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研究生:李錫勳
研究生(外文):ShyiShiun Lee
論文名稱:電漿波茲曼方程式之模擬
論文名稱(外文):Numerical Solution of the Boltzmann Equation
指導教授:李石頓李石頓引用關係楊照彥
指導教授(外文):Shih-Tuen LeeJaw-Yen Yang
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:機械工程學研究所
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:1999
畢業學年度:87
語文別:中文
論文頁數:175
中文關鍵詞:波茲曼方程式電漿福克-普蘭克碰撞效應加權不振盪算則
外文關鍵詞:Boltzmann equationplasmaFokker-Planckcollision effectweighted ENO
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波茲曼方程式是一個多變數的偏微分方程式,數學解析解不易獲得,若應用數值方法求解,計算程式需要大量記憶空間及耗費冗長計算時間。本文綜合應計算雙曲線型守恆律的高解析度算則,配合隱式法處理源項,發展一套數值方法可以較有效率地解析波茲曼方程式。
首先將此法運用於無碰撞效應的藍道阻尼、兩電子集束交互作用及多孔BGK結構的融合過程等問題上,在此類問題上本數值方法皆可得到滿意的結果,它克服了傳統特殊函數轉換法和粒子數模型法算則所產生的數值振盪現象,並且得到更細微的速度分佈函數圖形結構。其次也模擬具碰撞效應作用的電漿問題,結果發現,對於碰撞方程式,本法達到粒子數守恆的要求,而且能量守恆也在水準之上。計算也結果發現碰撞頻率的高低影響平衡分佈改變的快慢,但不會影響平衡最終的型態,在電場作用下,碰撞頻率越高時,電子動能的增加率也較大。
連續的速度分佈函數模型還是需要耗用大量的電腦記憶空間,因此本文也引進分離化波茲曼方程式模型,它能以較少方向的速度分佈函數,配合機率概念模擬碰撞效應作用,探討不同尺度下氣體的行為模式。在連續體情況下,紐森數越大所得震波厚度越厚,而稀薄氣體的流場情況比連續流體豐富。此模式可以容易地模擬化學反應作用,探討雙原子之分子氣體因震波作用解離成單原子氣體的過程。
本文發展的方法可以有系統的由一維問題推展到二維問題,面對流場尺度及碰撞效應的大小差異很大的情況下,程式都不需調整,經有系統的測試及驗證,本文所發展出的數值方法提供了模擬電漿行為一有用工具,可以揭露電漿非線性交互作用,碰撞效應對速度分佈函數變化的影響及不同尺度下氣體的行為模式。
Boltzmann equation is a partial differential equation with many variables; its mathematical analytic solution is hard to obtain. Large memory space and long computational time are also needed; if the numerical method is applied to solved it. In this study, the high-resolution numerical schemes for hyperbolic conservation law and the implicit method for treating the source term are combined to develop a numerical method that can solve the Boltzmann equation more efficientively.
First, the present method is used to simulate the Landau damping, two-stream instability and N hole Bernstein-Greene-Kruskal structure problems, the performances are satisfactory. Compared with these results obtained by transformation or PIC methods, it avoids numerical oscillatory phenomena and gives better resolution of the structure of the distribution functions. Second, The plasmas with collision effects are simulated. The requirement for conservation of number density is achieved and the conservation of kinetic energy is acceptable. It results also show that the collision frequency effects transition speed of the velocity distributions but doesn''t change the final equilibrium type. Plasmas with stronger collision effects, the kinetic energy of electrons increases faster under the action of the electric field.
The continuous velocity distribution function model also demands many computer memories, therefore the discrete Boltzmann equation model is also introduced. In the discrete model, the velocity distribution functions are based on a finite number of velocity directions and the collision effects are modelled by probability theory. This model is used to study problems with various characteristic lengths. The results indicate that the larger Knudsen number, the bigger the shock thickness is in the continuum flows. The flow types of rarefied flows are more abundant than those of the continuum flow. It can easily simulate the reacting flows, it shows that molecules dissociate into atoms after shock passing.
The method developed is easily expanded from one-dimensional to two-dimensional problem. It can model the flows with various characteristic lengths without modifying any variable in the computer program; it also can handle the strong and weak collision effects. The method is a good tool for simulating Boltzmann equation, it can explore the nonlinear effects in the collisionless plasma and the collision effects on the evolution
封面
目錄
誌謝
中文摘要
英文摘要
圖目錄
符號說明
目錄
第一章 概論
1.1 引言
1.2 文獻回顧
1.3 本文目的
1.4 本文內容
第二章 氣體動力論
2.1 速度分佈函數
2.2 波茲曼方程式
2.3 巨觀物理性質的計算
2.4 磁撞反應項的物理模型
第三章 數值方法
3.1 雙曲線型守恆律算則
3.2 福克-普蘭克碰撞方程式的計算
3.2 鬆弛效應雙曲線型系統的計算
第四章 波茲曼-維拉索夫方程式的數值模擬
4.1 藍道阻尼問題
4.2 空間中兩電子集束的交互作用
4.3 多孔BGK結構的融合過程
4.4 二維度藍道阻尼問題
第五章 波茲曼-福克-普蘭克方程式的數值模擬
5.1 兩電子集束碰撞過程的演變
5.2 馬克斯威爾鬆弛問題
5.3 電場作功下電子能量的變化
5.4 磁撞頻率對平衡過程的影響
第六章 分離化波茲曼方程式模型
6.1 統御方程式
6.2 巨觀物理性質的計算
6.3 平面與空間上分離速度模型
6.4 里曼問題的數值模擬
6.5 震波作用下解離氣體的傳播現象
第七章 結論與展望
參考文獻
附錄A
附錄B
附圖
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