Comp. Part. Mech. (2014) 1:3–13DOI 10.1007/s40571-014-0001-z

Coupled DEM-LBM method for the free-surface simulationof heterogeneous suspensions

Alessandro Leonardi · Falk K. Wittel ·Miller Mendoza · Hans J. Herrmann

Received: 3 December 2013 / Accepted: 9 January 2014 / Published online: 13 February 2014© Springer International Publishing Switzerland 2014

Abstract The complexity of the interactions between theconstituent granular and liquid phases of a suspensionrequires an adequate treatment of the constituents them-selves. A promising way for numerical simulations of suchsystems is given by hybrid computational frameworks. Thisis naturally done, when the Lagrangian description of parti-cle dynamics of the granular phase finds a correspondencein the fluid description. In this work we employ extensionsof the Lattice-Boltzmann Method for non-Newtonian rhe-ology, free surfaces, and moving boundaries. The modelsallows for a full coupling of the phases, but in a simplifiedway. An experimental validation is given by an example ofgravity driven flow of a particle suspension.

Keywords Suspensions · Lattice-Boltzmann method ·Discrete element method

1 Introduction

Free-surface flows of heterogeneous suspensions are abun-dant in nature and technical applications. In principle theyare multiphase materials composed of a mixture of a liquidand of solid grains of various size. A multitude of interactionmechanisms between these two phases renders the problemof their description rather difficult. For example very smallgrains are bounded to the liquid by electrostatic forces, whilebigger ones interact mainly by viscous forces [10]. Addition-ally, inter-grain interactions give rise to the typical complexbehavior of granular matter. Often grains have a broad sizedistribution spanning over several orders of magnitude. Two

A. Leonardi (B) · F. K. Wittel · M. Mendoza · H. J. HerrmannInstitute for Building Materials, ETH Zurich,Stefano-Franscini-Platz 3, 8093 Zurich, Switzerlande-mail: aleonardi@ethz.ch

well known examples are mixtures of mud with sand androcks as well as suspensions of Portland cement, sand, andlarger aggregates, also known as fresh concrete. While thelatter is used for construction purposes [13], the former givesrise to devastating debris flows [23].

The simulation of such materials is based either on con-tinuum [35,39] or on particle methods [7], depending onwhether the investigated effects arise from the physics of thefluid or the granular phase. Continuum models are appro-priate when the rheological behavior of the material canbe captured by rheometry techniques and phenomenologicalconstitutive laws. However, many physical phenomena areeluded by this approach such as size and phase segregation.Examples can be found in concrete casting, where impropermixing or vibration leads to inhomogeneities in the physi-cal properties of the hardened concrete. In debris flows, sizesegregation leads to locally changing flow properties. A flowfront rich of large grains with high destructive power is com-monly observed, followed by a fluid in a more homogeneoustail. Describing this situation by continuum methods is quitedifficult or even physically inappropriate [24]. With particlemethods, such as the Discrete Element Method (DEM), thesephenomena can be naturally captured, which makes them anideal tool for the study of the complex behavior of granularmaterials.

A complete simulation tool requires a combination of bothcontinuum and particle description, which poses serious chal-lenges from a computational point of view. If granular andfluid phases are fully coupled, grains represent an irregu-lar and discontinuous boundary for the fluid domain. Therelative motion of the phases complicates the picture further,because it requires the management of continuously evolvinginterfaces. For these reasons, traditional CFD solvers suchas the standard Finite Element or Finite Volume methods,have enormous difficulties to tackle the issue. An attractive

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4 Comp. Part. Mech. (2014) 1:3–13

alternative is given by the Lattice-Boltzmann Method (LBM)[20,42], because of its extreme flexibility in the treatment ofelaborate boundary conditions, its ease of implementationin parallel computing and its superior scaling when com-pared to traditional solvers. For these reasons, much efforthas been payed to develop a framework for particle–fluid sys-tems combining the advantages of LBM and DEM. The earlyworks in this field are due to Ladd [26–28], who first coupledLBM and boundaries with imposed velocity. The basic modelwas enhanced by the use of the Immersed Boundary Method[12,30,31,36], by the inclusion of turbulence modeling [11],and extended further to the simulation of non-Newtonian rhe-ology models [16,29,45] and for free-surface flows [25,46].

Drawbacks of an approach based on the LBM are its lim-itation to low-Mach and relatively low-Reynolds flows, andthe necessity to rely on a regular grid, since irregular grids areknown to produce a complicated formalism and sometimesto lower the accuracy [44].

The paper is organized as follows: First a classification ofparticle suspensions by scales and types of physical interac-tions is given, before we explain how the dynamics of thedifferent phases is addressed. In Sect. 3 we summarize theDEM approach for the granular phase, followed by a sec-tion with a comprehensive description of the LBM solver forthe fluid phase. Section 5 explains necessary extensions tothe LBM for the simulation of suspensions, like fluid–particleinteraction, non-Newtonian rheology, or the representation offree surfaces. The experimental validation of the describedmodel completes the manuscript in Sect. 6, followed by abrief summary.

2 Dynamics of suspensions

The contribution of grains to the mechanics of the mixturecan be of different nature depending, among other factors, onthe grain size distribution. For a phenomenological classifi-cation, we use the term small scale when electrostatic forcesare dominant, medium scale when viscous forces prevail, andlarge scale when inter-particle collisional forces dominate[10]. Note that the length-scales defined by grain size are byno means absolute, but depend on other parameters such asthe concentration of particles, the viscosity of the liquid, andthe state of the system, since the same material can exhibitdifferent behaviors when sheared at different rates.

Small scale grain dynamics is governed mainly by inter-actions of electrostatic nature, e.g. Van der Waals forces.This finer part of the grains, together with water, forms acolloidal dispersion. A complete description of this kind ofmaterial can be found elsewhere [38]. For practical purposes,the mechanics of colloidal dispersions is reproduced by con-tinuum methods. A non-Newtonian model, however, is gen-erally required, since colloidal dispersions can exhibit both

shear-thinning behavior and plastic properties. In this workwe choose to employ the Bingham plastic, a fluid model witha yield stress, well-known for its wide applicability [37,47].It is described as

{γ = 0 if fluid does not yield(σ < σy),

σ = σy + μpl γ if fluid flows(σ > σy),(1)

where γ is the magnitude of the shear rate tensor, and μpl ,σy denote plastic viscosity and yield stress. In analogy toNewtonian fluids, an apparent viscosity (from now on, simplycalled viscosity) can be locally defined as the ratio of shearrate and shear stress

μapp = σ/γ = μpl + σy

γ. (2)

As the shear rate γ approaches zero, the viscosity becomesinfinite, giving a simple but efficient way to model plasticbehavior.

Medium scale grains are sufficiently big to elude theeffects of microscopic electrostatic forces and therefore needa different numerical treatment. For them the hydrodynamiceffects due to the viscous nature of the fluid become domi-nant. In analogy to the smaller scale, grains can be homog-enized in the fluid. Obtaining an appropriate rheologicalbehavior of the final mixture is however more difficult. Whenexperimental data is not available, the value of the viscositycan be approximated by constitutive relations. A review ofthese models can be found in Ref. [41].

Large scale grain dynamics is dominated by collisions.When collisional effects are not damped by viscosity, grainsgive rise to collective phenomena, such as segregation, forcepercolation or shock waves [21]. Bagnold defined a dimen-sionless number as the ratio of grain collisional and viscousstresses [2]. It reads

Ba = ρsd2s λ

1/2s γ

μ f, (3)

where μ f is the dynamic viscosity of the liquid, γ themagnitude of the shear rate, ρs and ds denote densityand characteristic diameter of the grains, and λs their lin-ear concentration (function of the solid fraction Cs asλs = 1/[(Cs,max/Cs)

1/3−1]with Cs,max the maximum solidfraction). As illustrated in Fig. 1, the Bagnold number is usedto distinguish two different regimes, where different rheo-logical laws are observed [43]: Mixtures with Ba < 40 aredominated by viscosity and therefore the shear stress growsproportionally to the shear rate. Mixtures with Ba ≥ 450are dominated by collisional effects and grains cannot behomogenized into a continuum description without a loss inthe descriptive capabilities of the method. An intermediaterange exists, where both effects are not negligible [22].

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Comp. Part. Mech. (2014) 1:3–13 5

Fig. 1 A typical grain sizedistribution and its effect on thedynamics of the mixture. Biggrains fall in the collisionalregime, small ones in theviscous regime. A transitionzone exhibits hybridcharacteristics. The Bagnoldnumber is calculated withμ f = 1.0 Pa s,ρ = 1,000 kg/m3, λ = 1,γ = 100 s−1

0.0001 0.001 0.01 0.1 1 10 100

Grain size [mm]

0

200

400

600

800

Bag

nold

Num

ber

[/]

0

20

40

60

80

100

Cum

ulat

ive

Pas

sing

Vol

ume

[%] Bagnold number

Grain size distribution

ViscousRegime

CollisionalRegime

40

450Intermediate

regime

The proposed model follows this classification to effi-ciently simulate and investigate suspensions. Small andmedium scale grains are homogenized for a fluid formula-tion with a continuum Bingham model. Large scale grainsare represented by a discrete description. The advantage ofthis method is that only a small portion grains is explicitlyrepresented. This fraction is representative both in terms ofmass and influence on the rheology of system.

3 Dynamics of the granular phase with the DEM

The granular phase is represented by the DEM, a well-established method for granular systems [5]. Every grain pis characterized as a Lagrangian element, with translationxp and rotation φp as degrees of freedoms. It is subjectedto multiple interactions that lead to a resultant force Fp andmoment Mp. These interactions can either be due to col-lisions, hydrodynamics, or volumetric forces and are func-tions of position, orientation, and velocity of the particles:Fp = Fp

(xp, xp, φp, φp

), Mp = Mp

(xp, xp, φp, φp

).

In the simplest case, DEM particles have spherical shapeallowing for fast contact detection and calculation of the over-lap ξp,q between particles p and q namely

ξp,q = ||dp,q || − Rp − Rq . (4)

Here dp,q denotes the distance between the center of thespheres with radii Rp, Rq . Unfortunately, for most practicalapplications spheres can only be used as a first approximationof the real particle shape. Note that spheres exhibit artificialmixing and rolling behavior, which is absent in natural systemthat are not composed of spheres. To overcome these effectswe use composite elements, created by aggregating a set ofspherical particles. While preserving the simplicity of thecontact calculation, composite elements allow for a morerealistic representation of granular effects, in particular inthe limit of dense concentrations.

Particle–particle interactions are written as the outcome ofcollisional events between particles. Although particles aregeometrically described as rigid spheres, the overlap ξp,q

between particles p and q is used to calculate collisionalforces and to represent the elastic deformation. We use thelaw for elastic spheres,

Fnp,q = 2

3

Y√

Ref f(1 − ν2

)(

ξ3/2p,q + A

√ξp,q

dξp,q

dt

)np,q , (5)

where Y and ν are the Young modulus and the Poisson’s ratioof the material, A is a damping constant [6], Ref f the effectiveradius defined as Ref f = Rp Rq/(Rp + Rq) and np,q thenormal vector of the contact surface. The tangential contactforce is considered to be proportional to the component ofthe relative velocity of the two spheres laying on the contactsurface ut

rel as

Ftp,q = −sign

(ut

rel

) · min(η||ut

rel ||, μd ||Fnp,q ||

)tp,q , (6)

with the tangential shear viscosity coefficient η and thedynamic friction coefficientμd , thus including Coulomb fric-tion. The tangential unit vector tp,q is obtained normalizingthe tangential relative velocity. Wall contacts are calculatedin a similar fashion.

The time evolution of the system is solved by integratingNewton’s second law,

m pxp = Fp, Jpφp = Mp − φp × (Jpφp

). (7)

While the translational motion is naturally solved in systemcoordinates, the rotational motion requires additional consid-erations. We use quaternion algebra rather that Euler anglesto represent the orientation of the elements, and calculate andinvert rotation matrices without singularities [8]. Newton’sequations in the body-fixed reference frame produce 6 × Nscalar equation, where N is the number of elements. The Gearpredictor-corrector differential scheme is used to integrate

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6 Comp. Part. Mech. (2014) 1:3–13

them [14]. During the predictor step, tentative values for par-ticle position, orientation, and their derivatives are computed,using a Taylor expansion of the previous time step values. Thepredicted values are then used to check for contacts, computecollisional forces, and solve Newton’s equations of motion.For the corrector step, the difference between the predictedvalues for acceleration and their counterpart resulting fromNewton’s equation is computed. This difference is used tocalculate the new corrected values for position, orientationand their derivatives.

The DEM time step should be small enough to resolve theparticle contacts. If tc is the collision time, then tDEM � tc,usually tDEM < 0.1tc. The collision time can be estimatedfor a Hertzian contact as

tc = 2.5

(m2

e f f

k2||unrel ||

) 15

, with k = 815

Y1−ν2

√Ref f , (8)

where mef f = m pmq/(m p + mq

)is the effective mass and

unrel denotes the normal relative velocity at the contact point

at the beginning of the collision.

4 Fluid dynamics with the LBM

The DEM described in the previous section is coupled withthe LBM for solving the fluid phase. LBM has been evolv-ing very fast in the last two decades and is considered tobe one of the most attractive alternatives to traditional CFDsolvers, especially when problems feature complex boundaryconditions. It originates from the Boltzmann kinetic theoryfor the evolution of molecular systems [3,9]. The fluid isdescribed using a distribution function, f (x, c, t), defined asthe probability density of finding molecules with velocity cat a location x and at a given time t .

In the LBM, the velocity space is discretized by afinite number of velocity vectors, ci , such that fi (x, t) ≡f (x, ci , t). We choose to employ the D3Q19 lattice cell con-figuration (3 dimensions and 19 velocities, see Fig. 2), whichprovides the required symmetries to correctly recover the

incompressible Navier–Stokes equations. In this work, forsimplicity, we use dimensionless lattice units (δx,y,z = 1,

δt = 1).The reconstruction of macroscopic physical variables such

as density ρ f and velocity u f can be done at every locationx and time t by computing the first two moments of the dis-tribution function fi (x, t)

ρ f (x, t) =∑

i

fi (x, t), u f =∑

i

fi (x, t)ci/ρ f (x, t). (9)

The distribution function evolves according to the LatticeBoltzmann Equation (LBE), which is written

fi (x + ci , t + 1) = fi (x, t) + Ωi (x, t). (10)

Ωi represents the collision operator, which in our case cor-responds to the linear approximation given by Bhatnagar–Gross–Krook [4],

Ωi (x, t) = f eqi (u f , ρ f ) − fi (x, t)

τ (x, t), (11)

where τ is the relaxation time and f eqi is the equilibrium

distribution function. The relaxation time is directly relatedto the viscosity of the fluid

μ f (x, t) = τ(x, t) − 1/2

3. (12)

For a Newtonian fluid, τ is a constant and global para-meter. However, as stated in Sect. 2, in order to representmost suspensions, a non-Newtonian formulation should beemployed. To do this, the relaxation time is treated as a localvariable τ(x, t). The equilibrium distribution function f eq

iis an expansion in Hermite polynomials of the Maxwell–Boltzmann distribution in the limit of small velocities [40].Using the local macroscopic velocity u f and density ρ f , thisyields

f eqi (u f , ρ f )=ρ f wi

(1+3ci · u f + 9

2

(ci · u f

)2− 3

2u f · u f

),

(13)

Fig. 2 a The regular spacediscretization of the lattice. bThe 19 discrete velocitiesallowed in the D3Q19 latticeconfiguration

(a) (b)c17

(a) (b)c177

c12

c16 c18

c15

c11

c14

c10

c13

c8

c2

c9

c4

c6

c1

c7c3

c5

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Comp. Part. Mech. (2014) 1:3–13 7

where the weights wi are constants that ensure the recoveringof the first and second moments of the distribution function(Eq. 9). For the D3Q19 lattice configuration they are

wi =⎧⎨⎩

1/3 for i = 11/18 for i = 2, . . . , 71/36 for i = 8, . . . , 19.

(14)

To introduce an external force, we employ the schemedeveloped by Guo et al. [19], which consists in modifyingEq. 10 as

fi (x + ci , t + 1) = fi (x, t) + Ωi (x, t) + Fi (x, t), (15)

where Fi (x, t) is an additional distribution function due tothe force field F, which can be calculated in a similar fashionas the equilibrium distribution,

Fi (x, t) = wi

(1 − 1

2τ

) [3(ci − u f

) + 9ci(ci · u f

)]F.

(16)

With this technique, the computation of the macroscopicvelocity field in Eq. 9 also needs to be modified,

u f =(∑

i

fi (x, t)ci + F/2

) /ρ f (x, t). (17)

The described approach reproduces the Navier–Stokesequations in the incompressible limit. The pressure is directlycomputable from the density as

Pf (x, t) = ρ f (x, t) · c2s , (18)

where cs is the speed of sound of the fluid, which correspondsto cs = 1/

√3 (lattice units). The stability and accuracy of

the LBM are guaranteed for small Mach numbers, Ma ≡||u||/cs � 1.

For every time step one first calculates the macroscopicvariables, using Eq. 9, and the corresponding equilibriumdistribution, from Eq. 13. Then, one uses Eq. 10 to evolvethe distribution function, which provides the new density andvelocity of the fluid for the next time step. Being solvedmostly at a local level, the scheme can be easily implementedin a parallel environment [33].

5 Extensions of the LBM for the simulationof suspensions

To widen the range of applicability of the model to hetero-geneous suspensions, we need to incorporate a few morefeatures. First, we introduce no-slip moving boundaries, nec-essary for the coupling with the DEM, and second, in Sect.

5.2, we extend the model to simulate free surfaces. Finally,Sect. 5.3 describes the method for non-Newtonian formula-tions.

5.1 Coupling with particles

The coupling with the DEM and the treatment of no-slipboundary conditions are performed at a local level by modi-fying the LBE. Lattice nodes are divided into fluid and solidnodes, the latter ones representing particles and walls (seeFig. 3). Solid nodes are inactive, i.e. on them the LBE is notsolved. No-slip is performed with the so-called bounce-backrule: every time a distribution function fi (x, t) is streamingin the direction i towards a solid node, it gets reflected backin the opposite direction i ′. If the boundary is moving, thereflected distribution needs to be corrected as

fi ′(x, t + 1) = fi (x, t) − 6wiρ f uw · ci , (19)

where uw is the local velocity of the wall at the bounce-backlocation. If the wall represents the surface of a particle, thelocal velocity can be obtained as

uw = up + rw × ωp, (20)

where up and ωp are the linear and angular velocity of theparticle, and rw is the vector connecting its center of masswith the bounce-back location. The momentum exchangeexperienced by the reflected distribution can also be used tocompute the force exerted on the wall when integrated overall bounce-back locations,

Fp =∑ (

2 fi (x, t) − 6wiρ f uw · ci)

ci . (21)

Solid boundaries treated this way are located halfwaybetween solid and active nodes. This technique was devel-oped for moving boundaries by Ladd [26] and Aidun andLu [1].

Particles move over a fixed, regular grid. Of course thenode classification into fluid and solid is not fixed but needsto be updated. Following the particle motion, fluid nodes arecreated (deleted) in the wake (front) of moving particles. Themacroscopic density and the velocity of the newly creatednodes are calculated as the average over the values in theneighborhood as initial values for the distribution functionfi (x, t) through Eq. 13. Deleted fluid nodes are convertedto solid ones and therefore made inactive. Both processesintroduce small variations in the global mass and momen-tum. However, due to the fact that all our simulations areperformed in the incompressible limit (variation of densityare very small), and that fluid nodes close to a particle pos-sess nearly the same velocity as the particle, we expect thesevariations to be negligible. Another problem is the represen-tation of the particle boundaries on the regular lattice, whichleads to a zig-zag approximation of the spherical shapes. An

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8 Comp. Part. Mech. (2014) 1:3–13

Fig. 3 Sketch showing howparticles or solid objects arediscretized on the regular lattice.The free-surface is treated in asimilar way, with a special typeof nodes defining the interface

Solid nodes

Liquid nodes

Solid-liquid interface

Particle real border

Gas nodes

Liquid nodes

Gas-liquid interface

Real interface

Interface nodes

alternative way to overcome these problems is the use of theImmersed Boundary Method [12] or of a fictitious domain[17,18]. Both methods are more precise and smooth the illeffects of particles traveling though the lattice. At the sametime, they require additional computations and are thereforeavoided following the spirit of this paper.

When two particles approach each other, the distancebetween the surfaces can become smaller than the lattice nodespacing, resulting in an imprecise resolution of the collisionprocess. To overcame this problem, we use the lubricationtheory of Nguyen and Ladd [34]. In this theory, when twoparticle are moving with a relative velocity urel , the correc-tion force

Flubp,q = −6μ f ||un

rel ||R2e f f

(1/sp,q − 1/dlub

)np,q , (22)

is added, where sp,q = −ξp,q is the distance between theparticle surfaces and dlub denotes a cut-off distance abovewhich no force is computed.

5.2 Free surface representation

We employ the mass tracking algorithm described in Refs.[9,32] which, despite its simplicity, leads to a stable and accu-rate surface evolution. Fluid nodes are further divided intoliquid, interface and gas nodes: Liquid and interface nodesare considered active, and the LBE is solved. The remainingnodes are the gas nodes and are inactive, with no evolutionequation. Liquid and gas nodes are never directly connected,but through an interface node (see Fig. 3).

An additional macroscopic variable for the mass m f (x, t)stored in a node is required, defined as

⎧⎨⎩

m f (x, t) = ρ f (x, t) if the node is liquid,0 < m f (x, t) < ρ f (x, t) if the node is interface,m f (x, t) = 0 if the node is gas.

(23)

The mass is updated using the equation

m f (x, t + 1)=m f (x, t)+∑

i

αi [ fi ′(x+ci , t) − fi (x, t)] ,

(24)

where αi is a parameter determined by the nature of the neigh-bor node in the i direction,

αi =

⎧⎪⎪⎨⎪⎪⎩

12

[m f (x, t)+m f (x+ci , t)

]if the neighbor node is interface,

1 if the neighbor node is liquid,

0 if the neighbor node is gas.

(25)

When the mass becomes zero (m f (x, t) = 0), the interfacenode is transformed into gas, with all liquid nodes connectedto it becoming interface. Analogously, an interface nodewhose mass reaches the density (m f (x, t) = ρ f (x, t)) istransformed into liquid, and all connected gas nodes becomeinterface. However, due to the discrete integration, theseequalities are not in general satisfied. The surplus of mass isequally distributed to the neighboring interface nodes, con-serving the total mass of the system.

Because gas nodes are not active, there are no distribu-tion functions streaming from gas nodes to interface nodes.These missing distribution functions are computed from themacroscopic variables at the interface, atmospheric densityρatm and interface velocity uint , as

fi ′(x + ci ′ , t + 1) = f eqi (uint , ρatm) + f eq

i ′ (uint , ρatm)

− fi (x, t). (26)

Note that this implies that gas nodes have the same macro-scopic velocity as the connected interface nodes.

5.3 Bingham plastic rheology model

The presence of the small particle fraction in the fluid leads tonon-Newtonian behavior, that needs to be considered. For theLBM this implies that the relaxation time τ is not a globalparameter for the system, but rather τ = τ(x, t). A non-linear dependency of viscosity, and thus of τ , on the shear raterequires an explicit computation of the shear rate tensor. Thiscan be done with ease in the LBM from the non-equilibriumpart of the distribution functions,

γab(x, t) = 3

2τ(x, t − 1)

∑i

ci,aci,b(

fi (x, t) − f eqi (x, t)

).

(27)

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Comp. Part. Mech. (2014) 1:3–13 9

She

ar s

tres

s σ

μmax

μpl

μmin

Shear rate γ

Range of LBMacceptable μ

Bingham model

σy

Fig. 4 Representation of the rheology model employed for plastic flu-ids. The approximation of the Bingham model is limited by the max-imum and minimum acceptable values for the relaxation time τ andtherefore for the viscosity μ f

With the second invariant of the shear rate tensor

�γ (x, t) =∑

a

∑b

γabγab, (28)

the magnitude of the shear rate is calculated as

γ (x, t) =√

2�γ (x, t). (29)

This can be included in any constitutive equation for purelyviscous fluids. As outlined before (Sect. 2), we choose theBingham constitutive model and get a new form of Eq. 12for the explicit update of τ ,

τ(x, t) = 1

2+ 3

(μpl + σy

γ (x, t)

). (30)

The accuracy and stability of LBM are guaranteed onlyover a certain range of values for τ . This limits the applica-bility of Eq. 30, because τ diverges when γ → 0. Following

Švec et al. [46], we use a simple solution to this problem,imposing that τmin ≤ τ(x, t) ≤ τmax . Reasonable valuesfor τmin and τmax are, respectively, 0.501 and 3.5. The con-stitutive equation arising from this approach is that of a tri-viscosity fluid (see Fig. 4). If μ f,min < μpl the model repre-sents a bi-viscosity, and if μ f,max μpl , the approximationof the Bingham model is fair. With these extensions the modelis complete and we can address examples.

6 Experimental validation by a gravity-driven flow

The capabilities of the model are shown by comparing withan experiment, featuring a free-surface flow of a suspen-sion under the effect of gravity. We employ fresh concrete,since it poses all the challenges necessary to validate themethod: a non-Newtonian rheology and an irregular granu-lar phase. The cement paste is obtained with a commercialPortland cement of type CEM I 42.5N. Water is added untila water/cement ratio of 0.4 is reached. The rheology of theobtained paste is measured with a coaxial rotational viscome-ter Haake RV20. The measurement procedure consists in theuniform shearing of the paste at 200 s−1 for 120 s, followedby a shear rate continuous ramp from 0 up to 200 s−1 occur-ring over 120 s [15]. The obtained rheological curve is shownin Fig. 5.

The paste is mixed with 1,000 silica rounded pebbleswith radius R = 4.0 ÷ 8.0 mm. The total weight of thegrains is 2.629 kg, and the density ρs = 2,680 kg/m3. Thecomponents are mixed in a bowl until homogenization andthen vibrated for degassing. The final mixture is poured in a150 × 150 mm rectangular box, open on top and bottom andpositioned over a wooden board inclined at 15◦. The boardsurface is upholstered with sandpaper and wetted before thestart of the test. The test is performed by steadily lifting thebox, and letting the sample spread on the board under the soleeffect of gravity. The flow falls in the intermediate regime (see

Fig. 5 Rheology test on thefresh cement paste. A linearBingham approximation is usedto fit the data, obtaining μpland σy

0 40 80 120 160 2000

2

4

6

8V

isco

sity

[m2 /

s]

0

20

40

60

80

100

Stress (Rheometer)Stress (Bingham approx.)Viscosity (Rheometer)Viscosity (Bingham approx.)

ply

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10 Comp. Part. Mech. (2014) 1:3–13

Fig. 6 Experiment setup with an inclined plane of 933 × 700mm. Theinternal size of the box is 150×150 mm. The dark gray area representsthe surface covered by the sandpaper

Fig. 1). Collisional effects are therefore not dominant, but stillimportant. The geometry of the test is illustrated in Fig. 6,and Fig. 7a is a picture of the final deposition of the sample.

The same environment is set up with a simulation on anLBM lattice of 350 × 250 × 80 nodes, with the lattice spac-ing corresponding to 2.0 × 10−3 m in physical units. Theinitial configuration of the fluid is a cube with edge lengthof 0.15 m, corresponding to 75 × 75 × 75 liquid nodes. Thepebbles are represented with 1,000 discrete elements, each

composed of 4 spheres with tetrahedral structure. The totalnumber of spheres is 4,000. The box is represented by a set ofmoving walls and is set as solid boundary both for fluid andgranular solvers. The lifting speed of the box is 0.15 m/s.The properties of the fluid are obtained from the viscometerdata, as represented in Fig. 5. A good fit is obtained with aBingham model with plastic viscosity μpl = 0.15 Pa s andyield stress σy = 62 Pa. The model is imprecise for lowershear rates, which is one of the limitations of the chosenlinear approach. Fig. 8 shows the results of the simulationon the longitudinal cross section of the sample. The evo-lution of the shear rate and the particle distribution can betracked continuously. The final shapes of the experimentaland numerical solution are compared in Fig. 7c, showingexcellent agreement.

A good compromise between stability and speed isobtained with a time step of tLBM = 3.0 × 10−5 s.This sets the maximum allowable speed in the system as0.667 m/s. The parameters in lattice units are then viscosityμLBM

pl = 6.25×10−4, relaxation time τLBMy = 7.75×10−6,

and gravity ||gLBM|| = 4.41 × 10−6. The simulation isstopped when 95 % of the fluid has reached the maximumviscosity. The total simulation time is 63 hours, with a paral-lel run on 4 cores with an Intel Xeon E5-1620 processor at3.60 GHz.

Fig. 7 Final shape of theflowing mass. a, d Numericalshape; b, e Experimental shape.Fluid mass opacity in thenumerical shape is lowered for abetter visualization of particles;c, f Comparison of numericalshape (solid line) andexperimental shape (dashedline). The background grid has5 cm spacing

(c)

(d)

(f)

(e)

(b)(a)

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Comp. Part. Mech. (2014) 1:3–13 11

Fig. 8 Dynamic viscositycontour on the longitudinalcross section of the simulation.Particles are represented in lightgray and walls in dark gray. Theyielded region of the fluid growsfrom a small portion close to thebox walls to the whole sampleduring the first part of thesimulation. When a newequilibrium is reached, theyielded region reduces and theflow is slowed

0.00 sec

0.24 sec

1.92 sec

1.00 sec

0.48 sec

3.89 sec

Viscosity [Pa s].

0.00.051 0.050.0010.002

YieldedUnyielded

7 Summary

In this paper a model for the simulation of the flow of sus-pensions was proposed. The multiscale nature of the modelis justified by the different interaction mechanisms actingbetween the liquid and the granular phase. A practical meanof phenomenological classification of interactions is givenby the Bagnold number: small grains are considered to begoverned by the viscous nature of the liquid and are mod-eled as part of the fluid phase itself with the use of a plasticnon-Newtonian formulation. Grains with a sufficiently largesize are dominated by collisional mechanisms. This is mod-eled with a two-way coupling between fluid and grains, alongwith the resolution of particle contacts.

The problem was solved with a hybrid of the Discrete Ele-ment Method for grains and the Lattice-Boltzmann Methodfor fluids. A combination of the most successful advancesin these methods was employed. The mass-tracking algo-rithm allows an inexpensive way to simulate free surfaces,while the variable relaxation time formulation can repro-duce non-Newtonian constitutive laws. The hydrodynamicinteractions with the granular phase were fully solved with

the bounce-back rule for coupling non-slip moving bound-aries and fluid. The proposed model finds its best applicationin the simulation of real flows and in particular of hetero-geneous suspensions with a granular phase that features acomplete size distribution, due to its multiscale nature. Theintrinsic advantages of the Lattice-Boltzmann solver, withits high-level performance, and its relatively simple imple-mentation make it a good choice for the fast developmentof such methods. Moreover, the core of the solver worksat a local level, making the parallelization of the code easyand natural. Grain–grain interactions were solved with aDiscrete Element Method. We assured that the scaling ofthe particle solver was not too far from the almost linearperformances of the fluid solver. The Hertzian contact lawwas used, and a formulation for non-spherical particles wasincluded. The capabilities of the approach were shown bycomparing to an experimental free-surface flow of a freshconcrete sample. An excellent agreement between numeri-cal and experimental data was found in the comparison ofthe final shape of the sample. The results of the simula-tion can provide insight into the mechanics of the flow. Thespatial distribution of particles can be tracked, along with

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the variables of the flow: velocity, pressure, shear rate andviscosity.

Another challenging application of the model is theprediction of debris flows, which can hardly be assessedexperimentally. Future works will focus on the rheology ofdebris materials, and on the full simulation of events fordeeper physical understanding, and on techniques for thedesign of effective protection measures.

Acknowledgments The research leading to these results has receivedfunding from the European Union (FP7/2007-2013) under GrantAgreement No. 289911. The authors are grateful for the support ofthe European research network MUMOLADE (Multiscale Modellingof Landslides and Debris Flows).

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