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ARTICLE IN PRESSG ModelARTIC-730; No. of Pages 11

Particuology xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Particuology

jo ur nal home page: www.elsev ier .com/ locate /par t ic

ifferentially weighted direct simulation Monte Carlo method forarticle collision in gas–solid flows�

ongxiang He, Haibo Zhao ∗, Haoming Wang, Chuguang Zhengtate Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, China

r t i c l e i n f o

rticle history:eceived 13 December 2013eceived in revised form 28 April 2014ccepted 12 May 2014

eywords:irect simulation Monte Carloifferentially weighted methodas–solid flowarticle–particle collisionour-way coupling

a b s t r a c t

In gas–solid flows, particle–particle interaction (typical, particle collision) is highly significant, despitethe small particles fractional volume. Widely distributed polydisperse particle population is a typicalcharacteristic during dynamic evolution of particles (e.g., agglomeration and fragmentation) in spite oftheir initial monodisperse particle distribution. The conventional direct simulation Monte Carlo (DSMC)method for particle collision tracks equally weighted simulation particles, which results in high statis-tical noise for particle fields if there are insufficient simulation particles in less-populated regions. Inthis study, a new differentially weighted DSMC (DW-DSMC) method for collisions of particles with dif-ferent number weight is proposed within the framework of the general Eulerian–Lagrangian models forhydrodynamics. Three schemes (mass, momentum and energy conservation) were developed to restorethe numbers of simulation particle while keeping total mass, momentum or energy of the whole systemunchanged respectively. A limiting case of high-inertia particle flow was numerically simulated to validatethe DW-DSMC method in terms of computational precision and efficiency. The momentum conservation
scheme which leads to little fluctuation around the mass and energy of the whole system performedbest. Improved resolution in particle fields and dynamic behavior could be attained simultaneously usingDW-DSMC, compared with the equally weighted DSMC. Meanwhile, computational cost can be largelyreduced in contrast with direct numerical simulation.
© 2014 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy ofSciences. Published by Elsevier B.V. All rights reserved.

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ntroduction

Gas–solid flows are frequently found in industrial processesuch as fluidized beds, pneumatic conveying, boilers and furnaces,lectrostatic precipitators, and particle separation in cyclones.hese systems often involve complicated flow dynamics andnteraction between flow constituents and their surroundings. Aumber of numerical studies of gas–solid flow have been con-ucted focusing on one-way coupling (Deutsch & Simonin, 1991)nd two-way coupling (Squires & Eaton, 1990; Wang, Zhao, Guo,

Please cite this article in press as: He, Y., et al. Differentially weightegas–solid flows. Particuology (2014), http://dx.doi.org/10.1016/j.partic

e, & Zheng, 2013). One-way coupling that the influence of theolid particle on the continuous phase is neglected, is reasonablehen the particle fractional volume ˚v and mass loading ˚m are

� This paper is adapted from the presentation at the 4th UK-China Internationalarticle Technology Forum, October 15–19, Shanghai, China, as recommended byrof. Xiaoshu Cai and Dr. Jerry Heng, the co-chairs of the scientific committee.∗ Corresponding author. Tel.: +86 27 87545526; fax: +86 27 87545526.

E-mail addresses: [email protected], [email protected] (H. Zhao).

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ttp://dx.doi.org/10.1016/j.partic.2014.05.013674-2001/© 2014 Chinese Society of Particuology and Institute of Process Engineering, C

mall (e.g., ˚v < 10−6). However, with an increase in fractionalolume ˚v (e.g., 10−6 < ˚v < 10−3), the effect of the continuoushase on the dynamics of the dispersed phase and the feedbackf the dispersed phase on the continuous phase dynamics shoulde considered simultaneously. The conservation equations of con-inuous phase include appropriate source terms resulting from theispersed phase. The interaction between two phases is called two-ay coupling. While the volume fraction of particles ˚v continues

o increase (e.g., ˚v > 10−3), flows are referred as dense suspen-ion. The particle–particle collision plays an important role on therofiles of continuous phase and dispersed phase, the term four-ay coupling effect emerges. In fact, it is essential to considerarticle–particle collision even though the fractional volume ofhe particles ˚v is small (e.g., ˚v < 10−3), because the turbulentransport effect and the preferential concentration effect lead ton increase in inter-particle collision rates by a factor of 30 (Wang,

d direct simulation Monte Carlo method for particle collision in.2014.05.013

exler, & Zhou, 2000). Indeed, the average particle fractional vol-me is not the only measure of the importance of particle collision.esterle and Petitjean (1993) performed a developed horizontalhannel flow calculation in a non-dilute gas–solid suspension flow.

hinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

dx.doi.org/10.1016/j.partic.2014.05.013


http://www.sciencedirect.com/science/journal/16742001

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ARTICLE IN PRESSG ModelPARTIC-730; No. of Pages 11

2 Y. He et al. / Particuology x

Nomenclature

c particle velocity, m/sd diameter, mEts initial total energy of the whole systemFother external and inter-particle forcesg gravity acceleration, m/s2

l grid length, mMts initial total mass of the whole systemN total particle numbersNc average collision rate per unit volume, m−3

Nf total simulation particlesP the probability of one particle interacting with any

other particlePts initial total momentum of the whole systemS source termt time scale, s�t time step, sTe the integral Lagrange time scale, su fluid velocity, m/s; velocityVs volume of the whole system, m3

w number weight

Greek lettersˇ collision cross-section, m3/s� turbulent kinetic energy� effective transport tensorε turbulent kinetic energy dissipation rate, m2/s3

ı relative error time-accumulated relative error� fluid density, kg/m3; density, kg/m3

�p relaxation time scale of particle, s�k Kolmogorov time scale, s˚v particle volume fraction, m−3

Subscriptsa analytical valuee discarded particlei, j, k index of simulation particlek coordinate directionnew condition after dynamic eventold condition before dynamic eventp particle phaser property between two simulation particlest detected valuex, y, z coordinate directions1 indication of one of the decomposed particles2 indication of the other decomposed particleϕ the general transport variable

Superscriptsmax maximum valuemin minimum value— ensemble average value* particle velocity after collision event′ post-collision and pre-restoration; a new coordi-

nate system

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′ ′ post-collision and post-restoration


t was found that particle–particle collision has a strong influ-nce on the profile of the vertical particle concentration comparedith dilute flow simulation, revealing that particle–particle col-

ision plays a non-negligible role once the particle mass loading

i

cC

xx (2014) xxx–xxx

xceeds unity. In many cases, the particle loading is moderate toigh and hence particle–particle collisions significantly influencehe flow field. Therefore, the particle–particle collision and four-ay coupling between the continuous and discrete phases should

e processed for the purpose of accurate description and control inispersed systems.

Various models have been proposed to considerarticle–particle collision in the general framework ofulerian–Lagrangian models, where the continuous flow behaviors studied in Euler coordinates and the motion of dispersed phase isescribed by Lagrangian equation. Wassen and Frank (2001) clas-ify these models in two principally different ways: the trajectoryalculation (TC) method and simultaneous particle tracking (SPT)ethod. The inherent assumption of the TC method is that each

rajectory represents a constant flow of particles with identicalhysical properties, and macroscopic properties of the dispersedhase can be acquired by averaging all trajectories that cross the

ocal numerical cell. As a consequence, this method is limited topplications of steady flow. While the SPT method is inherentlynsteady because each particle represents a certain number of realarticles with identical physical properties, and the macroscopicroperties of the particulate phase are achieved by averaging allarticles in the same cell.

To take particle–particle collision into account in the frame ofhe TC method, Oesterle and Petitjean (1993) proposed an iter-tive technique with which particle–particle collision is treatedtochastically on the basis of the local concentration and veloci-ies obtained from the previous iteration. In the frame of the SPT

ethod, collisions can be computed either deterministically ortochastically. The most straightforward deterministic trajectoryethod, in which all particles have to be tracked simultaneously

hrough the flow field and the occurrence of particle–particle colli-ion can be judged by particle position and relative motion duringne time step, is based on molecular dynamics. Sundaram andollins (1996) used direct numerical simulation (DNS) method toeal with particle–particle collision (hard-sphere collisions, simi-

ar to the present study) to investigate the particle collision rate insotropic turbulent flows. Two different techniques (proactive andetroactive) are used for particle collisions in the DNS method. Theormer (proactive) technique anticipates all the collisions that willccur within one time step and then enacts them in order. While theatter (retroactive) looks for particle–particle overlaps at the end ofne time step, and then resolves these overlaps by enacting particleollisions retroactively in the order in which they occurred until allverlaps have been considered (Chen, Kontomaris, & McLaughlin,999; Sundaram & Collins, 1996). However, the DNS method isot able to handle multiple particle collisions that are of great

mportance for the modeling of quasi-static systems. The distinctlement method (DEM) first proposed by Cundall and Strack (1979)or granular dynamic simulation provides new possibilities for dis-rete particle simulation to calculate the dense phase flows using aoft-sphere collision model to deal with particle collision dynamics.suji, Kawaguchi, and Tanaka (1993) performed a two-dimensionalas-fluidized bed simulation where particle collisions are modeledy DEM and observed the phenomena of the formation of bubblesnd slugs and the process of particle mixing which is consistentith experiment. Note that the inherent computation cost of theeterministic trajectory method while choosing colliding particleairs is proportional to N2, where N is the number of traced parti-les. For this reason, the deterministic calculation is of no practicalelevance in some engineering applications such as high mass load-


ng impinging streams (Xu, Zhao, & Zheng, 2014).In fact, in the majority of the applications of the SPT method,

ollisions were treated stochastically. The direct simulation Montearlo (DSMC) method developed by Bird (1976, 1994) is a physically



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ccurate method for the computation of non-equilibrium gas flows.his technique is most useful in circumstances where there arensufficient numbers of collisions in the flow to maintain the equi-ibrium forms of the distribution functions describing the variousnergy modes of the gas. Several stochastic models were developedo reduce the numerical effort to a linear proportionality of theotal number of simulation particles. Bird (1976, 1989) proposedhe event-driven time-counter method and time-driven no-time-ounter (NTC) method for gas molecular collisions. In the frame ofhe NTC method, the number of collisions in each grid is calculatednd then an acceptance–rejection method is used to judge whether

collision event will occur. In addition, other stochastic modelsIllner & Neunzert, 1987; Ivanov & Rogasinsky, 1988; Koura, 1986;anbu, 1980) for selecting colliding pairs were also developed in

he dilute molecular gas dynamics. These stochastic models takedvantage of the analogy between the motion of discrete particlesn gas–solid flows and the motion of molecules in dilute gas flowo consider particle–particle collision in gas–solid flows. Tanaka,onemura, and Tsuji (1995) simulated a two-dimensional upwardas–solid flow in a rectangular domain with a periodic boundaryondition to study the effect of the particle property on the struc-ure of the particle cluster. Tsuji, Kawaguchi, and Tanaka (1998)imulated the riser of a circulating fluidized bed to investigatehe formation of particle aggregation compared with the two-fluid

odel. Furthermore, a stochastic particle–particle collision modelor particle-laden flow was suggested by Sommerfeld (2001) andesterle and Petitjean (1993), in which a generated simulation par-

icle, whose size and velocity are sampled from local probabilityensity functions of particle diameters and velocities, collides withhe selected particle during each time step. The post-velocity of theelected simulation particle is solved by the conservation equationor linear and angular momentum, while the properties of the sim-lation particle generated are of no further interest. The advantagef this model is that it does not require information on the positionf surrounding particles. However, it needs to conserve the particleize and velocity distribution of each grid.

The studies mentioned above focused on the collision of simu-ation particles with equal number weight. In many applicationsf interest, the traced particles possess only a small portion ofulti-species or polydisperse particle population, and the equallyeighted DSMC faces enormous difficulties or complex particleynamics in terms of either huge computational cost or signifi-ant statistical noise or even both. To circumvent this difficulty, acheme to apply different weight to simulation particles is required,epending on the species or polydisperse particle population. Aonservative weighting scheme (Boyd, 1996) was developed tomprove the resolution of the very small quantities in multi-pecies non-equilibrium gas flows. Nonetheless, the scheme doesot conserve linear momentum and energy explicitly at each colli-ion. Zhao, Kruis, and Zheng (2009) proposed a new differentiallyeighted time-driven Monte Carlo (MC) method, which could cap-

ure the coagulation dynamics in dispersed systems with low noisend track the size distribution over the full size range simul-aneously. So far, the differentially weighted scheme has beenuccessfully adopted for particle coagulation in the population bal-nce (Zhao & Kruis, 2014; Zhao, Kruis, & Zheng, 2010; Zhao, Kruis,

Zheng, 2011; Zhao & Zheng, 2009a,b, 2013); however, there areew reports on particle collision in gas–solid two-phase flows.

If differentially weighted (DW) simulation particles are trackedn gas–solid flows, the key issues include how to design a rulemodel) for a collision event between two differentially weighted


imulation particles, restoring the number of simulation particlesfter a simulation collision event, and coupling the DW-DSMC withhe Eulerian–Lagrangian model. A splitting and restoring procedureas proposed for particle collision with different number weight.

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xx (2014) xxx–xxx 3

he first stage was to split a simulation particle with larger weightnto two particles. Mass, momentum and energy conservationchemes were then adopted to restore the number of simulationarticles. The purpose of this study is to validate the DW-DSMCodel with different schemes for particle number restoration and

o set up a general simulation strategy.

odel description

imulation strategy

The equations governing continuous phase operation are con-eniently expressed in the Eulerian frame of reference. By focusingn a small finite control volume in the suspension, the conservationquations for the continuous phase operation can then be writtens (Lun & Liu, 1997):

∂(�ϕ)∂t

+ ∂(�ujϕ)∂xj

= ∂

∂xj

(�ϕ∂ϕ

∂xj

)+ Sϕ, (1)

here � is the fluid density; � � is an effective transport tensor;� is a ϕ-dependent source term that is the feedback influencef the dispersed phase on the continuous phase; and ϕ is a gen-ral transport variable that could be 1 (for mass conservation), ujfluid velocity for momentum conservation), and E (total energyor energy conservation). Considerable research has focused on S�

nd many models have been developed such as the Reynolds stressodel (Zhou, 1993).The position and velocity of every single particle can be

escribed as follows using Newton’s second law (Lun & Liu, 1997):

dxpi

dt= upi, (2)

dupi

dt= gi +

1�p

(ui − upi) + Fother, (3)

here xpi and upi are the particle position and velocity, respec-ively; the first term (gi) on the right-hand side of Eq. (3) ishe gravitational acceleration; the second term is the drag force;p = �pdp

2/(18�) is the particle relaxation time (where �p is thearticle density, dp is the particle diameter, and is the fluidiscosity); and the last term (Fother) expresses the external andnter-particle forces contributing to the particle transport.

Our particular focus here is on the coupling strategy of the dif-erentially weighted Monte Carlo and multiphase flow model, andhus the detailed models for multiphase flows are not discussed.s is known, most numerical strategies solving the gas–solid

wo-phase flow in the framework of Eulerian–Lagrangian modelre time-driven. Therefore, it is possible to embed the time-riven differentially weighted Monte Carlo method with theulerian–Lagrangian model into a single framework. By settingn appropriate time increment �t, it is thought that the threerocesses, fluid flow, particle motion, and particle collisions, arencoupled. As a result, the flow fields characterized by Eq. (1),he particle fields characterized by Eqs. (2) and (3), and parti-le collisions (see Section “Treatment of particle collisions”) arendependently solved, and four-way coupling is realized by thexchange of the environmental variables for the continuous andiscrete phases within �t. The numerical simulation flowchart forhe four-way coupling model is shown in Fig. 1.

he probability of particle–particle collision


A probabilistic collision rule was developed here for collidingarticles with different number weights, analogous to the proba-ilistic coagulation rule (Zhao et al., 2009; Zhao & Zheng, 2009a,b).




Fdelprcj

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P

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wP

ww

D

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�

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�

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Fig. 1. Flowchart of simulation strategy for four-way coupling.

or the simulation particle i and j with weight wi and wj, diameterpi and dpj, mass mpi and mpj, and velocity upi and upj, respectively,ach real particle from the simulation particle i undergoes a col-ision event with a probability of min(wi, wj)/wi, and each realarticle from j does so with a probability of min(wi, wj)/wj. As aesult, only min(wi, wj) real particles from i or j participate in realollisions on average. The collision rate of simulation particle i and

can be described as (Zhao et al., 2009; Zhao & Zheng, 2009a,b):

ij = �(dpi + dpj)2cr

4, (4)

′ij = ˇijwj�t

Vs

2wiwj(wi + wj) min(wi, wj)

= ˇijwj�t

Vs

2 max(wi, wj)(wi + wj)

, (5)

here Vs is the volume of the whole system, ˇij is the collisionross-section of particle i and j in unit time, cr is the relative velocityf particle i and j, and cr = |upi − upj|. Thus, the total collision rate ofimulation particle i with any other simulation particles is:

′i =∑Nf

j=1, j /= iP ′ij

Vs= 1

V2s

Nf∑j=1,j /= i

ˇijwj2 max(wi, wj)

(wi + wj), (6)

here Nf is the total number of simulation particles. Note that


′ij

= �(dpi + dpj)2crw�t/(4Vs), if wi = wj = w. Thus, the equally

eighted collision rule is only a special case of the differentiallyeighted probabilistic collision rule.

s

P

xx (2014) xxx–xxx

etermination of the time step

As noted above, the most important factor in the simulationtrategy for coupling the differentially weighted Monte Carlo forarticle collision with the Eulerian–Lagrangian model for hydro-ynamics is the appropriate time step within which the fluidransport, the particle transport, and the particle collision arencoupled and then separately simulated. Therefore, the time stept needs to be smaller than the particle relaxation time scale �p

nd the flow integral time scale Te to satisfy the general two-wayoupling condition. In reality, to narrow the scope of potential col-iding particle pairs, the maximum displacement in each directionf a simulation particle is constrained to be smaller:

t ≤ min

(lk

vmaxpk

), k = 1, 2, 3 (7)

here vpkmax and lk are the maximum velocity and grid length

long the kth direction, respectively. Furthermore, to couple thearticle–particle collision, each particle is restricted to participate

n, at most, one collision event. The waiting time between two suc-essive collision events of the simulation particle i can be obtained:

t ≤ tcollision = 1VsC ′

i

. (8)

To sum up, the time step �t is restricted as follows:

t ≤ min(�p, Te, min∀i,∀k

(lk/vmaxpi,k ), min

∀i(1/VsC

′i)), (9)

here vpi,kmax is the maximum velocity of particle i in the kth direc-

ion.

election of colliding particle pairs

Several stochastic models such as the no-time-counter (Bird,989) and Nanbu’s methods (Nanbu, 1980) have been proposeds described in the introduction section. In this study, modifiedanbu’s method is employed to deal with particle collision and itan be described as follows:

Step 1: generate a uniform random number R ∈ (0, 1) and the otherparticle probably involved in the collision event of a particle i isj = int(R × N) + 1 (int is the integer formulation, N is total particlenumber in the local grid);Step 2: calculate the probability of particle i and j according to Eq.(5);Step 3: particles i and j will collide with each other if R > (j/N −P ′ij), which will be introduced in Section “Treatment of particle

collisions”; otherwise, return to Step 1 until all simulation particlesare checked.

In fact, the modified Nanbu’s method is similar to thecceptance–rejection method. The computational cost of thisechnique is linearly proportional to the numbers of simulationarticles of the whole system, because the collision partner j isetermined by a mathematical formula. The time step �t is moreestricted because the modified Nanbu’s method divides the unitength into equal portions of N with each of 1/N shown in Fig. 2.

ith respect to time step, the following condition needs to be


atisfied:

′ij = ˇijwj�t

Vs

2 max(wi, wj)(wi + wj)

<1N. (10)



Y. He et al. / Particuology xxx (2014) xxx–xxx 5

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reatment of particle collisions

ollision dynamics between two equally weighted simulationarticles

For the collision event between two equally weighted simu-ation particles (the common number weight w = wi = wj), theumber of collision events between two real particles is equal tohe common number weight w. Because the simulation particle isn indicator of the real particles, the collision dynamics betweenwo equally weighted simulation particles is fully identical to thatf two represented real particles. A new coordinate system (x′, y′,′) is established at the collision point of the two particles, wherehe x′-direction is along the centerline from particle i and j; the′-direction is situated in the plane determined by the x′-directionnd the vector (upj − upi), and is perpendicular to the x’-direction;nd the z′-direction is always perpendicular to both the x′-directionnd y′-direction. The post-collision velocities of particles i and j are∗pi and u∗

pj , respectively.Using the law of conservation of momentum, the relationships

etween the pre- and post-collision velocities in the coordinateystem (x′, y′, z′) can be written as (Lun & Liu, 1997):

u∗pi,x′ = upi,x′ − Jx′

mpi; v∗

pi,y′ = vpi,y′ ; w∗pi,z′ = wpi,z′

u∗pj,x′ = upj,x′ + Jx′

mpj; v∗

pj,y′ = vpj,y′ ; w∗pj,z′ = wpj,z′

(11)

here the impulse Jx′ = (upj,x′ − upi,x′ )(2mpimpj)/(mpi + mpj). Rota-ional movement of particle is ignored in this study. However, theost-collision properties, i.e. linear and angular velocities, of simu-

ation particle j and 1 can be obtained by solving the conservationquation for linear and angular momentum, i.e., the impulse equa-ions in connection with Couloumb’s law of friction (Tsuji, Tanaka,

Yonemura, 1998), if particle rotation is considered. Therefore, thengular velocity of simulation particle 1 and j should be updatedfter the collision event. In conclusion, the DW-DSMC method cane extended to systems with the consideration of particle rotation.

The post-collision velocities of the two particles in the origi-al coordinate system (x, y, z) are obtained through transforminghe coordinate system (x′, y′, z′) into (x, y, z) since the impact posi-ion of the two particles at the moment of collision first needs toe determined. The impact point of the two particles can only be

ocated if two of the three angles are known. A stochastic processs used to determine this impact point. It is assumed that the prob-bility density of finding the impact point on the surface of onearticle is uniform. Thus the new coordinate system (x′, y′, z′) isransformed into the original coordinate system (x, y, z) and thenhe post-collision velocities of the two particles in (x, y, z) can bebtained.

ollision dynamics between two differentially weighted


imulation particlesThis study developed a simple binary collision model for two

ifferentially weighted simulation particles i and j. As describedn Section “The probability of particle–particle collision”, once a

fPtv

odified Nanbu’s method.

ollision between simulation particles i and j is considered, the realarticles of simulation particle i interact with the real particles ofimulation particle j. It is thought that only min(wi,wj) real particlesf simulation particles i and j undergo a real collision event. In otherords, the number of real collision events between real particles

s wj provided that wi > wj. Therefore, the simulation particle i isplit into two simulation particles: 1 with number weight wj and 2ith number weight (wi–wj). The other environmental parameters

f simulation particles 1 and 2 are inherited from the parent parti-le i. Simulation particle 1 interacts with simulation particle j usinghe hard-sphere model, which belongs to the variant of collisionynamics between two equally weighted simulation particles (seeection “Collision dynamics between two equally weighted simu-ation particles”), while simulation particle 2 does not participaten the dynamic process as illustrated in Fig. 3. Parametric variationuring the splitting process can be described as:

(w1)new = (wj)old; (m1)new = (mi)old; (up1)new = u∗pm

(w2)new = (wi)old − (wj)old; (m2)new = (mi)old; (up2)new = (upi)old

(wj)new = (wj)old; (mj)new = (mj)old; (upj)new = u∗pj

(12)

here subscripts “old” and “new” mean pre- and post-collision,espectively.

It is noticeable that each collision event between two differen-ially weighted simulation particles results in a net generation ofimulation particles. As a result, the increasing simulation parti-les have to be traced simultaneously, resulting in high CPU cost.ome measures need to be adopted to restore the number of sim-lation particles. Lin, Lee, and Matsoukas (2002) and Smith andatsoukas (1998) developed a Monte Carlo algorithm called con-

tant number Monte Carlo in which the number of simulationarticles remains constant throughout the simulation, regardlessf whether the actual growth process leads to a net loss (coagula-ion) or gain (fragmentation). In the light of the constant number

onte Carlo, we proposed mass, momentum and energy conserva-ion schemes to restore the number of simulation particles. Thesechemes are described as follows:

Mass conservation scheme: Once a collision event between twoifferentially weighted simulation particles has occurred, a sim-lation particle randomly chosen from the subsystem (here this

ndicates the grid in which a collision event occurs) is discarded.he number weight of the remaining simulation particles in theubsystem is recalculated to maintain the mass unchanged.

Momentum conservation scheme: Discard a simulation particleandomly as in the mass conservation scheme. The total momen-um is conserved to alter the number weight of the remainingimulation particles.

Energy conservation scheme: Different from mass conservationcheme, conserved energy of the subsystem is used as an indicatoro update the weight of the simulation particles.

Table 1 shows the variation of the subsystem properties at dif-


erent stages while dealing with the collision event, where Mts,ts, and Ets mean the total mass, momentum, and energy beforehe collision event (pre-collision), respectively; and we, mpe, andpe are the weight, mass, and velocity of the discarded particle.



6 Y. He et al. / Particuology xxx (2014) xxx–xxx

ess for

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M

wd(dt

ec

w

f

w

M

tc

N

wac�s

TV

Fig. 3. Schematic of collision proc

i, mpi, and vpi are the weight, mass, and velocity of the simula-ion particle i before the collision event, wi′ , mpi′ , and vpi′ are theeight, mass, and velocity of the simulation particle i after the col-

ision dynamics and before the restoration, and w′i′ , m′

pi′ , and v′pi′

re the weight, mass, and velocity of the simulation particle i afterhe collision dynamics and after the restoration. Here we simplydjust the particle number weight to keep the system conservationarameter (mass, or momentum, or energy) constant before andfter the restoration. Therefore, mpi = m′

pi = m′′pi; v′

pi = v′′pi, while

pi /= v′pi because of the collision dynamics. Variation in the individ-

al weights of the remaining simulation particles in the subsystemith the mass conservation scheme is described as follows:

′′ts =

Ns∑i=1

w′′i mpi =

Ns+1∑i=1

w′impi =

Ns∑i=1

w′impi + wempe

= M′ts

(= Mts =

Ns∑i=1

wimpi

), w′′

i = M′′ts + wempe

M′′ts

w′i, (13)

here Mts′ is the total mass after dealing with the collision


ynamics and before restoring the simulation particle numberpost-collision and pre-restoration), M′

ts′ means the total mass afteriscarding a randomly selected simulation particle and restoringhe sample (post-collision and post-restoration).

mtoa

able 1ariation in the subsystem properties at different stages dealing with particle collision.

System parameter Pre-collision Post-c

Total particle numbers Ns Ns + 1

Total particle mass Mts =Ns∑i=1

wimpi M′ts =

Total particle momentum Pts =Ns∑i=1

wimpivpi P ′ts =∑

Total particle energy Ets = 12

Ns∑i=1

wimpiv2pi

E′ts =

differentially weighted particles.

Therefore, the weight variation using the momentum andnergy conservation schemes can be described as:for momentumonservation scheme,

′′i = P ′′

ts + wempevpe

P ′′ts

w′i (14)

or energy conservation scheme,

′′i =(

1 + wempev2Pe

2E′′ts

)w′i (15)

odel validation

The average inter-particle collision rate, Nc, is a key parame-er that describes the numbers of particle collisions and can bealculated as follows:

c = Nct

Vs�t, (16)

here Nct is the detected collision events within one time step. Thenalytical average collision rate can be obtained in some limitedonditions such as the zero-inertia particle motion (�p � �k, wherek is the turbulent Kolmogorov time scale) in simple homogeneoushear flow (Wang, Wexler, & Zhou, 1998) and the free molecular


otion of high-inertia particles (�p � Te) (Abrahamson, 1975). Inhis study, the high-inertia particle flow is simulated to validateur model. As the particle motion is not affected by the gas flow,nalogous to that of molecular gas, the particle velocity is constant

ollision and pre-restoration Post-collision and post-restoration

NsNs∑i=1

w′impi + wempe M′′

ts =Ns∑i=1

w′′impi

Ns

i=1

w′imivpi + wempevpe P ′′

ts =Ns∑i=1

w′′impiv′

pi

12

Ns∑i=1

w′impiv2

pi+ 1

2wempev2pe E′′

ts = 12

Ns∑i=1

w′′impiv′2

pi



ology xxx (2014) xxx–xxx 7

wcmtto&mpapaait

C

rmpck

N

wut

Table 2Initial parameters of monodisperse high-inertia particle flow.

Initial parameter Heavy particles

Computational region 0 < x < 2�,0 < y < 2�,0 < z < 2�Boundary condition Periodical boundary in three directionNumber of cells 8 × 8 × 8Initial particle velocity distribution Maxwell distribution and 〈vp

2〉 ≈ 3Total particle numbers, N 105

Total fictitious particles, Nf 8 × 104

Particle diameter, dp 0.1Time step, �t (s) 0.0001

ipptrse(dptucc

Fr


ithin a time step increment �t. In addition, the high-inertia parti-le flows are simulated using the DNS and equally weighted DSMCethod for particle collisions as references. The technique used

o identify collisions in DNS is the retroactive method. To reducehe computational cost, an optimization technique is implementedn the time-sequenced descending order collision list (Chen, Liu,

Zheng, 2004; Sundaram & Collins, 1996). The particles are firstoved backward in time to the point of collision of the first overlap-

ing particle pair, the elastic collision between the pair is affected,nd the particles are then moved forward in time. Overlap com-utations are redone for collided particles and new overlaps aredded to the schedule list while nonexistent overlaps are deleted,fter which the entire list is re-sorted in descending order. Thesenstructions are carried out until there are no remaining overlapso consider.

ase 1: monodisperse high-inertia particle flow

Abrahamson (1975) developed a turbulent collision kernel withegard to high-inertia particles. As described above, the particleotion is similar to that of molecular gas. Thus, when the initial

article velocities follow a Maxwell distribution, the mean parti-le collision rate can be written as follows by applying molecularinetic theory:

ca = 0.5n2d2p

(16�

⟨v2

p

⟩3

)0.5

, (17)


here Nca is the analytical average collision rate per unit vol-me and unit time, n the particle number concentration, and 〈vp

2〉he mean square value of the particle fluctuation velocity, which

r

pp

ig. 4. Comparison of statistical parameters between different numerical methods: (a) theate per unit volume, and (c) the time-accumulated relative error of average collision rate

Time evolution, T (s) 0.01

s dependent on the particle fields. Table 2 presents the initialarameters of the monodisperse high-inertia particle flow case. Thearticle volume fraction (�v = Nfd

3p/48�2) is 16.7% to ensure mul-

iple collision events during each time step. The total number ofeal particles for the whole system is 105 and the total number ofimulation particles 8 × 104, hence the average number weight ofach simulation particle using an equally weighted DSMC is 1.25=N/Nf). In this case, to validate the differentially weighted scheme,ifferent number weights are applied artificially to the simulationarticles. The weight ratio of one half of the simulation particleso the other half is 1.5. Simulation particles are initially distributedniformly in the computational region using a stochastic game. Aonstant time step is used throughout the simulation because of theollision dynamic event and hard-sphere elastic collision model,


educing computational time for evolution in a time step.Fig. 4(a) shows the time evolution of the average collision rate

er unit volume predicted by the DW-DSMC, DSMC, and DNS, com-ared with analytical results according to Eq. (17). The results

average collision rate per unit volume, (b) the relative error in the average collision per unit volume.



8 ology x

oaopoaıstse

|

av aeteu

nfPtt

teMtiFscmp

fietssbuaratdls

F(


btained using the stochastic model for particle collisions fluctu-te up and down around analytical data, and are of the same orderf accuracy. It can be inferred that the variation in the averagearticle collision rate using DNS is larger than that using thether two numerical methods. We further calculated the rel-tive error ıNC in the average collision rate per unit volume,NC = |Nc − Nc,a|/Nc,a. We found that the relative error using thetochastic model for particle collisions fluctuated within 7.5% whilehe relative error using DNS is in the range of 0.048–18.5% ashown in Fig. 4(b). Furthermore, the time-accumulated relativerror for the average collision rate NC was calculated, NC(t) =t∫0

Ncdt − tNc,a|/tNc,a. Fig. 4(c) shows the evolution of the time-

ccumulated relative error for the average collision rate per unitolume. It was inferred that the time-accumulated relative errorNC(t) using each numerical method reached an asymptotic value

nd it was generally constrained within 3.7% for a relatively longvolution time (the 20th time step, t = 0.002 s). It was obvious thathe time-accumulated relative errors using the momentum andnergy conservation schemes were slightly larger than those whensing the mass conservation scheme.

Comparison of the proposed schemes for the restoration of theumber, mass, momentum, and energy of the simulation particles


or the total system at the end of each time step, expressed as Mi,i, Ei, respectively, was also observed. Fig. 5(a) and (c) presentshe time evolution of the normalized total system mass, momen-um, and energy. It was found that not only the momentum of the

aIa

ig. 5. Total system properties as a function of time for monodisperse high-inertia particb) momentum, and (c) energy.

xx (2014) xxx–xxx

otal system, but also the mass and energy were conserved duringach time step when using the momentum conservation scheme.eanwhile, the mass conservation scheme sustained high statis-

ical noise levels for the energy of the whole system, althought fitted the momentum of the whole system well as shown inig. 5(b). It was obvious that the mass and momentum of the wholeystem increased with time after 30 time steps using the energyonservation scheme, as shown in Fig. 5(a) and (b). Therefore, theomentum conservation scheme for the restoration of simulation

articles was adopted in this study.Quantitative comparisons of the detailed information of particle

elds such as particle number density and particle turbulent kineticnergy are also discussed. Fig. 6(a) and (b) shows the information onhe detailed particle fields at a specified time-point (the 50th timetep t = 0.05 s) at a specific two-dimensional section (in the centralection of the z-direction (z = �)). Irrespective of the particle num-er density or particle turbulent kinetic energy, the results obtainedsing DW-DSMC were in good agreement with those using DNSnd DSMC. Therefore, we concluded that DW-DSMC was able toeasonably predict not only the statistical parameters such as theverage collision rate per unit volume (shown in Fig. 4(a)), but alsohe temporal and spatial evolution of particle fields, which wereetermined by the ensemble averaging procedure over all simu-

ation particles in the subsystem, such as the number density at apecific time-point (shown in Fig. 6(a)).


The numerical differences between the results from DNS, DSMC,nd DW-DSMC can be ascribed to the following two factors. (1)nitialization of particle number weight and velocity. The actu-lly traced simulation particles for each simulation method were

le flow normalized by the value of the total system initial properties for: (a) mass,



Y. He et al. / Particuology xxx (2014) xxx–xxx 9

Table 3Total number of simulation particles and particle number weight for each speciesusing different simulation methods.

Numerical method dp

0.05 0.1

Nf, 0.05 w0.05 Nf, 0.1 w0.1

DNS 1000 1 9.9 × 104 1

nttcfaoDuD

C

tofotodsOtc

Fig. 6. Distribution of the detailed information of particle fields using differentnumerical methods undergoing particle collision in the monodisperse high-inertiaflcp

D

Fst

DSMC 100 10 9.9 × 103 10DW-DSMC 1000 1 9.9 × 103 10

ot the same. Thus, statistical noise would be present for the ini-ial velocity distribution at the start of each simulation despitehe same seed for random number generator used for all numeri-al simulations. (2) Strategies adopted for particle collision. In therame of the SPT method, collision using DNS that is deterministic isscertained based on particle trajectory crossing, while it dependsn the probability between selected particle pairs in DSMC andW-DSMC. Furthermore, the collision dynamics between two sim-lation particles are the dominant difference between DSMC andW-DSMC in the monodisperse high-inertia particle flow.

ase 2: bidisperse high-inertia particle flow

In this case, we conducted a bidisperse numerical experimento illustrate the advantage of DW-DSMC. It considered a collectionf 105 real particles and the initial particle velocity distributionollowed a Maxwell distribution. The total system consisted of 1%f the particles that had a diameter of 0.05 and 99% of the particleshat had a diameter of 0.1. Table 3 shows the weight and numberf initial simulation particles with diameters of 0.05 and 0.1 usingifferent numerical methods, where Nf,0.05 and Nf,0.1 mean the total


imulation particles with diameters of 0.05 and 0.1, respectively.ther initial simulation requirements were the same as those in

he monodisperse case. Note that the system initial experimentalonditions for simulation particles with a diameter of 0.1 using

t

sa

ig. 7. Distribution of the detailed information of particle fields using different numeripecific time (t = 0.05 s) at a specific two-dimensional section (in the central section in theurbulent energy of particles with a diameter of 0.1, (c) number density of particles with

ow at a specific time (t = 0.05 s) at the specific two-dimensional section (in theentral section in the z-direction (z = �)), for: (a) particle number density and (b)article turbulent kinetic energy.

SMC and DW-DSMC, and 0.05 using DNS and DW-DSMC, were


he same (see Table 3).Fig. 7(a) and (b) presents the distribution of the number den-

ity and turbulent energy of particles with a diameter of 0.1 at specified time-point (the 50th time step t = 0.05 s) at a specific

cal methods undergoing particle collision in the bidisperse high-inertia flow at a z-direction (z = �)), for: (a) number density of particles with a diameter of 0.1, (b)

a diameter of 0.05, and (d) turbulent energy of particles with a diameter of 0.05.




Table 4Computational cost for high-inertia particle flows using different numericalmethods.

Computational cost DSMC DNS DW-DSMC (momentumconservation scheme)

Case 1 (CPU t/s) 77.695 815.828 99.348

t(toadTwoaldDwn

mfldCiplct

C

faEfsriflm

(

(

(

(

trpepteulbattpwwsps

bet

cnr

A

FK

R

A

B

B

B

B

C

C

C

D

I

I

Koura, K. (1986). Null-collision technique in the direct-simulation Monte Carlo

Case 2 (CPU t/s) 3.0313 926.924 3.350

wo-dimensional section (in the central section in the z-directionz = �)). It was obvious that there was no any difference betweenhe results using DSMC and DW-DSMC, which were in the vicinityf those using DNS, because of the statistical noise as illustratedbove. Fig. 7(c) and (d) presents the distribution of the numberensity and turbulent energy of particles with a diameter of 0.05.he results obtained using DW-DSMC were in excellent agreementith those using DNS. However, considerable statistical noise was

bserved for both the distribution of the particle number densitynd particle turbulent energy using DSMC to deal with particle col-isions, because only a small number of simulation particles with aiameter of 0.05 lay in a grid. In conclusion, the results using DW-SMC were correct enough and captured the collision dynamicsith low noise in regions where there were insufficient particleumbers.

Table 4 shows the computational cost using different numericalethods for the monodisperse and bidisperse high-inertia particle

ows. All the simulations in this study were performed on the sameesktop PC equipped with a Pentium (R) Dual-Core E6700 @3.2 GHzPU, and 1.96 GB of memory. The reduction of computational cost

n the DSMC and DW-DSMC simulations were the result of lowerarticle numbers and the use of stochastic processes to capture col-

ision events. Compared with DSMC, the increased computationalost for the DW-DSMC method was because of the restoration ofhe simulation particle numbers.

onclusions

In this study, a new differentially weighted DSMC (DW-DSMC)or particle collisions with different weights is proposed, and

simulation strategy of coupling DW-DSMC with the generalulerian–Lagrangian models for hydrodynamics within the sameramework is developed. The mass, momentum and energy con-ervation schemes are proposed for simulation particle numberestoration using a constant number Monte Carlo approach. Lim-ting monodisperse and bidisperse cases for high-inertia particleows were numerically simulated to validate the DW-DSMCethod. From this work, we conclude the following:

1) The monodisperse high-inertia particle flow simulation con-firms that the DW-DSMC method is able to correctly predictboth the statistical parameters and temporal–spatial evolutionof particle fields.

2) The momentum conservation scheme for DW-DSMC, in whichthe total momentum is conserved while discarding a simula-tion particle randomly, shows little fluctuation in the mass andenergy of the whole system. Meanwhile, the mass and energyconservation schemes cannot conserve these properties simul-taneously.

3) Improved resolution can be obtained simultaneously usingDW-DSMC in the bidisperse particle flow case, in comparison


with the equally weighted DSMC in which considerable statis-tical noise in the particle properties (dp = 0.05) (e.g., numberdensity, turbulent energy) exists.

L

xx (2014) xxx–xxx

4) Computational cost can be largely reduced using a stochasticmodel (DSMC, DW-DSMC) for particle collisions compared withthe deterministic trajectory model (DNS).

In short, this study focused on the development of a differen-ially weighted method, different schemes for simulation particleestoration and a simulation strategy for four-way coupling, butaid little attention to the complex two-phase turbulent mod-ls and other particle systems involved with particle rotation andolydisperse size distribution, which is easily incorporated withhe DW-DSMC and require further study in future. The stochasticssence dealing with particle collision makes DW-DSMC methodnable to obtain particle force information in the process of col-

ision dynamics. Therefore, this method is limited to account forinary collision. As illustrated by Bird (1994), the statistical fluctu-tions associated with variable particle weight appear to be largerhan normally to be expected from equally weighted DSMC simula-ion because of the random walk effect. In reality, adequate particleroperties can be gained without consideration of variable weighthen the ratio of traced particles to numbers of particles in thehole system exceeds 0.1. Therefore, the variable particle weighted

cheme should only be used where absolutely necessary, for exam-le, ratio of traced particles to numbers of particles in the wholeystem should be 0.1 or less (Boyd, 1996).

It is also recommended that the total simulation particle num-ers at a small portion of particle number concentration should bequal to those with higher portion by modulating the weight of theraced particles. This approach can weaken the random walk effect.

Furthermore, investigation into the operability of the numeri-al method in simulating the complex particulate processes, and aew method for differentially weighted particle collision that couldetain the system properties mathematically, are underway.

cknowledgements

The authors were supported by the National Natural Scienceoundation of China (51276077 and 51390494) and the Nationaley Basic Research and Development Program (2010CB227004).

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