本文的主要目的是實現文獻[22]所提出的一種人造壓縮性直接施力沉 浸邊界法模擬二維非穩態流體與固體交互作用的行為，此方法結合了 人造壓縮性法和直接施力沉浸邊界法，並採用一種預測-修正策略。 我們依循[22]的主要想法，首先引入懲罰技術去弱化不可壓縮納維爾-史托克方程中的不可壓縮條件，同時引入一個只分布在固體上的虛擬力並將其附加到流體動量方程式上，用以處理沉浸固體上的無滑移邊界 條件。接著我們採用一階隱式尤拉法離散方程組中的時間變數，並使 用顯式一階的方法線性化其中的非線性對流項，然後運用直接施力沉 浸邊界法結合一種預測-修正策略求解時間離散後的方程組。關於方法中的空間離散方式，我們採用交錯網格的二階中央差分格式。本文執行了數個二維非穩態流體與固體交互作用的實驗來說明此演算法的效能，數值模擬結果顯示這個簡單的人造壓縮性直接施力沉浸邊界法對於流 固耦合問題可以取得相當合理的數值結果。;The main purpose of this thesis is to implement an artificial compressibility-immersed boundary method with direct forcing proposed in [22] for simulating 2-D unsteady flows interacting with rigid solid objects. This approach is based on the artificial compressibility method and the direct-forcing immersed boundary method combined with a prediction-correction strategy. Following the ideas in [22], we employ the penalty technique to weaken the incompressibility condition in the incompressible Navier-Stokes equations and introduce a virtual force distributed on the whole solid object and imposed to the fluid momentum equations to accommodate the no-slip boundary condition at the immersed solid boundary. We then use the first-order implicit Euler scheme to discretize the temporal variable in the resulting system of equations and apply the explicit first-order approximation to linearize the nonlinear convection term. After that, we employ a direct forcing immersed boundary method with a prediction-correction strategy to solve the system of time-discretized equations. For the spatial discretization in this approach, we take the second-order central differences on the staggered grids. We illustrate the performance of the algorithm by performing several 2-D numerical experiments of unsteady flow interacting with solid object. From the numerical results, we find that this simple artificial compressibility immersed boundary method with direct forcing can achieve reasonable results for 2-D fluid-solid interaction problems.
