發展一非結構性網格有限體積法對固體結構及流固耦合計算分析

Abstract

本文研究主要為發展一個非結構性網格有限體積法，應用於計算結構力學及流體力學來探討流固耦合之問題。在結構力學計算中，透過有限體積法，將方程式離散後整理成一組聯立常微分方程式，求解方式分成直接解法與虛擬時間法兩類，再藉由龍格庫塔法或預測修正法求解。結果發現在虛擬時間法下的求解方式較佳，其中又以預測修正解法有較快之收斂速度和較大之時間步階。此方法能有效的運用在典型的二維懸臂樑和固定樑問題，且分析之結果與理論之位移、自然頻率非常相近。由於固體結構的振動，在流場計算中採用移動網格，透過空間守衡的概念來滿足移動網格的質量守衡。方程式的離散同樣採用有限體積法，其中對流項採用上風法和中央差分法混合的高階方法。在處理動量與連續方程式間的耦合，則是藉由先預測再修正的PISO法則處理。在耦合測試問題中，第一個探討的是流體流經一垂直彈性平板之管流，平板因受流場壓力與剪力的影響而產生擺動；第二個探討的是一位於矩形塊體後的水平彈性平板，由於受到此塊體不穩定渦流的影響，造成平板的振動。在此二測試問題中，闡述了結構與流場間相互影響的關係。

A new unstructured-grid finite-volume method is developed for the structural dynamics and fluid-structure interaction. In structural calculations, the governing equations are discretized using the finite-volume method to obtain a system of ordinary differential equations. These system of ODEs are solved by either a direct method or a virtual time method using Runge-Kutta or predictor-corrector methods. It is shown that the virtual time method is superior to the direct method. Incorporating the predictor-corrector scheme with the virtual time method, larger time steps and faster convergence rates are obtained. The methods are tested for cantilever and built-in beams. The results are in good agreement with theoretical analysis in terms of displacements and natural frequencies. Due to the vibration of the structure, moving grids are employed in flow calculations. By incorporating the space conservation concept, the mass is conserved in each cell with moving grids. The equations are discretized also using the finite-volume method. The convection term is approximated by a hybrid scheme which mixes the central and upwind differences. The coupling between the momentum and continuity equations is tackled using the predictor-corrector method of the PISO algorithm. In the tests of fluid-structure interaction, the first case considered is a vertical plate swinging in a channel flow. In the second case, a horizontal plate is placed behind a rectangular block. The vortices caused by the flow over the block result in vibration of the plate. The interaction between the vibration of the structure and the vortex flow of the fluid is clearly delineated.