IBM-LBM-DEM Study of Two-Particle Sedimentation - MDPI

Citation: Li, X.; Liu, G.; Zhao, J.;

Yin, X.; Lu, H. IBM-LBM-DEM Study

of Two-Particle Sedimentation:

Drafting-Kissing-Tumbling and

Effects of Particle Reynolds Number

and Initial Positions of Particles.

Energies 2022, 15, 3297.

https://doi.org/10.3390/en15093297

Academic Editor:

Mehrdad Massoudi

Received: 31 March 2022

Accepted: 27 April 2022

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energies

Article

IBM-LBM-DEM Study of Two-Particle Sedimentation:Drafting-Kissing-Tumbling and Effects of Particle ReynoldsNumber and Initial Positions of ParticlesXiaohui Li 1, Guodong Liu 1,*, Junnan Zhao 1, Xiaolong Yin 2 and Huilin Lu 1

1 School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China;[email protected] (X.L.); [email protected] (J.Z.); [email protected] (H.L.)

2 Department of Petroleum Engineering, Colorado School of Mines, Golden, CO 80401, USA; [email protected]* Correspondence: [email protected]

Abstract: Particle sedimentation is a fundamental process encountered in various industrial applica-tions. In this study, we used immersed boundary lattice Boltzmann method and discrete elementmethod (IBM-LBM-DEM) to investigate two-particle sedimentation. A lattice Boltzmann method wasused to simulate fluid flow, a discrete element method was used to simulate particle dynamics, and animmersed boundary method was used to handle particle–fluid interactions. Via the IBM-LBM-DEM,the particles collision process in fluid or between rigid walls can be calculated to capture the informa-tion of particles and the flow field more efficiently and accurately. The numerical method was verifiedby simulating settling of a single three-dimensional particle. Then, the effects of Reynolds number(Re), initial distance, and initial angle of particles on two-particle sedimentation were characterized.A specific focus was to reproduce, analyze, and define the well-known phenomenon of drafting-kissing-tumbling (DKT) interaction between two particles. Further kinematic analysis to define DKTis meaningful for two-particle sedimentation studies at different particle locations. Whether a pair ofparticles has experienced DKT can be viewed from time plots of the distance between the particles(for kissing), the second-order derivative of distance to time (for drafting), and angular velocities ofparticles (for tumbling). Simulation results show that DKT’s signatures, including attraction, (near)contact, rotation, and in the end, separation, is only completely demonstrated when particles havenearly vertically aligned initial positions. Hence, not all initial positions of particles and Reynoldsnumbers lead to DKT and not all particle–particle hydrodynamic interactions are DKT. Whetherparticle–particle interaction is attractive or repulsive depends on the relative positions of particlesand Re. Collision occurs when Re is high and the initial angle is small (<20), almost independent ofthe initial distance.

Keywords: lattice Boltzmann method; immersed boundary method; discrete element method;sedimentation; drafting-kissing-tumbling

1. Introduction

Particle sedimentation exists in many natural and engineering systems and is a clas-sical problem in fluid mechanics and energy science. Resolved simulation of particlemovement in sedimentation is of great importance for in-depth understanding and mod-eling of particle sedimentation [1,2]. While both experiments and resolved simulationsmay be used to investigate the complex particle-fluid dynamics in sedimentation, informa-tion that can be extracted from experiments is usually limited to particles’ positions andvelocities and the velocity of the fluid phase when particle imaging velocimetry (PIV) isused. Particle-resolved numerical simulation, on the other hand, can provide many moredetails and is more suitable for studying the details of the flow field and the history offluid-particle acceleration.

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Energies 2022, 15, 3297 2 of 20

Sedimentation of one, two, and several particles has long been used to test the ac-curacies of particle-resolved simulations [3]. When particles are settling with moderateReynolds numbers, their pairwise drafting-kissing-tumbling (DKT) interaction, first dis-covered experimentally by Fortes et al. [4], is classical for particle-resolved simulations toreproduce. Here, the Reynolds number is defined as Re = ud/ν where u is particle velocity,d is particle diameter, and ν is the kinematic viscosity of the fluid. Feng et al. [5] useda two-dimensional finite element method to simulate the sedimentation of circular andelliptic particles in a Newtonian fluid and put forward five modes of sedimentation with Rebetween 0 and 600. Patankar et al. [6] simulated settling of two two-dimensional particlesusing a distributed Lagrange-multiplier-based fictitious-domain method (DLM). Followingthe simulation conditions of Patankar et al., several research teams used different numericalmethods to simulate DKT. Niu et al. [7] and Feng et al. [8] used an immersed boundarymethod-lattice Boltzmann method (IBM-LBM) scheme. The IBM coupled fluid and par-ticle motions and LBM was used over the entire computational domain. Jafari et al. [9]simulated DKT using an LBM combined with a smoothing-profile method. Wang et al. [10]simulated DKT using a two-dimensional multiple-relaxation-time (MRT) LBM, in whichthe solid boundary was explicitly implemented. The no-slip boundary condition was recov-ered using the bounce-back method of Ladd and Verberg [11]. Hu et al. [12] developed afinite-volume LBM with modified fluid–particle momentum exchange and simulated DKT.Cao et al. [13] studied the influence of the initial angle and the distance on the interactionbetween two spherical particles in a 3D channel by using the lattice-Boltzmann methodwith discrete external boundary force and pointed out that two particles existed in threestates, named repulsion, attraction, and transition (Re ≈ 100), and the smaller the initialangle the more likely the interaction is a repulsion. Liu et al. [14] proposed an improvedcurved boundary condition (CBC) for two-particle sedimentation in their latest study andcompared their results with Niu et al. [7] and Hu et al. [12]. The overall agreement wasgood, except that the particle separation distance in Liu et al. [14] was slightly higher thanthose in Niu et al. [7] and Hu et al. [12]. The above studies on two-particle sedimentationwere mostly based on the conditions of Patankar et al. [6], and the results of these studieswere quite close.

The dynamics of particles in two-dimensional simulations and that in three-dimensionalsimulations are different. Nie et al. [15] proposed that the extra dimension allows dif-ferent patterns of particle motion and new features of fluid–particle system to exist.Three-dimensional numerical simulation and limited experimental study presented byDash et al. [16] on the sedimentation of two inline spheres showed that 2D particles settlingin narrow channel experienced repeated DKT process that was not revealed in 3D study.Many scholars have made great efforts to study the influencing factors of the interactionbetween particles in the process of particle sedimentation. Among which, Dash et al. [16] be-lieved that normalized trajectories, settling velocities and hydrodynamic force coefficientsof sediment spheres are independent of Re (in the range of 10–60). It is also believed thatparticle size was a significant factor [15,17,18]. Nie et al. [19] found that the effect of particledensity ratio cannot be ignored due to the effect of inertia on the motion of the particles.Studies have also been carried out to characterize the role of various influencing factors ofthe settlement of two particles, including thermal characteristics of particles [18], particlesurface slip [20], cohesive forces during particle–particle interactions [21,22], and adhe-sion [23]. Liu et al. [24] proved that the settlement process is dominated by the viscous effectwhen Re is less than 60 and dominated by vortex interaction when Re becomes greater.

From this review, we note that in most previous studies on DKT, researchers paidattention to the properties of the particles on the interaction between particles. However,the dynamics of DKT are not only controlled by density ratio and Re, but also the initialpositions of the particles. The combined effect of the initial positions, density ratio, and Rerepresenting particle inertia and fluid viscous effects has not been thoroughly studied. Inaddition, as a classical phenomenon of the interaction between two particles, DKT definesthe motion process of two particles in tandem. Further kinematic analysis to investigate

Energies 2022, 15, 3297 3 of 20

similar DKT at any particle location is necessary to develop a DKT definition applicable todifferent particle locations.

From the research scale of particle fluid two-phase flow, there is macro-scale continuumtheory, mesoscale lattice Boltzmann method, and micro-scale molecular dynamics. Theconsumption of computing resources increases sharply with the decrease of the researchscale. The traditional numerical simulation used the solving of the Navier-Stokes equationcoupled with DEM to study the motion of particles in a fluid [25,26]. A recent studythat used this method was by Karvelas et al. [27]. They used the Navier-Stokes equationto calculate the fluid phase together with a Lagrangian model and track the particles inthe discrete phase to study the aggregation formation and magnetic driving of sphericalparticles. The LBM developed in recent years is different from the traditional method. Ithas both the efficiency in the order of macro hydrodynamics and the accuracy in the orderof micro molecular dynamics. Meanwhile, this method has a powerful parallel processingability and can improve the computational efficiency of the numerical simulation. Immersedboundary method of fluid–structure interaction allows for nonconforming discretizationsof the fluid and structure. This avoids the frequent re-meshing that is required by thebody-conforming discretization method [28]. In the IBM, the effect of a solid boundaryon fluid flow is reproduced by distributing force terms to the flow field, and the solidboundary is represented discretely by the Lagrange points. The flow equation is still solvedover the entire computational domain [29]. As a simple and efficient boundary treatmentmethod, IBM has a natural affinity with LBM due to the use of Euler nodes to simulate theflow field. Via the discrete element method (DEM) coupled with the IBM-LBM method, thecollision process between fluid or rigid walls can be calculated to capture the informationof particles and flow field more efficiently and accurately.

In this paper, we used two methods, LBM with an explicitly implemented solidboundary over which bounce-back is exercised (LBM-BB) and IBM with LBM and DEM(IBM-LBM-DEM), to simulate DKT of three-dimensional particles. In Section 2, we pre-sented the details of these two numerical methods. IBM-LBM-DEM coupling is realized bycompiling open-source codes. The main program takes the LBM C++ library (Palabos) andcalls the DEM open source library (LIGGGHTS) to solve the dynamics of fluid and that ofthe particles. In Section 3, we used LBM-BB and IBM-LBM-DEM to simulate single-particlesedimentation and DKT. By reproducing known and published results, the accuracy of thetwo methods was proved. In Section 4, a specific focus was to reproduce, analyze, and de-fine the well-known phenomenon of drafting-kissing-tumbling (DKT) interaction betweentwo particles. Furthermore, sedimentations of two particles were initiated at differentpositions and angles with different Re numbers. Section 5 summarized the contributions ofthis work.

2. Model Descriptions2.1. Lattice Boltzmann Method

In LBM, the fluid domain is divided into discrete Cartesian grids. In this paper, thelattice BGK (LBGK) method proposed by Chen and Qian [30,31] is used. The discretevelocity model is D3Q19, and the half-way bounce back method described in Ladd andVerberg [10] is used at the wall boundary.

In this paper, the evolution equation for the discrete velocities is:

fi(x + ciδt, t + δt)− fi(x, t) = − 1τ

[fi(x, t)− f eq

i (x, t)]

(1)

where fi(x, t) is the velocity distribution at spatial position x and time t in the direction i,τ is the relaxation time, Ω f =

(f eqi − fi

)/τ is the fluid collision operator in BGK model,

f eqi (x, t) is the equilibrium velocity distribution at x and t in the direction of i, ci is fi’s

corresponding lattice velocity, δt is the streaming time step, and fi(x + ciδt, t + δt) is the

Energies 2022, 15, 3297 4 of 20

distribution at spatial position x + ciδt and time t + δt in the direction of i. The equilibriumdistribution is

f eqi = wiρ f

1 +ci·u f

c2s

+

(ci·u f

)2

2c4s−

u2f

2c2s

(2)

The discrete velocity of the D3Q19 model is

ci = c

0 1 −1 0 0 0 0 1 −1 1 −1 1 −1 1 −1 0 0 0 00 0 0 1 −1 0 0 1 −1 −1 1 0 0 0 0 1 −1 1 −10 0 0 0 0 1 −1 0 0 0 0 1 −1 −1 1 1 −1 −1 1

(3)

The weighting coefficients of the D3Q19 model are:

cs =c√3

, wi =

1/3 c2

i = 01/18 c2

i = c2

1/36 c2i = 2c2

(4)

cs is referred to as the lattice sound velocity, c = dx/dt is the lattice speed, dx is thelattice length unit that represents the spacing between two lattice nodes, dt is the latticeBoltzmann time step, and wi is the weighting coefficient. The local density is ρ f = ∑

ifi and

the local fluid momentum is ρ f u f = ∑i

fici. The relaxation time parameter τ controls the

kinematic viscosity ν of the fluid:

ν = c2s

(τ − 1

2

)δ2

xδt

(5)

2.2. Discrete Element Method

In DEM, particles are discrete entities, each of which possesses its momentum andangular momentum equations [32]. In this study, forces and torques used to updateparticles’ velocity and angular velocity are from particle–particle contact, gravity, andparticle–fluid interaction [33]. The specific equations are:

mpdvp

dt= mpg + Fc + F f (6)

Ipωp = Tc + T f (7)

where mp is particle mass, vp is particle velocity, ωp is particle angular velocity, Ip is particlemoment of inertia, Fc is the net force from particle–particle contact, Tc is the net torque fromparticle–particle contact. F f , the net force from the fluid, and T f , the net torque generatedby fluid–solid interaction, will be presented in Section 2.3 as they are parts of the IBM.

Fc and Tc were calculated using the soft sphere model [34]. In this model, interactionsbetween spherical particles are modeled as a combination of elastic and frictional contacts.The normal force between particles is generated by a spring and a damper, and the tangen-tial force is generated by a spring, a damper, and a slider (Figure 1). A small overlap isallowed between the particles, based on which the normal force and the tangential forceare calculated. The soft sphere model is particularly effective in handling dense particleflows where multi-body contacts occur frequently.

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overlap is allowed between the particles, based on which the normal force and the tan-gential force are calculated. The soft sphere model is particularly effective in handling dense particle flows where multi-body contacts occur frequently.

Figure 1. Schematic diagram of the soft sphere collision model.

The net contact force is decomposed into the normal contact force ( nijF ) and the tan-

gential contact force ( tijF ) in the soft sphere model. A Hertz model [35] was adopted to

describe the elastic parts of normal and tangential forces generated during particle–parti-cle contacts.

( )1,

N n tc ij ijj j i= ≠

+F = F F (8)

The normal contact force between particle i and particle j is calculated as

,-nij n n n n ijk δ η ΔF = v (9)

where nδ is the length of overlap along the normal direction, kn is the coefficient of stiff-

ness in the normal direction, ηn is the normal damping coefficient, and ,n ijΔv is the rela-tive normal velocity between the particles. The tangential contact force is calculated simi-larly as

,tij t t t t ijk δ η− ΔF = v (10)

where tδ is the length of the tangential overlap. The production t tk δ produces tangen-

tial elastic restitution; tδ is specifically calculated by integrating the relative tangential velocity throughout the contact. kt is the coefficient of stiffness in the tangential direction, ηt is the tangential damping coefficient, and ,t ijΔv is the relative tangent velocity between the particles. The tangential overlap is truncated to fulfill t n

ij ij ijμ≤F F . The handling of particle and wall interactions requires special instructions. Bound

the simulation domain of a system with a frictional wall. All particles interact with the wall when they are close enough to touch it. The equation for the force between the wall and particles touching it is the same as the corresponding equation, in the limit of one of the two particles going to infinite radius and mass.

2.3. Immersed Boundary Method There are two different IB calculation schemes, called kinematic IB scheme and dy-

namic IB scheme, which were used to impose boundary conditions on immersed geome-try [36,37]. The kinematic IB method considered the influence of the object on the sur-rounding fluid through the forced term obtained from the solid boundary, and added its

Figure 1. Schematic diagram of the soft sphere collision model.

The net contact force is decomposed into the normal contact force (Fnij) and the tangen-

tial contact force (Ftij) in the soft sphere model. A Hertz model [35] was adopted to describe

the elastic parts of normal and tangential forces generated during particle–particle contacts.

Fc =N

∑j=1,j 6=i

(Fn

ij + Ftij

)(8)

The normal contact force between particle i and particle j is calculated as

Fnij = knδn − ηn∆vn,ij (9)

where δn is the length of overlap along the normal direction, kn is the coefficient of stiffnessin the normal direction, ηn is the normal damping coefficient, and ∆vn,ij is the relativenormal velocity between the particles. The tangential contact force is calculated similarly as

Ftij = ktδt − ηt∆vt,ij (10)

where δt is the length of the tangential overlap. The production ktδt produces tangentialelastic restitution; δt is specifically calculated by integrating the relative tangential velocitythroughout the contact. kt is the coefficient of stiffness in the tangential direction, ηt isthe tangential damping coefficient, and ∆vt,ij is the relative tangent velocity between theparticles. The tangential overlap is truncated to fulfill Ft

ij ≤ µijFnij.

The handling of particle and wall interactions requires special instructions. Bound thesimulation domain of a system with a frictional wall. All particles interact with the wallwhen they are close enough to touch it. The equation for the force between the wall andparticles touching it is the same as the corresponding equation, in the limit of one of thetwo particles going to infinite radius and mass.

2.3. Immersed Boundary Method

There are two different IB calculation schemes, called kinematic IB scheme and dy-namic IB scheme, which were used to impose boundary conditions on immersed ge-ometry [36,37]. The kinematic IB method considered the influence of the object on thesurrounding fluid through the forced term obtained from the solid boundary, and added itsinfluence to the fluid. This is a traditional method which can be accurately applied to the noslip boundary. Di [1], Feng [8], Dash [16], Yang [18], and others adopted this scheme whendealing with the geometry boundary. The dynamic IB scheme determined the contributionof hydrodynamic external force and structural internal force by Newton’s second law. Ithas more advantages for applying acceleration boundary conditions [38]. Therefore, thisformat was adopted in the research of Niu [7], Hu [12], and Cao [13]. Existing research

Energies 2022, 15, 3297 6 of 20

results showed that the two schemes agree well for low capillary numbers. The dynamicIB scheme has a stronger prediction ability in the geometry of inertial and non-inertialdeformation. Therefore, the dynamic IB scheme was selected in this paper.

For LBM-IBM in this paper, it is on the Eulerian points that the velocity distribution isdefined. Over and across the fluid–particle boundary immersed in lattice cells, the collisionoperator Ω becomes dependent on the local solid ratio ε (1 in solid cells, 0 in fluid cells, andboundary cells 0 < ε < 1). The solid ratio of the boundary cell is further solved by a latticedecomposition. As shown in Figure 2, a two-dimensional boundary cell is subdivided into5 × 5 decomposition cells. If the distance from a decomposition cell to the center of thesphere is greater than the radius of the sphere, the decomposition cell is considered as afluid cell; if the distance from the cell to the center of the sphere is less than the radius, thecell is considered as a solid cell. Using this method, the solid ratio of the boundary cellis determined.


influence to the fluid. This is a traditional method which can be accurately applied to the no slip boundary. Di [1], Feng [8], Dash [16], Yang [18], and others adopted this scheme when dealing with the geometry boundary. The dynamic IB scheme determined the con-tribution of hydrodynamic external force and structural internal force by Newton’s sec-ond law. It has more advantages for applying acceleration boundary conditions [38]. Therefore, this format was adopted in the research of Niu [7], Hu [12], and Cao [13]. Ex-isting research results showed that the two schemes agree well for low capillary numbers. The dynamic IB scheme has a stronger prediction ability in the geometry of inertial and non-inertial deformation. Therefore, the dynamic IB scheme was selected in this paper.

For LBM-IBM in this paper, it is on the Eulerian points that the velocity distribution is defined. Over and across the fluid–particle boundary immersed in lattice cells, the col-lision operator Ω becomes dependent on the local solid ratio ε (1 in solid cells, 0 in fluid cells, and boundary cells 0 < ε < 1). The solid ratio of the boundary cell is further solved by a lattice decomposition. As shown in Figure 2, a two-dimensional boundary cell is sub-divided into 5 × 5 decomposition cells. If the distance from a decomposition cell to the center of the sphere is greater than the radius of the sphere, the decomposition cell is con-sidered as a fluid cell; if the distance from the cell to the center of the sphere is less than the radius, the cell is considered as a solid cell. Using this method, the solid ratio of the boundary cell is determined.

Figure 2. A two-dimensional circular particle mapped on a two-dimensional lattice. Boundary cells on the lattice are then sub-divided into decomposition cells.

When solid ratio ε is zero, regular collision operator Ωf is used; when ε is unity, the non-equilibrium bounce-back put forward by Noble et al. [39] is used to calculate a solid collision operator Ωs.

( ) ( ) ( ) ( ), , , ,s eq eqi i i f f i f s iΩ f t f f f tρ ρ− −= − + −x u u x (11)

For lattices on the boundary with 0 < ε < 1,

( )1b s fi iΩ BΩ B Ω= + − (12)

where B is a weighting function related to the solid ratio and the relaxation time.

( ) ( )0.5, =

0.5B

ε τε τ

ε τ−

− + (13)

The hydrodynamic force Ff is the sum of the momentum transfer along all directions of the lattice on lattice cells with non-zero solid ratios (both solid cells and boundary cells).

Figure 2. A two-dimensional circular particle mapped on a two-dimensional lattice. Boundary cellson the lattice are then sub-divided into decomposition cells.

When solid ratio ε is zero, regular collision operator Ωf is used; when ε is unity, thenon-equilibrium bounce-back put forward by Noble et al. [39] is used to calculate a solidcollision operator Ωs.

Ωsi = f−i(x, t)− f eq

−i

(ρ f , u f

)+ f eq

i

(ρ f , us

)− fi(x, t) (11)

For lattices on the boundary with 0 < ε < 1,

Ωb = BΩsi + (1− B)Ω f

i (12)

where B is a weighting function related to the solid ratio and the relaxation time.

B(ε, τ) =ε(τ − 0.5)0.5− ε + τ

(13)

The hydrodynamic force Ff is the sum of the momentum transfer along all directionsof the lattice on lattice cells with non-zero solid ratios (both solid cells and boundary cells).

F f =n

∑j=1

Bj

18

∑i=0

Ωsi ci (14)

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The hydrodynamic torque Tf is calculated from a similar sum

T f =n

∑j=1

[Bj(xj − xs

)×

18

∑i=0

Ωsi ci

](15)

where xs is the center of mass of the solid particle, and xj is the coordinate of lattice cell j.

2.4. Phase Coupling

IBM-LBM-DEM coupling is realized by compiling open-source codes [40]. The mainprogram takes the LBM C++ library (Palabos) and calls the DEM open source library(LIGGGHTS) to solve the dynamics of fluid and that of the particles, respectively, wherethe Palabos library is a framework for general-purpose computational fluid dynamics, witha kernel based on the LBM [41]. Its programming interface is straightforward making itpossible to set up fluid flow simulations with relative ease or to extend the library with yourown models. LIGGGHTS is an Open Source Discrete Element Method Particle SimulationSoftware. During DEM calculation, hydrodynamic force and hydrodynamic torque remainunchanged. The time step for DEM is set as tDEM = tLBM × tconst, where tLBM is the timestep for LBM, tconst is a constant number which is defined as tconst = Dp/ut. Ten-timesteps of DEM calculation are carried out before results are passed to IBM-LBM. AfterDEM calculation is completed, updated particle positions and velocities are mapped tothe grid. After one cycle of collision and streaming steps, a new flow field is obtained.Updated hydrodynamic force and torque are then transferred to DEM for the next update.For IBM-LBM-DEM coupling methods, lubrication forces/torques are not included inthis paper.

Ladd and Verberg [11] presented a “pure” LBM for the simulation of particle-ladenflows. Each particle is a collection of solid points that moves as a rigid body. Boundariesof particles are explicitly mapped to the LBM grid. Velocity distributions that propagatetoward particles are bounced back. The momentum exchange associated with bounce-backis summed over the boundary of a particle to generate fluid-particle interactive force andtorque, based on which the translational and rotational motions of particles are updated. Inthis study, this method is referred to as LBM-BB with BB standing for bounce-back. ForLBM-BB methods, lubrication forces/torques are included in this paper.

3. Model Validation3.1. Single-Particle Sedimentation

The experiment of Cate et al. (2002) [34] as shown in Table 1was set up by placing asmall sphere of 0.015 m in diameter at a height of 0.12 m in the center of a container, the sizeof which is 0.1 m× 0.1 m× 0.16 m. The sphere was let to freely settle with terminal Re of 1.5,4.1, 11.6, and 31.9. In IBM-LBM-DEM, the simulation was carried out by directly adoptingthe parameters of the experiment (fluid density, particle density, fluid viscosity, particleterminal Re). In both LBM-BB and IBM-LBM-DEM, the container was resolved by 68 × 68× 108 grids. The particle was resolved by 10 lattices (resolution) across its diameter. Therelaxation time τ of LBM-BB was set to 0.6. The gravity acceleration is −9.8 m/s2. In single-particle sedimentation simulations, the boundary conditions of all surfaces are no-slip. InLBM-BB, there was a difference between the hydrodynamic diameter and the geometricdiameter. The hydrodynamic correction ∆H = 0.224 used in this study was establishedthrough drag calibration (Ladd and Verberg 2001). Coclite et al. [42] developed a dynamicIB scheme combined with LBGK and employed moving least squares reconstruction toaccurately interpolate the flow and force fields required to enforce boundary conditions onimmersed geometry. Therefore, the simulation results of Coclite et al. are also includedfor comparison.

Energies 2022, 15, 3297 8 of 20

Table 1. Setup of single-particle sedimentation experiments.

CaseNumber

ρf

(kg/m3)µf

(Ns/m2)u∞

(m/s) Re St

Case 1 970 373 0.038 1.5 0.19Case 2 965 212 0.060 4.1 0.53Case 3 962 113 0.091 11.6 1.50Case 4 960 58 0.128 31.9 4.13

Figure 3 shows velocities as functions of time for the four cases. Experimental data(from Cate et al. [43]) and simulation date from Coclite et al. [42] were also included forcomparison. In all simulations, the particle’s velocity increased until gravity, buoyancy,and drag forces reached a balance, after which the settling velocity became constant. Asthe particle approached the bottom wall, its velocity decreased and eventually becamezero. In all simulations, both LBM-BB, IBM-LBM-DEM, and Coclite et al.’s simulationresults agree with the experiments. By monitoring the settling velocity over time, differentmethods have been documented to be in excellent agreement. It can also be proved that theIBM-LBM-DEM method adopted in this paper has the same accuracy as other developedmethods. When Re is small, the deviation of settling velocities between IBM-LBM-DEM andthe experiment is greater than between the LBM-BB method and the experiment. However,at the three higher Re, the settling velocities by the LBM-BB method became greater thanthe experimental results, and the settling velocities by the IBM-LBM-DEM method arecloser to the experimental results. The higher the particle resolution, the more accuratethe calculation of particle motion information, but the larger the number of grids at thesame time. A grid effect on numerical results is obtained for the IBM-LBM-DEM method tochoose an accurate and appropriate resolution. We increased the lattice resolution of theparticle from 5 to 25 and re-simulated the case with Re = 31.9. Figure 4 shows that underthe simulation result the resolution is 5, and the sedimentation velocity is a little smallerthan others. Resolutions ranging from 10 to 25 have little effect on the simulation resultswith a particle Reynolds number of 31.9. It can be considered that the resolution of 10 issuitable for Re = 31.9. To improve the accuracy at higher Re (>31.9), however, in the nextsection we doubled the resolution to 20.


Table 1. Setup of single-particle sedimentation experiments.

Case Number fρ

(kg/m³) fμ

(Ns/m²) u∞

(m/s) Re St

Case 1 970 373 0.038 1.5 0.19 Case 2 965 212 0.060 4.1 0.53 Case 3 962 113 0.091 11.6 1.50 Case 4 960 58 0.128 31.9 4.13

Figure 3 shows velocities as functions of time for the four cases. Experimental data (from Cate et al. [43]) and simulation date from Coclite et al. [42] were also included for comparison. In all simulations, the particle’s velocity increased until gravity, buoyancy, and drag forces reached a balance, after which the settling velocity became constant. As the particle approached the bottom wall, its velocity decreased and eventually became zero. In all simulations, both LBM–BB, IBM–LBM–DEM, and Coclite et al.’s simulation results agree with the experiments. By monitoring the settling velocity over time, different methods have been documented to be in excellent agreement. It can also be proved that the IBM-LBM-DEM method adopted in this paper has the same accuracy as other devel-oped methods. When Re is small, the deviation of settling velocities between IBM-LBM-DEM and the experiment is greater than between the LBM–BB method and the experi-ment. However, at the three higher Re, the settling velocities by the LBM-BB method be-came greater than the experimental results, and the settling velocities by the IBM-LBM-DEM method are closer to the experimental results. The higher the particle resolution, the more accurate the calculation of particle motion information, but the larger the number of grids at the same time. A grid effect on numerical results is obtained for the IBM-LBM-DEM method to choose an accurate and appropriate resolution. We increased the lattice resolution of the particle from 5 to 25 and re-simulated the case with Re = 31.9. Figure 4 shows that under the simulation result the resolution is 5, and the sedimentation velocity is a little smaller than others. Resolutions ranging from 10 to 25 have little effect on the simulation results with a particle Reynolds number of 31.9. It can be considered that the resolution of 10 is suitable for Re = 31.9. To improve the accuracy at higher Re (>31.9), however, in the next section we doubled the resolution to 20.

Figure 3. Evolution in the settling velocity from Coclite et al. [42], Experiment [43] LBM-BB and IBM-LBM-DEM.

Figure 3. Evolution in the settling velocity from Coclite et al. [42], Experiment [43] LBM-BB andIBM-LBM-DEM.

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Figure 4. Evolution in the settling velocity from simulations with different resolutions (Re = 31.9).

3.2. Two-Particle Sedimentation In two-particle sedimentation, when the initial positions of the two particles are in-

line with gravity, the upper particle will catch up with the lower particle. After a collision, the pair turns horizontal and separates. This process is called drafting-kissing-tumbling, DKT. In this section, we used IBM-LBM-DEM to simulate DKT.

3.2.1. Simulation of DKT Two particles were placed in a domain 0.02 m × 0.02 m × 0.08 m in size. The diameter

of both particles is 0.002 m. The diameter of a single particle is resolved by 10 grids when Re ≤ 31.9 and 20 grids when Re > 31.9. The upper particle is named particle 1 and the lower particle is named particle 2. These two particles were released from position (0.01, 0.01, 0.072) and position (0.0099, 0.01, 0.069). A small offset in the x-positions of the particles was set to enhance the instability of the pair so that they can exercise DKT. In the simula-tion, the following boundary conditions are used for the computational domain. For the z-direction (settlement direction), the bottom wall is a non-slip wall, and both the x-direc-tion and the y-direction (horizontal direction) are periodic boundaries.

Figure 5 exhibits the distributions of vorticity (magnitude) around the two in-tandem particles, the terminal Re of which is 18.18, at several typical moments in the sedimenta-tion process. The particles are colored based on their angular momenta (magnitude). Par-ticle 1 is drafted by the wake of particle 2 and falls onto particle 2. The nearly in-line con-figuration of the two particles, however, is not stable. The two particles hence begin to rotate. After particle 1 moves from on top of particle 2 to the side of particle 2, they sepa-rate in the horizontal direction due to the opposite signs of their shed y-vorticity.

Figure 4. Evolution in the settling velocity from simulations with different resolutions (Re = 31.9).

3.2. Two-Particle Sedimentation

In two-particle sedimentation, when the initial positions of the two particles are in-linewith gravity, the upper particle will catch up with the lower particle. After a collision, thepair turns horizontal and separates. This process is called drafting-kissing-tumbling, DKT.In this section, we used IBM-LBM-DEM to simulate DKT.

3.2.1. Simulation of DKT

Two particles were placed in a domain 0.02 m × 0.02 m × 0.08 m in size. The diameterof both particles is 0.002 m. The diameter of a single particle is resolved by 10 grids whenRe≤ 31.9 and 20 grids when Re > 31.9. The upper particle is named particle 1 and the lowerparticle is named particle 2. These two particles were released from position (0.01, 0.01,0.072) and position (0.0099, 0.01, 0.069). A small offset in the x-positions of the particles wasset to enhance the instability of the pair so that they can exercise DKT. In the simulation, thefollowing boundary conditions are used for the computational domain. For the z-direction(settlement direction), the bottom wall is a non-slip wall, and both the x-direction and they-direction (horizontal direction) are periodic boundaries.

Figure 5 exhibits the distributions of vorticity (magnitude) around the two in-tandemparticles, the terminal Re of which is 18.18, at several typical moments in the sedimentationprocess. The particles are colored based on their angular momenta (magnitude). Particle 1is drafted by the wake of particle 2 and falls onto particle 2. The nearly in-line configurationof the two particles, however, is not stable. The two particles hence begin to rotate. Afterparticle 1 moves from on top of particle 2 to the side of particle 2, they separate in thehorizontal direction due to the opposite signs of their shed y-vorticity.

Figure 6 shows the velocities of the two particles in series in the x and z directions.Particle settling time is nondimensionalized as t∗ = tphy/

(Dp/ut

), and the dimensionless

velocities are velocities normalized by the terminal velocity of the particles. The fallingprocess of particles goes through three stages: drafting-kissing-tumbling, and the pressurealso changes in the whole process. This change leads to the interaction of attraction andrepulsion of particles, which is consistent with Bernoulli’s law of inviscid fluid. In theinitial period of t* < 2, the accelerations of the two particles suggest that they did not“feel” each other hydrodynamically. As settling continued, as Figure 7 shows, a zone ofpositive pressure was created on the foreside of particle 2, and a zone of negative pressurewas created on the aft side. Since particle 1 is in the negative pressure zone of particle

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2, the drag force acting on particle 1 is reduced, causing particle 1 to accelerate relativeto particle 2. The distance between two particles gradually approaches, and the fluidbetween particles is “extruded”. Therefore, at this time, the fluid pressure is transformedinto kinetic energy, which further causes the rise of fluid velocity between particles andthe proximity of two particles, that is, particle attraction. After particle 1 caught up withparticle 2 and collided, they settled together for a short period (5.5 < t* < 7). The asymmetryof the vorticity structure around the pair caused the pair to rotate. The foreside of particle 1became exposed to the incoming flow (3 s, or, t* = 13.5), and a high-pressure zone began todevelop on the foreside of particle 1. This high-pressure zone gradually separated particles1 and 2. Figure 6 suggests that the particles began to develop different x-velocities withseparation and the settling velocities of both particles decreased. After particles 1 and 2completely separated, their shed vorticities had no interactions with each other and theyagain reached identical settling velocities (22 < t* < 26).


Figure 5. Distribution of the magnitudes of fluid vorticity and particle angular momentum in the sedimentation process of two in-tandem particles with terminal Re = 18.18.

Figure 6 shows the velocities of the two particles in series in the x and z directions. Particle settling time is nondimensionalized as ( )* / /phy p tt t D u= , and the dimensionless ve-locities are velocities normalized by the terminal velocity of the particles. The falling pro-cess of particles goes through three stages: drafting–kissing–tumbling, and the pressure also changes in the whole process. This change leads to the interaction of attraction and repulsion of particles, which is consistent with Bernoulli’s law of inviscid fluid. In the initial period of t* < 2, the accelerations of the two particles suggest that they did not “feel” each other hydrodynamically. As settling continued, as Figure 7 shows, a zone of positive pressure was created on the foreside of particle 2, and a zone of negative pressure was created on the aft side. Since particle 1 is in the negative pressure zone of particle 2, the drag force acting on particle 1 is reduced, causing particle 1 to accelerate relative to parti-cle 2. The distance between two particles gradually approaches, and the fluid between particles is “extruded”. Therefore, at this time, the fluid pressure is transformed into ki-netic energy, which further causes the rise of fluid velocity between particles and the prox-imity of two particles, that is, particle attraction. After particle 1 caught up with particle 2 and collided, they settled together for a short period (5.5 < t* < 7). The asymmetry of the vorticity structure around the pair caused the pair to rotate. The foreside of particle 1 be-came exposed to the incoming flow (3 s, or, t* = 13.5), and a high-pressure zone began to develop on the foreside of particle 1. This high-pressure zone gradually separated parti-cles 1 and 2. Figure 6 suggests that the particles began to develop different x-velocities with separation and the settling velocities of both particles decreased. After particles 1 and 2 completely separated, their shed vorticities had no interactions with each other and they again reached identical settling velocities (22 < t* < 26).

Figure 5. Distribution of the magnitudes of fluid vorticity and particle angular momentum in thesedimentation process of two in-tandem particles with terminal Re = 18.18.


Figure 6. Velocity variation of two interacting particles.

Figure 7. Distributions of pressure during settling of the two intandem particles.

3.2.2. Comparison of 2D and 3D DKT In this section, we compare the evolutions in the distance between two particles from

2D and 3D DKT. Here, dx, dy, and dz represent the x-, y-, and z-distance between the particles, respectively. dr is the surface-to-surface distance and is defined as

2 2 2, pd phy sr dx dy dz D= + + − . The distances dx, dz, and dr as functions of time from IBM–

LBM–DEM and the 2D DKT of Wang et al. [10] are presented in Figure 8. In our simula-tion, since we only offset particles’ positions in x but not in y, dy is very small. Hence, dy is not included in Figure 8. The surface-to-surface x-distance between particle 1 and par-ticle 2 decreased before the collision, as expected. Consistent with the conclusions ob-tained by Nie et al. [15], the dynamics of particles in 2D simulations and that in 3D simu-lations are different. The most striking difference in the character of DKT, however, is that after tumbling 2D particles separated significantly more than 3D particles. The reason for this difference is that the hydrodynamic interaction between particles decreases with in-creasing separation much slower in 2D than in 3D.


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3.2.2. Comparison of 2D and 3D DKT In this section, we compare the evolutions in the distance between two particles from

2D and 3D DKT. Here, dx, dy, and dz represent the x-, y-, and z-distance between the particles, respectively. dr is the surface-to-surface distance and is defined as

2 2 2, pd phy sr dx dy dz D= + + − . The distances dx, dz, and dr as functions of time from IBM–

LBM–DEM and the 2D DKT of Wang et al. [10] are presented in Figure 8. In our simula-tion, since we only offset particles’ positions in x but not in y, dy is very small. Hence, dy is not included in Figure 8. The surface-to-surface x-distance between particle 1 and par-ticle 2 decreased before the collision, as expected. Consistent with the conclusions ob-tained by Nie et al. [15], the dynamics of particles in 2D simulations and that in 3D simu-lations are different. The most striking difference in the character of DKT, however, is that after tumbling 2D particles separated significantly more than 3D particles. The reason for this difference is that the hydrodynamic interaction between particles decreases with in-creasing separation much slower in 2D than in 3D.


3.2.2. Comparison of 2D and 3D DKT

In this section, we compare the evolutions in the distance between two particlesfrom 2D and 3D DKT. Here, dx, dy, and dz represent the x-, y-, and z-distance be-tween the particles, respectively. dr is the surface-to-surface distance and is defined asdrphy,s =

√dx2 + dy2 + dz2 − Dp. The distances dx, dz, and dr as functions of time from

IBM-LBM-DEM and the 2D DKT of Wang et al. [10] are presented in Figure 8. In oursimulation, since we only offset particles’ positions in x but not in y, dy is very small.Hence, dy is not included in Figure 8. The surface-to-surface x-distance between particle 1and particle 2 decreased before the collision, as expected. Consistent with the conclusionsobtained by Nie et al. [15], the dynamics of particles in 2D simulations and that in 3Dsimulations are different. The most striking difference in the character of DKT, however, isthat after tumbling 2D particles separated significantly more than 3D particles. The reasonfor this difference is that the hydrodynamic interaction between particles decreases withincreasing separation much slower in 2D than in 3D.


Figure 8. The variation of dx, dz, and drs in three methods.

4. The Effect of Particle relative Position and Rep on DKT The dynamics of DKT are not only related to the particle Reynolds number that de-

termines the intensity and the distribution of shed vorticity, but also to the initial relative positions (including angles and distances) of the two particles. In this section, we present a systematic study on the effect of particle relative position and Re on DKT. Table 2 shows the parameters studied. In all, we simulated 72 conditions of different Re, angles, and relative distances. We changed Re by changing the viscosity of the fluid, which means that the relax time is the variable parameter for a different case. Particle–fluid density ratio was not changed in the simulations, and dimension properties of fluid and particles and computational domain size are the same as those in 3.2.1. Particle surface-to-surface dis-

tance is presented non-dimensionally as ( )phyd * d /s p pr r D D= −. Particle center-to-center dis-

tance is presented non-dimensionally as phyd * d / pr r D= . The horizontal force that arises

from particle–particle interaction is presented non-dimensionally as ( )*, /x x phy pf f m g=

,

where pm g is particle’s gravity force. Lastly, the non-dimensional angular velocity is ( )* / 2phy tD uω ω= . Figure 9 displays the relative positions that were studied.

Table 2. Summary of simulation parameters exploring the effect of relative position and particle Re on DKT.

Parameter Value particle centroid distance 1.5 Dp, 2.0 Dp, 3.0 Dp, 4 Dp

initial angle, ° 0, 15, 30, 45, 60 and 90 Re 1.82, 18.18, 180.18

Figure 8. The variation of dx, dz, and drs in three methods.

4. The Effect of Particle relative Position and Rep on DKT

The dynamics of DKT are not only related to the particle Reynolds number that de-termines the intensity and the distribution of shed vorticity, but also to the initial relativepositions (including angles and distances) of the two particles. In this section, we present asystematic study on the effect of particle relative position and Re on DKT. Table 2 shows theparameters studied. In all, we simulated 72 conditions of different Re, angles, and relative

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distances. We changed Re by changing the viscosity of the fluid, which means that therelax time is the variable parameter for a different case. Particle–fluid density ratio wasnot changed in the simulations, and dimension properties of fluid and particles and com-putational domain size are the same as those in 3.2.1. Particle surface-to-surface distanceis presented non-dimensionally as drs∗ =

(drphy − Dp

)/Dp. Particle center-to-center dis-

tance is presented non-dimensionally as dr∗ = drphy/Dp. The horizontal force that arisesfrom particle–particle interaction is presented non-dimensionally as f ∗x = fx ,phy/

(mpg

),

where mpg is particle’s gravity force. Lastly, the non-dimensional angular velocity isω∗ = ωphyD/(2ut). Figure 9 displays the relative positions that were studied.

Table 2. Summary of simulation parameters exploring the effect of relative position and particle Reon DKT.

Parameter Value

particle centroid distance 1.5 Dp, 2.0 Dp, 3.0 Dp, 4 Dpinitial angle, 0, 15, 30, 45, 60 and 90

Re 1.82, 18.18, 180.18


Figure 9. Diagram of relative positions studied.

4.1. The Effect of Particle Relative Position Figure 10 shows the vorticity fields at 6 s and particles’ trajectories. The color of the

trajectories represents the magnitude of angular momenta of the particles. Both separation and Re are constant in Figure 10. Hence, the effect of angle is examined. At the angle of 0°, DKT took place but “tumbling” was slow, leading to late lateral separation. At 15°, after a short period of free settling, DKT was triggered. In this DKT, particle 1 was rapidly turned to the side of particle 2. The angular momenta of both particles increased. The two particles then repelled each other and separated. The two particles did not get very close to each other and the angular momenta of the particles were always low. Both the maxima of the angular momenta and the times of their occurrences decreased with increasing ini-tial angle. As particle 2 is no longer in particle 1′s wake, the attraction between the two particles diminished at higher angles. Instead, repulsion between the particles developed. The greater the angle, the stronger the repulsive force. This repulsive force separated the particles horizontally. As the horizontal distance between the two particles increases, the strength of the repulsive force decreases. In the end, the particles settled side-by-side with-out significant interaction. Kim et al. [8] studied the pressure distribution of two particles in static parallel and found a pressure difference between the pressures on both sides of the particles. The repulsive force between two moving particles is also for the same reason. The pressure of the flow field in the moving particle interval is greater than the pressure outside the particles, and the particles will repel each other under the push of this pressure difference.

Figure 9. Diagram of relative positions studied.

4.1. The Effect of Particle Relative Position

Figure 10 shows the vorticity fields at 6 s and particles’ trajectories. The color of thetrajectories represents the magnitude of angular momenta of the particles. Both separationand Re are constant in Figure 10. Hence, the effect of angle is examined. At the angleof 0, DKT took place but “tumbling” was slow, leading to late lateral separation. At15, after a short period of free settling, DKT was triggered. In this DKT, particle 1 wasrapidly turned to the side of particle 2. The angular momenta of both particles increased.The two particles then repelled each other and separated. The two particles did not getvery close to each other and the angular momenta of the particles were always low. Boththe maxima of the angular momenta and the times of their occurrences decreased withincreasing initial angle. As particle 2 is no longer in particle 1′s wake, the attraction betweenthe two particles diminished at higher angles. Instead, repulsion between the particlesdeveloped. The greater the angle, the stronger the repulsive force. This repulsive forceseparated the particles horizontally. As the horizontal distance between the two particlesincreases, the strength of the repulsive force decreases. In the end, the particles settledside-by-side without significant interaction. Kim et al. [8] studied the pressure distributionof two particles in static parallel and found a pressure difference between the pressures onboth sides of the particles. The repulsive force between two moving particles is also for thesame reason. The pressure of the flow field in the moving particle interval is greater thanthe pressure outside the particles, and the particles will repel each other under the push ofthis pressure difference.

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Figure 10. Total vorticity fields at 6 s from particles released with different angles. Re = 18.18 and particle center-to-center distance is 1.5 Dp.

If the two particles are treated as a single entity, this entity achieves the highest set-tling velocity when it is in line with the direction of gravity (the angle of the particles is 0°). However, DKT always increases the angle of the pair toward 90° and hence decreases the average settling velocity of the pair. In dilute suspensions with moderate Re, Yin and Koch [44] found that the average settling velocity rapidly decreased from the terminal velocity with increasing solid fraction. They attributed that rapid decrease to DKT. Con-versely, if the relative angle of the particles could be managed to be more in line with gravity in dilute and moderate Re suspensions, DKT could be used to accelerate particle sedimentation.

Figure 11 shows the dimensionless surface-to-surface distance of particles as func-tions of time. When the angle is 0°, a clear DKT took place. When the angle is 15°, the particles did not actually touch, however, clear drafting and separation due to tumbling can be noticed in the figure. One may hence infer 15° as an approximate boundary of DKT for Re = 18.18 and the initial separation of 1.5 Dp. When the angle is 30°, although weak drafting was present, the two particles never got very close to each other. DKT hence did not occur. Figure 12 shows the maxima of the dimensionless horizontal force that occurred during sedimentation and the dimensionless times that the maxima occurred. The cross-over of fx1 and fx2 between 15° and 30° indicates that the dominant force interaction be-tween the two particles changed from attractive in DKT to repulsive when the two parti-cles’ initial conditions became more side-by-side.

Figure 10. Total vorticity fields at 6 s from particles released with different angles. Re = 18.18 andparticle center-to-center distance is 1.5 Dp.

If the two particles are treated as a single entity, this entity achieves the highest settlingvelocity when it is in line with the direction of gravity (the angle of the particles is 0).However, DKT always increases the angle of the pair toward 90 and hence decreasesthe average settling velocity of the pair. In dilute suspensions with moderate Re, Yin andKoch [44] found that the average settling velocity rapidly decreased from the terminal veloc-ity with increasing solid fraction. They attributed that rapid decrease to DKT. Conversely, ifthe relative angle of the particles could be managed to be more in line with gravity in diluteand moderate Re suspensions, DKT could be used to accelerate particle sedimentation.

Figure 11 shows the dimensionless surface-to-surface distance of particles as functionsof time. When the angle is 0, a clear DKT took place. When the angle is 15, the particlesdid not actually touch, however, clear drafting and separation due to tumbling can benoticed in the figure. One may hence infer 15 as an approximate boundary of DKT forRe = 18.18 and the initial separation of 1.5 Dp. When the angle is 30, although weakdrafting was present, the two particles never got very close to each other. DKT hence didnot occur. Figure 12 shows the maxima of the dimensionless horizontal force that occurredduring sedimentation and the dimensionless times that the maxima occurred. The crossoverof f x1 and f x2 between 15 and 30 indicates that the dominant force interaction betweenthe two particles changed from attractive in DKT to repulsive when the two particles’ initialconditions became more side-by-side.

First-order and second-order derivatives of surface-to-surface distance to time asfunctions of time are depicted in Figures 13 and 14. The sign of the first-order derivative,negative or positive, is indicative of whether the two particles are approaching each otheror separating. Attraction and repulsion can be characterized by the second-order derivativeof the surface-to-surface distance of particles to time. When the second-order derivative ispositive, the interaction between particles is repulsive. On the contrary, if the second-orderderivative is negative, the interaction force between particles is attractive.

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Figure 11. Dimensionless surface-to-surface distance of particles released with different initial an-gles as functions of dimensionless particle settling time.

Figure 12. Maxima of the dimensionless horizontal force acting on particles 1 and 2 (right axis) and the dimensionless times that the maxima occurred.

First-order and second-order derivatives of surface-to-surface distance to time as functions of time are depicted in Figures 13 and 14. The sign of the first-order derivative, negative or positive, is indicative of whether the two particles are approaching each other or separating. Attraction and repulsion can be characterized by the second-order deriva-tive of the surface-to-surface distance of particles to time. When the second-order deriva-tive is positive, the interaction between particles is repulsive. On the contrary, if the sec-ond-order derivative is negative, the interaction force between particles is attractive.

Figure 11. Dimensionless surface-to-surface distance of particles released with different initial anglesas functions of dimensionless particle settling time.


Figure 11. Dimensionless surface-to-surface distance of particles released with different initial an-gles as functions of dimensionless particle settling time.

Figure 12. Maxima of the dimensionless horizontal force acting on particles 1 and 2 (right axis) and the dimensionless times that the maxima occurred.

First-order and second-order derivatives of surface-to-surface distance to time as functions of time are depicted in Figures 13 and 14. The sign of the first-order derivative, negative or positive, is indicative of whether the two particles are approaching each other or separating. Attraction and repulsion can be characterized by the second-order deriva-tive of the surface-to-surface distance of particles to time. When the second-order deriva-tive is positive, the interaction between particles is repulsive. On the contrary, if the sec-ond-order derivative is negative, the interaction force between particles is attractive.

Figure 12. Maxima of the dimensionless horizontal force acting on particles 1 and 2 (right axis) andthe dimensionless times that the maxima occurred.


Figure 13. Derivative of surface-to-surface distance to time as functions of time.

Figure 14. The second-order derivative of surface-to-surface distance to time as functions of time.

As can be seen from Figure 14, when the initial angle is 0° to 30°, the force between particles was initially attractive but became repulsive after the particles turned side-by-side. We note that in the data for 0°, the first very high positive second-order derivative of short duration is the result of particle–particle contact and is not due to hydrodynamics. When the initial angle is between 45° and 90°, repulsive force dominates throughout the interaction. The dynamics of particles released at 0° shows a strong drafting (negative force), a clear kissing (collision), and a strong repulsion after tumbling. That at 15° does not have a collision but still exhibits strong drafting and repulsion after tumbling. They therefore support our previous statement based on Figure 11—that particles released at 0° and 15° executed DKT. The second-order derivative plot, however, shows that the char-acteristics of the dynamics at 30° is not qualitatively different from that at 15°. Hence, if we focus on the characteristics of the second-order derivative and relax on the definition of kissing, the dynamics of 30° could also be counted as a DKT.

However, the plot of the magnitude of angular velocities, Figure 15, suggests that the dynamics of 30° lacks sufficient rotation (tumbling) and hence should not be counted as DKT. In Figure 15, the magnitudes of the angular velocities of particles are plotted as func-tions of time. The maxima of angular velocity decreased with increasing angle. For both particles, the maxima of 0° and 15° stand out. The angular velocities of other initial angles


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As can be seen from Figure 14, when the initial angle is 0° to 30°, the force between particles was initially attractive but became repulsive after the particles turned side-by-side. We note that in the data for 0°, the first very high positive second-order derivative of short duration is the result of particle–particle contact and is not due to hydrodynamics. When the initial angle is between 45° and 90°, repulsive force dominates throughout the interaction. The dynamics of particles released at 0° shows a strong drafting (negative force), a clear kissing (collision), and a strong repulsion after tumbling. That at 15° does not have a collision but still exhibits strong drafting and repulsion after tumbling. They therefore support our previous statement based on Figure 11—that particles released at 0° and 15° executed DKT. The second-order derivative plot, however, shows that the char-acteristics of the dynamics at 30° is not qualitatively different from that at 15°. Hence, if we focus on the characteristics of the second-order derivative and relax on the definition of kissing, the dynamics of 30° could also be counted as a DKT.

However, the plot of the magnitude of angular velocities, Figure 15, suggests that the dynamics of 30° lacks sufficient rotation (tumbling) and hence should not be counted as DKT. In Figure 15, the magnitudes of the angular velocities of particles are plotted as func-tions of time. The maxima of angular velocity decreased with increasing angle. For both particles, the maxima of 0° and 15° stand out. The angular velocities of other initial angles


As can be seen from Figure 14, when the initial angle is 0 to 30, the force betweenparticles was initially attractive but became repulsive after the particles turned side-by-side.We note that in the data for 0, the first very high positive second-order derivative ofshort duration is the result of particle–particle contact and is not due to hydrodynamics.When the initial angle is between 45 and 90, repulsive force dominates throughout theinteraction. The dynamics of particles released at 0 shows a strong drafting (negativeforce), a clear kissing (collision), and a strong repulsion after tumbling. That at 15 doesnot have a collision but still exhibits strong drafting and repulsion after tumbling. Theytherefore support our previous statement based on Figure 11—that particles released at0 and 15 executed DKT. The second-order derivative plot, however, shows that thecharacteristics of the dynamics at 30 is not qualitatively different from that at 15. Hence,if we focus on the characteristics of the second-order derivative and relax on the definitionof kissing, the dynamics of 30 could also be counted as a DKT.

However, the plot of the magnitude of angular velocities, Figure 15, suggests thatthe dynamics of 30 lacks sufficient rotation (tumbling) and hence should not be countedas DKT. In Figure 15, the magnitudes of the angular velocities of particles are plottedas functions of time. The maxima of angular velocity decreased with increasing angle.For both particles, the maxima of 0 and 15 stand out. The angular velocities of otherinitial angles are all significantly lower. For particle 1, both the maximum value and thetime of maximum decreased with increasing initial angle. Increasing the angle at constantseparation, therefore, made the disturbance to the leading particle occur more readily butwith less intensity. For particle 2, except for 90, at higher angles (30, 45, and 60) theangular velocities of particle 2 are much smaller. Figure 15 shows that the upper particlerotated much less than the lower particle in high-angle interactions. Hence, if we requireparticles to roll together in the “tumbling” phase, the dynamics at 30 is disqualified.

The contour map of dimensionless minimum separation for different angles and initialdistance is displayed in Figure 16. The dimensionless particle–particle minimum separationis defined as representing the minimum distance between particles during sedimentation.When s = 0, particles must have experienced collision. When 0 < s < 1, particles have beenpulled closer than their initial distance during sedimentation. Note that the minimum in scannot be greater than the initial s. Hence, s cannot be greater than one. The variation of shence is an indication of the intensity of attractive hydrodynamic interaction and whethersuch attractive hydrodynamic interaction led to a collision. For particles with small initialangles (<10), s is very close to 0 regardless of separation. It is an indication that for particles

Energies 2022, 15, 3297 16 of 20

that are vertically aligned, DKT can always bring them together. For particles with largerinitial angles, attraction weakens. Interestingly, dimensionless minimum separation firstdecreases and then increases with increasing initial surface-to-surface distance at someintermediate angles. This shows the complexity in which the shed vorticities from the twoparticles interact with each other during sedimentation.


are all significantly lower. For particle 1, both the maximum value and the time of maxi-mum decreased with increasing initial angle. Increasing the angle at constant separation, therefore, made the disturbance to the leading particle occur more readily but with less intensity. For particle 2, except for 90°, at higher angles (30°, 45°, and 60°) the angular velocities of particle 2 are much smaller. Figure 15 shows that the upper particle rotated much less than the lower particle in high-angle interactions. Hence, if we require particles to roll together in the “tumbling” phase, the dynamics at 30° is disqualified.

(a) (b)

Figure 15. Particle’s dimensionless angular velocities as functions of dimensionless time. (a) Particle 1; (b) Particle 2.

The contour map of dimensionless minimum separation for different angles and ini-tial distance is displayed in Figure 16. The dimensionless particle–particle minimum sep-aration is defined as representing the minimum distance between particles during sedi-mentation. When s = 0, particles must have experienced collision. When 0 < s < 1, particles have been pulled closer than their initial distance during sedimentation. Note that the minimum in s cannot be greater than the initial s. Hence, s cannot be greater than one. The variation of s hence is an indication of the intensity of attractive hydrodynamic interaction and whether such attractive hydrodynamic interaction led to a collision. For particles with small initial angles (<10°), s is very close to 0 regardless of separation. It is an indication that for particles that are vertically aligned, DKT can always bring them together. For par-ticles with larger initial angles, attraction weakens. Interestingly, dimensionless minimum separation first decreases and then increases with increasing initial surface-to-surface dis-tance at some intermediate angles. This shows the complexity in which the shed vorticities from the two particles interact with each other during sedimentation.

Figure 15. Particle’s dimensionless angular velocities as functions of dimensionless time. (a) Particle1; (b) Particle 2.


Figure 16. Contour maps of dimensionless minimum separation for different angles and initial sur-face-to-surface distances.

4.2. The Effect of Reynolds Number Reynolds number affects the degree of vorticity shedding and the fore-aft symmetry

of flow around the particle. The dynamics of particle pair settling and DKT are hence a function of Re.

Figure 17 shows the trajectories of particles at two different Reynolds numbers. The initial center-to-center distance is 3 Dp. In Section 4.1, we presented that at small initial angles, particles collide, rotate, and then separate and settle independently following DKT dynamics. At greater initial angles, the leading particle is pushed to the side and then overtaken by the trailing particle. After the relative positions of the two particles switch, the original leading but now trailing particle experiences an attraction (c.f. Figure 14). The two particles can therefore approach each other again and execute another switch in their leading/trailing positions. When the Reynolds number of particles is 18.18, particles’ tra-jectories crossed only once at 0°, 15°, and 30°. When the particle angle is 45°, particles’ trajectories crossed twice, which means that the leading/trailing positions of the particles switched again before they settled to the bottom of the domain. When Re of particles in-creased to about 180, DKT only occurred at 0°. Increased Reynolds number leads to de-creased influence of fluid viscosity compared to that of particle/fluid inertia, which, in turn, makes the changes in particles’ trajectories less pronounced.

(a) (b)

Figure 16. Contour maps of dimensionless minimum separation for different angles and initialsurface-to-surface distances.

4.2. The Effect of Reynolds Number

Reynolds number affects the degree of vorticity shedding and the fore-aft symmetryof flow around the particle. The dynamics of particle pair settling and DKT are hence afunction of Re.

Figure 17 shows the trajectories of particles at two different Reynolds numbers. Theinitial center-to-center distance is 3 Dp. In Section 4.1, we presented that at small initial

Energies 2022, 15, 3297 17 of 20

angles, particles collide, rotate, and then separate and settle independently following DKTdynamics. At greater initial angles, the leading particle is pushed to the side and thenovertaken by the trailing particle. After the relative positions of the two particles switch,the original leading but now trailing particle experiences an attraction (c.f. Figure 14).The two particles can therefore approach each other again and execute another switch intheir leading/trailing positions. When the Reynolds number of particles is 18.18, particles’trajectories crossed only once at 0, 15, and 30. When the particle angle is 45, particles’trajectories crossed twice, which means that the leading/trailing positions of the particlesswitched again before they settled to the bottom of the domain. When Re of particlesincreased to about 180, DKT only occurred at 0. Increased Reynolds number leads todecreased influence of fluid viscosity compared to that of particle/fluid inertia, which, inturn, makes the changes in particles’ trajectories less pronounced.


Figure 16. Contour maps of dimensionless minimum separation for different angles and initial sur-face-to-surface distances.

4.2. The Effect of Reynolds Number Reynolds number affects the degree of vorticity shedding and the fore-aft symmetry

of flow around the particle. The dynamics of particle pair settling and DKT are hence a function of Re.

Figure 17 shows the trajectories of particles at two different Reynolds numbers. The initial center-to-center distance is 3 Dp. In Section 4.1, we presented that at small initial angles, particles collide, rotate, and then separate and settle independently following DKT dynamics. At greater initial angles, the leading particle is pushed to the side and then overtaken by the trailing particle. After the relative positions of the two particles switch, the original leading but now trailing particle experiences an attraction (c.f. Figure 14). The two particles can therefore approach each other again and execute another switch in their leading/trailing positions. When the Reynolds number of particles is 18.18, particles’ tra-jectories crossed only once at 0°, 15°, and 30°. When the particle angle is 45°, particles’ trajectories crossed twice, which means that the leading/trailing positions of the particles switched again before they settled to the bottom of the domain. When Re of particles in-creased to about 180, DKT only occurred at 0°. Increased Reynolds number leads to de-creased influence of fluid viscosity compared to that of particle/fluid inertia, which, in turn, makes the changes in particles’ trajectories less pronounced.

(a) (b)

Figure 17. Particles’ trajectories at two different Reynolds numbers. (a) Re = 18.18; (b) Re = 181.8.

Figure 18 shows the dimensionless particle–particle minimum separation at variousinitial distance and angles and different Reynolds numbers. It is observed that two particleswill collide at an initial angle of 0, but not at an initial angle of 90, in the focused rangeof Re. The surface of s gradually transitions from steep in low-angle region to less steepin low-angle region from small Re to large Re, and the range of initial angle and particles’distance for attraction increases with increasing Re. The domain of attraction (s < 1) islimited to initial angles less than 55 for small Re of 1.82. When Re is 18.18, some limitedattraction can be noticed at short distances and high angles. Figure 18a,b shows thatat the two lower Reynolds numbers, the minimum distance increases with the increaseof the initial distance. Figure 18a shows that for pairs with short initial distances, theminimum separation distance varies significantly, with initial angle in the range of 20

and 55. For pairs with longer initial distances, the minimum separation distance variesgreatly with initial angle in the range of 10 to 35. The domain attraction is independentof initial distances when the initial angle in the range of 0 to 30 and the Re of 18.18as shown in Figure 18b. When Re is raised to 181.8, it is observed that the domain ofattraction is significantly expanded. The most striking and qualitative difference, however,is that particle pairs with initial angles in the range of 40–70 showed strong attractionsat high separations, shown in Figure 18c, likely due to inviscid potential flow interactionsthat dominate the interactions between pairs of high-Re and spherical gas bubbles, butnonetheless should also exist between high-Re solid particles when their viscous boundarylayers do not overlap [45–47].

Energies 2022, 15, 3297 18 of 20


Figure 17. Particles’ trajectories at two different Reynolds numbers. (a) Re = 18.18; (b) Re = 181.8.

Figure 18 shows the dimensionless particle–particle minimum separation at various initial distance and angles and different Reynolds numbers. It is observed that two parti-cles will collide at an initial angle of 0°, but not at an initial angle of 90°, in the focused range of Re. The surface of s gradually transitions from steep in low-angle region to less steep in low-angle region from small Re to large Re, and the range of initial angle and particles’ distance for attraction increases with increasing Re. The domain of attraction (s < 1) is limited to initial angles less than 55° for small Re of 1.82. When Re is 18.18, some limited attraction can be noticed at short distances and high angles. Figure 18a,b shows that at the two lower Reynolds numbers, the minimum distance increases with the in-crease of the initial distance. Figure 18a shows that for pairs with short initial distances, the minimum separation distance varies significantly, with initial angle in the range of 20° and 55°. For pairs with longer initial distances, the minimum separation distance varies greatly with initial angle in the range of 10° to 35°. The domain attraction is independent of initial distances when the initial angle in the range of 0° to 30° and the Re of 18.18 as shown in Figure 18b. When Re is raised to 181.8, it is observed that the domain of attrac-tion is significantly expanded. The most striking and qualitative difference, however, is that particle pairs with initial angles in the range of 40–70° showed strong attractions at high separations, shown in Figure 18c, likely due to inviscid potential flow interactions that dominate the interactions between pairs of high-Re and spherical gas bubbles, but nonetheless should also exist between high-Re solid particles when their viscous bound-ary layers do not overlap [45–47].

(a) (b)

(c)

Figure 18. Variation of s with initial distance and angle under different Reynolds numbers;(a) Re = 1.81; (b) Re = 18.2; (c) Re =181.8.

5. Conclusions

The present work employs an IBM-LBM-DEM method to analyze the settling of twoparticles with hydrodynamic interactions. Our method was verified by comparing single-particle settling velocities to those measured in experiments. Our main observations fromtwo-particle simulations are as follows:

1. Whether the hydrodynamic interaction between particles is attractive or repulsiveduring sedimentation depends on the relative location of the particles and theReynolds number.

2. Whether a pair of particles has experienced DKT can be viewed from time plots of thedistance between the particles (for kissing), the second-order derivative of distanceto time (for drafting), and angular velocities of particles (for tumbling). In a DKT,particles should approach each other along the vertical direction to a close distance,develop nearly equal and opposite angular velocities, and then separate along thehorizontal direction.

3. As expected, not all combinations of initial distance, angle, and Reynolds number leadto DKT. DKT only occurs when the initial positions of particles are nearly verticallyaligned. At high Reynolds numbers, due to high particle/fluid inertia, the influenceof fluid–particle force on particles’ trajectories becomes less. Particles hence couldgenerally get closer to each other at higher Reynolds numbers, regardless of whetherthe interaction is DKT or not.

Energies 2022, 15, 3297 19 of 20

4. When Re is low or moderate (O(1–10)), particle pairs with high initial angles do notexperience attraction. When Re is high (O(100)), however, our results show that thedomain of attractive interaction significantly expanded and particle pairs with initialangles in the range of 40–70 also experienced attraction at long initial distances (2.5to 3.0 Dp) likely attributed to inviscid potential flow interaction.

5. The IBM-LBM-DEM method is consistent with other numerical simulation results andexperimental data. Collectively, the proposed approach can be efficiently used forpredicting complex particle dynamics relevant flows.

Author Contributions: X.L.: Conceptualization, Methodology, Software, Writing—original draft,Investigation, Visualization. G.L.: Resources, Formal analysis, Validation, Methodology, Writing—review & editing, Supervision. J.Z.: Software, Validation, Writing—review & editing. X.Y.: Writing—review & editing, Supervision. H.L.: Writing—review & editing, Supervision. All authors have readand agreed to the published version of the manuscript.

Funding: This work was supported by Interdisciplinary Research Foundation of HIT IR2021228 andthe Natural Science Foundation of China through Grant No. 51776058.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Conflicts of Interest: The authors declare no conflict of interest.

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