起伏浮體流體動力之二維高階小板分析法

The Analysis on the Hydrodynamic Coefficients of the Heaving Floating Body Using the Two Dimensional High Order Panel Method

以往求解具自由液面浮體流體動力係數之相關問題，多以低階方式來離散物體幾何、源點強度和勢流等，在有限的線段離散下，無法真實地呈現其物理性質，為了不失去其特性，本文透過二維高階小板法建立浮體幾何，並求解起伏運動浮體之流體動力係數。本文將物體表面的源點強度分佈導入B-Spline概念，在求解邊界值問題中，基底函數為已知，其控制點為待求解之變數，當邊界值問題求解後，透過源點強度控制點可計算出源點強度、勢流和流體動壓力；進而可得到其流體動力係數─附加質量與阻尼。在邊界值問題中，必須將積分式離散，積分式當中包含了B-Spline曲線中的基底函數，本文則使用高斯積分的數值技巧來處理。高階小板法中，物體幾何必須以曲線描述，然而一般問題常以多點資訊描述而非曲線。假設待解物體的形狀變化不大，本文在已知幾何點位資訊下，以局部差分技巧重新建立幾何資訊，使其建立的曲線必會通過原本的幾何點位。在本文中，探討起伏運動浮體的流體動力係數，並與以往之方法比較，而離散積分所使用數值技巧中（高斯積分），透過數值實驗有找出適當的源點強度控制點數量，且有一定的計算準確性。 Traditionally for solving the hydrodynamic coefficients of a floating body, the concerned quantity distributions of the model are discretely approximated, which usually cannot get the generality sufficiently. Thus, the paper applies the high-order panel to formulate geometry and concerned quantity in spline sense without losing generality. In the present study, we cast the source intensity distribution over body in B-Spline sense. Since that, source intensity is transformed to be formulation related to product of basis function and control points. Then the basis function is well defined and unknown control points are solved by a boundaryvalue problem (BVP). After solving the BVP, the intensity of control points would lead to the solutions of source intensity, velocity potential and hydrodynamic pressure on body. Ultimately, the relevant hydrodynamic coefficients, i.e. added mass and damping coefficient, can be obtained. Besides, integration of the B-Spline source intensity in the present BVP is performed by the Gaussian quadrature technique. To implement high-order panel, the spline information for body geometry is essential. If the shape of target geometry is simple and its variation is small, we can apply the local interpolation technique, to construct a B-Spline curve precisely to pass through a set of geometric data. Through the comparisons with the computation results by tradition model, we can conclude that present high-order panel method works well through the limited control points in the spline intensity and integration point in the Gaussian quadrature technique.