On the collision rate of small particles in isotropic turbulence. II. Finite inertia case

PHYSICS OF FLUIDS VOLUME 10, NUMBER 5 MAY 1998

On the collision rate of small particles in isotropic turbulence. II. Finiteinertia case

Yong Zhou, Anthony S. Wexler, and Lian-Ping Wanga)

Department of Mechanical Engineering, 126 Spencer Laboratory, University of Delaware, Newark,Delaware 19716-3140

~Received 25 March 1997; accepted 21 January 1998!

Numerical experiments have been performed to study the geometric collision rate of heavy particleswith finite inertia. The turbulent flow was generated by direct numerical integration of the fullNavier-Stokes equations. The collision kernel peaked at a particle response time between theKolmogorov and the large-eddy turnover times, implying that both the large-scale and small-scalefluid motions contribute, although in very different manners, to the collision rate. Both numericalresults for frozen turbulent fields and a stochastic theory show that the collision kernel approachesthe kinetic theory of Abrahamson@Chem. Eng. Sci.30, 1371 ~1975!# only at very largetp /Te ,wheretp is the particle response time andTe is the flow integral time scale. Our results agree withthose of Sundaram and Collins@J. Fluid Mech.335, 75 ~1997!# for an evolving flow. A rapidincrease of the collision kernel with the particle response time was observed for smalltp /tk , wheretk is the flow Kolmogorov time scale. A small inertia oftp /tk50.5 can lead to an order ofmagnitude increase in the collision kernel relative to the zero-inertia particles. A scaling law for thecollision kernel at smalltp /tk was proposed and confirmed numerically by varying the particle size,inertial response time, and flow Reynolds number. A leading-order theory for smalltp /tk wasdeveloped, showing that the enhanced collision is mainly a result of the nonuniform particleconcentration that results from the interaction of heavy particles with local flow microstructures.© 1998 American Institute of Physics.@S1070-6631~98!01305-1#

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I. INTRODUCTION

Small solid particles and droplets are often disperstransported, and mixed by turbulent flow in many natural aindustrial processes. Turbulence can enhance the coagulrate among particles in a concentrated suspension in atthree ways. If the particles are sufficiently small in size atheir inertia relative to the fluid motion negligible, the locshear in turbulence determines completely the collision pcess~the shear mechanism!.1 For heavy particles in vigorouturbulence, particle inertia becomes significant and collisithen arise from the differing inertial response of polydispeparticles to local fluid motion~the accelerative mechanism!.2

The turbulent shear may also increase the collision efficcies of small particles by an order of magnitude throualteration of local relative motion between particles.3

In this paper, we consider only the geometric collisioof small, monodisperse particles in turbulence. We shallsume that the particles are small, with diameterdp typicallyon the order of or less than the Kolmogorov length scaleh[(n3/ e )1/4, wheren and e are fluid kinematic viscosity andthe average rate of energy dissipation per unit mass restively. In such cases, the local shear rate around a particassumed to be uniform and equal to the local velocity graent, any deviation from this assumption~namely, the size

a!Corresponding author: Department of Mechanical Engineering, 126 Scer Laboratory, University of Delaware, Newark, Delaware 19716-31Phone: ~302! 831-8160; Fax: ~302! 831-3619; Electronic mail:[email protected]

1201070-6631/98/10(5)/1206/11/$15.00

Downloaded 01 Jul 2005 to 128.117.47.188. Redistribution subject to AIP

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effect! is negligible as long as only the geometric collisiorate is of concern.4 The particle densityrp is much largerthan the fluid densityr so that the particle inertial responstime

tp5rp

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may be comparable to the flow Kolmogorov time scaletk

5(n/ e )1/2. For example, in cumulus clouds5 the average dis-sipation rate can be on the order of 200 cm2/s3, 40mm drop-lets would have a time scale ratiotp /tk'0.2 and a sizedp /h'0.06. In many engineering applications, the dissiption rate is usually much larger than in clouds, so thattp mayeven be comparable to the large-eddy time scale andparticle inertia becomes a key factor in the collision proceThe terminertia effectshall be used to represent all aspeof the particle motion in response to the changes of the lofluid motion in a finite timetp due to particle inertia.

There is another reason to believe that the inertial effeeven for smalltp , can enhance the collision rate to a levmuch larger than that described previously. It has beshown recently that the intense vortex tube structures, whcharacterize the dissipation-range dynamics in fully devoped turbulence, lead to a nonuniform particle concentrafield. This so-called inertial bias or preferential concentratin turbulence was first illustrated by Maxey6 and then dem-onstrated by Squires and Eaton,7 Wang and Maxey8 throughdirect numerical simulations. The inertial bias was found

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6 © 1998 American Institute of Physics

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1207Phys. Fluids, Vol. 10, No. 5, May 1998 Zhou, Wexler, and Wang

follow a Kolmogorov scaling, namely, being most effectiin producing a nonuniform concentration whentp /tk;1.Unlike the case of negligible particle inertia where the lostrain rate plays the dominant role, the inertial bias is cauby both the local vorticity and strain rate variations.

Consider a monodisperse system consisting ofNp par-ticles in a volumeV, the collision rate per unit volume,N c ,is given by

N c5Gn0

2

2, ~2!

provided thatNp@1, wheren0[Np /V is the average particle number concentration in the volume andG is the colli-sion kernel. Saffman and Turner9 first considered simultaneously the effects of the shear mechanism, the accelermechanism, and the gravitational mechanism on the colliskernelG under the assumptions thatdp!h, tp!tk , and theparticle concentration field was uniform. Their results shthat the accelerative mechanism gives a nonzero contributo the collision kernel only if two colliding particles havdifferent inertias. This work was followed by a numberstudies, all done in the context of stochastical theory of tbulence, where the effect of local flow structures on the pticle concentration was not explicitly considered. Theseclude Panchev10 for the case oftp spanning overtk ,Williams and Crane11 for intermediate-inertia particles withtk,tp,Te , and Abrahamson12 for very large particles withtp.Te , whereTe is the time scale of energy-containing edies. Yuu,13 Kruis and Kusters14 considered the combineeffect of local shear and unequal inertial response timesthe collision rate for arbitrarytp . The key in the stochasticatheory is to identify the proper range of scales of motionthe fluid turbulence which contribute most actively to trelative velocity between two colliding particles. In theefforts, closure assumption of one type or another was ually assumed in order to derive an expression for the aveparticle relative velocity. Also the particle concentration fiewas assumed to be uniform.

Our main objective here is to study the effect of tinertial bias on the collision rate, using numerical simutions and asymptotic analysis. In part 1 of this work,4 wehave clarified the formulation of Saffman and Turner9 for thezero-inertia case. We intend to extend this formulationfinite-inertia particles with a nonuniform concentration fieConsistent with recent studies,15,16 we find that the collisionkernel increases rapidly withtp for tp /tk,1. This will beshown to result mainly from the nonuniform particle concetration field due to the inertial bias. A scaling law for thcollision kernel will be proposed and examined numericaby varying particle size, inertial response time, and flReynolds number. Of significance is the observation thasmall particle inertia oftp /tk;0.5 can lead to an order omagnitude increase in the collision kernel. Most of the nmerical experiments in this work were performed for frozflow fields. We note that more results for the evolving flocase were reported by Sundaram and Collins.15 It is notedthat there is no quantitative difference incollision kernelbetween frozen and evolving flows under the two limiti


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cases, namely, very small particle inertial response timevery large inertia response time. Both the simulation resand analyses in the present paper focused on these limcases, and, therefore, are expected to be representative oevolving flow case as well.

The paper is organized as follows. The next section pvides the details of the flow simulation and particle trackinThe numerical collision kernels for arbitrarytp are presentedfirst in Sec. III. In Sec. IV we describe a scaling law ancompare it with numerical results. A leading order analyfor small tp /tk is developed in Sec. V and compared to tsimulation results. While most simulations utilized a frozturbulence field, the results in an evolving turbulencebriefly discussed in Sec. VI. Finally main conclusions adrawn in Sec. VII.

II. NUMERICAL SIMULATION

A. Flow field

A homogeneous and isotropic turbulent flow was genated by full numerical simulations using a pseudo-specmethod. The incompressible Navier-Stokes equations

]u

]t5u3v2¹S P

r1

1

2u2D1n¹2u1f~x,t !, ~3!

were solved along with the continuity equation¹•u50 in aperiodic box of side 2p. Herev[¹3u is the vorticity,P isthe pressure. The time evolution was computed by applya second-order Adams-Bashforth scheme to the nonlinterms and a second-order Crank-Nicholson scheme forviscous terms. The pressure was eliminated through thetinuity equation.

The flow was generated from rest by the random forcterm f(x,t) which is nonzero only at low wave numberuku,A8. Nonlinear interactions propagate energy from loto high wave numbers and eventually viscous dissipationcomes active, leading to a quasi-steady balance of the forenergy and the viscous dissipation.

For most discussions in this paper, the flow was frozafter the statistically stationary stage was reached, andticles were then introduced into the flow. The start of partirelease will be denoted ast50. This provides us an identicaflow microstructure for different runs with various particparameters. The case of evolving flow will be consideredSec. VI.

Since all the important flow scales are resolved in a fnumerical simulation, the grid resolution determines tscale separation, and thus the Reynolds number of the reing flow. Various grid resolutions from 323 to 1283 wereused to provide a range of Taylor microscale Reynolds nuber ~Table I!.

Table I lists the flow parameters~from top to bottom!:the component rms fluctuating velocityu8, average dissipa-tion ratee , viscosityn, Taylor microscale Reynolds numbeRl[u8l/n ~wherel is the transverse Taylor microscalel[u8/A^(]u1 /]x1)2&), Kolmogorov lengthh, time scaletk ,large-eddy turnover timeTe[u82/ e , the time scale ratioTe /tk , collision radiusR ~which is equal to the particle di


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1208 Phys. Fluids, Vol. 10, No. 5, May 1998 Zhou, Wexler, and Wang

ameter for a monodisperse system!, total number of particlesNp , and particle volume fractionf. Other details of thesimulated flows can be found in Wang and Maxey.8

B. Particle motion

We consider the motion of heavy spherical particles inonuniform turbulent flow. The particle is assumed tosmall in comparison with the Kolmogorov microscale of tturbulence and the loading dilute enough that the presencthe particles does not modify the base turbulence. Underassumption that the density of the particlerp is much largerthan the density of the fluidr, and that a quasi-steady Stokdrag can be used, the equation of motion for a heavy parbecomes

dV~ t !

dt5

u~Y~ t !,t !2V~ t !

tp, ~4!

whereV(t) andY(t) are the velocity and the center positioof a heavy particle, respectively. The body force is neglecsince we focus solely on inertia-induced collisions. The cobined effects of inertia and gravitational settling will be cosidered separately.

The location and velocity of each particle were advancwhile the flow field was either frozen or continued to evolin time. A fourth-order Adams-Bashforth method was usto integrate the particle equation of motion. The fluid veloity at the location of a particle was interpolated from tvalues at neighboring grids using a 6-point Lagrange inpolation. Typically 103;104 particles were introduced att50 into the computational domain at random initial potions with an initial velocity equal to the local fluid velocityAfter about 3tp , any effects of the initial conditions on thparticle motion became lost. The simulation was continufor at least 13tp or 4 to 5 large-eddy turnover times durinwhich collision counts and other statistical averages wtaken. It should be noted that the particle concentration fiand the local-in-time collision kernels~defined in Sec. II Cbelow! may not reach their asymptotic, statistically statioary stage att53tp since the local accumulation processaffected by large-scale fluid motion.8 On the other hand, inpractical applications one may be more interested in theticle collision statistics shortly after the particle release ratthan the asymptotic value. With this in mind and for t

TABLE I. Flow characteristics and parameters for the particle system.

Grid resolution 323 643 963 1283

u8 17.02 18.22 18.30 18.64

e 3568.8 3421.0 3554.7 3374.4

n 0.6000 0.2381 0.1387 0.09450Rl 24 45 59 75h 0.0882 0.0450 0.0294 0.0224tk 0.0130 0.00830 0.00625 0.00529Te 0.081 0.097 0.094 0.103Te /tk 6.23 11.7 15.0 19.5R/h 1.78 1.0/0.5 1.0/0.5 1.0/0.5Np 1024/2048 2048/3072 2640/4096 3072/512f(3104) 84/168 4.00/0.74 1.40/0.27 0.73/0.15


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purpose of obtaining a small numerical uncertainty, we tat53tp as the starting time for all statistical averages.change of the starting time tot52Te showed that the resultchange less than 20% and are unchanged qualitatively.

C. Collision detection

The collision detection algorithm was described in Parof this work4 and will not be repeated here. It was shown ththree different collision counting schemes could be appland each gave slightly different numerical collision kernfor zero-inertia particles. These schemes differ in the choof particle system used for collision counting. They are sumarized here:

Scheme 1. Particles were allowed to overlap in the sytem at the beginning of a time step and were not remofrom the system after collision. This scheme was shownPart 1 to be the only scheme that is consistent with the Sman and Turner9 formulation.

Scheme 2. At the beginning of each time step, the ovelapping particles were marked and excluded from collisdetection. Therefore, the actual number of particles usedcollision detection was less than the total number of particused and varied in time.

Scheme 3. Particles were removed immediately from thsystem when they collide. As a result, the total numberparticles decreased with time and particles remaining insystem were nonoverlapping at the beginning of each tstep. This scheme closely represents reality if both the cosion efficiency and coagulation efficiency are close to osince two particles upon collision will form a particle olarger size and as such will disappear from the current sgroup.

For each of the three schemes, one can define a locatime collision kernel for any time stepdt as

G i~ t ~n!!52VNc~ t ~n!→t ~n11!!

dt@Npi~ t ~n!!#2 , ~5!

where i 51, 2, and 3 denotes the individual schemeNc(t

(n)→t (n11))'N cV dt is the total collision count in thetime step t (n),t<t (n11), Npi(t

(n)) is the total number ofparticles participating in collision detection and is given feach scheme as

Npi~ t ~n!!5H Np5const, fori 51;

Np22N0~ t ~n!!, for i 52;

Np~ t ~n!!, for i 53;~6!

whereN0(t (n)) is the number of overlapping pairs att (n) inscheme 2,Np(t (n)) is the number of particles left in the system in scheme 3. It is assumed that particle volume fractis very small so that binary collisions dominate. These locin-time collision kernels were quitenoisy if Np is not verylarge. They can be improved by averaging over differerealizations of turbulence field and initial particle locationA further average over time gives the final collision kern^G i&.


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III. NUMERICAL RESULTS FOR ARBITRARY INERTIA

Figure 1 shows the numerical collision kernels, normized byR3/tk , as a function oftp /tk in a frozen turbulenceat Rl524. The error bars denote the 95% numerical codence intervals, which were estimated based on 21 runsindependent realizations of particle initial locations. We oserve that qualitatively the three collision counting schemyield similar results. The collision kernels increase very raidly for small tp /tk , reach a peak attp /tk'4 or tp /Te

'0.6, and then drop slowly with increasingtp /tk . Thesame qualitative behavior was found by SundaramCollins,15 although they assumed a perfectly elastic collisof nonoverlapping particles~their post-collision treatment isclose to, but not exactly the same as, our scheme 2!. Thepeak reflects both the effects of small and large scalefluid motion. The inertial bias produces a nonuniform pticle concentration. This small-scale effect8 can enhance thecollision kernel significantly~see Sec. V! and is scaled on theKolmogorov time scale. On the other hand, astp increases,particle velocities fail to correlate at collision and the largscale fluid motion becomes more relevant in determiningrelative velocity between two particles. This causes ancrease in the relative velocity between two particles anconcomitant increase in the collision kernel. This latter effis likely to scale with the integral time scale of the flow. Thquestion of how the location of this peak changes withflow Reynolds number remains to be examined with simutions at higher flow Reynolds numbers.

Schemes 1 and 2 result in almost the same collisionnel values, as expected for this low volume fraction syste4

For zero-inertia particles (tp50), scheme 1 gives the samvalue as the Saffman and Turner9 result. A close examinationseems to suggest that scheme 1 gives a slightly larger kefor tp /tk,7 while scheme 2 gives a larger kernel ftp /tk.7, but the difference is not significant statisticallScheme 3, on the other hand, yields a collision kernel 10%15% less than the other two schemes, due to a prefereremoval as discussed in Part 1.4

FIG. 1. Numerical collision kernels, normalized byR3/tk , as a function ofthe particle response time over the Kolmogorov time scale. For this sesimulations, parameters were set to:dt50.001,R50.8Dx, grid resolution is323. Np51024 for schemes 1 and 2, whileNp was increased to 2048 foscheme 3 to achieve comparable uncertainty intervals. The error barscate the 95% confidence intervals. The horizontal line in the insert denthe Saffman and Turner resultGtk /R351.294.


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Figure 2 replots the results in a manner that can be eacompared with the theory of Abrahamson12 for largetp /Te .Also included in this figure are results at higher flow Renolds number (643 grid simulations!. Herevp8[A^ViVi /3& isthe rms particle fluctuating velocity. We note th^G&/(vp8R

2) increases monotonically withtp /Te . This canbe explained as follows. Letv r

(1) andv r(2) be the longitudinal

velocities of the two particles at collision. Assuming the retive velocity wr[v r

(2)2v r(1) between two colliding particles

follows a Gaussian distribution with a standard deviationsw

and the particle concentration is uniform, thenûwr u&5A2/psw . The collision kernel for scheme 1 is

^G1&52pR2ûwr u&52A2pR2sw . ~7!

Defining the correlation coefficient of the particle velocitiasr12[^v r

(1)v r(2)&/(vp8)

2, we can expresssw as

sw5A^wr2&5A2vp8A12r12. ~8!

It follows that

^G1&

vp8R2 54ApA12r12. ~9!

This reduces precisely to the Abrahamson12 result if r1250.In general, we can argue that the correlation coefficientr12

decreases monotonically withtp , since particle velocitiesdepend more and more on their history of travel and thnonlocal fluid motion as tp increases. Consequently^G1&/(vp8R

2) increases monotonically. Of significance is ththe numerical collision kernels attp /Te52.5 are about 22%and 14% less than Abrahamson’s prediction forRl524 andRl545, respectively, implying that there is still a significavelocity correlation.

An estimate ofr12 as a function oftp /Te can be madeby treating the fluid velocity around a particle as a simpMonte-Carlo process~see the Appendix!. Substituting Eqn.~A8! into ~9!, we have

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FIG. 2. Numerical collision kernels over the particle fluctuation velocityvp8 ,as a function of the particle response time over the large-eddy turnoverTe . The error bars indicate the 95% confidence intervals. TAbrahamson12 prediction isG/(vp8R

2)54Ap57.09.


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vp8R2 52A2pF12expS 2

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whereu50.5tp /Te . This result may be viewed as an improvement over that of Abrahamson,12 and is plotted in Fig.2. Interestingly, Eqn.~10! shows a reasonable compariswith the numerical results fortp /Te.1.5. Furthermore Eqn~10! indicates thatr12 diminishes slowly~in an algebraicmanner, see~A9!! for large tp /Te . For example,A12r12

50.937 attp /Te510, thus the difference between Eqn.~10!and Abrahamson’s result is still noticeable attp /Te510. Fortp /Te,1.5, Eqn.~10! predicts a smaller value than the nmerical results, particularly for theRl545 case. The differ-ence can be viewed as the effect of the nonuniform partconcentration~see Sec. V!. Also shown are the predictions oKruis and Kusters,14 Williams and Crane11 ~see the Appen-dix!. Interestingly, all the theoretical results are similarshape. The present theory seems to be slightly betterother theories. However, one should keep in mind that indevelopment of the other theories, the flow Reynolds numwas assumed to be large. Furthermore, empirical constsuch as the one used to relate the Lagrangian time scaleTL toTe may be adjusted in the works of Kruis and Kusters14

Williams and Crane11 to better match the numerical resultThe ratio of fluid to particle kinetic energy is shown

Fig. 3 as a function oftp /Te . A simple stochastic analysiwould predict a linear curve~for example, Abrahamson,12

Williams and Crane11!. Figure 3 shows that the curves anot exactly linear, but rather the slope drops slowly astp /Te

increases. The average slope is very close to one at ltp /Te , in agreement with the previous theories.11,12 Alsoshown in the figure are our result, Eqn.~A4!, using a simplestochastic analysis, and the results of Abrahamson,12 Kruisand Kusters.14 They all give a reasonable although not veaccurate prediction.

To further examine the effect of collision detectioschemes on the numerical collision kernel, we compareFig. 4 results based on scheme 1 and two other scheScheme 44 is a more realistic scheme in which particles arelocated after collisions to represent generation of partiin the current size group due to the coagulation of sma

FIG. 3. The ratio of fluid to particle kinetic energy as a function oftp /Te .


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particles in a stationary system. In our simulations usscheme 4, we introduced more particles at the beginningthe simulations and only used a portion of the particlesthe collision calculation. Particles involved in a collisiowere replaced by those particles not previously used incollision calculation, so as to keep the system stationaryto maintain the preferential concentration at the same tiScheme 4 gives almost the same results at very smalllargetp /Te , but can possibly result in collision kernels 5%to 10% smaller than those of scheme 1 for intermediate pticle inertia. Also shown are results based on the hard-sphmodel used previously by Sundaram and Collins15 in which aperfectly elastic collision model without friction is usedcalculate the particle velocities after a collision. We note twhile there is essentially no difference between the hasphere model results and those of scheme 1 fortp /Te.0.1,the hard-sphere model can lead to a collision kernel mlarger than the Saffman and Turner9 prediction in the limit oftp /Te→0. This latter difference is due to artificial repeatecollisions in the hard-sphere model as noted by Sundaand Collins.15

IV. THE SCALING LAW FOR G AT SMALL tp

In the following, we shall focus on particles with smainertia, tp /tk,1. In this case, particles will respond to thchange of local large-scale fluid motion rather quickly. Asconsequence, we expect that the dissipation-range fluidnamics dominates the collision process. If, in addition, pticle size or the collision radiusR is much smaller than theintegral length scale of the turbulence and the flow Reynonumber is sufficiently large, we can argue that the colliskernelG depends only on the collision radius R, the averadissipation ratee , the kinematic viscosityn, and the particleresponse timetp . Then we can write

G5 f ~R, e ,n,tp!. ~11!

A dimensional analysis would lead to

FIG. 4. Numerical collision kernels, normalized by 2A2pu8R2, as a func-tion of the particle response time over the integral time scale. For this ssimulations, parameters were set to:dt50.001,R50.458Dx, grid resolutionis 643. Np52048. The error bars indicate the 95% confidence intervals.horizontal line in the insert denotes the Saffman and Turner reG/(2A2pu8R2) 50.0767.


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G

R3/tk5 f S tp

tk,R

h D , ~12!

where tk5(n/ e )1/2 is the Kolmogorov time scale,h5(n3/ e )1/4 is the Kolmogorov length scale.

Therefore, when we consider small inertia and small seffects, the dimensionless collision kernel is a function ofdimensionless particle response time~or Stokes number! andthe dimensionless collision radius. In the limit oftp /tk→0and R/h→0, Eqn. ~12! is consistent with the Saffman anTurner9 result for zero-inertia particles, withf (0,0)51.294.Eqn. ~12! is essentially a Kolmogorov scaling in highReynolds number turbulence.

To confirm the above scaling law, we performed a serof numerical simulations by varying the following three prameters one at a time: the Stokes numbertp /tk , the dimen-sionless collision radiusR/h, and the flow Reynolds numbeThe results are summarized in Fig. 5. Since in the last secwe showed that the results were similar for different collisicounting schemes, we shall use scheme 1 only for the rediscussions in this paper. Several interesting observatcan be made for Fig. 5.

First, the nondimensional collision kernels do not showdependence on the flow Reynolds numberRl for tp /tk

,0.6, even though the Reynolds numbers are not high insimulations. For the two higher Reynolds number cases,range of validity of the scaling law extends totp /tk,0.8.For largertp /tk , particles start to interact with a rangeflow time scales, including the large-scale motions whdepend on the flow Reynolds number. Since the fluid motat larger scales is more fully represented in higher Reynonumber flow, the contribution of larger scale fluid motionthe collision kernel is increased. As a result, the colliskernel increases withRl ~or equivalentlytp /Te) for largertp /tk . We speculate that the range of validity of the scalilaw would be extended further, should these simulationsdone at even higher Reynolds numbers. These observasupport the proposed scaling law.

Second, astp /tk→0, the numerically-derived collisionkernels all approach the Saffman and Turner9 prediction,G51.294R3/tk , independent of the flow Reynolds numbeNote that the Gaussian probability distribution for the velo

FIG. 5. Numerical collision kernels, normalized byR3/tk , as a function oftp /tk . The error bars indicate6 standard deviation. The important parameters for these runs are shown in Table I.


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ity gradient is a reasonable approximation for low Reynonumber flows and the finite size effect forR/h<1 isnegligible.4

Third, for non-zerotp /tk , the nondimensional collisionkernel decreases asR/h increases. This is expected since thistory effect, namely the relative particle motion before ttwo particles are brought to a distanceR apart, increasinglyinfluences the relative velocity at collision. Particles respoto scales of motion larger thanR. For small R in the fardissipation range, the relative fluid velocity scales asR,while in the inertial subrange it is scaled asR1/3 ~seeKolmogorov17!. The mean shear rate between two particthen decreases withR for a giventp . Therefore, the normal-ized collision kernel drops as the collision radius increas

The most significant observation is that the numericaderived collision kernels increase very rapidly with the pticle response time. Even for a smalltp /tk50.5 for whichthe scaling law is applicable, the collision kernel is abouorder of magnitude larger than the value for zero-inertia pticles whenR/h50.5. This rapid increase cannot be eplained by any of the previous theories. We shall study trapid increase analytically in the next section.

V. A LEADING-ORDER ANALYSIS FOR SMALL tp

Our objective here is to shed some light on the obserrapid increase ofG with tp for small tp . We will apply theresults of an asymptotic analysis developed by Maxey6 forsmall tp , that accounts for the nonuniform concentratieffect due to the inertial bias.

If we assume thattp /tk,1, tk represents the smallestime scale in the flow and the particles must respond vquickly to any local change of fluid motion. Consequentthe particle velocity is completely specified by its instanneous position, to leading order intp /tk . This allows us todefine a particle velocity fieldv(x,t) which is given by6

v~x,t !5u~x,t !2tpS ]u

]t1u•¹uD , ~13!

in the absence of body forces.Maxey6 pointed out that this particle velocity field, un

like the fluid velocity field, is not incompressible. The divegence field of the particle velocity is

¹•v52tp

]uj

]xi

]ui

]xj52tpS si j si j 2

v2

2 D , ~14!

wherev5¹3u andsi j 5(]ui /]xj1]uj /]xi)/2 are the localfluid vorticity and rate of strain field, respectively. It followthat particles will accumulate in regions of low vorticity anhigh strain rate. Maxey6 showed that the local particle number concentration,n(x,t), after a uniform release att50,would evolve as

n~x,t !5n0expH tpE0

tFsi j si j 2v2

2 G~Y~ t8;x,t !,t8!dt8J ,

~15!

where the integrand@si j si j 2 (v2/2)# in the above expression should be evaluated following the trajectory of a partiwhose position would be atx at time t. We note that the


dthtrith

to

io

xihrae.

th

ad

s,

ricalx-

calion

r the

for

.

theiveareen-ofheeryent

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clebe


integrand can take both positive and negative values ansuch may cancel in the integration over the history ofparticle trajectory. Therefore, we argue that the main conbution to the integral comes from the part of trajectory wt8't, or local in space relative tox. We thus propose thefollowing approximation to~15!:

n~x,t !'n0expH tpt f S si j si j 2v2

2 D J , ~16!

where t f is a history time scale to be determined, andleading order is only a property of the flow. Sincetp is verysmall, the leading order expansion forn(x,t) is

n~x,t !'n0H 11tpt f S si j si j 2v2

2 D J . ~17!

In the context of the scaling law discussed in the last sectwe expect thatt f be directly related totk . We will use Eqn.~17! as it is more consistent with the leading order appromation, however, Eqn.~17! cannot be applied to very higvorticity regions as it may lead to negative local concenttion for a givent f . To circumvent this, we simply set thlocal concentration to zero if Eqn.~17! becomes negativeEquations~16! and ~17! do reflect correctly the qualitativeconnection between the local particle concentration andlocal vorticity or strain rate.

We shall now combine the above results to derive a leing order approximation for the average collision kernel. Wstart by introducing a local-in-space collision kernelG(x,t)as

G~x,t !5N ~x,t !

n2~x,t !/25E

VR

~2wr2!dV. ~18!

The relative velocitywr has been partitioned into two parta positive part and a negative part, according to

wr25H 0, if wr>0,

wr , if wr,0; wr15H wr , if wr>0,

0, if wr,0. ~19!

Since

EVR

uwr udV5EVR

wr1dV2E

VR

wr2dV, ~20!

EVR

¹•wdV5EVR

w•dVW 5EVR

wr1dV1E

VR

wr2dV,

~21!

it follows that

G~x,t !5EVR

~2wr2!dV5

1

2EVR

uwr udV21

2EVR

¹•wdV,

~22!

whereVR is the volume of the sphere with radiusR centeredon x.

If the collision radiusR is small compared toh and theflow is locally isotropic, the first integral in~22! can be writ-ten as


asei-

n,

-

-

e

-e

1

2EVR

uwr udV'2pR3U]v1

]x1U

52pR3U]u1

]x12tp

]

]x1S ]u1

]t1u•¹u1D U, ~23!

using Eqn.~13!. The second term may be rewritten as

1

2EVR

¹•wdV51

2VR¹•w52

1

2VRtpR

]

]x1Fsi j si j 2

v2

2 G ,~24!

where the overbar denotes a local average over the sphevolume VR . Therefore the local collision kernel can be epressed as

G~x,t !52pR3U]u1

]x12tp

]

]x1S ]u1

]t1u•¹u1D U

11

2VRtpR

]

]x1Fsi j si j 2

v2

2 G . ~25!

The average collision kernel is related to both the locollision kernel and the local particle number concentratas

^G&5^N ~x,t !&

n02/2

5^G~x,t !n2~x,t !&

n02 , ~26!

where the angle brackets denote a spatial average oveentire computational domain, andn05^n(x,t)& is the aver-age concentration. Substituting~17! and ~25! into ~26! anddropping the local averaging as a first approximationsmall R, we obtain

^G&52pR3K U ]u1

]x12tp

]

]x1S ]u1

]t1u•¹u1D U

3F11tpt f S si j si j 2v2

2 D G2L 12pR3

3tpK R

]

]x1

3S si j si j 2v2

2 D F11tpt f S si j si j 2v2

2 D G2L . ~27!

Several remarks regarding Eqn.~27! can be made hereIn a direct numerical simulation, the full nonuniform flowfield is simulated, so all the terms in Eqn.~27! can be com-puted directly. The first part inside the absolute signs offirst term includes the leading order correction to the relatvelocity gradient. The second part enclosed by the squbrackets represents the effect of nonuniform particle conctration due to the inertial bias. If we neglect the effectnonuniform particle concentration or equivalently set tsecond part to one, the collision kernel will increase but vslowly with tp due to the first part or the effect of particlinertia on the particle relative velocity. The important poito note is that the two parts in the first term of~27! arepositivelycorrelated, namely, both high local concentratiand high local relative velocity are found in regions of higstrain rate. The combined effect is then a much higher avage collision kernel than what one would have if the particoncentration were uniform. Similar observations can


e

th

ion-

e

sne

-inn

l

ing-

ize

nedece

tsowl-e

g

w


made for the second term in~27!. We note, however, that thsecond term would not make any net contribution to^G&under the uniform concentration approximation sincevolume average of (si j si j 2 (v2/2)) is zero, but it does makea positive contribution when the nonuniform concentrateffect is considered. Finally, Eqn.~27! is essentially consistent with the scaling law~12! if the history characteristictime t f scales withtk . If the particle concentration weruniform, Eqn.~27! would become

^Gu&52pR3K U ]u1

]x12tp

]

]x1S ]u1

]t1u•¹u1D U L . ~28!

We can now fit the approximation~27! to the numericalresults in Fig. 5 for the regiontk /tp,0.5 for which thescaling law is justified. A least square procedure~minimizingthe square error between the analysis and the simulationsults in the regiontp /tk,0.5) was used to deduce the bevalue of t f . The comparisons of the numerical results athe approximation~27! are shown in Figs. 6, 7 and 8 for ththree flow Reynolds numbers. We conclude that Eqn.~27!predicts the shape of^G1& versustp observed in the numerical simulations despite all the approximations involvedderiving~27!. Also shown in these figures are the predictiounder the uniform concentration approximation, Eqn.~28!,which yield a much smallerG1& value than the numerica

FIG. 6. Comparison of the asymptotic analysis, Eqn.~27!, with numericalresults for small particle response timetp /tk,0.5. The error bars indicate6 standard deviation. Grid resolution is 643, Rl545.



e

re-td

s

results. Therefore, the nonuniform concentration resultfrom finite particle inertia is the dominant factor rapidly increasing the collision kernel at smalltp /tk .

Figure 9 shows the value oft f , normalized bytk , as afunction of the flow Reynolds numberRl and R/h. For agiven R/h, t f /tk is only weakly dependent ofRl . For therange ofRl covered in the simulations,t f /tk is changed byabout 16%, which is much less than theTe /tk variation~seeTable I!. Therefore, we may conclude thatt f scales withtk .The difference int f /tk between the twoR/h values is some-what larger, but part of this difference is due to the finite seffect which is not included in Eqn.~27!.

VI. RESULTS IN EVOLVING FLOW

The numerical results considered so far were obtaiusing turbulent fields that did not evolve in time. Here wpresent some preliminary results for an evolving turbulenfield at Rl'45 for smalltk . Figure 10 compares the resulwith those obtained in a frozen turbulence at the same flReynolds number. Note that for the evolving flow the Komogorov time scaletk varies in time so an average valuover time was used in Fig. 10. Fortp→0, the frozen andevolving flow fields yield the same collision kernel, implyinthat the formulation of Saffman and Turner9 is valid for thecollision of fluid elements in an evolving flow.4 However, astp /tk increases, the collision kernel in the evolving flo


FIG. 9. The history time scalet f , normalized by the Kolmogorov timescaletk , as a function of the Taylor microscale Reynolds numberRl .


aeoemityre

cot aal

th

thanof

nganulitiofiveivv

ityss

eItpaesotuthise

olli-sg of

rstiss

ed’s-

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icith

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the

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en


deviates from that in the frozen flow, and is always less ththe latter. At tp /tk51, the reduction is about 20%. Thphysical explanation is that heavy particles respond trange of eddies that evolve both in time and space, the tporal evolution of the flow tends to reduce the fluid veloccorrelation or persistence of a given local flow structuaround a particle. For smalltp /tk , this effect can lower thelevel of local particle accumulation and thus the averagelision kernel. The observed reduction, however, does noter the qualitative behavior of the collision kernel at smtp /tk .

We are in the process of collecting more data forevolving flow, covering a wider range oftp and the flowReynolds number. It should be noted that more results forevolving flow case are reported recently by SundaramCollins.15 Our preliminary results agree with thoseSundaram and Collins15 qualitatively in general. It is notedthat there is no quantitative difference incollision kernelbetween frozen and evolving flows under the two limiticases, namely, very small particle inertial response timevery large inertia response time. Both the simulation resand analyses in the present paper focused on these limcases, and, therefore, are expected to be representativeevolving flow case as well. There are significant quantitatdifferences for the intermediate inertia case, which will btopic of future study. It may be useful to note that a passscalar field advected by Gaussian, frozen, and evolvinglocity fields shows different spectra and dynamics.18,19

Therefore, the dynamical features of the fluid velocfield should be considered carefully for the collision proce

VII. SUMMARY

Numerical experiments were conducted to study the gmetric collision rate of heavy particles with finite inertia.was found that the collision kernel reached a peak at aticle response time larger than the Kolmogorov time but lthan the large-eddy turnover time. This indicates that bthe large-scale and small-scale fluid motion can contribalthough in very different manners, to the collision rate. Tis consistent with the observation of Sundaram and Collin15

on particle collision in an evolving flow. The ratio of thcollision kernel to particle fluctuating velocity,^G&/vp8 , on

FIG. 10. The numerical collision kernels in an evolving flow and a frozturbulence as a function oftp /tk . Grid resolution is 643, Rl545. The errorbars denote the 95% numerical confidence intervals.


n

a-

l-l-l

e

ed

dtsngtheeaee-

.

o-

r-she,s

the other hand, increases monotonically withtp , implyingthat the large scale effect dominates the change of the csion kernel for mosttp . Simulations at higher flow Reynoldnumbers are necessary to further clarify the proper scalinthe maximum collision rate.

In the limit of zero inertia (tp /tk→0), the analyticalresult of Saffman and Turner9 provides a useful estimate fothe collision kernel but a finite correction to their result mube made if a realistic collision counting schemeemployed.4 On the other hand, for very large particle(tp /Te@1), the kinetic theory of Abrahamson12 is expectedto apply. Our simulations show that the numerically-derivcollision kernel is still significantly less than Abrahamsonprediction attp /Te52.5, and that his prediction is only approached at extremely largetp /Te . This was also shown bya simple stochastic theory in which the fluid velocity onparticle was treated as a Monte-Carlo process.

For smalltp /tk , which is most relevant to atmosphercontexts, the collision kernel increases very quickly wtp /tk . A scaling law for the collision kernel for this limitwas proposed and confirmed by numerical simulations. Trapid increase of the collision kernel was shown, byasymptotic analysis, to result mainly from the nonuniforparticle concentration field due to the inertial bias. Of snificance is the observation that a small inertia withtp /tk

50.5 may lead to an order of magnitude increase incollision rate.

Most of the results were obtained in a frozen turbuleflow. Preliminary results for an evolving flow indicate ththe non-persistence of flow structure may reduce the cosion rate. This is expected, at least for smalltp /tk , since thelevel of local particle accumulation is somewhat reducedan evolving turbulence. Further work is necessary to systatically study this effect.

It should be noted that most previous studies shownon-zero contribution of the particle inertia to the collisiokernel only through unequal or differential inertia in a poldisperse system. Here we have demonstrated that evenmonodisperse system, the particle inertia must be considto accurately describe the collision kernel. For equal-sparticles, the inertial effect alters the collision rate in at lefour ways in addition to the shear mechanism:~a! by a re-sponse to the local fluid acceleration in addition to the lofluid velocity, through which the localspatial variation inthe fluid acceleration can modify the relative velocity~e.g.,Eqn.~13!!; ~b! by the lack of correlation of fluid velocity andfluid acceleration on the particle trajectories which can affthe relative velocity due to the combined effect of particinertia and both thespatial and temporal variationsof theflow field; ~c! by the local particle accumulation as a resultthe inertial bias, and~d! by different initial conditions withwhich the particles are released into the flow. For finite pticle inertia, all these are no longer a local phenomena,rather depend on both the spatial and temporal variationthe turbulence. The present numerical and analytical reswill help us to develop a better and more complete thewhich can combine all the above aspects and be appliearbitrary particle inertia.


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ACKNOWLEDGMENTS

This work was supported by the University of DelawaResearch Foundation, the IBM Watson Research Center,the State of Delaware. L.P.W. is grateful to Professors Mtin Maxey, Lance Collins, and Renwei Mei for several heful discussions in the course of this work.

APPENDIX: A SIMPLE STOCHASTIC ANALYSIS OFTWO-PARTICLE VELOCITY CORRELATION

Here we present a simple stochastic analysis to gaqualitative understanding of the two-particle velocity corlation coefficientr12 introduced in Sec. III. We shall treat thparticle motion as a succession of interactions with turbueddies, where each eddy has constant flow properties. Mspecifically, we consider a one-dimensional version ofparticle equation of motion

dv~ t !

dt5

u2vtp

, ~A1!

where the fluid velocityu is treated as a Monte-Carlo process with a fixed eddy life timeT.20 The fluid velocity willtake a Gaussian random value in each eddy with a standeviation equal to the rms fluid fluctuation velocityu8. It canbe shown that the particle velocity variance (vp8)

2 is relatedto the fluid velocity correlationRf(t)[û(t)u(t1t)& by21

~vp8!2[^v2~ t !&51

tpE

0

`

Rf~t!expS 2t

tpDdt. ~A2!

For a Monte-Carlo processRf is a triangle function20

Rf~t!5H u82S 12t

TD , for utu,T;

0, for utu>T.~A3!

Substituting~A3! into ~A2!, we obtain a relationship betweethe particle and fluid kinetic energy

~vp8!2

u82 512uF12expS 21

u D G , ~A4!

whereu[tp /T.Now we would like to estimate the velocity correlatio

between two colliding particles,v (1)v (2)&. If the particlesize is very small, the two particles must be found in a saeddy upon collision. We further assume that the two particenter the eddy at a same time,22 say,t50. Then by integrat-ing Eqn.~A1! the particle velocities can be written as

v ~ i !~ t !5ueF12expS 2t

tpD G1v ~ i !~0!expS 2

t

tpD ,

for 0,t,T, i 51,2; ~A5!

wherev ( i )(0) denotes the particle velocity att50, andue isthe eddy velocity. Before entering the eddy (t,0), the twoparticles interact independently with different eddies so t^v (1)(0)v (2)(0)&5^v (1)(0)ue&5^v (2)(0)ue&50. If the


ndr--

a-

tree

rd

es

t

probability distribution for the two particles to collide in thtime interval 0,t,T is uniform, then the velocity correlation is

^v ~1!v ~2!&51

TE0

T

^v ~1!~ t !v ~2!~ t !&dt. ~A6!

Substituting~A5! into ~A6!, we have

^v ~1!v ~2!&u82 5122uF12expS 2

1

u D G1u

2F12expS 22

u D G .~A7!

The essential physics is that the particle velocities mustpartially correlated due to interactions with a same eddy ribefore collision, and that the level of correlation dependshow quickly the particles can respond to the new fluid vlocity in the eddy. Finally, combining~A4! and ~A7!, weobtain

r12512u

2

F12expS 21

u D G2

H 12uF12expS 21

u D G J . ~A8!

Equation~A8! shows thatr12 decreases monotonically witu. In particular, the following asymptotic behaviors are otained

r1255 12u

2, for u!1;

2

3u, for u@1.

~A9!

In Sec. III, the results for arbitrary inertia were presentin terms oftp /Te , whereTe is the eddy turnover time.Te

may also be viewed as the integral time scale of the turlence. The form of the velocity correlation~A3! implies thatT/25Te , therefore, we set

u50.5tp

Te~A10!

when the above analysis was compared to the numericasults in Sec. III.

It should be noted that the above analysis does notclude the eddy size effect, which can modify the particeddy interaction time as the particles may traverse the ein a time less thanT. Also the shear mechanism is not cosidered.

Finally we cite here the recent results by Kruis aKusters14 when applied to equal size particles [email protected] gave the following expression for the ratio of the pticle and fluid kinetic energy

~vp8!2

u82 511~g11!z

~11z!~11gz!, ~A11!

where

z[2.5tp

Te, g[0.183S u8

vkD 2

. ~A12!


ize

d

-

a-nes

id

r i

op

llid

ded

e-

by

is-uid

nt

nt

u-

pic

nt

ic,

nece

leon,

ofFlu-

,’’

entn.

nt

s the

icalions

thate.


Equation~A11! differs from ~A4! as it is a function of twoindependent dimensionless parameters. Note that~A11! wasfirst derived by Williams and Crane11 but with a differentdefinition for g. Williams and Crane11 assumed thatg@1and used a simpler form under that limit. For equal sparticles, Kruis and Kusters’ result for the collision kernbecomes

G

vp854ApR2A gz2

~11z!~11gz!

3AS 12A112z

11zD S 21~g11!z

11~g11!zD . ~A13!

Note that ~A13! reduces to the result of Williams anCrane11 if g@1, which is

G

vp854ApR2A z

11zA12

A112z

11z. ~A14!

If, in addition, we assumez@1, then the Abrahamson’s result is recovered.

Finally, we note that for largetp /Te our theory ap-proaches Abrahamson’s resultGA in the following manner:

G5GAF122Te

3tpG , ~A15!

while both ~A13! and ~A14! yield

G5GAF12A Te

5tpG ~A16!

for largetp /Te . Therefore, all the theories predict that Abrhamson’s result will be approached in an algebraic manalthough our theory seems to approach Abrahamson’s remore quickly.

1T. R. Camp and P. C. Stein, ‘‘Velocity gradients and internal work in flumotion,’’ J. Boston Soc. Civil Eng.30, 219 ~1943!.

2T. W. R. East and J. S. Marshall, ‘‘Turbulence in clouds as a factoprecipitation,’’ Q. J. R. Meteorol. Soc.80, 26 ~1954!.

3P. R. Jonas and P. Goldsmith, ‘‘The collection efficiencies of small drlets falling through a sheared air flow,’’ J. Fluid Mech.52, 593 ~1972!.

4L. P. Wang, A. S. Wexler, and Y. Zhou, ‘‘On the collision rate of smaparticles in isotropic turbulence. Part 1. Zero-inertia case,’’ Phys. Flu10, 266 ~1998!.


el

r,ult

n

-

s

5J. C. Weil and R. P. Lawson, ‘‘Relative dispersion of ice crystal in seecumuli,’’ J. Applied Meteorol.32, 1055~1993!.

6M. R. Maxey, ‘‘The gravitational settling of aerosol particles in homogneous turbulence and random flow fields,’’ J. Fluid Mech.174, 441~1987!.

7K. D. Squires and J. K. Eaton, ‘‘Preferential concentration of particlesturbulence,’’ Phys. Fluids A3, 1169~1991!.

8L. P. Wang and M. R. Maxey, ‘‘Settling velocity and concentration dtribution of heavy particles in homogeneous isotropic turbulence,’’ J. FlMech.256, 27 ~1993!.

9P. G. Saffman and J. S. Turner, ‘‘On the collision of drops in turbuleclouds,’’ J. Fluid Mech.1, 16 ~1956!. Also Corrigendum, J. Fluid Mech.196, 599 ~1988!.

10S. Panchev,Random Function and Turbulence~Pergamon Press, NewYork, 1971!, pp. 301–309.

11J. J. E. Williams and R. I. Crane, ‘‘Particle collision rate in turbuleflow,’’ Int. J. Multiphase Flow9, 421 ~1983!.

12J. Abrahamson, ‘‘Collision rates of small particles in a vigorously turblent fluid,’’ Chem. Eng. Sci.30, 1371~1975!.

13S. Yuu, ‘‘Collision rate of small particles in a homogeneous and isotroturbulence,’’ AIChE. J.30, 802 ~1984!.

14F. E. Kruis and K. A. Kusters, ‘‘The collision rate of particles in turbulemedia,’’ J. Aerosol Sci.27, S263~1996!.

15S. Sundaram and L. R. Collins, ‘‘Collision statistics in an isotropparticle-laden turbulent suspension,’’ J. Fluid Mech.335, 75 ~1997!.

16C. Hu and R. Mei, ‘‘Effect of inertia on the particle collision coefficient iGaussian turbulence,’’ the7th International Symposium on Gas-ParticlFlows, Paper FEDSM 97-3608, ASME Fluids Engineering Conferen~1997!.

17A. N. Kolmogorov, ‘‘The local structure of turbulence in incompressibviscous fluid for very large Reynolds numbers,’’ Proc. R. Soc. LondSer. A 434, 9 ~1991! @in Russian in Dokl. Akad. Nauk. SSSR30, 301~1941!#.

18J. R. Chasnov, V. M. Canuto, and R. S. Rogallo, ‘‘Turbulence spectruma passive temperature field: results of a numerical simulation,’’ Phys.ids 31, 2065~1988!.

19M. Holzer and E. D. Siggia, ‘‘Turbulent mixing of a passive scalarPhys. Fluids6, 1820~1994!.

20L.-P. Wang and D. E. Stock, ‘‘Stochastic trajectory models for turbuldiffusion: Monte-Carlo process versus Markov chains,’’ Atmos. Enviro26A~9!, 1599~1992!.

21L.-P. Wang and D. E. Stock, ‘‘Dispersion of heavy particles by turbulemotion,’’ J. Atmos. Sci.50~13!, 1897~1993!.

22One can remove this assumption and allow the second particle entereddy at a different time, say, att1, where 0,t1,T. In that case, a double

integration over botht and t1 will have to be carried out to obtain the

average velocity correlation, which may not yield an algebraic, analytexpression. One may also assume more realistic probability distributfor the interaction time (t2t1) and the time lagt1. We had performednumerical integrations to test various possibilities, with the conclusionthe final result was not so sensitive to the assumptions we made her


On the collision rate of small particles in isotropic turbulence. II. Finite inertia case

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