基于总能形式的耦合的双分布函数热晶格玻尔兹曼数值方法

刘飞飞; 魏守水; 魏长智; 任晓飞

doi:10.7498/aps.64.154401

摘要

双分布函数热晶格玻尔兹曼数值方法在微尺度热流动系统中得到广泛的应用. 本文基于晶格玻尔兹曼平衡分布函数低阶Hermite展开式, 创新性地提出了包含黏性热耗散和压缩功的耦合的双分布函数热晶格玻尔兹曼数值方法, 将能量场内温度的变化以动量源的形式引入晶格波尔兹曼动量演化方程, 实现了能量场与动量场之间的耦合. 研究了考虑黏性热耗散和压缩功的和不考虑的两种热自然对流模型, 重点分析了不同瑞利数和普朗特数下流场内的流动情况以及温度、速度和平均努赛尔数的变化趋势. 本文实验结果与文献结果一致, 验证了本文数值方法的可行性和准确性. 研究结果表明: 随着瑞利数和普朗特数的增大, 方腔内对流传热作用逐渐增强, 边界处形成明显的边界层; 考虑黏性热耗散和压缩功的模型对流作用相对增强, 黏性热耗散和压缩功对自然对流的影响在微尺度流动过程中不能忽略.

关键词:

Abstract

Micro-scale flow is a very important and prominent problem in the design and application of micro-electromechanical systems. With the decrease of the scale, effects, such as viscous dissipation, compression work and boundary slip etc., which are ignored in a large-scale flow, play important roles in a microfluidic system. #br#With its certain advantages such as high numerical efficiency, easy implement, parallel algorithms etc., the lattice Boltzmann method is a powerful numerical technique for simulating fluid flows and modeling the physics in fluids. The double-distribution-function lattice Boltzmann method has been widely used in a micro-scale thermal flow system, since it utilizes two different distribution functions to take account of the viscous dissipation and compression work. However, most of the existing double-distribution-function lattice Boltzmann methods are “decoupling” models, and decoupling will cause the models to be limited to Boussinesq flows in which temperature variation is small. In order to overcome the above problem, based on the low-order Hermite expansion of the continuous equilibrium distribution function, we propose a coupling double-distribution-function thermal lattice Boltzmann method. This method introduces temperature changes into the lattice Boltzmann momentum equation in the form of the momentum source, which can affect the distribution of flow velocity and density, so as to realize the coupling between the momentum field and the energy field. In the process of fluid flow, the temperature change of the energy field includes two parts: one is for different times at the same lattice which can cause the change of the fluid characteristic parameters, such as the viscosity coefficient and the thermal diffusivity; the other is for the same time at different lattices which mainly affects the distribution of the velocity. In the collision and the migration processes, temperature change is introduced into the fluid flow to achieve the effect of temperature changes on the flow field and the coupling between the energy field and the momentum field. This method can break up the limitation of the Boussinesq flows and expand the application scope of the lattice Boltzmann method. #br#Two natural convection models (one takes into consideration the viscous dissipation and compression work, and the other does not) are studied in this paper to verify the effectiveness and accuracy of the coupling double-distribution-function thermal lattice Boltzmann method. Flow field and the changing trend in temperature, velocity and the averaged Nusselt number are analyzed emphatically at different Rayleigh number and Prandtl number. Results of this paper are excellently consistent with those in papers published, confirming the validity and accuracy of this method. Results also show that the convective heat transfer gradually enhances with increasing Rayleigh number and Prandtl number in the cavity, and the boundary layer is obviously formed in the regions very close to the walls; the heat transfer is greatly enhanced if viscous dissipation and compression work are considered; and these effects should not be neglected in the micro-scale flow system.

Keywords:

作者及机构信息

1.
山东大学, 控制科学与工程学院, 济南 250061;

2.
济南大学, 信息科学与工程学院, 济南 250002

基金项目: 国家自然科学基金(批准号: 51075243, 11002083)和山东省自然科学基金(批准号: ZR2014EEM003, ZR2014AM031)资助的课题.

Authors and contacts

1.
School of Control Science Engineering, Shandong University, Jinan 250061, China;

2.
school of Information Science and Engineering, University of Jinan, Jinan 250002, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 51075243, 11002083), and the Natural Science Foundation of Shandong Province(Grant Nos. ZR2014EEM003, ZR2014AM031).

参考文献

[1]	Jiang H Y, Ren Y K, Ao H R, Ramos A 2008 Chin. Phys. B 17 4541
[2]	Esfahanian V, Dehdashti E, Dehrouye A M 2014 Chin. Phys. B 23 084702
[3]	Alexander F J, Chen S, Sterling J D 1993 Phys. Rev. E 47 R2249
[4]	Gan Y, Xu A, Zhang G, Li Y 2011 Phys. Rev. E. 83 056704
[5]	Chikatamarla S S, Karlin I V 2008 Comput. Phys. Commun. 179 140
[6]	Gan Y, Xu A, Zhang G, Yu X, Li Y 2008 Physica A 387 1721
[7]	Pierre L, Luo L S 2003 Phys. Rev. E 68 036706
[8]	Pierre L, Luo L S 2003 Int. J Mod. Phys. B 17 41
[9]	Liu F F, Wei S S, Wei C Z, Ren X F 2014 Acta Phys. Sin. 63 1947041 (in Chinese) [刘飞飞, 魏守水, 魏长智, 任晓飞 2014 63 194704]
[10]	Li Q, He Y L, Tang G H, Tao W Q 2009 Phys. Rev. E 80 037702
[11]	Chen S, Tölke J, Krafczyk M 2009 Phys. Rev. E 79 016704
[12]	He X, Chen S, Doolen G D 1998 J. Comput. Phys. 146 282
[13]	Dixit H N, Babu V 2006 Int. J. Heat Mass Trans. 49 727
[14]	Wang C H, Yang R 2006 Appl. Math. Comput. 173 1246
[15]	Shi Y, Zhao T S, Guo Z L 2006 Comput. Fluids 35 1
[16]	Peng Y, Shu C, Chew Y T 2003 Phys. Rev. E 68 143
[17]	Li Q, He Y, Wang Y, Tang G 2008 Int. J. Mod. Phys. C 19 125
[18]	Guo Z L, Zheng C G, Shi B C, Zhao T S 2007 Phys. Rev. E 75 3654
[19]	Mo J Q, Cheng Y 2009 Acta Phys. Sin. 58 4379 (in Chinese) [莫嘉琪, 程燕 2009 58 4379]
[20]	Shan X, Yuan X F, Chen H 2006 J. Fluid Mech. 550 413
[21]	Hung L H, Yang J Y 2011 Ima J. Appl. Math . 76 774
[22]	Li Q, Luo K H, He Y L, Gao Y J, Tao W Q 2012 Phys. Rev. E 85 016710
[23]	Basu R, Layek G C 2013 Chin. Phys. B 22 054702
[24]	Sun D K, Zhu M F, Yang C R, Pan S Y, Dai T 2009 Acta Phys. Sin. 58 S285 (in Chinese) [孙东科, 朱鸣芳, 杨朝蓉, 潘诗琰, 戴挺 2009 58 S285]
[25]	Abdel R G, Khader M M, Megahed A M 2013 Chin. Phys. B 22 030202
[26]	Liu F F, Wei S S, Wang S W, Wei C Z, Ren X F 2014 J. Nanoengin. Nanosys. 228 189
[27]	Tang G H, Tao W Q, He Y L 2005 Phys. Rev. E 72 6435
[28]	Sun L, Sun Y F, Ma D J, Sun D J 2007 Acta Phys. Sin. 56 6503
[29]	Costa V A F 2005 Int. J. Heat Mass Tran. 48 2333
[30]	Barakos G, Mitsoulis E, Assimacopoulos D 1994 Int. J. Numer. Meth. Fl. 18 695
[31]	Cheng T S 2011 Int. J. Therm. Sci. 50 197
[32]	Arcidiacono S, Dipiazza I, Ciofalo M 2001 Int. J. Heat Mass Tran. 44 537
[33]	Chatterjee D, Biswas G 2011 Numer. Heat Tr. A-Appl. 59 421
[34]	MacGregor R, Emery A 1969 J. Heat Tran. 91 391
[35]	He Y, Yang W, Tao W 2005 Numer. Heat Tr. A-Appl. 47 917
[36]	Pesso T, Piva S 2009 Int. J. Heat Mass Tran. 52 1036
[37]	Karimipour A, Nezhad A H, D’Orazio A, Shirani E 2013 J Theor. App. Mech-pol 51 447
[38]	Kawamura H, Abe H, Matsuo Y 1999 Int. J. Heat Fluid Fl. 20 196
[39]	Dipiazza I, Ciofalo M 2000 Int. J. Heat. Mass Tran. 43 3027

施引文献

[1]	Jiang H Y, Ren Y K, Ao H R, Ramos A 2008 Chin. Phys. B 17 4541
[2]	Esfahanian V, Dehdashti E, Dehrouye A M 2014 Chin. Phys. B 23 084702
[3]	Alexander F J, Chen S, Sterling J D 1993 Phys. Rev. E 47 R2249
[4]	Gan Y, Xu A, Zhang G, Li Y 2011 Phys. Rev. E. 83 056704
[5]	Chikatamarla S S, Karlin I V 2008 Comput. Phys. Commun. 179 140
[6]	Gan Y, Xu A, Zhang G, Yu X, Li Y 2008 Physica A 387 1721
[7]	Pierre L, Luo L S 2003 Phys. Rev. E 68 036706
[8]	Pierre L, Luo L S 2003 Int. J Mod. Phys. B 17 41
[9]	Liu F F, Wei S S, Wei C Z, Ren X F 2014 Acta Phys. Sin. 63 1947041 (in Chinese) [刘飞飞, 魏守水, 魏长智, 任晓飞 2014 63 194704]
[10]	Li Q, He Y L, Tang G H, Tao W Q 2009 Phys. Rev. E 80 037702
[11]	Chen S, Tölke J, Krafczyk M 2009 Phys. Rev. E 79 016704
[12]	He X, Chen S, Doolen G D 1998 J. Comput. Phys. 146 282
[13]	Dixit H N, Babu V 2006 Int. J. Heat Mass Trans. 49 727
[14]	Wang C H, Yang R 2006 Appl. Math. Comput. 173 1246
[15]	Shi Y, Zhao T S, Guo Z L 2006 Comput. Fluids 35 1
[16]	Peng Y, Shu C, Chew Y T 2003 Phys. Rev. E 68 143
[17]	Li Q, He Y, Wang Y, Tang G 2008 Int. J. Mod. Phys. C 19 125
[18]	Guo Z L, Zheng C G, Shi B C, Zhao T S 2007 Phys. Rev. E 75 3654
[19]	Mo J Q, Cheng Y 2009 Acta Phys. Sin. 58 4379 (in Chinese) [莫嘉琪, 程燕 2009 58 4379]
[20]	Shan X, Yuan X F, Chen H 2006 J. Fluid Mech. 550 413
[21]	Hung L H, Yang J Y 2011 Ima J. Appl. Math . 76 774
[22]	Li Q, Luo K H, He Y L, Gao Y J, Tao W Q 2012 Phys. Rev. E 85 016710
[23]	Basu R, Layek G C 2013 Chin. Phys. B 22 054702
[24]	Sun D K, Zhu M F, Yang C R, Pan S Y, Dai T 2009 Acta Phys. Sin. 58 S285 (in Chinese) [孙东科, 朱鸣芳, 杨朝蓉, 潘诗琰, 戴挺 2009 58 S285]
[25]	Abdel R G, Khader M M, Megahed A M 2013 Chin. Phys. B 22 030202
[26]	Liu F F, Wei S S, Wang S W, Wei C Z, Ren X F 2014 J. Nanoengin. Nanosys. 228 189
[27]	Tang G H, Tao W Q, He Y L 2005 Phys. Rev. E 72 6435
[28]	Sun L, Sun Y F, Ma D J, Sun D J 2007 Acta Phys. Sin. 56 6503
[29]	Costa V A F 2005 Int. J. Heat Mass Tran. 48 2333
[30]	Barakos G, Mitsoulis E, Assimacopoulos D 1994 Int. J. Numer. Meth. Fl. 18 695
[31]	Cheng T S 2011 Int. J. Therm. Sci. 50 197
[32]	Arcidiacono S, Dipiazza I, Ciofalo M 2001 Int. J. Heat Mass Tran. 44 537
[33]	Chatterjee D, Biswas G 2011 Numer. Heat Tr. A-Appl. 59 421
[34]	MacGregor R, Emery A 1969 J. Heat Tran. 91 391
[35]	He Y, Yang W, Tao W 2005 Numer. Heat Tr. A-Appl. 47 917
[36]	Pesso T, Piva S 2009 Int. J. Heat Mass Tran. 52 1036
[37]	Karimipour A, Nezhad A H, D’Orazio A, Shirani E 2013 J Theor. App. Mech-pol 51 447
[38]	Kawamura H, Abe H, Matsuo Y 1999 Int. J. Heat Fluid Fl. 20 196
[39]	Dipiazza I, Ciofalo M 2000 Int. J. Heat. Mass Tran. 43 3027

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