mechanism of particle deposition in fully developed turbulent open channel flow _2003

PHYSICS OF FLUIDS VOLUME 15, NUMBER 3 MARCH 2003

Mechanisms of particle deposition in a fully developed turbulentopen channel flow

Chidambaram Narayanan and Djamel Lakehala)

Institute of Energy Technology, Swiss Federal Institute of Technology, ETH-Zentrum/CLT,CH-8092 Zurich, Switzerland

Lorenzo Botto and Alfredo SoldatiDipartimento di Energetica e Macchine, Universita’ degli Studi di Udine, Udine 33100, Italy

~Received 13 August 2002; accepted 17 December 2002; published 3 February 2003!

Particle dispersion and deposition in the region near the wall of a turbulent open channel is studiedusing direct numerical simulation of the flow, combined with Lagrangian particle tracking underconditions of one-way coupling. Particles with response times of 5 and 15, normalized using thewall friction velocity and the fluid kinematic viscosity, are considered. The simulations wereperformed until the particle phase reached a statistically stationary state before calculating relevantstatistics. For both response times, particles are seen to accumulate strongly very close to the wallin the form of streamwise oriented streaks. Deposited particles were divided into two distinctpopulations; those with large wall-normal deposition velocities and small near-wall residence timesreferred to as thefree-flight population, and particles depositing with negligible wall-normalvelocities and large near-wall residence times~more than 1000 wall time units!, referred to as thediffusional depositionpopulation. Diffusional deposition~deposition induced by the small residualturbulent fluctuations near the wall! is found to be the dominant mechanism of deposition for bothparticle response times. The free-flight mechanism is shown to gain in importance only fortp

1

515 particles. Fortp155 particles only 10% deposit because of free flight, whereas the fraction is

around 40% fortp1515 particles. This result runs counter to the widely held opinion that free flight

is the dominant mechanism of deposition in wall-bounded flows and clearly quantifies the relativeimportance of the two mechanisms. A simple relationship between the particle wall-normal velocityon deposition and the residence time for free-flight particles is presented. Particle depositionlocations over the period of the entire simulation reveal that, while diffusional deposition occursmostly along streamwise oriented lines below the near-wall particle accumulation patterns,free-flight particles deposit more evenly over the wall. ©2003 American Institute of Physics.@DOI: 10.1063/1.1545473#

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I. INTRODUCTION

Particle deposition in wall-bounded flows has receivconsiderable attention for more than four decades due tpractical relevance to many industrial applications. Onethe earliest models of deposition is the one by Friedlanand Johnstone,1 who proposed the so-called free-fligtheory. The essence of this model is that particles are trported by turbulent motions to within onestop-distanceofthe wall, where they acquire sufficient inertia to coathrough the viscous sublayer and deposit. This pioneemodel was further improved by the work of many researers. Cleaver and Yates2 suggested that the free-flight theoignores the structure of the near-wall turbulence, and deoped a model for the turbulent deposition process which csidered the effect of ‘‘sweep’’ events in carrying particlesthe wall.

Direct numerical simulation~DNS! of a channel flow

a!Author to whom correspondence should be addressed. Telephone:141~0!1 632 4613; fax: 141 ~0!1 632 1166. Electronic mail:[email protected]

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with Lagrangian particle tracking carried out bMcLaughlin3 showed that particles tend to gradually accmulate in the viscous sublayer and conjectured that this pcess of accumulation would continue for times much lonthan their simulation interval. Some evidence was also pvided on the enhancement of particle accumulation duethe addition of the Saffman lift force in the particle equatiof motion. However, the time interval of their simulation watoo short to obtain reliable results on deposition ratesother Eulerian statistics, as will be clarified later.

Rashidiet al.,4 describing an experiment in which paticles were released in an open-channel flow, underlinedimportance of sweep-ejection events in depositing and retraining particles. They too report an accumulation of pticles near the wall and observe that particles with radii lthan 0.5 wall units, approaching very close to the wall wiout depositing, are rarely lifted up by wall ejections. Tabove observations have been confirmed in another expment conducted by Kaftoriet al.,5 where the motion of par-ticles was found to be intimately related to the action of tquasi-streamwise vortices populating the near-wall regiona recent DNS study, Marchioli and Soldati6 have helped

© 2003 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp

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764 Phys. Fluids, Vol. 15, No. 3, March 2003 Narayanan et al.

identify the turbulent mechanisms which promote partiaccumulation near the wall. They report particle transmechanisms due to strong, coherent sweep and ejecevents, and specifically point out the effect of small streawise vortices very close to the wall in promoting particaccumulation under the low-speed streaks. Recently, sattempts to develop empirical models accounting fornear-wall phenomena have been presented.7

Brooke et al.8 employed DNS to study particle depostion in a channel flow with the view of evaluating the freflight theory of Friedlander and Johnstone.1 By looking at theprobability density function~PDF! of the near-wall particlewall-normal velocities, they point out that at any instant ona small fraction of particles have a high enough velocityexecute a free flight to the wall and to deposit. This fact isodds with the assumption of the original free-flight modwhere, at the stop-distance, all particles are supposemove on a free flight path to the wall.1 In a subsequent papeBrookeet al.9 make an interesting subdivision of the particflux into three components: The free-flight flux, the turbphoretic flux, and the diffusive flux. The turbophoretic flux10

accounts for the particle flux due to gradients in turbulenintensity, whereas the diffusive flux accounts for the partiflux due to concentration gradients. They found that deption was dominated by particles starting free flights towall at large distances from the wall. However, noting taccumulation of particles near the wall, they mentionpossibility of particle deposition due to diffusive processeven though in their simulation the diffusive deposition flwas reported to be insignificant.

An important point to be noted with regard to the abostudies3,8,9 is the fact that in all simulations the mean particconcentration remained in a state of continuous evoludue to the short simulation times. The wall-normal depotion velocities for most particles was found to be quite higimplying that deposition was predominantly caused by fflight. However, since only a small fraction of the total paticle flux directed towards the wall was seen to depositfree flight,8 it is natural to expect continued particle accumlation near the wall. In order to reach a steady state unsuch conditions, some additional mechanism of deposihas to arise to balance the accumulation. Hence, althoughdiffusive deposition flux was found to be negligible in thabove studies, it could become important at later times wa large number of particles have accumulated very closthe wall.

A statement by Brookeet al.9 in this regard acts as thmain motivation for the present work. They state that difsion is not likely to control the deposition fluxat any timesince most of the particles near the wall are trapped iregion of very small wall-normal velocity fluctuations. Verclose to the wall, the distance required by a particle toposit is very small, but the probability of having largenough momentum to carry the particle across that distais also extremely small. Brookeet al.9 hypothesize that particles residing in the near-wall region would need to moaway from the wall in order to acquire a high enough veloity to deposit. Present results for similar particle respo


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times show, to the contrary, that the diffusive flux is tdominant deposition flux.

The present study aims at investigating the mechaniof particle deposition in the wall region of an open channflow, using DNS for simulating the flow and Lagrangian paticle tracking under the condition of one-way coupling. Tfocus is on dilute suspensions of particles for which Browian effects can be ignored, but which interact strongly wthe turbulent structures. The study uses the same simulamethodology as van Haarlemet al.,11 who focus in detail onthe preferential accumulation phenomenon near the freesurface of an open channel, apart from presenting deposrate coefficients and near-wall variation of particle fluxeTheir method differs from previous numerical work in thatallows the particle field to reach a statistically stationary stby reintroducing deposited particles at the inflow plane. Athough the present study follows their work closely, it prvides new insight into the mechanisms of particle depositonto a flat wall for a fully developed particle field. Variouconflicting viewpoints exist in the current understandingdeposition mechanisms as pointed out in the previous pgraph. In this work, we clearly show the two dominamechanisms of deposition and quantify their relative imptance.

The outline of the paper is as follows: In the next sectiwe present the governing equations for the fluid and the pticle phases, followed by a section describing the numermethods used and the particle parameters chosen. In thetion on results, we present instantaneous particle concention patterns both near the wall and near the free-slip surfadeposition coefficients, and particle-phase mean and rmean-square~RMS! velocity profiles, in order to put thepresent study in perspective with respect to previous onThe last part of this section is devoted to studying depositvelocity statistics in different ways to bring out the dominamechanisms of deposition. Conclusions are presented aend.

II. GOVERNING EQUATIONS

A. Fluid equations

The fluid flow in an open channel is described by tNavier–Stokes equations under the assumptions that theis incompressible, isothermal, and Newtonian. The equatiare

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765Phys. Fluids, Vol. 15, No. 3, March 2003 Mechanisms of particle deposition

are imposed at the wall and at the upper boundary free-conditions are imposed in order to represent an open chaflow.12

B. Particle equations

The motion of particles is described by solving a setordinary differential equations for the particle velocity aposition at every time instant. Most calculations found in tliterature are based on the Maxey and Riley13 formulation forthe force acting on a rigid sphere in a nonuniform flow undthe following conditions: The diameter of the spheresmaller than the Kolmogorov length scale and the spherisolated and far from the boundaries~in this mannerparticle–particle interaction and particle–boundary intertion are excluded!. Moreover, the Reynolds number for threlative motion between the particle and the fluid has tosmall. The equation for the particle acceleration thuscludes the well-known forces such as buoyancy, fluid~due tothe pressure gradient and viscous stresses!, added-massStokes drag, and Basset forces.

For the case of particles much heavier than the fl(rp /r@1), Elghobashi and Truesdell14 have shown that theonly significant forces are the Stokes drag, the buoyancy,the Basset forces. Moreover, they found that the Basset fwas always an order of magnitude smaller than the dragbuoyancy forces. In the present work the effect of gravitynot accounted for either. With the above simplificationsfollowing Lagrangian equation for the particle velocityobtained

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whereCD is the drag coefficient given by

CD524

Rep~110.15Rep

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in which Rep is the particle Reynolds number (Rep5dpuup

2uu/n). The empirical correlation15 for CD is necessary becauseRep does not necessarily remain small, in particufor depositing particles.3 For particles strictly in the Stokeregime (Rep!1), Eq. ~3! simplifies to

dup

dt52

~up2u!

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wheretp(5rpdp2/18m) is the particle response time, whic

is a measure of the time required by a particle released atto reach velocity equilibrium with the surrounding fluid.

Other authors3,16 have also considered the Saffman lforce that could be important near the boundaries. This foacts in the wall-normal direction and is proportional to twall-normal gradient of the streamwise fluid velocity. Therfore, its contribution might be important near the wall acould influence the particle deposition rate.3 However, Wanget al.16 in their study of the role of the lift force in particledeposition have found that neglecting the lift force altogetresults in only a slight reduction in the deposition rate. Mo


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over, since many of the previous studies do not accountthis term, it has not been included in the present studyfacilitate direct comparison.

III. NUMERICAL PROCEDURE

A. Direct numerical simulation of the open channelflow

The fluid equations are solved using a pseudo-specmethod based on Fourier representations in the streamand spanwise directions and a Chebychev representatiothe wall-normal ~nonhomogeneous! direction. For timemarching, a two-level explicit Adams–Bashforth schemwas employed for the nonlinear terms and an impliCrank–Nicholson scheme for the viscous terms. Furthertails of the numerical procedure can be found in Lam aBanerjee.12

The dimensions of the computational domain are choto be l x54ph, l y52ph, l z52h, in the streamwise, spanwise, and normal directions, respectively. In wall units~i.e.,normalized using the kinematic viscosity and the friction vlocity! the dimensions are (l x

1 ,l y1 ,l z

1)5(1074,537,171). Agrid consisting of 64364365 nodes was used to perform thcomputations. A nonuniform distribution of collocatiopoints is used in the normal direction for the Chebychpolynomials, with the grid spacing varying fromDz1

50.10 near the wall toDz154.19 in the domain center. Thshear Reynolds number of the flow defined asRe*5u* h/n was chosen to be 85.5. Periodic boundary contions were imposed in the streamwise and spanwise ditions.

B. Lagrangian particle tracking

A Lagrangian particle tracking code17 has been used totrack particles in the flow field. The code interpolates fluvelocities at discrete grid nodes onto the particle positiand with this velocity the equations of motion of the particare integrated in time.

The code incorporates linear, cubic and fifth-order Lgrangian polynomials for interpolation yielding seconfourth, and sixth-order accuracy, respectively. For the tiintegration the module has the choice between secondfourth-order Runge–Kutta, and second-order AdamBashforth schemes. A parametric study18 was conducted tochoose the appropriate numerical methods for interpolatintegration, and the number of particles needed to obaccurate statistics. For the simulations presented h100 000 particles were tracked using fourth-order RungKutta time integration and fourth-order Lagrangian polynmial interpolation for an interval of 5436 wall time units.

At the start of the simulation, particles were distributhomogeneously over the computational domain. The ptions of the particles were chosen randomly and their inivelocity was set equal to the fluid velocity.

1. Particle-phase boundary conditions

When a particle leaves the domain across the outflplane or in the spanwise direction periodic boundary contions are applied for both the position and the velocity of t


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particle. The wall and free-slip boundaries are considerebe completely absorbing; a particle at a distance lessone particle radius from these boundaries is assumed todeposited and is removed.

Since the total number of particles has to be maintaiconstant in time to reach statistically stationary conditionsparticle is reintroduced in the domain at the inflow plane~atx150), whenever a particle deposits at the wall or the frslip boundary. The spanwise and normal coordinates ofreintroduced particle are chosen randomly and their velois set equal to the fluid velocity at that position. This procdure introduces a constraint wherein the velocities ofreintroduced particles are necessarily affected by theposed initial conditions for a certain amount of time.

According to the arguments presented by van Haaret al.11 the distance covered by atp

1515 particle before itsvelocity becomes independent of the inflow conditions isproximately ten times the height of the channel~which isequivalent to 1700 wall units in the present work!. As thislength is greater than the streamwise extent of the fluidmain, a longer domain has to be adopted for trackingparticles. The streamwise extent of this domain was se53 l x , whereas the spanwise and normal dimensions wkept unchanged. The dimensions of the computationalmain in which the particles were tracked were thus,Lx

55370,Ly5537, andLz5171 in the streamwise, spanwisand normal directions, respectively. The fluid velocity at eery grid point was obtained simply by a periodic extensionthe original flow in the streamwise direction. Moreover, onparticles located more than 1700 wall units away frominflow plane were considered for analysis. This method issame as that used by van Haarlemet al.11

This procedure offers a twofold advantage: First, itlows the particle-phase to reach a statistically stationary sdue to the reintroduction process, and second, particle sttics can be computed as a function of both the wall-normand the streamwise directions without any effect of theposed inflow conditions.

C. Particle parameters

Studies on particle deposition suggest that based onnondimensional particle response times three differentgimes of deposition can be defined.19 For very small particleswith tp

1,0.2 the deposition rate decreases astp1 increases.

In this regime, particle transport is well represented bygradient diffusion model accounting for turbulent diffusioin the bulk flow and Brownian diffusion in a thin regioadjacent to the wall.

For 0.2,tp1,20 a dramatic increase of several orders

magnitude in the deposition rate is observed as the partime constant increases. This regime is referred to asdiffusion-impactionregime, and the observed increasedeposition is mainly due to the strong interaction betweparticles and the turbulent eddies. In this regime transporparticles due to turbulence plays an important role. Inthird regime, known as theinertia-moderatedregime, par-ticles having very high inertia acquire sufficient momentufrom eddies in the turbulent core to reach the wall. He


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diffusion plays a very small role and deposition tendsdecrease with an increase in the particle time constant, asresponse of the particles to the turbulence becomes weak19

Two sets of particles withtp155 and 15 belonging to the

diffusion-impaction regime have been chosen in this stusince the aim is to understand the contribution of turbuleto particle deposition. The values are the same as those sied by van Haarlemet al.11 and have been chosen to faciltate comparison.

IV. RESULTS AND ANALYSIS

A. Preferential concentration

The phenomenon of preferential concentration of pticles is one of the most important aspects of this probleSeveral animations and snapshots of the particle field clereveal varied concentration patterns both in the bulk and nthe boundaries.

Starting with a uniform distribution, particles quickly assume an inhomogeneous distribution as they start movtowards both boundaries, due to the phenomenon of tuphoresis. This results in the accumulation of particles, pticularly in the near-wall region. The accumulation procecontinues for a long time until a sharp peak in the conctration is formed in the near-wall region, as shown in tfollowing section. Also, the distribution of particles near thwall is far from being homogeneous in the spanwise dirtion. The particles, in fact, accumulate in streamwisoriented streaks. Instantaneous correlation between parstreaks near the wall and the low-speed streaks in a turbu

FIG. 1. Particle accumulation patterns near the wall~a! tp155, ~b!

tp1515.


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boundary layer has been established previously.4–6,11 Instan-taneous near-wall particle concentration patterns are shin Fig. 1 for both particle response times, forz1,3. Bothsets of particles strongly accumulate in streamwise strea

Once the particles reach the region near the free-boundary, they are subjected to the large-scale structcharacteristic of free-surface turbulence such as upwellindown-drafts, and attached vortices.11 Typical particle concen-tration snapshots near the free surface (z1.150) are shownin Fig. 2. Particles are distributed in the form of roughcircular and elongated voids surrounded by thin regionshigh concentration, very similar to those obtained by vHaarlemet al.11 The behavior of particles near the free suface will not be discussed further, as the study primafocuses on near-wall deposition mechanisms.

B. Particle concentration profiles

Figure 3 presents the development of particle numconcentration along the streamwise direction. Statistics wobtained by dividing the domain into cross-stream bins200 wall units each. The mean concentration in each binormalized by the concentration in the case of uniform pticle distribution over the entire domain. As expected,concentration shows an overall decreasing trend whichcounts for particle deposition at both boundaries. A discrancy can be observed at the beginning of the domain, whmay be ignored, since the particles there are still affectedthe imposed conditions on reintroduction at the inflow plaThe periodic undulations in the concentration, though,quire further clarification.

FIG. 2. Particle accumulation patterns near the free surface~a! tp155, ~b!

tp1515.


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The periodic undulations seen in Fig. 3~a! are an obviousartifact of the periodic extension of the flow domain. Peodic patterns are formed immediately at the start of the simlation and the particles accumulated very close to the wretain a memory of this fact for a long time due to the qescent nature of the region and because of their small strewise velocity. Thus, the actual flow-through time requiredwipe out these undulations would be much higher thancurrent simulation period making it computationally prohibtive. Indeed, in Fig. 3~b!, showing the streamwise concentrtion profile obtained with the procedure described previoubut accounting only for particles located in the region,z1,171, the profile is almost linear, confirming the obsevation that the periodic features are mainly due to partiaccumulation patterns in the region very close to the w(z1,3). Figure 3~b! now clearly reveals the difference ithe rate of change of bulk concentration betweentp

155 andtp

1515 particles. It points to the fact thattp1515 particles

have a higher overall deposition rate and a higher concention in the bulk.

The variation of particle concentration along the wanormal direction at equilibrium~i.e., when the statistically

FIG. 3. ~a! Streamwise variation of the cross-section averaged concention, ~b! not accounting for particles accumulated very close to the w(z1,3).


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stationary state is achieved;t1.1000) is shown in Fig. 4. Inthis case, the bin height was kept constant atDz150.15. Thedistribution shown refers to a region located in the centethe computational domain (x152400– 2600), and the timeaveraged concentration in every slab has been normalizethe average concentration of particles in the region conered~i.e., setting the integral across the channel heightwidth equal to unity!. A large increase in particle concentrtion very close to the wall is observed. Peak values occuz150.3 for tp

155 particles and atz150.4 for tp1515 par-

ticles. Particle concentration is higher fortp155 particles

very close to the wall (z1,0.4), whereas the opposite is trufor z1.0.4. This behavior very close to the wall has nbeen reported in any of the previous studies.

Particle accumulation near the wall has also beenserved by other authors3,4,6,11 in both numerical simulationsand experiments. However, a great deal of ambiguity exbetween the values and the way the statistics were obtaiIn particular, van Haarlemet al.,11 who studied conditionssimilar to those considered here, do not report such hpeaks near the wall for the same particle response timTheir values are of the order of 10, normalized by the inituniform concentration. This is due to the larger bin size thhave used to calculate the average quantities. In fact, tfirst bin was ten times larger than the one used in the prestudy. Also, they could not have captured the variationdistances less than one wall unit mentioned in the previparagraph. In van Haarlem’s work, accumulation of particnear the wall is higher for the higher-response-time particThis trend is confirmed only up to a distance ofz1.0.4 inthe present calculations. Very close to the wall the trendinverted. A possible scenario to explain this observationpresented below.

Time evolution of the particle concentration~not shownhere! indicates clearly that turbophoresis plays an importrole in particle dispersion. Since the initial particle distribtion is homogeneous, the only mechanism at the beginncapable of inducing a net drift towards the wall is turbphoresis. Referring to the region of maximum particle cocentration as the accumulation zone~AZ! and looking at the

FIG. 4. Average particle concentration profile in the wall-normal directi


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fluxes towards and away from it at steady state, onediscern a balance between four main contributions:~i! Tur-bophoresis, where particles migrate from the bulk flowwards the AZ,~ii ! free-flight deposition flux, which represents a fraction of the turbophoretic flux passing directhrough the AZ and leading to deposition,~iii ! turbulent dif-fusionflux acting to smooth the concentration build up in tAZ, and ~iv! diffusional deposition flux which again acts tremove particles from the AZ by deposition due to thesidual turbulent fluctuations at the AZ. Note that, althoughstandard parlance the termturbulent diffusionaccounts forall the above transport mechanisms, here it is specificmeant to signify only the effect of turbulence to smooth oconcentration gradients.

Since the deposition oftp155 particles is mainly due to

diffusional deposition~as will be shown later!, one wouldexpect the concentration build up required to balanceturbophoretic flux to be higher. Also, the diffusional depotion for tp

155 is less efficient as compared totp1515 be-

cause of the lower level of particle normal-velocity fluctutions at the AZ~refer to Fig. 9!. This would strengthen thecase for a higher near-wall concentration oftp

155 particles.According to the above scenario, it would be logical to epect a higher concentration very close to the wall fortp

1

55 particles. However, the situation is quite complicatand a detailed study of the near-wall flux balance wouldrequired to resolve this issue.

It is interesting to note that there is also a slight accmulation of particles at the upper boundary for the caconsidered, with values two to three times larger thanbulk concentration. This phenomenon has also been repoby van Haarlemet al.11 and is again attributed to turbophoresis since the free-slip boundary condition generates a grent in the wall-normal turbulence intensity in the normdirection.

C. Deposition rate

Figure 5~a! shows the cumulative number of particleimpinging on the boundaries as a function of time. The sloof the curve reaches an asymptotic value after approximat1.1000. This reflects the fact that after a transient periodwhich particles redistribute in the domain, the numberparticles depositing every instant of time is almost constaThe deposition rate is a strong function of particle inertbeing larger fortp

1515 than fortp155.

The deposition of smaller particles on the wall is rmarkably low. The underlying reason is that a large numof these particles reside very close to the wall without depiting and keep continuously accumulating. These particneither deposit for long times nor are they significantly reetrained in the core flow on reaching the near wall regioThis phenomenon was observed by Kaftoriet al.,5 and itmainly characterizes experiments with small particles alow shear rate.

The deposition coefficient is defined by

Kd15

Jw

Cmu*, ~6!

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whereJw is the mass of particles reaching the surfaceunit area per unit time,Cm is the mean bulk concentration oparticles, andu* is the friction velocity. Note that the concentration can alternatively be expressed in terms of numor mass density, as the suspension is mono-disperse.

The nondimensional deposition coefficient is presenin Fig. 5~b! as a function of the streamwise coordinate. Heslabs ofDx15200 were used. Tests performed showed tthe calculated quantities were actually insensitive to theact slab thickness. The deposition rate is observed torather uniform along the streamwise direction. The depotion rates on the wall are reported in Table I compared tovalues found in the literature. The values of the present si

FIG. 5. ~a! Cumulative number of particles deposited at the lower and upboundaries.~b! Deposition coefficient at the lower and upper boundaries

TABLE I. Comparison of deposition coefficients from the present studyfrom previous works.

EXP/DNS St55 St515

McCoy and Hanratty@Exp. ~Ref. 21!# 0.0081 0.073Liu and Agarwal@Exp. ~Ref. 20!# 0.015 0.135van Haarlemet al. @DNS ~Ref. 11!# 0.0064 0.051Present~DNS! 0.0056 0.045


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lations are in general smaller than those reported in expments; the comparison shows, for instance, the deposrate to be three times lower than the data of Liu aAgarwal20 and 40%–60% lower than those of McCoet al.21 It can be argued that there is no general agreemamong authors on the exact values of the deposition rReportedly, experiments show a wide range of differedeposition rates for a given particle response time. Forample, one of the key differences could be due to the mconcentration profile existing when the deposition rate wmeasured. Brookeet al.9 report that in the experiment of Liuand Agarwal20 droplets were distributed uniformly in a vertical pipe flow. In the present case, the mean profile showsharp peak very close to the wall.

Other reasons for the differences could be as followThe inclusion of the Saffman lift force would enhance depsition as discussed in Sec. II B. Moreover, the near-wallcumulation of particles being very high, other effects, neasily reproducible by numerical simulation, could becoimportant in reality~e.g., particle–fluid interaction or twoway coupling, particle–particle interaction, etc.!. Differencesin the turbulence properties can also bring about significchanges. In fact, experimental databases have been obtin physical situations significantly different from those simlated here~e.g., pipe flow at higher Reynolds number!.

D. Velocity statistics

1. Mean velocity

Particles having finite inertia accumulate preferentiain the flow both in the bulk and in the near-wall regiondiscussed in previous sections. Evidently, the particle-phvelocity statistics would not be the same as those of the flFigure 6~a! shows the particle-phase mean streamwise veity as compared to the fluid. Particle velocity on averageseen to lag behind the fluid. This is surprising for a partifield in equilibrium with the flow, unless one accounts for tpreferential concentration of particles in specific regionsthe flow which could be characterized by lower averagelocities~in this case, the low-speed streaks!. A lower particle-phase streamwise velocity was also obtained by RousonEaton22 in their channel flow simulations for a similar paticle response time.

Figure 6~b! shows the mean streamwise velocity otained by considering the fluid velocity at the particle potions~FVPP! rather than the particle velocities themselvesthe particles were not preferentially distributed in the dmain, the profiles for the fluid and FVPP should coincide~towithin statistical uncertainty!. Therefore, the extent of deviation between these two quantities can be taken as a qutative estimate of the magnitude of preferential concention. Figure 6~b! shows that the region with the largedeviation between the particle streamwise velocity andfluid velocity corresponds to the region with higher prefeential concentration. One should also note that particles wtp

155 are slightly more preferentially concentrated thtp

1515 particles.As can be observed in Fig. 6~a!, tp

1515 particles exhibita slightly larger mean velocity thantp

155 particles, espe-

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cially in the region 5,z1,30. This behavior has not beenoticed in the work of van Haarlemet al.11 In their work, nodifference in the mean streamwise velocity was foundtween the two sets of particles. In reality, since both setsparticles accumulate in the low-speed streaks, a quantively lower accumulation would imply a higher mean veloity for the tp

1515 particles.It is interesting to note that in the experiments of Kafto

et al.23 for smaller response times than in the current stu(tp

150.065,0.51,4.41) the streamwise velocity defectcreases with increasing particle response time. RousonEaton22 obtain similar results as the present study fortp

1

58.6 but for larger response times (tp15117 and 810! the

particle streamwise velocity is greater than the fluid veloc~in this case, the gravitational acceleration along the strewise direction was also considered!. The reason for this varied behavior is the dependence of the extent of preferenconcentration on the particle response time.

Studies conducted for homogeneous isotroturbulence24 show that there exists a critical particle respontime which results in maximum preferential accumulati~the value for homogeneous turbulence being of the ordethe Kolmogorov time scale!. For response times higher anlower than this critical value preferential accumulationquantitatively lower, which in this particular case woutranslate into a higher mean velocity. Thus, for increas

FIG. 6. ~a! Mean particle streamwise velocity.~b! Mean streamwise fluidvelocity at particle positions.


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particle response times the mean streamwise velocityfirst see an increase in the lag up to the critical response tand from then on a trend in the opposite direction. Therefothe observed trend in the streamwise velocity would depon the position of particle response times relative to the ccal particle response time. Although it has been shown22,25

that the Kolmogorov scale remains an appropriate time sfor characterizing preferential concentration for chanflows, a more accurate quantitative estimate for this critiparticle response time is not available. As the Kolmogorscale for inhomogeneous turbulent flows also varies withsition, the exact behavior is difficult to quantify without futher detailed study.

Figure 7 presents the particle-phase mean wall-norvelocity. Even if the fluid has a zero wall-normal velocitthe particles do have a nonzero wall-normal velocity onerage. This is consistent with the fact that for depositionoccur, particles must have a mean drift velocity towardsboundaries. The velocity towards the wall is higher fortp

1

515 particles signifying a higher deposition rate. Qualitively similar results have been presented previously fopipe flow problem.19,26 However, since these studies useReynolds-averaged approach involving modeling the coplex turbulent transport mechanisms in both phases, thesults presented here are more reliable and can be looupon as a validation of previous results. Most of the othstudies have not reported the Eulerian particle-phase mwall-normal velocity which would be very useful for modevalidation.

2. Turbulence intensity

Turbulence intensity profiles in the streamwise, spawise, and normal directions are shown in Fig. 8. As canobserved in all cases, particles have a lower fluctuationtensity than the fluid except very close to the wall. The newall behavior will be discussed later. A comparison of tresults with the data of Brookeet al.9 is satisfactory, whereasthe comparison with those of van Haarlemet al.11 is not.However, the fact that the latter authors report RMS valufor tp

155 particles higher than the fluid is difficult to reconcile.

The particle phase turbulence intensity is lower thanfluid due to two mechanisms acting in tandem. The fi

FIG. 7. Mean particle wall-normal velocity.


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mechanism is preferential concentration of particles ingions with lower turbulence intensity~for example, lowspeed streaks! and the second one is the unresponsivenesa particle to high frequency or wave number fluid fluctutions due to its inertia. For Stokesian particles in homoneous turbulence, an expression relating the power specof the particle velocities to that of the fluid velocities canderived27

Ep~v!51

11tp12v2 Ef~v!, ~7!

where v is an angular frequency andEp and Ef are theparticle and the fluid velocity spectra along the particle pa

FIG. 8. Particle-phase turbulence intensity,~a! streamwise,~b! spanwise,and ~c! wall-normal directions.


-

of--m

,

respectively. This expression shows that even in the abseof preferential concentration, the particle fluctuating intensis suppressed.~We thank one of the reviewers for bringinthis point to our attention.! For ease of further discussion, wrefer to this effect asinertial filtering27 due to fact that theparticle energy spectrum can be obtained from the fluid sptrum through the action of a filtering function. Inertial filteing is the incomplete response of a single particle to its fltuating fluid environment whereas preferential concentratis related to pattern formation and loses meaning for anlated particle. Another difference between the two mecnisms is that while the inertial filtering effect increasmonotonically with particle inertia, preferential concentrtion has a more complex dependence on particle inertiamust be noted that both the above effects are due toinertia of the particles.

To quantify the contributions of these mechanisms,fluctuation intensity of the fluid velocities at the particle psitions ~FVPP! is also presented in Fig. 8~c! along with theparticle velocity fluctuation intensity. The fluctuation intesity of FVPP represents the average turbulence intensityby the particles and captures the effect of preferential ccentration on the particle velocity fluctuation intensity. Tfigure shows that preferential concentration does causlarge decrease in the particle velocity fluctuation intensityshows thattp

155 particles have a higher chance of beingregions of lower turbulence intensity as compared totp

1

515 particles. This is consistent with the previous obsertion that tp

155 particles are more preferentially concetrated thantp

1515 particles. Inertial filtering results in a further reduction in the particle velocity fluctuation intensiwith respect to the fluctuation intensity of FVPP becausethe inability of the particles to respond to the small-scfluctuations in the surrounding fluid.

The departure of the particle turbulence intensity frothe fluctuation intensity of FVPP is again a function of tparticle response time. Particles withtp

155 can follow thelocal fluid turbulent motion better thantp

1515 particles.Their turbulence intensity, therefore, is very close to the fltuation intensity of FVPP, whereas a significant differenbetween the two quantities exists fortp

1515. For large par-ticle response times the reduction in particle turbulencetensity can be attributed mainly to inertial filtering as prefeential concentration effects will be small. However, for tresponse times studied in this work, both the mechanisplay a significant part in reducing the particle turbulenintensity.

Very close to the wall, it can be observed from Fig.that the RMS of particle normal-velocity fluctuations remanonzero, withtp

1515 particles having a significantly higheRMS value as compared totp

155 particles. This implies thathe diffusional deposition process fortp

1515 particles wouldbe more efficient.

E. Mechanisms of particle deposition

In this section we first present the cumulative distribtion function of particle deposition velocities in Fig. 10~a!.Reading from right to left, it indicates the probability that


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particle deposits with a velocity higher than the value onabscissa. A large increase in probability around2Wp

'0.001 can be observed for both particle response timThe figure suggests the possibility of dividing the populatof sampled velocities into two groups: Population A with lodeposition velocities, and population B with high velocitie

FIG. 9. Near-wall variation of particle wall-normal turbulence intensity

FIG. 10. ~a! Cumulative distribution function.~b! Probability density func-tion of wall-normal velocity of depositing particles.


e

s.

.

The division is necessarily arbitrary, but the existence osignificant intermediate range of velocities where no depotion occurs~the flat portion of the curves! clearly impliessuch a separation.

For tp155 particles almost 90% deposit with velocitie

smaller than 0.001, and the remaining 10% deposit withlocities greater than 0.1. Fortp

1515 particles, the corre-sponding fractions are 60% and 40%, respectively. As mtioned by Brookeet al.,9 particles depositing with a velocityroughly equal to the fluid velocity fluctuations very closethe wall may be said to undergodiffusional deposition. Onthe other hand, particles depositing with velocities mularger than the near-wall velocity fluctuations may beferred to asfree-flight particles. Using this definition, thepresent study suggests that once the particle field has reaa statistically steady state, the dominant mechanism of desition is diffusional. The free-flight population, howeveconstitutes a significant fraction fortp

1515 particles. Figure10~b! shows the PDF of the normal velocities of the depoited particles. The peaks correspond to velocity valuesaround 131024 for tp

155 particles and of 231024 fortp

1515 particles, confirming the above fact.At this point, an interesting comparison can be ma

between the present study and some of the previous work3,8,9

mentioned earlier. Brookeet al.9 found that most of the depositing particles have a velocity much higher than the RMwall-normal velocity near the wall. This led them to coclude that deposition occurs predominantly because offree-flight process similar to the model proposed by Frielander and Johnstone.1 They also found that the number oparticles depositing by diffusion is small and that the diffsion flux is negligible. Thus, their conclusions run exaccounter to the results obtained in the present study. Thisis also observed in the probability density [email protected]~b!#, where, in the case of Brookeet al.,9 the value of themost probable deposition velocity is around 1000 timlarger for similar particle response times. This is understaable because, in their simulation, free flight was the oobserved mechanism of deposition.

The reason for this drastic discrepancy lies in the diffences between the simulation procedures. In their work,proximately 16 000 particles were released from a planez1540 and were tracked for 700 wall time units. The depoited particles were removed and not reintroduced intoflow, thus precluding the possibility of achieving a statiscally stationary state. Also, given the short simulation timthey report that the mean concentration profile continuesevolve throughout the simulation. In the present study,simulation was carried out for more than 5000 wall timunits, so that an acceptable steady state was reached focalculation of the deposition coefficient@refer to Fig. 5~a!#.At steady state, the concentration near the wall becomesenough for diffusional deposition to be dominant. Since oa small fraction of the particles arriving near the wall halarge enough velocities to deposit by free flight,8 a steadystate can be reached only by an increase in diffusional desition. Although this process of deposition is not very efcient because of the quiescent environment close to the wit is aided by the large accumulation of particles very close


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the wall. Moreover, a lower concentration away from twall results in a reduction in the number of possible freflight particles.

To analyze this claim further, a particle residence timanalysis has been conducted. The time spent by a parbefore deposition in a slab 3 wall units from the wall hbeen recorded. An algorithm was implemented such that,particle escapes from the slab before depositing~due to re-entrainment!, the time counter for this particle was reinitiaized to zero. Figure 11 shows a scatter plot of the partresidence time versus the particle wall-normal velocitydeposition. The two populations of diffusional and free-fligparticles can now be distinguished more clearly in combition with the residence time. The free-flight population cnow be defined as particles having both a high deposivelocity and a short residence time, and the diffusional desition particles are those with very small deposition veloties and very large residence times. Logarithmic scales hbeen deliberately adopted to clearly show the separationtween the two different populations.

1. Free-flight mechanism

The behavior of the deposition velocity versus residetime shown for the free-flight particles in Fig. 11 can

FIG. 11. Residence time of particles in the slabz1,3 versus depositionvelocity for ~a! tp

155, ~b! tp1515.


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ele

a

lent-

no--vee-

e

explained using a very simple analysis. Assuming thatparticle wall-normal velocity is much larger than the wanormal fluid velocity fluctuations in the near-wall regio(z1,3 for the residence time analysis!, the equation for theparticle wall-normal velocity can be approximated by

dWp

dt52

Wp

tp1 , ~8!

the solution of which is obtained as

Wp5Wp,@z153# expS 2t

tp1 D . ~9!

The wall-normal velocity at deposition,Wdep, therefore is

Wdep5Wp,@z153# expS 2t res

tp1 D , ~10!

where t res is the residence time of the particle. Solving fthe position of the particle such thatz153 at t150 andz150 at t15t res, the following condition is obtained:

35Wp,@z153#tp1FexpS 2t res

tp1 D 21G . ~11!

Eliminating Wp,@z153# from Eqs. ~10! and ~11!, a relation-ship betweent res andWdep is obtained

tp1WdepF12expS t res

tp1 D G53, ~12!

where the number 3 on the right-hand side is the slab hechosen for the residence time analysis.

This expression matches very well the actual behavobtained by the DNS for particles withtp

155 as well as 15,as shown in Fig. 11. As the velocity of the particles enterthe slab becomes smaller and comparable to the RMS flvelocity in the region~,0.01!, it is no longer appropriate toneglect the effect of the fluid velocity fluctuations on thparticle path, and the assumption of free flight breaks doParticles now do not have sufficient momentum to depodirectly and remain in the slab for longer periods of timuntil they deposit by a random process due to the residfluctuations near the wall.

2. Preferential deposition

Another interesting statistic is the location where pticles tend to deposit on the wall. At every time step, onlyfew particles deposit, hence no preferential zones can becerned when taking instantaneous snapshots of deposlocations. Figure 12 shows the locations where particles hdeposited on the wall over the whole simulation intervThis can be considered as the probability that a particleposits at a certain position on the wall. In the figure, tdiffusional deposition population is shown in gray.

An examination of the distribution leads to the conclsion that the diffusional deposition population deposits prerentially in streamwise oriented streaks, while the free-fliones are more evenly distributed over the whole plane. Tis not surprising, since diffusional deposition particles cofrom locations very close to the wall, where particles ha


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already accumulated in such streaks. These near-wall strewise particle streaks have a long lifetime because they every close to the wall in a particularly quiescent region, amove with very small streamwise velocity. Consequenparticle distribution at the deposition position clearly reflethe near-wall particle distribution. On the other hand, freflight particles arrive from regions further away from thwall, where larger scale fluid motions project particles moevenly towards the wall. This result has not been presenin previous studies.

V. SUMMARY AND CONCLUSIONS

Direct numerical simulation of a turbulent open channflow was combined with Lagrangian particle trackingstudy the mechanisms of particle deposition onto the wParticles with inertial response times of 5 and 15 wtracked under the assumption of one-way coupling. TStokes drag force, corrected for higher particle Reynonumbers, was assumed to be the only force acting onparticles. Particles were removed on coming within onedius of the boundaries and reintroduced at the inflow plana random location. This procedure allowed the eventualvelopment of a statistically stationary particle field.

FIG. 12. Particle deposition patterns,~a! tp155, ~b! tp

1515.


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eed

l

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Particle concentration patterns were found to reflectflow characteristics in the different regions of the flow. Nethe wall, particles accumulate in streamwise oriented strecorrelated with the so-called low-speed streaks in wall turlence. Near the free-slip boundary they form large circuand elongated voids surrounded by thin regions of high ccentration, consistent with the large scale structure of turlence near a free surface. A large increase in particle contration very close to the wall is observed. The peak valuelocated aroundz1'0.3– 0.4. This flux of particles towardthe wall is driven by the process of turbophoresis. Very cloto the wall, particle concentration is higher fortp

155 par-ticles than fortp

1515 particles.The deposition rates presented in the paper compare

sonably well with nominal experimental and numerical rsults presented previously. The deposition rate fortp

1515particles was found to be significantly higher than fortp

1

55 particles. The particle-phase mean streamwise velocishown to be smaller than that of the fluid for both setsparticle response times. This is attributed to the accumulaof particles in the low-speed regions of the flow. Preferenconcentration was quantified by examining the mean strewise velocity of the fluid at the particle positions. The resushow thattp

1515 particles are slightly less preferentialconcentrated in the region 5,z1,30. The particle-phasemean wall-normal velocity is nonzero, even though the fluhas zero mean wall-normal velocity. The velocity towarthe wall was found to be higher fortp

1515 particles, con-sistent with their higher deposition rate. The particle phaturbulence intensity was found to be significantly lower ththe fluid phase due to two mechanisms working in conjution with each other. The first mechanism is the preferenconcentration of particles in regions with lower turbulenintensity and the second one is the lack of response oparticle to small-scale turbulent fluctuations due to its iner

Studies on particle deposition had so far indicated tfree flight is the dominant mechanism for particle depositin wall-bounded flows. One of the main findings of this stuis the fact that diffusional deposition~deposition induced bythe small residual turbulent fluctuations near the wall! ofparticles strongly concentrated near the wall is the dominmechanism for particle deposition. This is clearly suggesby the cumulative distribution functions of the wall-normvelocities of depositing particles. Almost 90% of thetp

155particles deposit due to this mechanism. The free-flimechanism is shown to gain in importance fortp

1515,where it accounts for 40% of the deposited particles. Tfact is further clarified by looking at the deposition velocivis-a-vis the residence time of the particles in a thin sadjacent to the wall before deposition. A clear distinctibetween diffusional and free-flight particles was revealedwas also shown that free-flight particles deposit more uformly over the wall as compared to the diffusional particlthat deposit in streamwise oriented streaks coinciding wthe near-wall accumulation patterns.

ACKNOWLEDGMENTS

The authors appreciate the help and support of MaFulgosi and Professor Yadigaroglu. L.B. gratefully acknow


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edges the financial support by the Nuclear EngineerLaboratory, ETH-Zurich. Part of the computations were pformed on the NEC SX-5 at the Swiss Center for ScientComputing~CSCS! in Manno, Switzerland.

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mechanism of particle deposition in fully developed turbulent open channel flow _2003

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