Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1259840, 11 pages

https://doi.org/10.1155/2017/1259840

## Computer Simulation of Three Particles Sedimentation in a Narrow Channel

Institute of Fluid Mechanics, China Jiliang University, Hangzhou 310018, China

Correspondence should be addressed to Deming Nie

Received 3 December 2016; Revised 3 February 2017; Accepted 28 February 2017; Published 15 March 2017

Academic Editor: Claudia Adduce

Copyright © 2017 Rongqian Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The settling of three particles in a narrow channel is simulated via the lattice Boltzmann direct-forcing/fictitious domain (LB-DF/FD) method for the Reynolds number ranging from 5 to 200. The effects of the wall and the Reynolds number are studied. It is interesting to find that at certain Reynolds numbers the left (right) particle is settling at 0.175 (0.825) of the channel width irrespective of its initial position or the channel width. Moreover, numerical results have shown that the lateral particles lead at small Reynolds numbers, while the central particle leads at large Reynolds numbers due to the combined effects of particle-particle and particle-wall interactions. The central particle will leave the lateral ones behind when the Reynolds number is large enough. Finally the effect of the Reynolds number on the trajectory of the lateral particles is presented.

#### 1. Introduction

Particle sedimentation is one of the most common phenomena in nature and industrial production, such as fluidized beds, gas filtration technology, and sand deposition, which always presents a classical problem in hydrodynamic fields. It is important to understand the particle-particle interactions as well as the particle-fluid interactions to provide an insight into the behavior of particles suspended in fluids. To study the mechanism of particle sedimentation, a direct numerical simulation (DNS) is usually adopted. In a DNS method, the fluid flow and particle motion are coupled to study the dynamics of individual particles suspended in fluids, which is the highest-resolution numerical method without any empirical model.

It is very important to deal with the curve boundary of particles in the simulation via the DNS method. Roughly speaking, there are two kinds of schemes which are usually adopted to simulate the particulate flows: the immersed boundary method [1, 2] and the fictitious domain method [3, 4]. The essence of these two schemes is similar. In both schemes, a Lagrangian mesh is used to deal with the solids while an Eulerian mesh is used to treat the fluids. The nonslip condition is enforced by applying a forcing term into the Navier-Stokes equations. In comparison with conventional numerical methods such as the finite volume method and finite-element method, the computational efficiency is improved significantly because there is no requirement of remeshing procedure for the above-mentioned methods. The lattice Boltzmann direct-forcing/fictitious domain (LB-DF/FD) method, which has been proposed by Nie and Lin [5], has been successfully applied to simulate the motion of two-dimensional and three-dimensional particles [6–8].

So far there exists some research work which focused on the study of the settling particle. Feng et al. [9] used the DNS method to simulate the sedimentation of circular and elliptical particles in a Newtonian fluid. They revealed five different falling patterns depending on the Reynolds number: steady motion with and without overshoot and weak, strong, and irregular oscillations. Feng et al. [9] also reported the occurrence of Drafting-Kissing-Tumbling (DKT), which is an important phenomenon in the particle sedimentation. Xia et al. [10] adopted a lattice Boltzmann method (LBM) to simulate the sedimentation of an elliptical particle in a narrow channel. They found five distinct modes according to the blockage ratio: oscillating, tumbling along the wall, vertical sedimentation, horizontal sedimentation, and an inclined mode. The situation may become more complex when multiple particles are taken into account. Aidun and Ding [11] simulated the sedimentation of two circular particles in an infinite channel by using LBM. Their work showed that the particles go through a complex transition to reach a low-dimensional chaotic state [11] due to the interaction between particles and the interaction between particles and walls. El Yacoubi et al. [12] simulated the sedimentation of multiple particles at an intermediate Reynolds number by using an immersed interface method. Some interesting falling patterns of particles were observed in their study. The middle particle is always leading in the case of odd-numbered particles, and the falling pattern is concave-up, whereas it is concave-down in the case of even-numbered particles [12]. Similarly, Nie et al. [13] studied the sedimentation of multiple particles via the LB-DF/FD method. They focused on the effect of the interaction between particles on the DKT process. Wang et al. [14] used the LBM to simulate the DKT phenomenon of two nonidentical particles. They [14] demonstrated that the effect of the diameter difference on the DKT process was significant. The two particles may not undergo the DKT process if the smaller particle is initially located above the larger one [14]. Recently, Nie et al. [15] reported the grouping behaviors of multiple particles settling along their line-of-centers in a narrow channel. They showed that the settling particles separated into several groups resulting from the particle-particle interaction, with each group settling at the same velocity. Furthermore, their work demonstrated that this type of grouping behavior strongly depended on the number of particles and the Reynolds number [15]. Verjus et al. [16] studied the sedimentation of two circular particles in a narrow channel and revealed some new features of the settling behavior of particles. They established the link between the terminal Reynolds number and the nondimensional driving force which displays various behaviors. More recently, Amiri Delouei et al. [17] developed a direct-forcing immersed boundary non-Newtonian lattice Boltzmann method to simulate the motion of particles in shear-thinning and shear-thickening fluids. They studied the DKT motion in non-Newtonian fluids for the first time, which shows that increasing the shear-thickening behavior of fluid leads to a significant increase in the kissing time.

However, the attention paid to the effect of particle-wall interaction on the motion of particles is very limited. It is unclear how the particles interact with each other when the hydrodynamic interaction between particles as well as particle-wall interaction is taken into account. This motivates the present work, because it is expected that the effect of walls on the falling pattern of particles is significant. Furthermore, this problem could be more complicated and unexpected if the effect of inertia is taken into account. We may wish to inquire whether the middle particle is always leading in the case of odd-numbered particles. We may also wish to know under what circumstances the particles are falling together in the channel. Therefore, the purpose of this work is to simulate the sedimentation of three circular particles in a vertical channel by using the LB-DF/FD method. We focus on the effects of the Reynolds number as well as the walls on the falling pattern of particles and intend to provide a better understanding of the combined effects of particle-particle and particle-wall interactions on the stable or unstable motion of particles by considering the effect of inertia. We also aim to state that the particle sedimentation system is simple but rich in dynamics and worthy of extensive examination.

#### 2. Numerical Model

The details of the LB-DF/FD method are presented elsewhere [5]. The physical model is shown in Figure 1, where the computational region is , and the density and diameter of the particles are and , respectively. The distance between the lateral particles (the left particle and the right one) and the side wall is . The three particles in a horizontal arrangement with zero initial velocity are released in a vertical channel. The velocity scale in the simulations is expressed as follows: where is gravity and is the density ratio of solid particles and fluids, which is . Therefore, the time scale and Reynolds number are expressed as and , respectively, where is the kinematic viscosity of the fluid. For convenience of description, some nondimensional parameters are introduced: , , and . The horizontal and vertical coordinates of particles are expressed as follows: and . The parameters are chosen as follows: , , and (in lattice unit). The height of the channel is fixed at . The simulation starts at away from the top wall and ends when the distance between the center particle and the bottom wall is . For convenience no-slip boundary conditions are set on all four fixed walls of the domain. This avoids specifying the far-field boundary conditions on a finite computational domain. The arrangement of Lagrangian points inside the circular particle is shown in Figure 2, with one point at the particle’s center and points on the th ring.