正交曲線座標擬似三維水理模式於彎道水流之模擬研究

Abstract

本研究以垂直水平分離(VHS)演算的方法發展一擬似三維水理模式，探討彎 道流場之水理現象。在水平部份，利用深度平均的控制方程式計算水位與水平 兩方向的水深平均流速值；在垂直部份，先假設任一深度的流速等於水深平均 流速加上一擾動量，帶入三維控制方程式後再扣除深度平均控制方程式即可得 水平兩方向擾動量在垂向(水深方向)的控制方程式；綜合水平與垂直的計算結 果，即可計算不同深度之水平兩方向的流速值。在座標系統上，水平方向與垂 直方向分別採用正交曲線座標與σ 座標系統，使模式能便利處理不規則的渠道 與底床邊界。在計算步驟上，水平部份採用雙階分割操作趨近法，將水理控制 方程式分割成延散步驟(包含移流項與擴散項)和傳播步驟(包含底床剪應力項與 壓力項)求解，增進模式在實用上的彈性。數值差分上，水平與垂直部分皆採用 隱式法求解，使模式可用較大之演算間距。最後，本研究分別採用緩彎、急彎 及連續彎來模擬彎道流場，並與實驗值與水深平均二維模式模擬結果比較，展 示本模式在不同彎道型態、彎道長度、二次流強度及彎道反曲的情況下，皆能 模擬出合理的結果。

This study develops a semi-3D model based on a vertical-horizontal splitting (VHS) method to analyze the flow in open-channel bends. In horizontal, the surface elevation and depth-averaged velocity components are computed by 2D depth-averaged model. In vertical, assume the 3D velocity profile of Navier-Stokes equations is equal to the depth-averaged velocity plus the deviation of velocity profile, and then the vertical governing equations can be derived by subtracting the 2D depth-averaged equations to the 3D Navier-Stokes equations. In order to fit the complex geometry in both side wall and bed slope of channel, the orthogonal curvilinear coordinate system is used in horizontal gird, and sigma coordinate system is used in vertical grid. As for the numerical solution procedure, the two-step split-operator approach, which includes dispersion process (advection and diffusion terms) and propagation process (bed shear stress and pressure terms), is adopted to solve the 2D depth-averaged flow equations to improve the application flexibility. Implicit difference methods are adopted to relax the time step restriction allowing large time steps. Finally, three sets of experimental data including mildly curved, sharply curved and meandering channel are used to demonstrate the capability and accuracy of the semi-3D model, and the results of 2D depth-averaged model are also compared. The simulation results of semi-3D model show well agreement with experimental data considering different curved channels, bend lengths, secondary current and transverse mixing conditions.