水下長時間尺度之底床變形研究

A study on the slowly deforming bed form under water

郭遠錦(Yuan-Ching Kuo)

指導教授 ： 黃良雄

國立臺灣大學/工學院/土木工程學研究所/博士(2011年)

https://doi.org/10.6342/NTU.2011.02655

摘要

本研究的主旨在釐清時間尺度以探討二維緩慢底床變形。在本研究中，底床之控制方程式乃修改自Biot (1956)所提出之多孔彈性介質動量方程式。本研究先以量階分析修改Biot方程式使其適用於研究緩慢底床變形，再搭配Hsieh et al. (2001) 所提出之簡化邊界條件，以解析方式探討水波、流及底床變形間之時間尺度差異。由時間尺度分析顯示，此系統中存在四個明顯不同之時間尺度:水波、砂波、水流及底床變形 (砂波即彈性波中的雷利波 (Rayleigh wave))。時間尺度差異表示系統中之各種運動間存在速度差異並顯示量階分析的可行性，即水波、流與底床之交互作用並不需迭代計算，僅需計算主要驅動模式後以修正方式計算次要項即可。採用量階方析之主要優點在於避免以數值方法模擬緩慢底床變形時，因迭代計算無法收斂而導致之數值發散問題。
本研究接著以首階方程式及邊界積分方程式法建立一數值水槽，以數值水槽模擬李鴻源等(1991) 之緩慢底床變形試驗並得到合理的一致性。數值水槽之建立係採用邊界積分方程式法，邊界積分方程式法以積分方式將控制方程式離散於邊界點上，此一數值方法同時兼具計算效率及適應不規則邊界。
總結本研究之主要貢獻有三:一、本研究以量階分析探討Biot(1956)式於緩慢底床變化中的量階項並以Hsieh et al. (2001) 所提出之簡化邊界條件建立可用於探討緩慢底床變形之首階控制方程式與邊界條件。二、本研究發現緩慢底床變形之時間尺度比。三、本研究以量階分析及邊界積分方程式法建立一數值水槽，此數值水槽克服了江百祥(1996)及施宛平(1998)所提出之數值困難並模擬了緩慢底床變形。

關鍵字

緩慢底床變形 ； 量階分析 ； 長時間尺度 ； 首階控制方程式 ； 邊界積分方程式法 ； 數值水槽

並列摘要

The primary object of this study aims at clarifying the time scales for analyzing the two dimensional slowly deforming bed forms. In the study, we modified the momentum equations of Biot (1956) with an order of magnitude analysis and made the equations suitable for modeling the slowly deforming beds. Together with the simplified boundary conditions proposed by Hsieh et al. (2001), we analyzed the time scales of the water waves, flows, sand waves and the slowly deforming bed analytically and we discovered that there are four different time scales, i.e. the time scales of water waves, sand waves, water flows and the deformation of bed (the sand waves are the Rayleigh waves of the elastic waves). The differences in time scales imply the differences between each phenomenon and validated the feasibility of the order of magnitude analysis. With the order of magnitude analysis, the iteration between the computation of water and the bed are no longer required and the computations may be done by calculating the major driven terms and modified the minor terms and hence the numerical approaches to the slowly deformation of bed forms shall no longer be suffering from the numerical divergences.
After we established the leading order analysis for the slowly deforming bed, we constructed a numerical water tank. Moreover, we verified the model with the flume experiment of Li et al. (1991) and obtained reasonable agreement. The numerical water tank was constructed with the boundary integral equation method which transforms the governing equations into discrete grids on the boundaries with integral equations and the boundary integral equation method is efficient and flexible for irregular boundaries.
The major achievements of the present study are threefold: 1. we conducted an order of magnitude analysis to the equations of Biot (1956) and adopted the simplified boundary conditions of Hsieh et al. (2001) for establishing the suitable formulations for modeling the slowly deforming beds. 2. We discovered the time scale ratio of the slowly deforming bed. 3. We constructed a numerical water tank with the leading order formulations and the boundary integral equation method and the numerical water tank conquered the numerical difficulties reported by Chiang (1996) and Shih (1998) and modeled the slowly deforming bed forms.

並列關鍵字

Slowly deforming bed form ； order of magnitude analysis ； long time scale ； leading order formulations ； Boundary Integral Equation Method ； numerical water tank

參考文獻

39. Huang F.Y., 2003, A study in boussinesq equations for wave transformations on porous beds, Master’s thesis, National Taiwan University (in Chinese)

3. Besio G., Blondeaux P., and Frisina P., 2003, “A note on tidally generated sand waves”, Journal of Fluid Mechanics, Vol. 485, pp.171-190.

4. Best J., 2005, “The fluid dynamics of river dunes: A review and some future research directions”, Journal of Geophysical Research, Vol. 110, F04S02

5. Becker A.A., 1992, The Boundary Element Method in Engineering – A complete course, McGraw-Hill Book Company.

6. Biot M.A. 1941, “General theory of three-dimensional consolidation”, Journal of Applied Physics, Vol.12, pp.155-164.