a lagrangian modeling approach with the direct simulation monte-carlo method for inter-particle...

Advanced Powder Technol., Vol. 18, No. 4, pp. 395–426 (2007)© VSP and Society of Powder Technology, Japan 2007.Also available online - www.brill.nl/apt

Original paper

A Lagrangian modeling approach with the directsimulation Monte-Carlo method for inter-particle collisionsin turbulent flow

CHIH-HUNG HSU 1 and KEH-CHIN CHANG 2,∗1 Institute of Information Science, Academia Sinica, Taipei 115, Taiwan, Roc2 Department of Aeronautics and Astronautics, National Cheng Kung University,

Tainan 701, Taiwan, Roc

Received 18 May 2006; accepted 23 October 2006

Abstract—A Lagrangian modeling approach, which combines the direct simulation Monte-Carlo(DSMC) method and a Reynolds-averaged Navier–Stokes model to account for inter-particle colli-sions and turbulence characteristics of the carrier fluid, respectively, is proposed. The wall-boundedturbulent particle-laden flows in which the experimental data are available are chosen as the testproblems for demonstration. Results obtained with the deterministic method accounting for inter-particle collisions are used as a basis for validating the proposed stochastic Lagrangian model. Goodagreement between the predictions obtained separately with the deterministic and DSMC methods isachieved. The benefit of saving computational expenditure when using the DSMC method becomesmore remarkable than the deterministic method as the number of particles loaded in the flow is in-creased. In addition, the study demonstrates that τP/τC is a proper parameter to monitor the role ofinter-particle collisions in the physical processes of particle-laden flows.

Keywords: Direct simulation Monte-Carlo method; particle-laden turbulent flow; inter-particlecollision.

NOMENCLATURE

b displacement (m)b∗

ij non-dimensional Reynolds stress tensor1b∗

ij linear part of b∗ij

2b∗ij ,

3b∗ij nonlinear parts of b∗

ij

CB, CT, Cε1, Cε2 model coefficients of NLEVM

∗To whom correspondence should be addressed. E-mail: [email protected]

396 C.-H. Hsu and K.-C. Chang

CD drag coefficientCLR model coefficient of Magnus lift forceCTV model coefficient of viscous shear torqueCε1, Cε2 model coefficients of turbulence modulation in ε equationDP,i diameter of the ith particle (m)eP, ew restitution coefficients of inter-particle and particle–wall

collisions, respectivelyF

g↔Pi,n surface force acting on the carrier fluid due to the nth

sampling particle (kg m/s2)f model coefficient of Saffman lift forcefB, fr1, fr2, fS1, fS2, ft,fw1, fw2, fw, fε, fμ model functions of NLEVMfC mean inter-particle collision frequency (1/s)Gi Gaussian random numberIP,i particle inertia (kg m2)

J total number of nodes along the wall-normal directionJn, Jt normal and tangential components of impulse,

respectively (kg m/s)k turbulence kinetic energy (m2/s2)Mm number of real particles represented by the mth sampling

particlemP,i particle mass (kg)Ncell number of particles in a specified cellNP particle numbernp particle number density (1/m3)NLEVM non-linear eddy-viscosity modelp gas pressure (kg/s2 m)P ′ collision probability between two real particlesPl collision probability of the lth sampling particlePlm collision probability between a lth sampling particle and

all real particles represented by the mth samplingparticle

Pθ probability of the collisions lain underneath the contactangle θ

PDF(·) probability density functionRC radius of collision cylinder (m)Ri anisotropic correlation functionRm, Rθ, Rϕ random numbers in the range of 0 � Rm, Rθ, Rϕ < 1RP,i particle radius (m)Rt turbulent Reynolds numberReP particle Reynolds numberReR rotating particle Reynolds numberS∗, S∗∗, S∗

ij , S∗∗ij nondimensional strain rates of NLEVM

Inter-particle collisions in particle flow 397

Sij strain rate (1/s)SPk,n turbulence modulations of k (m2/s3)SPui ,n source term accounting for fluid-particle interaction

in momentum equation (m/s2)SPε,n turbulence modulations of ε (m2/s4)St Stokes numberT1, T2 starting and ending time of sample collection,

respectively (s)TLi Lagrangian integral time scale for the ith

component (s)TM max(τT ) (s)t time (s)Ugi , ugi , u

′gi mean, instantaneous and fluctuating gas (ith)

velocity component, respectively (m/s)U+

g , U+P mean streamwise gas and particle velocities

normalized by friction velocity on wallUm mean gas velocity in the channel (m/s)u′giu

′gj Reynolds stress tensor (m2/s2)

u′gu

′g+, v′

gv′g+, w′

gw′g+, u′

gv′g+

Reynolds stresses normalized by the squaredfriction velocity

u′+P , v′+

P fluctuating streamwise and transverse particlevelocities normalized by friction velocity

uτ friction velocity, uτ ≡ √τw/ρg (m/s)

W ∗, W ∗∗, W ∗kj , W

∗∗ij non-dimensional vorticities of NLEVM

Wij vorticity (1/s)yn, y

∗n dimensional (m) and non-dimensional normal

distances from the grid node to wall, respectively

Greek

β model coefficient of Saffman lift forceθ contact angleδij Kronecker delta functiont time step (m/s)tC free time scale of particles (m/s)Vcell volume of the computational cell (m3)ε dissipation rate of turbulence kinetic energy (m2/s3)μg gas viscosity (kg/s m)μP, μw slip-friction coefficients of inter-particle and particle–wall collisions,

respectivelyν, νt molecular and turbulent kinematic viscosities of gas, respectively (m2/s)


ξ prescribed constantρg, ρP gas and particle densities, respectively (kg/m3)σk, σε turbulent diffusion coefficients of k and ε, respectivelyτ, τd non-dimensional time scales of NLEVMτw viscous shear stress on wall, τw ≡ μg(

∂Ug

∂y)w (kg/s2 m)

τC particle mean free time between collisions (s)τK, τT Kolmogorov and turbulence time scales, respectively (s)τP particle relaxation time (s)ϕ orientation of the contact point on the collision plane

Bold

FD, FG, FM, FS drag, gravity, Magnus lift and Saffman lift force vectors (kg m/s2),respectively

J impulse (kg m/s) vectorn normal unit vectorr relative position (m) vector between two particlest tangential unit vectorTV viscous shear torque (kg m2/s2) vectorug, uP gas and particle velocity (m/s) vectors, respectivelyu∗

Pl post-collision velocity (m/s) vector of particleuP,rel relative velocity (m/s) vector between two real particlesuP,rel,F relative velocity (m/s) vector between contact points on two

particlesωg vorticity (1/s) vectorωP angular velocity (1/s) vector of particleω∗

Pl post-collision angular velocity (1/s) vector of particle

1. INTRODUCTION

Particle-laden flows are frequently encountered in engineering applications, suchas powder classifiers, powder mixing devices, fluidized beds, pneumatic conveyingsystems, solid rocket plumes, etc. In such two-phase flows, the motion of carrierfluid is usually turbulent. Thus, the interactions between turbulence and particleshave to be taken into consideration in the physical modeling in addition to thecomplicated turbulence phenomena of the carrier fluid. Interactions betweenturbulence and particles are 2-fold. One is the turbulence dispersion of the particles,i.e. the effects of turbulence on particles. The other is the modulation of thecarrier-fluid turbulence due to particles, i.e. the effects of the presence of particleson the turbulence structure. Another physical process in the particle-laden flowsis inter-particle interaction (collisions). However, the importance of inter-particleinteraction in the physical processes is heavily dependent on the particle number


density (concentration) of the particle-laden flows. Based on the classificationdefined by Crowe et al. [1], a dilute flow is a flow in which the particle motionis controlled by the surface and body forces on the particle. In a dense flow, theparticle motion is controlled primarily by inter-particle interaction. Tsuji [2] furtherclassified the particle-laden flows into three categories, in view of the extent of inter-particle interaction, as follows:

(i) Collision-free flow (dilute case).

(ii) Collision-dominated flow (non-dense case).

(iii) Contact-dominated flow (dense case).

Sommerfeld [3] suggested a quantitative criterion in terms of the parameter τP/τCto distinguish the dilute and dense regimes of particle-laden flows as follows.

τP/τC < 1, dilute case,(1)

τP/τC > 1, dense case.

Two-phase flow models have developed along two parallel paths, i.e. Eulerian–Eulerian (or two-fluid) and Eulerian–Lagrangian frameworks, depending on themanner in which the dispersed phase is treated [1]. A prerequisite basis ofcontinuum on the Eulerian formulation of the dispersed phase is always a stringentchallenge for the two-fluid models as applied to the dilute two-phase flows. Incontrast, the Eulerian–Lagrangian approach has been successfully applied to allregimes, from dilute to dense, of the two-phase flows. In the Eulerian–Lagrangianapproach, also named the discrete particle simulation by Tsuji [2], the dispersed-phase field is represented by the trajectories of the dispersed-phase entities obtainedfrom their equations of motion in the Lagrangian models.There are two main approaches in modeling inter-particle collisions under the

frame of the Eulerian–Lagrangian formulation. The most straightforward approachis the deterministic method in which the search for particle collision pairs is madedeterministically. This requires that all the particles have to be tracked simultane-ously throughout the flow field. However, the computational expenditure using thedeterministic method is proportional to the square of the total number of tracked par-ticles in the flow. This prevents application of the deterministic method to the flowsladen with relatively high particle number densities. The other is the stochastic inter-particle collision model in which collisions are treated probabilistically. Computa-tions of the non-dense (i.e. medium particle number density) particle-laden flowsusing the stochastic collision models are cost-effective. The Tsuji/Tanaka group[4–6] applied the direct simulation Monte-Carlo (DSMC) method, which was orig-inally developed for the study of molecular dynamics by Bird [7], to the studies ofthe phenomena of cluster formation and the flow instability induced by particles innon-dense particle-laden laminar flows. Recently, Sommerfeld [8] developed a sto-chastic inter-particle collision model, which relies on the generation of a fictitiouscollision partner and accounts for a possible correlation of the velocities of collid-ing particles in turbulent flows. Sommerfeld validated his stochastic inter-particle


collision model in particle-laden, homogeneous, isotropic turbulence. Later, Lainet al. [9] applied this stochastic inter-particle collision model to the computationof a particle-laden channel flow using the second-moment closure model. Since thestochastic inter-particle collision model does not require searching for possible col-lision partners in the vicinity of the considered particle as the deterministic methoddoes, the use of this stochastic model is economic with regard to computational ex-penditure. However, the stochastic model developed by Sommerfeld is limited tothe cases for inertia particles with sizes larger than the dissipation scale of turbu-lence. A review of the numerical studies of particle-laden flows in the past can befound in the current paper by Sommerfeld [3].Theoretically, direct numerical simulation (DNS) or large eddy simulation (LES)

can readily solve the carrier fluid flow under the Eulerian frame. However, thecurrent computer capacity permits the DNS computations of turbulence to be doneat rather low Reynolds numbers. The Reynolds-averaged Navier–Stokes (RANS)approach continues to play a principal approach for simulating the turbulent flows ofpractical engineering applications, which are usually associated with high Reynoldsnumbers. The LES approach seems to be a trade-off between the DNS andRANS approaches. However, in addition to more computational effort required forturbulence simulation by the LES approach as compared to the RANS approach,Sommerfeld [8] further pointed out one restriction in the application of LES to two-phase flow simulation, i.e. the LES approach is merely suitable for the cases inwhich the particle motion is dominantly controlled by the most energetic turbulenteddies. In other words, the particle relaxation time cannot be very much smaller thanthe integral time scale of turbulence. The RANS approach, which is applicable forthe whole range of turbulent flows, is thus used to solve the carrier-fluid turbulencein this study.An approach combining an available RANS turbulence model developed currently

with the DSMC method is introduced and, then, tested with the particle-ladenchannel flows experimentally conducted by Kulick et al. [10]. The performanceof the proposed approach (RANS+DSMC) accounting for inter-particle collisionsin turbulence will be validated by comparing the computed results with both whatis obtained using the same RANS model, but in association with the deterministicmethod for inter-particle collisions, and the experimental data of Kulick et al. [10].A sensitivity study of how to obtain a statistically significant solution by couplingwith the DSMC procedure is also performed in the study.

2. TEST PROBLEM

The experimental work on fully developed downward channel flows laden withparticles, which is schematically shown in Fig. 1, serves as the test problem.The present computation domain is the same as made in the numerical study byYamamoto et al. [11]. The Reynolds number based on the channel half-width andfriction velocity is 644, which is equivalent to the Reynolds number based on the


Figure 1. Computational flow configuration. This figure is published in color onhttp://www.ingenta.com

Table 1.Properties of the carrier fluid and two tested particles

Air Glass particles Copper particles

Density, ρ (kg/m3) 1.1766 2500 8800Particle diameter, DP (μm) 50 70Kinematic viscosity, ν (m2/s) 1.52 × 10−5

Restitution coefficient, e 0.95 0.95Skin friction coefficient, Cf 0.4 0.4

channel width and mean flow velocity of 24 326 [12]. Two flows laden individuallywith 70-μm uniform copper particles and 50-μm uniform glass particles, but at thesame mass loading ratio of 0.2, are tested in the study. The properties of the carrierfluid, which is air, and the tested particles are summarized in Table 1.

3. PHYSICAL MODELING

3.1. Flow of carrier fluid

A low-Reynolds-number, non-linear eddy-viscosity turbulence model (NLEVM)currently developed by Abe et al. [13], which is capable of accounting for theanisotropy of turbulence structure in the flow type considered in Fig. 1, is used todetermine the turbulent flow field of the carrier fluid. Since the volumetric fractionsoccupied by the copper particles and glass particles of the tested problems are nolarger than the order of 10−4, the void fraction of the carrier fluid can be reasonably


taken as unity in the whole computational domain. The instantaneous velocity ugi

can be decomposed into the mean (Ugi ) and the fluctuating (u′gi) components in

the turbulent flow. The Reynolds-averaged governing equations coupled with theturbulence model are given below.

Continuity:

∂Ugj

∂xj

= 0. (2)

Momentum:

DUgi

Dt= ∂

∂xj

[ν∂Ugi

∂xj

− u′giu

′gj

]− 1

ρg

∂p

∂xi

+Ncell∑n=1

SPui ,n. (3)

Turbulent kinetic energy:

Dk

Dt= ∂

∂xj

[(ν + νt

σk

)∂k

∂xk

]− u′

giu′gj

∂Ugi

∂xj

− ε +Ncell∑n=1

SPk,n. (4)

Dissipation rate of turbulent kinetic energy:

Dε

Dt= ∂

∂xj

[(ν + νt

σε

)∂ε

∂xj

]− Cε1

ε

ku′giu

′gj

∂Ugi

∂xj

− Cε2fε

ε2

k+

Ncell∑n=1

SPε,n. (5)

The Reynolds stress tensor u′giu

′gj shown in (3)–(5) is evaluated by an anisotropic

tensor b∗ij defined by [13]:

b∗ij ≡ CT

(u′giu

′gj

2k− δij

3

)= 1b∗

ij + 2b∗ij + 3b∗

ij . (6)

Here b∗ij is decomposed into one linear (1b∗

ij ) part and two nonlinear (2b∗

ij ,3b∗

ij ) partsin which the modifications for strong anisotropy in the near-wall region have beenincluded in the two nonlinear parts. These three parts are given by:

1b∗ij = −CB

{1 + fS1[1 − fw(26)]}S∗

ij , (7)

2b∗ij = −

{2CB[1 − fw(26)] + CTfw(26)(1 − f 2

r1)fw1

(τd

CTτ

)2}

× (S∗ikW

∗kj − W ∗

ikS∗kj ), (8)

3b∗ij =

{2CB[1 − fw(26)](1 + fS2) + CTfw(26)(1 − f 2

r1)fw2

(τd

CTτ

)2}

× (S∗ikS

∗kj − δij

3S∗2), (9)


where the relevant turbulent model functions and coefficients are summarized asfollows:

fw(ξ) = exp

[−

(y∗

n

ξ

)2], fS1 = 15fr1fr2(W

∗2 − S∗2),

fS2 = −fr1fr2[1 + 7(W ∗ − S∗)],fr1 = W 2 − S2

W 2 + S2, fr2 = S2

W 2 + S2,

fw1 = 5

12(1 + S∗∗W ∗∗), fw2 = 1

2(1 + S∗∗2),

τd = [1 − fw(15)]kε

+ fw(15)

√ν

ε, τ = νt

k, νt = 0.12fμ

k2

ε,

fμ ={1 + 35

R3/4t

exp

[−

(Rt

30

)3/4]}[1 − fw(26)],

y∗n = ynε

1/4ν−3/4, Rt = k2

νε,

Cε1 = 1.45, Cε2 = 1.83, σk = 1.2/ft, σε = 1.5/ft,

CT = 0.8, CB =[1 + 22

3W ∗2 + 2

3(W ∗2 − S∗2)fB

]−1

,

ft = 1 + 5fw(5), fB = 1 + 100(W ∗ − S∗),

fε ={1 − 0.3 exp

[−

(Rt

6.5

)2]}[1 − fw(3.3)],

Sij = 1

2

(∂Ugi

∂xj

+ ∂Ugj

∂xi

), Wij = 1

2

(∂Ugi

∂xj

− ∂Ugj

∂xi

),

S = √SmnSmn, W = √

WmnWmn,

S∗ij = CTτSij , W ∗

ij = CTτWij ,

S∗ = √S∗

mnS∗mn, W ∗ = √

W ∗mnW

∗mn,

S∗∗ij = τdSij , W ∗∗

ij = τdWij ,

S∗∗ = √S∗∗

mnS∗∗mn, W ∗∗ = √

W ∗∗mnW

∗∗mn.

The instantaneous interaction term for the fluid–particle momentum exchange isgiven by:

SPui ,n = − 1

ρg

MFg↔Pi,n

Vcell. (10)

The fluid–particle interaction term SPk,n for the k-equation is determined by:

SPk,n = ugiSPui ,n − UgiSPui ,n, (11)


while SPε,n for the ε-equation is modeled by following the approach of Mostafa andMongia [14] as:

SPε,n = Cεpε

kSPk,n. (12)

Kulick et al. [10] found Cεp ≈ Cε2 in their experimental study of particle-ladenturbulent channel flow, and Cεp = 1.83 is thus used in the study.

3.2. Particle motion

Equations of ith particle motion consist of translation and rotation, which areexpressed, respectively, as follows.

Translation:

mP,i

duP

dt= FD + FS + FM + FG. (13)

Rotation:

IP,i

dωP

dt= TV. (14)

Note the virtual mass, Basset and buoyancy forces have been neglected due to a factof the particle (copper or glass) density being much larger than the carrier fluid (air)density.The quasi-steady drag force FD is determined using the empirical drag coefficient

in terms of the particle Reynolds number ReP suggested by Schiller and Naumann[15] as:

FD = π

8ρgCDD2

P,i |ug − uP|(ug − uP), (15)

where

CD = 24

ReP(1 + 0.15ReP), ReP ≡ DP,i |ug − uP|

νg.

The Saffman lift force FS is induced by the velocity gradient of the surroundingfluid, which becomes remarkable in the near-wall region. It is determined using thefitted expression of Mei [16] as:

FS = f · 1.61D2P,i(μgρg)

1/2|ωg|−1/2[(ug − uP) × ωg], (16)

where:

ωg = ∇ × ug,

f ={

(1 − 0.3314β1/2) exp(−Rep/10) + 0.3314β1/2 if ReP � 40,0.0524(βReP)1/2 if ReP > 40,

β = DP,i

2|ug − uP| |ωg|.


The Magnus lift force FM accounts for the effect of particle rotation and isdetermined using the empirical lift coefficient in terms of ReP and the rotationalReynolds number ReR suggested by Oesterle and Bui Dinh [17] as:

FM = π

8ρg|ug − uP|CLRD2

P,i

[(ug − uP) × (ωp − (1/2)ωg)

(ωp − (1/2)ωg)

], (17)

where

CLR = 0.45 +(RePRer

− 0.45

)e−0.05684Re0.4R Re0.3P , ReR ≡ DP,i |ωp − (1/2)ωg|

4ν.

The viscous shear torque is acted by the viscous shear force on the rotating particleand is determined by [18, 19]:

TV = −CTVρ

2

(DP,i

2

)5∣∣∣∣ωp − 1

2ωg

∣∣∣∣(

ωp − 1

2ωg

), (18)

where the coefficient CTV is a function of ReR expressed by:

CTV = CT1√ReR

+ CT2

ReR+ CT3ReR. (19)

The values of CT1, CT2 and CT3 at various ranges of ReR are listed in Table 2.Note only the mean quantity of the carrier fluid velocity is determined in the

RANS approach. The fluctuating velocity of the carrier fluid along the particletrajectory is generated using the so-called Langevin equation model [20] as follows:

(u′gi )

t+t = Ri(u′gi )

t + Gi

√u′2gi

√1 − R2

i , (20)

where Gi is the Gaussian random number with a mean value of zero and a standarddeviation of unity. u′2

gi can be calculated from the turbulence model used in the study.The first term on the right-hand side in (20) represents the correlated part, while thesecond term contributes the random fluctuation of the velocity. The anisotropiccorrelation function Ri is given by:

Ri = exp(−t/TLi ), (21)

Table 2.Values of CT1, CT2 and CT3 in various ReR ranges [29]

ReR range CT1 CT2 CT3

ReR < 1 0.0 16π 0.01 < ReR < 10 0.0 16π 0.041810 < ReR < 20 5.32 37.2 0.020 < ReR < 50 6.44 32.2 0.050 < ReR 6.45 32.1 0.0


where the Lagrangian integral time scale TLi is determined from [21]:

TLi = 4

7

u′2gi

ε. (22)

3.3. Inter-particle collisions

Two steps are used to search for the particle collision pair. First, all particles areadvanced to the next time step (t + t) through solving the equations of motionwithout taking into account inter-particle collisions. Second, the occurrence ofinter-particle collisions within the time step t is examined by using either thedeterministic approach or the stochastic approach in this study. When the collisionoccurs, only the particle velocity and angular velocity are updated to the post-collision values, which are obtained by the hard-sphere model with the assumptionof spherical particles.

Deterministic method. Inter-particle collision partner is conditioned by (23),which states the distance (r) between two particles (say, particles l and m) equalto the sum of their radii (RP,l + RP,m) during a time step t in the deterministicmethod:

|rt + λ(rt+t − rt )|2 = (RP,l + RP,m)2. (23)

If there exist two real roots, λ1 and λ2 (λ1 � λ2), in (23) and 0 < λ1 � 1, a collisiontakes place between pairs at t + λ1t as schematically shown in Fig. 2 and theirvelocities are then replaced by the post-collision velocities.

DSMC method. In contrast, the DSMC method generates a collision by aprobability between two real particles (say, particles l and m) in a computationalcell Vcell during time interval t given by:

P ′ = π(RP,l + RP,m)2|uP,rel|t/Vcell. (24)

Being aware of a sampling particle representing a number of real particles in theDSMC method, the collision probability between a lth sampling particle and thenumber of real particles represented by the mth sampling particle (Mm) is given by:

Plm = MmP ′lm. (25)

Finally, it yields the collision probability of the lth sampling particle in a cell as:

Pl =Ncell∑m=1

Plm. (26)

No information is required about the actual positions of the particles as it is in theDSMC method.


Figure 2. Search for a collision pair in the deterministic method.

Determination of the collision between a pair of sampling particles follows themodified Nanbu method [22]. The procedure is briefly described as follows.A ‘candidate’ collision partner of the lth sampling particle (say, the mth samplingparticle) is arbitrarily selected from the subtotal sampling particles in the specifiedcell through the equation of:

m = INT[Rm · Ncell] + 1, (27)

where Rm is a random number obtained from a generator in the range 0 � Rm < 1and INT[Rm ·Ncell] is an intrinsic Fortran function which denotes the integer part ofRm · Ncell. If the condition

Rm >m

Ncell− Plm, (28)

occurs, it indicates that the lth sampling particle can collide with the mth samplingparticle during the time step t ; and the velocity of the lth sampling particle is,thus, replaced by the post-collision velocity.Once a collision takes place, any point on the half-spherical surface of the lth

particle in the direction of the relative velocity between the collision pair is apossible contact position. Consider a search for the collision between a specifiedparticle, say lth, and the mth particles. Figure 3 shows schematically the collisioncylinder with the radius RC (=RP,l + RP,m) along the trajectory of the lth particle.The location of the collision point on the collision plane of the collision cylindercan be specified in terms of the displacement b (or the contact angle θ , wheresin θ = b/RC) and the orientation. With an assumption of uniform collidingprobability inside the collision cylinder, the fraction of collision possibility withinthe annular zone at the position b with db thickness (or at the contact angle θ withdθ thickness) is equal to

uP,relt(2πb db)

uP,reltπR2C

= 2b db, (29)


Figure 3. Collision cylinder used in the DSMC method.

where b ≡ b/RC. Hence, the probability density function of b is defined as

PDF(b) = 2b, (30)

or can be expressed in terms of the contact angle θ , through introducing the relationof sin θ = b, as

PDF(θ) = sin 2θ. (31)

Thus, the probability of the collisions (Pθ ) laying underneath the contact angle θ

can be determined by

Pθ =∫ θ

0PDF(θ)dθ = sin2 θ. (32)

The Pθ value (0 � Pθ � 1) can be obtained from a random number generator in theMonte-Carlo process, say Rθ . This leads to:

θ = sin−1√

Rθ. (33)

Note only one random number is required to generate the sampling contact angle,while it needs to generate two random numbers for specifying a sampling contactangle in the stochastic model developed by Sommerfeld [8]. The orientation of thecontact point on the collision plane is randomly sampled from a uniform distributionin the circumferential angle between 0 and 2π, i.e.:

ϕ = 2πRϕ, (34)

where Rϕ (0 � Rϕ � 1) is another random number to be generated in the Monte-Carlo process.

Hard sphere model. After the occurrence of a particle collision by checking witheither the deterministic method or the DSMC method, the post-collision velocityand angular velocity, u∗

Pl and ω∗Pl , respectively, of the colliding lth particle are

calculated using the hard sphere model [23] as follows:

u∗Pl = uPl + J

mP,l

, (35)


ω∗Pl = ωPl + RP,l

IP,l

n × J, (36)

where n is the normal unit vector from the center of the lth particle to thecontact point and J is the impulsive force acting on the lth particle which can bedecomposed into the normal and tangential components as:

J = Jnn + Jtt. (37)

Here the tangential unit vector t, directed toward the relative velocity of the collisionpairs, is defined by:

t = uP,rel,F/|uP,rel,F|, (38)

where:

uP,rel,F = n × uP,rel × n. (39)

Using the momentum conservation concept and adopting Coulomb’s law of frictionfor the tangential component of the impulsive force, Jn and Jt are given by:

Jn = 1

2mP,l(1 + eP)n · uP,rel (40)

Jt = min

[−μPJn,

1

7mP,l|uP,rel,F|

]. (41)

The coefficients of restitution eP and slip friction μP are set to be 0.95 and 0.4,respectively, in the following computations.

3.4. Particle–wall collisions

Wall roughness is an important factor in the particle–wall collision process [3].However, a smooth wall boundary condition with sliding collisions, in which theparticles rebound from the wall as inelastic spheres (ew = 0.95, μP = 0.4) bouncingoff the wall, is adopted in the study due to lack of wall roughness information in theexperiment of Kulick et al. [10].

4. NUMERICAL METHOD

The carrier fluid governing equations are solved by the finite volume method ina collocated grid mesh. The unsteady term is discretized using a second-orderbackward difference scheme. A third-order MUSCL scheme [24] is used fordiscretizing the convective terms, and a second-order central difference schemefor discretizing the diffusion, pressure gradient and other source terms. Thecoupling between velocity and pressure is treated using the SIMPLEC algorithm[25] associated with the pressure-weighted interpolation method (PWIM) [26]. Theconvergent criteria is set as the sum of the absolute values of residuals being less


than 10−4 in each time step for each carrier fluid-dependent variable. In the wall-normal direction, the grid layout is generated by the function of

yj = 0.5 + 0.5tanh[α((j − 1)/(J − 1) − 0.5)]

tanh[α/2] . (42)

The parameter α is set equal to 4.87 in the study. In contrast, uniform griddistributions are used in the other directions.The equations of particle motion for both translation and rotation are solved

by iteratively integrating the non-linear first-order ordinary differential equationsto an acceptable tolerance in a specified time step. To avoid the full quadraticexpense of naive collision detection via a search of the entire computationaldomain using the deterministic method, the domain is split into a number of non-overlapping subdomains. Search for the potential partners for collisions are, then,limited to the particles residing within the same subdomain of the target particleunder consideration and the other 26 adjacent subdomains. Incorporating with themarching time step t as determined in (43), the subdomains are set as the same asthe grid sizes. Following this search of collision pairs, the computational cost forinter-particle collisions using the deterministic method can be effectively reduced.To assure an unconditionally stable computation, the marching time step used in

both the fluid solver and particle tracking process is dynamically decided by:

t = CFL · min

(xi

Ugi

∣∣∣∣i=1,2,3

,xi

uPi

∣∣∣∣i=1,2,3

, τT, τP, tC

), (43)

where xi/Ugi is the convective time of the carrier fluid and xi/uPi is theresidence time of particles in the cell. The value of the Courant number (CFL)is set to be 0.8. The turbulent time scale τT is evaluated by:

τT = max

(k

ε, τK

). (44)

Here the Kolmogorov time scale τK is set equal to 4√

ν/ε [27]. The particle relax-ation time τP represents the time a particle requires to respond to the surroundingfluid and is estimated by the following expression:

τP = ρPD2P

18μg. (45)

The free time scale of particles in a cell tC is evaluated as follows. Accordingto (26):

Pl =Ncell∑m=1

MπD2P|uP,rel|t/Vcell � NcellMπD2

P|uP,rel|maxt/Vcell

� NcellMπD2P2|uP|maxt/Vcell, (46)


where |uP|max is the maximum speed of particles in the cell. To assure the conditionof Pl � 1, it requires:

NcellMπD2P2|uP|maxt/Vcell � 1, (47)

and this leads to:

tC = Vcell/[2NcellMπD2P|uP|max]. (48)

5. RESULTS AND DISCUSSION

Although the investigated problem is a fully developed flow, a three-dimensional(3-D) flow model is used to account for the 3-D nature of the inter-particlecollisions for the translating and rotating particles. Due to the fully developedcharacteristics, the flow quantities vary primarily along the wall-normal directiononly in the computational domain. Periodic boundary conditions are thus appliedin the streamwise (x) and spanwise (z) directions for both gas and particles. Theno-slip condition in association with kw = 0 and εw = 2ν(∂

√k

∂y)2wall is imposed on

the wall [transverse (y) direction] for the carrier gas, while smooth wall boundaryconditions together with sliding particle–wall collisions is adopted due to lack ofwall-roughness information in the experiment of Kulick et al. [10].

5.1. Grid-independence test

Two grid meshes of 16 × 100 × 16 (x × y × z) and 32 × 100 × 32 are used forthe computations of a benchmarked case of fully developed channel flow withoutparticle loading in which the DNS data [12] is available. The computed results of themean streamwise velocity and all turbulence quantities (k, ε, u′

giu′gj ) using the two

meshes are presented in Fig. 4 and compared with the DNS data of Abe et al. [12].Two observations can be made from the comparisons shown in Fig. 4. First, a nearlygrid-independent solution can be achieved by using a rather coarse mesh of 16 ×100 × 16. This mesh is thereafter used in the following computations. In contrast,the LES computation of the same test problem conducted by Yamamoto et al. [11]required significantly finer meshes of 128 × 128 × 128 and 64 × 48 × 64 in thesingle-phase and two-phase cases, respectively. It demonstrates an advantage ofcomputational savings in use of the RANS approach for the turbulence computation.The other is that the performance made by the employed low-Reynolds-numberNLEVM is generally satisfactory in comparison with the DNS data.

5.2. Validation of DSMC computation

In the calculations of the flows laden with particles, particles are randomly locatedinside the channel at the beginning. The particles’ velocities are initially set as thesame as the carrier gas at the same positions. The mean quantities of the particleproperties at a specified grid cell are determined by use of the ensemble-averaging


Figure 4. Comparison between the predictions obtained with two grid meshes and the DNS data forthe particle-unladen flow at Reτ = 640.

concept. In view of the fully developed characteristic of the investigated flow, themean quantities of the particle properties at a specified lateral (say, j th) gird cell isstatistically calculated by:

φp(j) =∫ T2

T1

I∑i=1

K∑k=1

Ncell∑n=1

φp(n, t) dt/ ∫ T2

T1

I∑i=1

K∑k=1

Ncell dt, (49)

where I and K are the total numbers of cells in the streamwise (x) and spanwise(z) directions. The starting time of sample collection, T1, is chosen when the meancollision frequency has approached quasi-steadiness. The sample collecting timeinterval, T2−T1, must be sufficiently long to assure a statistically significant solutionand this issue will be elaborated later.The particle number density of the investigated copper particle-laden flow (con-

sisting of 15 900 real particles) is rather low. The stochastic Lagrangian modelwas usually not recommended to model such a dilute flow [11]. Nevertheless, thecomputations using both the deterministic and DSMC methods are purposely per-formed in this study. First, the numbers of sampling particles are set to be the sameas those of real particles of the investigated cases and the sample collecting timeinterval (T2 − T1) used in the statistical calculation of (49) is set equal to 10TM

where TM = max(τT) in the flow field. The results of the probability distribution


Figure 5. (a) Probability distribution of sampled contact angle and (b) distributions of mean inter-particle collision frequency of a particle and mean void fraction for the flow laden with 70-μm copperparticles.

of sampled contact angles and the distributions of the mean inter-particle collisionfrequencies and void fractions for the flow laden with 70-μm copper particles areshown in Fig. 5(a and b, respectively). Here, the mean inter-particle collision fre-quency is determined by:

fC = 2 × (number of collision pairs in the cell)

(mean number of particles in the cell) × (collecting time). (50)

The probability of the sampled contact angle θ obtained with the DSMC methodfollows a sinusoidal distribution as the theoretical result does. Furthermore, theprobability of the contact angle determined with the deterministic method alsoexhibits the same sinusoidal distribution. Both the lateral distributions of the meaninter-particle collision frequency and mean void fraction, obtained separately withthe DSMC and deterministic methods, are well coincident each other as shown inFig. 5(b). Sommerfeld [8] followed the derivation of Abrahamson [28], which is alimiting case of the kinetic theory for St → ∞ in addition to the neglect of external(such as gravitational) forces, and obtained an approximation solution of the meaninter-particle collision frequency for the monodispersed particles as:

fC = 4π1/2nPD2P|u′2

P|1/2. (51)

The theoretical values of fC are calculated with the aid of the predicted nP and u′2P

in the simulation, and are plotted in Fig. 5(b). The comparison shows that slightunderpredictions of the approximate solution, in comparison with the two presentfC predictions, are observed except in the very near-wall region in which the factorsof the particle–wall collision and the sharp gradient of mean streamwise velocity ofthe carrier gas were not considered in the derivation of (51). Overall, it validatesthat even the DSMC method is applicable to dilute particle-laden turbulent flowssuch as the investigated one with 70-μm copper particles.Figure 6(a) shows the predicted (using the deterministic method) and measured

[10] streamwise components of the mean velocity and r.m.s. of the fluctuating


Figure 6. Comparison of the predicted (using the deterministic method) and measured streamwisecomponents of the mean velocity and r.m.s. of the fluctuating velocity for (a) the carrier gas and70-μm copper particles and (b) the carrier gas and 50-μm glass particles.

velocity for the carrier gas and particles in the flow laden with 70-μm copperparticles, while Fig. 6(b) shows the predicted and measured results in the flowladen with 50-μm glass particles. Generally speaking, the agreement between thepredictions and measurements shown in Fig. 6(a and b) is satisfactory except for theproperties of the 70-μm copper particles. The mean streamwise particle velocitiesare remarkably overpredicted in the whole section, as compared to the measureddata, while the r.m.s. values of the fluctuating particle velocities are remarkablyunderpredicted in the core region for the 70-μm copper particles (see Fig. 6a). Thesame inconsistency between the predictions and measurements in the flow ladenwith 70-μm copper particles has been reported in the study by Yamamoto et al. [11]by using the LES incorporated with the deterministic method.Figure 7 shows the distributions of the Stokes number (St) and the ratio of τP/τC

for the two investigated particle-laden flows. Here, τC = 1/fC and St is defined by:

St = τP

τT. (52)

Approaching the wall, the turbulence is affected more by the viscous force andfinally reaches its smallest (Kolmogorov) scale as implicitly expressed in (44). This


Figure 7. Distributions of the Stokes number and ratio of τP/τC for the flow laden with (a) 70-μmcopper particles and (b) 50-μm glass particles.

explains the trend of St distributions shown in Fig. 7(a and b), i.e. the St valuesincrease when approaching the wall. It is known that the larger St value is, the lessthe influence of the carrier fluid on the particles. Due to the conditions of St � 1that occur in the near-wall region for both the investigated flow cases, it yields moredifferences of the flow properties between the carrier gas and particles in this regionthan those in the core regions, as exhibited in Fig. 6. Furthermore, the differencesof the flow properties between the carrier gas and particles in the flow laden with70-μm copper particles are larger than in the flow laden with 50-μm glass particles(cf. Fig. 6a and b) since the St values for the flow laden with 70-μm copper particlesare generally one order of magnitude larger than those for the flow laden with 50-μmglass (lighter than copper) particles (cf. Fig. 7a and b).

5.3. Effects of inter-particle collisions

The predicted mean streamwise velocity and r.m.s. of u′P and v′

P of the 70-μm copperparticles, using separately the deterministic and DSMC methods, are presented inFig. 8. Hereafter, the symbol DSMC(M) denotes the prediction using the DSMCmethod with a sampling particle representing M number of real particles. Inaddition, the predictions without considering inter-particle collisions (i.e. the two-way coupling reported by Crowe et al. [1]) are also presented in Fig. 8 for thepurpose of comparison. Figure 9(a and b) compares two snapshots of particledistributions near the wall (y+ < 5) in the flow laden with 70-μm copper particleswith and without considering inter-particle collisions. It can be seen that moreparticles are concentrated in the near-wall region for the case without consideringinter-particle collisions than the case when considering inter-particle collisions.This observation is consistent with what was reported in the study by Yamamotoet al. by using LES method. As revealed from the results of Figs 8 and 9, andnoting that the values of τP/τC shown in Fig. 7(a) are much larger than unity, itbecomes clear that the effect of inter-particle collisions cannot be ignored in themodel formulation of the present case according to the criterion indicated in (1).


Figure 8. Comparison of the predictions with [using separately the deterministic and DSMC(1)methods] and without inter-particle collisions for the flow laden with 70-μm copper particles (withthe sample collecting time of 10TM).

(a) (b)

Figure 9. Two snapshots of particle distributions near the wall (y+ < 5) in the flow laden with 70-μmcopper particles (a) with and (b) without considering inter-particle collisions.

As seen in Fig. 8, the results predicted separately with the DSMC and determin-istic methods differ slightly from each other, whereas the results predicted with-out considering inter-particle collisions differ greatly from those considering inter-particle collisions, as anticipated from the result of τP/τC > 1 shown in Fig. 7(a).It demonstrates that the quantitative criterion of (1) is a proper measure to monitor


the importance of inter-particle collisions in the physical processes of particle-ladenflows.

5.4. Relation between sampling particle number and sample collecting time inDSMC computation

Two more DSMC computations for the flow laden with 70-μm copper particles,but using less sampling particles, i.e. M = 10 and 100 (equivalent to 1590 and159 sampling particles, respectively), are performed for the investigated steadyflow. Their results are presented in Fig. 10 and compared with those obtained withM = 1. The statistical shot noise is obviously observed in the results of the twocomputations with M = 10 and 100, particularly in the near-wall region wherevery fine grids are located. Other runs using longer statistical sample collectingtimes of T2 − T1 = 100TM for the three cases discussed in Fig. 10 are furtherperformed, and their results are presented in Fig. 11. Two observations can bemade as follows. First, it shows that the extent of statistical shot noise shown inFig. 10 can be alleviated using a longer collecting time in the particles’ statisticsof (49). Second, the result obtained with the DSMC(1) method using the statisticalsample collecting time of 100TM remains nearly invariant as compared with thatusing the shorter collecting time of 10TM (cf. Figs 10 and 11). This indicates that a

Figure 10. Comparison of the DSMC predictions obtained separately with M = 1, 10 and 100 forthe flow laden with 70-μm copper particles (with the sample collecting time of 10TM).


Figure 11. Comparison of the DSMC predictions obtained separately with M = 1, 10 and 100 forthe flow laden with 70-μm copper particles (with the sample collecting time of 100TM).

statistically significant solution has been achieved in the DSMC(1) computation byusing the statistical sample collecting time of 10TM for the flow laden with 70-μmcopper particles. However, statistical shot noise still appears in the results obtainedwith M = 100. It implies that a longer collecting time than 100TM is needed topursue a statistically significant solution for the computation with M = 100, but nosuch effort is made in the study to find this answer.It is known that the computational expenditure varies with the employed numer-

ical model, M value (number of real particles represented by a sampling particle)and sample collecting time for particles’ statistics. Figure 12(a) presents the totalCPU times required in the computations made separately with the two-way cou-pling model (without considering inter-particle collisions) and with the determinis-tic, DSMC(1), DSMC(10) and DSMC(100) methods, versus the sample collectingtime for particles’ statistics. Note that the symbol ‘×’ shown in Fig. 12 indicates thesampling times required to obtain the statistically significantly solutions for eachtested cases. All the computations were performed with a single Pentium 4 PCunit in this study. The total CPU time is made up of two parts: the carrier gascomputation and the particle computation. The CPU time required for the particlecomputation (see Fig. 12b) can be further distributed to the computations of theequations of particle motion, i.e. (13) and (14), the inter-particle collisions and theparticle-wall collisions. The computations made with the two-way coupling model,


Figure 12. (a) Total CPU and (b) CPU for particle computation versus the sample collecting timefor particle statistics in various computations of the flow laden with 70-μm copper particles. (Herethe symbol ‘×’ indicates the sample collecting times required to obtain the statistically significantsolutions for each tested cases.)

the deterministic method and the DSMC(1) method were performed by tracking allthe real particles in the tested flow as the sampling particles. Thus, the difference ofCPU time between using the two-way coupling model and using the deterministicor DSMC(1) method shown in Fig. 12(b) represents the computational expenditureaccounting for inter-particle collisions, which is the main expense in the presentcomputation of the turbulent particle-laden flow. Although an efficient approach forsearching the particle collision pairs has been used in this study (see Section 4),the results shown in Fig. 12(b) indicate that the computation of inter-particle colli-sions using the deterministic method is more expensive than the computation usingthe DSMC(1) even in this rather dilute case. The difference of CPU time betweenFig. 12(a) and (b) for a specified method/model represents the computational ex-penditure for the flow solution of carrier gas. It shows that little expense is requiredfor the carrier gas computation except for the cases of DSMC(10) and DSMC(100),which were performed with noticeably less sampling particles. It is known that thecomputational expenditure for the flow solution of carrier gas (in Eulerian formula-tion) is directly dependent on the number of grid nodes employed in the computa-tion. Moreover, the present computations, based on the RANS turbulent model, canbe accurately performed using a rather coarse grid mesh. This is one main benefit(i.e. in terms of computational economics) in adopting the RANS approach in themodeling. Since the computations made with the two-way coupling model, the de-terministic method and the DSMC(1) method were performed using all real particlesas the sampling particles, the required sample collecting times for particle statistics,i.e. (T2 − T1) in (49), are basically the same and equal to O(10TM). Although theuse of less sampling particles in the computation can greatly reduce the expenseaccounting for the particles’ computation, it needs a longer sample collecting timeto attain the particles’ statistics as discussed before, which, in turn, increases therequired CPU time as shown in Fig. 12. For example, DSMC(10) required a samplecollecting time of O(102TM) to attain the statistically significant solution (cf. Figs 10and 11). It is noted that the marching time step t used in the present computations


was determined through (43) in which one time scale is calculated by xi/uPi . Inpractice, there exists a higher probability to generate a larger instantaneous particlevelocity uPi when using more sampling particles in the computation. It leads to theoccurrence of small t in the computation with small M number (M: number ofreal particles represented by sampling particles), which, in turn, increases the CPUtime required to reach the final solution in the computation. This explains why theCPU time required for the attainment of the statistically significant solution of par-ticle flow properties by adopting the DSMC(1) method is slightly longer than thatby the DSMC(10) as revealed in Fig. 12.

5.5. Further application to the case with higher number density

Next, the flow laden with 50-μm glass particles, but at the same mass loading ratioof 0.2 as the previous test flow laden with 70-μm copper particles, is investigated.Since the glass particles are lighter and smaller than the copper particles, the secondtest flow consists of 154 200 real glass particles, which is one order of magnitudelarger than the particle number in the flow laden with 70-μm copper particles. Thepredicted mean streamwise particle velocity and r.m.s. of u′

P and v′P, using the

models in which one does not consider inter-particle collisions (two-way coupling)and the other two consider inter-particle collisions but incorporated separately withthe deterministic and DSMC(1) methods, are presented in Fig. 13. Note the numberof sampling particles used in these three computations is the same as that of the realparticles. The sample collecting time used in (49) is set equal to 1TM, which is oneorder of magnitude less than the sample collecting time required for the attainmentof a statistically significant solution in the DSMC(1) computation of the flow ladenwith 70-μm copper particles. Once more, the results predicted separately with thedeterministic and DSMC methods agree well with each other, as observed for theprevious test flow laden with 70-μm copper particles (see Fig. 8).Note the values of τP/τC shown in Fig. 7(a) are much larger than unity, which

hints, according to the criterion indicated in (1), that the effect of inter-particlecollisions cannot be ignored in the modeling. In contrast, the values of τP/τC forthe flow laden with 50-μm glass particles (Fig. 7b) are comparably smaller thanthose for the flow laden with 70-μm copper particles (Fig. 7a), which implies thatthe mechanism of inter-particle collisions plays less of a role in the flow ladenwith 50-μm glass particles than that laden with 70-μm copper particles at the sameloading ratio of 0.2. Furthermore, it can be also observed that the St value of the50-μm glass particles (Fig. 7b) is generally one order of magnitude smaller than thatof the 70-μm copper particles at the same y position, which implies that the 50-μmglass particles not only have a longer time to interact with the carrier gas and, then,to recover from the post-collision state before occurrence of the next collision, butalso possess more tendency to follow the motion of the carrier gas. These argumentscan explain why the deviations between the two particle flow properties predictedseparately with and without considering inter-particle collisions in the flow laden


Figure 13. Comparison of the predictions with [using separately the deterministic and DSMC(1)methods] and without inter-particle collisions for the flow laden with 50-μm glass particles (with thesample collecting time of 1TM).

(a) (b)

Figure 14. Two snapshots of particle distributions near the wall (y+ < 5) in the flow laden with50-μm glass particles (a) with and (b) without considering inter-particle collisions.

with the 50-μm glass particles (Fig. 13) are less remarkable than those in the flowladen with 70-μm copper particles.Nevertheless, although the effects of inter-particle collisions become less remark-

able in the flow laden with the 50-μm glass particles, there remains apparent differ-ences between two snapshots of the near-wall particle distributions obtained withand without considering inter-particle collisions as displayed in Fig. 14. This indi-cates the significance of inter-particle collisions in modeling the present problem.


Figure 15. Comparison of the DSMC predictions obtained separately with M = 1, 10 and 100 forthe flow laden with 50-μm glass particles (with the sample collecting time of 1TM).

Two additional DSMC computations using less sampling particles, i.e. M = 10and 100 (equivalent to 15 420 and 1542 sampling particles, respectively), areperformed with the statistical sample collecting time of 1TM for the second testflow. Their results are presented in Fig. 15 and compared with those obtained withM = 1. The statistical shot noise is obviously observed in the case of M = 100,while it becomes less noticeable in the case of M = 10. Further runs using alonger statistical sample collecting time of 10TM with M = 1, 10 and 100 areperformed, and their results are presented in Fig. 16. It shows that a statisticallyinvariant solution can be now achieved using the DSMC(10) in association with thesample collecting time of 10TM, as compared to the solution of DSMC(1), whereasthere still appears slight shot noise in the solution obtained with DSMC(100).Figure 17(a and b) compares the total CPU time and the CPU time required for

the computation of particles versus the sample collecting time, respectively, for thefive computations which are made separately with the two-way coupling model andwith the deterministic, DSMC(1), DSMC(10) and DSMC(100) methods. Note thatthe symbol ‘×’ shown in Fig. 17 indicates the sampling times required to obtainthe statistically significantly solutions for each tested cases. Two observationscan be made after comparing Figs 12 and 17. One is the remarkable increaseof the computational expenditure for obtaining a statistically significant solutionin between the flow laden with 154 200 glass particles (which is marked with a


Figure 16. Comparison of the DSMC predictions obtained separately with M = 1, 10 and 100 forthe flow laden with 50-μm glass particles (with the sample collecting time of 10TM).

Figure 17. (a) Total CPU and (b) CPU for particle computation versus the sample collecting timefor particle statistics in various computations of the flow laden with 50-μm glass particles. (Herethe symbol ‘×’ indicates the sample collecting times required to obtain the statistically significantsolutions for each tested cases.)

crossline symbol in Fig. 17) and the one laden with 15 900 copper particles (whichis marked with a crossline symbol in Fig. 12) no matter what method/model isadopted in the computation. The other is that the gap between the two CPUtimes, which are required for the particle computations using the deterministicand DSMC(1) methods separately, is significantly enlarged as the tracked particle


number increases from 15 900 (the case of copper particles, Fig. 12b) to 154 200(the case of glass particles, Fig. 17b). This indicates that the DSMC methodbecomes more computationally inexpensive than the deterministic method as thereal particle number in the investigated flow is increased. Again, the results inFig. 17 show that the CPU time required for the attainment of the statisticallysignificant solution of particle flow properties by adopting the DSMC(10) methodis slightly shorter than that by the DSMC(1), which has been disclosed in Fig. 12for the flow laden with 70-μm copper particles. It has to be noted that the use of arather coarse grid mesh in the computation, which has been shown to be capable ofyielding an accurate solution of carrier gas for the present fully developed channelflow, used little CPU time in the computational procedure of each marching timestep. For the complex flows frequently encountered in the engineering practiceswhich have to be numerically solved using a much finer grid mesh than the presentone, the computational expenditure for the flow solution of carrier fluid would benoticeably increased. Thus, the choice of less sampling particles, which must beaccompanied with a longer statistical sample collecting time to attain a statisticallysignificant solution, in the computation of the practical cases of particle-laden flowmight not have such benefit of less computational expenditure as indicated in thisstudy.The total CPU costs for calculating the flow laden with 15 900 copper particles

by using the deterministic, DSMC(1), DSMC(10) and DSMC(100) methods for thesample collecting time 10TM are 66.98, 34.82, 4.255 and 1.214 min, respectively,as shown in Fig. 12(a). The computational expenditures of the DSMC(1),DSMC(10) and DSMC(100) methods are 52.0, 6.35 and 2.17% of the computationalexpenditure using the deterministic method, respectively. The CPU costs forcalculating the flow laden with 154 200 glass particles by using the deterministic,DSMC(1), DSMC(10) and DSMC(100) methods for the sample collecting time1TM are 519.3, 70.66, 3.271 and 0.538 min, respectively, as shown in Fig. 17(a).The computational expenditures of the DSMC(1), DSMC(10), and DSMC(100)methods are 13.6, 0.63 and 0.10%, respectively, of the computational expenditureusing the deterministic method for the calculation of particle motion. It is thusconcluded that the saving of computational expenditure is more remarkable withincreasing particle number.

6. CONCLUSIONS

An Eulerian–Lagrangian turbulent model accounting for inter-particle collisions,based on a RANS turbulence model for the carrier fluid and a stochastic Lagrangianmodel for the particles, is proposed. The mean quantities of the carrier fluidproperties are computed using the low-Reynolds-number NLEVM developed byAbe et al. [13], while the Langevin model generates the fictitious fluctuations ofthe carrier fluid velocity. The stochastic Lagrangian modeling approach followsthe DSMC method. For validating the proposed stochastic Lagrangian model, the


results obtained with the deterministic method are used to serve as a comparisonbasis. It has demonstrated that the proposed stochastic inter-particle collision modelcan perform as well as the deterministic inter-particle collision model does, butnoticeably saves computational expenditure, even for the case loaded with low-number-density particles such as the investigated flow laden with 70-μm copperparticles.A parametric sensitivity study on how to obtain statistically significant solution

of the particle properties is performed by changing either the number of samplingparticles or the time interval for sample collection used in the statistics of (49).It is shown that the lower the number of sampling particles tracked in the DSMCprocess, the longer the statistical sample collecting time in the statistical calculationrequired to attain a statistically significant solution of the particle flow properties.The benefit of saving computational expense in use of the DSMC method becomesmore remarkable, in comparison with the use of the deterministic method, as thenumber of the particles loaded in the flow is increased.In addition, it demonstrates that the criterion in terms of τP/τC, i.e. (1), is a

proper measure to monitor the importance of inter-particle collisions in the physicalprocesses of particle-laden flow.

Acknowledgments

The National Science Council of the Republic of China financially supports thiswork under grant NSC94-2212-E006-071.

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a lagrangian modeling approach with the direct simulation monte-carlo method for inter-particle...

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