par

a

tion

l of

ed

ows, plays an important role for improving the design ofengineering systems and control the particle transport.

form concentration distribution in the ow eld, but theparticles with the intermediate Stokes number of the orderof unity concentrate in the outer boundary regions of thelarge-scale vortex structures with highest diusivity. There-after, there are many related studies that obtain the similarconclusions (Longmire, 1994; Wicker and Eaton, 2001;

* Corresponding authors.E-mail addresses: zjulk@zju.edu.cn (K. Luo), fanjr@zju.edu.cn (J.R.

Fan).

Available online at www.sciencedirect.com

International Journal of Multiphase1. Introduction

Due to its particular ow characteristics, turbulent planejet has attracted extensive attention for a long time. Thefundamental study on it is helpful to the overall under-standing of the turbulent ow dynamics. The particle-ladenturbulent jet is also of practical interest because of its pres-ence in a broad range of engineering applications such ascombustion, aerosol reaction, jet propulsion and air pollu-tion control. In these processes, the inter-particle collision,which is one of the most interesting problems in two-phase

There have been many experimental and numerical stud-ies of gassolid two-phase turbulent jet ows over the pastdecades. Melville and Bray (1979) investigated the gassolid two-phase jet ows and presented a simple model tocharacterize them. To predict the particle dispersion in par-ticle-laden ows, Crowe et al. (1985, 1988) rst proposedthe use of the particle Stokes number which is the ratioof the particle aerodynamics response time to the time scaleassociated with the large-scale organized vortex structuresin the free shear ows. They found that the particles withthe smaller and the larger Stokes numbers have the uni-Abstract

To investigate the behaviour of inter-particle collision and its eects on particle dispersion, direct numerical simulation of a three-dimensional two-phase turbulent jet was conducted. The nite volume method and the fractional-step projection algorithm were usedto solve the governing equations of the gas phase uid and the Lagrangian method was applied to trace the particles. The deterministichard-sphere model was used to describe the inter-particle collision. In order to allow an analysis of inter-particle collisions independent ofthe eect of particles on the ow, two-way coupling was neglected. The inter-particle collision occurs frequently in the local regions withhigher particle concentration of the ow eld. Under the inuence of the local accumulation and the turbulent transport eects, the var-iation of the average inter-particle collision number with the Stokes number takes on a complex non-linear relationship. The particledistribution is more uniform as a result of inter-particle collisions, and the lateral and the spanwise dispersion of the particles consideringinter-particle collision also increase. Furthermore, for the case of particles with the RosinRammler distribution (the medial particle sizeis set d50 = 36.7 lm), the collision number is signicantly larger than that of the particles at the Stokes number of 10, and their eects oncalculated results are also more signicant. 2008 Elsevier Ltd. All rights reserved.

Keywords: Inter-particle collision; Hard-sphere model; Gassolid two-phase turbulent jet; Direct numerical simulation; Particle dispersionDirect numerical simulation ofjet considering inte

Jie Yan a, Kun Luo a,*, Jianren FaState Key Laboratory of Clean Energy Utiliza

bDepartment of Mechanophysics Engineering, Graduate Schoo

Received 7 September 2007; receiv0301-9322/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijmultiphaseow.2008.02.004rticle dispersion in a turbulent-particle collisions

n a,*, Yutaka Tsuji b, Kefa Cen a

, Zhejiang University, Hangzhou 310027, China

Engineering, Osaka University Suita, Osaka 565-0871, Japan

in revised form 6 December 2007

www.elsevier.com/locate/ijmulow

Flow 34 (2008) 723733

f MUchiyama and Naruseb, 2003; Luo et al., 2006). Hardalu-pas et al. (1989) investigated the velocity and particle uxcharacteristics of a turbulent particle-laden jet and foundthe mass mixture ratio has a little eect on the particle con-centration distribution. The dispersion of non-evaporatingdroplet in a round jet was studied experimentally by Long-mire and Eaton (1992). They found that particles of Stokesnumber which is between 1 and 10 concentrate largely inthe high-strain-rate and low-vorticity regions. This non-uniform particle distribution is known aspreferential con-centration which was also observed by Eaton and Fessler(1994). Yuu et al. (1996) performed direct simulation of agas-particle turbulent free jet with a low Reynolds number.They obtained relatively good agreement with their LDAexperimental data.

All the above-mentioned simulations are based on theassumptions of dilute ow and the inter-particle collisionsare neglected. But some studies showed that the inter-par-ticle collisions may play an important role in the dilutetwo-phase turbulent ows. Tanaka and Tsuji (1991) simu-lated the gassolid two-phase ow in a vertical pipe withaccounting for inter-particle collision. They describedinter-particle collision using the hard-sphere model withinter-particle collision calculated by a deterministic methodand found that the inter-particle collision has large eecton the diusion of particles even in dilute conditions inwhich particle volume fraction is O(104). Sommerfeldand Zivkovic (1992) considered the inter-particle collisionin dilute phase pneumatic conveying through pipe systemsby applying a stochastic collision model. They revealedthat inter-particle collision plays a non-negligible role evenat low overall mass loading for the development of the par-ticle concentration proles. Oesterle and Petitjean (1993)and Sommerfeld (1995) performed respectively the simula-tion of gassolid ow in a horizontal pipe and channel anddemonstrated the importance of inter-particle collisions.Thereafter, many researchers presented the similar conclu-sions (e.g., Lun and Liu, 1997; Sakiz and Simonin, 2001;Sommerfeld, 2003). Besides, there is some work on colli-sion frequency in homogeneous isotropic turbulence. Sun-daram and Collins (1997) performed one-way coupledDNS of isotropic particle-laden turbulent suspension tostudy the collision frequency. They found the collision fre-quency involves a complex interplay of the particle concen-tration and mechanisms responsible for relative motionbetween the particles. Wang et al. (1998a,b) performednumerical experiments to study the geometric collision rateof nite-size particles in isotropic turbulence and comparedtheir results with theoretical predictions by Saman andTurner (1956) and by Abrahamson (1975). It was foundthat although in very dierent manners, both the large-scale and small-scale uid motion can contribute to the col-lision rate. Sommerfeld (2001) applied the stochastic colli-sion model for inter-particle collisions in homogeneousisotropic turbulence and obtained consistent results.

724 J. Yan et al. / International Journal oAlthough there are some studies on the inter-particlecollision in the homogeneous isotropic turbulence and theows in pipes and channels, there is not the study ofinter-particle collision in the spatially-developing turbulentjet to the authors knowledge. In this study, we combine theDNS method and the hard-sphere model to study the inter-particle collisions for particles at dierent Stokes numbersin a three-dimensional spatially evolving plane jet. Themain objective is twofold. Firstly, to reveal the characteris-tics of inter-particle collision for particles at dierentStokes numbers. Secondly, to investigate the eects of theinter-particle collision on the particle dispersion and pro-vide references for the application in related industries.

2. Mathematical descriptions

2.1. Flow conguration and boundary conditions

Fig. 1 shows the ow conguration of a three-dimen-sional gassolid two-phase turbulent plane jet. The uidis injected into the domain through the whole slot nozzle,but the particles are just injected through a square regionlocated in the center of the slot with side length d. The uidReynolds number based on inow velocity U0 and nozzlewidth d is 3000. To include the largest vortex structuresin the simulations, the length of the calculation region inthe streamwise(x) and spanwise(z) directions 20d and 6.4din the lateral direction(y). Initially, a shear layer with thevelocity prole in the region with the nozzle width d atthe inlet boundary is calculated according to the followingtypical top-hat shape velocity prole (Klein et al., 2003a,b):

u U 02

U 02

tanhy2h0

v 0w 0

1

in which U0 is the initial streamwise inow velocity, h0 isthe initial momentum thickness selected as 0.05d. u, v andw are, respectively, the streamwise, lateral and spanwisevelocities. Outside the region of the nozzle width d, allthe uid velocities are set as zero. At the streamwise out-ow boundary, Neumann boundary conditions for velocityand pressure are used and the pressure is also corrected. Atthe lateral boundaries, the pressure is set as zero. In thespanwise direction, the periodic boundary conditions areapplied.

2.2. Fluid motion

To calculate the uid ow, direct numerical simulation(DNS) was performed. The gas phase is regarded as anideal, Newtonian uid. And, two-way coupling eects arenot considered in this study, which is a valid approach toanalyse solely the inter-particle collision behaviour and toevaluate the eects of inter-particle collisions. The samemodel was used by Tanaka and Tsuji (1991) and Sommer-

ultiphase Flow 34 (2008) 723733feld (2003). We will take into account two-way couplingbetween the uid and particle phase in the future work.

ensi

of MWhen the body force is not included, the ow eld can becalculated according to the following non-dimensional gov-erning equations:

Continuum equation :ouioxi

0 2

Momentum equation :ouiot

ouiujoxj

1qopoxi

1Re

ooxj

ouioxj

oujoxi

3

in which, ui, uj and p are non-dimensional velocity andpressure; q is the density of the uid; Re is the uid Rey-nolds number given by Re = U0*d/m, where U0 is inowvelocity, d is nozzle width and m is kinematic viscosity ofuid. And U0 and d are the characteristic velocity andlength scales, respectively. The reference density is the den-sity of the air under the normal temperature and pressure.

The general requirements to simulate the turbulenceusing DNS are that the numerical techniques can providethe high accuracy in both space and time and can also solvethe full NS equations eciently. The above governingequations are solved by the nite volume method and thefractional-step projection technique (Chorin, 1968). In thestreamwise and spanwise directions, the uniform staggeredgrids are used; in the lateral direction, the uniform stag-gered grids are set in the region of 2.25d < y < 2.25d, out-side this area the grids are stretched. The total grid points

Fig. 1. Flow congure of the three-dimJ. Yan et al. / International Journal200 64 128 are used along x, y and z directions. Cen-tral dierences are used for the spatial discretization toensure the second-order precision of the solutions in spaceand a low-storage, third-order RungeKutta scheme isapplied for time integration. The Poisson equation forpressure is solved using a direct fast elliptic method. Fordetailed numerical strategies, please refer to references(Klein et al., 2003a,b).

2.3. Particle motion

In the present work, the calculation of particle motion isdivided into two progresses by using the uncoupling tech-nique developed by Bird (1976). Firstly, the inter-particlecollision is not to be considered at rst and the motionsof all particles are calculated by the governing equationsmentioned later. Secondly, the occurrence of inter-particlecollisions during the rst step is examined for all particles.If a particle is found to collide with another particle, themotion of the collision pair is re-calculated using thepost-collision velocities and their position is assumed tobe changeless.

2.3.1. Particle motion without collision

To focus on the inter-particle collision and its eect onthe particle dispersion, the inuence of particles on uidis neglected. All individual real particles are tracked inthe Lagrangian framework based on one-way coupling.Due to the material density of particles is far larger thanthe density of the uid, the minor force terms such as thevirtual mass, the buoyancy, the Basset and the Magnusforces can be neglected (Stock, 1996; Vojir and Michae-lides, 1994). Considering only the Stokes drag, the non-dimensional governing equations for determining the parti-cle momentum and position are, respectively,

dupdt

fStuf up 4

dx

dt up 5

where up is the particle velocity vector and uf is the uidvelocity vector at the position of the particle. fis the mod-ied factor for the Stokes drag force, which can be de-

onal gassolid two-phase turbulent jet.

ultiphase Flow 34 (2008) 723733 725scribed by f 1 0:15Re0:687p when Rep 6 1000 (Cliftet al., 1978). The particle Reynolds number Re-

p = |uf up|dp/m (dp is particle diameter, m is kinematic vis-cosity of uid). The particle Stokes number St is dened as

St qpd2p=18ld=U0 (qp is the density of the particle, l is the uiddynamics viscosity). The velocity and position of the parti-cle can be obtained by integrating Eqs. (4) and (5). Theuid velocity at the position of particle is obtained by usingthe third-order Lagrangian interpolating polynomials.

2.3.2. Inter-particle collision

In the simulation, the numerical algorithm of inter-par-ticle collision is a key issue, because it greatly aects thewhole computation time. Considering the particle numberdensity is so low in the present work that a statistical

at time t and rRDt is the relative position vector after Dt.The relative position vector of the collision point betweenthe two particles is given by rRc = rR0 + k(rRDt rR0),where k represents a time scale factor. Inter-particle colli-sion has occurred during this time step when the followingequation for k has two real roots k1 and k2(k1 < k2,0 6 k1 < 1):

jrR0 krRDt rR0j2 d2p 6

where |rR0 + k(rRDt rR0)| is the relative distance betweenthe two particles. The potential inter-particle collisionsmentioned above are located by evaluating the roots ofEq. (6).

When inter-particle collision occurs, the post-collisionvelocities u0p of the two particles i and j can be obtainedaccording to the equations of impulsive motion:

u0pi upi J=mp 7u0pj upj J=mp 8

In the above equations, upi and upj are the pre-collisionvelocities of the two particles i and j, mp is the particlemass, J is the impulsive force exerted on particle i. Based

Fig. 3. Sketch of the relative motion between two particles.

f Mmethod to represent the inter-particle collision (Yonemuraet al., 1993) is not appropriate, so we use the deterministichard-sphere model (Tanaka and Tsuji, 1991). This model isbased on the following ideas: (a) particles are rigid spheres(i.e. any persistent particle deformation during a collision isabsent); (b) particles can interact each other only throughbinary collisions because particle concentration is su-ciently low that binary collisions are overwhelming; (c)inter-particle collisions are considered as instantaneousinelastic ones.

It should be noted that the foundation stone of the hard-sphere model is the law of conservation of momentum.This approach for treatment of inter-particle collision isbased on kinematics, in which what would arise during col-lision processes is not considered in detail. Instead, thismodel focuses on what have changed after the collisions.Thus, the solution for the collision is carried out basedon just a gure of geometrical relative displacements andany other kinetic relative parameters, like apparent mass,are not considered. Moreover, energy loss during the colli-sion process is taken into account by the coecient of res-titution e which is included by this model. The modelapplied for the present study is appropriate, because theparticle concentration is very low and the collision processis instantaneous, the particles are viewed as rigid ones andthe occurrence of many-body collision is scarce, which isvalidated by Tanaka and Tsuji (1991) and Yamamotoet al. (2001). The main procedures are as follows.

2.3.2.1. Grid index method. There are Np particles in thecomputational domain, and each individual particle maycollide with any other particle theoretically. Then identi-cation of particle collisions requires examiningNp(Np 1)/2 particle pairs, therefore, it is terribly ine-cient to go through the entire list of particles in order toidentify colliding particles of a given particle. Moreover,the time step Dt in the present numerical algorithm is sosmall that each particle can only collide with neighboringparticles during one time step. Consequently, we can exam-ine a reduced subset of possible colliding particles andimprove the computing eciency by using optimum grid-index method. The specic method is as follows: in thecomputational domain a uniform grid spacing with theside-length of the grids in three directions li(i = 1, 2,3)P ui p maxDt is used, where u

ip max is the maximum com-

ponent in three directions of the particle velocity. This sug-gests the potential collision particles for a given particle inany grid are found in the neighborhood comprising thatgrid and the 26 grids that surround the central gridABCDEFGH, as shown in Fig. 2. More details can befound in the literature (Sundaram and Collins, 1996).

2.3.2.2. Collision detection. Now let us identify whetherinter-particle collision occurs during one time step Dt. Eachparticle is assumed to move with a constant velocity during

726 J. Yan et al. / International Journal othis time step. Fig. 3 shows the relative motion particle j to iwhen they collide, where rR0 is the relative position vectorFig. 2. Sketch of the 27 neighborhood cells.

ultiphase Flow 34 (2008) 723733on the above assumptions, J can be expressed from (Tana-ka and Tsuji, 1991):

J Jnn J tt 9Jn 1 eMc n 10

J t min lfJn;2

7M jcfcj

11

Jnn and Jtt are the normal and the tangential componentsof the impulsive force J. n is the normal unit vector ofthe relative position vector of the collision point betweenthe two particles, t is the tangential unit vector in the direc-tion of the relative velocity of the particle j to i, e is thecoecient of restitution (e = 0.9 in present work),M = mp/2 for inter-particle collision and lf is the coe-cient of friction. The pre-collision relative velocity of theparticle j to i is given c = upj upi cfc is the tangentialcomponent of the pre-collision relative velocity c.

The motion of particles of the collision pair in the timestep Dt is re-calculated using the post-collision velocitieswhich are obtained by Eqs. (7) and (8).

To study the dependence of the inter-particle collisionon the particle size, eight kinds of particles at the Stokesnumber of 0.01, 0.1, 0.5, 1, 5, 8, 10 and 100 are traced. Fur-thermore, the particle cluster, whose size distribution iscorresponding with RosinRammler distribution function

(Macas-Garca et al., 2004; Giuliano et al., 2007), is alsochosen as tracers to demonstrate the collision among parti-cles with dierent sizes. For each case, 462 particles areinjected into the ow-eld every six time intervals. Initially,the particles are distributed uniformly in the nozzle and thevelocity equal to the velocity of the local uid.

3. Numerical results and discussions

3.1. Spatial distribution of the inter-particle collisions

Fig. 4 shows the spatial distribution of the inter-particlecollisions at the Stokes number of 0.01, 0.1, 1, 10 and 100in the ow eld at the non-dimensional time t = 42, as wellas the distributions of particles. In the gure, the red pointsrepresent the particles and the little green spheres representthe positions of inter-particle collisions. To observe thevortex structures interacting with the dispersed phase, thecorresponding three-dimensional lateral vorticity are alsoshown in Fig. 4a. By comparison, the eects on dispersionof the particles having dierent Stokes number by vortexstructures can be observed in detail. A brief summary ofthe particle dispersion patterns is provided here to explain

J. Yan et al. / International Journal of Multiphase Flow 34 (2008) 723733 727Fig. 4. Spatial distribution of inter-particle collision for particles at dieren(c) St = 0.1, (d) St = 1, (e) St = 10, (f) St = 100.t Stokes numbers at the time t = 42. (a) Lateral vorticity, (b) St = 0.01,

tures from large-scale to small scale in the ow eld, theparticles move towards down stream at higher speed andform the considerable ellipse prole. However, the numberof particles which disperse towards two sides is less andmost particles move downstream through the vortex coreregions. The reason is that the characteristic time scale ofthe particles is substantially larger than that of the vortexstructures during the turbulence transition. Then, the par-ticles are inuenced less and less by vortex structures. Asa result, the inter-particle collision positions are mainly in

Fig. 5. Spatial distribution of vorticities and inter-particle collision forparticles at the Stokes number of 10 at dierent times. (a) t = 18,(b) t = 26, (c) t = 34, (d) t = 50, and (e) t = 74.

f Mthe spatial distribution of the inter-particle collisions. How-ever, the interested reader is referred to Luo et al. (2006)for additional details. At time t = 42, due to the coexistenceof various vortex structures with dierent time scales andspace scales in the ow eld, the particle dispersion exhibitsselective characteristic, i.e. the local particle concentrationcorrelates well with the ratio of the aerodynamic responsetime of particles to the characteristic time of the local vor-tex structures. So, the particles having dierent Stokesnumber concentrate in dierent places (Chung and Troutt,1988; Uchiyama and Naruseb, 2003). For particles havingSt = 0.01, because of the smaller aerodynamic responsetime scale, they can response quickly to uid motion andare distributed uniformly in the ow eld. Some particlescan even disperse into the vortex core regions and form asimilar uniform distribution to the vortex structures. Thisdispersion is consistent with the results of previous studiesby Fan et al. (2004). Therefore, the phenomenon of inter-particle collisions for this kind of particles is not very obvi-ous, as shown in Fig. 4b. The particles at the Stokes num-ber of 0.1 can not disperse into the core regions any longer,most particles also distribute uniformly, but part of themconcentrates in the inner boundaries of the large-scalestructures. The inter-particle collisions are obvious in theseregions, as shown in Fig. 4c. For St = 1, the particles con-centrate largely in the outer boundaries of the large-scalevortex structures. In these regions where the local-focusingphenomenon happens, the inter-particle collisions can befound, as shown in Fig. 4d. Most particles at Stokes num-ber of 10 move downstream through the vortex structures,and part of them disperses towards the upper and lowersides with some angle. So the inter-particle collisions notonly happen in the center regions of the jet but also in bothsides of the dispersion, as shown in Fig. 4e. When theStokes number reaches 100, the particles are inuenced lessby the vortex structures and most particles directly movethrough the center regions of the jet towards down streamwith nearly rectilinear paths. Consequently, the inter-parti-cle collision positions are mainly in the center regions ofthe jet, as shown in Fig. 4f.

Fig. 5 shows the spatial distribution of collisions for par-ticles at the Stokes number of 10 in the ow eld at dier-ent times. Also, the corresponding three-dimensionallateral vorticities are shown to observe the vortex struc-tures interacting with the dispersed phase. In the initialstage of the jet, due to the larger inertia, this kind of parti-cles has not responded rapidly to the uid motion and thelateral dispersion is small with local concentration regions.In these regions, the inter-particle collisions are prominent,as shown in Fig. 5a. The large-scale vortex structures aredominant at the non-dimensional time t = 26 and 34. Thelateral dispersion of the particles can be observed in thedown stream of the jet, some particles disperse towardsthe upper and lower sides with some angle. In this case,besides in the center regions of the jet, the local inter-par-

728 J. Yan et al. / International Journal oticle collisions happen in both sides of dispersion, as shownin Fig. 5b and c. With the transition of the vortex struc-ultiphase Flow 34 (2008) 723733the center regions of the jet, as shown in Fig. 5d and e.These results suggest that the phenomenon of the inter-par-

ticle collision correlates well with the dispersion patterns ofparticles and the local particle concentration. Though theaverage concentration of the particles is lower in wholeow eld, the inter-particle collisions are remarkable inthe regions of local higher particle concentration.

3.2. Collision number

To further quantitatively investigate the eect on theinter-particle collision by the particle Stokes number, animportant parameter, the average inter-particle collisionnumber and its variation with the Stokes number isdepicted in Fig. 6. Here the average inter-particle collisionnumber is dened by N a NDt, where N is the total collision

J. Yan et al. / International Journal of Mnumber during the time step Dt. It is clear that the averageinter-particle collision number is not the simple linear rela-tionship with the Stokes number. At St = 0.1, the averageinter-particle collision number reaches a local peak value.After this, the average inter-particle collision numberdecreases and then turns to increase gradually with theincreasing of the Stokes number. This complex behaviouris related to two foundational factors, both of which causethe intersection of particle trajectories and the occurrenceof inter-particle collision: (i) particles tend to collect inregions of low vorticity leading to the changes of the localparticle concentration and (ii) particle inertia aects the rel-ative motion between neighboring particles thus alteringtheir relative velocity. In this paper, the inuence of bothfactors on the inter-particle collision is, respectively,termed the accumulation eect and the turbulent transporteect, followed Wang et al. (2000). Both of these eects willbe presented.

As mentioned previously, the dispersion of particleshaving dierent Stokes number exhibits selective character-istic. In general, particles tend to assemble in the low-vor-ticity and high-strain regions (Chung and Troutt, 1988;Uchiyama and Naruseb, 2003; Fan et al., 2004; Luoet al., 2006). Particles with moderate Stokes number com-pared to the larger Stokes number are more susceptibleto accumulation. Hence, they spend more time in the local

10-2 10-1 100 101 1020

500

1000

1500

2000

Ave

rage

inte

r-pa

rtic

le c

ollis

ion

num

ber

StFig. 6. Variation of the average inter-particle collision number with theStokes number.regions of accumulation. For these particles, the inter-par-ticle collision is mainly determined by the local accumula-tion eect and the relative motion between them is ofsecond importance. Especially, the particles havingSt = 0.1 are more susceptible to accumulation due to theaerodynamics response time scale of them may be in thesame order of the characteristic time scale of small-scalevortex structures and are driven predominantly by thesmall-scale vortex structures. The eect due to the localaccumulation (i.e. the preferential concentration eect)reaches a maximum for the particles having St = 0.1 withthe transition of vortex structures from large-scale to smallscale, and thus, the local concentration of particlesincreases signicantly with the development of jet and theinter-particle collisions are remarkable in these regions oflocal higher particle concentration. As a result, the averageinter-particle collision number reaches a local peak value atSt = 0.1. For St = 1, the aerodynamics response time scaleof particles is close to the characteristic time scale of large-scale vortex structures. So the local concentration eectoccurs only in the initial stage of the jet (a shorter time),when the large-scale vortex structures are in dominant.But with development of the ow eld, the preferentialconcentration of the particles having St = 1 does not occurprominently and the concentration distribution of the par-ticles also becomes uniform when large-scale structureschange to small-scale. Consequently, the average inter-par-ticle collision number in the whole stage of jet becomessmall. However, the particles with larger Stokes numbers(e.g. St > 5) are inuenced less and less by the uid andunable to follow the uid motion due to the larger inertia.They tend to collide more frequently because their inertiakeeps them moving towards each other despite drag. Inother words, the eect due to the local accumulationdecreases for particles with larger Stokes numbers, andthe turbulent transport eect starts to be the dominant fac-tor contributing to the occurrence of inter-particle colli-sion. Additionally, most of the particles having St = 5 ormore directly move through the center regions of the jettowards down stream with small dispersion and thus thedistance between particles decrease, which can easily causethe intersection of particle trajectories and thus causes theoccurrence of inter-particle collision. Therefore, these par-ticles collision number increases evidently.

To investigate the evolution of the inter-particle colli-sion with time, Fig. 7 shows the time history of the collisionnumber and maximum particle number in the cell for typ-ical particles at dierent Stokes numbers. The large-scalevortex structures are dominant in the early stage of thejet, the maximum particle number for each kind of particlesin the cell increases linearly with t. So the inter-particle col-lision number also increases linearly with time during thisperiod. After that, because the large particles for St = 10and 100 are inuenced less by the uid, most particles movetowards down stream with higher velocity. Along with this

ultiphase Flow 34 (2008) 723733 729trend, the inter-particle collision number in the ow eldalso changes with the same patterns. But for the particles

f M0200400600800

1000

St=0.01

400600800

1000

St=0.1

the

cell

730 J. Yan et al. / International Journal owith St = 0.01, 0.1 and 1, their dispersion are inuencedgreatly by the vortex structures. Due to the coexistence ofvarious vortex structures with dierent time scales andspace scales in the down stream of the jet, these particlesadjust their dispersion patterns to exhibit selective disper-sion characteristic. Then the preferential concentrationeect occurs and the distribution of particles becomesnon-uniform. Consequently, the maximum particle numberin the cell for these particles is obviously uctuant and

0200400600800

1000

St=1

0200400600800

1000

St=10

0 100 200 300 4000

200400600800

1000

St=100

0200

0

40

80

120

St=10

0 100 200 300 4000

40

80

120St=100

0

40

80

120

St=1

0

40

80

120

St=0.1

0

40

80

120

St=0.01

t

Max

imum

par

ticle

num

ber i

n

t

Col

lisio

n nu

mbe

r in

the

flow

fiel

d

Fig. 7. Time history of the collision number and maximum particlenumber in the cell for particles at dierent Stokes numbers. (a) Maximumparticle number in the cell; (b) collision number in the ow eld.reaches several evident peak values during the period oftime t = 200300, as shown in Fig. 7a. The collision num-ber for these particles in the ow eld also displays the cor-responding variation. It is also found from Fig. 7 thatthough the concentration of particles with Stokes numberof 100 in the cell is lower and uctuant with little change,its collision number is higher. One explanation is the dis-persion of large particles is lower and the relative motionsof these particles become relatively signicant due to theinuence of large-scale energetic eddies, the turbulenttransport eect is the dominant factor leading to the higherinter-particle collision number. This is in agreement withthe conclusion showed in Fig. 6.

3.3. Eect of collision on particle dispersion

We regarded particles at Stokes number of 10 as repre-sentatives to quantitatively examine the eects of inter-par-ticle collision. Cases with and without collision werecalculated. The root mean square function of particle num-ber per cell Nrms(t), the ratio of the particle number whichis greater than 2 in single cell to the total particle number aare dened as

N rmst Pnt

i1nit2nt

s12

a XnNi1

N=NtN > 2 13

where nt is the total number of computational cells in the sta-tistical domain, ni(t) is the number of particles in the ith cellat the same time. N is the number of particles in certain cell,n(N) is the number of cells with the particle numberN,N(t) isthe total number of particles in the ow eld at the time t.The function Nrms(t) can reect the non-uniform of particledistribution in the ow eld; the value of a can also reectconcentration distribution characteristics of particles.

To facilitate a comparison in later section, the develop-ments of the root mean square function of particle numberper cell and a with time for particles at the Stokes numberof 10 are presented in Figs. 10 and 11 (in Section 3.4),respectively. The inter-particle collisions aect the proles,i.e. the calculated results with inter-particle collisions aresmaller than those without collisions. This indicates the dis-tribution of particles in the ow eld and in the cellbecomes more uniform due to inter-particle collision. Espe-cially in the initial stage of the jet, the concentration distri-bution of particles is the most non-uniform, and the inter-particle collision occurs very frequently due to the inuenceof the local accumulation and the turbulent transporteects. Consequently, the eect of the collisions on the sta-tistic results is more obvious in the period of time t = 2050, as shown in Fig. 10. The reason is that inter-particlecollision causes the changes of the position and instanta-

ultiphase Flow 34 (2008) 723733neous velocity of particles and leads to more uniform par-ticle distribution in the ow eld.

3.4. Inter-particle collision with dierent Stokes numbers

In the above investigation, the inter-particle collisionwith the same Stokes number in each case was shown. Inthis section, the inter-particle collision is studied for theparticle cluster with dierent Stokes numbers, that is, eachof the particle may be a dierent size. Particle-size distribu-tion is given by applying the most common RosinRamm-ler distribution function, which is expressed as

d m

statistic results in the case of particles with dierent Stokesnumbers becomes more signicant compared with theeects of that in the case of particles at the Stokes numberof 10. It is mainly associated with the more frequent inter-

t0 100 200 300 400

0

20

40

60

Col

lisio

n nu

mbe

r in

th

Fig. 9. Comparison of collision number for the particles according to theRosinRammler distribution (d50 = 36.7 lm) and the particles at Stokesnumber of 10.

t0 100 200 300 400

0.3

0.4

0.5

0.6

0.7

with collision,different sizesno collision,different sizesSt=10 (with collision)St=10 (no collision)

Nrm

s(t

)

Fig. 10. Development of the root mean square function of particlenumber per cell for particles.

t0 100 200 300 400

0.7

0.75

0.8

0.85

0.9

0.95

1

with collision,different sizesno collision,different sizesSt=10 (with collision)St=10 (no collision)

Fig. 11. Development of a for particles.

J. Yan et al. / International Journal of MF d 1 exp d50

14

where F(d) is the distribution function, d is the particle size,d50 is the particle size corresponding with F(d50) = 0.5,namely the medial particle size, and m is a measure ofthe spread of particle sizes. Here d50 and m are the adjust-able parameters for dierent characteristics of the distribu-tion. In this section, the particles with the size distributiondemonstrated in Fig. 8 as weight percent amount in eachsize are traced. Here d50 = 36.7 lm, corresponding to theparticles at the Stokes number of 10.

Fig. 9 shows the comparison of collision number for theparticles with the RosinRammler distribution(d50 = 36.7 lm) and the particles at the Stokes number of10. It can be seen that both the variations of the collisionnumber with time are basically the same, but the averagecollision number of the particles with the RosinRammlerdistribution is signicantly larger than that of the particlesat the Stokes number of 10. This is because the particleswith dierent sizes have dierent velocities, which easilycause the intersection of particle trajectories and makethe particles to collide frequently.

The eects of inter-particle collision on particle distribu-tion for these particles with dierent sizes are also investi-gated. Figs. 10 and 11 show the development of the rootmean square function of particle number per cell and awith time separately. The inter-particle collision aectsthese proles and the calculated results considering inter-particle collision are remarkably smaller than those with-out at the same time. This indicates that inter-particle col-lision makes the distribution of particles in the ow eldbecome more uniform. These results are in good agreementwith those corresponding to the particles having St = 10.Furthermore, the eects of inter-particle collision on the

01.2 3.7 36.7 52 116

10

20

30

40

50

60

particle size (micron)

Wei

ght (

%)Fig. 8. Particle size distribution according to the RosinRammlerdistribution function.80

100

120

Rosin-Rammler distributionst=10

e flo

w fi

eld

ultiphase Flow 34 (2008) 723733 731particle collision when a particle size distribution isapplied. Because the velocities and trajectories of particles

are various when there is a size distribution, small particlesare more likely to collide with big particles than particleshaving the same size. As a result, the inter-particle collisionis easier to happen for these particles having the dierentsizes.

Moreover, in order to quantitatively examine the eectsof inter-particle collision on dispersion characteristics ofparticles along the lateral and the spanwise directions whenthere is a size distribution, the lateral dispersion functionand the spanwise dispersion function are dened as

Dyt XNti1

yit yi02vuut =Nt 15

Dzt XNti1

zit zi02vuut =Nt 16

where N(t) is the total number of particles in the ow eldat the time t. yi(t) and zi(t) are, respectively, the lateral andspanwise displacements of the ith particle. and zi0 are theinitial displacements of the ith particle when it is injectedinto the jet. The lateral dispersion function Dy(t) and the

Figs. 12 and 13 show the time history of the lateral andspanwise dispersion function for particles having the dier-

distribution is more uniform and the lateral and the span-wise dispersion of the particles also are enhanced. For the

0.002

0.003

0.004with collision,different sizesno collision,different sizes

Dy(t)

732 J. Yan et al. / International Journal of Mt0 100 200 300 400

0.001

Fig. 12. Time history of the lateral dispersion function for the particleswith dierent Stokes numbers.

t0 100 200 300 400

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

with collision,different sizesno collision,different sizes

D z(t)Fig. 13. Time history of the spanwise dispersion function for the particleswith dierent Stokes numbers.particles according to the RosinRammler distribution (themedial particle size is set d50 = 36.7 lm, corresponding tothe particles at the Stokes number of 10), their collisionnumber is signicantly larger than that of the particles atthe Stokes number of 10, and their eects on the statisticalresults are also more signicant. According to the aboveconclusions, the inter-particle collision cannot be neglectedeven in the dilute turbulent jet.

Acknowledgement

The present work is supported by the National Natureent sizes. The values of the lateral and spanwise dispersionfunctions with inter-particle collision are both greater thanthose without. This demonstrates the lateral and the span-wise dispersion of the particles increase due to the eect ofinter-particle collision in general. This agrees with the con-clusions mentioned above that inter-particle collisionmakes the distribution of particles become more uniformin the ow eld.

4. Conclusions

In summary, a detailed study of the inter-particle col-lision and its eects on the particle dispersion in a three-dimensional spatially evolving plane jet was presentedbased on direct numerical simulation and the determinis-tic hard-sphere model. The spatial distribution of instan-taneous inter-particle collision for particles at thedierent Stokes numbers and the development of inter-particle collision number with the Stokes number andtime were visualized. It is found that the occurrence ofthe inter-particle collision correlates well with the localparticle concentration, though the average concentrationof the particles is lower in the whole ow eld. The rela-tion between the average inter-particle collision numberand the Stokes number is not the simple linear, buthas a local maximum. First, the average inter-particlecollision number increases to a local peak value, thenturn to decline, and nally increases again gradually withthe increasing of Stokes number. For particles at thesame Stokes number, the variation of the collision num-ber with time in the ow eld is basically accordant withthat of the maximum particle number in the cell withtime.

Due to the eect of inter-particle collision, the particlespanwise dispersion function Dz(t) denote the dispersion le-vel of particles along the lateral and spanwise directiondeviated from the center of jet, respectively.

ultiphase Flow 34 (2008) 723733Science Foundation of China (Grant Nos. 50506027 and50736006). We are grateful to that.

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Direct numerical simulation of particle dispersion in a turbulent jet considering inter-particle collisionsIntroductionMathematical descriptionsFlow configuration and boundary conditionsFluid motionParticle motionParticle motion without collisionInter-particle collisionGrid index methodCollision detection

Numerical results and discussionsSpatial distribution of the inter-particle collisionsCollision numberEffect of collision on particle dispersionInter-particle collision with different Stokes numbers

ConclusionsAcknowledgementReferences