楊照彥Yang, Jaw-Yen臺灣大學:應用力學研究所石育炘Shi, Yu-HsinYu-HsinShi2010-06-022018-06-292010-06-022018-06-292008U0001-2107200820470800http://ntur.lib.ntu.edu.tw//handle/246246/184684Euler 方程式及Navier-Stokes 方程式可分別藉由以Chapman-Enskog 展開古典波茲曼方程式後的零階及一階近似做積分矩得到。相同的概念可以應用到半古典波茲曼方程式而得到在Euler 極限及Navier-Stokes 極限的半古典流動力方程組。由半古典波茲曼方程式所描述的量子氣體流動包涵了在古典波茲曼方程式下未考慮的粒子簡併效應。 本研究發展求解半古典波茲曼方程式在平衡及非平衡極限下對應的流體動力方程組的氣動力數值方法。 為了簡化半古典波茲曼方程式的散射效應而採用了BGK 鬆弛模型。本研究使用推導的非平衡分佈函數是對於BGK 鬆弛模型的Chapman-Enskog 一階展開而得到，並推導了鬆弛模型下對應的物理鬆弛時間。首先在發展求解平衡極限下的量子動力射束法並把藉由不同案例來測試一維及二維的公式。 並推導量子氣動力BGK 法則用來求解及測試非平衡的流況。數值方法的數值驗證主要藉由震波管來測試並模擬了流體在高度簡併極限及古典極限下的行為。結果顯示，在古典熱力條件下，量子氣動力法將可以回到古典氣動力BGK 方法所得到的結果。採用了WENO、TVD 變數外插及一般座標的數值技巧來改善並推廣所用的氣動力數值方法。Euler equations and Navier-Stokes equations can be obtained by taking moments to zero and first order approximations of Chapman-Enskog expansions to classical Boltzmann equation, respectively. The same idea can be implemented to semiclassical Boltzmann equation and one obtains the semiclassical hydrodynamic equations in the limit of Euler and Navier-Stokes orders. The flows of quantum gases described by semiclassical Boltzmann equation includehe effects of particles degeneracy which are not considered in classical Boltzmann equation. Kinetic numerical methods for solving equilibrium and non-equilibrium flows in hydrodynamic limits of semiclassical Boltzmann equation are developed in this study. The BGK relaxation model is used to simplify the scattering term of the semiclassical Boltzmann and the non-equilibrium distribution used in this study was obtained through the first order approximation of Chapman-Enskog expansion to BGK relaxation model. The physical relaxation time in the relaxation model is also derived. First, the quantuminetic beam scheme is developed for solving the flow in equilibrium limit. Formulations to both one and two dimensionality are tested in different cases. In non-equilibrium flow, the quantum gas-kinetic BGK is alsoerived and tested. The numerical validations of the schemes are tested with shock tube. The flow in highly degenerate limit and classical limit condition are simulated. It is shown that in classical thermodynamic conditions the result of classical gas-kinetic BGK scheme can be recovered from the quantum gas-kinetic BGK scheme. Numerical techniques such as WENO, TVD variable extrapolation, and generalized coordinates are adopted to improve and generalized current kinetic schemes.Contents Introduction 1.1Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Semiclassical Boltzmann equation . . . . . . . . . . . . . . . . 3.3 Kinetic Numerical Methods . . . . . . . . . . . . . . . . . . . 5.4 Contents of Thesis . . . . . . . . . . . . . . . . . . . . . . . . 8 The Equilibrium Distribution of Quantum Particles 9.1 Indistinguishableness of Particles . . . . . . . . . . . . . . . . 10.2 Quantum Gases in the Grand Canonical Ensemble . . . . . . . 13.3 D-dimensional space . . . . . . . . . . . . . . . . . . . . . . . 15.4 Phase-Space Evolution of the Distribution Functions . . . . . 17.4.1 Liouville’s equation . . . . . . . . . . . . . . . . . . . . 18.4.2 Semiclassical Boltzmann Equation . . . . . . . . . . . . 19.5 Equilibrium Distribution, H-theorem . . . . . . . . . . . . . . 20.6 BGK RelaxationModel . . . . . . . . . . . . . . . . . . . . . . 22 Semiclassical Hydrodynamic Equations 25.1 Hydrodynamic Conservation Laws . . . . . . . . . . . . . . . . 25.1.1 Local Equilibrium Limit: Euler limit . . . . . . . . . . 27.1.2 Chapman-Enskog Expansion of BGKmodel . . . . . . 29.1.3 First Order Solution: Navier-Stokes limit . . . . . . . . 32.2 Comparisons on Quantum and Classical Hydrodynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.3 Classical Limit of Quantum Gases . . . . . . . . . . . . . . . . 37 Kinetic Numerical Methods 40.1 QuantumKinetic BeamScheme . . . . . . . . . . . . . . . . 42.1.1 Formulations to Two dimension . . . . . . . . . . . . . 46.2 Quantum Kinetic Flux Vector Splitting Scheme . . . . . . . . 49.2.1 First Order Quantum Kinetic Flux Vector Splitting . . 50.2.2 Second order KFVS . . . . . . . . . . . . . . . . . . . . 52.2.3 KFVS for Navier-Stokes equation . . . . . . . . . . . . 55.3 Second-order Quantum BGK Scheme . . . . . . . . . . . . . 57.3.1 Initial Distribution f0 . . . . . . . . . . . . . . . . . . 59.3.2 The Equilibrium Distribution g . . . . . . . . . . . . . 60.3.3 The Solution Distribution Function of QBGK . . . . . 61.3.4 Relaxation Time of QBGK Scheme . . . . . . . . . . . 63.4 Multidimensional Quantum BGK Scheme . . . . . . . . . . . 65 Numerical Techniques 69.1 Implementation of WENO Schemes . . . . . . . . . . . . . . . 69.1.1 Implement WENO Methods to Beam scheme . . . . . 71.2 Generalized Coordinates . . . . . . . . . . . . . . . . . . . . . 72.2.1 Kinetic Beam schemes in Generalized Coordinates . . . 72.2.2 QBGK scheme in Generalized Coordinate . . . . . . . 73.3 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . 75.3.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . 75.3.2 Initial Conditions . . . . . . . . . . . . . . . . . . . . . 76.4 Non-Dimensional Units . . . . . . . . . . . . . . . . . . . . . . 77 Numerical Results 80.1 Kinetic Beam Scheme . . . . . . . . . . . . . . . . . . . . . . . 82.1.1 One Dimensional Shock Tube Test . . . . . . . . . . . 82.1.2 Two Dimensional Numerical Experiment . . . . . . . . 101.2 Gas-Kinetic QBGK Scheme . . . . . . . . . . . . . . . . . . . 113.2.1 One Dimension Shock Tube Test . . . . . . . . . . . . 113.2.2 Two Dimensional QBGK . . . . . . . . . . . . . . . . . 131 Conclusion & Future Work 135.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 135.2 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.2.1 Mean Field Potential . . . . . . . . . . . . . . . . . . . 137.2.2 Kinetic Boundary Condition . . . . . . . . . . . . . . . 138ibliography 139 Bose and Fermi functions 144.1 Bose-Einstein function . . . . . . . . . . . . . . . . . . . . . . 144.2 Fermi-Dirac function . . . . . . . . . . . . . . . . . . . . . . . 145.3 Lists ofMoment Integrations . . . . . . . . . . . . . . . . . . . 145.3.1 Integrations to Vector Splitting . . . . . . . . . . . . . 145.3.2 Moments to Peculiar Velocity . . . . . . . . . . . . . . 145.4 Fugacity Equation in d Dimension . . . . . . . . . . . . . . . . 146application/pdf4541475 bytesapplication/pdfen-US半古典波茲曼方程式動力數值方法BGK 模型理想量子氣體簡併效應semiclassical Boltzmann equationkinetic numerical methodsBGK modelideal quantum gasesdegeneracythesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/184684/1/ntu-97-F92543072-1.pdf