本文旨在用數值方法模擬潰壩波通過孔隙結構的運動，以探討潰壩波與透水性孔隙結構物互制時，液面變化及內部流場的運動特性。首先中文以黏性流體運動方程式(Navier-Stokes方程式)當做基礎，推導出巨觀型態(macroscopic)的黏性流體運動方程式。對於流體在孔隙中受到的阻力則參考Sollitt and Cross (1972)所提出的阻力方程式。 經由這兩者的結合，可以得到適用於描述透水結構物內部與外部的控制方程式。又在考慮模式的實用性與執行效率後，將以流體體積數值方法發展的三維模式(吳, 2004)修改成符合新推得的控制方程式，同時搭配大渦流模擬(large eddy simulation)之紊流模式與移動固體的技術以達到控制閘門的目的。
本模式亦經由與理論解析解及相關實驗數據的比較來證明模式本身的可信度與準確度。待驗證成功後，本模式即用來模擬不同的物理問題。在潰壩流流經乾底床的問題中，初始潰壩過程中長波理論的解析解不能適切描述自由液面的準確位置，而是在經潰壩波傳遞約11倍特徵時間後，解析解方能與實際狀況吻合。此外，在考慮底床的黏滯效應後發現自由液面會略微抬升。若考慮閘門的抽取效應，則發現潰壩波最終波形並不受閘門抽取所影響，但會造成波前到達的時間不同。由與解析解相比較後，顯示潰壩波前速度將不受有限壩體長度而影響，且不論考慮底床為滑動邊界條件或是不可滑移邊界條件，該結論皆相同。
在潰壩流流經濕底床的問題上，其結果顯示壩潛水深是決定潰壩後的波浪型態最重要因素。此外，閘門抽取會對整體的渦度分佈與自由液面有很嚴重的影響。
考慮潰壩波對孔隙結構的互製作用，其結果顯示孔隙結構物會影響整體流場的演變且有效減低因不同閘門抽取所產生的擾動。
對於潰壩波撞擊孔隙結構物問題中，作用於孔隙結構上的正向作用力也會隨之不同。孔隙率高的結構物受到的正向作用力較小。根據本研究在孔隙率為0.2時可減低20%衝擊力；而孔隙率為0.8時可達到70%的改善。此外潰壩波對於孔隙結構內的影響也不能小覷，根據本研究，在撞擊前後孔隙結構內(孔隙率為0.5)的水位亦可以達到50% 的初始壩體水深。
本文亦針對局部潰壩的問題進行數值之校驗與模擬，並研究局部潰壩波撞上固體結構物與孔隙結構物的交互作用. 對於三維的局部潰壩的應用來說，由於其破壞力不若全潰壩的強大，因此在僅在固體結構物能提供遮蔽效應的區域才有較大的差異。
整體而言，使用本模式可以得到合理的結果，也說明了本模式適合用以描述潰壩波的運動行為以及潰壩波與結構物的交互作用。

A new set of model equations is developed to investigate the interactions of the dam-break wave and porous media. The unsteady three-dimensional macroscopic Navier-Stokes type governing equations with embedding the resistance force formula proposed by Sollitt and Cross (1972) are first derived. The numerical model is then developed by modifying the extended Truchas (Wu, 2004) based on the macroscopic Navier-Stokes type governing equations. In addition, a moving solid algorithm is incorporated to study the effects caused by the gate motion on the evolution of dam-break wave.
To examine whether the present numerical model is suitable for describing dam-break wave hydrodynamics and the interaction of dam-break wave and porous media, present numerical model is validated against the analytical solution of Ritter (1892) and several available experiments. It is found that the numerical results for the surface elevation are in good agreement with available analytical solution and experimental data.
For the dam-break wave moving along a dry bed, the time when the analytical solution matches the computational surface elevation is obtained. We find that bore front will reach the speed of 2√gho in which ho is impoundment depth when the free-slip bottom boundary condition is applied and it reduces to √gho when the no-slip condition is employed. Moreover, the speed of the backward wave front and its reflected wave are found to be √gho. Therefore, the bore front velocity will not be influenced by the finite length of impoundment. Additionally, if we consider the bottom viscous effect, the surface elevation at x=0 will be slightly shifted up about z/ho=0.0176 comparing with the theoretical prediction.
For the dam-break propagating over a wet bed, the different ambient water depths give the different behavior of dam-break wave. Also, the distribution of vorticity and interface between the impoundment and the ambient water will be strongly affected by the different gate motions.
Finally, interaction of the dam-break wave and porous media are investigated. The results show that the flow filed of the dam-break wave will be affected by the porosity of structures. During the impact, the structure with high porosity will suffer smaller net force. Also, the water table in the porous media will be oscillated to reach the maximum of level as z/ho~0.5 The presence of the dam-break wave transmission will reduce the normal force about 70% for the porosity is 0.8 and 20% for the porosity is 20 %. Conclusively, the presence of porous media will greatly reduce the normal impact.
Besides, the present numerical model is also employed to study the three-dimensional partial dam-break wave. The applications to describe the interaction of partial dam-break wave and solid/porous square structures are carried out. Because the partial dam-break wave is weaker than full dam-break wave dose, the obvious discrepancy on the surface elevation and pressure is found at the area where downstream of the solid structure. In other words, the scanning effect provided by the solid structure can greatly decrease the impact by the partial dam-break wave.
Overall, the computational results show reasonably good agreement with the experimental data, suggesting the present model can describe the interaction of dam-break wave and structures well.

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Appendix A Least Square Linear Reconstruction (LSLR) Method
Consider a discrete arbitrary scalar data at cell centroid in the cell. Its first order derivative/gradient on a two dimensional or three dimensional structured/unstructured meshes can be evaluated by a least square method (Barth, 1994). Barth (1994) first did the linear and quadratic reconstruction of discrete data on unstructured meshes by using the least square algorithms. By doing so, the second and higher order accuracy has been found on the highly random triangular meshes. The algorithm of reconstruction of discrete data on unstructured meshes is called "least square linear reconstruction" (LSLR) Method.
In the LSLR algorithm, the discrete scalar data on the neighbor cells at centroid which the subscript denotes the immediate neighbor cell can be approximated from the reference discrete scalar data at centroid in the cell by the Taylor series expansions to find the unknown components of . With the aid of Taylor series expansions, the immediate neighbor discrete scalar data can be approximated as :
(A-1)
Note that the immediate neighbor cells should share at least one vertex with the reference cell (the cell). Only the first derivative terms are retained in the expansion to consider a linear reconstruction.
(A-2)
If there are total immediate neighbors around the reference cell, then minimized the sum of the square of the residual that is defined as over all neighbors immediate can be expressed as:
(A-3)
where and record the scalar data and cell centroid of neighbor cell around the reference cell (the cell), respectively. The subscript could be an integer from to . The following condition for the residuals to be a minimum is that
(A-4)
Let define the number of unknowns for or the dimensionality of the system as . For the cell, Eq. (A-4) will yield the equations in which . If the system is undetermined, if the system is solvable, and if the number of dimensions, the system is over-determined. Generally, the system is over-determined, a minimizing solution must be obtained according to Eq. (A-4). Equation (A-4) can be expressed as the matrix form:
(A-5)
Where ,
(A-6)
and the solution vector ,
(A-7)
Where , , and are , , and components of the spatial gradients. And ,
(A-8)
For a linear system, the least square solutions of Eq. (A-4) can be robustly found by the weighted least square method:
(A-9)
in which is a diagonal by matrix. The diagonal entries in are geometric weight defined as ,
(A-10)
Equation (A-9) yields by linear system which can be easily solved with decomposition method such as LU decomposition.
We remark here that LSLR method can be employed to obtain if it is not known. The reference data in Eq. (A-1) is not necessarily the same as that at each cell centroid. Specifically, it can be anywhere inside the cell or on the face of cell. If the reference value is on the cell face centroid which the subscript denotes the cell face, then Eq. (A-2) can be modified as:
(A-11)
where is the reference data at the cell face centroid in cell. Since the in Eq. (A-11) is unknown, Eq. (A-4) is derived by performing partial derivatives with respect to in addition to the components of . Also, Eq. (A-5) can be modified and rewritten as follows:
(A-12)

(A-13)

(A-14)
Similarly, according to Eq. (A-5) with Eq. (A-12) to Eq. (A-14) can be solved by the weighted least square method with a diagonal by matrix. Then the resulting by linear system can be solved by the conventional direct decomposition methods.
The LSLR method is powerful for the following reasons: First, it can be applied to any mesh topology including one, two, or three dimensional structured and unstructured meshes. To obtain the gradient terms, only the known discrete data values at cell centroid and the physical location of cell centroid are needed. To solve the unknown discrete data values at face centroid, the physical location of face centroid is the only variable needed to be set. Second, there are no constraints when performing the minimization. For example, or norms might also be minimized rather than the norm as above (Ollivier-Gooch, 1997). Third, this method has been demonstrated to provide second (and higher) order accuracy on highly irregular meshes (Barth, 1994). With these reasons, this study adopts LSLR method for solving the gradient of scale variables at cell centroid or the variable at cell face centroid.