擬似三維分層積分淺水波模式之發展

Abstract

本研究發展一交錯網格有限體積擬似三維分層積分淺水波模式，提出二次形狀函數近似層內速度分布以確保層介面之流速與剪力連續，毋需二維水深平均子模式決定水面高程，可有效降低模式計算量。模式之驗證首先透過封閉水池風吹之垂直環流假設案例與解析解比較，除觀察模式對層數之敏感度外，亦檢視模式對環流現象水深方向流速剖面發展過程並反映水面坡降變化之機制，及垂直水深方向黏滯性分布改變對流速剖面形狀之影響；此外，由於底床坡降與粗糙度密切影響明渠流況，經由明渠流假設案例之功能測試檢驗，模式除能應用本研究假設均勻流條件推導渦流滯性與水深、粗糙度、剪力速度之關係，模擬出水深方向均勻紊流強度之拋物線流速剖面外，若應用實驗觀測水深方向紊流強度拋物線分布亦能模擬出對數型流速剖面。

A novel semi-three-dimensional (semi-3D) layer-integrated approach was proposed in this study for the shallow water wave computation. A quadratic shape function proposed to approximate the velocity distribution in layer ensures the continuity of velocities and shear stresses at interfaces. To discretize the governing equations, the finite volume formulation with staggered grid is used. As the two-dimensional (2D) sub-model for locating the free surface is not needed in this approach, the computational time consumption has been dramatically reduced. The model was verified through the wind-induced circulation in a closed basin by comparing the velocity profiles to the analytical solution. The formation of the velocity profiles due to change of viscosity distribution and the evolution process of water surface slope were also investigated. Further, the eddy viscosity is estimated by a function related to discharge, water depth, and Manning’s n under the assumption of parabolic distribution of longitudinal velocity along the vertical direction for open channel flows. Two designed hypothetically cases were conducted to examine the proposed eddy viscosity relation and to demonstrate the capabilities of the proposed model.