不可壓縮黏性流中圓柱之終端速度與地板耦合距離模擬研究

Abstract

本實驗主要利用陳立杰學長所開發的沉浸邊界法流固耦合模擬軟體，研究二維圓柱在不可壓縮黏性流場中，圓柱受重力自由沉降時的終端速度。分析各個物理參數(通道寬度、流固密度比與阿基米德數)對終端速度的影響，並歸納出終端速度對各項物理參數的經驗方程式。同時探討圓柱以終端速度接近地板時的地板耦合距離，分析通道寬度、密度比以及阿基米德數對地板耦合距離的影響，最後歸納出地板耦合距離對各項物理參數的經驗方程式。 圓柱的終端速度研究發現，終端速度對密度比與解析解具有相似的形式，而意外發現終端速度對阿基米德數在半對數圖中，具有很好的線性關係。最後擬合出終端速度的經驗方程式，此經驗方程式的方均根誤差為0.039。在地板耦合距離的研究，發現地板耦合距離對密度比與阿基米德數具有相似的影響，在全對數圖中皆有很好的線性關係，而有趣的發現地板耦合距離隨著通道寬度的增加而增加，但同時終端速度亦隨著通道寬度的增加而增加，而有終端速度越快而的地板耦合距離越大的現象，亦歸納出地板耦合距離經驗方程式，其方均根誤差為0.089。最後結合上述兩個經驗方程式討論史托克數對地板耦合距離的影響。 (適用範圍：Re: 0.14~126.94, 側牆寬度: 4D~16D) D: 圓柱直徑

We investigate the terminal velocity and coupling distance of a free falling cylinder in viscous and incompressible flow, with the help of software simulator based on the immersed boundary method from previous work. We analyze the influence different physical parameters (Ex. density ratio, channel width, Archimedes number) have on terminal velocity and try to integrate an empirical equation of terminal velocity. We also observe the relation between coupling distance and above mentioned physical parameters in a simulation where the cylinder approaches the floor at terminal velocity, and integrate its empirical equation. From these simulations, we observe that the relation between terminal velocity and density ratio remains similar to the analytical solution, even in a situation previously thought unfit for the analytical solution (at a higher Reynolds number.) We also observe that the terminal velocity and the Architecture number almost have a perfect linear relationship. The fitted empirical equation of terminal velocity has a root-mean-square error of 3.9%. Regarding coupling distance, we can observe from the log-log graphs that density ratio and Architecture number have the same linear relationship with coupling distance. More importantly, we find that coupling distance increases with channel width, which induces the positive correlation between terminal velocity and coupling distance. The fitted empirical equation of coupling distance has a root-mean-square error of 8.9%. At last, we combine the empirical equations to analyze the relation between Stokes number and coupling distance.