非靜水壓模式模擬自由液面波之發展

Development of a Non-hydrostatic Model for Simulating Free-surface Waves

楊智傑(Chih-Chieh Young)

指導教授 ： 林正芳

國立臺灣大學/工學院/土木工程學研究所/博士(2009年)

https://doi.org/10.6342/NTU.2009.01472

摘要

本研究之目的為發展一套高精度的三維非靜壓sigma模式模擬自由液面波由深水向淺水傳遞之特性。本模式採用隱式有限差分法同時求解非穩態的納維-史托克斯方程式（Navier-Stokes equation）及自由液面邊界條件。本研究提出高精度的表層非靜水壓處理法使得本模式可精確地模擬延散波（dispersive wave）的傳遞。遇到非線性波時，使用sigma座標轉換在計算壓力梯度上會產生極大的數值誤差。因此，本模式採用四階精度空間離散方法來處理水平壓力梯度以克服數值誤差。直接求解三維的水波問題會產生很高的數值計算量，故本模式將一個三維的問題分解成一系列的垂直二維問題。於每個垂直二維問題中，所有的未知數可以水平速度來代換使得整個系統變成水平速度的塊狀七對角矩陣。此種矩陣系統，不需要迭代求解，只需使用解矩陣的副程式（如：雙掃法）就可以直接有效地求解。本研究首先檢視此非靜壓模式的特性包含延散性與非線性。接著，將此模式應用到一系列自由液面波的問題，例如:波流共存、非線性深水波群與流過不規則底床的沿岸波。水波與水流、水波與水波以及水波與底床間的交互作用亦被仔細地探討。本研究證實非靜壓模式利用少許的垂直格網（如：二至五層）能精確有效率地模擬自由液面波問題。

關鍵字

非靜水壓 ； sigma座標 ； 延散性 ； 非線性 ； 波-流交互作用 ； 波-波交互作用 ； 波-底床交互作用

並列摘要

A higher-order 3D non-hydrostatic model in a sigma-coordinate system is developed for simulating free-surface wave propagation from deep to shallow waters. The model using an implicit finite difference scheme on a staggered grid simultaneously solves the unsteady Navier–Stokes equations and the free-surface boundary condition. A higher-order top-layer pressure treatment is proposed to resolve dispersive wave propagation. To capture non-linear waves, a 4th-order spatial discretization is utilized to approximate the large horizontal pressure gradient. Based on a domain decomposition method, the 3D system matrix is decomposed into a series of 2D vertical plane problems. An efficient direct solver is developed to solve the resulting block hepta-diagonal sub-system matrix. Model’s characteristics including linear wave dispersion and non-linearity are critically examined. The model is then applied to examine a range of free-surface wave problems including the co-existence of waves and currents, non-linear deep-water wave group and near-shore wave propagating over irregular bottom. Features of wave-current, wave-wave, and wave-bottom interactions are carefully discussed. Overall, good agreement between the model results and experimental data shows that the newly developed non-hydrostatic model using a few vertical layers (e.g. 2-5) is capable of accurately and efficiently resolving various wave phenomena.

並列關鍵字

non-hydrostatic pressure ； sigma co-ordinate ； dispersion ； non-linearity ； wave-current interactions ； wave-wave interactions ； wave-bottom interactions

參考文獻

1. Anthonio, S.L., Hall, K.R., 2006 High-order compact numerical schemes for non-hydrostatic free surface flows. International Journal for Numerical Methods in Fluids 52, 1315-1337.

2. Badiei, P., Namin, M., Ahmadi, A., 2008. A three-dimensional non-hydrostatic vertical boundary fitted model for free-surface flows. International Journal for Numerical Methods in Fluids 56 (6), 607-627.

4. Bateman, W.J.D., Swan, C., Taylor, P.H., 2001. On the efficient numerical simulation of directionally spread surface water waves. Journal of Computational Physics 174, 277-305.

5. Battjes, J.A., 2006. Developments in coastal engineering research. Coastal Engineering 23 (2-3), 121-132.

6. Beam, R.M., Warming, R.F., 1978. An implicit factored scheme for the compressible Navier–Stokes equations. AIAA Journal 16, 393-402.