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横向磁场下侧壁加热方腔熔化的数值模拟研究

孙思睿 张杰 倪明玖

孙思睿, 张杰, 倪明玖. 横向磁场下侧壁加热方腔熔化的数值模拟研究. 力学学报, 2022, 54(9): 2377-2386 doi: 10.6052/0459-1879-22-155
引用本文: 孙思睿, 张杰, 倪明玖. 横向磁场下侧壁加热方腔熔化的数值模拟研究. 力学学报, 2022, 54(9): 2377-2386 doi: 10.6052/0459-1879-22-155
Sun Sirui, Zhang Jie, Ni Mingjiu. The numerical simulation of melting process in a lateral heated cavity under transverse magnetic fields. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(9): 2377-2386 doi: 10.6052/0459-1879-22-155
Citation: Sun Sirui, Zhang Jie, Ni Mingjiu. The numerical simulation of melting process in a lateral heated cavity under transverse magnetic fields. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(9): 2377-2386 doi: 10.6052/0459-1879-22-155

横向磁场下侧壁加热方腔熔化的数值模拟研究

doi: 10.6052/0459-1879-22-155
基金项目: 国家自然科学基金(11872296, 51927812)资助项目
详细信息
    作者简介:

    张杰, 教授, 主要研究方向: 磁流体力学、计算流体力学、界面流体力学. E-mail: j_zhang@xjtu.edu.cn

  • 中图分类号: O361.3

THE NUMERICAL SIMULATION OF MELTING PROCESS IN A LATERAL HEATED CAVITY UNDER TRANSVERSE MAGNETIC FIELDS

  • 摘要: 磁场下的固−液相变过程在电磁冶金和增材制造等工程应用中广泛存在, 其中的熔化过程和流动机理尚未完全探究清楚. 方腔熔化是研究固−液相变过程的基础模型, 具有良好的普适性, 研究磁场对其流动的影响可以为其他复杂相变过程提供参考. 本文基于焓方法开展了固−液相变的数值模拟研究, 得到了垂直主环流方向的横向磁场对侧壁加热方腔中流动、传热和熔化过程的影响. 首先, 对于无磁场时的方腔熔化问题, 通过与已有的实验结果和数值结果进行对比, 证实了文献中方腔宽度对固−液界面的形状及位置影响不能忽视的结论. 随后, 对小磁场情况下的三维工况进行了直接数值模拟, 发现此时磁场效应主要表现为对混乱三维流动产生整流作用, 使流动趋于二维化. 但由于固−液界面的存在, 主流区的速度在趋于一致的同时也会反作用于界面, 其形状随磁场增强而逐渐转变为二维结构. 最后, 本文采用准二维模型分析了更强磁场时的情形, 讨论了不同参数对传热效率及界面形状的影响, 并发现了横向磁场作用下的垂直最大速度仍满足磁对流中的无量纲参数标度律关系.

     

  • 图  1  物理模型示意图

    Figure  1.  Sketch of the physical model

    图  2  固−液界面形状及位置比较. 空心点为文献[7]的实验结果, 虚线为文献[12]的数值模拟结果, 实线为本文的计算结果

    Figure  2.  Profile comparison of the melting fronts. The hollow dots are measured by Ref. [7] experimentally, the dashed lines are the numerical results by Ref. [12], while the solid lines are the present numerical solutions

    图  3  $ Ra=1\times {10}^{5} $, 不同$ Ha $下位于$ x=0.05 $, $ y=0.5 $线段处的平均竖直速度($ \overline{{u}_{y}} $)分布图. 点状线为文献[28]的计算结果,实线为本文的计算结果

    Figure  3.  The distribution of mean vertical velocity ($ \overline{{u}_{y}} $) along $ x=0.05 $, $ y=0.5 $ for $ Ra=1\times {10}^{5} $ with different $ Ha $. Dotted lines are the results simulated by Ref. [28], solid lines are the present numerical results

    图  4  $ Fo=1.6, x=0.05 $, $ y=0.5 $处的瞬时竖直速度$ {u}_{y} $(黑线)和洛伦兹力分量$ {Fl}_{y} $(蓝线)的分布

    Figure  4.  The distribution of the instantaneous vertical velocity (black lines) and y-component Lorentz force (blue line) at $Fo=1.6$, $ x=0.05 \;{\rm{and}}\; y=0.5 $

    图  5  $ Fo=1.6, x=0.05 $处的瞬时竖直速度云图, 分别对应$ Ha=0 $ ($z\leqslant 0.5$)和$ Ha=50 $($z\geqslant 0.5$). 图中矢量表示$ z-y $平面内的速度

    Figure  5.  The contour maps of the instantaneous vertical velocity ($ {u}_{y} $) for $ Ha=0 $($z\leqslant 0.5$) and $ Ha=50 $($z\geqslant 0.5$) at $ x=0.05 $. The vectors present the velocity in the $ z-y $ plane

    图  6  $ Ha=0 $$ Ha=50 $时, 全场总动能随时间的变化情况

    Figure  6.  Evolution of total kinetic energy in the whole domain for $ Ha=0 $ and $ Ha=50 $

    图  7  不同$ Ha $下, (a)液相体积分数随时间的变化和(b)$ Nu $随时间的变化

    Figure  7.  (a) Time evolution of the liquid volume fraction and (b) time evolution of $ Nu $ at different $ Ha $

    图  8  $ Fo=1.6 $, 界面形状及流线图比较, 界面上云图的颜色表示 $ x $轴坐标值

    Figure  8.  Comparison of the solid-liquid interface and the streamlines at $Fo=1.6 $ The contours on the interface are the $ x $ coordinate

    图  9  $ Ha= $ 400时, 不同$ Ra $下的温度云图及流线图比较(从左至右依次为$ Ra=1\times {10}^{4} $, $ Ra=1\times {10}^{5}, \;Ra=1\times {10}^{6} $)

    Figure  9.  Comparison of the temperature contour and the streamlines at different instants ($ Ra=1\times {10}^{4} $, $ Ra=1\times {10}^{5},\;Ra=1\times {10}^{6} $ from left to right) at $ Ha= $ 400

    9  Ha = 400时, 不同Ra下的温度云图及流线图比较(从左至右依次为$Ra=1\times {10}^{4} $, $Ra=1\times {10}^{5},\; Ra=1\times {10}^{6} $)(续)

    9.  Comparison of the temperature contour and the streamlines at different instants ($Ra=1\times {10}^{4} $, $Ra=1\times {10}^{5},\;Ra=1\times {10}^{6} $ from left to right) at $Ha= $ 400 (continued)

    图  10  $ Ra=1\times {10}^{6} $, $ Fo=2 $固−液界面随$ Ha $增大的变化趋势

    Figure  10.  Comparison of the solid-liquid interfaces at $ Ra=1\times {10}^{6} $ and $ Fo=2 $ when Ha is varied

    图  11  不同$ Ha $下, (a)液相体积分数和(b)左侧壁面$ Nu $随时间的变化, 从左至右依次为$ Ra=1\times {10}^{4} $, $ Ra=1\times {10}^{5}, \;Ra=1\times {10}^{6} $

    Figure  11.  Time evolution of (a) the liquid volume fraction and (b) the Nusselt number at different Hartmann number, while the sub-panels correspond to $ Ra=1\times {10}^{4} $, $ Ra=1\times {10}^{5} $ and $ Ra=1\times {10}^{6} $ from left to right

    图  12  熔化过程中的最大速度($ {u}_{y,\mathrm{m}\mathrm{a}\mathrm{x}} $)与无量纲参数组合 $ HaPr/Ra $的关系

    Figure  12.  The relation between the maximum velocity (${u}_{y,{\rm{max}}}$) during the melting process and the combination of dimensionless numbers of $ HaPr/Ra $

    表  1  物性参数表

    Table  1.   Material physical properties

    Physical properties of galliumUnitsValues
    density (liquid) $ {\rho }_{\text{l}} $$ {\text{kg/m}}^{\text{3}} $6095
    density (solid) $ {\rho }_{\text{s}} $$ {\text{kg/m}}^{\text{3}} $5907
    specific heat ${c}_{{\rm{p}}}$${ {\rm{J} }/}({{\rm{kg}}\cdot {\rm{K}}}^{-1})$397.6
    thermal expansivity $ \beta $${{\rm{K} } }^{-1}$1.27 × 10−4
    dynamic viscosity $ \mu $$ \text{kg/}\left(\text{m}\cdot \text{s}\right) $1.92 × 10−3
    thermal conductivity $ \lambda $$ \text{W/}\left(\text{m}\cdot \text{K}\right) $31.24
    electrical conductivity (liquid) $ {\sigma }_{\text{e,l}} $$ \text{S/m} $3.85 × 106
    electrical conductivity (solid) $ {\sigma }_{\text{e,s}} $$ \text{S/m} $6.64 × 106
    thermal diffusivity (liquid) $ {\kappa }_{\text{l}} $$ {\text{m}}^{\text{2}}\text{/s} $1.29 × 10−5
    thermal diffusivity (solid) $ {\kappa }_{\text{s}} $$ {\text{m}}^{\text{2}}\text{/s} $1.33 × 10−5
    latent heat $ L $$ \text{J/kg} $80160
    下载: 导出CSV

    表  2  不同$ Ha $下的$ \overline{Nu} $与文献[28]的计算结果比较

    Table  2.   Comparison of $ \overline{Nu} $ against the numerical results provided by Ref. [28] at different Ha

    $ Ha $2550100
    $ \overline{Nu} $ for 3D (Ref. [28]) 3.257 3.279 3.165
    $ \overline{Nu} $ for 3D (present) 3.157 3.186 3.039
    $ \overline{Nu} $ for 2D (Ref. [28]) 3.147
    $ \overline{Nu} $ for 2D (present) 3.016
    下载: 导出CSV
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出版历程
  • 收稿日期:  2022-04-11
  • 录用日期:  2022-06-01
  • 网络出版日期:  2022-06-02
  • 刊出日期:  2022-09-18

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