利用沉浸邊界法於三維多球體沉降的數值模擬

Development of an immersed boundary method to the numerical simulation of settling of multiple spheres

簡鈺霖(Yu-Lin Chien)

指導教授 ： 周逸儒

國立臺灣大學/工學院/應用力學研究所/碩士(2019年)

https://doi.org/10.6342/NTU201903754

摘要

本研究以拉格朗日-尤拉法的方式使用沉浸邊界法(immersed boundary method)於直接數值模擬(direct numerical simulation)中，模擬多顆球體於流體中的運動行為。其目的主要是驗證並改善直接施力(direct forcing)沉浸邊界法[5]，並採用拉格朗日點向內收縮方法[8]，且同時考慮軟球及硬球碰撞模型，來模擬單球體和多顆球體於流場中的複雜運動行為。驗證和改善可區分為兩部分，第一部分是模擬穩態條件下靜止球體於均勻流場所受阻力並與前人模擬數據結果做比較。其中直接施力沉浸邊界法的力回授採用狄拉克正規化脈衝函數，此方法會增加阻力係數值，由本研究結果顯示球體直徑與網格大小的比值為8、雷諾數Re = 50 時，模擬出來的阻力係數為1.805，比較文獻[24]模擬出來的阻力數值1.574來的大，故採用於拉格朗日點向內收縮的方法，此方法在不同流場網格解析度下，可以得到一Re、內縮值和網格解析度的修正式，本研究結果指出在同樣解析度下，Re = 50時可以得到的阻力係數為1.582，此值趨近於上述文獻模擬結果。而第二部分則是模擬多顆球體於流場中的運動行為，首先模擬和驗證單一球體自然沉降行為，而後模擬多顆球體於不同高度進行自由沉降，讓多個球體產生碰撞，最後參考前人文獻[8]評估此不同條件下的運動行為。

關鍵字

直接數值模擬 ； 沉浸邊界法 ； 球體運動

並列摘要

This study presents the direct numerical simulation with an immersed boundary method (IBM) to simulate the motion of multiple spheres in flow field. The purpose of this study is to verify and improve the direct forcing immersed boundary method [5] and to adopt the inward contraction of Lagrangian points [8]; Simultaneously, we also consider the soft-sphere and hard-sphere collision model to simulate the collision behavior between sphere to sphere and sphere to wall respectively. The verification and improvement process consists of two parts. The first is to simulate the stationary sphere of steady state in the uniform flow and compare with the results of the simulation data of previous literature. The forced calculation of direct forcing immersed boundary method uses the regularized Dirac delta function which increases the drag coefficient value. When the results of this study show that the ratio of the sphere diameter to the grid size is 8 and Reynolds number (Re) is 50, the simulation is performed that the drag coefficient (Cd) is 1.805 which is larger than the literature [24]. Therefore, this study uses inward contraction of Lagrangian points. It can obtain a correction formula in form of Re, the contraction value and the grid resolution in different flow field. The simulating results of using this method indicate that the Cd can be obtained with Re = 50 is 1.582 at the same grid resolution. This value approaches the result of the literature [24]. The second part is to simulate the motion of multiple spheres in the flow field. This study is to simulate and verify the natural settlement behavior of a single sphere firstly, and then it simulates the settling of multiple spheres at different heights. The process of settling generates collision of spheres by variation of the flow field and the spherical velocity.

並列關鍵字

Direct numerical simulation ； Immersed boundary method ； the motion of spheres

參考文獻

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[3] Roma, A. M., Peskin, C. S., & Berger, M. J. (1999). An adaptive version of the immersed boundary method. Journal of computational physics, 153, 509-534.

[4] Fadlun, E. A., Verzicco, R., Orlandi, P., & Mohd-Yusof, J. (2000). Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. Journal of computational physics, 161, 35-60.

[5] Uhlmann, M. (2005). An immersed boundary method with direct forcing for the simulation of particulate flows. Journal of Computational Physics, 209, 448-476.