聲子熱傳輸與理想量子氣體動力學之高解析算則

High resolution schemes for phonon heat transfer and ideal quantum gas dynamics

謝澤揚(Tse-Yang Hsieh)

指導教授 ： 楊照彥

國立臺灣大學/工學院/應用力學研究所/博士(2007年)

https://doi.org/10.6342/NTU.2007.01754

摘要

物質是由原子所構成。在宏觀的尺度下，物質可視為連續體，並適用宏觀之主導方程 (Governing equation)，如Fourier定律、Euler方程、Navier-Stokes方程…等。當流場之特徵長度與分子之平均自由徑 (Mean free path) 逐漸接近，尺寸效應已不能再忽略，此時必須由分子的運動行為考量問題。
在非平衡狀態問題，本文探討微尺度半導體材料之聲子 (Phonon) 熱傳輸問題，聲子熱傳輸須以粒子運動的圖像描述。聲子分布函數之主導方程為聲子Boltzmann方程，直接求解非線性積微分Boltzmann方程不易，利用鬆弛時間近似將Boltzmann方程簡化為BGK (Bhatnagar-Gross-Krook) 方程，並使用離散座標法 (Discrete ordinate method)，以一組離散速度點之聯立線性方程組取代空間、速度與時間連續的BGK方程，再利用計算流力發展之高解析算則求解，最後由分布函數取矩 (Take moment) 得溫度和熱通量。對於薄膜厚度等於或小於聲學厚度 (Acoustic thickness) 時，因方向離散造成射線效應 (Ray effect)，以修正離散座標法 (Modified discrete ordinate method) 消除並搭配高解析算則使用。為了解決計算量龐大的問題，使用平行計算 (Parallel computing) 減少計算時間並分散記憶體需求。本文探討單層薄膜和薄膜、線型、粒型超晶格 (Superlattice) 之熱傳導特性。
在準平衡狀態問題，本文探討理想量子氣體動力學問題，由分子之角度出發分析依循量子統計之理想氣體的運動行為，因不考慮黏滯效應，其主導方程為量子Euler方程。本文發展氣體運動算則 (Gas kinetic scheme) 計算理想量子氣體之運動行為，氣體運動算則利用Boltzmann方程取矩可得宏觀方程之性質，由分布函數隨時間的演化計算宏觀量之數值通量。理想量子氣體之速度分布依循Bose-Einstein或Fermi-Dirac分布，在已知氣體分布函數的情況下，對分布函數取矩可得宏觀量及其通量，本文即利用此概念發展適用於理想量子氣體之氣體運動算則，並探討量子效應造成之影響，本文探討理想量子氣體之震波管、楔形體和圓柱震波繞射問題。

關鍵字

聲子 ； 波茲曼方程 ； 理想量子氣體 ； 氣體運動算則

並列摘要

It is universally acknowledged that substance is composed of molecules. In macroscopic scale, substance can be considered as continuum, and the transport phenomena can be described by macroscopic governing equations, e.g. Fourier law, Euler equation, Navier-Stokes equation, etc. The size effect can no longer be neglected as the characteristic length of the flow filed is comparable with the molecule mean free path. We need to consider motions and interactions of the individual molecules.
In non-equilibrium problems, we investigate phonon heat transfer in micro-scale semiconductor materials. The macroscopic governing equations are no longer valid as the length of the material is comparable with the phonon mean free path. The special heat transport phenomena must be described with the behavior of individual phonons. In micro-scale problems, the governing equation for the phonon distribution function is the phonon Boltzmann equation. The nonlinear integral-differential Boltzmann equation is very difficult to solve in general. Under the relaxation time approximation, Boltzmann equation can be simplified to BGK (Bhatnagar-Gross-Krook) equation. With the discrete ordinate method, the original space, velocity and time continuous equation can be transformed to a set of equations which are continuous in physical space and time only and point-wise in velocity space. The set of equations are hyperbolic partial differential equations with source terms, so they can be solved by utilizing the high resolution conservation law scheme. In the transient thin film problem, we apply the modified discrete ordinate method to eliminate ray effects and cooperate with the high resolution scheme to eliminate false scattering. To reduce the computation time and separate the memory requirement, we adopt the parallel computing strategy. The heat conduction properties of thin film, thin film type, wire type and particle type superlattice are investigated in this thesis.
In quasi-equilibrium problems, we analyze the gases that obey quantum statistics from the molecule’s point of view. The governing equation is quantum Euler equation when the viscosity effect is not considered. We develop a kinetic scheme that suitable for describing ideal quantum gas dynamics. Kinetic schemes utilize the property that macroscopic governing equations can be derived from taking moments to Boltzmann equation and calculate the numerical flux from the distribution function. The ideal quantum gases obey Bose-Einstein or Fermi-Dirac statistics and the flux can be derived from the known distribution function. In this sense, we develop the kinetic schemes that suitable for ideal quantum gas and investigate the influence of the quantum effect. In this thesis we discuss shock tube problems and shock diffraction by wedges and shock diffraction by cylinders for the ideal quantum gases.

並列關鍵字

Phonon ； Boltzmann equation ； Ideal quantum gas ； Gas kinetic scheme

參考文獻

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[4] G. Chen, “Thermal Conductivity and Ballistic-Phonon Transport in the Cross-Plane Direction of Superlattices”, Physical Review B, 57, pp.14958-14973, 1998.

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