本文發展數值模式求解二維非穩態雷諾平均方程式 (RANS)及紊流模式( model)，以模擬不同波浪入射至真實底床上的海堤後，在海堤上溯升及越波的現象。本文應用等位函數法 (Level Set Method)及質點等位函數法(Particle Level Set Method)求出波浪溯升及越波過程中，複雜的自由液面變化及其波形演變。此外，為了能夠在卡氏座標網格下滿足不規則結構物上的邊界條件，本文提出了新的邊界處理方法，稱為質量守恆邊界法(Mass-Conserved Boundary Method，簡稱MCBM)，讓數值模式能在卡氏座標網格下模擬波浪與不規則結構物之交互作用。有別於過去研究中以切割網格法(Cut-cell method)計算質量通率所造成的不便，MCBM乃是採用虛擬網格法(Ghost Fluid Method)分別求算平行及垂直固液交界面的邊界流速，以同時滿足固液邊界網格中連續方程式與非穿透及無滑移的固體邊界條件。本模式藉由等位函數描述固體邊界的優勢，可容易且正確地獲得固液交界面附近的邊界流速。
為了確保數值模式的準確性，本文利用多項前人實驗或理論的研究成果逐一驗證本模式的計算結果，其中包括缺口圓盤問題 (Zalesak’s problem)、拉穴流問題 (cavity flow problem)及均勻流通過圓柱等。經由數值模式模擬波浪於不規則結構物上的溯升及越波，可觀察出數值結果與實驗結果在自由液面變化、流體速度及越波量的比對上相當一致。
在驗證數值模式的準確性後，本文進行兩項研究。第一，為了瞭解孤立波溯升時的流場特性，數值模式模擬不同孤立波入射至不同坡度的斜坡時自由液面的演變，並探討流場、波壓及底床剪應力等物理現象。由斜坡堤趾前方底床邊界層內的流場可見，當孤立波溯升速度減緩時，邊界層外緣的流體會朝向岸方向流動，而邊界層內的流體則朝離岸方向前進。同樣的現象也會發生在斜坡面上，此靠近底床的逆向流稱為水下逆流 (undertow)。此外，本文亦同時探討在溯升及溯降期間，孤立波在斜坡附近所造成的波壓及底床剪應力。
第二，為了預測颱風季節期間風浪在實際海堤附近溯升與越波的情形，本文模擬孤立波入射至宜蘭、花蓮及台東等地區的真實海堤。文中先利用統計方法求得該地區於五種不同重現期時的預測暴潮位及預測最大波高作為入射波條件。數值模擬結果顯示，孤立波於實際地形及海堤上前進時，會產生碎波、淺化、溯升、溯降及越波等現象。為了將結構物後方的溢淹結果量化，本文亦探討在不同重現期及相異地區下的越波量時序列分佈。本模式可應用於預測各種波浪所造成的越波量，作為風險評估的參考。

In this study, numerical schemes were developed to solve the unsteady, two-dimensional Reynolds Averaged Navier-Stokes (RANS) equations and the turbulence model ( model) for simulating run-up and overtopping of waves at real seawalls. The level set method and the particle level set method were adopted to capture the complex free surface and its evolution involved in the run-up and overtopping processes. A novel Cartesian grid method, called the Mass-Conserved Boundary Method (MCBM), was proposed to satisfy the boundary conditions at the irregular boundaries under the Cartesian grid system. This will enable the simulation of wave and irregular structure interaction easily.
To avoid the complexity of the cut-cell method, the ghost fluid method was employed in MCBM to determine the boundary velocities in the tangential and normal directions of the fluid-solid interface. Thus, the impermeable, no-slip boundary conditions are enforced at the fluid-solid interface, and the continuity equations are guaranteed at the cells containing the immersed boundary. By taking the advantage of the level-set representation of the solid boundary, the boundary velocities near the fluid-solid interface can be evaluated easily and precisely.
To ensure the accuracy of the numerical model, several tests have been made and numerical solutions have been compared with either analytical solutions or experimental data. The tested cases include the Zalesak’s problem, cavity flow problem, and a uniform flow past through a circular cylinder. Furthermore, the free-surface profiles and the flow velocities on a sloping beach and the discharge of overtopped flow at a seawall obtained by the present numerical model compared favorably with the experimental data.
After having verified the accuracy of the present numerical model, two major topics were investigated in this study. First, the numerical model was applied to simulate the flow field on a sloping beach for various incident solitary waves. The free-surface evolution, the flow fields, the wave pressure on the sloping beach and the bed shear stress were discussed to reveal the characteristics of the wave and flow fields. After examining the boundary layer flows in front of the beach toe, we found that the fluid particles outside the boundary layer move in the direction of wave propagation, while the fluid particles inside the boundary layer move in the reversed direction as the wave run-up decelerates. This type of reverse flow occurs also near the bottom of the sloping beach and was usually referred to as undertow. The total pressure on the slope and the bed shear stress during the run-up and run-down phases were also calculated.
Second, to provide information on the wave run-up and overtopping at seawalls during the typhoon period, this model was applied to determine the surface evolution as a solitary wave attacks real seawalls at Yilan, Hualien and Taitung of eastern Taiwan. The storm surge and the maximum wave height were obtained from statistical analysis based on five return periods. These data were adopted as the still water depth and the local incident waves in the numerical computation. Wave breaking, shoaling, run-up, run-down and overtopping were all observed in the numerical results. To estimate the overflow rate, time series of the overtopping discharge were analyzed under various topography and return periods. The wave model has been demonstrated to be a promising tool for forecasting the wave overtopping. This information can be used to appraise the risk of coastal structures from wavy damage.

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