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

ZHANG Chuang; GUO Zhaoli

doi:10.7498/aps.74.20250694

Abstract

Multiscale particle transport problems are universally existent in the fields of precision manufacturing, nanomaterials, energy and power, national defense and military. Such issues involve large-scale length and time scales, posing great challenges to physical modeling and numerical simulation. In order to study multiscale particle transport problems, cross-scale numerical simulation based on the Boltzmann transport equation has become an effective method. However the nonlinear, multi-scale, and high-dimensional characteristics of the equation pose significant challenges to the stability, compatibility, computational efficiency/accuracy, and asymptotic preserving property of numerical methods. In recent years, many multiscale kinetic methods applicable to any Knudsen numbers have been developed, and one of them is the discrete unified gas kinetic scheme. Unlike 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 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, therefore, the multiscale particle transport problems from the ballistic to diffusive limit can be adaptively and efficiently simulated. A large number of numerical results show that the present scheme has good numerical stability and low numerical dissipation, and it is not limited by the Knudsen number or Mach number. Based on the framework of the finite volume method, this method has been successfully applied to micro/nano scale fluid flow and heat transfer, hypersonic aircraft flows, solid-material thermal conduction, radiation, plasma, and turbulence. This paper mainly reviews the method and discusses its future prospects in the field of multi-scale heat conduction in solid materials, including applications in phonon transport, electron-phonon coupling, phonon hydrodynamic heat conduction, and thermal management of electronic equipment.

Keywords:

Authors and contacts

ZHANG Chuang¹,
GUO Zhaoli²

1.
Institute of Energy, School of Sciences, Hangzhou Dianzi University, Hangzhou 310018, China
2.
Institute of Interdisciplinary Research for Mathematics and Applied Science, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Corresponding author: GUO Zhaoli, zlguo@hust.edu.cn

Funds: Project supported by the Interdisciplinary Research Support Program of Huazhong University of Science and Technology, China (Grant No. 2023JCYJ002).

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References

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Cited By

图 1 DUGKS在多尺度粒子输运领域的发展^[76], 涵盖气体分子^[96,97]、声子^[98]、电子^[99]、光子^[100]和等离子体^[101]等　(a) 稀薄高超声速流动^[102,103], 其中位置和动量空间均采用非结构网格离散; (b) 电子设备导热^[104,105]: 瞬态温度与热流分布; (c) 等离子体输运: 非平衡分布函数在位置和动量空间的分布; (d) 辐射输运^[100]; (e) 不可压低速渗流^[106]: 温度分布与速度流线; (f) 可压缩衰减湍流^[107]: 同一时刻, 涡量的模/涡量的均方根为 2的等值面

Figure 1. Development of DUGKS in the field of multiscale particle transport^[76] covers gas molecules^[96,97], phonons^[98], electrons^[99], photons^[100] and plasma^[101], etc: (a) Rarefied hypersonic flow^[102,103], where both position and momentum space are discretized using unstructured meshes; (b) thermal conduction in electronic devices^[104,105]: transient temperature and heat flux distribution; (c) plasma transport: distribution of nonequilibrium distribution functions in position and momentum space; (d) radiative transport^[100]; (e) incompressible low-speed seepage flow^[106]: temperature distribution and velocity streamlines; (f) compressible decaying turbulence^[107]: isosurfaces of the vorticity modu-lus/vorticity root mean square equals 2 at the same time.

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图 2 不同介观方法因其数值建模过程的差异, 在不同的适用范围或时空尺度上展现其各自的优越性^{[2,18,20,30,66,76,115,121,124]}

Figure 2. Different mesoscopic methods demonstrate their respective advantages in different scopes of application or spatiotemporal scales due to differences in their numerical modeling processes^{[2,18,20,30,66,76,115,121,124]}.

DownLoad: Full-Size Img PowerPoint

图 3 常用的声子BTE模型或近似处理^{[29,41,47,123,126]}, 非灰与灰体模型的区别在于是否考虑声子色散关系或频率依赖特性, 动量空间: 各向异性或各向同性; 平衡态分布: 采用非线性的Bose-Einstein平衡态分布或引入比热作线性化近似处理

Figure 3. Commonly used phonon BTE models or approximations^{[29,41,47,123,126]}, the difference between non-gray and gray models lies in whether the phonon dispersion relation or frequency dependence is considered. Momentum space: anisotropic or isotropic; equilibrium distribution: using nonlinear Bose-Einstein equilibrium distribution or introducing specific heat for linear approximation.

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图 4 常温下硅材料中不同平均自由程的声子模式占比, 其中声子物性参数由第一性原理计算得到^[123]

Figure 4. The proportion of phonon modes with different mean free paths in silicon materials at room temperature, where the phonon physical properties are obtained by first-principles calculations^[123].

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图 5 有限体积框架下DUGKS沿着特征线方向重构网格界面处的分布函数^{[21,78,98,100,131]}

Figure 5. DUGKS reconstructs the distribution function along the characteristic line direction at the cell interface in the finite volume framework^{[21,78,98,100,131]}.

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图 6 多尺度粒子输运模拟流程^{[102,118,130–133]}, 即前处理-DUGKS求解器-后处理. 前处理: 动量空间和位置空间均可采用结构/非结构/自适应网格等, 输入参数—弛豫时间和色散关系等, 可以通过第一性原理计算、实验或经验公式等方式获取; DUGKS求解器: 分布函数在时间和位置空间的演化过程; 后处理: 通过分布函数求矩得到宏观量并计算相关等效参数, 不局限于经典的宏观本构关系

Figure 6. Multiscale particle transport simulation process^{[102,118,130–133]}: Pre-processing-DUGKS solver-post-processing. Pre-processing: Both momentum space and position space can use struc-tured/unstructured/adaptive meshes, etc. Input parameters such as relaxation time and dispersion rela-tions can be obtained through first-principles calculations, experiments, or empirical formulas; DUGKS solver: The evolution of the distribution function in time and position space; Post-processing: Macro-scopic quantities are obtained by taking the moments of the distribution function and calculating related effective parameters, not limited to classical macroscopic constitutive relations.

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图 7 (a), (c) 声子流体动力学导热: 热涡^[141,144]与热波涟漪^{[125,149,150]}; (b), (d) Fourier导热

Figure 7. (a), (c) Phonon hydrodynamic heat conduction: thermal vortices^[141,144] and thermal wave ripples^{[125,149,150]}; (b), (d) Fourier heat conduction.

DownLoad: Full-Size Img PowerPoint

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