均勻流通過傾斜階梯產生波動的理論及數值解析

Abstract

本文探討在二度空間中，理想流體的均勻流通過在底部的傾斜階梯在自由表面產生穩態波動的問題。本解析方式為基於King and Bloor (1987)求解垂直階梯情狀的方法，使用廣義的史瓦茲-克里斯多福轉換，轉換在原物理平面的流場成為有一沉流的上半平面，獲得流場控制方程式，在斜坡長度遠小於上游水深條件下以傅立葉轉換求出線性解，此線性解顯示波動特性及阻力與階梯抬升量和福祿數有關，但與斜坡角度無關。 為求非線性波動的數值解，以有限差分法離散化完整的非線性方程式成一組封閉型式的非線性方程式，以馬夸特-李文柏格演算法迭代技巧求出可靠的數值解。 當上游為超臨界流時，下游水位抬升，斜坡所受阻力與水流方向相同，但上游為亞臨界流時，下游產生波動且平均水位下降，斜坡所受阻力與水流方向相反。水位變化量與階梯抬升量和斜坡角度呈現正相關。阻力只正相關於階梯抬升量，但與斜坡角度無關。在階梯抬升量較大時，線性解的自由表面與水平夾角和阻力比數值解小。

The problem of a steady and uniform free-surface flow of an ideal fluid propagating over a sloping step on the bottom is investgated in this thesis. Based on the method of King and Bloor (1987) for the case of a vertical step, the generalized Schwarz-Christoffel transformation is used to map the physical plane onto the upper half-plane where the flow field is a sink at the origin. The nonlinear governing equations can be linearized for the case of small step and be solved by the help of Fourier transformation. The linear solution shows that the free-surface and drag force on the step respond to the height of the step and Froude number, but is independent of the slope of the inclined step. A closed form of nonlinear equations is obtained by the finite-difference method discretizing governing equations. The numerical solution with high accuracy are iteratively calculated by the Linear solutions, based upon small step height are solved by Fourier transform. As the step height is increased, solutions to the exact nonlinear equations are obtained by using finite-difference methods and Marquardt- Levenberg algorithm. For a supercritical flow the free-surface rises in the downstream and the drag on the step is in the same direction of the uniform flow. Contrarily, the free-surface falls down in the downstream and the drag on the step is in the opposite direction of the uniform flow for a subcritical flow. The numerical solution of free-surface depends on both the height and slope of the step. However, drag force on the step only depends on the height of the step, but is unconnected from the slope of the step.