理論與數值模擬研究次磁音速的寬頻Kelvin-Helmholtz不穩定波動成長率與非線性發展過程

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徐峻彥(Chun-Yen Hsu) 查詢紙本館藏

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太空科學與工程研究所

論文名稱

理論與數值模擬研究次磁音速的寬頻Kelvin-Helmholtz不穩定波動成長率與非線性發展過程
(Theoretical and Simulation Study of the Growth Rates and the Nonlinear Evolutions of the Submagnetosonic Broadband Kelvin-Helmholtz Instabilities)

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摘要(中)

速度切不穩定又稱作Kelvin-Helmholtz (K-H) 不穩定是自然界一種常見的流體能量與動量交換過程。自然界中所觀測到的K-H不穩定，擾動層厚度會不斷增加，但是早期理論研究卻假設擾動層的厚度是固定不變的，因此可以找到一個成長率最高的擾動波波長，此波長的波被視為該速度切條件下的最不穩定波。較新的理論研究(張益偉, 2016) 發現，給定速度切大小與擾動波的波長，隨著擾動層厚度增加，不穩定波的成長率也會隨之增加，直到達到該波長的最大擾動層厚度。最大擾動層厚度會隨著波長增加而增厚。本論文利用高階的二維磁流體數值模擬碼，模擬研究磁流體電漿中快波馬赫數小於1的K-H不穩定事件。我們成功找到了一種初始條件讓模擬結果可以驗證張益偉 (2016) 的理論解。我們比較數值模擬結果與理論解發現：K-H不穩定過程中的最不穩定波，並不見得是系統中最主要的擾動波。因為最不穩定波的有效擾動擾動層厚度比較薄，所以達到飽和的時間通常早於較長的擾動波。反之，較長擾動波的有效擾動層厚度比較厚，所以可以從背景電漿流中獲得較多的能量，故達到飽和的時間晚，且振幅較大。因此系統中的擾動波會往長波發展，使得擾動層厚度不斷增加。我們的模擬結果與理論解相比，在振幅與相位的空間分佈上都相當一致，但是模擬中的耗散項使得振幅的成長率略低於理論值。

摘要(英)

Velocity shear instability, which is also called the Kelvin-Helmholtz (K-H) instability, provides an efficient mechanism for energy exchange and momentum exchange of different flows in nature. The thickness of the surface perturbation usually increases with time in the observed K-H instability in the natural environment. However, in early theoretical studies of K-H instability, the surface perturbation was assumed to be confined in a boundary layer. The halfwidth of the boundary layer is less than ten times the initial thickness of the velocity shear layer. As a result, one could always find a wave of a particular wavelength that has the highest growth rate in the confined boundary layer. This wave was considered to be the most unstable mode in the system. Recently, the theoretical study by Chang (2016) showed that the location of the maximum perturbation boundary varies with the tangential wavelength of the surface wave. The submagnetosonic K-H instabilities in a magnetohydrodynamic (MHD) plasma are studied by a higher-order two-dimensional MHD simulation in this thesis. We successfully find one type of initial conditions which allow broadband surface disturbances to grow simultaneously. As a result, our simulation results can verify the theoretical solutions obtained by Chang (2016). We analysis the amplitude distributions and the phase distributions of the broadband waves generated by the K-H instability and find the simulation results are in good agreement with the theoretical solutions. However, the growth rates obtained in our simulation are less than the corresponding growth rates found in the theoretical solutions due to the additional dissipation terms added in our simulation model. Our simulation results also show that the most unstable mode in the K-H instability will be the dominant mode only during the initial phase of the K-H instability. When the perturbed boundary grows beyond the outermost edge of the most unstable mode, the growth rate of the most unstable mode begins to decrease. Whereas, waves with longer wavelengths have a more extended outermost boundary for them to grow linearly. These waves can receive more energy from the flow field and continuously grow to higher amplitudes. Thus, these waves with longer wavelengths eventually become the dominant modes in the late phase of the K-H instability.

關鍵字(中)

★ 電漿數值模擬
★ 速度切不穩定

關鍵字(英)

★ Kelvin-Helmholtz instability
★ plasma physics

論文目次

中文摘要 ……………………………………………………... i
英文摘要 ……………………………………………………... ii
致謝 ……………………………………………………... iv
目錄 ……………………………………………………... v
圖目錄 ……………………………………………………... vii
表目錄 ……………………………………………………... x
符號說明 ……………………………………………………... xi
一、簡介 ………………………………………………... 1
二、基本方程式………………………………………... 3
2-1 基本方程式介紹…………………………………... 3
2-2 歸一化的處理……………………………………... 4
三、平衡態結構 ………………………………………... 6
四、數值方法 …………………………………………... 10
4-1 高階積分法 ………………………………………... 10
4-2 高階微分法 ………………………………………... 11
4-3 耗散項………………………………………........... 11
五、 K-H不穩定成長率理論分析…………………….. 13
5-1 表面波的線性頻散關係式……………………….. 13
5-2 K-H不穩定成長率理論解……………………….. 20
六、模擬邊界條件、初始條件、與參數設定……….. 25
6-1 邊界條件 ………………………………………….. 25
6-2 初始擾動的選擇 ………………………………….. 25
6-3 模擬參數 ………………………………………….. 28
七、模擬結果與分析 ………………………………….. 29
7-1 數值模擬的結果 ………………………………….. 29
7-2 模擬結果的分析 ………………………………….. 34
7-2-1 表面波波速………………………………………………… 34
7-2-2 擾動波分析………………………………………………… 36
7-2-3 理論解的不穩定波成長率………………………………… 47
7-3 比較數值模擬與理論解的結果…………………... 51
7-4 非線性飽和………………………………………... 58
八、總結與討論………………………………………... 59
參考文獻 …………………………………………………….. 62
Appendix A 週期性初始擾動的模擬結果…………………….. 65

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指導教授

呂凌霄(Ling-Hsiao Lyu)

審核日期

2020-6-3

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