顯式有限解析法模式結合預測-­修正數值法於超亞臨界混合流之研究

Abstract

本研究延續Hsu and Yeh (1996)之一維顯式有限解析法(explicit finite analytic method，簡稱EFA)模式，發展出適於超臨界流與超亞臨界混合流流況之數值模式。EFA求解之特點，乃在於求解水流動量方程式時，以特性法觀念解得其中變量(流量與通水斷面積)之局部解析解，並且遵守可蘭穩定性條件；邊界處理方面則透過水流之連續方程式與動量方程式，利用特性法觀念求解邊界處之變量；而在超亞臨界混合流況之內部邊界，根據福祿數來判斷水躍發生之位置；內部相鄰計算點的水位高程，先透過預測­修正數值法(MacCormack scheme)來得到超臨界流區域的流量及通水面積，接著再將預測­修正法所得流量及通水面積以特性法觀念求解超臨界流區域之下游邊界水深。此方式可有效減少地形驟變對數值計算之影響，並可解決因流況改變所引起數學上屬於奇異點(singularity)之問題。本文針對實驗室水槽及實際河川陡緩坡交替之情況進行數模與評估。

This study extends Hsu and Yeh’s (1996) one-dimensional explicit finite analytic model (EFA) for simulating supercritical and mixed supercritical and subcritical flows. The essence of the EFA is the adoption of the concept of method of characteristics to the momentum equation for solving the local analytic solution of the dependent variables (i.e., discharge and cross-section area of flow). To ensure stability of the scheme, Courant condition should be obeyed. The dependent variables at the upstream and downstream boundaries are obtained through the method of characteristics. For the interior boundary condition at mixed supercritical and subcritical flows, the locations of the occurrences of hydraulic jumps are determined according to the values of Froude numbers. And water depths for supercritical regime at downstream boundaries were calculated. This was done through the method of MacCormack scheme and method of characteristics, by utilizing the water surface elevations of the interior neighboring computational points. The mixed supercritical and subcritical flow fields in laboratory flumes and natural rivers will be simulated and evaluated by the proposed model.