Gas-Kinetic Schemes for Semi-Classical Boltzmann Equation

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Project title

Code/計畫編號

MOST103-2221-E019-017-MY3

Translated Name/計畫中文名

半古典Boltzmann方程式之氣體動力算則

Project Coordinator/計畫主持人

Juan-Chen Huang

Funding Organization/主管機關

National Science and Technology Council

Co-Investigator(s)/共同執行人

楊照彥

Department/Unit

Department of Merchant Marine

Website

https://www.grb.gov.tw/search/planDetail?id=11276433

Year

2015

Start date/計畫起

01-08-2015

Expected Completion/計畫迄

31-07-2016

Bugetid/研究經費

623千元

ResearchField/研究領域

機械工程

"基於 Maxwell-Boltzmann 統計分佈的經典氣體動力學，依據Boltzmann 方程式的Chapmann-Enskog 展開階數，平衡態或近平衡態氣體模式包括Euler 方程式、Navier-Stokes 方程式、Burnett 和Super Burnett 方程等，以及非平衡態氣體模式包括BGK、Shakhov 或ES 模式等，相關之氣體動力學數值計算技術發展已逐漸完備。類比於經典的Boltzmann 方程式，Uehling & Uhlenbeck 提出表現費米子(fermions) 和玻色子(bosons)傳輸現象的半經典(semi-classical) Boltzmann 方程式與其模型方程式的量子氣體動力學數值計算技術則尚有待開發。本計畫參考台大楊照彥教授 2006 年至2013 年關於半經典量子氣體Boltzmann-BGK 方程式數值計算技術的研究，研擬提出一系列半經典量子氣體動力算則，可適用於平衡態與非平衡態量子氣體流動計算與模擬，探討基於各物理參數(馬赫速、雷諾數、Knudsen 數等)影響之量子流體動力學特性，以及在相同的流體動力問題上研究三種粒子統計(Maxwell-Boltzmann、Bose-Einstein、Fermi-Dirac)分佈的影響。全計畫分三年，第一年目標將發展適用於平衡態連續流域量子氣體動力學BGK 算則(Quantum BGK scheme; Q-BGK)。第二年目標將發展非平衡稀薄流域量子氣體動力學守恆分立座標法(Quantum Conservative Discrete Ordinate Method; Q-DOM)。第三年目標將發展適用於全流域(平衡態連續流域至非平衡稀薄流域)半經典量子氣體統一氣體動力算則(Quantum Unified Gas Kinetic Scheme; Q-GKS)。預期本計畫成果可應用於半導體領域電子和聲子傳輸現象的模擬與分析，以及廣泛應用於微型和納米尺度能量傳輸研究，國防科技與產業發展等均能有所助益。""Based on the classical Maxwell-Boltzmann statistical distribution, the class of equations for equilibrium and near equilibrium gases covered includes the solution of respectively, the Euler, Navier–Stokes, Burnett and super-Burnett equations, as well as for non-equilibrium gases like as BGK, Shakhov and ES model equations, depending on the order of the Chapman–Enskog expansion of the Boltzmann equation. The associated gas kinetic methods for the above models are almost well developed. Analogous to the classical Boltzmann equation, Uehling & Uhlenbeck proposed semi-classical Boltzmann equation for quantum gases to describe fermions and bosons transport phenomena. The numerical methods for semi-classical Boltzmann equation and its model equations are now to be developed. This project follows the works of Professor Yang in NTU from 2006 to 2013 on the numerical methods and for semi-classical quantum gas Boltzmann-BGK equation. The purpose of this project is to developed a class of numerical methods for equilibrium and non-equilibrium quantum gas flow, and to explore the wide range of physical parameters, for examples Mach number, Reynolds number and Knudsen number, of the semiclassical Boltzmann-BGK equation and second to examine the effects of particle statistics, that is Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac, under the same flow problems. The period for this project is three years. The working iterms included are (1) the multi-dimensional Q-KFVS, Q-KWPS and Q-BGK scheme for equilibrium quantum gas, (2) the conservative discrete ordinate scheme for semi-classical Boltzmann-BGK equation in non-equilibrium flow regime, (3) the boundary condition models for constant temperature diffusive wall, (4) the unified quantum gas kinetic scheme Q-GKS for all flow regimes, It is expected that the results of this project can be applied to the field of semiconductor for simulation and analysis electronic and phonon transport phenomena, and is widely used in micro-and nano-scale energy transmission research. Also, it is helpful for the development of defense and industrial technology."

Keyword(s)

半經典Boltzmann-BGK 方程式
量子氣體
守恆分立座標法
氣體動力學算則
Q-KFVS 算則
Q-BGK 算則
Q-GKS 全流域算則
semi-classical Boltzmann-BGK equation
quantum gas
conservative discrete ordinate method
gas kinetic scheme
quantum KFVS scheme
quantum BGK scheme
unified quantum gas-kinetic scheme Q-GKS

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Gas-Kinetic Schemes for Semi-Classical Boltzmann Equation

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