理想二維流場的處理可在任何座標系統上，離散Laplace方程式後求解流函數 。本文建議利用微分幾何的關係，將流線以及某條與 軸的平行線分別轉換至一通用座標系統(ξ,η)，經過轉換後的Laplace方程式求解變數即變成流線的位置y(ξ,η)。若指定x=x(ξ),ψ=ψ(η)，使用「擴充Von Mises轉換」即能得到(ξ,η)座標系統，因為流線可適當地貼合固體邊壁所以又稱為貼壁流線座標(Boundary-fitted Coordinate)。特定的ξ線( =constant)即代表一條流線，沿此線可反算流線之位置；若給予適當邊界條件與位置，即能由許多 滿足Laplace方程式以獲得內部流線分佈與速度場。

基於流線座標之特性，可將其應用至穩態明渠流，進而推導出包含垂直速度分量之能量方程式。若忽略黏性，該方程式無論於何種底床形狀，皆能以某流線作為參考位置來求解自由液面；如選擇與底床貼合之流線其 ，藉由底床與水面兩流線間的定值關係，可計算水面位置。

本文首先以均勻流通過固定鈍體的問題作為模式測試工具，與解析解結果比較可精確分析模式誤差。接著探討明渠亞臨界流通過底床障礙物，其下游為亞臨界流或超臨界流之流況；本文建議以能量方程式的根是否有重根或最小臨界流量作為判定臨界流況之條件，經特殊的疊代程序獲得自由液面位置。得知自由液面後，求解內部流線可迅速獲得詳細的流場資訊，並繪製全域流線分佈與動壓力分佈，使此類明渠問題之分析更完整。

Ideal 2D flow field can be described by stream function ψ(x,y) as the solution of Laplace’s equation. In this paper, the Extended von Mises transformation is applied to transform the equation from the Cartesian coordinates to the Streamline coordinates (ξ,η)system and solve the vertical coordinates y(ξ,η) along a streamline η(ψ)=constant as x(ξ) and ψ are given. In this system, one obtains the pressure distribution from the Bernoulli equation. For the application to steady open-channel flow, some difficulties may appear when one tries to find the unknown position of the free surface. The complete form of energy equation involving the vertical velocity head is proposed as the free surface dynamic condition here. Last, after the calculation of free-surface elevation for the given locations of inner streamlines, one can re-calculate the locations of inner streamlines iteratively. In this study, to validate the model a uniform flow past an obstacle without the free surface is first calculated in comparison with the corresponding analytical solutions. Next, a steady open-channel flow over a bottom hump is considered when such a flow transits from the subcritical upstream flow to either subcritical or supercritical downstream flow. The detailed numerical investigation is carried out here to determine the location where the critical condition is applied, if a double-root solution of the free surface profile exits in this flow field. The calculated result is discussed and some future study is recommend in this thesis.

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