离散统一气体动理学格式及其多尺度导热应用

张创; 郭照立

doi:10.7498/aps.74.20250694

摘要
基于Boltzmann输运方程的数值模拟已成为研究多尺度粒子输运问题的一个有效方法，但是该方程的非线性、多尺度、高维度等特征对数值方法的稳定性、相容性、计算效率/精度、渐近保持性质提出了巨大挑战。近些年发展了诸多适用于任意努森数的多尺度动理学方法，离散统一气体动理学格式便是其中之一。不同于传统直接数值插值格式，离散统一气体动理学格式通过动理学方程在时间和位置空间上的特征解重构网格界面处的分布函数，从而在一个数值时间步长尺度上耦合、累积和计算粒子输运和碰撞效应。基于将物理方程演化信息融入到数值方法构造过程中的思想，该方法的网格尺寸和时间步长不再受限于粒子平均自由程和弛豫时间，能够自适应地高效模拟从弹道到扩散极限的多尺度粒子输运问题。该方法基于有限体积法框架，已经成功应用于微纳尺度流动传热、高超声速飞行器、固体材料导热导电、辐射、等离子体和湍流等领域。本文主要针对该方法在多尺度热传导领域的发展进行综述及展望。

关键词:
多尺度粒子输运 /

Boltzmann输运方程 /

介观数值方法 /

离散统一气体动理学格式 /

热传导
Abstract
Multiscale particle transport problems are widely present in the fields of precision manufacturing, nanomaterials, energy and power, national defense and military. Such problems involve large-scale length and time scales, which pose great challenges to physical modeling and numerical simulations. To study multiscale particle transport problems, cross-scale numerical simulation based on the Boltzmann transport equation has become an effective method, but the nonlinear, multi-scale, and high-dimensional characteristics of the equation pose great challenges to the stability, compatibility, computational effciency/accuracy, and asymptotic preservation of numerical methods. In recent years, many multiscale kinetic methods suitable for arbitrary Knudsen numbers have been developed, and the discrete unified gas kinetic scheme is one of them. Different from the traditional direct numerical interpolation scheme, the discrete unified gas kinetic scheme reconstructs the distribution function at the cell interface through the characteristic solution of the kinetic equation in both time and position space, thereby coupling, accumulating, and calculating particle transport and collision effects on a numerical time step scale. Based on the idea of incorporating the evolution inschemeion of physical equations into the construction process of numerical methods, the cell size and time step of this method are no longer limited by the mean free path and relaxation time of particles, and can adaptively and effciently simulate multiscale particle transport problems from the ballistic to diffusive limit. A large number of numerical results show that the present scheme has good numerical stability and low numerical dissipation, not limited to Knudsen number and Mach number. Based on the framework of finite volume method, this method has been successfully applied to micro/nano scale fluid flow and heat transfer, hypersonic aircraft, solid material thermal conduction, radiation, plasma and turbulence. This paper mainly reviews and prospects the development of this method in the field of multi-scale heat conduction in solid materials, including the application in phonon transport, electron-phonon coupling, phonon hydrodynamic heat conduction and thermal management of electronic equipment.

Keywords:
Multiscale particle transport /

Boltzmann transport equation /

Mesoscopic numerical methods /

Discrete unified gas kinetic scheme /

Heat conduction
施引文献

[1]	Stettler M A, Cea S M, Hasan S, Jiang L, Keys P H, Landon C D, Marepalli P, Pantuso D, Weber C E 2021 IEEE T. Electron Dev. 685350
[2]	陈锦峰, 朱林繁2024物理学报73095201
[3]	中国科学院2022中国学科发展战略: 电子设备热管理(北京: 科学出版社)
[4]	Bird G A 1994 Molecular Gas Dynamics And The Direct Simulation Of Gas Flows (Oxford University Press)
[5]	段文晖, 张刚2017纳米材料热传导. 21世纪理论物理及其交叉学科前沿丛书(科学出版社)
[6]	Chen G 2021 Nat. Rev. Phys. 3555
[7]	Chen J, Xu X, Zhou J, Li B 2022 Rev. Mod. Phys. 94025002
[8]	罗天麟, 丁亚飞, 韦宝杰, 杜建迎, 沈翔瀛, 朱桂妹, 李保文2023物理学报72234401
[9]	曹炳阳2023纳米结构的非傅里叶导热(北京: 科学出版社)
[10]	Tang Z L, Shen Y, Cao B Y 2025 IEEE T. Electron Dev. 721907
[11]	吴志鹏, 张创, 胡世谦, 马登科, 杨诺2023物理学报72184401
[12]	赵瑾, 孙向春, 张俊, 唐志共, 文东升2022航空学报4389
[13]	沈青2003稀薄气体动力学(国防工业出版社)
[14]	樊菁2013力学进展43185
[15]	陈伟芳, 赵文文2017稀薄气体动力学矩方法及数值模拟(上海: 科学出版社)
[16]	靳旭红, 黄飞, 张俊, 王学德, 程晓丽, 沈清2024航空学报456
[17]	Jin S 1999 SIAM J Sci. Comput. 21441
[18]	Xu K 2015 Direct Modeling for Computational Fluid Dynamics: Construction and Application of Unified Gas-Kinetic Schemes (WORLD SCIENTIFIC)
[19]	Guo Z, Li J, Xu K 2023 Phys. Rev. E 107025301
[20]	Adams M L, Larsen E W 2002 Prog. Nucl. Energ. 403
[21]	Guo W, Bai B, Sawin H H 2009 J. Vac. Sci. Technol. A 27388
[22]	Chen G 2005 Nanoscale energy transport and conversion: A parallel treatment of electrons, molecules, phonons, and photons (Oxford University Press)
[23]	Plimpton S 1995 J. Comput. Phys. 1171
[24]	Giannozzi P, Baroni S, Bonini N, Calandra M, Car R, Cavazzoni C, Ceresoli D, Chiarotti G L, Cococcioni M, Dabo I, Dal Corso A, de Gironcoli S, Fabris S, Fratesi G, Gebauer R, Gerstmann U, Gougoussis C, Kokalj A, Lazzeri M, Martin-Samos L, Marzari N, Mauri F, Mazzarello R, Paolini S, Pasquarello A, Paulatto L, Sbraccia C, Scandolo S, Sclauzero G, Seitsonen A P, Smogunov A, Umari P, Wentzcovitch R M 2009 J. Phys-condens. Mat. 21395502
[25]	Broido D A, Malorny M, Birner G, Mingo N, Stewart D A 2007 Appl. Phys. Lett. 91231922
[26]	Kim H, Son D, Myeong I, Kang M, Jeon J, Shin H 2018 IEEE T. Electron Dev. 654520
[27]	Zhang C, Lou Q, Liang H 2025 Int. J. Heat Mass Transfer 236126374
[28]	Joseph D D, Preziosi L 1989 Rev. Mod. Phys. 6141
[29]	Kovács R 2024 Phys. Rep. 10481
[30]	Succi S, Karlin I V, Chen H 2002 Rev. Mod. Phys. 741203
[31]	Kaviany M 2008 Heat transfer physics (Cambridge University Press)
[32]	Kubo R T M, N H 1991 Statistical Physics II Nonequilibrium Statistical Mechanics. Springer Series in Solid State Sciences (Springer, Berlin, Heidelberg)
[33]	Sydney Chapman C C T G Cowling 1995 The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases. Cambridge Mathematical Library, 3rd edn. (Cambridge University Press)
[34]	Cheremisin F 1985 USSR Comp. Math. Math. Phys. 25156
[35]	吴雷, 张勇豪, 李志辉2017中国科学: 物理学力学天文学4728
[36]	Li W, Carrete J, Katcho] N A, Mingo N 2014 Comput. Phys. Commun. 1851747
[37]	Guyer R A, Krumhansl J A 1966 Phys. Rev. 148766
[38]	Hardy R J, Albers D L 1974 Phys. Rev. B 103546
[39]	Murthy J Y, Narumanchi S V J, Pascual-Gutierrez J A, Wang T, Ni C, Mathur S R 2005 Int. J. Multiscale Computat. Eng. 35
[40]	许爱国, 张玉东2022复杂介质动理学(北京: 科学出版社)
[41]	Callaway J 1959 Phys. Rev. 1131046
[42]	曾嘉楠, 李琪, 吴雷2022空气动力学学报401
[43]	Bhatnagar P L, Gross E P, Krook M 1954 Phys. Rev. 94511
[44]	Dimarco G, Pareschi L 2014 Acta Numer. 23369-520
[45]	Mazumder S 2022 Annu. Rev. Heat Transfer 2471
[46]	Struchtrup H 2006 Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory. Interaction of Mechanics and Mathematics (Springer Berlin Heidelberg)
[47]	Guo Y, Wang M 2015 Phys. Rep. 5951
[48]	华钰超, 曹炳阳, 过增元2015科学通报602344
[49]	Sendra L, Beardo A, Torres P, Bafaluy J, Alvarez F X, Camacho J 2021 Phys. Rev. B 103 L140301
[50]	Burnett D 1936 P. Lond. Math. Soc. s2-40382
[51]	Cattaneo C 1948 Atti Sem. Mat. Fis. Univ. Modena 383
[52]	Grad H 1949 Commun. Pur. Appl. Math. 2331
[53]	Guyer R A, Krumhansl J A 1966 Phys. Rev. 148778
[54]	Struchtrup H, Torrilhon M 2003 Phys. Fluids 152668
[55]	Wu L, Gu X J 2020 Adva. Aerodyn. 22
[56]	E W, Han J, Zhang L 2021 Phys. Today 7436
[57]	Han J, Ma C, Ma Z, E W 2019 Proc. Natl. Acad. Sci. 11621983
[58]	Zhao J, Zhao W, Ma Z, Yong W A, Dong B 2022 Int. J. Heat Mass Transfer 185122396
[59]	Chen L, Zhang C, Zhao J 2024 Phys. Rev. E 110025303
[60]	Bird G A 1963 Phys. Fluids 61518
[61]	Klitsner T, VanCleve J E, Fischer H E, Pohl R O 1988 Phys. Rev. B 387576
[62]	Peterson R B 1994 J. of Heat Transfer 116815
[63]	Mazumder S, Majumdar A 2001 J. Heat Transfer 123749
[64]	Peraud J P M, Landon C D, Hadjiconstantinou N G 2014 Annu. Rev. Heat Transfer 17205
[65]	Shen Y, Yang H A, Cao B Y 2023 Int. J. Heat Mass Transfer 211124284
[66]	Wollaber A B 2016 J. Comput. Theor. Trans. 451
[67]	Pareschi L, and G R 2000 Transp. Theory Stat. Phys. 29415
[68]	Liu C, Zhu Y, Xu K 2020 J. Comput. Phys. 401108977
[69]	Fei F, Zhang J, Li J, Liu Z 2020 J. Comput. Phys. 400108972
[70]	Baker L L, Hadjiconstantinou N G 2005 Phys. Fluids 17051703
[71]	Péraud J P M, Hadjiconstantinou N G 2011 Phys. Rev. B 84205331
[72]	Carrete J, Vermeersch B, Katre A, [van Roekeghem] A, Wang T, Madsen G K, Mingo N 2017 Comput. Phys. Commun. 220351
[73]	Pathak A, Pawnday A, Roy A P, Aref A J, Dargush G F, Bansal D 2021 Comput. Phys. Commun. 265108003
[74]	Yang R, Chen G, Laroche M, Taur Y 2005 J. Heat Transfer 127298
[75]	舒昌, 杨鲤铭, 王岩, 吴杰2022南京航空航天大学学报54801
[76]	Guo Z, Xu K 2021 Adva. Aerodyn. 36
[77]	Romano G 2021 arXiv:2106.02764
[78]	Chen S, Chen H, Martnez D, Matthaeus W 1991 Phys. Rev. Lett. 673776
[79]	Qian Y H, d’Humières D, Lallemand P 1992 EPL 17479
[80]	Chen H, Chen S, Matthaeus W H 1992 Phys. Rev. A 45 R5339
[81]	Mittal A, Mazumder S 2011 J. Comput. Phys. 2306977
[82]	Loy J M, Murthy J Y, Singh D 2012 J. Heat Transfer 135011008
[83]	Hao Q, Zhao H, Xiao Y 2017 J. Appl. Phys. 121204501
[84]	Hua Y C, Shen Y, Tang Z L, Tang D S, Ran X, Cao B Y 2023 Adv. Heat Transf. 56355
[85]	Xu K, Huang J C 2010 J. Comput. Phys. 2297747
[86]	刘畅, 徐昆2020空气动力学学报38197
[87]	Kopp H J 1963 Nucl. Sci. Eng. 1765
[88]	Alcouffe R E 1977 Nucl. Sci. Eng. 64344
[89]	Zhang C, Guo Z, Chen S 2017 Phys. Rev. E 96063311
[90]	Frisch U, Hasslacher B, Pomeau Y 1986 Phys. Rev. Lett. 561505
[91]	Guo Z, Shu C 2013 Lattice Boltzmann method and its applications in engineering, vol. 3(World Scientific)
[92]	Succi S 2001 The Lattice Boltzmann Equation for Fluid Dynamics and Beyond (Oxford University Press)
[93]	Ali S A, Kollu G, Mazumder S, Sadayappan P, Mittal A 2014 Int. J. Therm. Sci 86341
[94]	Lathrop K D 1968 Nucl. Sci. Eng. 32357
[95]	Chai J C, Lee H S, Patankar S V 1993 Numer. Heat Transf. Part B 24373
[96]	Xu K, Huang J C 2010 J. Comput. Phys. 2297747
[97]	Guo Z, Xu K, Wang R 2013 Phys. Rev. E 88033305
[98]	Guo Z, Wang R, Xu K 2015 Phys. Rev. E 91033313
[99]	Guo Z, Xu K 2016 Int. J. Heat Mass Transfer 102944
[100]	Lian M, Zhang C, Guo Z, Lü J T 2024 Phys. Rev. E 109065310
[101]	Song X, Zhang C, Zhou X, Guo Z 2020 Adva. Aerodyn. 23
[102]	Liu H, Quan L, Chen Q, Zhou S, Cao Y 2020 Phys. Rev. E 101043307
[103]	Yuan R, Zhong C 2020 Comput. Phys. Commun. 247106972
[104]	Yuan R, Liu S, Zhong C 2021 Commun. Nonlinear Sci. 92105470
[105]	Zhang C, Wu L 2022 Phys. Rev. E 106014111
[106]	Zhang C, Xin Z, Lou Q, Liang H 2025 Appl. Therm. Eng. 271126293
[107]	Liu P, Wang P, Jv L, Guo Z 2021 Commun. Comput. Phys. 29265
[108]	Qi Y ff, Chen T ff, Wang L P ff, Guo Z ff, Chen S ff 2022 Phys. Fluids 34116101
[109]	Liu W, Feng Y, Li R, Bai C, Niu B 2024 Comput. Phys. Commun. 299109157
[110]	Harter J R, Hosseini S A, Palmer T S, Greaney P A 2019 Int. J. Heat Mass Transfer 144118595
[111]	Fiveland V A, Jessee J P 1996 J. Thermophys. Heat Transfer. 10445
[112]	Loy S R M James M, Murthy J Y 2015 J. Heat Transfer 137012402
[113]	Randrianalisoa J, Baillis D 2008 J. Heat Transfer 130072404
[114]	Ran X, Wang M 2022 J. Heat Transf. 144082502
[115]	Mieussens L 2000 Math. Models Methods Appl. Sci. 101121
[116]	Chacón L, Chen G, Knoll D A, Newman C, Park H, Taitano W, Willert J A, Womeldorff G 2017 J. Comput. Phys. 33021
[117]	Zhu Y, Zhong C, Xu K 2016 J. Comput. Phys. 31516
[118]	Su W, Wang P, Liu H, Wu L 2019 J. Comput. Phys. 378573
[119]	Zhang R ff, Liu S ff, Chen J ff, Zhuo C ff, Zhong C ff 2024 Phys. Fluids 36016114
[120]	Zhang C, Guo Z, Chen S 2019 Int. J. Heat Mass Transfer 1301366
[121]	Zhang C, Chen S, Guo Z, Wu L 2021 Int. J. Heat Mass Transfer 174121308
[122]	Zhang C, Huberman S, Song X, Zhao J, Chen S, Wu L 2023 Int. J. Heat Mass Transfer 217124715
[123]	Hu Y, Shen Y, Bao H 2024 Fundam. Res. 4907
[124]	Hu Y, Jia R, Xu J, Sheng Y, Wen M, Lin J, Shen Y, Bao H 2024 J. Phys-condens. Mat. 36025901
[125]	Succi S 2015 EPL 10950001
[126]	Hao Q, Chen G, Jeng M S 2009 J. Appl. Phys. 106114321
[127]	Jeong J, Li X, Lee S, Shi L, Wang Y 2021 Phys. Rev. Lett. 127085901
[128]	Huberman S C 2018 Thermal transport at the nanoscale: from fourier diffusion to phonon hydrodynamics. Ph.D. Dissertation, Massachusetts Institute of Technology
[129]	Huberman S, Duncan R A, Chen K, Song B, Chiloyan V, Ding Z, Maznev A A, Chen G, Nelson K A 2019 Science 364375
[130]	Zhang C, Guo Z 2019 Int. J. Heat Mass Transfer 1341127
[131]	Zhu L, Chen S, Guo Z 2017 Comput. Phys. Commun. 213155
[132]	Karzhaubayev K, Wang L P, Zhakebayev D 2024 Comput. Phys. Commun. 301109216
[133]	Zhang Q, Wang Y, Pan D, Chen J, Liu S, Zhuo C, Zhong C 2022 Comput. Phys. Commun. 278108410
[134]	Zhang F, Wang Y, Zhang R, Guo J, Zhao T, Liu S, Zhuo C, Zhong C 2025 Adv. Eng. Softw. 201103840
[135]	Luo X P, Yi H L 2017 Int. J. Heat Mass Transfer 114970
[136]	Huberman S, Zhang C, Haibeh J A 2022 arXiv:2206.02769
[137]	Luo X P, Guo Y Y, Wang M R, Yi H L 2019 Phys. Rev. B 100155401
[138]	Zhang C, Guo R, Lian M, Shiomi J 2024 Appl. Therm. Eng. 249123379
[139]	Lee S, Broido D, Esfarjani K, Chen G 2015 Nat. Commun. 66290
[140]	Cepellotti A, Fugallo G, Paulatto L, Lazzeri M, Mauri F, Marzari N 2015 Nat. Commun. 66400
[141]	Machida Y, Martelli V, Jaoui A, Fauqué B, Behnia K 2024 Low Temp. Phys. 50574
[142]	Zhang C, Chen S, Guo Z 2021 Int. J. Heat Mass Transfer 176121282
[143]	Zhang C, Wu L 2025 Appl. Phys. Lett. 126032201
[144]	Tur-Prats J, Gutiérrez-Pérez M, Bafaluy J, Camacho J, Alvarez F X, Beardo A 2024 Int. J. Heat Mass Transfer 226125464
[145]	Shang M Y, Zhang C, Guo Z, Lü J T 2020 Sci. Rep. 108272
[146]	Guo Y, Zhang Z, Nomura M, Volz S, Wang M 2021 Int. J. Heat Mass Transfer 169120981
[147]	Raya-Moreno M, Carrete J, Cartoixà X 2022 Phys. Rev. B 106014308
[148]	Liu C, Wang Y, Yang Y, Duan Z 2016 Sci. China Phys. Mech. Astron. 59684711
[149]	Sýkora M, Pavelka M, Restuccia L, Jou D 2023 Phys. Scr. 98105234
[150]	Zhang C, Guo Z 2021 Int. J. Heat Mass Transfer 181121847
[151]	Qian X, Zhang C, Liu T H, Yang R 2025 Phys. Rev. B 111035406
[152]	Shan B, Wang P, Zhang Y, Guo Z 2020 Phys. Rev. E 101043303
[153]	Simoncelli M, Marzari N, Mauri F 2022 Phys. Rev. X 12041011
[154]	Zhang Y, Zhang C, Xinliang S, Zhaoli G 2025 Commun. Comput. Phys. 37383
[155]	Karniadakis G E, Kevrekidis I G, Lu L, Perdikaris P, Wang S, Yang L 2021 Nat. Rev. Phys. 3422
[156]	Weinan E 2021 Not. Am. Math. Soc. 68565
[157]	Liu S, Xu K, Zhong C Acta Mech. Sin. 38122123
[158]	Liu H, Yang X, Zhang C, Ji X, Xu K 2025 arXiv:2505.09297
[159]	Raissi M, Perdikaris P, Karniadakis G 2019 J. Comput. Phys. 378686
[160]	Lin Q, Zhang C, Meng X, Guo Z 2024 arXiv:2505.09297

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离散统一气体动理学格式及其多尺度导热应用

Discrete unified gas kinetic scheme and its application in multi-scale heat conduction

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离散统一气体动理学格式及其多尺度导热应用

Discrete unified gas kinetic scheme and its application in multi-scale heat conduction

摘要

Abstract

施引文献

计量

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作者中心

审稿中心

关于期刊

关于我们