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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°).
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Keywords:
- thermal convection /
- direct numerical simulation /
- bifurcation /
- inclination angle /
- high-accuracy
[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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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 — 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 — 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 — 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 — 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 — -
[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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