EFFECT OF ASPECT RATIO ON THERMOCAPILLARY CONVECTION INSTABILITY OF GAAS MELTL IQUID BRIDGE
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摘要: 本文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. 我们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.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.
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图 2 Mac随高径比(As)的变化(Pr = 0.068)
Figure 2. The variation of Mac versus the aspect ratio As for Pr = 0.068.
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)
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
图 4 高径比As = 0.48, Pr = 0.068的临界状态(Mac = 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. The disturbance energy balance (a) and distribution of the local disturbance kinetic energy extreme value (b) under critical condition(Mac = 1936.3) at As = 0.48, Pr = 0.068.
图 5 高径比As = 0.48, Pr = 0.068的临界状态(Mac = 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 (Mac = 1936.3) at As = 0.48, Pr = 0.068.
图 6 高径比As = 0.77, Pr = 0.068的临界状态(Mac = 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. The disturbance energy balance (a) and distribution of the local disturbance kinetic energy extreme value (b) under critical condition (Mac = 1083.9) at As = 0.77, Pr = 0.068.
图 7 高径比As = 0.77, Pr = 0.068的临界状态(Mac = 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 (Mac = 1083.9) at As = 0.77, Pr = 0.068.
图 8 高径比As = 1.11, Pr = 0.068的临界状态(Mac = 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. The disturbance energy balance (a) and distribution of the local disturbance kinetic energy extreme value (b) under critical condition (Mac = 1273.6) at As = 1.11, Pr = 0.068.
图 9 高径比 As = 1.11, Pr = 0.068的临界状态(Mac = 1273.6)下扰动温度(云图)和扰动速度矢量(箭头): (a)自由液面; (b) z = 0.20截面; (c) z = 0.51截面; (d) z = 0.74截面
Figure 9. The disturbance temperature (colored contour) and disturbance velocity vector (arrow) under the critical condition (Mac = 1273.6) for the case of As = 1.11 and Pr = 0.068 at: (a) the free surface; (b) the z = 0.20 section; (c) the z = 0.51 section; (d) the z = 0.74 section
图 10 高径比As = 1.43, Pr = 0.068的临界状态(Mac = 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. The disturbance energy balance (a) and distribution of the local disturbance kinetic energy extreme value (b) under critical condition (Mac = 531.9) at As = 1.43, Pr = 0.068.
图 11 高径比As = 1.43, Pr = 0.068的临界状态(Mac = 531.9)下扰动温度(云图)和扰动速度矢量(箭头): (a)自由液面; (b) z = 0.20截面; (c) z = 0.51截面; (d) z = 0.74截面
Figure 11. The disturbance temperature (colored contour) and disturbance velocity vector (arrow) under the critical condition (Mac = 531.9) for the case of As = 1.43 and Pr = 0.068 at: (a) the free surface; (b) the z = 0.20 section; (c) the z = 0.51 section; (d) the z = 0.74 section
图 12 高径比As = 1.43, Pr = 0.009临界状态(Mac = 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 (Mac = 531.9) for the case of As = 1.43 and Pr = 0.009
图 13 高径比 As = 2.00, Pr = 0.068的临界状态(Mac = 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. The disturbance energy balance (a) and distribution of the local disturbance kinetic energy extreme value (b) under critical condition (Mac = 234.6) at As = 2.00, Pr=0.068.
图 14 高径比As = 2.00, Pr = 0.068的临界状态(Mac = 234.6)下扰动温度(云图)和扰动速度矢量(箭头): (a)自由液面; (b) z = 0.20截面; (c) z = 0.51截面; (d) z = 0.74截面
Figure 14. The disturbance temperature (colored contour) and disturbance velocity vector (arrow) under the critical condition (Mac = 234.6) for the case of As = 2.00 and Pr = 0.068 at: (a) the free surface; (b) the z = 0.20 section; (c) the z = 0.51 section; (d) the z = 0.74 section
图 15 As = 2.00 Pr = 0.009时临界状态(Mac = 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 (Mac = 234.6) for the case of As = 2.00 and Pr = 0.009
图 16 失稳模式I在临界状态下的基态流函数(云图)、基态温度场(等值线): (a) As = 0.48, Mac = 1936.3; (b) As = 0.77, Mac = 1083.9; (c) As = 1.11, Mac = 1273.6.
Figure 16. The basic flow (colored contour) and temperature (isotherm) of instability mode I under critical condition for case of: (a) As = 0.48, Mac = 1936.3; (b) As = 0.77, Mac = 1083.9; (c) As = 1.11, Mac = 1273.6.
图 17 失稳模式II在临界状态下的基态流函数(云图)、基态温度场(等值线): (a) As = 1.43, Mac = 531.9; (b) As = 2.00, Mac = 234.6(续)
Figure 17. The basic flow (colored contour) and temperature (isotherm) of instability mode II under critical condition at (a) As = 1.43, Mac = 531.9; (b) As = 2.00, Mac = 234.6. (continued)
17 失稳模式II在临界状态下的基态流函数(云图)、基态温度场(等值线): (a) As=1.43, Mac=531.9; (b) As=2.00, Mac=234.6
17. The basic flow (colored contour) and temperature (isotherm) of instability mode II under critical condition at (a) As = 1.43, Mac=531.9; (b) As = 2.00, Mac=234.6.
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(Mac)和波数kc
Table 1. The critical Marangoni number (Mac) and wave number kc for the instability of thermocapillary flow with Pr = 0.068 under different mesh resolution conditions
mesh resolution 41 × 46 49 × 55 57 × 64 Mac / kc 1179.7 / 3 1182.4 / 3 1182.7 / 3 
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表 2 计算失稳临界参数(Rec、fc和kc)与文献结果[14]对比
Table 2. The critical values(Rec, fc, kc)from the present computations against the reference results [14]
Pr Rec / fc /kc(Present) Rec / fc /kc [14] relative error of Rec/ fc 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(Mac和kc)与文献结果[15]对比
Table 3. The critical values(Mac and kc)from the present computations against the reference results[15]
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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 Mac fc kc 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
失稳模式I
(振荡失稳)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
失稳模式II1.33 691.3 0.0 2 1.43 531.9 0.0 2 1.54 483.6 0.0 2 1.67 350.5 0.0 1 (静态失稳)
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 IV1 IV2 IV3 IV4 IV5 IV4 + IV5 MZ Mφ MZ + Mφ 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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