均勻降雨下植生地表水流之解析解

Abstract

植生的存在不僅影響水流流況，還會增加水流之阻抗，遲滯水流進而減緩地表沖刷。爰此，為瞭解植生對於均勻降雨之地表水流流速和水深之影響，以Navier-Stokes 方程式和多孔介質層流模式為理論架構，以期研究成果在環境和生態工程之設計及地表水的管理上能有所助益。 本文從水力學的觀點出發，將流場分成均質水層、水-草混合層和透水土層(以下簡稱水層、草層、土層)，且將透水土層分為無限厚及有限厚兩類型進行討論，以Navier-Stokes 方程式描述水層之流況，將草、土二層中之流體視為孔隙介質流的運動，且引入宋(1993)所建立多孔介質層流模式加以描述之，最後搭配Runge-Kutta 法而求得其半解析解。 文中依水深與草層高度之相對關係將水理現象分成非浸沒型(水深不及草高)、浸沒型(水深漫過草高)與混合型三部分討論，並與楊(2010)裸露地上均勻降雨之情況進行比較，結果顯示植生存在之地表，其水流流速明顯地下降。此外探討坡度、降雨強度、土層厚度和比滲透係數等因子對流況之影響，及在各種狀況下植生阻力之變化，發現增加坡度、減緩降雨強度、增加土層厚度和減少草層植生疏密度皆可調降地表水深。其中以坡度之影響最為顯著，係控制水面剖線型式的主要參數，而土層厚度在浸沒狀態時對水深之影響極小，而在非浸沒型態時則可看出其差異，但水面剖線還是隨著土層厚度的增加而下降。增大植株直徑會使草層承受水流之拖曳力下降，而增加降雨強度和坡度則會使得該拖曳力上升。植株直徑與株高之比值較大時，坡度對草層拖曳力的影響小；而植株直徑與株高之比較小時，坡度對草層拖曳力有顯著的影響。

The existence of vegetation which increases the resistance to flow and retards the flow changes the surface water flow velocity and water depth, and thus reduces the surface erosion. To understand the effect of vegetation on flow velocity and water depth, a semi-analytical approach was proposed to study surface water flow passing over vegetated areas under a uniform rainfall event. A one-dimensional hydraulic model was constructed by using the Navier-Stokes equations and the theory of poro-elasticity. The results are believed to be helpful in environmental and ecological engineering design and surface water management. In this study, the flow region was divided into three layers-- homogenous water layer, water/vegetation layer and permeable soil layer (briefly denoted by water layer, vegetation layer, and soil layer) and the permeable soil layer was divided into two types of infinite thickness and finite thickness. The flow of the water layer was governed by the Navier-Stokes equations. The vegetation and soil layers were both regarded as porous media and Song’s (1993) laminar model modified from Biot’s theory of poroelasticity was applied. The semi-closed solutions were then obtained via the Runge-Kutta method. Three cases--submerged vegetation, emergent vegetation and submerged-emergent vegetation were discussed. The results were compared with those of the unplanted ground case (Yang, 2010) and showed that the flow velocity obviously decreases with increasing vegetation density. Additionally, the effect of slope, rainfall intensity, thickness of soil layer and permeability on the flow and the resistance resulting from vegetation were also discussed. The water depth decreases with increasing the slope and soil thickness, decreasing the rainfall intensity, and reducing the vegetation density. The slope factor affects the flow most significantly. The soil thickness factor has an effect on the flow depth in the emergent condition, but it doesn’t in the submerged condition. The drag force decreases with increasing the plant diameter, but increases with increasing the rainfall intensity and slope. When the ratio of plant diameter to plant height is large, the slope factor has little effect on the drag force of the grass layer. However, when the ratio is small, the slope factor becomes very significant.