局部極值消減法應用於二維水深平均模式之研究

Abstract

本研究旨在以局部極值消減法及配合有限體積法與Runge-Kutta多時階法求解二維水深平均方程式。本研究採用局部極值消減法，其具有最大值不會再增加與最小值不會再減少的特性，可以有效抑制數值震盪。同時，為了提升數值的穩定度，在時間離散化上採用4階Runge-Kutta時階法。

本研究以潰壩模擬、穿臨界流流經底床突起之模擬、束縮段產生交波之模擬等案例，探討控制方程式之慣性項、重力項及摩擦項對模擬精度之影響，最後本模式也實際應用於現場案例東埔蚋溪之模擬。另外，針對消散黏滯係數之指數β值的探討，本研究採用交波模擬案例作敏感度分析，並以模擬值與實驗值之效率係數、均方根誤差、最大值之相對差異百分比三種指標作比較。

經過計算值與解析解及實驗數據的比對，本研究所發展的局部極值消減法具有更佳的求解精度，而且就東埔蚋溪現場應用來看，局部極值消減法亦能處理穿臨界流等複雜流況。

The purpose of this study is to use the Local Extremum Diminishing (LED) scheme integrated with finite volume method and Runge-Kutta time stepping method for solving 2D depth-averaged equations. The LED scheme preserves the properties of that the local maxima would not increase and the local minima would not decrease to restrain the numerical oscillation efficiently. Meanwhile, the fourth order Runge-Kutta method is also adopted in time difference term to improve the numerical stability.

In this study, the dam-break flow, the transcritical flow with hump and the cross-wave flow with contraction are simulated to discuss the accuracy affected from inertia force term, gravity force term and bed friction term in the governing equations receptively. Furthermore, the proposed model is also applied to the practical field simulation of Tung-Pu-Ruei creek. The discussions for the exponent β of coefficient of artificial viscosity with cross-wave simulation are presented. The sensitivity analysis of β is analyzed by using Coefficient of Efficiency (CE), Root-Mean-Square Error (RMSE), and Error for the Peak Value (EVP), which are analyzed on the basis of comparing the simulation and experimental data.

By comparing with analytic solution and experimental data, the computed results from proposed model with LED scheme can provide adequate accuracy. In addition, for the practical application of Tung-Pu-Ruei creek, the LED scheme can also handle the complex flow.