Development of Gas-Particle Euler-Euler LES Approach: A Priori Analysis of Particle Sub-Grid Models...

Flow Turbulence Combust (2010) 84:295–324DOI 10.1007/s10494-009-9233-z

Development of Gas-Particle Euler-Euler LESApproach: A Priori Analysis of Particle Sub-GridModels in Homogeneous Isotropic Turbulence

Mathieu Moreau · Olivier Simonin · Benoît Bédat

Received: 11 December 2007 / Accepted: 29 July 2009 / Published online: 25 September 2009© Springer Science + Business Media B.V. 2009

Abstract A new large eddy simulation (LES) approach for particle-laden turbulentflows in the framework of the Eulerian formalism for inertial particle statisticalmodelling is developed. Local instantaneous Eulerian equations for the particlecloud are first written using the mesoscopic Eulerian formalism (MEF) proposed byFévrier et al. (J Fluid Mech 533:1–46, 2005), which accounts for the contribution ofan uncorrelated velocity component for inertial particles with relaxation time largerthan the Kolmogorov time scale. Second, particle LES equations are obtained byvolume filtering the mesoscopic Eulerian ones. In such an approach, the particulateflow at larger scales than the filter width is recovered while sub-grid effects needto be modelled. Particle eddy-viscosity, scale similarity and mixed sub-grid stress(SGS) models derived from fluid compressible turbulence SGS models are presented.Evaluation of such models is performed using three sets of particle Lagrangianresults computed from discrete particle simulation (DPS) coupled with fluid directnumerical simulation (DNS) of homogeneous isotropic decaying turbulence. The twophase flow regime corresponds to the dilute one where two-way coupling and inter-particle collisions are not considered. The different particle Stokes number (basedon Kolmogorov time scale) are initially equal to 1, 2.2 and 5.1. The mesoscopicfield properties are analysed in detail by considering the particle velocity probabilityfunction (PDF), correlated velocity power spectra and random uncorrelated velocitymoments. The mesoscopic fields measured from DPS+DNS are then filtered to

M. Moreau · O. Simonin · B. Bédat (B)INPT, UPS, IMFT (Institut de Mécanique des Fluides de Toulouse), Université de Toulouse,Allée Camille Soula, 31400 Toulouse, Francee-mail: [email protected]

M. Moreau · O. Simonin · B. BédatCNRS, IMFT, 31400 Toulouse, France

296 Flow Turbulence Combust (2010) 84:295–324

obtain large scale fields. A priori evaluation of particle sub-grid stress models givescomparable agreement than for fluid compressible turbulence models. It has beenfound that the standard Smagorinsky eddy-viscosity model exhibits the smaller corre-lation coefficients, the scale similarity model shows very good correlation coefficientbut strongly underestimates the sub-grid dissipation and the mixed model is on thewhole superior to pure eddy-viscosity model.

Keywords Gas-particle flow · Turbulence · Large Eddy simulation

Nomenclature

F(k)p force acting on the particle k

ε f fluid turbulent kinetic energy dissipationηK Kolmogorov length scaleL simulation box lengthLt integral length scalenp particle number densityNp total number of particlesmp particle massq2

f fluid turbulent kinetic energyq fp fluid particle covarianceq fp filtered fluid particle covarianceq2

p mean particle kinetic energyq2

p mean particle correlated kinetic energyq2

p filtered mean particle correlated kinetic energyδq2

p mean particle uncorrelated kinetic energyqp,SGS sub-grid particle correlated kinetic energyRet turbulent Reynolds numberSt Stokes number based on Lagrangian time scale (τp/τ

tf )

StK Stokes number based on Kolmogorov time scale (τp/τK)τ t

f Lagrangian turbulent timeτK Kolmogorov time scaleτp particle relaxation timeδθp random uncorrelated energy, RUEup particle correlated velocityup filtered particle correlated velocityV(k) velocity of particle kδV(k) random uncorrelated velocityX(k) position of particle kδRp,ij particle uncorrelated velocity second order momentδQp,m particle uncorrelated velocity third order momentSp,ij correlated strain rate tensorTp,ij particle sub-grid stress tensor�p particle sub-grid production of RUEQp,m particle sub-grid diffusion of RUE

Flow Turbulence Combust (2010) 84:295–324 297

Acronym

DNS direct numerical simulationDPS discrete particle simulationHIT homogeneous isotropic turbulenceLES large eddy simulationMVF mesoscopic velocity fieldRUE random uncorrelated energyRUV random uncorrelated velocitySGS sub-grid stress

1 Introduction

Particle or droplet-laden turbulent flows are an inherent part of industrial processessuch as fluidized bed in chemical engineering or spray atomisation in aeronautic orautomotive engines. Different numerical approaches exists to simulate such flows. Inparticular, Direct numerical simulations (DNS) of turbulent fluid flows coupled withdiscrete particle simulation (DPS) are extensively used to study dilute gas-particleturbulent flows [2–5]. In such an approach, Lagrangian particle equations for thecenter mass position X(k)

i and velocity V(k)

i of each k−particle are written:

dX(k)

i

dt= V(k)

i (1)

dV(k)

i

dt= − F(k)

p,i

mp(2)

where F(k)

p,i the force acting on the k−particle of mass mp. The fluid phase follows theNavier-Stokes equations with extra terms accounting for interphase transfers [6, 7].In DNS, all the spatial and time scales of the turbulence have to be computed sosuch an approach is restricted to low Reynolds number flows and simple geometries.These restrictions are overcome in large eddy simulation (LES) where only theflow at large scales is computed while the effect of the unresolved eddies has to bemodelled. Several DPS+LES studies exist of particle laden homogeneous isotropicflows [7–9], channel flows [10] or sprays. Works have also been devoted to thedevelopment of fluid sub-grid models accounting for the particle presence [11] andto the study of turbulence small scale effect on particle statistical properties [12].

However, due to the huge number of particle involved, such Euler-Lagrange ap-proach is very difficult to handle in industrial configurations. A numerical simulationapproach based on Eulerian transport equations for dispersed phase and Navier-Stokes equations for the fluid, coupled through interphase exchange terms mightbe an effective alternative. Dispersed phase Eulerian equations can be obtained bydifferent ways. Druzhinin [13] use a spatial averaging of the instantaneous dispersedphase equations over a scale of the order of Kolmogorov length scale, assumed tobe much larger than particle diameter and inter-particle distance, to derive Euleriantransport equations for the dispersed phase concentration and velocity. Computa-tions of Eulerian fields were performed for bubbles and particles with relaxation


time much smaller than Kolmogorov time scale [14, 15]. By a Taylor expansion of theparticle velocity in term of small response time, the particle velocity can be expressedin term of fluid velocity time and spatial derivative quantities [13, 16]. Ferry andBalachandar [17] have extended this particle equilibrium approach to account foradded mass, Saffman lift and Basset history forces. Validation from Lagrangianquantities was performed in homogeneous isotropic turbulence (HIT), homogeneousturbulent shear flow and in channel configuration [18, 19]. Recently, the equilibriumapproach was extended to LES [20]. The equilibrium approach is well adapted forsmall particle relaxation time, but fails with increasing particle inertia [21].

Moreover, a crucial assumption of equilibrium model, the particle velocity distri-bution uniqueness, is not verified for particle Stokes number based on Kolmogorovtime scale greater than unity [17]. As shown by Fevrier et al. [1], in the case of inertialparticles with response time larger than the Kolmogorov time scale, the Eulerianapproach for the dispersed phase must account specifically for the effects of crossingbetween the particle trajectories. By introducing an ensemble averaging over particlephase realizations conditioned by a given fluid turbulent flow realization [1], particlestatistical properties may be derived in the frame of a probability density function(PDF) approach. The discrete particle velocities may be separated into an Eulerianfield, the mesoscopic velocity field (MVF) and a Lagrangian random distribution, therandom uncorrelated velocity (RUV). The conditioned particle velocity PDF followsa Boltzmann-type kinetic equation accounting for forces acting on particles andinterparticle collisions. The moments of the PDF are mesoscopic Eulerian quantitieswhich obey transport equations obtained by integration of the Boltzmann-type equa-tion. Recently, Kaufmann et al. [22] have computed an Eulerian simulation with theparticle number density, correlated velocity and random uncorrelated energy (RUE)coupled with standard Navier-Stokes equations for the fluid and compared particleEulerian quantities with DNS+DPS results in decaying HIT. They obtained a verygood agreement between the mesoscopic Eulerian and Lagrangian predictions forvery small particle Stokes number and found a negligible RUE effect. In contrast, forincreasing particle inertia, the dispersion effect induces by the RUE contribution inthe momentum equation becomes crucial for the accurate prediction of preferentialconcentration.

As shown in the present paper, the dispersed phase moment equation derived inthe frame of the MEF may be filtered to obtain LES-type transport equations withsub-grid terms which need to be modeled. Therefore, particle sub-grid stress termmodels are proposed and evaluated by performing a priori test.

Since the pioneer work of Clark et al. [23], a priori estimation of sub-grid modelshas been widely used in monophasic fluid flows: a DNS is performed, then thesefields are spacially filtered leading to the large scale fields. The sub-grid modelsare calculated using the large scale quantities and compared to the exact sub-gridquantities. In order to apply such an approach to the dispersed phase, local instanta-neous mesoscopic Eulerian fields are measured from particle Lagrangian simulationscoupled to fluid DNS. Then these fields are filtered and a priori estimation of sub-gridmodels is carried out.

A DNS+DPS database for particles suspended in a decaying homogeneousisotropic turbulent is generated. The computational domain is a cubic box of lengthL = 2π discretized into 643 grid points. Flow parameters are listen in Table 1. Theonly force acting on the particle is the Stokes drag (particle relaxation time τp).


Table 1 Dimensionless fluidflow parameters at particleinjection time (t = 0)

Turbulent kinetic energy q2f 0 0, 0107

Energy dissipation rate ε f 0 7, 51.10−4

Integral length scale Lt0 0.804Eddy turnover time Tu0 = Lt0√

23 q2

f 0

9.51

Kolmogorov length scale ηK0 0.0493Kolmogorov time scale τK0 1.48Lagrangian turbulent time [24] τ t

f 0 6.87

Turbulent Reynolds number Ret0 = (Lt0ηK0

)43 40.2

Two-way coupling and interparticle collisions are not taken into account in thepresent study. Three two-phase flow configurations with differents particle Stokesnumbers are simulated, St0

K = 1, 2.2 and 5.1 based on the initial Kolmogorov timescale or St0 = 0.21, 0.47, 1.1 based on a Lagrangian turbulent time macro-scale τ t

f

[24] and corresponding to regimes where particle prefential concentration and RUVeffects are not negligible. Lagrangian particles were randomly embedded in a devel-oped fluid turbulence field with a velocity equal to the fluid one at their position. Atthe end of simulation after 3.5 initial macro-scale turbulent time, StK and St are equalto 0.37 and 0.7 times the initial Stokes number (Fig. 1), respectively. Without collisionand two-way coupling effects, performing one simulation with a large number ofLagrangian particles is equivalent to simulate several different particle realizationsof fewer particles for the same fluid turbulent flow realization. Moreover, a particlenumber per cell of approximately 305 is simulated (Np = 8 × 107 particles for eachcase) to insure the accuracy of particle local statistic computation. The needed verylarge number of particle per cell forced to simulate low Reynolds number casesonly. A run cost approximately 1100 cpu hours on IBM-Power4 supercomputer, Thebackup file size is about 10GB per time step. MPI-parallelized numerical code usessixth order spectral-like compact scheme on carthesian grid [25], third order Runge

Fig. 1 Time evolutions of theStokes numbers during thesimulation. StK/St0K : solid line;St/St0: dashed line

0 1 2 3 4

t/τf

t(t=0)

0

0.2

0.4

0.6

0.8

1

1.2

Sto

kes(

t)/in

itial

Sto

kes


Kutta time stepping [26] and third order Lagrangian polynom interpolation scheme[27].

In Section 2, the local instantaneous particle Eulerian equations [1] are recalledand the mesoscopic quantities are evaluated using the DNS+DPS database. Then inSection 3, the particle Eulerian LES equations are derived and unclosed terms aremodelled by analogy with standard fluid compressible turbulence approach. Subgridmodels for the particle sub-grid tensor and a priori testing results are presented inSection 4.

2 Local Instantaneous Particle Eulerian Equations

2.1 Mesoscopic equations

Following Février et al. [1], let us introduce fp(cp, x, t|H f ), the particle probabilitydensity function for a given fluid flow realization H f . fp(cp, x, t|H f )dcp is the meannumber of particles at time t, with mass center position x and having a translationvelocity V(k) ∈ [cp, cp + dcp] for the considered fluid flow realization H f . Thus, fp

obeys the following Boltzmann like kinetic transport equation:

∂

∂tfp + ∂

∂x jcp, j fp + ∂

∂cp, j

Fp, j

mpfp =

( ∂

∂tfp

)coll

(3)

where Fp is the force acting on a particle of mass mp and the rhs. term is the PDFmodification by inter-particle collisions. Considering spherical particle with diameterdp smaller than the Kolmogorov length scale (dp/ηK0 ≈ 5 × 10−3, here) and highparticle to fluid density ratio (ρ f /ρp � 1) without gravity effect, the particle motionis governed by the drag force [28, 29] which can be taken as the Stokes drag force forthe small particle Reynolds number values considered here (< 0.03):

Fp,i = −mp

τp

(cp,i − u f @p,i

)with τp = ρpd2

p

18μ f(4)

where u f @p is the fluid velocity “viewed” by the particle. For sufficiently dilute flows,the collision effect is negligible, (3) is then closed and is written:

∂

∂tfp + ∂

∂x jcp, j fp = ∂

∂cp, j

(cp, j − u f @p, j

τpfp

)(5)

Particle mesoscopic Eulerian quantities such as number density np, correlated veloc-ity up,i, and random uncorrelated velocity (RUV) second order moment δRp,ij aregiven by integration of the conditioned PDF fp over particle velocity space:

np(x, t|H f

) =∫

fp(cp, x, t|H f

)dcp (6)

up,i(x, t|H f

) = 1

np

∫cp,i fp

(cp, x, t|H f

)dcp (7)

δRp,ij(x, t|H f

) = 1

np

∫ (cp,i − up,i

) (cp, j − up, j

)fp

(cp, x, t|H f

)dcp (8)


For sake of simplicity, the notation |H f is forgotten in the rest of the paper.The random uncorrelated energy (RUE) δθp is defined as half the trace of δRp,ij andδR∗

p,ij is the deviatoric of the RUV second order moment:

δθp = 1

2δRp,ii and δR∗

p,ij = δRp,ij − 2

3δθpδij (9)

Transport equations of particle number density, correlated momentum and RUE arederived by integration of the Boltzmann-type equation (5):

∂

∂tnp + ∂

∂x jnpup, j = 0 (10)

∂

∂tnpup,i + ∂

∂x jnpup, jup,i = np

τp(u f @p,i − up,i) − ∂

∂x jnp

(2

3δθpδij + δR∗

p,ij

)(11)

∂

∂tnpδθp + ∂

∂x jnpup, jδθp = −2

np

τpδθp − np

(2

3δθpδij + δR∗

p,ij

)∂up,i

∂x j

− ∂

∂x jnpδQp, j (12)

where δQp, j is the third order moment of the RUV:

δQp, j = 1

2np

∫ (cp, j − up, j

) (cp,i − up,i

) (cp,i − up,i

)fp

(cp, x, t

)dcp (13)

The first terms of the rhs of (11) and (12) take into account the drag force effects,and the last terms represent the effect of uncorrelated velocity moment and needto be modelled. According to Février et al. [1], in homogeneous turbulence, theparticle-fluid interaction generates particle correlated motion, schematically. Then,correlated kinetic energy is transfered to RUE through δR∗

p,ij term to be dissipatedby drag. In addition, Février et al. [1] pointed out that the RUE acts like a pressureterm, Pp = 2/3npδθp, in the particle correlated momentum equation and inducesdispersion of particles which limits the preferential concentration effect.

2.2 Measurement of particle Eulerian quantities from DNS+DPS results

Instantaneous Lagrangian dispersed phase quantities, particle position X(k)

i (t) andvelocity V(k)

i (t) are post-processed to obtain continuous Eulerian fields of meso-scopic quantities. Particle mesoscopic number density and velocity are calculated byprojection in the framework of the particle in cell (PIC) approximation from theLagrangian position to a carthesian regular grid of node spacing �P (see Kaufmannet al. [30]). Mesoscopic Eulerian quantities are computed from Lagrangian resultsby:

np(x, t) =∫

H�′P(x − x′)δ(x′ − X(k)(t))d3x′ (14)

np(x, t)up,i(x, t) =∫

H�′P(x − x′)δ(x′ − X(k)(t))V(k)

i (t)d3x′ (15)


Where �′P is the characteristic length of the projector H�′

Pand is chosen smaller

than the fluid Kolmogorov length scale. Numerical tests [22] show that projectionerrors are due to the finite particle per cell number (statistical bias) and spatialdiscretisation. To limit these effects, the truncated Gaussian projector kernel is used:

H�′P(x − x′) = H

(�P

2− |x − x′|

)1

erf(√

6)3

(6

π�P

) 32

exp

(− 6|x − x′|2

�P

)(16)

where H(x) is the Heaviside function. The characteristic length of the so-definedGaussian projector is approximately the projecting node spacing (�′

P ≈ �P). Thecorrelated velocity field up is then interpolated at each particle location and the RUVcomponents for each particle is obtained by subtraction:

δV(k)

i (t) = V(k)

i (t) − up,i(X(k)(t), t) (17)

So the RUV second order moment field is finally computed by projection on theEulerian mesh:

np(x, t)δRp,ij(x, t) =∫

H�(x − x′)δ(

x′ − X(k)p

)δV(k)

i δV(k)

j d3x′ (18)

In the following, the Lagrangian properties measured at different simulation timesare projected on a 643 cartesian grid. Due to particle segregation effect, it is possiblethat the particle density is zero for few cells (less than 0.5%). In such cases (at theend of simulations St0

K = 1 and 2.2 for example), the correlated velocity is taken tobe the neighboring node average value, the RUV quantities are assumed to be zero.This regularization does not affect the Favre mean quantities and just insure thecontinuity of the correlated velocity field.

The distribution of the Lagrangian and of the correlated velocities is found to beGaussian for all the Stokes number considered as shown on Fig. 2. This characteristichas been already found by Lavieville [8] for the Lagrangian velocities in HIT.For the particle uncorrelated velocities and for the lightest particle, the pdf is nonGaussian and when the Stokes number increases the pdf tends to be Gaussian. Thischaracteristic can be explained by the fact that when particle inertia increases, theparticle velocities will be less and less correlated which will have as consequence tobroadened the PDF.

2.3 Eulerian statistics

The mean correlated and uncorrelated kinetic energies are written:

q2p = 1

2〈up,iup,i〉p (19)

δq2p = 〈δθp〉p (20)

where the brackets 〈.〉p is the Favre spatial average over the computational domainweighted by the particle number density np.The mean particle kinetic energy q2

p =1

2Np

∑Np

k=1 V(k)

j V(k)

j in the Lagrangian framework is equivalently the sum of the


Fig. 2 Particle velocity PDF:full velocities V(k)

i (a),correlated velocities up,i(X(k))

(b) and uncorrelated velocitiesδV(k)

i (c) at time t/τ tf 0 = 1.

St0K = 1: solid line; St0K = 2.2:dotted line; St0K = 5.1: dashedline; Gaussian distribution:circles

(a)

velocity/standard deviation

10−3

10−2

10−1

100

PD

F

(b)


10−3

10−2

10−1

100

PD

F

(c)

−4 −2 0 2 4

−4 −2 0 2 4

−4 −2 0 2 4


10−3

10−2

10−1

100

101

PD

F


correlated and uncorrelated energy in the Mesoscopic Eulerian Formalism [1]. Atthe initialisation, the particle correlated kinetic energy equals the particle total andthe fluid kinetic energies since it has been chosen that the particle velocities equalthe fluid velocity. Another consequence of the initialization is that the mean RUE isequal to zero. Then, as shown by Fig. 3, the RUE, δq2

p, is increasing due to a transferfrom the correlated particle kinetic energy, q2

p, which is decreasing more quickly thanthe mean particle kinetic energy, q2

p. After about one turbulent characteristic timemacroscale, the RUE is also decreasing due to the dissipation by the drag force. Themaximum mean RUE values for the three simulations are 2%, 13% and 38% of themean particle kinetic energy q2

p for the cases St0K = 1, 2.2 and 5.1 respectively. These

results are in quantitative agreement with the ones obtained by Février et al. [1] inforced HIT.

Power spectra of the correlated velocity are presented on Fig. 4 for time t/τ tf 0 = 1.

These spectra are continuous. At large scale, they approximately follow the fluidkinetic energy spectrum, while at smallest scale their level are superior to the fluidone. At large scale, increasing the particle inertia leads to decrease the particlecorrelated power spectrum level while at small scale, the power spectrum levelincreases. This effect is not due to local fluid-particle interaction (fluid energy beingnegligible at small scales), but results from the mesoscopic equation set, which leadsto the formation of (shock-like) particle cluster structures. Moreover, the correlationbetween the fluid and the particle correlated velocities are evaluated by the powerspectra of fluid-particle correlated velocities defined by

E fp(k) = 1

4

∫ ∫ ∫ (uFT

f,i (k)uFT∗p,i + uFT∗

f,i (k)uFTp,i (k)

)δ(k − |k|)dk (21)

where the superscripts .FT and .∗ mean Fourier transform and complex conjugatedquantities. For the smallest particle inertia, the fluid-particle correlated velocitiesspectrum is identical than the fluid one as shown on Fig. 5. By increasing theparticle inertia, deviation from the fluid spectrum is observed at small scales. Thisindicates that the small scale of the correlated velocity fields for heavy particle arenot influenced by the fluid turbulence.

The range of Stokes number considered corresponds to particle preferential con-centration regime. The parameter 〈n2

p〉/〈np〉2, where 〈.〉 stands for spatial averagingover the domain, is used to quantified this phenomenon. At the initialisation, theparticle number density is homogeneous (see Fig. 6) and the parameter equals unity.The particle segregation increases almost with the same rate for the three casesconsidered at beginning of the simulation. It remains increasing for the two ligthercases and does not reach a plateau at the end of the simulation. This plateau isobserved for the heavier case. In forced turbulence, the maximum concentrationis obtained for a Stokes number based on the Kolmogorov scale equals unity [31].In freely decaying turbulence, where turbulence characteristics evoluate with timeand so the Stokes number, the preferential concentration is difficult to analyse sinceparticles do not respond instantaneously to the modification of the velocity fields.

Snapshot of particle number density (grayscale field) and particle correlatedvelocity divergence isocontour lines for case St0

K = 1 at the time t/τ tf 0 = 0.5, where

the fluid turbulence interaction with particles is the most effective, are presented


Fig. 3 Time development ofthe mean fluid turbulentkinetic energy (circle), meanparticle kinetic energy q2

p(solid line), mean correlatedenergy q2

p (dotted line) andmean RUE δq2

p (dashed line)

St0K = 1.

t/τf

t(t=0)

0

0.2

0.4

0.6

0.8

1

ener

gy/q

f2 (t=

0)

St0K = 2.2

0 1 2 3 4

0 1 2 3 4

t/τf

t(t=0)

0

0 1 2 3 4

0.2

0.4

0.6

0.8

1

ener

gy/q

f2 (t=

0)

St0K = 5.1

t/τf

t(t=0)

0

0.2

0.4

0.6

0.8

1

ener

gy/q

f2 (t=

0)


Fig. 4 Power spectra of thefluid velocity (circle) andparticle correlated velocity attime t/τ t

f 0 = 1. St0K = 1: solid

line; St0K = 2.2: dotted line;St0K = 5.1: dashed line

10−1

100

kηK

10−6

10−5

10−4

10−3

10−2

10−1

100

spec

trum

/qf2 (t

=0)

on Fig. 7a. As pointed out by Maxey [16], the particle preferential concentrationis referred to the compressibility of the particle velocity field and the particleconcentration is larger in negative particle correlated velocity divergence regions.Moreover, using the particle equilibrium approach, the particle velocity divergencefield can be predicted in function of fluid velocity derivative for sufficiently lightparticles (StK < 1) in frozen fluid velocity field as:

∂up,k

∂xk≈ I f = −τp

∂u f,i

∂x j

∂u f, j

∂xi(22)

Fig. 5 Power spectra of thefluid velocity (circle) andfluid-particle correlatedvelocities at time t/τ t

f 0 = 1.

St0K = 1: solid line; St0K = 2.2:dotted line; St0K = 5.1: dashedline

10−1

100

kηK

10−6

10−5

10−4

10−3

10−2

10−1

100

spec

trum

/qf2 (t

=0)


Fig. 6 Time development ofthe particle preferentialconcentration parameter〈n2

p〉/〈np〉2. St0K = 1: solid line;St0K = 2.2: dotted line;St0K = 5.1: dashed line

0 1 2 3 4

t/τf

t(t=0)

1

1.5

2

2.5

3

<n p2 >

/<n p>

2

The divergence of the particle velocity field is positive in regions of fluid highvorticity or low strain rate, and negative in regions with large strain rate or low fluidvorticity. On Fig. 7b, a snapshot of the I f function (grayscale field) is compared tothe measured particle divergence field in the case St0

K = 1 at time t/τ tf 0 = 0.5. As

expected, the I f function reaches high value (region with high fluid vorticity valuesare represented by white regions) where the particle correlated velocity divergencefield is positive (dashed line contour plot). The negative divergence regions are alsowell correlated with the distribution of negative values of I f in the case of particleswith low inertia.

As expected since the parameter I f has been developed for light particles, thecorrelation coefficients1 between I f and the correlated particle velocity divergence(displayed on Fig. 8) decreases with particle inertia.

2.4 Model of the random uncorrelated velocity correlation tensor

As proposed by Simonin et al. [33], the deviatoric part of the RUV second ordermoment δRp,ij can be modelled by a Boussinesq hypothese leading to the RUVviscosity model:

δR∗p,ij = −τp

3δθpS∗

p,ij (24)

where S∗p,ij is the trace-free strain rate tensor expressed using the particle correlated

velocity up:

S∗p,ij = ∂up,i

∂x j+ ∂up, j

∂xi− 2

3

∂up,k

∂xkδij (25)

1Correlation coefficients [32] between fields A and B are calculated following:

c(AB) = 〈AB〉 − 〈A〉〈B〉√(〈A2〉 − 〈A〉2)(〈B2〉 − 〈B〉2)

(23)


Fig. 7 Snapshot of particlenumber density (a) and fluidpreferential concentrationindicator function I f τ

tf 0 (see

(22)) (b). The lines are contourplots of the correlated velocitydivergence ( ∂up, j

∂x jτ t

f 0 = 0.34:dashed line and∂up, j∂x j

τ tf 0 = −0.34: solid line).

Case St0K = 1 at timet/τ t

f 0 = 0.5

Such a RUV viscosity model may be derived by assuming that the anisotropy tensor(δR∗

p,ij/δθp) is in transport equilibrium and remains weak. PDF of the measured

RUV dissipation (δR∗p,ij

∂up,i∂x j

) and of the model predictions are shown for the threecases at time t/τ t

f 0 = 1 on Fig. 9. The measured term is mainly negative (0.3%, 3%and 25% of positive values in the case St0

K = 1, 2.2 and 5.1 respectively). The viscositymodel is able to predict accurately the RUV dissipation PDF shape in the casesSt0

K = 1 and 2.2, but is can not, by construction, take into account the positive valuesof the term for the heaviest particle case (case St0

K = 5.1).


Fig. 8 Time development ofcorrelation coefficient betweenI f (see (22)) and thedivergence of the particlecorrelated velocity field.St0K = 1: solid line; St0K = 2.2:dotted line; St0K = 5.1: dashedline

0 1 2 3 4

t/τf

t(t=0)

−0.2

0

0.2

0.4

0.6

0.8

corr

elat

ion

The correlation between the exact and modelled RUV dissipation, referred asdissipation correlation coefficient are displayed on Fig. 10. The dissipation corre-lation coefficients decrease with particle inertia. For the cases St0

K = 1, 2.2, thecorrelation values are almost constant during the simulation and are superior to 0.8.In contrast, for the case St0

K = 5.1, the correlation coefficient is not constant duringthe simulation. The maximum error is reached approximately at the time 1.5 and lateron the correlation coefficient increases. This behavior is explained by the decrease ofthe Stokes number during the simulation due to the absence of forcing. These resultsshow the abilities of the viscosity model to predict the deviatoric part of the RUVsecond moment when the ratio between the RUE and the correlated energy is smallbut not negligible. Third order RUV correlations are modelled by a diffusion term[34] similar to the temperature Fick law.

npδQp,m = −κRUV∂

∂xmδθp (26)

where κRUV is calculated by analogy with the RANS two-fluid approach (κRUV =1027τpnpδθp).

3 Filtered Particle Mesoscopic Eulerian Equations

3.1 General equation

In the frame of LES approach, the general definition of a filtering process is written:

φ(x) =∫

G�F (x − x′)φ(x′)dx′ (27)

where G�F is the filter kernel of characteristic length �F and the integral is overthe flow domain. The filtering process is linear and conserve the mean quantities.The particle number density is not homogeneous for some regimes (e. g. preferential


Fig. 9 PDF of RUVdissipation term (δR∗

p,ij∂up,i∂x j

)measured from DNS+DPS(solid line) and predictionsusing the RUV viscosity modelassumption (dashed line) givenby (24) at time t/τ t

f 0 = 1

St0K = 1

−0.4 −0.2 0 0.2RUV Dissipation

10−4

10−3

10−2

10−1

100

101

102

PD

F

St0K = 2.2

−1 −0.5 0 0.5 1RUV Dissipation

10−3

10−2

10−1

100

101

102

PD

F

St0K = 5.1

−2 −1 10 2RUV Dissipation

10−3

10−2

10−1

100

101

102

PD

F


Fig. 10 Contractedcorrelation coefficientsbetween δR∗

p,ij∂up,i∂x j

measuredfrom DNS+DPS and RUVviscosity model assumption(see (24)). St0K = 1: solid line;St0K = 2.2: dotted line;St0K = 5.1: dashed line

0 1 2 3 4

t/τf

t(t=0)

0

0.2

0.4

0.6

0.8

1

corr

elat

ion

coef

ficie

nt

concentration regime), so mesoscopic transport equations are written in term ofFavre filtered (or particle number density-weighted) quantities (denoted by an hat):φp = npφp/np. Applying the spatial filter G� f to the governing equations (10), (11)and (12) leads to the particle filtered mesoscopic equations:

∂

∂tnp + ∂

∂x jnpup, j = 0 (28)

∂

∂tnpup,i + ∂

∂x jnpup, jup,i = np

τp(u f @p,i − up,i)

− ∂

∂x jnp

(2

3δθ pδij + δR

∗p,ij + Tp,ij

)(29)

∂

∂tnpδθ p + ∂

∂x jnpup, jδθ p = −2

np

τpδθ p − np

(2

3δθ pδij + δR

∗p,ij

)∂up,i

∂x j+ np�p

− ∂

∂x jnp

(δQp, j + Qp, j

)(30)

where

Tp,ij = up,iup, j − up,iup, j (31)

�p = −(

δRp,ij∂up,i

∂x j− δRp,ij

∂up,i

∂x j

)(32)

Qp, j = up, jδθp − up, jδθ p (33)


The sub-grid stress (SGS) tensor Tp,ij represents the effect of the unresolvedmesoscopic velocity field on the predicted one, �p is the sub-grid production andQp,m the sub-grid diffusion of the particle RUE respectively. These sub-grid termshave to be modelled in term of computed variables.

One main specificity of the modelling approach is that two different contributionsto the filtered mesoscopic momentum equation (29) are present. Filtered RUV termsaccount for uncorrelated particle velocity effects and is analogous to the filteredpressure and to the viscous laminar contributions in the fluid LES equation. Thesecontributions are expected to slightly depend on the filtering length scale �F . Incontrast, the correlated sub-grid contribution is analogous to the one of the fluidLES approach and will strongly depend on �F . This approach differs from theone proposed by Pandya and Mashayek [35], where LES particle equations areobtained by direct volume filtering of the kinetic equation for single realization ofthe particulate flow, the sub-grid term contains implicitely both RUV and correlatedSGS contributions.

3.2 Magnitude of terms in particle LES equations

In this work, a spherical top hat filter in physical space is used:

G�F (x − x′) = 143π

(�F2

)3 H(

�F

2− |x − x′|

)(34)

This filter is chosen for its simplicity and the fact than the filtered equations are stillGalilean invariant. The filter characteristic length scale must be larger than the pro-jection one �P. Particle mesoscopic Eulerian fields are filtered using four differentlength scales �F = 2�P, 4�P, 6�P and 8�P. The filtered quantities are measuredon the particle projection grid. The SGS stresses and models are computed usinginstantaneous particle mesoscopic data. The considered flows being homogeneous,averaging is performed over the computational domain.

The magnitude of the mean filtered RUE, 〈δθp〉, and sub-grid correlated energies,qp,SGS = 1/2Tp,ii are compared for the different filter length scales at time t/τ t

f 0 = 1(Fig. 11). The behavior differs largely between these two quantities. As expected, thesub-grid energy increases almost linearly with the filter length. In contrast, the levelof the filtered RUE is almost constant because it corresponds to filtered quantitiesinstead to residue of the filtering. Another point to highlight is that the levels of thesetwo quantities are of the same order of magnitude, and in term of modeling, none ofthem can be neglected.

Instantaneous filtered RUE and sub-grid energy fields are shown (Fig. 12) for thecase StK = 2.2 at time t/τ t

f 0 = 1, and a filtering length scale of �F = 4�P. This caseis chosen because the mean RUE and mean SG energy are almost equal. Contourlines of the filtered number density field for np/〈np〉 = 1 and 1.5 are drawn on thefigure to show how the different energies are distributed with respect to the particleclusters. The RUE is mainly located in high particle number density regions, whilethe sub-grid energy is mainly at the interface of the clusters. The uncorrelated andthe sub-grid correlated energies have different behavior and need different physical-based modelling approach.


Fig. 11 Comparison betweenmean filtered RUE (dashedline) and particle correlatedsub-grid energy (solid line) fordifferent filter length scales �Fat time t/τ t

f 0 = 1 obtained for

particle inertia: a St0K = 1.,b St0K = 2.2, and c St0K = 5.1

(a)

2 4 6 8ΔF/ΔP

2 4 6 8ΔF/ΔP

2 4 6 8ΔF/ΔP

0

0.1

0.2

0.3

0.4

ener

gy/q

p2 (t)

(b)

0

0.1

0.2

0.3

0.4

ener

gy/q

p2 (t)

(c)

0

0.1

0.2

0.3

0.4

ener

gy/q

p2 (t)


Fig. 12 Snapshot of particleRUE (a) and sub-grid energy(b). Value arenondimentionalized by theirmean values. The lines arecontour plots of the filterednumber density fornp/〈np〉 = 1: solid line andnp/〈np〉 = 1.5: dashed line.Case St0K = 2.2 at timet/τ t

f 0 = 1, �F = 4�P

The mean filtered correlated energy (q2p = 1

2 〈up, jup, j〉 p) budget is obtained bymultiplying (29) by up,i and averaging over the computational domain. Using theperiodic boundary condition, this budget equation reads

ddt

q2p = − 1

τp

(2q2

p − q fp

)+

⟨δRp,ij

∂up,i

∂x j

⟩

p

+⟨Tp,ij

∂up,i

∂x j

⟩

p

(35)

where q fp is the fluid-particle large scale covariance: q fp = 〈u f, jup, j〉 p, where 〈.〉 p isthe spatial average over the computational domain weighted by np. The first term ofthe RHS is the drag force effect at large scales, the second and the third terms are thefiltered RUV and SGS “dissipations” respectively. The mean pressure-like dilatation


Fig. 13 Particle filteredcorrelated energy budget.dq2

p/dt: solid line; drag term:circle; RUV dissipation:dashed line and SGSdissipation: dotted line. CaseSt0K = 5.1, �F = 4�P

0 1 2 3 4

t/τf

t(t=0)

−0.8

−0.6

−0.4

−0.2

0

0.2

term

/(q f2 (t

=0)

/τft (t

=0)

)

terms remained at least one order of magnitude smaller than the dissipation inducedby the deviatoric part of the RUV second order moment, so this term is included inthe “dissipation” term definitions for sake of simplicity. The time evolutions of thedifferent terms of the budget are shown on Fig. 13 for the case St0

K = 5.1 and a filterlength �F = 4�P. The mean budget is closed with an error of 5% of the maximumterm. At the beginning of the simulation t/τ t

f 0 < 1, the two-phase flow evoluates fromthe initial condition, where the correlated kinetic energy, q2

p equals the fluid kineticenergy q2

f , mainly under the action of the RUV and SGS dissipation. Later on, q2p is

produced by the drag force and always dissipated by both RUV and SGS effects.The relative importance of both dissipations found previously depends of the filter

size. The RUV and SGS dissipation value function of the filter length, for the threecases considered, are compared at the time t/τ t

f 0 = 1 on Fig. 14. These terms are ofsame order in the three different cases and need to be accounted for simultaneously.Nevertheless, the SGS term is preponderant one for the lightest particle while theopposite is found for the heavier particle. Being of the same order in the three cases,the two terms need to be accounted for. The large scale RUV dissipation decreasesby increasing the filter length scale indicating that some RUV dissipation occurs atsmall scale. For the largest considered filter, the SGS dissipation decreases, probablybecause �F is comparable to the energetic length scale.

4 Particle Sub-grid Stress Tensor Models

4.1 Models

In this section, three widely used SGS models for fluid compressible flows areadapted to particle mesoscopic Eulerian LES: two alternatives of the Smagorinsky


Fig. 14 Comparison betweenRUV dissipation (dashedlines) and SGS dissipation(dotted lines) for different filterlength scales �F at timet/τ t

f 0 = 1 obtained for particle

inertia: a St0K = 1., bSt0K = 2.2, and c St0K = 5.1

(a)

ΔF/ΔP

ΔF/ΔP

ΔF/ΔP

0

0.01

0.02

0.03

diss

ipat

ion/

(−q p2 (t

)/τ p)

(b)

0

0.02

0.04

0.06

0.08

diss

ipat

ion/

(−q p2 (t

)/τ p)

(c)

2 4 6 8

2 4 6 8

2 4 6 8

0.05

0.1

0.15

0.2

diss

ipat

ion/

(−q p2 (t

)/τ p)


model, a scale similarity model and a mixed model. The first model is a compressibleversion of the well-known Smagorinsky model [36] given by

T∗p,ij = −C2

S�2F |S∗

p|S∗p,ij (36)

with S∗p,ij the trace-free strain rate tensor of the filtered correlated velocity, its norm

is defined by |S∗p|2 = 1

2 S∗ij S

∗ij. We can notice, that, in a similar way, Shotorban and

Balachandar [19], in the framework of the particle equilibrium approach, have sug-gested to model the particle sub-grid scale effect by a Smagorinsky model expressedin term of fluid velocity field leading to the so-called fluid Smagorinsky model:

T∗p,ij = −CS�

2F |S f |S f,ij (37)

where S f,ij is the filtered fluid strain rate tensor.In incompressible single phase flow, the trace of the SGS tensor is not modelled

but in fact is incorporated into the filtered pressure. In the context of dispersed phase,the trace of Tp,ij will have the same dispersive characteristic of the trace of δRp,ij andits modelling will be crucial to predict particle segregation.

To complete the modelling of SGS tensor, when using Smagorinsky model for thedeviatoric part, a Yoshizawa model may be adapted to the dispersed phase:

qp,SGS = CY�2F |S∗

p|2 (38)

In the three simulation cases, the mean filtered RUE is of the same order than themean sub-grid energy (Fig. 11), indicating that both terms have to be modelled.Moreover, δθ p and qp,SGS are not located in the same flow regions (Fig. 12), sothe sub-grid energy could not a priori been neglected and modelled by the sameapproach in particle LES equations.

In the scale similarity model, in term of a “refiltered” velocity field, the SGS tensoris written:

Tp,ij = CB

(up,iup, j − up,i up, j

)(39)

A value of CB = 1 is necessary to insure Galilean invariance [37]. By definition thesimilarity model only accounts for the contribution of the filtered variables to thesub-grid stresses. In fluid flows, this model is theoriticaly able to take into accountbackscatter effect, ie. inverse cascade of energy from small to larger scales but it isknown to provide “high correlation” with the exact SGS tensor but to underestimatethe dissipation leading to unstable LES computation.

This drawback of the scale similarity model can be fixed by using the mixed modelwhich is accurate to predict the interscale transport effect and the dissipation. Thismodel has been introduced in fluid compressible flows by Erlebacher et al. [38]. Inthe framework of the particle mesoscopic Eulerian LES, this model is written as

Tp,ij = up,iup, j − up,i up, j − C′2S �2

F |S∗p|S∗

p,ij +2

3C′

Y�2F |S∗

p|2δij (40)


Speziale et al. [39], using a-priori test of weak compressible HIT propose to neglectthe Yoshizawa part of the mixed model (C′

Y = 0). Due to the high compressibility ofthe particle cloud found, this term can not be neglected.

4.2 A priori tests

The proposed models are tested using particle mesoscopic Eulerian fields obtainedby projection of the Lagrangian results. It is known [40, 41] that a priori tests arenot sufficient to validate the performance of SGS model in true LES computation.However, they are a first step in the appraisal of SGS models.

Test results are only shown for the case St0K = 5.1, where the correlated kinetic

spectrum presents a constant decrease rate of approximatly k−2 from k = 6 to k = 32.Despite the lack of comprehension of energy transfert in such particulate system, itis argued that such a constant slope can indicate an “universal” behavior. Filteringtypical wave number is chosen to be in the range of the velocity power spectrumconstant slope.

Correlation between the measured SGS tensor deviatoric part (T∗p,ij) and model

predictions are shown on Fig. 15. At the direct level, fluid or particle Smagorinskymodel performances are equivalent with very low correlation coefficients (< 0.5),while similarity and mixed models provide higher correlations (> 0.8). However,the correlations measured on the contracted product of the tensor with the velocitygradients, are the more representative of the model behavior. Fluid Smagorinskymodel contracted correlation coefficient remains very low (≈ 0.6), while the particleSmagorinsky correlation reaches a higher value of 0.8, which is inferior than theones obtained with scale similarity or mixed model which are slightly over 0.9.Concerning the sub-grid energy (Fig. 16), the Yoshizawa, scale similarity and mixedmodels provide all high direct (sub-grid energy) and contracted (sub-grid energytimes velocity divergence) correlation coefficients with mean values superior to 0.8and 0.9.

PDF of the exact particle correlated sub-grid energy and the sub-grid dissipationare compared to the predictions of particle Smagorinsky+Yoshizawa, similarity andmixed models (Fig. 17). Smagorinsky and mixed model coefficients are chosen tomatch the sub-grid energy and dissipation and the Scale similarity coefficient is takento be equal to unity (CB = 1). The Yoshizawa model overestimates the level of thesub-grid energy at its low values while the scale similarity model underestimates itby approximately a factor 2.2. In contrast, Mixed model predicts quite well the exactdata. The PDF of the measured sub-grid dissipation T∗

p,ij∂up,i

∂x jis mainly negative, but

has some positive values (backscater effect). This phenomenom can not be taken intoaccount by the Smagorinsky model, and is underestimated by both the scale similarity(CB = 1) and the mixed models. The negative part of the sub-grid dissipation PDF isunderestimated by the Smagorinsky and the similarity models while the mixed modelis in very good agreement.

As said previously, the prediction of the preferential concentration effects de-pends of the modelling of the RUV second order moment and of the SGS tensors.If one uses an isotropic viscosity model then the trace of the SGS tensor acts like apressure term and is purely dispersive. The work of Fede and Simonin [12] shows


Fig. 15 Correlationcoefficients between deviatoricSGS tensor T∗

p,ij measuredfrom DNS+DPS and modelpredictions at the direct (a)and contracted (b) levels.Particle eddy-viscosity model:solid line; similarity model:dotted line; mixed model:dashed line; fluideddy-viscosity model: circle.Case St0K = 5.1, �F = 4�P

(a)

0 1 2 3 4

t/τf

t(t=0)

0

0.2

0.4

0.6

0.8

1

corr

elat

ion

(b)

0 1 2 3 4

t/τf

t(t=0)

0

0.2

0.4

0.6

0.8

1

corr

elat

ion

that the particles interaction with sub-grid fluid turbulence can be anti dispersivewhen the particle relaxation time is inferior to the sub-grid fluid time scale. Then onecan expect that isotropic viscosity model can not predict correctly this effect.

Coefficients of the particle Smagorinsky and Yoshizawa models are evaluated bymatching the a priori modeled SGS dissipation and sub-grid energy to the exact ones.They are practically independent of the filtering length scale (Table 2). A valueof CS = 0.16 is found for the Smagorinsky constant. This value is very closed tostandard fluid turbulence values: 0.2 or 0.18 in incompressible HIT [42, 43], 0.13 incompressible mixing layer [44] or 0.1 in channel flow [45]. The Yoshizawa coefficientis evaluated to be approximately 0.051. This value is quite higher than Yoshizawa


Fig. 16 Correlationcoefficients between sub-gridenergy measured fromDNS+DPS and modelpredictions at the direct (a)and contracted (b) levels.Eddy-viscosity model: solidline; similarity model: dottedline; mixed model: dashed line.Case St0K = 5.1, �F = 4�P

(a)

t/τf

t(t=0)

0

0.2

0.4

0.6

0.8

1

corr

elat

ion

(b)

0 1 2 3 4

0 1 2 3 4

t/τf

t(t=0)

0

0.2

0.4

0.6

0.8

1

corr

elat

ion

original proposal (CY = 0.039). As expected, the similarity model is not able toprovide enough dissipation and a value of CB = 2.2 is needed to match the accuratedissipation level but this coefficient value violates the Galilean invariance property.Coefficient values for the Smagorinsky and the Yoshizawa part of the mixed modelare C′

S = 0.12 and C′Y = 0.025, respectively. The C′

S value is 20% larger than thestandard fluid value (C′

S = 0.092). In contrast, C′Y is much larger than the fluid one

(0 ≤ C′Y ≤ 0.0033). From the knowledge of such modelling, it is difficult to stand

that the constant values are universal. It is know that their gaseous counterpartsdepend on the flow configuration. As initially proposed by Moin et al. [41], a moregeneral approach should be to determine dynamically such coefficients using a scalesimilarity-type hypothesis.


Fig. 17 PDF of the sub-gridenergy (a) and sub-grid

dissipation T∗p,ij

∂up,i∂x j

(b). Exactterm: squares; Eddy-viscositymodel: solid line; similaritymodel: dotted line; mixedmodel: dashed line. CaseSt0K = 5.1, �F = 4�P

(a)

10−2

10−1

100

101

subgrid energy/mean subgrid energy

10−2

10−1

100

101

PD

F

(b)

−15 −10 −5 0 5 10

subgrid dissipation/|mean subgrid dissipation|

10−5

10−4

10−3

10−2

10−1

100

101

PD

F

Table 2 Time average model coefficients evaluated for different filtering length scales �F fromsimulation case St0K = 5.1

Model �F = 2�P 4�P 6�P 8�P

Eddy-viscosity CS 0.19 0.16 0.16 0.15CY 0.055 0.043 0.053 0.053

Mixed C′S 0.13 0.12 0.13 0.12

C′Y 0.021 0.020 0.029 0.032


5 Conclusion

In this paper, a new Eulerian LES approach for inertial particles suspended ina turbulent fluid flow is presented. The derivation of the LES equations for theparticulate phase is based on a two step methodology: first a conditioned averageover particle realizations, second a standard LES filtering of the transport equations.Then, the terms to be closed are of two different types: ones arising from theconditioned averaging are modelled by analogy with the kinetic theory of particleflows and other due to spatial filtering are modeled in the frame of fluid compressibleturbulence LES approach. Compressible SGS models (eddy-viscosity, similarity andmixed) extended to particle phase are proposed.

Validation of the approach is done using Lagrangian results of a very large numberof individual particles in decaying HIT interpreted in term of particle mesoscopicEulerian fields such as particle number density, particle correlated velocity andrandom uncorrelated energy (RUE). Three cases with different particle relaxationtimes leading to measurable preferential concentration effects have been performed.

Correlated velocity power spectrum are continuous. At large scales, they moreor less follow the fluid energy spectrum. In contrast, their levels at small scales areupper than the fluid one. Increasing particle inertia leads to a decrease of correlatedvelocity power spectrum level at large scales and an opposite effect at small scales.

A priori evaluation of Smagorinsky, scale similarity and mixed sub-grid stressmodels gives comparable results than for fluid compressible turbulence flows.Smagorinsky model exhibits the smaller correlation coefficients and scale similaritymodel even through very good correlation coefficient is not able to predict the rightsub-grid dissipation, while mixed model is on the whole superior to pure eddy-viscosity model. Moreover, particle Smagorinsky and mixed SGS models coefficientsare of the same order than standard fluid values.

Futhermore, the development of particle mesoscopic SGS models requires highReynolds number data of variety of flows and a posteri test in true LES computa-tion. Two Fluid LES of more complex flows (confined jets) is under investigationand shows encouraging results [46]. The two-fluid mesoscopic LES has also beencoupled to evaporation and combustion models and evaluated in a realistic annularaeronautical gas turbine [47]. Modelling the filtered RUE (filtered temperature-like)sub-grid terms remains also a challenging issue.

Acknowledgements The numerical simulations were performed on the IBM-Power4 supercom-puter using time made available by the Institut du Développement et des Ressources en InformatiqueScientifique (IDRIS). Financial support was provided by the LESSCO2 european community projectNo. NNE5-2001-00495.

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Development of Gas-Particle Euler-Euler LES Approach: A Priori Analysis of Particle Sub-Grid Models...

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