰动能量平衡条状图, (b) 局部扰动动能极值图

Figure 10. (a) The disturbance energy balance and (b) distribution of the local disturbance kinetic energy extreme value under critical condition (Ma_c = 531.9) at As= 1.43, Pr = 0.068

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图 11 高径比As = 1.43, Pr = 0.068的临界状态(Ma_c = 531.9)下扰动温度(云图)和扰动速度矢量(箭头)

Figure 11. The disturbance temperature (colored contour) and disturbance velocity vector (arrow) under the critical condition (Ma_c = 531.9) for the case of As = 1.43 and Pr = 0.068

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图 12 高径比As = 1.43, Pr = 0.009临界状态(Ma_c = 531.9)下自由液面处扰动温度(云图)和扰动速度矢量(箭头)

Figure 12. The disturbance temperature (colored contour) and disturbance velocity vector (arrow) at the free surface under the critical condition (Ma_c = 531.9) for the case of As = 1.43 and Pr = 0.009

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图 13 高径比 As = 2.00，Pr = 0.068的临界状态 (Ma_c = 234.6) 下: (a)扰动能量平衡条状图, (b)局部扰动动能极值图

Figure 13. (a) The disturbance energy balance and (b) distribution of the local disturbance kinetic energy extreme value under critical condition (Ma_c = 234.6) at As= 2.00, Pr = 0.068

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图 14 高径比As = 2.00, Pr = 0.068的临界状态(Ma_c = 234.6)下扰动温度(云图)和扰动速度矢量(箭头)

Figure 14. The disturbance temperature (colored contour) and disturbance velocity vector (arrow) under the critical condition (Ma_c = 234.6) for the case of As = 2.00 and Pr = 0.068

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图 15 As = 2.00 Pr = 0.009时临界状态(Ma_c = 234.6)下自由液面处扰动温度(云图)和扰动速度矢量(箭头)

Figure 15. The disturbance temperature (colored contour) and disturbance velocity vector (arrow) at the free surface under the critical condition (Ma_c = 234.6) for the case of As = 2.00 and Pr = 0.009

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图 16 失稳模式I在临界状态下的基态流函数(云图)和基态温度场(等值线)

Figure 16. The basic flow (colored contour) and temperature (isotherm) of instability mode I under critical condition

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17 失稳模式II在临界状态下的基态流函数(云图)和基态温度场(等值线)

17. The basic flow (colored contour) and temperature (isotherm) of instability mode II under critical condition

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图 17 失稳模式II在临界状态下的基态流函数(云图)和基态温度场(等值线) (续)

Figure 17. The basic flow (colored contour) and temperature (isotherm) of instability mode II under critical condition (continued)

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表 1 基于不同网格分辨率得到的热毛细对流(Pr = 0.068)失稳临界Marangoni数(Ma_c)和波数k_c

Table 1. The critical Marangoni number (Ma_c) and wave number k_c for the instability of thermocapillary flow with Pr = 0.068 under different mesh resolution conditions

Mesh resolution	41 × 46	49 × 55	57 × 64
Ma_c / k_c	1179.7/3	1182.4/3	1182.7/3

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表 2 计算失稳临界参数(Re_c, f_c和k_c)与文献[14]结果对比

Table 2. The critical values (Re_c, f_c, k_c) from the present computations against Ref. [14]

Pr	Re_c/ f_c/k_c (present)	Re_c/ f_c/k_c^[14]	Relative error of Re_c/f_c
0.068	17379/205.3/3	17229/206/3	0.87% / 0.36%
0.183	14021/409.1/3	14276/415/3	1.78%/1.42%
1.0	2544/64.9/2	2551/65/2	0.27%/0.15%

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表 3 计算失稳临界参数(Ma_c和k_c)与文献[15]结果对比

Table 3. The critical values (Ma_c and k_c) from the present computations against Ref. [15]

As	Ma_c/k_c (present)	Ma_c/k_c^[15] (LSA)	Ma_c/k_c^[15] (DNS)
0.6	32.27/3	32.21/3	34.73/3
0.8	23.12/2	23.18/2	24.71/2
1.0	16.76/2	17.00/2	19.77/2

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表 4 Pr = 0.068时不同高径比下的失稳临界参数

Table 4. The critical values at different aspect ratios for Pr = 0.068

As	Ma_c	f_c	k_c	Type
0.40	2033.1	±490.7	6
0.43	1920.0	±493.6	5
0.48	1936.3	±507.4	5
0.53	1770.8	±504.7	4
0.59	1838.4	±536.1	4
0.67	1988.8	±233.0	4	type I (oscillatory instability)
0.71	1278.0	±200.3	4
0.77	1083.9	±201.7	4
0.91	995.3	±155.2	3
1.00	1182.4	±205.3	3
1.05	1246.8	±63.6	2
1.11	1273.6	±62.7	2
1.18	1393.6	±36.7	2
1.20	1348.4	0.0	2
1.25	1058.6	0.0	2	type II (stationary instability)
1.33	691.3	0.0	2
1.43	531.9	0.0	2
1.54	483.6	0.0	2
1.67	350.5	0.0	1
1.82	262.8	0.0	1
2.00	234.6	0.0	1
2.22	248.4	0.0	1
2.50	326.0	0.0	1

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表 5 Pr = 0.068时不同高径比能量分析结果

Table 5. The energy analysis results at different aspect ratios for Pr = 0.068

As	I_V1/%	I_V2/%	I_V3/%	I_V4/%	I_V5/%	(I_V4 + I_V5)/%	M_z/%	M_φ/%	(M_z + M_φ)/%
0.40	2.45	−7.90	−1.44	51.69	47.92	99.61	3.54	3.67	7.21
0.43	2.17	−12.31	−1.45	59.63	44.46	104.09	4.04	3.34	7.38
0.48	4.39	−6.12	−1.81	48.98	48.38	97.36	3.11	2.99	6.10
0.53	4.14	−11.26	−1.84	58.69	44.07	102.76	3.55	2.53	6.08
0.59	6.69	−3.11	−2.53	44.19	49.81	94.00	2.59	2.22	4.81
0.67	0.63	9.55	−1.11	18.19	63.87	82.06	2.57	6.17	8.74
0.71	−0.18	10.44	−2.67	20.40	62.95	83.35	2.49	6.49	8.98
0.77	0.85	12.58	−4.16	19.41	63.65	83.06	1.86	5.75	7.61
0.91	0.33	9.57	−4.29	22.39	59.06	81.45	3.36	6.50	9.86
1.00	7.10	16.23	−5.75	14.43	61.53	75.96	1.48	5.04	6.52
1.05	−1.95	−23.35	−1.05	71.59	26.65	98.24	18.95	8.40	27.35
1.11	2.49	−16.15	2.00	65.21	23.27	88.48	15.51	7.06	22.57
1.18	9.54	−6.57	6.34	55.90	19.22	75.12	10.95	4.36	15.31
1.20	10.29	−2.22	7.53	50.60	20.41	71.01	9.73	3.66	13.39
1.25	4.91	1.52	7.66	44.28	27.33	71.61	9.14	4.93	14.07
1.33	−2.44	1.75	8.78	43.12	29.86	72.98	10.72	8.18	18.90
1.43	−4.71	1.34	9.22	43.80	28.81	72.61	10.88	10.64	21.52
1.54	−4.84	2.51	9.18	43.27	27.71	70.98	10.04	12.14	22.18
1.67	5.33	−23.21	−13.15	117.74	8.12	125.86	16.78	−11.67	5.11
1.82	5.42	−19.76	−13.34	118.69	7.33	126.02	15.32	−13.65	1.67
2.00	4.22	−14.91	−12.64	108.46	11.71	120.17	16.04	−12.88	3.16
2.22	2.39	−9.98	−11.03	93.18	17.04	110.22	17.92	−9.51	8.41
2.50	−0.11	−5.68	−8.57	77.13	19.57	96.70	20.53	−2.87	17.66

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