分佈式拉格朗日乘數/虛擬區域法模擬雙分散懸浮沉降之行為

Abstract

在本論文中，我們主要研究方向為：Boycott effect、輕、重粒子所構成之雙分散系統。對於二維-固液二相系統，我們考慮液體為黏性且不可壓縮之牛頓流體、固體為圓柱之剛體。根據Glowinski、Pan、Chang等學者所發展之分布式拉格朗日乘數法/虛擬區域（DLM/FD），分析粒子於傾斜容器之沉降行為，有別於傳統分析方法，我們考慮粒子-粒子、粒子-壁面及粒子-流體之相互關係，可詳細地描述每顆粒子之位置、速度及角速度。 首先，對於Boycott effect之問題，已有許多學者透過實驗、理論及模擬分析，且大多數學者皆根據簡單PNK模型，作為預測界面沉降速度之依據。在本論文中，我們將透過更精確地數值計算，考慮不同體積分率（10%, 15%, 20%）及傾斜角度 0°、10°、20°、30°、40°、50°及60°，可得粒子沉降軌跡、界面沉降速度、整體渦流及局部二次渦流於論文之數值結果中，並比較界面沉降與PNK模型之關係，說明低傾斜角度時，流場會產生不穩定之局部渦流，進而影響沉降界面，此時PNK將不適用。 其次，對於雙分散系統之問題，截至目前為止，其數值模擬研究尚未有詳細介紹，為了瞭解此系統之沉降行為，我們考慮輕、重粒子數各50顆，且總體積分率為20%，分別沉降於傾斜角度 0°、20°、30°、40°及50°之情況。利用相同概念，我們亦解釋輕、重粒子於傾斜容器之流場變化。由數值結果可知，初始沉降時，輕、重粒子密度不同，造成傾斜管內產生局部正、負小渦流，此時流場之對流為雜亂且不穩定；當沉降持續發展後，局部渦流逐漸結合成較大之渦流，此時流場之對流為一穩定之順時鐘方向，與容器傾斜方向有關。另外，我們修改PNK模型分別討論輕、重懸浮區重疊及分離之界面沉降。 對於傾斜管沉降之問題而言，在傳統分析上，可知流場內存在一主要整體對流，但對於局部渦流影響界面沉降之部分，卻無法加以說明。因此我們藉由DLM/FD之數值方法，可以成功觀察到局部渦流，並透過本文之研究，可以了解局部渦流於傾斜管之沉降過程中，扮演了一個很重要角色。

In this thesis, we have studied the Boycott effect for heavy particles and bidisperse suspension that contains light and heavy particles. In these two-dimensional solid-liquid two-phase systems, the fluid is a viscous incompressible Newtonian fluid and the particles are rigid disks. We have applied a distributed Lagrange multiplier/fictitious domain method, developed by Glowinski, Pan, Chang, et al., to simulate the motion of particles during the sedimentation process in an inclined closed channel. We have considered the particle-particle, particle-wall and particle-fluid interactions, and calculated all particle positions, settling velocities, and angular velocities via the direct numerical simulation. In the most of study related to the Boycott effect, continuum theory and the PNK model have been used to predict the interface between the clear fluid and the suspension. In this thesis, we have computed the fluid velocity field and the trajectories of particles with different solid fractions (10%, 15%, 20%) and tilted angles (0°, 10°, 20°, 30°, 40°, 50° and 60°). The global convection and local secondary vortexes are observed in our simulation results (in an inclined channel). We have compared our results with the PNK model, and found that the local vortexes affect the interface settling velocity in a low tilted channel. For the bidisperse suspension problem, we have considered a suspension of light and heavy particles of equal size in a tilted channel with the total solid fraction of 20% and the angle of 0°, 20°, 30°, 40° and 50°, respectively. We have calculated the light and heavy particle trajectories and the fluid velocity field via the direct numerical simulation. We have found that smaller local vortexes merge and form a clockwise rotation of a global convection in time. The simulation results have been compared with the prediction of the interfaces of the light and heavy particles, respectively, based on a modified PNK model when both types of particles are either fully mixed or completely separated. In conclusion, although the continuum theory can explain global convection associated with the sedimentation in a tilted channel, it still couldn't describe the local vortex occurrence. In this thesis, we have shown how the local smaller vortexes have a strong effect on the interface settling velocity in a low tilted channel.