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系統識別號 U0026-2708201806444700
論文名稱(中文) 均勻電漿中的自由電子流之模擬
論文名稱(英文) Simulation of free-streaming electrons in a uniform plasma
校院名稱 成功大學
系所名稱(中) 太空與電漿科學研究所
系所名稱(英) Institute of Space and Plasma Sciences
學年度 106
學期 2
出版年 107
研究生(中文) 楊宗桓
研究生(英文) Tsung-Huan Yang
學號 LA6041048
學位類別 碩士
語文別 英文
論文頁數 106頁
口試委員 指導教授-張博宇
口試委員-藍永強
口試委員-談永頤
中文關鍵字 自由電子流  伏拉索夫解法器  動能理論  雙流不穩定性 
英文關鍵字 Free-streaming electrons  Vlasov solver  Kinetic regime  Two-stream instability 
學科別分類
中文摘要 本論文旨在透過數值模擬來探討均勻電漿中的自由電子流的不穩定性。自由電子流可以被我們正在建造的平行板電容(Parallel-plate capacitor bank)脈衝系統(Pulsed-power system)產生,而均勻電漿可以利用輝光放電產生。伏拉索夫解法器通常是種用動力學理論來模擬電漿現象的方法,因此,我們使用伏拉索夫解法器來模擬自由電子流。我們先透過模擬雙流不穩定性來驗證我們開發的伏拉索夫解法器是否正確,雙流不穩定性的能量守恆會被確認,而模擬中的不穩定成長率也會被拿來和理論比較。我們所模擬的自由電子流的熱速度為vth = 1,電子束平均速度vb 分別設為vb = 2、3、4、5,自由電子流與背景電子數量密度的比值γ 則分別設為γ = 0.5、1、2。因此,會有12 種不同的初始條件。我們發現,不穩定性的成長會發生在vb = 3 和vb = 4 的時候,vb = 2 和vb = 5 時則不會。在vb = 2 時,分佈函數因受熱運動影響而擴散開,在vb = 5 時,背景電漿和電子流間的相對速度太大以至於無法互相影響。vb = 3和vb = 4 時的最高成長率發生在γ = 1 時,但是成長率在不同的γ 中都未超過10%。
英文摘要 This thesis is to study the free-streaming electrons moving in a uniform background plasma in numerical simulation. Free-streaming electrons can be generated in our parallel plate capacitor bank (PPCB) system under construction. Uniform plasma can be generated by glow discharge. The simulation code of free-streaming electrons is implemented via Vlasov solver. Vlasov solver is usually used to simulate plasma phenomena in kinetic regime. Two-stream instability is used to benchmark the code. The energy conservation is checked and energy growth rate of two-stream instability calculated from simulation results and theory are compared. The free-streaming electrons is simulated with thermal velocity vth = 1, beam velocity vb = 2, 3, 4, 5, respectively. Ratios between the number density of the free-streaming electrons and the background electron density γ = 0.5, 1, 2, respectively. Therefore, there are 12 different initial conditions were simulated. Instabilities occur when beam velocity vb = 3 and vb = 4. No instability occurs for vb = 2 and vb = 5. In vb = 2, instability vb is diffused by electron of thermal motions. In vb = 5, the relative velocity between electrons and the background plasma is too large so that they don’t interact with each other. The growth rate is the highest when γ = 1. Nevertheless, the growth rate in different γ doesn’t change over 10% between different γ we simulated.
論文目次 摘要 .................................................................................................................. i
Abstract .................................................................................................................................. ii
Acknowledgment .................................................................................................................. iii
Table of Contents .................................................................................................................. iv
List of tables ......................................................................................................................... vi
List of figures ...................................................................................................................... vii
Chapter 1 Introduction ................................................................................................... 1
1.1 Parallel plate capacitor bank .......................................................................... 2
1.2 Free-streaming electrons................................................................................ 6
1.3 Vlasov-Piosson system .................................................................................. 7
Chapter 2 Vlasov solver ................................................................................................. 8
2.1 Basic equations .............................................................................................. 8
2.2 Normalization ................................................................................................ 9
2.3 Simulation structure ..................................................................................... 12
2.4 Simulation grids ........................................................................................... 14
2.5 Numerical methods and verification............................................................ 17
2.5.1 Initial condition ................................................................................... 17
2.5.2 Boundary conditions ............................................................................ 18
2.5.2.1 Set of boundary conditions ........................................................... 18
2.5.2.2 Benchmark of boundary conditions .............................................. 20
2.5.3 Density of electrons ............................................................................. 22
2.5.3.1 Using trapezoidal method to solve numerical integration ............ 22
2.5.3.2 Benchmark of number density ...................................................... 24
2.5.4 Poisson’s equation ............................................................................... 24
2.5.4.1 Numerical methods of solving Poisson’s equation ....................... 24
2.5.4.2 Benchmark of Poisson’s equation................................................. 34
2.5.5 Electric field and acceleration ............................................................. 36
2.5.5.1 Method of numerical differentiation ............................................. 36
2.5.5.2 Benchmark of the subroutine calculating electric field and
acceleration .................................................................................................. 37
2.5.6 Advection equations and Vlasov equation .......................................... 38
2.5.6.1 Finite volume method ................................................................... 39
2.5.6.2 Operator splitting scheme ............................................................. 40
2.5.6.3 Piecewise linear method ............................................................... 42
2.5.6.4 Piecewise parabolic method ......................................................... 48
2.5.6.5 Benchmark of solving advection equation by using Godunov’s
scheme 51
2.6 Benchmark by simulating two-stream instability ........................................ 55
2.6.1 Theory of two-stream instability ......................................................... 55
2.6.2 Benchmark of Vlasov solver using two-stream instability .................. 59
Chapter 3 Free-streaming electrons.............................................................................. 74
3.1 Initial condition of free-streaming electrons ............................................... 74
3.2 Simulation results of free-streaming electrons ............................................ 77
3.2.1 γ = 0.5 .................................................................................................. 78
3.2.2 γ = 1 ..................................................................................................... 86
3.2.3 γ = 2 ..................................................................................................... 94
3.3 Discussion of results .................................................................................. 102
Chapter 4 Conclusion and summary .......................................................................... 105
Reference ............................................................................................................................ xiii
Appendix A Calculation of integration in PPM .................................................................. xiv
Appendix B Figures of the electrical energy of free-streaming electrons ........................ xviii
Appendix C List of data films of Vlasov solver .................................................................. xx
參考文獻 [1]Balsara, Dinshaw S., Higher-order accurate space-time schemes for computational astrophysics—Part I: finite volume methods. Living Rev. Comput. Astrophys. 3:2, 2017.
[2]Burden, Richard L., Faires, J. Douglas, Numerical Analysis (9th ed.). Brooks/Cole, 2011.
[3]Cheng, C. Z., Knorr, G., The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22, 330-351, 1976.
[4]Colella P., Woodward P., The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys. 54, 174–201, 1984.
[5]Colella P., Sekora Michael D., A limiter for PPM that preserves accuracy at smooth extrema. J. Comput, Phys, 227:7069, 2008.
[6]Filbet, Francis, Sonnendrücker, Eric, Comparison of Eulerian Vlasov Solvers. J. Comput. Phys. 150:3, 247-266, 2002.
[7]Godunov, S. K., A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations. Math. Sbornik. 47, 271–306, 1959
[8]LeVeque, Randall J., Finite Volume Methods for Hyperbolic Problems. Cambridge university press, 2004.
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[10]van Leer, B., Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136, 1979.
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