動量與質量傳輸時間尺度差異之檢驗

An Examination of the Time Scales Difference Between Momentum and Mass Transports

黃泊源(Po-Yuan Huang)

指導教授 ： 黃良雄

國立臺灣大學/工學院/土木工程學研究所/碩士(2010年)

https://doi.org/10.6342/NTU.2010.01010

摘要

本研究的主旨為分析並檢驗一動量與質量傳輸問題的時間尺度關係，其中動量傳輸的選擇為振盪流場(co-oscillating flow field)，而質量傳輸模式的選擇為傳流擴散方程式(advection-diffusion equation)。
為了數學解析上的可行性與分析結果的代表性，文中設計了一個數學模型，由一維駐波流場以及一維傳流擴散方程式所構成。針對流場，採用線性波理論來求得。針對傳流擴散方程式，利用映像法(method of image)以及有限傅立葉正弦轉換(finite Fourier sine transform)，將原來的變係數二階偏微分方程式(即傳流擴散方程式)轉換成變係數一階常微分聯立方程組。接著，採用龍格庫塔法(Runge-Kutta method)來進行數值計算，以求得轉換之半解析解。最後，利用有限傅立葉正弦反轉換(finite Fourier sine inverse-transform)來求得傳流擴散方程式之半解析解。
對於數學模型的控制方程式，即一維駐波流場控制方程式和一維傳流擴散方程式，利用無因次技巧進行分析，然後藉由近似的估計過程，以及滿足限定條件的前提之下，來獲得一近似之時間尺度關係式(time scales expression)。最後，透過上述半解析解的數值驗證來證明此時間尺度關係式的合理性，以及進行物理和數學上的討論。
本文的重要貢獻共有三部分：第一，找到特定情況下傳流擴散方程式的半解析解；第二，估計出適用於感潮河段之時間尺度關係；第三，發現時間尺度關係會隨時間而改變。

關鍵字

時間尺度 ； 振盪流場 ； 傳流擴散方程式 ； 有限傅立葉正弦轉換 ； 龍格庫塔法

並列摘要

The purpose of this thesis is to examine the time scales relation between the momentum transports and the mass transports, where the selected momentum transports is an co-oscillating flow field, and the selected model of mass transports is the advection-diffusion equation.
For the feasibility in the mathematical analysis and for the representativeness of the investigation results, in the contents we design a mathematical model, which contains 1-D standing wave flow field and 1-D advection-diffusion equation. For 1-D standing wave flow field, we solve it by linear wave theorem. And for 1-D advection-diffusion equation, by using the method of image and finite Fourier sine transform, we can transform the 2nd order variable coefficient partial differential equation (i.e. advection-diffusion equation) to 1st order coupled variable coefficient ordinary differential equations. Next, adopting Runge-Kutta method to proceed numerical evaluations, by which we can obtain the semi-analytical transformed solutions, and get the semi-analytical solution for advection-diffusion equation by inverse finite Fourier sine transform.
Using dimensionless technique for the governing equations for 1-D standing wave flow field and 1-D advection-diffusion phenomenon, and by an approximate estimation procedure, we can obtain an approximate time scales expression. Finally, by the numerical verifications of the semi-analytical solutions, we can confirm the rationality of the time scales expression, and carry out some discussions based on physics and mathematics.
The major contributions of this thesis are as follows: the first one is that we find the semi-analytical solution of advection-diffusion equation under specified conditions; the second one is that we estimate the time scales expression which is suitable for tidal channel problems; the third one is that we find that time scales relation can change with time.

並列關鍵字

time scales ； oscillating flow field ； advection-diffusion equation ； finite Fourier sine transform ； Runge-Kutta method

參考文獻

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