二維剛體與流體雙向耦合運動模擬工具之研發

Abstract

流固雙向耦合是工業應用及大自然現象中常遇到的課題，複雜而難解。本研究旨在開發一個二維流場數值模擬工具，其可計算任意形狀之剛體在不可壓縮黏性流體中之運動行為。使用的空間離散方法是具幾何彈性且準確性階數易於調整之寬頻元素法(spectral element method)，並搭配使用沉浸邊界法(immersed boundary method)以及懲罰模型(penalization model)來捕捉固體運動及滿足流固邊界條件。為因應剛體可為任意形狀之要求，除一套不隨時間更改的尤拉(Eulerian)全域網格供解全域方程式外，我們另備一套拉格朗(Lagrangian)網格固定於固體上，再結合本文所提出之「分胞法」(cut-cell method)，可合理並在滿足動量守恆條件下進行內差，以取得流固邊界附近之物理量(速度、密度、壓力等)，且有效率完成任何物理量在固體面積內之積分。 文中我們測試有圓柱、橢圓柱、方柱、三角柱以及棒狀物在重力作用下之自由漂移行為。在圓柱的模擬中我們發現在低雷諾數且固流密度比低時，擁有相當準確的結果；準確性隨雷諾數或固流密度比增高而降低，穩定性亦然。橢圓柱、方柱、三角柱於低固流密度比下模擬所得之運動行為與前人觀察到的現象皆相當吻合。棒狀物的模擬因為固流密度比高，則有不同的結果。總結初步模擬方法可行，未來研究仍須朝準確性及穩定性的改善進行。

Complicated fluid structure interaction (FSI) problems commonly appear in industrial applications and in the nature. This work is aimed at developing a 2D numerical tool for simulating the two-way interactions between a rigid body of arbitrary shape and a viscous incompressible fluid flow. We apply the spectral element method for the spatial discretization because of its geometric flexibility and its high accuracy. In order to capture the arbitrary shape of the rigid body, a Lagrangian grid is generated and attached to the rigid body in addition to an Eulerian grid for the whole flow field. The no-slip boundary condition at the rigid body surface is enforced by the penalty method, in which a penalization term is added in the momentum equation inside the solid domain. Besides, we propose a so-called “cut-cell method” to calculate the FSI force. This method guarantees the momentum conservation and enhances the efficiency of the integration over the solid domain. The freely-falling motion of a circular, elliptic, square, triangular, or rod type cylinder under gravity is simulated. In the circular cylinder case, the simulation results confirm the validity and the accuracy of the presently proposed simulator at low Reynolds numbers and low solid-to-fluid density ratios. The elliptic cylinder case shows the same phenomenon observed by others. The simulation results of the elliptic, square, and triangular cylinder cases at low density ratios are also in accordance with those in the literature. The simulation result of the rod-type cylinder case however does not agree with the result in the literature, mostly probably because it has a density ratio as high as 2.