摘要
重力流的研究在理論解析方法上，通常是發展一套數學模式(即所謂的淺水方程式shallow-water equation, SWE)，並求其解來模擬分析重力流的相關現象。然而這套方程式並沒有考慮任何來自外圍流體的阻力，使得其解與許多實驗觀察結果不符合，無法正確預測重力流之流動行為。為了使這套模型能被應用合宜，先前學者們便在重力流的前緣處套用一個波前條件來解SWE。雖然這個方法已被廣泛並有效地應用在許多重力流的研究上，但由於波前條件僅適用於二維流場，因此使得淺水方程式一直無法用來預測三維流場的行為，此為其一大缺點。另外一方面，雖然波前條件的使用已經導致理論分析上很大成功，但卻沒有人曾解釋過為何會成功的原因。從數學上看來，使用這個波前條件來解一套雙曲線型的方程式是不嚴謹的，所以「使用波前條件」被視為只是一個邏輯性的妥協，並非正規或嚴謹的方法。 本論文的主要目的，是發展一套由基本統御方程式出發的完整數學模型，並把影響重力流動力行為甚鉅的形狀阻力(form drag)包含在方程式中，使得本模型可以不用波前條件便足以適當的描述重力流之動力行為。因此本模型可被擴充和應用至三維流場的分析，這是重力流理論分析的一大進展。更特別的是，我們利用微擾法中一種很有用的方法－貼合漸近展開法，可將重力流分成靠近頭部的內場區域和遠離頭部的外場區域，分別解得各區域的近似解，最後再貼合起來成為全解。結果顯示，我們所獲得的外場方程式便等於SWE，而我們的內場分析竟然可以回復成波前條件。最終我們便可由此結果清楚地解釋為何波前條件解淺水方程的方法可以奏效的原因。 研究結果亦顯示，重力流前緣的阻力主要來自於將外圍流體排開之形狀阻力。但當密度比很小時，形狀阻力式微，此時黏滯阻力才變得重要。因此對密度比不很小的重力流而言，本理論模型可以適當地描述出重力流主要的行為。
並列摘要
To analyze the motion of gravity currents, a common approach is to solve the hyperbolic shallow water equations (SWE) together with the boundary conditions at both the current source far upstream and the current front at the downstream margin. The use of the front condition aims to take the resistance from the ambient fluid into account, however, which is absent in the SWE. In the present study, we re-derive the SWE by taking the ambient resistance into account, and end up with the so-called modified shallow water equations (MSWE) in which the ambient resistance is accounted for by a nonlinear term, so that the use of the front condition becomes unnecessary. These highly nonlinear equations are approximated by perturbation expansions to the leading order, and the resultant singular perturbation equations are solved by an inner-outer expansion. For constant-flux and constant-volume gravity currents, the outer solutions turn out to be exactly the same as the previous ones obtained by solving SWE with the front condition. The inner solution gives both the profile and the velocity of the current head and also leads to the recovery of the front condition while is in a more general form. The combination of inner and outer solutions gives a composite solution for the whole current, which was called by Benjamin (1968) a “formidably complicated” task. To take the turbulent drag on the current into account, we introduce the semi-empirical Chezy drag term into the MSWE and end up with a result comparable with experimental data. The result implies that the ambient resistance is contributed mainly by the inviscid form drag, and the viscous drag dominates only when the density ratio is small. Furthermore, the MSWE can be extended for three-dimensional viscous currents, while will become more complicate that present analytical approach may not be feasible.
參考文獻
延伸閱讀
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