International Journal of Multiphase Flow xxx (2009) xxx–xxx

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International Journal of Multiphase Flow

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Review

Physics and modelling of turbulent particle deposition and entrainment: Reviewof a systematic study

Alfredo Soldati *, Cristian MarchioliDipartimento di Energetica e Macchine and Centro Interdipartimentale di Fluidodinamica e Idraulica, Università degli Studi di Udine, Udine, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 July 2008Received in revised form 16 February 2009Accepted 19 February 2009Available online xxxx

Keywords:Turbulent boundary layerCoherent structuresParticle dispersionDirect numerical simulationLagrangian particle tracking

0301-9322/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijmultiphaseflow.2009.02.016

* Corresponding author. Tel.: +39 0432 558020.E-mail address: soldati@uniud.it (A. Soldati).

Please cite this article in press as: Soldati, A., Mastudy. Int. J. Multiphase Flow (2009), doi:10.10

Deposition and entrainment of particles in turbulent flows are crucial in a number of technological appli-cations and environmental processes. We present a review of recent results from our previous works,which led to physical insights on these phenomena. These results were obtained from a systematicnumerical study based on the accurate resolution – Direct Numerical Simulation via a pseudo-spectralapproach – of the turbulent flow field, and on Lagrangian tracking of particles under different modellingassumptions. We underline the multiscale aspect of wall turbulence, which has challenged scientists todevise simple theoretical models adequate to fit experimental data, and we show that a sound renderingof wall turbulence mechanisms is required to produce a physical understanding of particle depositionand re-entrainment. This physical understanding can be implemented in more applied simulation tech-niques, such as Large-Eddy Simulation. Our arguments are based also on the phenomenology of coherentstructures and on the examination of flow topology in connection with particle preferential distribution.Starting from these concepts, reasons why theoretical predictions may fail are examined together withthe requirements which must be fulfilled by suitable predictive models.

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1. Introduction

In the context of turbulent dispersed flows (gas–solid, gas–li-quid and liquid–solid), a key information for practical applicationsis the rate at which the dispersed phase (e.g. particles, droplets andaerosols; from now on referred to as particles for sake of simplic-ity) is transported to, deposited at, and re-entrained from a solidboundary by turbulence. In relation to the many and complex phe-nomena involved, an elemental physical insight was given byFriedlander and Johnstone (1957) in their early paper: the ‘‘rateof particle transfer is always less than or equal to the transfer rateof the common gases which follow approximately the Reynolds anal-ogy”. This remark provides the essentials for a model: first it is nec-essary to understand wall turbulence, second it is necessary tounderstand the coherent flow motions (instantaneous realizationsof the Reynolds stresses), and third it is necessary to model theintrinsic inadequacy of the adverb approximately both in qualita-tive terms – phenomena – and quantitative terms – models. Anumber of later studies had emerged to investigate on the sameproblem, but it is beyond the scope of this paper to make a com-plete literature review; rather we just cite a few experimentalstudies (among others: Kaftori et al., 1995a,b; Niño and Garcia,1996) and theoretical developments (from Caporaloni et al.,1975; Cleaver and Yates, 1975; to Cerbelli et al., 2001; Slater

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rchioli, C. Physics and modellin16/j.ijmultiphaseflow.2009.02.0

et al., 2003). Upon examination of most of the published works,it emerges that the main difficulty of the modelling approachesis associated with the inability of particles to follow turbulent vor-tices: due to inertia, they cross through vortices and accumulateinto specific flow regions where they tend to stay long time. Forthis reason, particles do not experience fully the Eulerian statisticsof the turbulent flow field; rather they sample it only preferentially(see Fessler et al., 1994; and references therein). The response to alack of satisfactory theories produced an effort to collect experi-mental data sources for model benchmarking: examples, mostlyreferring to fully-developed turbulent flow in straight verticaltubes or ducts, can be retrieved in the paper by Young and Leeming(1997) or, more recently, in the report by Sippola and Nazaroff(2002). An interesting feature of these data collections is the inac-curacy affecting measurements in the so-called ‘‘diffusion–impac-tion” regime. In this regime, particles are large enough for theirinertia to be influential on their motion and small enough to be-come rather quickly independent of the strong motions character-izing the regions away from the walls. In fact, the inertia of theseparticles is just but sufficient to influence their motion in the wallregion.

In our previous papers (Marchioli and Soldati, 2002; Marchioliet al., 2003, 2006; Picciotto et al., 2005) it was shown that thereis a strong correlation between coherent wall structures, local par-ticle segregation and subsequent deposition phenomena, whichhave been subdivided into several steps, all quantified from a sta-tistical viewpoint. Specifically, it was demonstrated that, in the dif-

g of turbulent particle deposition and entrainment: Review of a systematic16

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fusion–impaction regime, particles deposition process is initiallydominated by inertia-induced segregation and accumulation intospecific flow regions close to the walls. Only afterwards particlesare driven to the walls. Modelling this physical mechanism isnon-trivial, especially with numerical methods coarser than DNS.The complicacy lies in the complex interaction between particleinertia and the non-homogeneous structure of turbulence in thewall-normal direction and justifies the lack of physically-basedaccurate correlations for particle deposition flux (Oliemans et al.,1986; Soldati and Andreussi, 1996).

In this paper, we will review the most relevant findings ob-tained at our computational laboratory through systematic investi-gation of particles–turbulence interaction in boundary layer bymeans of Eulerian–Lagrangian numerical simulations. Specifically,we will refer to Direct Numerical Simulations (DNS) and comple-mentary Large-Eddy Simulations (LES) of fully-developed gas–solidchannel flow under the pointwise particle approximation. Also, wewill focus on particles which possess inertia, emphasizing resultsmore than techniques. Despite the limitations to the modellingcapabilities introduced by the point-particle approximation, thisapproach can still provide the proper level of description to extractphysical knowledge from a complex two-phase system. Accordingto our experience, this approach is fully representative of the mainqualitative features characterizing particles–turbulence interactionin boundary layer in the limit of small particles and dilute flowconditions.

The paper is organized as follows. First, we will describe thenumerical methodology (Section 2). Then, we will overview themechanisms by which particles are deposited at the wall, trappedinside the boundary layer and entrained into the outer flow region(Section 3). Finally we will review the statistical tools that can beused to quantify and characterize these phenomena, also identify-ing current limitations of LES and possible lines of development(Section 4).

2. Physical modelling and numerical methodology

The reference geometry consists of two infinite flat parallelwalls: the origin of the coordinate system is located at the centerof the channel and the x; y and z axes represent streamwise, span-wise and wall-normal directions, respectively (see Fig. 1). Periodicboundary conditions are imposed on the fluid velocity field in thehomogeneous directions (x and y), no-slip boundary conditions areimposed at the walls (z ¼ �h and z ¼ h in Fig. 1). The size of thecomputational domain is Lx � Ly � Lz ¼ 4ph� 2ph� 2h. We con-sider non-reactive, isothermal and incompressible (low Machnumber) flow and monodispersed micrometer-size particles. Cal-culations are done using variables in dimensionless, representedby the superscript + and expressed in wall units, but to focus onapplication we consider air with density q ¼ 1:3 kg m�3 and kine-matic viscosity m ¼ 15:7� 10�6 m2 s�1, and pointwise heavy parti-cles with density qp ¼ 10

3 kg m�3 and diameters ranging from 10to 100lm (see Tables 1 and 2).

2.1. Equations for the fluid phase and flow solver

For the DNS, the governing balance equations for the fluid indimensionless form read as (Soldati and Banerjee, 1998; Soldati,2000):

@ui@xi¼ 0; ð1Þ

@ui@t¼ �uj

@ui@xjþ 1

Res

@2ui@x2j� @p@xiþ d1;i; ð2Þ

Please cite this article in press as: Soldati, A., Marchioli, C. Physics and modellinstudy. Int. J. Multiphase Flow (2009), doi:10.1016/j.ijmultiphaseflow.2009.02.0

where ui is the ith component of the velocity vector, p is the fluctu-ating kinematic pressure, d1;i is the mean pressure gradient thatdrives the flow and Res � ush=m is the Reynolds number based onthe shear (or friction) velocity, us, and on the half channel height,h. The shear velocity is us �

ffiffiffiffiffiffiffiffiffiffiffisw=q

p, where sw is the mean shear

stress at the wall. The superscript + is dropped from Eqs. (1) and(2) for ease of reading.

For the LES, the governing balance equations are smoothed witha filter function of width D. Accordingly, all flow variables aredecomposed into a resolved (large-scale) part and a residual (sub-grid scale) part as uðx; tÞ ¼ �uðx; tÞ þ duðx; tÞ. The filtered equationsfor the resolved scales are then:

@�ui@xj¼ 0; ð3Þ

@�ui@t¼ ��uj

@�ui@xjþ 1

Res

@2�ui@x2j� @

�p@xiþ d1;i �

@sij@xj

; ð4Þ

where sij ¼ uiuj � �ui�uj represents the subgrid scale (SGS) stress ten-sor. The large-eddy dynamics is closed once a model for sij is pro-vided. In our studies, the dynamic SGS model of Germano et al.(1991) was adopted.

The flow solver used to perform the numerical simulations isbased on a pseudo-spectral method that transforms the field vari-ables into wave space to discretize the governing equations. In thehomogeneous directions (x and y), all the quantities are expressedby Fourier expansions using kx and ky wavenumbers. In the wall-normal non-homogeneous direction, they are represented byChebyshev polynomials. The solution, represented spectrally inall three flow directions, has the general form:

uðkx; ky;nÞ ¼X

kx

Xkx

Xn

ûðkx; ky;nÞeiðkxxþkyyÞTnðzÞ; ð5Þ

in which TnðzÞ � cos½n � cos�1ðz=hÞ� is the nth order Chebyshev poly-nomial. By using the orthogonality property of eiðkxxþkyyÞ, the equa-tions for the Fourier coefficients ûðkx; ky;nÞ can be obtained. Allthe differential equations to be solved are of Helmholtz type withDirichlet boundary conditions specified at the walls. Equations aretime advanced using a two-level explicit Adams–Bashforth schemefor the non-linear convection terms and an implicit Crank–Nicolsonmethod for the diffusion terms. All calculations are carried out inwave space except for the non-linear terms, which are computedin the physical space and then transformed back to wave space. Thisnumerical scheme is standard for direct simulation of turbulence indomains of simple geometry, such as rectangular channels (Soldatiand Banerjee, 1998; Soldati, 2000).

2.2. Equations for the dispersed phase and Lagrangian particle tracking

In the Lagrangian framework, the motion of particles is de-scribed by a set of ordinary differential equations for particle po-sition, xp, and velocity, up. These equations in vector form readas:

dxpdt¼ up; ð6Þ

dupdt¼ ðu@p � upÞ

spð1þ 0:15Re0:687p Þ; ð7Þ

where u@p is the fluid velocity at the particle position, andsp � qpd

2p=18l is the particle relaxation time (dp and l being the

diameter of the particle and the dynamic viscosity of the fluid,respectively). The Stokes drag coefficient is computed using a stan-dard non-linear correction required when the particle Reynoldsnumber, Rep ¼ ju@p � upjdp=m, does not remain small (i.e. suffi-ciently large inertia).

g of turbulent particle deposition and entrainment: Review of a systematic16

Table 1Particle parameters for the Rels direct numerical simulations.

Stl ¼ StjRels slp ðsÞ d

þp dp ðlmÞ V

þs ¼ gþ � St Re

þp ¼ V

þs � d

þp =mþ

0.2 0:227� 10�3 0.068 9.1 0.0188 0.001281 1:133� 10�3 0.153 20.4 0.0943 0.014435 5:660� 10�3 0.342 45.6 0.4717 0.1613225 28:32� 10�3 0.765 102.0 2.3584 1.80418125 1:415� 10�1 1.71 228 11.792 20.1643

Table 2Particle parameters for the Rehs direct numerical simulations.

Sth ¼ StjRehs shp ðsÞ d

þp dp ðlmÞ V

þs ¼ gþ � St Re

þp ¼ V

þs � d

þp =mþ

1 0:283� 10�3 0.153 10.2 0.0118 0.002754 1:132� 10�3 0.306 20.4 0.0472 0.014445 1:415� 10�3 0.342 22.8 0.0590 0.0201820 5:660� 10�3 0.684 45.6 0.2358 0.1612925 7:075� 10�3 0.765 51.0 0.2948 0.22552100 28:30� 10�3 1.530 102.0 1.1792 1.80418

L y

L x

z,w

x,uy,v

O

z=−h

z=hvortex

regionLow−speedstreak

Clockwisevortex

No−slip Walls

Counter−clockwise

High−speed

In−sweepEjection

In−sweep

Fig. 1. Particle-laden turbulent gas flow in a channel: sketch of the computational domain and minimal schematics of near-wall turbulent coherent structures. Strong causalrelationship links low-speed streaks to ejections generated by quasi-streamwise vortices, which also generate in-sweeps of high streamwise momentum fluid to the wall inthe high velocity regions.

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To calculate individual particle trajectories, a Lagrangian track-ing routine is coupled to the DNS/LES flow solver. The routinesolves for Eqs. (7) and (6) under the following assumptions: (i) par-ticles are pointwise, non-rotating rigid spheres (point-particle ap-proach); (ii) particles are injected into the flow at concentrationlow enough to consider dilute system conditions: the effect of par-ticles onto the turbulent field is neglected (one-way coupling ap-proach) as well as inter-particle collisions. The equations ofparticle motion are time advanced by a 4th-order Runge–Kuttascheme: at the beginning, particles are randomly distributed overthe computational domain and their initial velocity is set equalto that of the fluid at the particle initial position. In our study, fluidvelocity interpolation is performed using 6th-order Lagrangianpolynomials (near the wall, the interpolation scheme switches toone-sided). The timestep for particle tracking was chosen equalto that of the fluid, dtþ ¼ 0:045, more than four times smaller thanthe non-dimensional response time of the smallest particle tracked

Please cite this article in press as: Soldati, A., Marchioli, C. Physics and modellinstudy. Int. J. Multiphase Flow (2009), doi:10.1016/j.ijmultiphaseflow.2009.02.0

ðStl ¼ 0:2Þ and fully adequate to represent precisely particledynamics (Marchioli et al., 2008c). Periodic boundary conditionsare imposed on particles moving outside the computational do-main in the homogeneous directions. Eq. (7) does not includenear-wall hydrodynamic effects which may complicate the actualmechanism of deposition when the particle-to-wall distance be-comes small compared to particle size: perfectly-elastic collisionsat the smooth walls are assumed when the particle center is at adistance lower than one particle radius from the wall (Marchioliand Soldati, 2002).

For the purposes of performing a phenomenological study ofturbulent particle dispersion and investigating the fundamentalphysics of the deposition and entrainment phenomena, a basesimulation was undertaken in which the setting is kept as sim-plified as possible and the number of degrees of freedom is min-imized. Subsequent inclusion of additional forces (gravity and liftin our problem) can be added to single out their specific effecton particles and to analyze possible changes to the physical sce-nario depicted by the base simulation. However, several previousworks (Marchioli and Soldati, 2002; Arcen et al., 2006; Marchioliet al., 2007) have demonstrated that, within the range of param-eters examined – particle dimension (see Tables 1 and 2 for de-tails), density, and concentration – the effect of these forces justadds quantitative corrections (Marchioli and Soldati, 2002; Arcenet al., 2006; Marchioli et al., 2007). In the limit of the dilute flowassumption, the two-way coupling – particles feedback onto theflow field – will also add just quantitative corrections and theweak modulation of the flow (Kaftori et al., 1995a,b; Pan andBanerjee, 1996) will not modify substantially the quality of themodel (Soldati, 2005).

2.3. Database and repositories

The results presented in this paper are relative to two values ofthe Reynolds number: Res ¼ 150 (Rels hereinafter) corresponding toa shear velocity uls ¼ 0:11775 ms�1, and Res ¼ 300 (Re

hs hereinafter)

corresponding to a shear velocity uhs ¼ 0:2355 ms�1. Average (bulk)Reynolds numbers are thus Relb � ulbh=m ¼ 2100, where ulb ’1:65 ms�1 is the average (bulk) velocity; and Rehb � uhbh=m ¼ 4200,where uhb ’ 3:3 ms�1, respectively. The size of the computational

g of turbulent particle deposition and entrainment: Review of a systematic16

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domain in wall units is Lþx � Lþy � L

þz ¼ 1885� 942� 300 for the Re

ls

simulations and Lþx � Lþy � L

þz ¼ 3770� 1885� 600 for the Re

hs sim-

ulations. In DNS, the computational domain was discretized inphysical space with 128� 128� 129 grid points (128� 128 Fou-rier modes and 129 Chebyshev coefficients in the wave space) forthe Rels simulations and with 256� 256� 257 grid points(256� 256 Fourier modes and 257 Chebyshev coefficients in thewave space) for the Rehs simulations to maintain the grid spacingfixed. In LES, two computational grids were considered: a coarsegrid made of 32� 32� 65 nodes and a fine grid made of64� 64� 65 nodes. Only the lower value, Rels, of the shear Rey-nolds number was considered.

For the DNS, the number of grid points in each direction waschosen to ensure that the grid spacing is always smaller than thesmallest flow scale1 and the requirements imposed by the point-particle approach are satisfied. Assuming that particle motions dueto strain are negligible, these requirements deal primarily with thesize of the particle, which must be much smaller than the grid cellto justify the approximation that the velocity u@p used in Eq. (7) isthe (undisturbed) fluid velocity at the center of the particle. Thisvelocity is obtained by interpolation and an accurate estimate re-quires a grid cell significantly larger than the particle. The accuracyof the fluid flow simulation, however, also requires a grid cell signif-icantly smaller than the fluid scales to solve: if the particles aremuch smaller than the smallest relevant flow scale, than thepoint-particle restriction is satisfied. In the case of DNS, this meansthat particles must be much smaller than the Kolmogorov lengthscale ðdp

Fig. 2. Cross-section of the flow field and front view of particles in the region of particle accumulation (a) and front view of particles and structures in the region of particleaccumulation (b).

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pension, since particles can only leave the wall layer through fluidejections. Particles driven to the wall by a sweep and not re-en-trained to the outer flow by an ejection are bound to remain inthe viscous wall layer for long times, slowly diffusing to the walldue to turbophoresis (Reeks, 1983; Narayanan et al., 2003). Clearly,the mechanism of particle re-entrainment from the viscous sublay-er is influenced by inertia. QSV control particle resuspension viathe ejections they generate. However, one can intuitively arguethat larger particles will require larger momentum than smallerparticles. To characterize the influence of inertia on particle resus-pension it is thus important to establish a quantitative link be-tween the inertia of the re-entrained particle and the strength,e.g. size and turnover time, of the structure responsible for re-entrainment. This type of analysis is also useful to understand ex-actly where resuspended particles come from, namely to under-stand whether particles have been just swept to the wall layer orhave been sitting there for a long time.

Fig. 3(a) shows the probability density function (PDF) of thenon-dimensional particle residence time, tþres, in the viscous sublay-er – threshold fixed at zþ ¼ 5. Profiles for all the particle sets ofTable 1 are shown. To compute tþres a time counter is started foreach particle entering the viscous sublayer, the time counter isthen stopped when the particle exits the viscous sublayer. A shortresidence time indicates that a particle penetrating the viscoussublayer may exit by being transported on the same vortical struc-ture which brought it inside in the first place. All curves in Fig. 3(a)follow a similar trend for tþres > 40, so our analysis is focused onshorter residence times. Each curve has a rather well defined peak:the PDF reaches its maximum value at tþres ’ 7 for the St

l ¼ 0:2 andStl ¼ 1 particles; between 7 and 13 for the Stl ¼ 5 particles; andabout tþres ’ 20 for the larger St

l ¼ 25 and St ¼ 125l particles. It is

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20 25 30 35

(a)

40

PDF(

t+ r

es)

t +res

St = 0.2 1 5

25 125

+

Fig. 3. PDF of the particle residence time, tþres , in the viscous sublayer (a); and local

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interesting to compare these peak values with the characteristictimescale of the turbulent structures in the near-wall region. Thistimescale, given here in terms of local dimensionless eddy turn-over time, tþeddy � 1=xþx (where xþx is the streamwise fluid vortici-ty), is shown in Fig. 3(b) as a function of the wall distance, zþ.Considering that tþeddy scales linearly with z

þ within the viscoussublayer and decreases progressively as the structures lie closerto the wall, we first observe that large particles may exit the vis-cous sublayer quickly only if re-entrained by large structures. Sec-ond, re-entrainment mechanisms for small particles aredominated by those structures with turnover time tþeddy ’ 7 in wallunits.

The transport mechanisms described so far are summarized inthe schematics of Fig. 4. First, particles segregate and form coher-ent clusters in regions of the buffer layer where in-sweeps can en-train them: segregation into clusters is thus the first mechanismcharacterizing the deposition process. Particles entrained in asweep experience a net drift toward the near-wall accumulationregion, where particle concentration reaches its maximum. In thephysical situation under investigation, the main mechanism capa-ble of inducing such drift is turbophoresis (Narayanan et al., 2003;Marchioli et al., 2003). Once in the accumulation region, which islocated well into the viscous sublayer, particles may either depositat the wall or be re-entrained toward the outer flow by ejections.Two main deposition mechanisms can be identified (Portelaet al., 2002; Narayanan et al., 2003): particles that have acquiredenough momentum may coast through the accumulation regionand deposit by impaction directly at the wall; otherwise, after along residence time, particles can deposit under the action of tur-bulent fluctuations which are strictly zero only at the wall and,due to turbulence non-homogeneity, are always stronger in driving

0

1

2

3

4

5

6

7

8

0 5 10 15

(b)

20

ted

dy

z +

eddy turnover time, tþeddy, as a function of the wall-normal coordinate, zþ (b).

g of turbulent particle deposition and entrainment: Review of a systematic16

[z ~ O(d )]+ + p

Ejections

In−sweepsDepositionby impaction

Depositionby diffusion

Solid wall

Segregation Region

Walldistance

Deposition Region

particle fluxOutward

Near−Wall AccumulationRegion (Turbophoretic Peakof Particle Concentration)

Wallwardparticle flux

(Buffer Layer)

Fig. 4. Near-wall driving mechanisms, responsible for particle concentration build-up in the near-wall accumulation region.

6 A. Soldati, C. Marchioli / International Journal of Multiphase Flow xxx (2009) xxx–xxx

ARTICLE IN PRESS

particles to the wall. The relative importance of these two mecha-nisms depends on particle inertia.

3.3. Statistical description of particle preferential distribution,segregation and deposition

As discussed in the previous sections, particle deposition can beregarded as the outcome of a multi-step process: segrega-tion! accumulation! deposition. In the following a number ofstatistical tools that can be used to characterize from a statisticalviewpoint all these steps will be surveyed. Our analysis will restrictto the physical situation in which, while velocity statistics for bothphases are at stationary state, particle distribution is still develop-ing and much longer simulation times would be required to obtainsteady-state concentration profiles (Marchioli et al., 2008c). Therationale for this choice is that, in a number of industrial applica-tions including separation (Soldati and Banerjee, 1998; Soldati,2000) and droplet-laden flows (Soldati and Andreussi, 1996), par-ticle distribution never reaches equilibrium. From an engineeringviewpoint, statistically-developing particle concentration is thusthe most probable (and the most interesting) situation toinvestigate.

3.3.1. Quantification of particle preferential distribution in the viscoussublayer

In recent years, several studies (e.g. Picciotto et al., 2005; Rou-son and Eaton, 2001; and references therein) were conducted tocorrelate particle distribution in the viscous sublayer with coher-ent structures based on global statistical identifiers. A methodol-ogy was thus developed to correlate non-homogeneous particleaccumulation to coherent flow structures, according to their topo-logical properties. A general flow topology classification was pro-posed (see Blackburn et al. (1996) for details) to group allelementary three-dimensional flow patterns with respect to thethree invariants, P; Q and R, of the velocity gradient tensor, ui;j.For incompressible flow fields:

P � ui;i ¼ 0; ð8Þ

Q � 12ðui;iÞ2 � ui;juj;ih i

¼ 12ðXijXij � SijSijÞ; ð9Þ

R � �k1k2k3; ð10Þ

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and D ¼ ð27=4ÞR2 þ Q3 is the discriminant that determines theeigenvalues k1; k2 and k3 of ui;j. The quantities Xij � 12 ðui;j � uj;iÞand Sij � 12 ðui;j þ uj;iÞ are the antisymmetric and symmetric compo-nents of the velocity gradient, respectively. Four regions can beidentified across the curves D ¼ 0 and R ¼ 0: two vortical flow re-gions, the so-called stable focus/stretching ðIÞ and unstable focus/compressing critical nodes ðIIÞ, and two convergence regions, theso-called stable node/saddle/saddle ðIIIÞ and unstable node/saddle/saddle critical nodes ðIVÞ. Further critical points can be identifiedalong the Q-axis and the D ¼ 0 line. The reader is referred to the pa-pers by Cantwell and co-workers for a complete treatment (see Cha-cin and Cantwell (2000) or Blackburn et al. (1996), for instance).

The structural classification scheme employed by Blackburnet al. (1996) eliminates the arbitrary choice of threshold values re-quired by other schemes and, therefore, it can be convenientlyused to elucidate the relationship between particle distributionand near-wall vortices. In their DNS study, Rouson and Eaton(2001) focused their attention on moderate- to high-Stokes num-ber particles (St ¼ 8:6, 117, and 810) throughout the entire chan-nel. This range includes particles too big to match the ever-decreasing turbulent flow scales which particles encounter whenapproaching the wall. In a subsequent DNS study (Picciotto et al.,2005), we tried to complement the analysis of Rouson and Eaton(2001) by considering low- to moderate-Stokes number particles,which are more ‘‘respondent” to the near-wall flow timescales.Since we are interested in providing a statistical description of par-ticle preferential distribution very near the wall, in Fig. 5 we showthe joint probability density function (JPDF) of Q and R in the vis-cous sublayer only. JPDFs were calculated over 400 realizationsof the flow field (covering 1800 dimensionless time units) to con-sider only those events with significant statistical occurrence.

The JPDF sampled for the fluid at grid points (Fig. 5(a)) showsthat the most probable value of Q and R is zero for all the instants,and that the preferred quadrants correspond to the stable focus/stretching ðIIÞ and the unstable node/saddle/saddle ðIVÞ topologies.Also, the lines of constant JPDF asymptote toward the D ¼ 0 curve,which represents the tail of the tear-drop shaped ðQ ;RÞ-distribu-tion. As expected, the JPDF sampled at the position of the smallerparticles (Stl ¼ 0:2 in Fig. 5(b) and Stl ¼ 1 in Fig. 5(c)) is similarto that of the fluid and shows weak preferential distribution. Thedegree of particle preferential sampling increases monotonicallyup to the intermediate-size particles: the area covered in theðQ ;RÞ-plane by the JPDFs becomes smaller (see Fig. 5(d) and (e)),indicating that these particles avoid the strongest vortical regionsin quadrants I and II as well as the strongest vortex-stretching re-gions (quadrant IV) along the positive-R, zero-D curve. Only con-vergence regions ðIIIÞ are characterized by particle preferentialsampling. A broader JPDF is obtained again for the Stl ¼ 125 parti-cles, as shown in Fig. 5(f). This demonstrates the effect of theStokes number on particle preferential sampling. Results from per-cent particle counts for each topological quadrant and for each par-ticle set (reported in Picciotto et al., 2005) corroborate theconclusion that a larger proportion (more than 70%) of particlestends to occupy convergence regions in the viscous sublayer, whilethe strongest vortical regions are depleted of particles due to thecentrifuge-like effect of the eddies in the near-wall region (Rousonand Eaton, 2001).

Particles may remain long time or short time in the accumula-tion regions (Narayanan et al., 2003). Yet, the analysis in the ðQ ;RÞ-space may not discriminate between long- and short-term accu-mulation in convergence regions. A different identification crite-rion specifically valid in the wall proximity is thus required. Onepossibility is to exploit the relationship between the fluctuatingcomponents of the wall shear stress, s0ij, and the fluctuating compo-nents of the velocity gradient tensor evaluated at the wall, u0i;jjw.Limiting the analysis to incompressible flow, the only non-vanish-

g of turbulent particle deposition and entrainment: Review of a systematic16

Q

0.01

0

-0.01

-0.02

D = 0

III

III IV

Q

0.01

0

-0.01

-0.02

D = 0

III

III IV

0.01

0

-0.01

-0.02

D = 0

III

III IV

Q

R

0.01

0

-0.01

-0.02-0.001 0 0.001

D = 0

R

0.01

0

-0.01

-0.02-0.001 0 0.001

D = 0

R

0.01

0

-0.01

-0.02

0.01

-0.001 0 0.001

D = 0

Fig. 5. Viscous sublayer ðzþ < 5Þ joint PDF of Q, R conditionally sampled for fluid at grid points (a) and at particle positions (b–f). (b) Stl ¼ 0:2; (c) Stl ¼ 1; (d) Stl ¼ 5; (e)Stl ¼ 25; (f) Stl ¼ 125. Isoline values are: –– JPDF = 10, -- - JPDF = 1, � � � JPDF = 0.1, -�- JPDF = 0.01.

A. Soldati, C. Marchioli / International Journal of Multiphase Flow xxx (2009) xxx–xxx 7

ARTICLE IN PRESS

ing components of this tensor at the wall are u0x;zjw ¼ @u0=@zjw andu0y;zjw ¼ @v 0=@zjw. Now, the low-speed streaks are ejection-likeenvironments that correlate with lower-than-mean wall shearstress regions ðs0xzjw � l � @u0=@zjw < 0Þ, whereas the high-speedstreaks are associated with higher-than-mean wall shear stress re-gions ðs0xzjw > 0Þ. More specifically, two distinct near-wall flow re-gions can be identified with respect to the wall-normal direction: asweep-like inflow region, characterized by s0yzjw � l � @v 0=@zjw ¼ 0associated to s0xzjw > 0, and an ejection-like outflow region, charac-terized by s0yzjw ¼ 0 associated to s0xzjw < 0.

Fig. 6 shows the instantaneous joint correlations of non-vanish-ing components of u0i;jjw. Correlations were computed for both thefluid and the particles following the same procedure as in Picciottoet al. (2005). Visual inspection indicates that, regardless of particlesize, particles in the viscous sublayer accumulate preferentially inthe negative @u0=@zjw semiplane, which corresponds to ejection-like low-speed regions near the wall. This is also confirmed bythe percentage of sample points falling in each @u0=@zjw semiplane(percent figures are included in Fig. 6). Note that even the fluid(Fig. 6(a)) shows preferential distribution: a clear consequence ofthe mechanism by which fluid velocity streaks are generated. Ajet of fluid directed to the wall generates the sweep and also thehigh-speed region; then the jet of fluid, by continuity, is deflectedby the wall and generates the ejection. Due to the entrainment ofsurrounding fluid, the sweep is more intense and concentratedwhile the ejection spreads over a wider cross-section and has low-er momentum. Low-speed, low-shear regions, where s0w < 0 (i.e.@u0=@zþjw < 0Þ, appear much wider than high-speed, high-shear re-gions, where s0w > 0 (i.e. @u0=@zþjw > 0Þ. Thus, grid points necessar-ily sample @u0=@zþjw < 0 regions more often. The correlationsshown in Fig. 6 are similar for all the flow field realizations westudied (the same that were used to obtain Fig. 5). Percent datafor samples falling in each @v 0=@zjw semiplane (not shown) also

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indicate that the preferred near-wall flow regions correlate wellwith values of @v 0=@zjw close to zero, thus leading to the conclusionthat particle concentration build-up occurs preferentially in theproximity of a near-wall outflow region (Picciotto et al., 2005).

The physics described in Fig. 6 can be examined also in Fig. 7.This figure shows the instantaneous distribution of the Stl ¼ 25particles, chosen for their relatively larger tendency to preferentialsampling (see discussion of Fig. 5), together with the contours ofs0xzjw in the ðx; yÞ-plane. The behavior of @u0=@zþjw along the span-wise direction at a fixed streamwise location (identified with thedash-dotted AA-line) is also shown on top of Fig. 7. Dark grayspheres represent particles with positive spanwise velocityðv > 0Þ, moving from left to right; light gray spheres represent par-ticles with negative spanwise velocity ðv < 0Þ, moving from rightto left. Dark gray contours indicate high positive values of s0xzjw,white contours indicate low negative values. Black solid lines con-nect points where s0yzjw is equal to zero. From Fig. 7, it is apparentthat particles arriving at the wall are initially found in high-speed,high-shear regions (white contours), which are convergence flowregions where @u0=@zjw attains a local maximum. Particles staybriefly in the high-speed regions: they are swept away from theseregions and clusters begin to split along the @v 0=@zjw ¼ 0 lines,which thus mark the position of Short-Term Accumulation ðSTAÞregions. Particles move in the spanwise direction toward low-speed, low-shear regions (dark gray contours), where @u0=@zjw at-tains a local minimum – i.e. the low-speed streaks. In these regions,particles line up and form persistent clusters flanking the@v 0=@zjw ¼ 0 lines, which now mark the position of Long-TermAccumulation ðLTAÞ regions.

3.3.2. Quantification of particle segregationIn the previous sections, we have discussed some statistical

tools that can provide a quantitative description of near-wall phe-

g of turbulent particle deposition and entrainment: Review of a systematic16

Fig. 6. Viscous sublayer ðzþ < 5Þ instantaneous joint correlations of non-vanishing components of the fluctuating velocity gradient tensor, conditionally sampled at the fluidgrid points (a) and at particle positions projected onto the wall (b–f). (b) Stl ¼ 0:2, (c) Stl ¼ 1, (d) Stl ¼ 5, (e) Stl ¼ 25, (f) Stl ¼ 125. Joint correlations demonstrate that particlesare mostly concentrated in the ejection-like environments. Percent figures indicate the number of particles in a region of positive/negative @u0=@zjw .

8 A. Soldati, C. Marchioli / International Journal of Multiphase Flow xxx (2009) xxx–xxx

ARTICLE IN PRESS

nomena such as particle trapping/re-entrainment (Section 3.2) andparticle preferential distribution (Section 3.3.1). In the following,we will present additional statistical tools that can be used to char-acterize particle segregation and particle deposition. These toolshave been used also to characterize and optimize particle disper-sion in jets (Campolo et al., 2008; Sbrizzai et al., 2009, 2004; Cam-polo et al., 2005).

The relative tendency of particles to segregate in a turbulentflow field can be quantified in terms of maximum deviation fromrandomness, D (Fessler et al., 1994), also referred to as segregationparameter, Rp (Février et al., 2005). The maximum deviation fromrandomness is defined as:

D � r� rPoissonl

; ð11Þ

where r is the standard deviation for the measured particle numberdensity distribution and rPoisson is the standard deviation for a Pois-son distribution (i.e. a purely random distribution of the same aver-age number of particles). The parameter l is the mean particlenumber density. According to Eq. (11), D ¼ 0 corresponds to a ran-dom distribution, D < 0 corresponds to a uniform distribution, andD > 0 indicates segregation of particles. In this latter case, largervalues of D correspond to stronger segregation. The maximum devi-ation from randomness was applied to homogeneous isotropic tur-bulence (Février et al., 2005) and to the centerline of a turbulentchannel flow (Fessler et al., 1994), focusing on two-dimensional re-gions of nearly homogeneous flow to observe particle response tosmall scale turbulent motions. In our channel flow simulations,the use of D was extended to the near-wall region, where inhomo-geneities arise in the direction perpendicular to the wall. The parti-cle number density distribution is thus computed on a three-

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dimensional grid containing Ncell cells of volume Xcell covering theentire computational domain. This grid is independent of the Eule-rian grid used by the flow solvers, and the volume Xcell is varied bychanging the streamwise and the spanwise lengths of the cellwhereas the wall-normal length is maintained to a uniform sizeto avoid an additional averaging scale in the wall-normal direction.The value calculated for D depends on the cell size. Because of thisdependency, the segregation parameter can not provide an abso-lute, clearcut quantification of particle segregation; rather it shouldbe used just to identify and compare different trends. Taking thisinto account, the cell size dependency can be partially overcomeby computing the particle number density distribution for severalvalues of Xcell and keeping only the largest value of D (Sbrizzaiet al., 2009; Picciotto et al., 2005; Février et al., 2005). This choiceis justified by the following reason: the cell size for which D is max-imum indicates the length scales of particles clustering.

Fig. 8 shows the maximum deviation from randomness, Dmax(black circles), as a function of the particle Stokes number, Stl, inthe viscous sublayer ð0 < zþ < 5Þ. Values indicate that the maxi-mum segregation is obtained for the Stl ¼ 25 particles, which alsoexhibit the strongest tendency to sample preferentially the flowfield (see Fig. 5). This indicates that particle dynamics in the vis-cous sublayer is controlled by flow structures with non-dimen-sional timescale sþf ’ 25. Considering that sþf scales linearly withwall distance and decreases progressively as the turbulence struc-tures lie closer to the wall, we can infer that this value correspondsto the circulation time of the turbulence structures in the bufferlayer ð5 < zþ < 30Þ.

It is helpful to complement the analysis on particle segregationby providing a single quantitative measure rather than the twonumbers, D and the length scale for that value of D. Several possi-

g of turbulent particle deposition and entrainment: Review of a systematic16

Fig. 7. Instantaneous Stl ¼ 25 particle distribution in the viscous sublayerðtþ ¼ 6500; zþ < 5Þ. The computational window is 400 wall units long and 250wall units wide in the ðx; yÞ-plane. The mean flow is directed top down.

0

0.5

1

1.5

2

2.5

3

0.1 1 10 100

Dm

ax

Stl

Dmaxν

1.5

1.6

1.7

1.8

1.9

2

ν

Fig. 8. Maximum deviation from randomness, Dmax (black circles), and correlationdimension, m (open circles), in the viscous sublayer as a function of the particleStokes number, Stl . Data are relative to the no-gravity, no-lift case. Values of Dmaxare instantaneous (computed at tþ ¼ 6500, corresponding to the final instant in thecomputation of the statistics shown in Figs. 5 and 6). Values of m are time-averaged(over tþ ¼ 1800 time units).

1e-05

1e-04

0.001

0.01

0.1

1

10

0.001 0.01 0.1 1 10 100

Ener

gy, E

(ω) [

m2 /

s2]/[

rad/

s]

Frequency, ω [rad/s]

z+=25

St=0.2St=1St=5St=25St=125

Fig. 9. One-dimensional (streamwise) frequency spectrum for turbulent channelflow computed at zþ ¼ 25 for two different Reynolds numbers: Rels (�) and Re

hs ().

A. Soldati, C. Marchioli / International Journal of Multiphase Flow xxx (2009) xxx–xxx 9

ARTICLE IN PRESS

bilities to measure segregation have been identified (see, for in-stance, Calzavarini et al., 2008). Here, we will compute the correla-tion dimension, introduced by Grassberger and Procaccia (1983). Inits three-dimensional formulation, needed to study non-isotropic

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flows, this parameter can be computed by choosing one base par-ticle and counting the fraction, NpðrÞ, of particles within a distancer from the base particle. The correlation dimension, m, is defined asthe slope of NpðrÞ as a function of r in a log–log plot. The probabilitydistribution of the distance between the neighboring particles andthe base particle is obtained repeating this count for all possiblevalues of r, thus removing any dependence on the length scaleused. In general, NpðrÞ will scale with rm: if particles are uniformlydistributed in the volume surrounding the base particle, NpðrÞ willscale with r3 (namely with the volume of the sphere centered onthe base particle); if particles are uniformly distributed over a sur-face, NpðrÞwill scale with r2 (namely with the area of the circle cen-tered on the base particle), whereas if particles are concentratedinto a line, NpðrÞwill scale with linearly with r. Thus smaller valuesof m indicate greater preferential concentration. To compute resultssignificant from a statistical perspective, the procedure can be re-peated for different randomly chosen base particles and differenttimes, averaging the results. The correlation dimension calculatedfor the particles in the base DNS at Rels is shown in Fig. 8 (open cir-cles). The correlation dimension is always smaller than 2, indicat-ing that, regardless of their size, particles never attain a uniformspatial distribution. It is confirmed that, while nearly random dis-tribution is observed for the smaller particles, preferential concen-tration is maximum for particles with Stokes numbers around 25.In particular, the minimum value m ’ 1:53 indicates that the pref-erential accumulation of these particles mainly occurs in elongatedstructures.

3.3.3. Influence of the Reynolds number on particle segregationThe results shown in Fig. 8 are relative to the case of turbulent

channel flow at moderate Reynolds number ðRelsÞ. If higher valuesof Res are to be considered, then Reynolds number effects on par-ticle dispersion may become significant because the characteristiclength and time scales of the particle change with respect to thoseof the fluid when the flow dynamics change: in particular, thehigher the Reynolds number the smaller the particle response timefor a given value of the Stokes number. This point can be furtherelucidated considering Fig. 9, where the one-dimensional (stream-wise) frequency spectrum, EðxÞ, computed for the DNS at Rels iscompared with the frequency spectrum computed for the DNS atRehs at the z

þ ¼ 25 location inside the buffer layer. Also shown (so-lid vertical lines) are the estimated response frequencies, propor-tional to 1=sp, for each particle size. It is apparent that, in the Rehsflow (i) the turbulent kinetic energy budget spreads over a widerrange of frequencies, representing smaller flow timescales with

g of turbulent particle deposition and entrainment: Review of a systematic16

0

0.02

0.04

0.06

0.08

0.1

0.12

0.1 1 10 100 1000

Dm

ax

(a)

DNS - Reτ=150DNS - Reτ=300

0

0.5

1

1.5

2

2.5

3

0.1 1 10 100 1000

Dm

ax

St

(b)

DNS - Reτ=150DNS - Reτ=300

Fig. 10. Maximum deviation from randomness, Dmax , versus particle Stokesnumber, St, in turbulent channel flow at two different Reynolds numbers: Rels (�)and Rehs (). Panels: (a) channel centerline ð145 6 zþ 6 150Þ, (b) near-wall regionð0 6 zþ 6 5Þ.

10 A. Soldati, C. Marchioli / International Journal of Multiphase Flow xxx (2009) xxx–xxx

ARTICLE IN PRESS

which the particles may interact, and (ii) a given value of frequencycorresponds to higher values of the turbulent kinetic energy. Inprinciple, these observations should lead to the conclusion thatsimulation techniques like LES, requiring models for the filteredsubgrid fluid scales, must incorporate a dependency on the flowReynolds number to recover the correct amount of SGS turbulentkinetic energy. In fact, the need to include Reynolds number effectsshould be carefully assessed being based on the knowledge of howparticle preferential concentration scales with this parameter.Numerical investigations of the scaling properties of particle pref-erential concentration with the Reynolds number were performedin a synthetic turbulent advecting field by Olla (2002) and inHomogeneous Isotropic Turbulence (HIT) by Collins and Keswani(2004) and by Yeung et al. (2006).

Here, we investigate on the same effect in turbulent channelflow.

To introduce our scaling argument, we remark that the same(dimensional) value of the particle response time corresponds todifferent values of the Stokes number according to the followingexpression:

shp ¼ slp ! Sth � shf ¼ St

l � slf !Sth

Stl¼

slfshf¼ u

hs

uls

� �2¼ Re

hs

Rels

!2; ð12Þ

where Sth and Stl represent the particle Stokes number in the Rehssimulation and in the Rels simulation, respectively; andðRehs=Re

lsÞ

2 ¼ 4 in our case. If the shear velocity is the proper scalingparameter then the coupling between particles and fluid in the re-gime where particles preferentially concentrate is expected to obeyto Eq. (12). In Fig. 10 particle segregation in the center of the chan-nel (Fig. 10(a)) and in the near-wall region (Fig. 10(b)) is quantifiedby the Dmax parameter for the two DNS simulations. Black symbolsrepresent the values of Dmax for the particles in the DNS fields at Re

ls,

whereas open symbols are used for the particles in the DNS fields atRehs . Two observations can be made: first, lower segregation occursat higher Reynolds number for a given value of the particle Stokesnumber; second, the degree of segregation is indeed nearly thesame for particle Stokes numbers and shear Reynolds numbersmatching the condition given in Eq. (12), as indicated by the arrows.This is particularly true in the near-wall region. The above resultsseem to indicate that particle preferential concentration scales pro-portionally to the flow Reynolds number and that it is possible toparameterize Reynolds number effects simply by imposing a qua-dratic dependence of the particle Stokes number on the shear Rey-nolds number. These scaling effects appear to be consistent withother observations, most of which refer to the Kolmogorov scalingargument (Collins and Keswani, 2004; Yeung et al., 2006), that pre-dict statistical saturation only at Reynolds numbers higher thanthose considered here.

3.3.4. Quantification of particle deposition ratesAccording to our schematism of the deposition process, the de-

gree of particle segregation influences quantitatively the rate atwhich particles deposit (i.e. their deposition flux). This was ex-plained in the previous sections considering the chain of physicalmechanisms by which particles are transferred to and away fromthe wall. It is thus consequential to combine the quantitativedescription of particle segregation to the quantitative prediction ofparticle deposition rate. Virtually all the experimental data on thedeposition rate were obtained in turbulent pipe flow. However, be-cause deposition is mainly controlled by the near-wall turbulence,which is a local wall-dominated phenomenon not influenced bythe largest scales of the flow, calculations for channel flow are thesimplest setting for model validation. The deposition rate of non-interacting particles is proportional to the ratio between the particlemass transfer rate on the wall (flux of particles per unit deposition

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area), J, and the mean bulk concentration of particles (mass of parti-cles per unit volume), C. According to this definition the constant ofproportionality, named deposition coefficient kd � �J=C, representsa deposition velocity (Young and Leeming, 1997). Fig. 11 shows thenon-dimensional values of the deposition coefficient, kþd , computedas function of the Stokes number in the Rels simulation (Picciottoet al., 2005). The trend is similar to that seen in Fig. 8, which refersto the same simulation time span: Again, the Stl ¼ 25 particles exhi-bit the highest deposition rate. From our previous discussion, it isnow easy to argue that this happens because these particles arethe most responsive to near-wall turbulence in terms of segregationand preferential distribution. Particles with smaller or larger inertiaare not able to respond in this optimal way, either because they be-have more like tracers with strong stability against non-homoge-neous distribution and near-wall concentration (the Stl ¼ 0:2particles, in particular), or because they are too big to respond opti-mally to the fine turbulence structures in the buffer layer (theStl ¼ 125 particles, for instance). In general terms, the degree of par-ticle responsiveness to segregation and preferential distribution in-duced by the flow structures is strongly (and directly) correlated tothe rate at which particles deposit.

4. Modelling perspectives for Large-Eddy Simulation ofturbulent dispersed flows

In this section, we will try to answer the issue of the minimalphysics required to model particle motion at the subgrid level. This

g of turbulent particle deposition and entrainment: Review of a systematic16

10-4

10-3

10-2

10-1

10-1 100 101 102

k d+

Stl

Fig. 11. Particle deposition coefficients, kþd , as a function of the particle Stokesnumber, Stl . Data are relative to the base no-gravity, no-lift simulation and are time-averaged from tþ ¼ 4700 to tþ ¼ 6500.

0

0.02

0.04

0.06

0.08

0.1

0.12

1 10 100

Dm

ax

(a)

DNS (GRID 128x128x129)

A-POSTERIORI LES (GRID 32x32x65)

A-POSTERIORI LES (GRID 64x64x65)

0

0.5

1

1.5

2

2.5

3

1 10 100

Dm

ax

Stl

(b)DNS (GRID 128x128x129)

A-POSTERIORI LES (GRID 32x32x65)

A-POSTERIORI LES (GRID 64x64x65)

Fig. 12. Maximum deviation from randomness, Dmax , versus particle Stokesnumber, Stl: comparison between DNS (), a-posteriori LES on the fine64� 64� 65 grid (�) and a-posteriori LES on the coarse 32� 32� 65 grid (M).Panels: (a) channel centerline ð145 6 zþ 6 150Þ, (b) near-wall region ð0 6 zþ 6 5Þ.

A. Soldati, C. Marchioli / International Journal of Multiphase Flow xxx (2009) xxx–xxx 11

ARTICLE IN PRESS

level is not solved by LES, preventing tracked particles from beingexposed to flow scales which may influence their motion. Specifi-cally, we will try to answer in a quantitative way to the followingquestions: How does the SGS turbulence affect particle dispersion?How should one model these SGS effects to predict accurately theselective response of different-inertia particles? Having in mindthe phenomenological model illustrated in Section 3, we addressthese issues by analyzing first the effect of the subgrid fluid turbu-lence on particle segregation. Fig. 12 compares the DNS results forDmax against those obtained from real a-posteriori LES (Marchioliet al., 2008a). Here, we have adapted to our channel flow at Relsthe computational procedure used by Fede and Simonin (2006)for the case of heavy colliding particles in HIT. LES fields were com-puted as discussed in Section 2. Particle tracking was performedwith no extra model. In this way, the scales not resolved by LESare simply non-existing for the particle dynamics and it is possibleto judge if some of the filtered scales effects should be included assuggested previously (Fede and Simonin, 2006; Kuerten, 2006). Re-sults shown in Fig. 12 indicate that particle segregation is signifi-cantly influenced by the subgrid fluid turbulence, even when theparticle response time is much larger than the flow timescalesnot resolved in LES. Both in DNS and in LES, the behavior of the seg-regation parameter is qualitatively similar: a peak of Dmax occursaround Stl ¼ 25 and preferential concentration falls off on eitherside of this optimum value. However, due to filtering, segregationis always underestimated by LES, except for the Stl ¼ 125 particles.For these particles filtering leads to an increase of segregation, inagreement with the behavior observed by Fede and Simonin(2006). As expected, the quantitative inaccuracy of LES is particu-larly evident in the near-wall region (Fig. 12(b)), where the degreeof misprediction depends strongly on the ratio of the particle sizeto the filtered spatial scales.

The limits of LES in providing an accurate quantitative estimateof local particle segregation have an effect on particle depositionfluxes and, in turn, on particle near-wall accumulation. This canbe observed in Fig. 13, where different instantaneous particle con-centration profiles (taken at time tþ ¼ 1350 of the simulations atRels) are compared. Concentration is computed as particle numberdensity distribution, C, normalized by its initial value, C0 (seeMarchioli et al. (2008a) for details). Only the intermediate-size par-ticles are considered. For these particles, LES underpredicts bothqualitatively and quantitatively particle accumulation at the wall.

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Compared to the DNS results, different-shape profiles and lowerpeaks of particle concentration inside the viscous sublayer arefound. Note that the degree of underprediction also depends onthe spatial resolution of the Eulerian grid. These findings are in linewith previous LES applications to particle dispersion in turbulentwall-bounded flows (Kuerten and Vreman, 2005; Kuerten, 2006).In our opinion, Figs. 12 and 13 are important since they demon-strate that the inaccuracy of LES in predicting near-wall accumula-tion is a direct consequence of filtering, which removes bothenergy and flow structures from the LES turbulent flow field. LESmay thus produce an inaccurate rendering of the near-wall vorti-ces responsible for trapping particles in the viscous sublayer (Kuer-ten and Vreman, 2005). When these structures are smeared out ofthe flow field, their interactions with particles can not be fully cap-tured and representation of local segregation phenomena becomesinadequate, in turn influencing macroscale collective phenomenaas deposition and overall particle distribution (Marchioli et al.,2008a).

In the light of the above considerations, challenges for multi-phase LES are in the specific modalities by which the SGS effectsare recovered and incorporated into models. In recent years, sev-eral numerical studies have tried to clarify these modalities. Someauthors (Kuerten and Vreman, 2005; Shotorban and Mashayek,2005; Kuerten, 2006; Shotorban et al., 2007; Marchioli et al.,2008b) proposed a closure model for the equations of particle mo-tion, which uses the approximate deconvolution method (Stolzet al., 2001) to reconstruct the filtered fluid velocity and add its ef-fects in the particle equations. This model was assessed in different

g of turbulent particle deposition and entrainment: Review of a systematic16

0.5

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100

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cle

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cent

ratio

n (C

/C0)

Wall-Normal Coordinate (z+)

A-POSTERIORI (GRID 32x32x65)A-POSTERIORI (GRID 64x64x65)

DNS (GRID 128x128x129)

0

20

40

60

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(b)

100

0.1 1 10 100

Impact

Parti

cle

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cent

ratio

n (C

/C0)

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DNS (GRID 128x128x129)

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100

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cle

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cent

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n (C

/C0)

Wall-Normal Coordinate (z+)

A-POSTERIORI (GRID 32x32x65)A-POSTERIORI (GRID 64x64x65)

DNS (GRID 128x128x129)

Fig. 13. Particle concentration in a-posteriori LES without SGS modelling in theparticle equation of motion: comparison between DNS (), a-posteriori LES on thefine 64� 64� 65 grid (�) and a-posteriori LES on the coarse 32� 32� 65 grid (M).Panels: (a) Stl ¼ 1 particles, (b) Stl ¼ 5 particles, (c) Stl ¼ 25 particles. The verticalsolid line in each diagram indicates the position where the particles hit the wall(impact): note that impact for the St ¼ 1 particles occurs at zþ ¼ 0:034, outside thezþ-range covered in panel (a).

12 A. Soldati, C. Marchioli / International Journal of Multiphase Flow xxx (2009) xxx–xxx

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flow configurations (from homogeneous turbulent isotropic andshear flows to channel flow) and a general improvement in the pre-diction of particle-related statistics was observed. Obviously,deconvolution can be used to model the effect of the resolvedscales only and subgrid scales, virtually non-existing in the LEScontext, can not be retrieved. These scales, however, do influencethe instantaneous particle behavior (Kuerten, 2006). Hence, evenwhen ad hoc closures are used in the particle equations, theremight be situations in which prediction of local particle segrega-

Please cite this article in press as: Soldati, A., Marchioli, C. Physics and modellinstudy. Int. J. Multiphase Flow (2009), doi:10.1016/j.ijmultiphaseflow.2009.02.0

tion and, in turn, of near-wall accumulation are judged inaccuratefrom a quantitative viewpoint because a quantitative replica of thesubgrid turbulent flow scales is not ensured (Marchioli et al.,2008a). In our view, a model for subgrid particle dynamics maynot be obtained by the information provided via a LES computa-tion, rather it must be obtained via an independent physical rea-soning. A modelling strategy that goes along this direction wasdeveloped recently by Minier and co-workers (Guingo and Minier,2008; Chibbaro and Minier, 2008), who propose a Lagrangian mod-el for particle deposition based on a stochastic description of theeffects due to near-wall coherent structures. The idea behind thismodel is to account explicitly for the interactions of particles withthe coherent sweep/ejection events by including the relevant geo-metrical features of the near-wall region. From an engineeringviewpoint, this seems to be a very interesting and promising at-tempt to supply the particle equations with an adequate renderingof the flow field by capturing the relevant physics involved in theparticle deposition process.

5. Conclusions

In this paper, we have reviewed the physical mechanismsresponsible for deposition and entrainment of inertial particles inturbulent dispersed flow and have provided modelling perspec-tives for the numerical simulation of such processes. Interpretativemodels for the deposition and entrainment mechanisms were pro-posed and discussed on the basis of direct numerical simulations ofturbulence and Lagrangian tracking of inertial particles. Simula-tions were performed adopting the pointwise particle approachand assuming that the interaction between the particle and thesurrounding fluid is represented through a force located at the po-sition of the particle center. This keeps the level of modelling to aminimum, provided that the inter-particle distance is large andparticles are small compared with the smallest relevant flowscales. Even if the point-particle approach introduces some limita-tions to the modelling capabilities of the numerical methodology,and even if particle modelling is the simplest possible, simulationsprovide a fully representative three-dimensional and time-depen-dent description of the phenomena under investigation in the limitof small particles and dilute flow conditions.

Through the systematic application of this numerical methodol-ogy to a well-known archetypal instance of wall-bounded flow(particle-laden channel flow), we have provided a detailed viewof the particle transfer mechanisms in turbulent boundary layer.From this perspective, the role of the near-wall coherent structureson each space and time scale involved, as well as the dispersion ofparticles characterized by different values of the inertia parameterhas been considered carefully. In particular, we have pinpointedthat particle deposition to a wall is quintessentially linked to par-ticle accumulation into specific regions in the buffer layer. Thesefindings are of particular importance for the improvement of exist-ing deposition models and for the implementation of simulationtechniques of broader engineering significance, such as LES. Resultsindicate that, since LES prediction of local segregation is inaccurate,also inevitably inaccurate will be the macroscale prediction of par-ticle deposition and overall distribution. Thus, current capabilitiesof LES in Eulerian–Lagrangian studies of dispersed flows arestrongly limited by the modelling of the subgrid scale turbulenceeffects on particle dynamics. These effects should be taken into ac-count in order to reproduce accurately the physics of particle dis-persion since the LES cut-off filter removes not only energy but,most important, flow structures from the turbulent flow field. Thisobservation leads to the conclusion that recovering the subgrid en-ergy by reconstruction of the correct amount of fluid and particlevelocity fluctuations is not enough to reproduce the effect of SGSturbulence on particle near-wall accumulation.

g of turbulent particle deposition and entrainment: Review of a systematic16

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Acknowledgements

The work presented in this paper has required a vast amount ofsupport of all kinds. In particular, we are grateful to the followinginstitutions for funding over the years: the Italian Ministry of Re-search (under PRIN and FIRB programs), the Italian Ministry of Pro-ductive Activities, and the Regional Authority of Friuli VeneziaGiulia. C.M. also thanks the University of Udine for financial sup-port. CINECA supercomputing center (Bologna, Italy) and HPCEuropa Transnational Access Program are also gratefully acknowl-edged for generous allowance of computer resources.

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Physics and modelling of turbulent particle deposition and entrainment: Review of a systematic studyIntroductionPhysical modelling and numerical methodologyEquations for the fluid phase and flow solverEquations for the dispersed phase and Lagrangian particle trackingDatabase and repositories

DiscussionPhenomenology of particle deposition, entrainment and trapping phenomenaStatistical description of particle trapping and resuspension mechanismsStatistical description of particle preferential distribution, segregation and depositionQuantification of particle preferential distribution in the viscous sublayerQuantification of particle segregationInfluence of the Reynolds number on particle segregationQuantification of particle deposition rates

Modelling perspectives for Large-Eddy Simulation of turbulent dispersed flowsConclusionsAcknowledgementsReferences