cUnite

Article history:Received 28 August 2014Received in revised form 30 December 2014Accepted 4 February 2015Available online 28 February 2015

Keywords:CFDDEMCoarse grainingMulti-scale modeling

Nielsen, 1992; Sifferman et al., 1974; Yang, 1998). In particular, thiswork is concerned with the dense-phase regime, where both theuidparticle interactions and the inter-particle collisions playimportant roles. For examples, the dense particle-laden ows inuidized beds in chemical reactors and the sheet ows in coastalsediment transport both fall within this regime.

tions for the twoscretizatiorces betweepresented

TFM formulation, although the particle phase properttake into account certain particle characteristics. Therefocomputational cost of the two-uid model is relatively low, andthus this method is widely used in industrial applications, wherefast turnover times are often critical requirements. However, thephysics of the particle or granular ows are fundamentally differ-ent from that of uids. Among many other difculties associatedwith the TFM, a critical issue with this approach is that a universalconstitutive relation for the particle phase that is applicable to

Corresponding author. Tel.: +1 540 231 0926.E-mail addresses: sunrui@vt.edu (R. Sun), hengxiao@vt.edu (H. Xiao).

International Journal of Multiphase Flow 72 (2015) 233247

Contents lists availab

l

l seand engineering, e.g., sediment transport in rivers and coastaloceans, debris ows during ooding, cuttings transport inpetroleum-well drilling, as well as powder handling andpneumatic conveying in pharmaceutical industries (Iverson, 1997;

sets of mass and momentum conservation equaphases are solved with mesh-based numerical dicoupling terms accounting for the interaction fophases. Particles are not explicitly resolved or rhttp://dx.doi.org/10.1016/j.ijmultiphaseow.2015.02.0140301-9322/ 2015 Elsevier Ltd. All rights reserved.n, withen thein theies dore, theIntroduction

Particle-laden ows: Physical background and multi-scale modeling

Particle-laden ows occur in many settings in natural science

Various numerical simulation approaches have been proposedfor particle-laden ows in the past few decades. Among the mostestablished and most commonly used is the Two-Fluid Model(TFM) approach, which describes both the uid phase and the par-ticle phase as inter-penetrating continua (Sun et al., 2007). The twoIn this work, a coarse-graining method previously proposed by the authors in a companion paper basedon solving diffusion equations is applied to CFDDEM simulations, where coarse graining is used toobtain solid volume fraction, particle phase velocity, and uidparticle interaction forces. By examiningthe conservation requirements, the variables to solve diffusion equations for in CFDDEM simulations areidentied. The algorithm is then implemented into a CFDDEM solver based on OpenFOAM and LAMMPS,the former being a general-purpose, three-dimensional CFD solver based on unstructured meshes.Numerical simulations are performed for a uidized bed by using the CFDDEM solver with thediffusion-based coarse-graining algorithm. Converged results are obtained on successively renedmeshes, even for meshes with cell sizes comparable to or smaller than the particle diameter. This is acritical advantage of the proposed method over many existing coarse-graining methods, and would beparticularly valuable when small cells are required in part of the CFD mesh to resolve certain owfeatures such as boundary layers in wall bounded ows and shear layers in jets and wakes. Moreover,we demonstrate that the overhead computational costs incurred by the proposed coarse-grainingprocedure are a small portion of the total computational costs in typical CFDDEM simulations as longas the number of particles per cell is reasonably large, although admittedly the computational overheadof the coarse-graining procedure often exceeds that of the CFD solver. Other advantages of thediffusion-based algorithm include more robust and physically realistic results, exibility and easyimplementation in almost any CFD solvers, and clear physical interpretation of the computationalparameter needed in the algorithm. In summary, the diffusion-based method is a theoretically elegantand practically viable option for practical CFDDEM simulations.

2015 Elsevier Ltd. All rights reserved.a r t i c l e i n f o a b s t r a c tDiffusion-based coarse graining in hybridApplications in CFDDEM

Rui Sun, Heng Xiao Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA 24060,

International Journa

journal homepage: www.eontinuumdiscrete solvers:

d States

le at ScienceDirect

of Multiphase Flow

vier .com/ locate / i jmulflow

of Mdifferent ow regimes seems to be lacking despite much researchon this topic (Sun and Sundaresan, 2011). This difculty stemsfrom the fact that unlike the ow of real continuum uids (gasesor liquids) where strong separation of scales justies the contin-uum description, in granular ows the scale separation is weak(Glasser and Goldhirsch, 2001), i.e., the representative volumeelement can be of a similar order of magnitude to particlediameters. As such, a continuum description of the particle phasewould suffer from these intrinsic difculties. Other drawbacks ofthe TFM approach include the difculty in representing particleswith a continuous distribution of diameters or densities and thereliance on empirical models of uidparticle interactions, amongothers (Sun et al., 2009).

On the other hand, direct numerical simulations based onLattice-Boltzmann method (Yin and Koch, 2008) or by solvingNavierStokes equations with uidparticle interfaces fullyresolved (e.g., via immersed boundary method (Kempe et al.,2014)) are computationally expensive. The DNS methods are cur-rently limited to systems of O103 particles in spite of sustainedrapid growth of available computational resources in the past dec-ades. Interestingly, this difculty is also due to the fact that multi-ple scales do exist in the particle-laden ow problem, although thescale separation is weak, as explained above. That is, the scales ofconcern are several orders of magnitude larger than the particlediameter dp, and thus the simulated system may contain a largenumber of particles. It is expected that DNS will not be affordablefor simulating realistic dense particle-laden ows in the nearfuture, where the number of particles can be O106 or even more.

In view of the multi-scale nature of and the weak scale-separation in dense particle-laden ows, the continuumdiscreteapproach seems to be a natural choice. In this approach, continuummodel is used to describe the uid phase, while the particle phaseis described by the Discrete Element Method (DEM), where parti-cles are tracked individually based on Newtons second law in aLagrangian framework. DEM was rst used to model granular owwithout interstitial uids in geotechnical engineering in the 1970s(Cundall and Strack, 1979). The hybrid CFDDEM approach tomodel particle-laden ows was attempted in the 1990s (Tsujiet al., 1993). Traditionally the locally averaged NavierStokesequations are adopted as the continuum model (Anderson andJackson, 1967), leading to a hybrid method commonly referred toas CFDDEM (Computational Fluid DynamicsDiscrete ElementMethod). Recently, Large Eddy Simulation (LES), a CFD technologybased on the solution of ltered NavierStokes equations, has beenused as the continuum uid model, leading to hybrid LESDEMsolvers (Zhou et al., 2004). Other variations in the category ofcontinuumdiscrete solvers include those using Smooth ParticleHydrodynamics (SPH) or Lattice-Boltzmann for the uid ow(Han et al., 2007; Sun et al., 2013).

Coarse graining in continuumdiscrete particle-laden ow solvers

In all these continuumdiscrete particle-laden ow solversmentioned above including CFDDEM and LESDEM, one needsto bridge the continuum-based conservation equations for the uidphase and the discrete description of the particle phase.Specically, the presence and the dynamic effects of the particleson the uid are taken into account in the uid continuity andmomentum equations through the macroscopic quantities of theparticle phase, e.g., solid volume fraction es, solid phase velocityUs, and soliduid interphase forces F

fp. These Eulerian eld quan-tities are not solved for in the continuum-scale solver, but need tobe obtained from the discrete particle information (i.e., individual

234 R. Sun, H. Xiao / International Journalparticle locations x, particle velocities u, interaction forces onindividual particles f fp). The process of obtaining macroscopicquantities from particle-scale quantities is referred to as coarsegraining in this work.

In CFDDEM or LESDEM solvers the uid equations arediscretized with mesh-based numerical methods such as nitevolume for nite element methods. From here on we focus ourdiscussions on CFDDEM for brevity. However, note that thediscussions presented and the methods proposed in this work shallbe equally applicable to LESDEM solvers, and may be useful forother continuumdiscrete methods such as SPHDEM andLB-DEM for particle-laden ows. Another method that is closelyrelated to CFD/LESDEM is the Particle-in-Cell (PIC) method, whichis widely used in plasma simulations (Dawson, 1983),where individual physical particles (electrons, ions, etc.) orsuper-particles that represent a number of physical particles ofsimilar properties are tracked in a Lagrangian framework. Theinteractions among the particles are computed not in a pair-wiseway but via electric and magnetic elds that are Eulerian eldquantities computed from the particle distribution data. The coarsegraining is an important ingredient in the PIC method, and theproposed method can be of relevance there.

Details on how the solid phase quantities interact with theuid phase quantities in CFDDEM will be presented inSection Methodology after the mathematical formulation for themethod is introduced. In CFDDEM solvers the Eulerian eldquantities of the uid phase become cell-based quantities afternumerical discretization. Therefore, to bridge continuum-baseddescription of the uid phase and the discrete description of theparticle phase, we simply need to obtain cell-based representationof the Eulerian eld quantities (e.g., solid volume fraction es,Eulerian velocity Us, and uidparticle interaction forces F

fp) ofthe solid phase. A straightforward and probably the most widelyused method to link particle quantities and cell quantities is theParticle Centroid Method (PCM). The PCM utilizes the uid meshfor coarse graining by summing over all particle volumes in eachcell to obtain cell-based solid volume fraction es, and similar proce-dures are followed for other variables such as Us and F

fp. Thismethod is very straightforward to implement in almost any CFDsolvers, but it can lead to large errors when cell size to particlediameter ratios are small. Consequently, various alternatives havebeen proposed to improve the accuracy of PCM. The DividedParticle Volume Method (DPVM), rst proposed and implementedby Wu et al. (2009a,b), is such an example. In this method, thevolume of a particle is divided among all cells that it overlaps withaccording to the portion of the volume within each cell, and is notonly distributed entirely to the cell its centroid resides in as inPCM. As a consequence, the DPVM at least guarantees that the solidvolume fraction es in any cell should never exceed one, effectivelypreventing very large gradients in the obtained es eld. DPVMworks for arbitrary meshes, structured or unstructured, with anyelements shapes as long as any edge of the cell has a length largerthan the particle diameter. Comprehensive comparisons betweenDPVM and PCM recently performed by Peng et al. (2014) suggestthat DPVM has signicantly improved performance over PCM.Another idea, recently proposed by Deb and Tafti (2013), is touse two separate meshes for the CFD discretization and the coarsegraining. While the improved variants do outperform the PCM interms of accuracy, the implementations of these sophisticatedmethods are often signicantly more complicated, especially inCFD solver based on unstructured, non-Cartesian meshes.

In our efforts to develop a CFD/LESDEM solver with a parallel,three dimensional CFD code based on unstructured meshes witharbitrary cells shapes, we found that none of existing coarse-grainingmethods is able to satisfy the requirements of easy imple-

ultiphase Flow 72 (2015) 233247mentation and good accuracy simultaneously. The difcultiesmotivated us to develop a coarse-graining method that is suitable

no CFDDEM simulations were performed) have been presentedin Sun and Xiao (2014). Specically, the companion paper (1) com-

l of Mprehensively reviewed and compared existing coarse-grainingmethods in the literature, including PCM, DPVM, two-gridformulation, and statistical kernel methods, (2) motivated and pro-posed a diffusion-based coarse-graining method, (3) demonstratedthe equivalence (up to the mesh discretization accuracy) betweenthe currentmethod and the statistical kernel-based coarse-grainingmethodwith Gaussian kernel, and (4) evaluated the performance ofthe diffusion-based method by comparing it with existing methodsin various scenarios, with both structured and unstructuredmeshes, and both in the interior domain and near wall boundaries.While maintaining all the merits of its theoretically equivalentcounterpart such as mesh-independence, the diffusion-basedmethod is much easier for practical implementations in general-purpose CFDDEM solvers, and provides a unied framework fortreating interior particles and particles that are located nearboundaries.

The present work is a companion paper of Sun and Xiao (2014).The objective is to explore the theoretical and practical issues ofapplying the diffusion-based coarse-graining method in a general-purpose CFDDEM solver, and to evaluate its performance in prac-tical uidized bed simulations. Specically, in this paper (1) theconservation characteristics of the diffusion-based coarse-grainingmethod are studied, based on which the variables to solve diffusion

equations for are identied (i.e., es; esUs, efFfp), (2) the algorithm isimplemented into a CFDDEM solver and tested in uidized bedsimulations, highlighting the improved mesh-convergence behav-ior compared to the PCM, and (3) the choice of diffusion bandwidthis justied based on physical reasoning. The issues discussed in thepresent work (e.g., the in situ performance of the proposed coarse-graining method in CFDDEM solvers, as well as the choice of vari-ables to solve diffusion equations for and the diffusion bandwidth)are specic to the application of the diffusion-based method inCFDDEM simulations. These issues are not trivial and warrantthorough investigations.

The rest of the paper is organized as follows.Section Methodology introduces the mathematical formulationof the CFDDEM approach, gives a summary of the diffusion-basedcoarse-graining method, and then discusses their numericalimplementations and the numerical methods used in the sim-ulations. In Section Numerical simulations CFDDEM simulationsare conducted by using the proposed coarse-graining method, andthe results are discussed and compared with those obtained withPCM. The overhead computational costs associated with thecoarse-graining procedure are investigated in a series of cases withdifferent ratios of particle and cell numbers. The physical basis ofchoosing the bandwidth parameter in the diffusion-based methodand possible extensions to spatialtemporal averaging arediscussed in Section Discussion. Finally, Section Conclusionconcludes the paper.

Methodology

Mathematical formulations of CFDDEM

Due to the large number of symbols and subscripts used in thisfor practical implementation in general-purpose CFD/LESDEMsolvers, while maintaining the theoretical rigor and excellentaccuracy.

The general motivation, description, and derivation of the diffu-sion-based coarse-graining method as well as a priori tests (where

R. Sun, H. Xiao / International Journapaper, it is benecial to establish certain conventions in the nota-tions before proceeding to the presentation of the particle and uidphase equations. Unless noted otherwise, superscripts are used tocategorize the physical background associated with a quantity,e.g., col for collision, fp for uidparticle interactions, etc. Thesesuperscripts should be relatively self-evident. Phase subscriptsare used to denote quantities associated with solid phase (s), uidphase (f), individual particles (p), and individual cells (c). Indexsubscripts i and k are used as indices for particles and cells,respectively. To avoid further cluttering of indices, vector notationsare preferred to tensor notations throughout the paper. When aquantity has both the indices (i or k) and phase subscripts(s; f ; p, or c), they are separated by a comma. The particle-levelvelocities and the forces associated with individual particles aredenoted as u and f, respectively; and the velocities and forces inthe continuum scale are denoted as U and F, respectively.

Discrete element method for particlesIn the CFDDEM approach, the translational and rotational

motion of each particle is described by the following equations(Cundall and Strack, 1979; Ball and Melrose, 1997; Weber et al.,2004):

mdudt

fcol f fp mg; 1a

IdWdt

Tcol Tfp; 1b

where u is the particle velocity; t is time; m is particle mass; fcol isthe force due to collisions and enduring contacts with other parti-

cles or wall boundaries; f fp denotes the forces due to uidparticleinteractions, e.g., drag, lift, and buoyancy; g denotes external bodyforces. Similarly, I and W are angular moment of inertia and angular

velocity, respectively, of the particle; Tcol and Tfp are the torques dueto particleparticle interactions and uidparticle interactions,respectively. For the purpose of computing collision forces and tor-ques, the particles are modeled as soft spheres with interparticlecontact represented by an elastic spring and a viscous dashpot.Further details can be found in the literature (e.g., Cundall andStrack, 1979; Tsuji et al., 1993; Xiao and Sun, 2011).

Locally-averaged NavierStokes equations for uidsThe uid phase is described by the locally averaged incompress-

ible NavierStokes equations. Assuming constant uid densityqf , the continuity and momentum equations for the uid are(Anderson and Jackson, 1967; Kafui et al., 2002):

r esUs efUf 0; 2a

@ efUf @t

r efUfUf 1

qfrpr R efqfg Ffp

; 2b

where es is the solid volume fraction; ef 1 es is the uid volumefraction; Uf is the uid velocity. The four terms on the right handside of the momentum equation are pressure (p) gradient, diver-gence of stress tensor R (including viscous and Reynolds stresses),gravity, and uidparticle interactions forces, respectively. Sincethe equations are formulated in the Eulerian framework, all vari-ables herein are continuum quantities, i.e., they are mesh-basedwhen discretized numerically. As explained in Section Particle-laden ows: Physical background and multi-scale modeling, the

solid phase quantities es, Us; Ffp are not explicitly solved for, butare instead obtained from the particle information via coarse-graining procedures.

Fluidparticle interactions

ultiphase Flow 72 (2015) 233247 235While the uidparticle interaction force Ffp consists of many

components including buoyancy Fbuoy, drag force Fdrag , and Basset

of Mhistory force, among others, here we focus on the drag term for thepurpose of illustrating the bridging between the continuum anddiscrete scales. Other forces can be coarse grained in a similarway. In PCM-based coarse graining, the particle drag on the uidis obtained by summing the drag over all particle in a cell. The dragon an individual particle i is generally formulated as:

fdragi Vp;ief ;ies;i

bi up;i Uf ;i

; 3

where Vp;i, and up;i are the volume, and the velocity, respectively, ofparticle i; Uf ;i is the uid velocity interpolated to the center of par-ticle i; bi is the drag correlation coefcient. Various correlationshave been proposed for b in dense particle-laden ows (Di Felice,1994; Syamlal et al., 1993; Wen and Yu, 1966), which account forthe presence of other particles when calculating the drag on a par-ticle by incorporating es in the correlation forms. The correlation ofSyamlal et al. (1993) is adopted in this work. However, the specicform of the correlation is not essential for the present discussion,and is thus omitted here for brevity. It sufces to point out thatregardless of the specic form of the drag correlations, the solid vol-ume fraction es;i and the uid velocity uf ;i local to the particle areneeded to calculate b. Both es;i and uf ;i are Eulerian mesh-basedquantities interpolated to the centroid location xi of particle i.

To summarize, in CFDDEM simulations the following Eulerianmesh-based quantities need to be obtained by coarse graining theparticle data:

1. Solid volume fraction es.2. Solid phase velocity us.

3. Fluidparticle interaction force Ffp.

These elds are needed in solving the continuity and momentumEqs. (2a) and (2b) for the uid phase. Eulerian eld quantities thatneed to be interpolated to particle locations include es; us, and uf ,which are needed for the calculation of uid forces on individualparticles. It can been seen that the coarse graining and inter-polation are of critical importance for modeling the interactionsbetween the continuum and discrete phases.

Diffusion-based coarse-graining method

Summary of the diffusion-based methodThe proposed algorithm is built upon the particle centroid

method, which is the coarse-graining method used in mostCFDDEM solvers. Therefore, here we rst introduce the PCMalgorithm in detail. To calculate solid volume fraction eld es withPCM, we loop through all cells to sum up all the particles volume totheir host cells (dened as the cell within which the particlecentroid is located), thus obtaining the total particle volume ineach cell. The solid volume fraction for cell k is then obtained bydividing the total particle volume in the cell by the total volumeof the cell Vc;k. That is,

es;k Pnp;k

i1Vp;iVc;k

; 4

where np;k is the number of particles in cell k, which implies thatPNck1np;k Np. The es eld obtained with the PCM procedure above

(denoted as e0 for reasons that will soon be evident) may haveunphysically large values for some cells and consequently verylarge spatial gradients, which can cause instabilities in CFDDEMsimulations or lead to numerical artifacts. To address this issue, in

236 R. Sun, H. Xiao / International Journalthe diffusion based method we proposed in the companion paper(Sun and Xiao, 2014), a transient diffusion equation for esx; s issolved with initial condition e0 and no-ux boundary conditions:@es@s

r2es for x 2 X; s > 0; 5a

esx; s 0 e0x; 5b

@es@n

0 on @X; 5c

where x x; y; zT are spatial coordinates; X is the computationaldomain; r2es @2es=@x2 @2es=@y2 @2es=@z2 in Cartesiancoordinates; e0x is the solid volume fraction eld obtained withthe PCM; s is pseudo-time, which should be distinguished fromthe physical time t in the CFDDEM formulation. Finally, @X is theboundary of X; n is the surface normal of @X. The diffusionEq. (5a) is integrated until time s T with the initial conditionEq. (5b) and boundary condition Eq. (5c), and the obtained eldesx; T is the solid volume fraction eld to be used in theCFDDEM formulation. The end time T is a physical parametercharacterizing the length scale of the coarse graining. It was demon-strated that the diffusion-based method above is equivalent to thestatistical kernel function-based coarse graining with the followingGaussian kernel:

hi hx xi 1b2p3=2

exp x xiTx xib2

" #; 6

where hi is the kernel associated with particle i, which is located atxi. The equivalence between the two is established with b

4T

p.

Moreover, for particles located near boundaries (e.g., walls), thekernel-based methods need to be modied to satisfy conservationrequirements with methods such as method of images (Zhu andYu, 2002). It was further demonstrated that the diffusion-basedcoarse-graining method above satises conservation requirementsautomatically, and thus interior and near-boundary particles aretreated in a unied framework. In fact, with the no-ux boundaryconditions the diffusion-based method is equivalent to the methodof images proposed by Zhu and Yu (2002).

The solid phase velocity Us;k and the uidparticle interaction

force Ffpk per unit mass in cell k are computed in a similar way.That is, the initial elds are rst obtained by using PCM:

Us;k Pnp;k

i1qsVp;iup;iPnp;ki1qsVp;i

Pnp;k

i1qsVp;iup;iqses;kVc;k

; 7

Ffpk Pnp;ki1 f fpiqf ef ;kVc;k

; 8

and then, diffusion equations are solved for the elds esUs and efFfp.After the coarse graining is performed on esUs and efFfp, the coarse-grained elds are divided by es and ef (i.e., 1 es), respectively, toobtain Us and F

fp. Note that solving diffusion equations directly

for Us;k and Ffpk would violate conservation requirements. While

es;k seems to be the intuitive and natural choice to solve diffusionequations for, the choices of esUs and efFfp as the variables to solvediffusions for are not straightforward. Justications are thus pro-vided below, and detailed proofs are presented in Appendix A.

Conservation characteristics and choice of diffusion variablesIt is critical that any coarse-graining algorithm should conserve

the relevant physical quantity in the coarse-graining procedure.Specically in the context of CFDDEM simulations thesequantities include total particle mass, particle phase momentum,and total momentum of the uidparticle system. The con-

ultiphase Flow 72 (2015) 233247servation requirement implies that the total mass computed fromthe coarse-grained continuum eld should be the same as the totalparticle mass in the discrete phase. Similarly, when calculating

4. Easy implementation in CFD solvers with almost arbitrary

(Plimpton, 1995). This hybrid solver was originally developed by

l of Mmeshes and ability to produce smooth and mesh-independentcoarse-grained elds on unfavorable meshes.

5. Easy parallelization by utilizing the existing infrastructure inthe CFD solver.

Potential limitations of the proposed method are summarizedbelow:

1. Rigorous equivalence between the proposed method andstatistical kernel-based methods only holds theoretically, i.e.,when the CFD mesh is innitely ne. Numerical diffusions canoccur (compared with the results of the statistical kernelmethods) on meshes with large cells, particularly when the dif-fusion bandwidth is smaller than the cell size. This shortcomingcan be mitigated by setting the diffusion constant in the regionswith large cells to very small values locally, effectivelydegenerating it to PCM in these regions.

2. The computational overhead associated with the diffusion-based method is signicant, often exceeding the computationalcost of the CFD solver. However, in practical simulations wherethe number of particles per cell is reasonably large, thecomputational expense of the DEM solver dominates, and theoverhead incurred by the coarse-graining procedure onlyaccounts for a small fraction of the total computational cost.

3. Diffusing the uidparticle drag forces in the proposed methodsolid phase velocity Us, the total momentum of the particles shouldbe conserved before and after the coarse graining; nally, toconserve momentum in the uidparticle system, the total particleforces on the uid exerted by the particles should have the samemagnitude as the sum of the forces on all particles exerted bythe uid but with opposite directions.

The PCM-based coarse-graining schemes as in Eqs. (4), (7), and(8) are conservative by construction. Specically, it conserves totalparticle mass, total particle momentum, and total momentum ofthe uidparticle system. The proposed coarse-graining algorithmconsists of two steps: (1) coarse graining using PCM, and (2)solving diffusion equations for the appropriate quantities. Theconservation requirements above dictate that diffusion equationsshould be solved for the following three quantities:

es; esUs; and efFfp: 9The physical meaning of the three quantities above are particle

mass, particle phase momentum, and uidparticle interactionforces per unit volume, respectively. Detailed derivations are pre-

sented in Appendix A. For vector elds such as esUs and efFfp, diffu-sion equations are solved for each component of the vectorindividually, leading to seven diffusion equations in total for thethree eld variables. Conservation requirements are thus met forall components.

Merits and limitations of the diffusion-based coarse-graining methodThe advantages of the diffusion-based coarse-graining method

are extensively discussed and demonstrated in Sun and Xiao(2014) via a priori tests. The merits are summarized as follows:

1. Sound theoretical foundation with equivalence to statisticalkernel-based methods.

2. Unied treatment of interior and near-boundary particleswithin the same framework.

3. Guaranteed conservation of relevant physical quantities in thecoarse-graining procedure.

R. Sun, H. Xiao / International Journamakes it difcult to linearize and implicitly treat the uidparticle momentum exchange terms in the uid momentumequations.the second author and his co-workers to study particle segregationdynamics (Sun et al., 2009). The solver was later used as a test bedfor evaluating coarse graining and sub-stepping algorithms inCFDDEM (Xiao and Sun, 2011). Recently, we have improved theoriginal solver signicantly by enhancing its efciency in the cou-pling of OpenFOAM and LAMMPS, its parallel computing capabili-ties, and the coarse-graining algorithm, the last of which is thesubject of the current work.

The uid equations in (2) are solved in OpenFOAM with thenite volume method (Jasak, 1996). The solution algorithm ispartly based on the work of Rusche (2003) on bubbly two-phaseows. The discretization is based on a collocated grid, i.e., pressureand all velocity components are stored in cell centers. PISO(Pressure Implicit Splitting Operation) algorithm is used to preventvelocitypressure decoupling (Issa, 1986). A second-order upwindscheme is used for the spatial discretization of convection terms. Asecond-order central scheme is used for the discretization of thediffusion terms. Time integrations are performed with a second-order implicit scheme.

The solution of the particle motions including their interactionsvia collisions and endured contacts are handled by LAMMPS. The

uid forces f fp on the particles are computed in OpenFOAM andsupplied into LAMMPS and for its use in the integration of particlemotion Eq. (1). The particle forces on the uid are computed inOpenFOAM according to the forces on individual particles via acoarse-graining procedure.

The coarse-graining method used in this work involves solvingtransient diffusion equations. Solution procedures of these equa-tions are implemented based on the OpenFOAM platform, takingadvantage of existing infrastructure (e.g., discretization schemes,linear solvers, and parallel computing capabilities) available inOpenFOAM. The diffusion equations are solved on the same meshas the CFD mesh. A second-order central scheme is used for thespatial discretization of the diffusion equation; the CrankNicolson scheme is used for the temporal integration, whichguarantees the stability and allows for large time step sizes forthe solution of the diffusions equations to minimize computationaloverhead associated with the coarse graining.

Numerical simulations

In the companion paper (Sun and Xiao, 2014), a priori tests havebeen performed to highlight the merits of the diffusion-basedcoarse-graining method by calculating the coarse-grained solidvolume eld of a given particle conguration. The purpose of thepresent numerical tests is to examine the performance of thenew coarse-graining method in the context of a CFDDEM solverapplied to uidized bed ows.

The CFDDEM solver used in this study has been validatedextensively by the second author and his collaborators (GuptaReaders are referred to the companion paper (Sun and Xiao,2014) and the rest of the present paper for details.

Solver implementation and numerical methods

A hybrid CFDDEM solver is developed based on two state-of-the-art open-source codes in their respective elds, i.e., a CFDplatform OpenFOAM (Open Field Operation and Manipulation)developed by OpenCFD (2013) and a molecular dynamics sim-ulator LAMMPS (Large-scale Atomic/Molecular Massively ParallelSimulator) developed at the Sandia National Laboratories

ultiphase Flow 72 (2015) 233247 237et al., 2013, 2011a,b, 2012, 2011c; Sun et al., 2009; Xiao and Sun,2011), some of which were conducted within an EU-funded projectPARDEM (PARDEM, 2009-2013). Here we present only a brief

validation of the current solver with CFDDEM simulations in theliterature based on the same experimental setup used in this work.Then, the mesh-convergence tests are performed on the CFDDEMsolver with the diffusion-based coarse-graining method. This is afollow-up investigation of the a priorimesh-convergence tests pre-sented in the companion paper (Sun and Xiao, 2014). The purposeof this test is to demonstrate the capability of the diffusion-basedcoarse-graining algorithm in yielding mesh-converged results inCFDDEM simulation, which has been a major challenge so far,particularly when the cell sizes are small compared to the particles.Finally, numerical tests are performed by using the CFDDEMsolvers with the diffusion-based coarse graining and with thePCM-based coarse graining. This is to highlight the advantages ofthe diffusion-based coarse graining both in terms of producingmesh-independent results and in representing correct physicalmechanisms in dense particle-laden ows.

The numerical tests are set up based on the uidized bedexperiments (Mller et al., 2008, 2009). In the experiments, thedimensions of the uidized bed were 44 mm 1500 mm10 mm (width, height, and transverse thickness, aligned with thex; y, and z axes, respectively, in our coordinate system). Thedomain geometry is shown in Fig. 1 along with the coordinatesystem used here. In their experiments, magnetic resonance wasused to measure the volume fraction ef of the uid (i.e., air)(Mller et al., 2009) and the velocity of particles (Mller et al.,2008). The supercial inlet velocity of the air is 0:9 m/s, and the ini-tial bed is approximately 30 mm in height, consisting of 9240

DEM solver have favorable agreement with the experimentalresults and the numerical simulations of Mller et al. (2009).However, it is noted that both solvers predicted higher uid volumefraction ef near the boundaries (slippery walls) compared withexperimental measurements. This is particularly prominent forlocation y 31:2 mm. As pointed out by Mller et al. (2009), thediscrepancy in the ef proles is due to the over-prediction of thewidth of the bubbles in the CFDDEM solvers, which consequentlyleads to higher ef values near the wall. Since this over-prediction isattributed to the difculty of the CFDDEM framework in modelingthewall effects, and it is not directly related to our solver in particu-lar, further discussions are not pursued here.

The vertical component Uy of the time-averaged particle phasevelocity are shown in Fig. 3, with comparisons among the currentsimulations, the experimental measurement, and the numericalsimulations of Mller et al. The comparisons are presented fortwo cross-sections at the heights y 15 mm and y 25 mm, as

238 R. Sun, H. Xiao / International Journal of Mpoppy seed particles. Other parameters of the experiments aresummarized in Table 1. To reduce the computational costs withoutcomprising the accuracy of the numerical simulations, the heightof the bed is taken as 120 mm following Mller et al. (2008).Although the poppy seeds are kidney shaped and thus are notexactly spherical particles, for simplicity they are consideredspherical in the simulations here. Slip conditions are applied atthe boundaries in the z-direction, and no-slip boundary conditionsare applied in the y-direction boundaries. The time step in the DEMFig. 1. Geometry of the 3D computational domain for the simulation.simulations is taken as 4:0 106 s. To get the time-averaged pro-les of uid volume fraction and particle velocity, the simulationsare averaged for 18 s, which is approximately 135 ow-throughtimes, and is long enough to achieve statistically converged time-averaging elds (Mller et al., 2009). The choices of computationalsetup and parameters outlined above are consistent with previousnumerical validations of this set of experiments (Mller et al.,2008, 2009). The bandwidth b used in the diffusion-based coarse-graining method is 4dp (with dp being the particle diameter).

Solver validations

The purpose of this validation is to show the results obtainedfrom the proposed CFDDEM solver are consistent with the experi-mental measurements and numerical simulations in the literature.The setup and parameters used in this validation test are detailedin Table 1. The mesh resolution used in this simulation isNx Ny Nz 36 100 8, where Nx; Ny, and Nz are the num-bers of cells in the width (x-), height (y-), and transverse thickness(z-) directions, respectively.

The measurements of uid volume fraction ef in the experi-ments (Mller et al., 2009) were taken on two cross-sections atthe heights y 16:4 mmand y 31:2 mm. To facilitate comparisonwith the experimental measurements, the proles of ef at the twoheights are extracted in the present simulations. The comparisonwith the experimental data and the CFDDEM simulations ofMller et al. (2009) are presented in Fig. 2. As can be seen fromthe gure, overall the ef proles predicted by the present CFD

Table 1Parameters used in the CFDDEM simulations of the uidized bed ow.

Bed dimensionsWidth Lx 44 mmHeight Ly 120 mmTransverse thickness Lz 10 mm

Particle propertiesTotal number 9240Diameter dp 1.2 mmDensity qs 1:0 103 kg=m3Elastic modulus 1:2 105 PaPoissons ratio 0.33Normal restitution coefcient 0.98Coefcient of friction 0.1

Fluid propertiesDensity qf 1:2 kg=m3

Viscosity 1:8 105 kg=m sSupercial inlet velocity 0:9 m/s

ultiphase Flow 72 (2015) 233247these are the locations where experimental measurements wereperformed (Mller et al., 2008). It can be seen from Fig. 3 thatgenerally speaking both CFDDEM solvers give good predictions

ulamm and (b) y 31:2 mm.

l of M(a) y = 16.4 mm

Fig. 2. Proles of time-averaged uid volume fraction obtained from the present simMller et al. (2009). The results are presented at two vertical locations: (a) y 16:4

R. Sun, H. Xiao / International Journaof the particle phase velocity Uy for both locations, although argu-ably the prediction quality of our solver seems to be slightly better.Specically, the simulations of Mller et al. tend to over-predictthe particle velocities at both locations; while our simulations donot seem to have this issue except for a very minor over-predictionnear the centerline (between x 0:015 m and 0:025 m) for theprole at y 15 mm. On the other hand, the particle velocity Uynear the centerline is under-predicted for y 25 mm. Overall,the magnitude of time-averaged particle velocity predicted bythe present simulation using diffusion-based algorithm is smallerthan that in the simulation of Mller et al. A possible explanationis the different solid volume fraction es elds used in the two sol-vers. The es eld computed with the diffusion-based method issmoother and is thus free from very large es values. Since thecomputed drag force on a particle increases dramatically withthe increase of es, over-prediction of the drag forces on particlesdue to high es is more likely when the PCM coarse graining is usedcompared to the diffusion-based method. It is worth noting thatone should not be deceived by the very similar averaged es prolesfor both simulations in Fig. 2. In fact, the instantaneous es eldsobtained with PCM have much more very larger values, as has beendemonstrated in the a priori tests in Sun and Xiao (2014). Althoughpresented only in very few cells, these large values can have asignicant impact on the particle velocities, since the drag forceis a highly nonlinear function of es. Consequently, this leads to lar-ger uid drag forces on the particles in the PCM-based CFDDEMsolvers, and thus larger averaged solid phase velocities in theresults.

(a) h = 15 mm

Fig. 3. Proles of time-averaged vertical velocity of particles obtained from the preseexperiment (Mller et al., 2008). (a) Particle velocity at the height of 15 mm, (b) particl(b) y = 31.2 mm

tions compared with the experimental measurements and numerical simulation of

ultiphase Flow 72 (2015) 233247 239In summary, although some discrepancies exist between theprediction of ef by the present CFDDEM solver and the experi-mental results, the overall agreement is favorable. Moreover, theuid volume fraction and particle phase velocities obtained inthe present simulations are in very good agreement with the pre-vious CFDDEM simulations of the same case (Mller et al., 2008).Hence, the validations are deemed successful, and furtherinvestigations using the CFDDEM solver are pursued below.

Mesh-independence study of CFDDEM simulations

In the a priori simulations presented in the companion paper(Sun and Xiao, 2014), it has been demonstrated that the diffu-sion-based method yields mesh-independent solid volume fractionelds. Here we further demonstrate that the CFDDEM simulationswith diffusion-based coarse graining give mesh-independentresults. In particular, average particle phase velocities and uidvolume fraction at various locations are studied extensively andare compared among simulations performed on ve successivelyrened meshes. To characterize the sizes of arbitrarily shaped cells,we use the parameter cell length scale Sc dened as:

Sc Vc

3p

; 10

where Vc is the volume of a cell. Five meshes AE are used in themesh-independence studies here with the effective lengthsSc 6dp (mesh A), 4dp (mesh B), 2dp (mesh C), dp (mesh D), and0:5dp (mesh E), arranged with increasing mesh resolution. Larger

(b) h = 25 mm

nt simulation. The measurements are taken at different heights according to thee velocity at the height of 25 mm.

Sc values indicate larger cell sizes and thus coarser meshes. Detailsof the mesh parameters are presented in Table 2.

As with the validation studies above, the uid volume fractionsef proles at two cross sections at heights y 16:4 mm andy 31:2 mm are shown in Fig. 4(a) and (b). Moreover, the prolesof uid volume fraction ef at two vertical cross-sections atx 11 mm and x 22 mm (located at a quarter width locationand at the centerline of the domain, respectively) are shown inFig. 4(b) and (d). Although experimental data are not available atthe two vertical cross-sections, this does not impair the objectiveof the mesh-convergence study, since we are mainly concernedwith the comparison of results obtained with meshes withdifferent coarseness levels, and not with the agreement betweennumerical predictions and experimental measurements.Examining the proles at a few vertical cross-sections in additionto the horizontal cross-section locations helps shed light on thebehavior of the result in the entire domain. To facilitate visualiza-tion, the locations of the corresponding cross-sections are

indicated in the insets in the upper left corner of each plot alongwith the ef proles. The shaded regions in the insets indicate theinitial particle bed. It can be seen that mesh convergence isachieved in the prediction of ef as all meshes with Sc=dp smallerthan 4 (e.g., meshes B, C, D, and E) give identical results. As tothe ef proles obtained by using mesh A with Sc=dp 6, someminor discrepancies with the converged results are observed,particularly in the region near the bottom inlet in Fig. 4(c) and(d). Since this mesh resolution is probably inadequate, the minordiscrepancies are expected.

Similarly, the vertical component Uy of the time-averaged par-ticle phase velocity obtained at different horizontal and verticalcross-sections are shown in Fig. 5. Again, the general observationhere is that the four ner meshes (BD) all give identical results,indicating excellent mesh-convergence behavior, while some dis-crepancies are found in the results frommesh A (Sc=dp 6). In con-trast to the uid volume fraction proles shown in Fig. 4, thediscrepancies between the results from mesh A and the convergedresults from meshes BE occur mostly in the middle of the domain(e.g., between x 0:01 and x 0:03 in Fig. 5(a) and (b), and aroundy 0:25 in Fig. 5(c) and (d)).

Performance comparison with PCM

The mesh-convergence studies above demonstrated excellentmesh-convergence behavior of the CFDDEM solver with diffu-sion-based coarse-graining method. To highlight this advantage,the same mesh-convergence study is performed on the CFDDEMsolver with PCM-based coarse-graining. All simulation setup andparameters are kept the same except for the coarse-graining

Table 2Parameters in mesh-independence study of the CFDDEM solvers with diffusion-based and PCM-based coarse-graining methods. The study using the solver withdiffusion-based method covers a wider range of mesh resolutions, while meshes (e.g.,D and E) with small cells cannot be used in the PCM-based solver due to instabilities.

Mesh Sc=dp Nx Ny Nz Used in

A 6 6 16 1 Diffusion-based; PCMB 4 9 25 2 Diffusion-based; PCMB

03 12 32 3 PCM only

C 2 18 50 4 Diffusion-based; PCMD 1 36 100 8 Diffusion-based onlyE 0.5 72 200 16 Diffusion-based only

oars

240 R. Sun, H. Xiao / International Journal of Multiphase Flow 72 (2015) 233247(a) y = 16.4 mm

(c) x = 11 mm

Fig. 4. Mesh convergence study of the CFDDEM solver with the diffusion-based c

horizontal cross-sections located at (a) y 16:4 mm and (b) y 31:2 mm, respectiverespectively. Insets in the panels show the location of the cross-section corresponding toobtained based on ve consecutively rened meshes are compared.method and the meshes used. Here only four meshes A(Sc=dp 6), B (Sc=dp 4), B0 (Sc=dp 3), and C (Sc=dp 2) are used

(b) y = 31.2 mm

(d) x = 22 mm

e-graining method, showing the uid volume fraction ef ( 1 es) proles on two

ly, and two vertical cross-sections located at (c) x 11 mm and (d) x 22 mm,each prole. The shaded regions in the inset indicates the initial particle bed. Results

se-g

l of M(a) y = 15 mm

(c) y = 11 mm

Fig. 5. Mesh convergence study of the CFDDEM solver with the diffusion-based coar

R. Sun, H. Xiao / International Journain the mesh-convergence study with PCM, since it is not possible toobtain meaningful solid volume fraction eld on meshes D(Sc=dp 1) and E (Sc=dp 0:5), which have cell length scales Scequal or smaller than the particle diameter. Even by capping thees eld at 0.7 (detailed below), we were not able to completesimulations on meshes D or E without being interrupted byinstabilities.

The proles of time-averaged uid volume fraction ef obtainedby using the same CFDDEM solver with PCM-based coarse-graining method are shown in Fig. 6. Proles at the same fourcross-section as in Fig. 4 are presented. As can be seen fromFig. 6, at all cross-sections the ef proles obtained with meshes B(Sc=dp 4) and B0 (Sc=dp 3) are the same, indicating an approxi-mate mesh-independence. As with the results from the diffusion-based coarse-graining in Fig. 4, the results obtained with mesh A(Sc=dp 6) have some discrepancies with the other results due tothe inadequate mesh resolution. While this is not evident inFig. 6(a) and (b), it can be seen at several locations in Fig. 6(c)and (d). Note that the deviations do not necessarily occur nearthe bottom boundary as in Fig. 4(c) and (d), but at random loca-tions instead. Perhaps the most striking difference in the resultsin Fig. 6 is that the convergence is not achieved when the meshis further rened, as is evident from the fact the results of meshC (Sc=dp 2) are different from those of meshes B and B0. Withmesh C, unphysically large es values are frequently encounteredduring the simulations, which cause instabilities. To address thisissue associated with the PCM, a frequently used technique thatis adopted here is capping, i.e., for all cells with solid volumefraction es larger than a certain threshold value, e.g.,ethreshold 0:7, the es in these cells are capped to be ethreshold. Thisis done at each uid time step after the es eld is calculated. Thistechnique improves the robustness of CFDDEM simulations with

particle velocity obtained on two horizontal cross-sections located at (a) y 16:4 m(c) x 11 mm and (d) x 22 mm, respectively. Insets in the panels show the locationsconsecutively rened meshes are compared.(b) y = 25 mm

(d) y = 22 mm

raining method, showing the proles of vertical component Uy of the time-averaged

ultiphase Flow 72 (2015) 233247 241PCM-based coarse graining, but it may impair the accuracy of theuid drag calculation and the accuracy of the entire simulation,as an articially set es value is used instead of the physical valuesin these cells. The capping technique may have caused the failureof mesh-convergence observed in Fig. 6. The particle phase veloci-ties obtained on meshes A, B, B

0, and C with the PCM-based solver

are presented in Fig. 7. Similar to what was observed in Fig. 6,mesh-convergence is not achieved here, mainly due to the fact thatmesh C produces result different from those of meshes B and B

0.

The discrepancies are most evident from Fig. 7(c) and (d). InFig. 7(d), it can be seen that even the results obtained with meshesB and B

0are different.

To further compare the different effects of the two coarse-grainingmethods on the CFDDEM results, snapshot sequences of particlelocations during a bubble evolution cycle are presented in Fig. 8.The results from mesh C (Sc=dp 2) are presented here, since themesh-convergence study above seems to suggest this to be asuitable mesh for both cases. The snapshots at t 0:00 s corre-spond to the beginning of the cycle. The volume fraction contourscorresponding to es 0:28 are overlaid on top of the particle loca-tion plots to separate regions of lower and higher solid volumefractions, which facilitate visualization of the bubble shapes andlocations. This contour value 0.28 is chosen to be one half of themaximum solid volume fraction in the initial bed conguration.From Fig. 8(a), the bubble formation (t 0:04 s), growth (0:08 s),and burst (0:12 s) can be clearly identied. At t 0:12 s, whenthe upper bubble bursts, a small bubble formed near the bottomof the bed. In the subsequent snapshots from 0:16 s to 0:24 s, thebubble rises and grows until it reaches the top of the bed at0:28 s, and bursts at 0:32 s eventually. The bubble dynamicsobserved here is physically reasonable as conrmed in previousexperiments (Mller et al., 2008) and numerical simulations

m and (b) y 31:2 mm, respectively, and two vertical cross-sections located atof the cross-section corresponding to each prole. Results obtained based on ve

(a) y = 16.4 mm (b) y = 31.2 mm

(c) x = 11 mm (d) x = 22 mm

Fig. 6. Mesh convergence of PCM-based coarse graining, showing the proles of uid volume fraction ef ( 1 es) on cross-sections at (a) y 16:4 mm, (b) y 31:2 mm,(c) x 11 mm, and (d) x 22 mm. Results from four meshes A, B, B0 , and C are compared. Refer to Fig. 4 for detailed caption and for comparison.

(a) y = 15 mm (b) y = 25 mm

(c) x = 11 mm (d) x = 22 mm

Fig. 7. Mesh convergence of PCM-based coarse graining, showing the proles of vertical component Uy of the time-averaged particle velocity at (a) y 15 mm,(b) y 25 mm, (c) x 11 mm, (d) x 22 mm. Results from four meshes A, B, B0 , and C are compared. Refer to Fig. 5 for detailed caption and for comparison.

242 R. Sun, H. Xiao / International Journal of Multiphase Flow 72 (2015) 233247

n-b

PC

ion,0:28

l of M(Peng et al., 2014). In contrast, in the results obtained by using thesame CFDDEM solver but with PCM-based coarse-graining, whichis shown in Fig. 8(b), the cycle of bubble formation, growth, andbursting are not observed as clearly, although the sequence from0.04 s to 0.12 s does vaguely show a similar bubble evolutiondynamics as in Fig. 8(a). Moreover, the bubble shapes in the PCM

(a) Diusio

(b)

Fig. 8. Snapshots of particle locations during a cycle of bubble formation and evolutand (b) PCM coarse-graining methods. The overlaying contours corresponds to es origin (t 0 s) corresponds to the beginning of the cycle.R. Sun, H. Xiao / International Journaresults are much more irregular than those observed in thediffusion-based results. To further illustrate the different bubbledynamics, snapshots of solid volume fractions of the same cycleare shown in Fig. 9(a) and (b) for the CFDDEM solvers withdiffusion-based and PCM-based coarse graining, respectively. InFig. 9(a) the same sequence of bubble dynamics as explained aboveare observed. The bubbles can be clearly identied from thesnapshots of solid volume fractions. On the other hand, in thePCM results shown in Fig. 9(b), the bubbles are not as easilyidentied, which is partly due to the large local variations (i.e.,spatial gradients) in the es eld.

In summary, with the same mesh, computational setup,parameters, and CFDDEM solver but only with different coarse-graining methods, the two carefully designed test cases clearlysuggest the superiority of the diffusion-based method comparedwith the PCM-based method in obtaining CFDDEM simulationresults with mesh-convergence and in capturing the physics ofthe bubble dynamics in the uidized bed.

Computational overhead of diffusion-based coarse graining

In a typical CFDDEM solver, the computational cost of DEMsimulation dominates due to the high computational costs inresolving particle collisions and the small time-scales in these par-ticle-scale processes. In contrast, the number of cells in the CFDmesh and the associated computational cost for the CFD simulationare generally moderate, since the cells must be large enough tocontain a sufcient number of particles for the locally averagedNavierStokes equations to be valid. Since the diffusion equationsare solved on the CFD mesh, the computational cost associatedwith the diffusion-based coarse graining is of the same order asthat of the CFD simulation. Therefore, the additional computationalcosts incurred by solving the diffusion equation are unlikely to be asignicant portion of the entire CFDDEM simulation. In LESDEM,however, the number of cells in the LES mesh may be large, andthus the computational overhead due to the coarse-grainingprocedure based on solving diffusion equations may become a con-

ased method

M

showing the results obtained by using the CFDDEM solver with (a) diffusion-based, which is half of the maximum es value in the initial bed conguration. The time

ultiphase Flow 72 (2015) 233247 243cern. To minimize the computational overhead, an implicit timestepping scheme should be used to guarantee stability, whichallows for large time step sizes to be used in the solution of thecoarse-graining diffusion equations. In Sun and Xiao (2014) wehave shown that with any reasonable bandwidth the diffusionequation can be solved with one time step to obtain sufcientaccuracy, although some minor uctuations are still present inthe coarse-grained eld. If the diffusion equation is solved withthree time steps, the obtained eld is sufciently smooth, at leastcompared with those obtained by using PCM and DPVM.

To investigate the computational costs associated with differentparts of a typical CFDDEM simulation, a series of four numericalsimulations are performed with the ratio Np=Nc between the num-ber Np of particles and the number Nc of CFD cells varying from 2 to16. In these cases the number of particles Np is kept constant at

3:32 105, while the numbers of CFD cells Nc vary from2:1 104 to 1:7 105. The computational cases are constructedaccording to those presented in Section Numerical simulations,except that the computational domain is enlarged to 264 mm120 mm 60 mm (in the width x-, height y-, and transversethickness z-directions, respectively) to allow for a realistically largenumber of cells and particles to be used. As in the experiment, theinitial bed height is set to 30 mm to retain the same bed dynamicsas in the experiment, but note that both the width and the trans-verse thickness have been increased by six times. Since the CPUtime needed for each time step is relatively constant throughoutthe entire simulation, only 100 uid time steps are simulated inthese tests.

The CPU times consumed by different parts of the CFDDEMsimulations, including the CFD part, the DEM part, and the coarsegraining part (mainly the solution of the diffusion equations), are

ba

of M(a) Diusion-

244 R. Sun, H. Xiao / International Journalpresented in Table 3 for the four cases studies. It can be seen fromthe table that the total time spent on the DEM part do not varymuch among the four cases, which is expected since the total num-ber of particles are the same in all the cases. On the other hand, thetime spent on the CFD part and that on the coarse grainingincreases as the number of CFD cells increases, at a rate slightlyhigher than linear. It is worth noting that in all these cases the timespent on coarse graining is slightly longer than the CFD part. This isbecause the coarse graining diffusion equations are performed forseven elds associated with three variables es; ui, Fi in each uidtime step. Three times steps are used each time when these equa-tions are solved. However, note that the percentage of CPU timespent on the coarse graining decreases with increasing ratioNp=Nc . Even for case 1 with Np=Nc 2, which indicates that thereare only two particles for each cell on average, the coarse grainingaccounts for only 28% of the total computational cost. Consideringthat the number of particles relative to the number of cells is verysmall, this percentage is not really discouraging. Peng et al. (2014)pointed out that a cell size of 1:63dp is needed to satisfy theCFDDEM governing equations. A rule of thumb for CFDDEM

(b) PC

Fig. 9. Snapshots of solid volume fraction es during a cycle of bubble formation and evolubased and (b) PCM coarse-graining methods. The time origin (t 0 s) corresponds to th

Table 3Breakdown of computational costs associated with different parts of CFDDEMsimulations. The computational costs are presented for four cases with the samenumber of particles Np 3:3 105 and different numbers Nc of CFD cells. The CPUtimes presented here are normalized by the time spent on the CFD part of case 4,which has the smallest number of CFD cells.

Case Nc Np=Nc CFD DEM Coarse graining

1 1:7 105 2 12 23% 26 49% 14:8 28%2 8:3 104 4 5 13% 27 70% 6:4 17%3 4:1 104 8 2 7% 25 83% 2:8 10%4 2:1 104 16 1 3% 27 92% 1:5 5%sed method

M

ultiphase Flow 72 (2015) 233247simulations is that on average a typical cell should containapproximately nine particles (Mller et al., 2009; Xiao and Sun,2011). In view of these observations, cases 3 and 4 are probablymore realistic in terms of Np=Nc ratios. In the two cases, the diffu-sion-based coarse-graining procedure accounts for only 10% and5%, respectively, of the total cost of the CFDDEM simulations,which we believe are rather moderate. Admittedly, the diffusionequations incur more computational costs than the CFD solver.This is in stark contrast to PCM and DPVM, which are expectedto incur negligible computational costs. The computational over-head of the diffusion-based method may be signicant andundesirable when ne meshes are used, i.e., when high-resolutionsolvers such as LES are used in the CFD part. In these cases, onemay consider further increasing the time step size used in solvingthe diffusion equations, e.g., solving the diffusion equation in onlyone time step with an implicit scheme (see Sun and Xiao, 2014).Alternatively, one can recast the continuity Eq. (2a) to eliminatethe term esUs (see Eq. (1a) in Wu et al. (2014)), so that the numberof diffusion equations to solve is reduced from seven to four.

A few additional observations need to be made when examiningthe computational overhead due to the diffusion-based coarse-graining procedure. First, we have demonstrated in the companionpaper (Sun and Xiao, 2014) that even on unfavorable meshes thediffusion-based coarse-graining method lead to smooth solid vol-

ume fraction es elds (as well as Us and Ffp). As observed by Penget al. (2014), compared to the non-smooth coarse grained elds(e.g., es) such as those obtained by using the PCM, a smooth es eldcan signicantly accelerate convergence in the CFD solver, and thuseffectively reduces overall computational costs of the CFDDEMsimulation. Second, the parallelization of the diffusion-basedmethod is straightforward and can very effectively take advantageof exiting infrastructure in the CFD solver, which is often highlyoptimized. The two factors partly offset the moderate

tion, showing the results obtained by using the CFDDEM solver with (a) diffusion-e beginning of the cycle.

coarse-graining procedure. On the other hand, as demonstrated

2014). Since the support of the kernel is approximately 3b, this

times. This spatial and temporal symmetry is desirable from atheoretical point of view. However, in CFDDEM solvers a kernelfunction with the symmetry in time cannot be implemented,since at time t when we need to compute esx; t the particle dis-tributions at future times are not known but need to be solved.If coarse graining or averaging in time is indeed desirable ornecessary (e.g., when the ow elds have high-frequencyuctuations such as in LESDEM), an alternative to obtain a eldwith timevolume averaging, e.g., hesx; ti, is to use a single-sided averaging scheme as follows (Meneveau et al., 1996;Xiao and Jenny, 2012):

hesx; ti Z t0esx; t0Wt t0dt0 11

with Wt t0 1bt

exp t t0=bt ;

where es is the solid volume fraction with volume averaging only,Wis the exponential kernel function, and bt is the temporal averagingbandwidth with similar interpretation to b in Eq. (6). Comparedwith the Gaussian kernel function, a convenient feature of theexponential kernel function is that hesi as dened in Eq. (11) isthe solution of the following differential equation:

dhesidt

1btes hesi; 12

l of Mchoice of bandwidth corresponds to a kernel support of approxi-mately 1218 particle diameters. This is indeed of the same orderof magnitude as that suggested by the wake sizes shown in Fig. 10.In a polydispersed system with a range of particle diameters,although changing bandwidth b (or equivalently diffusion timespan T) adaptively according to relative particle velocities wouldbe unrealistic and probably unnecessary as well, it is possibleand easy to accommodate different particles sizes (and thus wakesizes) by using a spatially varying diffusion coefcient in Eq. (5a).Currently, a diffusion coefcient that is uniformly one is usedthroughout the domain.

Implementation of timevolume averaging in CFDDEM solversin previous studies (Wu et al., 2009a, 2006; Xiao and Sun, 2011),linearization and implicit treatment of the uidparticle momen-tum exchange terms are effective methods for accelerating theconvergence of the PISO algorithm by increasing the diagonaldominance in the matrix of the discretized linear equationssystems. Diffusing the uidparticle drag forces as performed inthe proposed method makes it difcult, if not impossible, toperform such linearizations. This is a potential limitation of the dif-fusion-based coarse-graining method if linearization and implicittreatment are essential, e.g., in such challenging cases as granularRayleighTaylor instability problems (Wu et al., 2014).

Discussion

Choice of bandwidth in kernel functions for coarse graining

In coarse-graining procedures used to link microscopic andmacroscopic quantities, the choice of parameters (e.g., the band-width b in the kernel functions) remains an open question (Ltzelet al., 2000; Zhu and Yu, 2002). Here we argue that in CFDDEMsimulations the bandwidth b should be chosen based on the sizeof the wake of the particles in the uid ow, which in turn dependson the particle diameter and the particle Reynolds number, amongother parameters (Wu and Faeth, 1993). While coarse graining maybe performed for different theoretical and practical purposesdepending on the physical context, in CFDDEM the main reasonfor the coarse graining is to compute the interactions forcesbetween the uid and the particle phases. Specically, for example,the solid volume fraction es is needed in the drag force calculationto account for the effects of a particle on the drag experienced byother particles. It is also needed in the continuity and momentumequations of the uid phase. In the latter, in addition to es and Us,which are obtained via coarse graining, the force of a particle onthe uid needs to be distributed to an appropriate volume of theuid. In light of this observation, from a physical perspective thesupport of the kernel function should extend approximately tothe same distance as the wake of the particles. The wake sizedepends on the particle Reynolds number, which in turn dependson the relative velocity between the particle and the uid. Tocompute the wake size of the particles (indicated by the velocitydefect along the particle centerline) in the case studied inSection Numerical simulations, a few representative relativevelocity values are used, and the results are displayed in Fig. 10.

A bandwidth between b 4dp and 6dp was used in the sim-ulations in the current work (Section Numerical simulations)and in the a priori tests in the companion paper (Sun and Xiao,computational overhead associated with the diffusion-based

R. Sun, H. Xiao / International JournaAs with most previous CFDDEM works, in this study we con-sider only volume averaging in the coarse graining procedure.That is, the kernel Gx is a spatial function, and it operates onparticle distributions at a particular time only. Accordingly, thediffusion equation as in Eq. (5a) is solved only to smoothenthe coarse-grained elds at each time uid step. Note that thetime variable s in the diffusion equation is the pseudo-time,and not the physical time. In principle it is possible to includetime in the kernel function as well, i.e., G Gx; t, with the nor-malization condition revised to be

RR4Gx; tdxdt 0. This is

indeed what was proposed in Zhu and Yu (2002). However,the kernel function Gx; t is symmetric both spatially and tem-porally. Essentially, the solid volume fraction es at any locationis contributed to by particles in its surroundings in all directions,with closer particles having more contributions. Similarly, esdepends on the particle distributions at both past and future

Fig. 10. Velocity defect u^ U1 u normalized by the free-stream velocity U1 inthe wake of a spherical particle along the imaginary streamline passing through thecenter of the sphere, illustrating the size of the wake of a particle at different owregimes and particle Reynolds numbers, which are dened as Rep U1dp=mf (wheremf is the uid kinematic viscosity). Results are obtained according to the empiricalformulas in Wu and Faeth (1993) and are presented for a range of typical particleReynolds numbers. To obtain these Rep values, particleuid relative velocities U1ranging from 0.005 m/s to 5 m/s, a particle diameter of dp 1 mm, and a uidviscosity of mf 1:0 106 m2=s are used.

ultiphase Flow 72 (2015) 233247 245which can be approximated to the rst order by the followingrelation:

with a 1 Dt=bt ; 14

exceeded that of the CFD solver in all cases, the additional

of Mcomputational costs are not signicant (less than 10% of the totalcosts) if the number of particles per cell is large, i.e., when thecomputational costs of the simulations are dominated by theDEM part. Therefore, the diffusion-based method is a theoreticallyelegant and practically viable option for coarse graining in general-purpose CFDDEM solvers.

Acknowledgments

The computational resources used for this project wereprovided by the Advanced Research Computing (ARC) of VirginiaTech, which is gratefully acknowledged. We thank the anonymousreviewers for their comments, which helped improving the qualityof the manuscript.

Appendix A. Conservation characteristics of the diffusion-basedmethodwhere n and n 1 are the time indices of the present and the pre-vious time steps, respectively. The scheme in Eq. (13) suggests thatto compute hesi for the current step, one only needs es of the currentstep and hesi of the previous time step. This would lead to reducedstorage requirements compared with a literal implementation of Eq.(11), where es at many previous time steps need to be stored. Whilethe above-mentioned characteristics of exponential kernel functionare exploited in the averaging in turbulence ow simulations(Meneveau et al., 1996; Xiao and Jenny, 2012), we are not awareof any such attempts in the context of coarse graining inCFDDEM simulations. Hence, we chose to present it here for thecompleteness of the diffusion-based coarse-graining algorithmand for the readers reference.

Conclusion

In this work we applied the previously proposed coarse-graining algorithm based on solving diffusion equations (Sun andXiao, 2014) to CFDDEM simulations. The conservation require-ments are examined and satised by properly choosing variablesto solve diffusion equations for. Subsequently, the algorithm isimplemented into a CFDDEM solver based on OpenFOAM andLAMMPS, the former being a general-purpose, three-dimensional,parallel CFD solver based on unstructured meshes. The imple-mentation is straightforward, fully utilizing the computationalinfrastructure provided by the CFD solver, including the parallelcomputing capabilities. Simulations of a uidized bed showed thatthe diffusion-based coarse-graining method led to more robustsimulations, physically more realistic results, and improved abilityto handle small CFD cell-size to particle-diameter ratios. Moreover,the mesh convergence characteristics of the diffusion-basedmethod are dramatically improved compared with the PCM. It isdemonstrated in the current work that the diffusion-based methodlead to mesh-independent results in CFDDEM simulations,conrming the conclusions drawn from the a priori tests in thecompanion paper (Sun and Xiao, 2014). The choice of thecomputational parameter, i.e., the bandwidth, has clear physicaljustications. The computational costs of the proposed methodwere carefully investigated. Results suggest that, although thecomputational overhead due to the diffusion-based coarse graininghesin 1 aens ahesin1 131

246 R. Sun, H. Xiao / International JournalThe conservation requirements for the particle mass, particlemomentum, total momentum in the uidparticle system asspecied in Section Conservation characteristics and choice of dif-fusion variables are summarized as follows:

qsXNck1

es;k Vc;k XNpi1

qs Vp;i; A:1a

qsXNck1

es;k Vc;kUs;k XNpi1

qs Vp;iup;i; A:1b

XNck11 es;kqf Vc;k F

fpk

XNpi1

f fpi ; A:1c

where the density qs is assumed to be constant for all particles; Ncis the number of cells in the CFD mesh; Np is the number of particlesin the system; Vc;k is the volume of cell k;Us;k is the Eulerian solidphase velocity in cell k; up;i is the Lagrangian velocity of particle

i; Ffpk is the force per unit uid mass exerted on uid cell k by all

particles; f fpi is the uid force on particle i.Multiplying both sides of the equations in the PCM coarse-

graining procedure, i.e., Eqs. (4), (7), and (8), by Vc;k; qses;kVc;k,and qf ef ;kVc;k, respectively, and taking summation over all cells,the conservation requirements in Eq. (A.1) can be recovered.Therefore, the PCM-based coarse-graining schemes as in Eqs. (4),(7), and (8) are conservative by construction. The proposedcoarse-graining algorithm consists of two steps: (1) coarse grainingusing PCM; and (2) solving diffusion equations for the quantities

es; esUs and efFfp. Hence, for the proposed algorithm to be con-servative, the diffusion step must also conserve the required physi-cal quantities in the domain, i.e.,

particle mass : qsXNck1

es;k Vc;k; A:2a

particle momentum : qsXNck1

ek Vc;kUs;k; A:2b

total momentum of fluidparticle system : qfXNck1

ef ;k Vc;k Ffpk : A:2c

For the diffusion equation @/=@t r2/ of a generic variable /on domain X with no-ux condition @/=@n 0 on the boundaries@X, integrating the equation on X yields:ZX

@/@t

dX ZXr2/dX

Z@Xr/ dS 0; A:3a

or equivalently;@

@t

ZX/dX 0; A:3b

which suggests that the conserved quantity that is implied by thediffusion equation @/=@t r2/ isZX/dX; A:4

or

XNck1

/iVc;k A:5

on a discretized nite volume mesh. Comparing Eq. (A.5) with thequantities in Eq. (A.2) suggests that diffusion equations should besolved for the following three quantities:

es; esUs; and efFfp A:6

ultiphase Flow 72 (2015) 233247to satisfy the conservation requirements for particle mass,particle momentum, total momentum in the uidparticle system,respectively.

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Diffusion-based coarse graining in hybrid continuumdiscrete solvers: Applications in CFDDEMIntroductionParticle-laden flows: Physical background and multi-scale modelingCoarse graining in continuumdiscrete particle-laden flow solvers

MethodologyMathematical formulations of CFDDEMDiscrete element method for particlesLocally-averaged NavierStokes equations for fluidsFluidparticle interactions

Diffusion-based coarse-graining methodSummary of the diffusion-based methodConservation characteristics and choice of diffusion variablesMerits and limitations of the diffusion-based coarse-graining method

Solver implementation and numerical methods

Numerical simulationsSolver validationsMesh-independence study of CFDDEM simulationsPerformance comparison with PCMComputational overhead of diffusion-based coarse graining

DiscussionChoice of bandwidth in kernel functions for coarse grainingImplementation of timevolume averaging in CFDDEM solvers

ConclusionAcknowledgmentsAppendix A Conservation characteristics of the diffusion-based methodReferences