A study on the slowly deforming bed form under water
Date Issued
2011
Date
2011
Author(s)
Kuo, Yuan-Ching
Abstract
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.
Subjects
Slowly deforming bed form
order of magnitude analysis
long time scale
leading order formulations
Boundary Integral Equation Method
numerical water tank
Type
thesis
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