以數值方法探討壁面效應對粒子沉降行為之影響

Abstract

對於兩顆粒子間牛頓流體中的交互運動，在雷諾數30~100的範圍間，已更很 多學者透過實驗、理論及模擬分析作一系列的研究，Joseph(1987)等學者探討了此雷諾數區間下粒子運動的基礎控制機制，並以拖曳、接觸、翻滾(DKT)來描述這些交互機制。在本論文中，我們的主要研究方向在於：本研究中將粒子放於L/D=3~7等五種不同寬度之無限長通道，受重力沉降。分別討論兩顆圓形粒子、橢圓粒子與dipole 長體粒子在牛頓流體中的異常沉降行為，此異常沉降行為多發生在黏彈性非牛頓流體中的兩圓形粒子沉降，落後粒子會因受拉曳影響，使得兩粒子接觸而串連，如果流體的彈力夠強則粒子無翻滾與分開之運動。對於此二維的固液二相流系統，我們考慮液體為黏性且不可壓縮之牛頓流體，固體則為剛體圓形粒子、橢圓粒子與dipole 粒子，其密度與液體十分接近。本論文研究中以Navier-Stokes equation 來描述流體行為以及Euler-Newton equation 來描述剛體粒子運動，並利用Glowinski與Pan學者所發展出的分佈式拉格朗日乘數/虛擬區域法(DLM/FDM)來模擬流體與粒子的運動。在本研究中，我們定義臨界雷諾數(Recr)，在雙粒子沉降系統中，此臨界雷諾數為不同寬度通道中能使兩粒子維持串連且垂直沉降的最大雷諾數。此外，為了與雙粒子所串連而成的長體沉降比較，臨界雷諾數在橢圓與dipole 長體粒子的沉降中則定義為能使長體粒子長軸平行於通道中心線並維持在通道中心垂直沉降的最大雷諾數。分析三種沉降系統下不同寬度通道中的臨界雷諾數，我們得到三種系統的臨界雷諾數會隨通道寬度增加而變小。進一步地，我們找出雷諾數之倒數(1/Recr)與通道寬度(L/D)之關係，發現在雙粒子沉降中，兩者之間更良好的線性關係，在橢圓與dipole 粒子中，兩者在較寬通道中的線性關係較為明顯。可以得知在窄通道中，壁面效應較大，即在較低黏滯力、較高慣性作用下，粒子仍可藉由壁面作用力的影響使兩顆粒子保持串連或使長體粒子的長軸平行於中心線並維持在通道中心垂直沉降。在寬通道中，壁面效應減弱，慣性作用增強，粒子則必頇在低雷諾數時，更較高的黏滯力才可以得到類似的結果。比較不同粒子系統的臨界雷諾數，在窄通道中，三種系統臨界雷諾數接近；而在寬通道中，長體粒子所受慣性作用較大，故其能保持垂直沉降的雷諾數範圍較小。

Many researcher have studied the interaction between two particles in a Newtonian fluid at the Reynolds number 30~100. Joseph et al. (1987) studied the basic mechanism controlling the motion and interactions of spherical at this Reynolds number interval. They described these motions as drafting, kissing and tumbling (DKT). When two particles settling in a viscoelastic non-Newtonian fluid, the drafting effect makes the trailing particle accelerate to kiss the leading one, as well as they chain together without tumbling or separating if the effect of elasticity is strong enough. In this thesis, our study focus on the abnormal settling of two particles system and long body, such as ellipse particle and dipole particle due to the gravity in Newtonian fluid in an infinite channel of different width L/D, including 3, 4, 5, 6, 7, respectively, where D is the characteristic length. For two-dimensional solid- fluid two-phase system, we consider the fluid as a viscous and incompressible Newtonian fluid, and the solid as rigid circular particle, elliptic particle, or dipole particle, whose density is slightly heavier than the fluid. In this study, we use Navier-Stokes equation to model the fluid flow and Euler-Newton equation to model the rigid body motion. Furthermore, we use the distributed Lagrangian multiplier/ fictitious domain method(DLM/FDM), which was developed by Glowinski and Pan, to simulate the fluid flow and the particle motion directly. In this thesis, we define a critical Reynolds number (Recr) as the max Reynolds number which can make two particles keep chaining and settling vertically for two particle system. To compare with the settling of the long body formed by two particles, we define the critical Reynolds number for ellipse and dipole as the max number which can make the long axis of the long body parallel to the central line of the channel and settle vertically in the middle of infinite channel. Analyzing the critical Reynolds number of these three systems, we have gotten the result that the critical Reynolds number decreases as the channel width increases. In addition, we found the relation between 1/Recr and L/D. The relation is linear for two particle settling. For ellipse and dipole cases, the relations are linear in wider channel. The simulation results indicate that in the narrow channel, the wall effect is stronger so that with a lower viscous force and higher inertia effect, two disks can chained together and a long body can settle vertically in the middle of the channel with its long axis parallel to the central line of the channel. In wide channel, the wall effect gets weaker, and the inertia effect gets stronger. The particle won’t behave similarly as that in the narrow channel unless the viscous force is large. In comparison of critical Reynolds number among different particle systems, in narrow channel, the three Reynolds numbers of the three systems are close; in wide channel, the inertia effect imposing on long body is larger so that the range of Reynolds number, which can make the particles settle vertically, is smaller.