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倾角对方腔内热对流非线性演化与分岔的影响

尹慧 赵秉新

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倾角对方腔内热对流非线性演化与分岔的影响

尹慧, 赵秉新

Effect of inclination on nonlinear evolution and bifurcation of thermal convection in a square cavity

Yin Hui, Zhao Bing-Xin
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  • 倾斜封闭腔内对流换热问题是非线性非平衡系统中研究的热点问题之一. 本文采用高精度数值方法对倾斜方腔内流体热对流进行了直接数值模拟, 研究了腔体倾角在$0^\circ— 180^\circ$之间变化时, 倾角的不同变化过程对流场非线性演化、传热效率以及流动分岔的影响. 所考虑的Rayleigh数范围为$10^3— 10^6$. 结果表明: 表征传热效率的Nusselt 数对Rayleigh数、Prandtl数及倾斜角度均具有较强依赖性, 在较高Rayleigh数时, Nusselt数会在80°和100°附近产生较大幅度的变化; 高Rayleigh 数下流场及温度场的演变更为复杂, 腔体内存在1—3个对流强度不等的涡卷; 低Rayleigh数下腔体倾角接近90°时流动状态为热传导状态. 当腔体倾角介于$70^\circ— 110^\circ$之间时, 在Rayleigh数$Ra\in(4949,314721)$内存在解的两条稳定分支.
    Heat transfer of natural convection in inclined cavities is one of the hot research topics in nonlinear non-equilibrium systems. In this paper, direct numerical simulations of natural convection in an inclined square cavity are carried out by using a high-accuracy numerical method. The effects of the different trends of inclination angle in a range of 0°–180° on the nonlinear evolution of flow field, heat transfer efficiency, and bifurcation are investigated. The Rayleigh number varies in a range from 103 to 106. The results show that the heat transfer efficiency characterized by Nusselt number is highly dependent on the Rayleigh number, Prandtl number, and the inclination angle. When the Rayleigh number is high, the Nusselt number will have a small jump near the inclination angle in a range of 80°–100°. The evolution of the flow field and temperature field are more complicated at high Rayleigh number. There are one to three vortices of different intensities in the cavity. At low Rayleigh number and inclination angle of the cavity being close to 90°, the flow state is composed mainly of heat conduction state. In addition, it is found that there exist two stable branches of solutions in a range of Rayleigh number (4949, 314721) when the inclination angle is in the interval of (70°, 110°).
      通信作者: 赵秉新, zhao_bx@nxu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11662016)、宁夏自然科学基金(批准号: 2020AAC03056, NZ16005)和宁夏大学生创新创业项目(批准号: 2019107490082)资助的课题
      Corresponding author: Zhao Bing-Xin, zhao_bx@nxu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11662016), the Natural Science Foundation of Ningxia Hui Autonomous Region, China (Grant Nos. 2020AAC03056, NZ16005), and the College Students’ Innovation and Entrepreneurship Project of Ningxia Hui Autonomous Region, China (Grant No. 2019107490082)
    [1]

    Batchelor G K 1954 Q. Appl. Math. 12 209Google Scholar

    [2]

    Eckert E R G, Carlson W O 1961 Int. J. Heat Mass Transfer 2 106Google Scholar

    [3]

    Patterson J C, Armfield S W 1990 J. Fluid Mech. 219 469Google Scholar

    [4]

    Xin S, Quéré P L 1995 J. Fluid Mech. 304 87Google Scholar

    [5]

    Das D, Roy M, Basak T 2017 Int. J. Heat Mass Transfer 106 356Google Scholar

    [6]

    Arnold J N, Catton I, Edwards D K 1976 J. Heat Transfer 98 67Google Scholar

    [7]

    John P, Jorg I 1980 J. Fluid Mech. 100 65Google Scholar

    [8]

    Khezzar L, Siginer D, Vinogradov I 2012 Heat Mass Transfer 48 227Google Scholar

    [9]

    Dider S, Abdelmadjid B, François P 2012 Exp. Therm Fluid Sci. 38 74Google Scholar

    [10]

    Torres J F, Henry D, Komiya A, Maruyama S 2014 J. Fluid Mech. 756 650Google Scholar

    [11]

    Torres J F, Henry D, Komiya A, Maruyama S 2015 Phys. Rev. E 92 023031Google Scholar

    [12]

    Miroshnichenko I V, Sheremet M A 2018 Renewable Sustainable Energy Rev. 82 40Google Scholar

    [13]

    徐丰, 崔会敏 2014 力学进展 44 201403

    Xu F, Cui H M 2014 Adv. Mech. 44 201403

    [14]

    Hamady F J, Lloyd J R, Yang H Q, Yang K T 1989 Int. J. Heat Mass Transfer 32 1697Google Scholar

    [15]

    Kuyper R A, Meer T H V D, Hoogendoorn C J 1994 Chem. Eng. Sci. 49 851Google Scholar

    [16]

    Rasoul J, Prinos P 1997 Int. J. Numer. Methods Heat Fluid Flow 7 438Google Scholar

    [17]

    Janssen R J A, Armfield S 1996 Int. J. Heat Fluid Flow 17 547Google Scholar

    [18]

    Varol Y, Oztop H F 2008 Build. Environ. 43 1535Google Scholar

    [19]

    Corcione M 2003 Int. J. Therm. Sci. 42 199Google Scholar

    [20]

    Wang H, Hamed M S 2006 Int. J. Therm. Sci. 45 782Google Scholar

    [21]

    Armfield S W, Janssen R 1996 Int. J. Heat Fluid Flow 17 539Google Scholar

    [22]

    Zhao B X, Tian Z F 2016 Int. J. Heat Mass Transfer 98 313Google Scholar

    [23]

    Sheremet M A, Pop I, Mahian O 2018 Int. J. Heat Mass Transfer 116 751Google Scholar

    [24]

    Boudjeniba B, Laouer A, Laouar S, Mezaache E H 2019 Int. J. Heat Technol. 37 413Google Scholar

    [25]

    Wang Q, Xia S N, Wang B F, Sun D J, Zhou Q, Wan Z H 2018 J. Fluid Mech. 849 355Google Scholar

    [26]

    Wang Q, Wan Z H, Yan R, Sun D J 2018 Phys. Rev. Fluids 3 113503Google Scholar

    [27]

    Wang Q, Chong K L, Stevens R J A M, Verzicco R, Lohse D 2020 J. Fluid Mech. 905 A21Google Scholar

    [28]

    Wang Q, Wan Z H, Yan R, Sun D J 2019 Phys. Fluids 31 025102Google Scholar

    [29]

    Wang Q, Verzicco R, Lohse D, Shishkina O 2020 Phys. Rev. Lett. 125 074501Google Scholar

    [30]

    Sugiyama K, Ni R, Stevens R J A M, Chan T S, Zhou S Q, Xi H D, Sun C, Grossmann S, Xia K Q, Lohse D 2010 Phys. Rev. Lett. 105 034503Google Scholar

    [31]

    Tian Z F, Liang X, Yu P X 2011 Int. J. Numer. Methods Eng. 88 511Google Scholar

    [32]

    Davis G D V 1983 Int. J. Numer. Methods Fluids 3 249Google Scholar

    [33]

    Kalita J C, Dalal D C, Dass A K 2001 Phys. Rev. E 64 066703Google Scholar

    [34]

    Tian Z F, Ge Y B 2003 Int. J. Numer. Methods Fluids 41 495Google Scholar

    [35]

    Yu P X, Tian Z F 2012 Phys. Rev. E 85 036703Google Scholar

  • 图 1  带边界条件的倾斜方腔示意图

    Fig. 1.  Schematic diagram of the inclined square cavity with boundary conditions.

    图 2  Nusselt数随Rayleigh数的变化($ \beta = 0^\circ $)

    Fig. 2.  Variation of Nusselt number as a function of Rayleigh number ($ \beta = 0^\circ $).

    图 3  Nusselt数随Prandtl数的变化($ \beta = {\rm{0}}^\circ $)

    Fig. 3.  Variation of Nusselt number as a function of Prandtl number ($ \beta = {\rm{0}}^\circ $).

    图 4  不同Rayleigh数下Nusselt数随倾斜角度的变化($ Pr = 0.71 $)

    Fig. 4.  Variation of Nusselt number with inclination angle for different Rayleigh numbers ($ Pr = 0.71 $).

    图 5  不同Prandtl数下Nusselt数随倾斜角度的变化(以过程B为例)

    Fig. 5.  Variation of Nusselt number with inclination angle for different Prandtl numbers (Process B).

    图 6  $Ra=10^{3}$时流线与等温线图

    Fig. 6.  Streamlines and isotherms for $Ra=10^{3}$.

    图 8  $ Ra = 10^{5} $时流线与等温线图

    Fig. 8.  Streamlines and isotherms for $ Ra = 10^{5} $.

    图 7  $Ra=10^{4}$时流线与等温线图

    Fig. 7.  Streamlines and isotherms for $Ra=10^{4}$.

    图 9  $ Ra = 10^{6} $时流场结构随倾角的变化

    Fig. 9.  Variation of flow field with inclination angle for $ Ra = 10^{6} $.

    图 10  分岔点附近Nusselt数随倾角的变化 (a) Ra = 4949; (b) Ra = 4950

    Fig. 10.  Variation of Nusselt number with inclination angle near the bifurcation point: (a) Ra = 4949; (b) Ra = 4950.

    图 11  分岔区间上界附近Nusselt数随倾角的变化曲线 (a) $Ra=314720 $; (b) Ra = 314721

    Fig. 11.  Variation of Nusselt number with inclination angle near the upper bound of bifurcation interval: (a) $Ra=314720$; (b) Ra = 314721.

    图 12  $ \beta = 80^\circ $时分岔点处过程B与过程C的对流斑图

    Fig. 12.  Flow field of process B and process C at the bifurcation point for $ \beta = 80^\circ $.

    表 1  与其他文献结果的对比($ Pr = 0.71 $, $ \beta = {\rm{0}}^\circ $)

    Table 1.  Comparison of the results by different numerical methods for $ Pr = 0.71 $ and $ \beta = {\rm{0}}^\circ $.

    文献 $ \left| \psi \right|_{\rm {max}} $ $ \left| {\psi _{\rm {mid}} } \right| $ $ Nu_{0} $ $ \overline{Nu} $ 文献 $ \left| \psi \right|_{\rm {max}} $ $ \left| {\psi _{\rm {mid}} } \right| $ $ Nu_{0} $ $ \overline{Nu} $
    $ Ra=10^{5} $ $ Ra=10^{6} $
    本文 9.615 9.115 4.520 4.522 本文 16.807 16.383 8.815 8.827
    [32] 9.612 9.111 4.509 4.519 [32] 16.750 16.320 8.817 8.800
    [33] 9.123 4.512 4.522 [33] 16.420 8.763 8.829
    [34] 9.6173 9.1161 4.5195 [34] 16.8107 16.3863 8.8216
    [35] 9.6202 9.1194 4.5214 [35] 16.8411 16.4183 8.8091
    下载: 导出CSV

    表 2  $ Pr = 0.71 $, $ \beta = {\rm{0}}^\circ $, $ Ra = 10^{6} $下的网格检验结果

    Table 2.  Grid test results for $ Pr = 0.71 $, $ \beta = {\rm{0}}^\circ $ and $ Ra = 10^{6} $.

    网格尺寸 $\left| \psi \right|_{\rm {max}}$ 误差/% $\left| {\psi _{\rm {mid}} } \right|$ 误差/% $ Nu_0 $ 误差/%
    $ 31\times31 $ 16.460 2.086 16.118 1.631 9.293 5.301
    $ 61\times61 $ 16.830 0.119 16.410 0.148 8.798 0.315
    $ 91\times91 $ 16.802 0.051 16.385 0.002 8.786 0.445
    $ 121\times121 $ 16.807 0.017 16.383 0.014 8.815 0.119
    $ 241\times241 $ 16.810 16.386 8.825
    下载: 导出CSV

    表 3  $ Pr = 0.71 $, $ \beta = {\rm{45}}^\circ $, $ Ra = 10^{6} $下的网格检验结果

    Table 3.  Grid test results for $ Pr = 0.71 $, $ \beta = {\rm{45}}^\circ $ and $ Ra = 10^{6} $.

    网格尺寸 $\left| \psi \right|_{\rm {max}}$ 误差/% $\left| {\psi _{\rm {mid}} } \right|$ 误差/% $ Nu_0 $ 误差/%
    $ 31\times31 $ 32.400 3.276 27.974 3.306 9.077 9.345
    $ 61\times61 $ 33.252 0.734 28.707 0.771 8.332 0.381
    $ 91\times91 $ 33.438 0.176 28.874 0.195 8.301 0.001
    $ 121\times121 $ 33.477 0.062 28.911 0.068 8.304 0.039
    $ 241\times241 $ 33.498 28.931 8.301
    下载: 导出CSV

    表 4  $ Pr = 7.01 $, $ \beta = {\rm{0}}^\circ $, $ Ra = 10^{6} $下的网格检验结果

    Table 4.  Grid test results for $ Pr = 7.01 $, $ \beta = {\rm{0}}^\circ $ and $ Ra = 10^{6} $.

    网格尺寸 $\left| \psi \right|_{\rm {max}}$ 误差/% $\left| {\psi _{\rm {mid}} } \right|$ 误差/% $ Nu_0 $ 误差/%
    $ 31\times31 $ 18.625 5.075 17.873 5.021 9.548 3.514
    $ 61\times61 $ 19.634 0.067 18.838 0.110 9.195 0.310
    $ 91\times91 $ 19.609 0.059 18.814 0.020 9.206 0.193
    $ 121\times121 $ 19.612 0.044 18.812 0.029 9.221 0.037
    $ 241\times241 $ 19.621 18.818 9.224
    下载: 导出CSV

    表 5  $ Pr = 7.01 $, $ \beta = 45^\circ $, $ Ra = 10^{6} $下的网格检验结果

    Table 5.  The grid test results for $ Pr = 7.01 $, $ \beta = {\rm{45}}^\circ $ and $ Ra = 10^{6} $.

    网格尺寸 $\left| \psi \right|_{\rm {max}}$ 误差 $\left| {\psi _{\rm {mid}} } \right|$ 误差 $ Nu_0 $ 误差
    $ {\rm{31}} \times {\rm{31}} $ 38.233 6.689% 34.649 6.739% 9.791 7.723%
    $ {\rm{61}} \times {\rm{61}} $ 40.665 0.752% 36.858 0.793% 9.114 0.271%
    $ {\rm{91}} \times {\rm{91}} $ 40.902 0.174% 37.090 0.167% 9.089 0.001%
    $ {\rm{121}} \times {\rm{121}} $ 40.950 0.057% 37.131 0.058% 9.092 0.025%
    $ {\rm{241}} \times {\rm{241}} $ 40.973 37.152 9.089
    下载: 导出CSV
  • [1]

    Batchelor G K 1954 Q. Appl. Math. 12 209Google Scholar

    [2]

    Eckert E R G, Carlson W O 1961 Int. J. Heat Mass Transfer 2 106Google Scholar

    [3]

    Patterson J C, Armfield S W 1990 J. Fluid Mech. 219 469Google Scholar

    [4]

    Xin S, Quéré P L 1995 J. Fluid Mech. 304 87Google Scholar

    [5]

    Das D, Roy M, Basak T 2017 Int. J. Heat Mass Transfer 106 356Google Scholar

    [6]

    Arnold J N, Catton I, Edwards D K 1976 J. Heat Transfer 98 67Google Scholar

    [7]

    John P, Jorg I 1980 J. Fluid Mech. 100 65Google Scholar

    [8]

    Khezzar L, Siginer D, Vinogradov I 2012 Heat Mass Transfer 48 227Google Scholar

    [9]

    Dider S, Abdelmadjid B, François P 2012 Exp. Therm Fluid Sci. 38 74Google Scholar

    [10]

    Torres J F, Henry D, Komiya A, Maruyama S 2014 J. Fluid Mech. 756 650Google Scholar

    [11]

    Torres J F, Henry D, Komiya A, Maruyama S 2015 Phys. Rev. E 92 023031Google Scholar

    [12]

    Miroshnichenko I V, Sheremet M A 2018 Renewable Sustainable Energy Rev. 82 40Google Scholar

    [13]

    徐丰, 崔会敏 2014 力学进展 44 201403

    Xu F, Cui H M 2014 Adv. Mech. 44 201403

    [14]

    Hamady F J, Lloyd J R, Yang H Q, Yang K T 1989 Int. J. Heat Mass Transfer 32 1697Google Scholar

    [15]

    Kuyper R A, Meer T H V D, Hoogendoorn C J 1994 Chem. Eng. Sci. 49 851Google Scholar

    [16]

    Rasoul J, Prinos P 1997 Int. J. Numer. Methods Heat Fluid Flow 7 438Google Scholar

    [17]

    Janssen R J A, Armfield S 1996 Int. J. Heat Fluid Flow 17 547Google Scholar

    [18]

    Varol Y, Oztop H F 2008 Build. Environ. 43 1535Google Scholar

    [19]

    Corcione M 2003 Int. J. Therm. Sci. 42 199Google Scholar

    [20]

    Wang H, Hamed M S 2006 Int. J. Therm. Sci. 45 782Google Scholar

    [21]

    Armfield S W, Janssen R 1996 Int. J. Heat Fluid Flow 17 539Google Scholar

    [22]

    Zhao B X, Tian Z F 2016 Int. J. Heat Mass Transfer 98 313Google Scholar

    [23]

    Sheremet M A, Pop I, Mahian O 2018 Int. J. Heat Mass Transfer 116 751Google Scholar

    [24]

    Boudjeniba B, Laouer A, Laouar S, Mezaache E H 2019 Int. J. Heat Technol. 37 413Google Scholar

    [25]

    Wang Q, Xia S N, Wang B F, Sun D J, Zhou Q, Wan Z H 2018 J. Fluid Mech. 849 355Google Scholar

    [26]

    Wang Q, Wan Z H, Yan R, Sun D J 2018 Phys. Rev. Fluids 3 113503Google Scholar

    [27]

    Wang Q, Chong K L, Stevens R J A M, Verzicco R, Lohse D 2020 J. Fluid Mech. 905 A21Google Scholar

    [28]

    Wang Q, Wan Z H, Yan R, Sun D J 2019 Phys. Fluids 31 025102Google Scholar

    [29]

    Wang Q, Verzicco R, Lohse D, Shishkina O 2020 Phys. Rev. Lett. 125 074501Google Scholar

    [30]

    Sugiyama K, Ni R, Stevens R J A M, Chan T S, Zhou S Q, Xi H D, Sun C, Grossmann S, Xia K Q, Lohse D 2010 Phys. Rev. Lett. 105 034503Google Scholar

    [31]

    Tian Z F, Liang X, Yu P X 2011 Int. J. Numer. Methods Eng. 88 511Google Scholar

    [32]

    Davis G D V 1983 Int. J. Numer. Methods Fluids 3 249Google Scholar

    [33]

    Kalita J C, Dalal D C, Dass A K 2001 Phys. Rev. E 64 066703Google Scholar

    [34]

    Tian Z F, Ge Y B 2003 Int. J. Numer. Methods Fluids 41 495Google Scholar

    [35]

    Yu P X, Tian Z F 2012 Phys. Rev. E 85 036703Google Scholar

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出版历程
  • 收稿日期:  2020-09-10
  • 修回日期:  2021-02-07
  • 上网日期:  2021-05-20
  • 刊出日期:  2021-06-05

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