discrete element modeling of non-linear submerged particle collisions

Granular Matter (2013) 15:759–769DOI 10.1007/s10035-013-0442-8

ORIGINAL PAPER

Discrete element modeling of non-linear submerged particlecollisions

Ingrid Tomac · Marte Gutierrez

Received: 13 December 2012 / Published online: 27 August 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract Coupling of the Discrete Element Method withComputational Fluid Dynamics (DEM–CFD) is a widelyused approach for modeling particle–fluid interactions.Although DEM–CFD focuses on particle–fluid interaction,the particle–particle contact behavior is usually modeledusing a simple Kelvin–Voigt contact model which may notrepresent realistic interactions of particles in high viscos-ity fluids. This paper presents an implementation of a newuser-defined contact model that accounts for the effects oflubrication of fluid between two approaching particles whilemaintaining all other DEM–CFD particle–fluid interactionphenomena. Theoretical model that yields a non-linear resti-tution coefficient for submerged particle collisions, whichwas developed by Davis et al. (J Fluid Mech 163:479–497,1986), is implemented in a DEM–CFD code. In this model,the behavior of particles at a contact depends on fluid prop-erties, particle velocities and distance between particle sur-faces. When two particles approach each other in a fluid,their kinetic energy decreases gradually because of a lubri-cation effect associated with the thin fluid layer between theparticles. Particle post-collision behavior is governed by asimplified elastic contact law. With lubrication, it is possiblethat particles are not able to rebound if the approaching veloc-ity is completely damped by lubrication, and in this case theparticles agglomerate in the fluid. Tangential surface friction-slip forces are activated as in the case of dry particle contact.The lubrication model represents an advanced submergedparticle collision approach that permits improved accuracywhen modeling problems with high particle concentrationsin a fluid. An application of the new model is shown in asimple sediment transport problem.

I. Tomac (B) · M. GutierrezCivil and Environmental Engineering, Colorado School of Mines,Coolbaugh 318, 1012 14th St, Golden, CO 80401, USAe-mail: [email protected]

Keywords DEM–CFD coupling · Submerged particlecollisions · Lubrication force · Restitution coefficient ·Sediment transport

1 Introduction

The Discrete Element Method (DEM) has been used innumerical modeling to predict particulate media interactionsfor over three decades, starting with the work of Cundall andStrack [1]. DEM is a method for solving the motion and theinteractions of particles in granular media. Individual parti-cle motion in the system is solved computationally using thefinite difference method. Today, several DEM codes are avail-able, and this study used the Particle Flow Code (PFC) thatwas developed by Itasca [2]. Particle trajectories are deter-mined by applying explicitly the Law of Motion for eachparticle in the system. Forces and moments that act on aparticle at each time step depend on the particle–particle orparticle–wall collisions, and, alternatively, interactions withfluid, gravity or other distance forces. DEM was originallydeveloped for dry particle collisions, and it has been extendedover the years for different applications. Contact model rep-resents the part of DEM where the forces and moments arecomputed as the consequence of particle collisions. The sim-plest dry particle contact model is the Kelvin–Voigt model,but Maxwell model and other more specific ones that accountfor molecular forces, Van der Waals forces and others are inuse today. For the purpose of fluid–solid interactions, DEMhas been coupled with the Computational Fluid Dynam-ics (DEM–CFD) to model solid–fluid interaction problems[3,4]. However, there has not been enough attention given tothe contact behavior for particles submerged in viscous fluid.Contact models that have been used so far in CFD–DEMare soft sphere models [1,5–11]. Using coupled DEM–CFD

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760 I. Tomac, M. Gutierrez

Fluid flow (CFD cells)

Particles interactions

(contact models)

Two-way fluid-particle coupling

Fig. 1 Fluid and particles coupling in DEM–CFD code for two-wayinteractions between particles and fluids

codes along with the dry-collision particle contact model, it ispossible to account for behavior of dilute suspensions or gas-particle coupled behavior. However, when particle collisionsare dominant and occur at a very frequent rate in a viscousfluid, contact models need to follow the specific non-linearsubmerged particle collisions behavior.

The study presented in this paper focuses on the mod-ification of the DEM–CFD method by the introduction ofnew contact model for submerged particle collisions in flu-ids. Figure 1 shows a scheme for the two-way fluid–particlemomentum coupling that is implemented in PFC. DEM–CFDcoupling in PFC is a geometrically simplified scheme thatuses a fixed rectangular grid setup aligned with the Carte-sian axes for modeling fluid dynamics. At each computa-tional time-step the forces that act on the particle motionfrom the fluid are computed, and the fluid motion in com-putational cells is updated using not only the boundary andinitial conditions but the particle motion in each computa-tional cell. Fluid motion is modeled using modified Navier–Stokes equations that include the effects of particles in thefluid. The method has capacity to account for two-way solid–fluid dynamic coupling, which is very important for modelingdense-phase flows with high particle concentrations in fluid[12]. Two-way coupling means that the code updates bothfluid and particle motion within a time step. More impor-tantly, the fluid exerts forces on the particles, and vice versa.In the dense-phase flow, particle motion is controlled by col-lisions [13]. A qualitative estimate of the nature of the flowcan be made by comparing the ratio of momentum responsetime of a particle to the time between collisions [12]. The par-

ticle momentum response time relates to the time requiredfor a particle to respond to a change in fluid velocity. In otherwords, in particle–fluid flows that are characterized as a denseflow, particles do not have time to respond completely to thechanges in fluid velocity between two collisions, and the timebetween collisions is relatively small.

Motivated by the importance of accurate modeling ofsubmerged particle collisions, a new user-defined model isdeveloped in PFC. The new model is based on a theoreticalformulation which accounts for the collision energy that isdissipated by a thin fluid layer between particles. This paperpresents a direct implementation of the lubrication forcethat develops on particle–particle and particle–wall sub-merged collisions. Theoretical and experimental predictionsand verifications of the model are shown using an examplethat involves the determination of the nonlinear coefficientof restitution of a particle dropped on a wetted horizontalsurface [14,15]. In addition, a simple sediment transportproblem, which applies the implemented lubrication contactmodel, is analyzed to illustrate the importance of lubrica-tion between particles in multi-particle systems submergedin fluids.

2 Model developments

2.1 Theoretical background

Lubrication force is introduced at the contact between par-ticles using elasto-hydrodynamic theory in order to improvethe simulation of submerged particle collision behavior inDEM–CFD models. A new user-defined contact model iscoded and implemented in PFC2D based on a theoreti-cally developed expression for lubrication force, which wasdeveloped and verified experimentally in previous research[14,15]. Davis’ lubrication model defines the behavior oftwo-particles collision submerged in viscous fluid. Themodel is based on the criteria for predicting whether twosolid spherical particles submerged in fluid will stick togetheror rebound from each other subsequent to impact. Figure 2shows the interaction of two colliding spheres submerged ina viscous fluid.

Lubrication force is active for small deformation, i.e.,when w � x , where w is the elastic deformation of thesphere and x is the distance between two un-deformed spheresurfaces. The un-deformed surfaces of the spherical particlescan be approximated by a parabolic curve:

h(r, t) = x(t) + r2

2a+ w(r, t) (1)

where h(r, t) is the distance between spheres surfaces at thetime t, r is the tangential distance from the line that con-nects the sphere centers and the position where h(r, t) is

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Coupling of the DEM–CFD 761

r

x

w(r , t)

h (r , t)

w(r , t)

Fig. 2 Deformation of two colliding elastic spheres in viscous fluid(x axes is normal to particle surfaces at contact, r is the direction per-pendicular to the line that connects sphere centers, w(r, t) is the elas-tic deformation of the sphere, h(r, t) is the distance between spheresurfaces)

calculated, a is the sphere radius, and w(r, t) is the elasticdeformation of the sphere surface. The deformation w(r, t)can be determined following Hertz contact theory of linearelasticity by integrating the surface stress distribution multi-plied by a Green function over the area subjected to the stress.The appropriate Green function is the fundamental solutionof the linear-elasticity equations for an applied point force ofunit magnitude over a solid surface (“Appendix” in [15]).

The surface deformation integral can be written as:

w(r, t) = 4θ

∞∫

0

f (y, t)φ(r, y)dy (2)

θ = 1 − υ21

π E1+ 1 − υ2

2

π E2(3)

φ(r, y) = y

y + rK

[4r y

(r + y)2

](4)

where ν is the Poisson’s ratio, E is the Young’s modulus,the subscripts 1 and 2 refer to the two particles, f (y, t) is thestress distribution over the solid surface at the time t, φ(r, y)

is the Green function kernel and K is the complete ellip-tic integral of the first kind [15]. The symbol y denotes thenormal axis to the particle surface for the area where theGreen function is applied. Surfaces are very close to oneanother, and the fluid flow between them is described by thelubrication equation of fluid dynamics. The hydrodynamicforce on the spheres can be obtained from the derivation byDavis et al. [15]:

dh

dt= 1

12μr

d

dr

[rh3 dp

dr

](5)

where p(r, t) is the pressure distribution in the fluid layerbetween particles which does not vary at any given timeacross the width of the narrow gap, μ is the fluid dynamicviscosity, and h(r, t) is defined by Eq. (1). The kinematic

equations which describe the relative motions of the un-deformed surfaces of the solid spheres are given by:

dx

dt= −v(t), m

dv

dt= −F(t) (6)

where v(t) is the relative velocity of the center of masses oftwo spheres, m is the reduced mass, and F(t) is the force onthe spheres.

It is assumed that for w � x , the elastic deformation ofthe sphere is negligible in Eq. (1). For w = 0 and solvingEq. (5) for the pressure profile yields:

p(r, t) = 3μav(x + r2

2a

)2 (7)

The hydrodynamic force on the spheres, F(t), is thenobtained by integrating Eq. (7) over a particle surface fromr = 0 to r = a as in [15]:

F(t) = 6πμa2v

x(8)

The lubrication model involves only the normal compo-nent of the contact forces, and the tangential/shear compo-nent is modeled similarly as for a dry system. This approachis validated by the observation of Joseph and Hunt [16] thatthe tangential component of collisions of particles in fluidsare similar to a dry system but with a lowered friction coef-ficient due to the lubrication effects. However, for collisionsof rough particles at increasing tangential velocities, the fric-tion coefficient increases to a value that approximates the dryfriction coefficient [16]. Therefore, in the lubrication contactmodel, tangential component of immersed collision is sim-plified and is modeled such as for a dry collision while usinga lower friction to account for lubrication.

The contact model developed here is suitable for problemswith spherical particles that are submerged in viscous fluid.The thin layer of fluid between the particles becomes moreeffective for damping as the fluid viscosity increases. Anexample is the lubrication of bearings, where high viscousoil is used. Fluids with very low viscosities, such as gas orair, do not necessarily form a thin layer that resists pressureof two particles approaching each other. Kobayashi et al.[17] investigated behavior of gas–solid particle systems usingDEM model and experiments on fluidized bed problem. Inthe case of gas–solid particle interactions, DEM model wasable to successfully replicate experimental results withoutintroducing lubrication force [17]. Application of lubricationforce depends on the particle size, particle surface roughnessand fluid viscosity. Lubrication force decreases in the case ofsignificant particle roughness because the fluid continuum isnot sustained between rough sphere surfaces [18].

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2.2 Fluid and particles coupling scheme

DEM is coupled with CFD for modeling interactions of rigidparticles and Newtonian fluids. The fluid motion is definedwith solution of the Navier–Stokes equations over a fixedrectangular grid for modeling the fluid field. The fluid flowfield is averaged over a fixed spatial fluid cell and updated ateach relatively small computational time-step �t . The sizeof the computational fluid cell is always larger than the solidparticle in the cell. The effect of the particles on the fluidmotion in the fluid cell is also averaged based on the localtemporary porosity of the cell. Therefore, the flow field is notfully resolved around the particles by the fluid flow equations.The particles effects are introduced to the numerical schemeof the Navier–Stokes’s equations in terms of the porosity andthe coupling force (Bouillard et al. 1989). For the porositiese < 0.8 the following fluid flow equation is used in PFC2D:

ρ f∂e�v∂t

+ ρ f �v · ∇(e�v) = −e∇ p + μ∇2(e�v) + �fb (9)

∂e

∂t+ ∇(e�v) = 0 (10)

where ρ f is the density of the fluid, �v is the fluid velocity, t isthe time, ρ is the particle density, ∇ is the del operator indi-cating the partial spatial derivative operator (with respect tox and y directions in two-dimensional space), fb is the bodyforce per unit volume, p is the fluid pressure and μ is the fluiddynamic viscosity. Equation (9) gives the fluid momentumequation with included effects of varying porosity, and Eq.(10) is the conservation of mass (continuity) equation for anincompressible fluid in a porous medium. It can be seen thatboth time and space derivatives of the porosity are present inthe formulation. The porosity e is:

e = 1 − Vp

V(11)

where Vp is the total volume of all the particles containedwithin the fluid element and V is the volume of the fluidelement.

The interaction between fluid and particles is achieved inboth ways. The fluid flow field exerts a drag force to theparticles’ center of mass but without the rotational momentthat might be caused by the fluid shear motion. The body forceterm �fb has unit of force per unit volume, and is applied toeach particle proportional to the volume of each particle. Thedrag force applied to an individual discrete particle is:

�fdrag = 4

3πr3

�fb

(1 − e)(12)

where �fdrag is the fluid drag force on the particle and r is par-ticle radius. An opposite drag force is applied by the particlesto the fluid in each fluid element/cell defined as:

�fb = β �U (13)

where �fb is the drag force per unit volume, β is the coefficientdefined by Eqs. (15) and (16), and �U is the average relativevelocity between the particles and the fluid, defined as:

�U = �u − �v (14)

where �u is the average velocity of all particles in a given fluidelement and �v is the fluid velocity. Different expressions forthe coefficient β are given for porosities with values higherand lower than 0.8 (Bouillard et al. 1989) as:

β = (1−e)

d2e2(150(1−e)μ+1.75ρ f d

∣∣∣ �U∣∣∣); e < 0.8 (15)

β = 4

3Cd

∣∣∣ �U∣∣∣ ρ f (1 − e)

de1.7; e ≥ 0.8 (16)

where d is the average diameter of the particles in the element,and Cd is the turbulent drag coefficient defined in terms ofparticle Reynolds number Rep:

Cd ={

24(1+0.15Re0.687p )

Rep; Rep < 1,000

0.44; Rep > 1,000(17)

Rep =∣∣∣ �U

∣∣∣ ρ f ed

μ(18)

DEM–CFD coupled scheme is established for modelingmulti-particle systems, where the particle radius is at leastseveral times smaller than the fluid cell in order to achievecomputational stability and accuracy of results. Particle–fluidcoupling enables building of quite accurate models for diluteflows of particles. Higher particles concentrations in fluidform dense system, where particle collisions between eachother or with boundary impose an additional forces on theparticle motion and, consequently, fluid flow field. Densephase solid–fluid flow is reached when the time betweenparticle collisions is shorter than the time elapsed for fullmobilization of the fluid drag on the particle [12]. Therefore,it is important to consider viscous fluid lubrication on par-ticles collisions. Viscous fluid lubrication dissipates kineticenergy of two particles that approach each other, and maydisturb their motion, decrease particles velocities and evenresult in agglomeration.

2.3 Implementation of the lubrication contact modelin a DEM code

By applying the lubrication contact logic in DEM, it is possi-ble to extend the application of the lubrication model for thetwo-sphere collision to the system of multiple particles colli-sions. Specifically, the model can be used to study problemscharacterized by dense granular phase flow with frequent par-ticle collisions. There are currently several DEM codes thatare available for the discrete element simulation of granular

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and particulate material behavior in different areas of appli-cation. In this study, the two-dimensional Particle Flow Code(PFC2D) developed by Itasca [2] is used. PFC2D provides asimplified CFD capability to model solid–fluid interactionsby coupling a finite difference solution of the Navier–Stokesequations for fluid flow in porous media with DEM. A regularfinite difference grid on which the Navier–Stokes equationsare numerically solved for the fluid flow domain is superim-posed on top of the DEM domain (Fig. 1). Fluid drag forcesare solved in the CFD model and added to other forces act-ing on particles in the DEM code. In turn, fluid pressures arechanged by the movement of the particles in the DEM modeland the change in volume of the space occupied by fluidsbetween the particles.

The elasto-hydrodynamic contact model described aboveis implemented as a User Defined Model (UDM) in PFC2D

to account for the effect of fluid viscosity and lubricationas an additional independent drag force between particlesimmersed in a fluid. The lubrication force F(t) is directlyintroduced in the coupled DEM–CFD code for both particle-to-particle as well as particle-to-wall interactions. Lubrica-tion force is a distance force, which means that it gets acti-vated even before the particle surfaces get in contact. Theimplementation of this distance force is achieved by intro-ducing an apparent particle radius that is slightly offset fromthe real particle surface (Fig. 3). When two particles con-tact each other at their apparent radii, lubrication force isactivated and affects the subsequent particle motion towardseach other. This contact model is simplified and is imple-mented based on the following assumptions:

(a) The deformable solid particle surfaces (and the wall incase of particle-to-wall interaction) are assumed to besmooth and to be separated by a thin, incompressibleNewtonian fluid layer that behaves as a continuum.

(b) The spheres rebound as a result of repulsion along theline connecting their centers.

(c) The spheres are ideal elastic and the elastic deformationw(r, t) is negligible in the outer region where fluid inertiais important.

RT ( j)RT (i)RI ( j)RI (i)

rij

Fig. 3 The scheme of the apparent (RI) and real (RT) radii and theapproaching distance ri j

(d) Friction between particles is mobilized if the particlesare elastically deformed after contact (i.e., if the crite-ria for the rebound behavior are met). Inertia terms offluid motion equations are assumed to be small if thegap between particles, x0, is small, and x0/a is assumedmuch less than unity, because inertia of the fluid maybe neglected in the analysis even when the Reynold’sNumber Re is not small.

(e) Deformation is determined only for the instantaneousstress distributions, and elastic oscillations are neglectedand the duration of the impact is large compared to theperiod of the oscillation.

The elasto-hydrodynamic model is a step forward fromthe DEM default that mostly use the Kelvin–Voigt (spring-dashpot) contact model because it introduces the contact cri-teria for stick or elastic rebound of two spheres at their con-tact based on the surrounding fluid properties rather than onlydamped elastic rebound of the particles. The thin layer of vis-cous fluid between two particle surfaces before the contactacts as a cushion that slows down the initial particles veloc-ities and decreases the kinetic energy of the particles. Thelubrication force changes its magnitude as the gap betweenthe surfaces gets smaller (Eq. 8). If the balance of the lubri-cation force and the fluid approaching velocities is such thatthe particle slows down enough to near zero, the particlesmay stick next to each other, get trapped with the fluid, or ifthe contact never happens, the friction between the surfacesdoes not have to be mobilized. In short, it can be arguedthat agglomeration or sticking contact behavior is dependenton the lubrication force of the thin fluid layer between thecolliding particles.

The elastic rebound depends on the overlap of two par-ticles, if they are in contact with the real radii (ri j < rc,where rc is the distance between centers of two particleswhen their surfaces contact), and the lubrication dampingforce acts upon contact when it is activated. The active parti-cle radius is represented in DEM by the apparent radius thatis offset from the particle (Fig. 3). This implies a slight cor-rection of the particle density parameter. The apparent radiusenables the activation of the contact and the contact force isinvoked when the particles approach each other at a closedistance. The lubrication force is activated at that moment,and its magnitude depends both on the approaching veloc-ity and the distance between particles (or overlap). Duringthe time-stepping procedure, if the particles are close enoughthat they overlap with their real radii, then the elastic reboundand friction are activated.

The contact force logic is implemented using the followingsteps. First, the approaching distance ri j is calculated as:

ri j = ri + r j − di j (19)

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where di j is the overlap of the particles. The lubrication con-tact force Fc is then calculated depending on the approachingdistance ri j . If ri j ≥ 2rc = cri t :

Fc = ma = −6πμa2 vi j

xi j= −lub

vi j

(ri j − 2rc)(20)

If ri j < 2rc = cri t :

Fc = ma = k(2rc − ri j ) + cvi j (21)

where crit is the critical distance which is assumed to be twicethe real particle radius (i.e., cri t = 2rc), k is the springnormal contact stiffness, lub is the lubrication constant, cis the dashpot constant, and vi j is the relative approachingvelocity of two particles. The particle rebound can be elastic(only k) or it can have some damping included (k and c), buteither behavior only occurs upon contact and the collision ofthe particles. The link of Eqs. (20) and (21) with Eq. (8) isthe lubrication force. In physical representation the lubrica-tion force is transmitted from fluid to particle surface whentwo particles approach each other at small distance. In orderto avoid this complicated particle–fluid interaction, the thinlayer of fluid responsible for the lubrication effect is mod-eled by introducing an outer shell contact layer around theparticle, which gets activated before the true particle surfacescome in contact.

The logic for the lubrication model was implemented inPFC2D using its FISH programming language and defininga new contact model within the source code using C++. Therequired model parameters are the dynamic fluid viscosity,particle radius, thickness of the outer particle shell respon-sible for the lubrication effect, particle surface friction coef-ficient, and particle contact normal and shear stiffness. Therelative normal velocity between two approaching particlesand the distance between particles radii are directly calcu-lated by the DEM code.

3 Model validation

The lubrication contact model implemented in PFC2D is val-idated by determining the coefficient of restitution that themodel predicts for the rebound of particles dropped on asmooth horizontal surface. The coefficient of restitution er

of two colliding objects is defined as the fractional value rep-resenting the ratio of speeds of the particles after and beforean impact taken along the line of the impact. In other words,er is a measure of the damping between two particles. First, aparticle is dropped with gravitational acceleration in PFC2D

from several different heights towards a smooth horizontalwall and allowed to rebound back up into the air. Second,the maximum height of particle rebound was measured. Themeasured heights were used to calculate the coefficient ofrestitution er as:

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Coe

ffic

eint

of

rest

itut

ion,

e(-

)

Particle impact velocity, vimp (m/s)

1cp5cp10cp0.00510cp

FLUID

Fig. 4 Particle drop test results for a granular particle with diameterd = 0.8 mm and different fluid viscosities

er = v2

v1=

√2gh2√2gh1

=√

h2

h1(22)

where v2 is the velocity of particle after the impact with walland v1 is the velocity of the particle before the impact with thehorizontal wall, g is gravity, h1 is the initial particle heightand h2 is the height at which the particles stops after rebound.

It was assumed that both the particle and wall surfaces arewetted with fluid in the model. Figure 4 shows the results ofthe coefficient of restitution measurements in PFC2D. Sincelubrication has the effect of damping the particle collisionswhile the particle is approaching the wall and outer apparentradius is active, it is important to adjust the time-steppingprocedure manually. The time-step of the model was set upto meet a criterion of being sufficiently small in order toallow particle collisions to happen across several time-steps.Figure 4 shows the results of a series of particle drop testsfor a medium sand particle with radius r = 0.4 mm using theuser-defined contact model. DEM simulation was run for fivesets of dynamic fluid viscosities (μ = 0.001 to 0.050 Pa s).The coefficient of restitution shown on the ordinate in Fig. 4was calculated using Eq. (22) and particle impact velocitiesshown on the abscissa were obtained from the original dropheight and gravity.

The user-defined contact model yields non-linear coeffi-cient of restitution curves that differ for each fluid viscos-ity. The non-linearity of the particle response after contact iscaused by applying the Eq. (8) to the model. Lubrication forcemagnitude is proportional to the particle approaching veloc-ity and to fluid dynamic viscosity, and increases as the dis-tance between surfaces decreases. Because lubrication actsbefore the particle surface contacts the wall, it changes theoriginal impact velocity. Therefore, for a very small value ofthe impact velocity the lubrication force is able to completelystop the particle motion and to significantly affect the coef-ficient of restitution. However, for a particle dropped from

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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.0 10.0 100.0 1000.0

Coe

ffic

ient

of

rest

itutio

n, e

(-)

Collision Stokes number, St (-)

DEM model (0.001 Pa s)





Yang and Hunt (2006)

Joseph et al. (2001)

Zhang et al. (2005) (a)

Zhang et al. (2005) (b)

Zhang et al. (2005) (c)

Fig. 5 PFC2D results using the user-defined lubrication contact modelfor wall and particle (r = 0.4 mm) after the theory developed by Daviset al. [15] and compared with the experimental results of Yang andHunt [20] for identical spheres impact (steel, glass, Delrum), Josephet al. [19] for particle–wall impact (glass particle on zerodur wall inwater–glycerol mixture) Zhang et al. [18] for the study of effect ofparticle surface roughness: (a) hmin/h0 = 1/5; (a) hmin/h0 = 1/10;(a) hmin/h0 = 1/20; where hmin is surface asperity height due toroughness and h0 is particle radius

a larger height, the lubrication does not completely stop theparticle but only decreases its kinetic energy. As expected,the impact energy gets more efficiently dissipated at higherthe fluid viscosity. For low fluid viscosity lubrication effectis almost negligible, while for higher viscosity particle wasable to sustain the rebound significantly. In comparison, thecoefficient of restitution of the commonly used Kelvin–Voigtcontact model in DEM does not depend on fluid propertiesand impact velocity. In other words, the coefficient of resti-tution has a constant value of 1 for any value of approachingvelocity if the damping is not part of the model, or it can havesome other value less than one, but constant, depending ofthe damping value. We chose the restitution coefficient andthe Stokes number as parameters for the evaluation of lubri-cation incorporation in DEM. Those parameters provide asufficient insight into the particles contact behavior and theyhave been previously used in both theoretical and experimen-tal work for lubrication evaluation.

The validity of the predicted non-linear coefficient of resti-tution and its dependency with fluid viscosity is verified bycomparing the results shown in Fig. 4 with experimental andtheoretical data. Individual particle-to-particle collisions andparticle-to-wall collisions of particles immersed in fluid havebeen examined over the past two decades and some of theresults are summarized in Fig. 5 and compared to the PFC2D

model results. These experimental results are in addition tothe theoretical development of the collision model [15,16]that was used in this study. Results are presented in Fig. 5 arein form of the coefficient of restitution as function of Stokesnumber St, which is defined as:

St = mv1

6πμr2 (23)

where m is the particle mass and v1 is the particle approach-ing velocity. Thus, the coefficient of restitution curve is nownormalized with the values of fluid viscosity, impact veloc-ity, particle mass and particle radius. Using the Stokes num-ber enables direct comparison of PFC2D simulation resultswith previous theoretical and experimental results on sphereswith different sizes and materials. Stokes number representsa relation between particle inertia forces and fluid forces, andin case of the single-particle Stokes number, the numeratorprovides a measure of available momentum in the solid phasethat sustains particle motion through the liquid.

The non-linear dependence of the coefficient of restitutionand Stokes number is apparent in the Fig. 5, and the trendis verified for a range of dynamic fluid viscosities. Experi-mental results of drop tests with spheres made of differentmaterials show an agreement with theoretical expression. Inthe context of coefficient of restitution observations, a criti-cal value of Stokes number represents the borderline case atwhich the sphere has enough kinetic energy to rebound fromthe horizontal wall. For situations where the Stokes numberis below the critical value, the spherical particle rests on thewall surface after the impact, and conversely, it rebounds backif the Stokes number is higher. For very high Stokes number,coefficient of restitution reaches the maximum value of 1.0.Local elastic damping parameter is set to zero in the elas-tic part of the code (spring-dashpot model) and the sphererebounds after the impact to the almost same initial height.The elastic contact behavior is predominant in the latter case,and lubrication damping plays a minor role in post-impactsphere behavior.

The PFC2D model results using a range of dynamic fluidviscosities show smaller critical Stokes number compared toexperimentally obtained results for spheres collisions glass,steel and Delrum particles. From the PFC2D model, this valueis St ≈ 4, which corresponds to theoretical value used toformulate the lubrication contact model. In comparison thisvalue is approximately St = 10 for the experimental work(Fig. 5) on both particle–particle and particle–wall impact.Physically this discrepancy means that particles in the DEMmodel will start to rebound at little bit lower velocities thanis observed experimentally. The discrepancy between theexperimental and theoretical results are for critical Stokesnumber at zero coefficient of restitution was investigated inprevious research, and was found to be due to the elasticproperties and asperity interactions of the spheres [16,18,19].Zhang et al. [18] investigated theoretically and numericallythe effect of surface asperities to the restitution of coeffi-cient and their numerical results that expand previous the-ory given by Davis et al. [15] are superimposed in Fig. 5.PFC2D model results satisfactorily match the theory pro-posed by Zhang et al. [18]. However, the surface roughnesshas not yet been directly introduced to the PFC2D model, andthis aspect of modeling will be investigated in the future. In

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addition, although Zhang et al. provided an important contri-bution regarding the applicability of lubrication for gas–solidsystems, they were focused only on a single particle–wallcollision using a numerical solution. However, they did notperform DEM simulations to obtain the results presented intheir paper and did not look at the system of the particles.In contrast, the model presented in this paper is capable ofinvestigating the entire system of a dense multi-particle fluidsolid coupled flow, which is a new contribution.

The discrepancy between theoretical development of thelubrication effect by Davis et al. [15] and the correspondingexperimental results is reported in papers of Yang and Hunt[20], and Joseph et al. [19]. The intent of the authors was notto develop a new lubrication theory for particles, but insteadit was to build a more accurate tool for investigating a densesolids flow and transport in viscous fluid, which is differ-ent from the case investigated by Zheng et al. [18] for gas–solid system. The authors think that the coefficient of restitu-tion versus the Stokes number functional curve matches theexperimental results well, except for the shift of the parti-cle rebound value at the critical Stokes number. Capturingthe non-linear collision behavior with our new DEM con-tact model, compared to the standard Kelvin–Voigt model, ismore important than the small difference in particles reboundvalue.

In PFC2D, the same contact model is used for particle–particle and particle–wall collision. Yang and Hunt [20]investigated experimentally particle–particle immersed col-lision behavior, and they concluded that the same correlationbetween coefficient of restitution and Stokes number can befound for both particle–wall and particle–particle collisions[20]. As can be seen, the trend predicted by the model agreessatisfactorily with experimental and theoretical data. It can beconcluded that the new user-defined contact model as imple-mented in PFC2D captures the desired non-linear energy dis-sipation behavior typical for immersed particle–particle andparticle–wall collisions.

4 Example application of the model

Application of the lubrication model for submerged parti-cle collisions using the coupled DEM–CFD capability inPFC2D is demonstrated using a multi-particle sediment trans-port problem. The purpose of this model is to demonstrate thecapability of the proposed model to simulate the formation ofparticle clumps in a moving fluid while they are settling andare being transported along the bottom of a smooth horizontalbed. Clumps of particles settle slower than single particles,and the formation of clumps is expected to be more pro-nounced in a high viscosity fluid than in lower viscosity ones.In this example, the fluid velocity is set low enough to preventparticle lift-off after they have once settled at the bottom.

Smooth horizontal bed

v

v

Fig. 6 Initial and boundary conditions of 400 particles with d = 2 mmsubmerged in a flowing fluid above a horizontal bed, where v is theboundary fluid velocity

Figure 6 shows the model with initial random particledistribution, and superimposed numerical grid used to solvethe fluid flow equations. The particles represent sand grains(number of particles = 200) with uniform radius of r =1.0 mm which were randomly distributed inside a quadraticarea. A smooth horizontal wall was modeled at the bottomunder the particles to represent the channel bed above whichthe sediments are flowing. The fluid boundary conditions arezero velocity along the bottom of the model and a constantvelocity of 15 cm/s from the left to the right boundary. TheCFD mesh is superimposed on the particles and two-wayparticle–fluid coupling is active. Three different fluids withviscosities of 0.001 Pa s (water), 0.010 Pa s and 0.025 Pa swere used to show the effect of fluid viscosity on sedimenttransport and particle interactions behavior.

Three examples of particle transport in a fluid were per-formed using the coupled DEM–CFD model with the newuser-defined lubricated particle collision model. Figure 7shows particle settling and transport in water (μ=0.001 Pa s).The model was run for 7 s with a lower particle concentration(200 particles) and lower fluid velocity (v f = 0.15 m/s).Blue arrows represent cell fluid velocities, and black arrowsare particle velocity vectors. It can be seen in Fig. 7 thatalmost all the particles settled after 7 s. The reason for this isthat gravity forces dominate particle transport, and fluid dragforces play a smaller role. Particle collisions are damped, andparticles do not rebound from the bottom or from each other.Some of the particles formed clumps just before they settledat the base. Two-way fluid–particle coupling means that fluidmotion is affected by particles motion in each fluid cell andthe opposite. All the fluid cells that contain particles havedecreased velocity and this affects neighboring cells as well.As a result, more fluid flows above the particles where cellscontain only fluid than below where particles partly hinderfluid flow.

Figures 8 and 9 show the result of particle transport influid that has a higher viscosity (μ = 0.01 Pa s) than waterin the previous example. The model was run for 7 s withlower particle concentration (200 particles) and lower fluidvelocity (v f = 0.15 m/s), and for 2.3 s with higher particleconcentration (400 particles) and higher fluid velocity (v f =

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Fig. 7 Medium sand transport in water, d = 2 mm, v f = 0.15 m/safter t = 7 s

Fig. 8 Medium sand transport, 200 particles in a μ = 0.01 Pa s fluid,d = 2 mm, v f = 0.15 m/s after t = 7 s

Fig. 9 Medium sand transport, 400 particles in a μ = 0.01Pa s fluid,d = 2 mm, v f = 0.3 m/s after t = 2.3 s

0.3 m/s). Particle transport dominates particle motion, andformation of particle clumps is still not very pronounced.Particles that fell on the bottom are clumped next to each otherand do not rebound. Other particles are still being transportedwith the fluid. The effect of the DEM–CFD coupling and fluidflow around the particle pack can be observed here as well.

Figures 10 and 11 show results of particle motions in a highviscosity fluid (μ = 0.025 Pa s). The model was run for 20 swith lower particle concentration (200 particles) and lowerfluid velocity (v f = 0.15 m/s) and for 2.5 s with higher par-ticle concentration (400 particles) and higher fluid velocity(v f = 0.3 m/s). In high viscosity fluid, forward motion isslow and this is an expected effect of fluid viscosity. Com-pared to the model with water, fluid drag dominates over set-tling for particles. Formation of particle clumps is more visi-ble than in the examples shown in Figs. 8 and 9. The increase

Fig. 10 Medium sand transport, 200 particles in a μ = 0.025 Pa sfluid, d = 2 mm, v f = 0.15 m/s after t = 20 s

Fig. 11 Medium sand transport, 400 particles in a μ = 0.025 Pa sfluid, d = 2 mm, v f = 0.3 m/s after t = 2.5 s

Fig. 12 Medium sand transport, 400 particles in a μ = 0.025 Pa sfluid, d = 2 mm, v f = 0.3 m/s after t = 1.5 s

of the initial particle concentration causes more pronouncedformation of particle clumps in Fig. 11. Several particlesform a clump in the fluid and the clump is transported withthe fluid. Blue arrows that denote fluid cell velocities showthat fluid tend to flow around the newly formed clumps.

In order to compare the results of sediment transport mod-eling using soft sphere dry contact model with Kelvin–Voigtspring-dashpot behavior and the newly developed contactmodel that accounts for lubrication, the 200, 300 and 400 par-ticle model in high viscosity fluid was run for 3 s. Figure 12shows the results using lubricated particle collisions with 400particles, where clumping and agglomeration of particles isvisible, while on Fig. 13 no agglomeration of particles hap-pened. Along with agglomeration and formation of clumpsgoes the disturbance in fluid flow, accentuated by the high

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Fig. 13 Medium sand transport, 400 particles in a μ = 0.025 Pa sfluid, d = 2 mm, v f = 0.3m/s after t = 1.5 s using Kelvin–Voigtcontact model

Fig. 14 Cumulative averaged kinetic energy of 200 particles in a μ =0.025 Pa s fluid, d = 2 mm, v f = 0.3m/s after t = 2.5s using Kelvin–Voigt and lubrication contact model

velocity streams around particle packs. In Fig. 13, fluid hasaverage velocity with particles distributed over the fluid grid.

Kinetic energy of sand particles motion in fluid was mon-itored during the transport and sedimentation calculations.Figures 13, 14 and 15 show the increase of particle kineticenergy with time and averaged over the number of particles inthe simulation. As particles settlement develops forward, thekinetic energy curves approach the horizontal. Finally, afterall the particles have fallen to the bottom line bed, there is nomore increase in kinetic energy, and the kinetic energy curvebecomes horizontal at the end of the particles transport. Acomparison was made between simulations with the Kelvin–Voigt and the lubrication contact models for fluid dynamicviscosities of 0.001 to 0.025 Pa s. The use of lubrication con-tact model introduces additional kinetic energy dissipationinto the system, compared with the use of the Kelvin–Voigtcontact model. Higher fluid viscosities and higher particlesconcentrations increase the kinetic energy dissipation thatcan be seen as the difference between two model resultscurves in Figs. 13, 14, 15 and 16. Statistical comparisonbetween obtained kinetic energies of both contact modelsis shown in Fig. 17 for the simple physical system simulated

Fig. 15 Cumulative system-averaged kinetic energy of 300 particlesin a μ = 0.025 Pa s fluid, d = 2 mm, v f = 0.3 m/s after t = 2.5 susing Kelvin–Voigt and lubrication contact model

Fig. 16 Cumulative system-averaged kinetic energy of 400 particlesin a μ = 0.025 Pa s fluid, d = 2 mm, v f = 0.3 m/s after t = 2.5 susing Kelvin–Voigt and lubrication contact model

Fig. 17 Cumulative system-averaged kinetic energy obtained from thecontact model with lubrication shown as a percentage of the equivalentdata obtained using the Kelvin–Voigt contact model

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in this study. For the majority of the obtained data usingnumerical model for flow, transport and sedimentation ofa small batch of sand particles in viscous fluid, the usageof the contact model with lubrication exhibit approximately15–25 % decrease of system-averaged kinetic energy, com-pared to the data obtained using the Kelvin–Voigt contactmodel.

5 Conclusions

A new contact model for coupled DEM–CFD simulationof granular materials submerged in fluids was presented. Inthis new contact model, an additional lubrication force wasintroduced to account for the dependency of particle contactbehavior on the fluid properties, particle velocities and dis-tance between particle surfaces. The lubrication effect stemsfrom the fact that when two particles approach each other ina fluid, their kinetic energy decreases gradually because of alubrication effect associated with the thin fluid layer betweenthe particles. If the kinetic energy of approaching particlesis dissipated completely by lubrication, they remain attachedwith each other in a fluid forming a clump. Otherwise, if thekinetic energy of the particles is larger and not completelydamped by lubrication, the particles will contact each otherwith decreased approaching velocity.

The importance of a lubrication force in coupled DEM–CFD simulation of particle–particle and particle–wall inter-actions was demonstrated in two problems: (1) the coeffi-cient of restitution of particle–wall impact, and (2) sedimenttransport in a flowing fluid in a horizontal bed. The proposedlubrication contact model was validated against experimen-tal and theoretical data on the coefficient of restitution ofcolliding particles submerged in viscous fluid. It was shownthat the lubrication force was successfully implemented inPFC2D and yielded a coefficient of restitution that followsthe same nonlinear trend as the one theoretically calculatedand experimentally confirmed in previous research.

The simulation of the sediment transport problem hasshown the lubrication contact force effect in the context of amulti-particle system. The effect of lubrication is pronouncedfor flow of higher particle concentrations and higher fluid vis-cosities, showing 15–25 % decrease in the overall averagedkinetic energy of the system. High fluid viscosity causesdamped collisions and particles to stick to each other notrebound but stay in approximate vicinity as they are beingdragged by the fluid. Particles collide but do not reboundwhile being transported with the fluid. The formation of par-ticle clumps is more pronounced as the particle concentrationincreases, causing the fluid to flow around those clumps. Theresults show the importance of including lubrication effectson particle collisions in modeling dense phase flows.

Acknowledgments Financial support provided by the U.S. Depart-ment of Energy under DOE Grant No. DE-FE0002760 is gratefullyacknowledged. The opinions expressed in this paper are those of theauthors and not the DOE.

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discrete element modeling of non-linear submerged particle collisions

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