Comparison between Lagrangian and Eulerian
mesoscopic modelling approaches for inertial
particles suspended in decaying isotropic
turbulence
A. Kaufmann a,1 M. Moreau b O. Simonin b J. Helie b,2
aCERFACS, 42 Av. G. Coriolis, 31057 Toulouse, France, [email protected]
bIMFT,UMR 5502 CNRS/INPT/UPS, Toulouse, France, [email protected]
Abstract
The purpose of this paper is to evaluate an unsteady Eulerian approach from a
direct comparison with a Lagrangian approach for the simulation of an ensemble
of non-colliding particles interacting with a decaying homogeneous isotropic turbu-
lence given by DNS. Comparison between both approaches is is carried out for two
different particle inertia from global instantaneous quantities such as particle kinetic
energy, local instantaneous Eulerian quantities such as particle number density and
spectral quantities such as spectral particle kinetic energy. The proposed Eulerian
approach yields instantaneous three dimensional velocity and number density fields
which are in very good agreement, at large scales, with the discrete particle simu-
lation results. However, appreciable discrepancies between the two approaches are
measured at small scales for the particles with larger inertia. Indeed, for Stokes
number value close to unity, segregation effects are very important and due to the
stiff gradients in number density the Eulerian method is limited by spatial grid res-
Preprint submitted to J. Comp. Physics 15 May 2006
olution. To overcome this numerical difficulty, a simulation approach using a bulk
viscosity is proposed.
Key words: particle laden flows, eulerian methods, preferential concentration,
compressibility
1 Introduction
A true direct numerical simulation (DNS) of dispersed two phase flows includes
the simulation of the carrier phase around each single particle (or droplet),
taking into account its deformation, to compute exactly the forces of the car-
rier phase acting upon it. Unfortunately such true DNS are today limited to
few particles [1,2]. When studying collective physical phenomena such as par-
ticle segregation or turbulence modulation occurring in industrial applications
or environmental flows, the full particle number cannot be accounted for and
requires approximations to limit the numerical cost. For very small particle
sizes compared to the typical energy containing scales, the momentum trans-
fer with the carrier phase can be modeled by some semi-empirical interaction
force expression that reduces to a drag law in the case of large density ra-
tio. The characteristics of the particles can be described by several means.
One possibility is the discrete particle simulation (DPS) approach in which
every particle is treated in an Lagrangian framework. An alternative method
is to consider only some collective characteristics of the particles in an Eule-
rian framework such as particle number density and mean particle velocity.
1 Present address : Siemens VDO, Regensburg Germany, An-
[email protected] Present address : Siemens VDO, Toulouse France, [email protected]
2
Equilibrium Eulerian approaches [3,4] consider a transport equation for the
particle number density and compute the particle velocity using the carrier
phase velocity and a Taylor expansion for the equation of particle motion.
The expansion parameter is written in terms of the particle relaxation time to
carrier phase characteristic time ratio and such approaches are well suited for
small Stokes number values but lead to increasing error with increasing Stokes
number. Computation of particle laden flow for particle relaxation time larger
than the Kolmogorov time scale requires a transport equation for the particle
velocity in order to account accurately for inertia effects.
The purpose of the present paper is to compare predictions using an Eulerian
approach, with a transport equation for the particle velocity, to the reference
results obtained from discrete particle simulation (DPS) in a homogeneous
isotropic decaying gaseous turbulence given by DNS. Following Fevrier et
al. [5], a probability density function (PDF) f (1)p (cp;xp, t|Hf ) depending on
the particle velocity cp, the particle location xp at time t may be defined for a
given carrier flow realization Hf . This function gives the local instantaneous
probable number of particles with the given translation velocity up = cp and
obeys a Boltzmann-type kinetic equation, which may account for momentum
exchange with the carrier fluid and inter-particle collisions. Alternatively, in
analogy with the derivation of the Navier-Stokes equations in the frame of
kinetic theory [6], transport equations of the first order moments (such as
particle number density, mean velocity and fluctuant kinetic energy) may be
derived directly by averaging from the PDF kinetic equation [7]. Such an ap-
proach has been evaluated in previous a priori test from DPS coupled with
DNS or LES of forced homogeneous isotropic turbulence [8] and fully devel-
oped channel flow [9].
This study is the first attempt to solve the set of equations of the mesoscopic
3
Eulerian approach allowing ”a posteriori” comparison of the model predictions
with DPS results. In the present study, interaction forces are limited to Stokes
drag and two-way coupling effects are neglected, that is the carrier phase is
considered as uninfluenced by the presence of the dispersed phase. Particles
are of identical size and the gravity force is not considered. However, the ex-
tension to evaporating droplets, gravity force, modelling non-linear effect in
the drag force and other interaction forces is not in conflict with the derivation
methodology of the Eulerian field equations. Also the introduction of inter-
particle collisions is without major difficulties [10]. The drastic assumptions
are chosen in order not to obscure the effects related to particle turbulence
dynamic interaction.
Comparisons between Eulerian an Lagrangian simulation results are performed
for three levels. First the integral properties of the dispersed phase such as vol-
ume averaged particle kinetic energy and particle-fluid velocity correlation are
investigated. The second comparison concerns the local instantaneous field
properties such as particle number density np, correlated velocity up,i and
random uncorrelated kinetic energy δθp. The third comparison concerns the
spectral properties such as the particle kinetic energy spectrum Ep(k) and
the fluid-particle correlation spectrum Efp(k) computed from the correlated
particle velocity field.
To allow these comparisons a spatial filtering procedure is used to project
Lagrangian quantities on a grid and thus to obtain continuous fields from
the DPS results. To validate such a procedure, several projection methods
are tested on a one-dimensional synthetic case. In the case of non-interacting
particles, the number of particles per cell for a given carrier phase realization
is not limited, a gaussian projector is then validated using DPS results with
different simulated particle numbers at the kinetic energy spectrum level. The
4
ˆEp(k) ˆEfp(k) Ep(k) Efp(k)
Eulerian Field Variables
np up,i δθp
Mesoscopic Field Variables
np up,i δθp
DPSEulerian
Simulation
q2p qfp q2
p qfp δq2p⇐⇒(1)
⇐⇒(2)
⇐⇒(3)
⇓Volume Filtering m
⇑ ⇑
⇓ ⇓
Fig. 1. Methodology for comparison of DPS to Eulerian Simulation Approaches
particle number densities np and the correlated velocity up obtained by the
volume filtering procedure can then be compared to it’s counterpart from the
mesoscopic Eulerian Simulation (fig. 1).
The paper is organised as follow : the mesoscopic eulerian approach is pre-
sented in section 2, then the projection procedure to obtain Eulerian field
from Lagrangian results is described and validated (section 3), the section 4
concerns the numerical test case, finally Eulerian simulation predictions are
discused and compared with DPS results in section 5.
2 The mesoscopic Eulerian approach
In turbulent particle laden flow the dispersed phase cannot be regarded as
independent of the carrier phase turbulence [11]. Therefore several statistical
5
approaches rely on joint carrier-phase, dispersed-phase probability functions.
In particular, the joint fluid-particle PDF equations may be used to establish
transport equations for carrier phase and dispersed phase quantities such as
number densities, momentum and energies as well as fluid-particle correla-
tions [7,12]. Since the interaction of particles and turbulence is rather com-
plicated, modelling of the unclosed terms is a difficult and challenging issue.
Due to the statistical character of such approaches, three dimensional un-
steady effects, as they can also be found in single phase turbulent flow, cannot
be represented. One important example among others are segregation effects
which need a two-point PDF approach [13].
To derive local instantaneous Eulerian equations in dilute flows (without tur-
bulence modification by the particles), Fevrier et al. [5,8,10] introduce an
ensemble average over dispersed-phase realizations conditioned by one carrier-
phase realization. Such an averaging procedure leads to a conditional velocity
PDF for the dispersed phase,
f (1)p (cp,x, t; Hf ) =
⟨W (1)
p (cp,x, t) |Hf
⟩. (1)
W (1)p is the fine grid PDF of the realizations of position and velocity in time
of any given particle [14] and Hf is the given carrier flow realization. Ignoring
particle collisions, the conditional PDF equation takes the form :
∂
∂tfp +
∂
∂xj
cp,j fp +∂
∂cp,j
[Fp,j
mp
fp
]= 0 (2)
This PDF equation can be used to derive transport equations for the moments
(number density np, mesoscopic velocity up,...) of the dispersed phase following
kinetic theory methodology [6].
6
2.1 Transport equations for particle properties
Instead of resolving directly the Boltzman-type kinetic equation (2) one may
solve the first order moments as done in kinetic theory of dilute gases when
deriving the Navier Stokes equations [6]. Integration over the particle velocity
phase space yields statistical “mesoscopic” properties of the dispersed phase.
The “mesoscopic” particle number density is written
np =∫
f (1)p (cp,x, t; Hf ) dcp (3)
Other averaged properties of the dispersed phase are obtained by integration
of the product of the quantity with the conditional PDF.
Φ = 〈Φ|Hf〉p =1
np
∫Φ (cp) f (1)
p (cp,x, t; Hf ) dcp (4)
With this definition one may define a local instantaneous particulate veloc-
ity field, which is here named “mesoscopic” or “correlated” Eulerian particle
velocity field.
up (x, t; Hf ) =1
np
∫cpf
(1)p (cp,x, t; Hf ) dcp. (5)
For simplicity, the dependence of the above variables on the single carrier
phase realization Hf is not shown explicitly. The velocity of every individual
particle V(k)i can be decomposed into its correlated up,i and uncorrelated δv
(k)i
components :
V(k)i = up,i + δv
(k)i (6)
7
This uncorrelated part of the particle velocity is here referred to as Random
spatially Uncorrelated Velocity component (RUV) 3 . By construction, δv(k)i
is spatially uncorrelated. Indeed the two point correlation
δRp,ij(x,x′) =∫ ∫
[cp,i − up,i(x, t)][c′p,j − up,j(x
′, t)]f (2)
p
(cp, c
′p,x,x′, t; Hf
)dcpdc
′p(7)
is zero when |x−x′| > 0. This uncorrelated velocity distribution has similarity
with the peculiar velocity distribution in the framework of kinetic theory of
dilute gases which obeys the molecular chaos assumption. Nevertheless it has
to be pointed out that the particle position distribution is not as random due
to the segregation mechanism and the uncorrelated velocity distribution is not
gaussian.
Application of the conditional-averaging procedure to the kinetic equation
(eq. (2)) governing the particle PDF leads directly to the transport equations
for the first order moments consisting of number density and mesoscopic Eu-
lerian velocity,
∂
∂tnp +
∂
∂xi
npup,i = 0 (8)
np∂
∂tup,i + npup,j
∂
∂xj
up,i =− np
τp
[up,i − ui] +∂
∂xj
npδσp,ij (9)
The first term on the right hand side of eq. (9) represents the drag force,
where ui is the carrier phase velocity at the particle position. τp is the particle
3 Random spatially Uncorrelated Velocity (RUV) has been referred to as Quasi
Brownian Motion (QBM) in previous publications. We agree that the expression
Quasi Brownian is misleading since the physical interpretation of the uncorrelated
motion is not of Brownian nature.
8
relaxation time modeled by Stokes drag :
τp =ρpd
2
18µf
(10)
Here ρp is the density of the particle, d is the particle diameter and µf is the
dynamic viscosity of the carrier phase. The stress term in eq. (9) arises from
the averaging of the convective term in the PDF transport equation :
npδσp,ij =−∫
(cp,i − up,i) (cp,j − up,j) f (1)p (cp,x, t; Hf ) dcp (11)
=−np〈δup,iδup,j|Hf〉p. (12)
This term account for the transport of mesoscopic momentum due to the un-
correlated part of the particle velocity. The time dependant fields of the parti-
cle number density and mesoscopic velocity can be predicted using eqs (8) and
(9). Concerning the momentum conservation equation it is however necessary
to specify an expression for the RUV stress tensor δσp,ij. One possibility is
to use transport equations for the components of the velocity variance tensor.
This shifts the difficulty however in finding an expression for the third order
tensor appearing in the transport equation and increases the numerical cost
because of six additional transport equations. Alternatively, one can model
directly the velocity variances using the computed moments as done in the
Navier-Stokes equations when deriving the viscosity assumption in the frame
of kinetic theory.
2.2 The uncorrelated velocity variance tensor δσp,ij
When the Euler or Navier-Stokes equations are derived from kinetic gas the-
ory, the trace of 〈δup,iδup,j〉p is interpreted as temperature (ignoring the Boltz-
9
mann constant and molecular mass) and defined as the uncorrelated part of
the molecular kinetic energy. By analogy, the Random Uncorrelated particle
kinetic Energy (RUE) is defined as half the trace of δσp,ij :
δθp =1
2〈δup,iδup,i|Hf〉p. (13)
In the frame of kinetic theory of dilute gas, pressure is linked to density and
temperature by an equation of state. In the same manner, a Random Uncor-
related Pressure (RUP) may be defined by the product of uncorrelated kinetic
energy and particle number density :
Pp = np2
3δθp (14)
With such a definition, the velocity stress tensor may be separated into an
isotropic part corresponding to the RUP and a deviatoric trace-free term δτp,ij,
npδσp,ij = −Ppδij + δτp,ij (15)
Then, the momentum transport eq. (9) is written,
np∂
∂tup,i + npup,j
∂
∂xj
up,i = − np
τp
[up,i − ui]−∂
∂xi
Pp +∂
∂xj
δτp,ij (16)
and, in first approximation, the deviatoric stress tensor is written in terms of
the mesoscopic rate-of-strain tensor and a dynamic viscosity,
δτp,ij = µp
(∂up,i
∂xj
+∂up,j
∂xi
− 2
3
∂up,k
∂xk
δij
)(17)
In such an approach the particle dynamic viscosity is written µp = 1/3npτpδθp
[10] where τp is the particle relaxation time. This expression can be obtained
using the transport equation for the deviatoric part of the velocity variance
10
tensor 〈δup,iδup,j〉 and supposing weak shear [7]. This closure model for the
dynamic viscosity corresponds to a characteristic mixing length proportional
to λp ∝ τp
√2/3δθp which is the stopping distance due to drag of a particle with
relative instantaneous velocity δup =√
2/3δθp. The closure model (eq. (17))
and the dynamic viscosity µp require the knowledge of the RUE δθp. Modelling
approaches for this quantity are developed in the next section.
2.3 Models for Random Uncorrelated kinetic Energy (RUE)
From the PDF equation (eq. 2) a general transport equation for RUE can be
obtained in the same way than temperature equation in the frame of kinetic
theory [6].
∂
∂tnpδθp +
∂
∂xj
npup,jδθp =−2np
τp
δθp + npδσp,ij∂up,i
∂xj
− ∂
∂xj
np
2〈δup,iδup,iδup,j|Hf〉p (18)
The first rhs. term account for the RUE dissipation by drag force, the second
is the production due to shear, the last term is the diffusion by RUV.
In a first approach, transport of RUE and production of RUE by divergence of
the mesoscopic velocity field are assumed to be dominant and the other contri-
butions being negligible. In a second approach it is attempted to solve directly
the transport eq.(18) with a closure model for the triple velocity correlations.
2.3.1 Isentropic Approach for δθp
The first approach is assuming a quasi isentropic behavior of the dispersed
phase leading to an algebraic expression for δθp depending on the particle
11
number density 4 :
δθp = 〈A〉 n2/3p (19)
In homogeneous turbulence 〈A〉 is only dependent on time and can be deter-
mined from the spatial average of δθp to insure the correct value for the mean
RUE.
The spatial average (over the computational domain) is defined by {φ} =
1/V∫V φdV . The particle weighted averages is defined by {φ}p = {npφ} / {np}.
This allows to define A = {np} δq2p/{n5/3
p
}. The mean uncorrelated particle
kinetic energy is defined as δq2p =
{δθp
}p.
DPS performed by P. Fevrier [8] in stationary homogeneous isotropic turbu-
lence suggest that the mean uncorrelated kinetic energy δq2p depends on the
resolved dispersed phase kinetic energy q2p = 1/2 {up,kup,k}p, the fluid-particle
correlation qfp = {ukup,k}p where uk is the carrier phase velocity, and the car-
rier phase kinetic energy weighted by the particle presence q2f@p = 1/2 {ukuk}p.
This equilibrium expression is written,
δq2p = q2
p
(4q2
pq2f@p
q2fp
− 1
)(20)
2.3.2 Transport equation for δθp
The second approach uses the full transport equation for random uncorre-
lated kinetic energy [10] and requires the closure of the triple correlation. The
equation is written,
∂
∂tnpδθp +
∂
∂xj
npup,jδθp = −2np
τp
δθp (21)
4 This quantity can be obtained by first order analysis [6,15]
12
+
[−Ppδij + µp
(∂up,i
∂xj
+∂up,j
∂xi
− 2
3
∂up,k
∂xk
δij
)]∂up,i
∂xj
+∂
∂xj
[κp
∂
∂xj
δθp
]
In this equation, the stress tensor δσp,ij from eq. (18) has been replaced by
the approach already presented for the momentum equation (eq. (15)). The
interpretation of the triple correlation as a diffusive flux motivates the gradient
law approximation :
np
2〈δup,iδup,iδup,j|Hf〉p = −κp
∂
∂xj
δθp (22)
The diffusivity coefficient is modelled assuming κp = 5/3npτpδθp. This closure
is equivalent to the Fick’s law for the heat flux in the Navier-Stokes equations.
3 Computation of Eulerian mesoscopic fields from Discrete Parti-
cle Simulation results
In DPS approach every single k-particle follows its individual trajectory X(k)i (t)
and has its proper particle velocity V(k)i (t). Discrete particle position and ve-
locity are given by the simple set of differential equations.
d
dtX
(k)i = V
(k)i (t) (23)
d
dtV
(k)i =
1
τp
[ui(X
(k)(t), t)− V(k)i (t)
](24)
Special care has to be taken when evaluating the carrier phase velocity u(X(k), t)
at the particle location for the computation of Stokes drag law. In the present
study high order interpolation methods (fourth order Lagrange polynomes)
are used to ensure a minimal numerical error [16].
13
Table 1
Projection procedure definitions in 3D
Control volume Weight function Characteristic length
Projection Vc w(X(k) − x) 2σ
Box (∆x)3 1 ∆x
Large box (2∆x)3 1 2∆x
Volumic (∆x)3 Π3i=1(2− 2 |X(k)
i −xi|∆x/2 )
∆x√2
Large volumic (2∆x)3 123 Π3
i=1(1−|X(k)
i −xi|∆x )
√2∆x
Gaussian (2∆x)3 (2∆x)3
erf(√
6)3( 6
π∆x2 )32 exp(−6|X(k)−x|2
∆x2 ) (1− 2√
6e−6√
πerf(√
6))∆x ≈ ∆x
The Eulerian model presented above is based, in theory, on an ensemble av-
erage over all particle flow realizations for a given fluid flow realization. But,
without inter-particle influences (directly by collisions, or through a modi-
fication of the fluid flow due to particles), an average over the Lagrangian
quantities of a large number of particle is equivalent to the statistical condi-
tional average on several particle realizations of a few number of particles. In
addition, interpretation of Lagrangian results in term of Eulerian mesoscopic
fields such as particle number density, correlated velocity or RUE require the
use of a projection procedure based on volume filtering method.
In this study, the DPS is performed with a large number of particles and
field properties are obtained by projection on a regular grid with a given cell
size ∆x. Such projection procedure from Lagrangian to Eulerian quantities
are widely used to handle the two-way coupling in DPS approach [11,17–
19]. The salient feature of the configuration is the discontinuous distribution
of the Lagrangian velocities due to the particle RUV. A control volume Vc
14
around a computational node at the location x is defined. Vc should be chosen
small enough with respect to the characteristic length scale of variation of
the mesoscopic variables but large enough to have sufficiently particles for the
averaging. Mesoscopic fields such as particle number density np, correlated
velocity up and random uncorrelated energy δθp are measured in DPS as :
np(x, t) =1
Vc
∑k
w(X(k)(t)− x) (25)
np(x, t)up,i(x, t) =1
Vc
∑k
w(X(k)(t)− x)V(k)i (t) (26)
np(x, t)δθp(x, t) =1
2
1
Vc
∑k
w(X(k)(t)− x)[V(k)i (t)]2
−1
2np(x, t)u2
p,i(x, t) (27)
where w(X(k)(t)−x) is a weight function. Definitions of several different projec-
tions are presented in Table 1. The top-hat and volumic (based on Schoenberg
M2 spline) projectors are written for two different widths, a gaussian projector
is also presented. We restrict our attention to second order space accurate pro-
jectors for two reasons. Firstly, higher order projector weight function should
have negative loops leading for unphysical negative particle number when few
particles are present. Secondly, the corresponding convergence rate in terms
of the particle number is slower [19]. Integral of the different projector kernels
over the control volume is unity to ensure the globaly (ie. for infinite par-
ticle number) quantity mean conservation by projection. It must be noticed
that the box, large box and large volumic projector are also locally mean-
conservative for finite particle number densities [17]. The characteristic length
scale of the projector is evaluated by twice the standard deviation of the weigth
function σ :
15
Fig. 2. Weight function of the different projection in 1D
σ2 =1
Vc
∫Vc
|X− x|2w(X− x)dX1dX2dX3 (28)
Leading to the characteristic length scale classification of the retained projec-
tors :
Large box > Large volumic > Box ≈ Gaussian > Volumic (29)
3.1 Validation on 1D synthetic case
Tests of these projection procedures have been performed on one-dimensional
synthetic case. The box, volumic and gaussian 1D projection kernels are pre-
sented on fig. 2. Np particles are randomly distributed on a one dimensional
grid with an equidistant node spacing ∆x of N nodes. The mean particle
number per cell is Np/N = 〈np〉∆x in 1D. The particle distribution is pro-
jected on the grid (eq. 25) to obtain the mesoscopic particle number density.
The conservation of mean quantities by gaussian and volumic projection is
16
Fig. 3. Evolution of the conservation of the projected particle number with the mean
particle number per cell (error = (〈np〉 − 〈np〉)2/〈np〉2) for the 1D synthetic case.
evaluated for the mean projected number density 〈np〉 (fig. 3). As expected
the associated error is decreasing when the mean particle number per cell is
increasing. It can easily be showed that this bias is avoided in the case of
equally spaced particles and is due to non homogeneous particle repartition
in every projection grid cell.
We focuss now our attention on kinetic quantities. A turbulent-like velocity
field (up) and a gaussian uncorrelated white noise (δv(k)) are given to the
randomly placed particles :
V (k) = up(X(k)) + δv(k) (30)
The velocity field follows a Passot-Pouquet type spectrum defined on 32 modes
(presented in the study), but tests using a −5/3 decrease rate spectrum show
similar results. The energy of the noise has been chosen to be 30% of the
correlated energy, which correspond to the maximum of the particle random
uncorrelated energy measured by Fevrier et al. [5] in stationnary homoge-
17
neous isotropic turbulence. Particle properties are then projected on a one-
dimensional 64 node grid of length 2π. Substituting eq. 30 in eq. 26 leads
to :
npup =1
Vc
∑k
w(X(k) − x)up(X(k)) +
1
Vc
∑k
w(X(k) − x)δv(k) (31)
npup = npup + npδv (32)
To identified the projected velocity up to the correlated velocity up, the pro-
jection procedure must be able to supprim the noise δv and must not affect
the correlated velocity field. The difference between the projected velocity
fields and the non projected correlated velocity distribution is evaluated by
the quadratic error :
Error(φ) =〈(φ(x)− up(x))2〉
〈up(x)2〉(33)
where φ is the projected velocity with added noise up or without noise up. The
same Lagrangian field is projected with the different projections. To obtain
statistical convergence, average is performed on 5000 different realizations of
the velocity field. This procedure is repeated for several values of the particle
number per cell of the projection grid, from one to thousand. Fig. 4 shows
the dependence of the velocity projection error (computed from eq. 33) on
the particle number per cell for the different projection procedures. The error
decreases when increasing the particle number, but a systematic error occurs
even for a large particle number density. For more than 10 particles per cell,
the related error arises in decreasing order from the large box projector, the
large volumetric filter, the box filter and the volumetric filter. In the noisy case
(fig. 5) more than 100 particles per cell are needed to reach the systematic pro-
jector error level, but the projector efficiency classification remains the same.
18
Fig. 4. Dependence of velocity quadratic error (given by eq. 33) due to the pro-
jection procedure on the particle number per cell for 1D synthetic case. Particle
velocity distribution is given from the interpolation of a continuous turbulent field
on the particle position. Upper figure : comparison between between box, large box,
volumic, large volumic projections. Lower figure : comparison between box, volumic
and gaussian projections at the bottom.
19
Fig. 5. Same caption as fig. 4. Particle velocity is the velocity of the continuous field
at their location with an added gaussian white noise of 30% energy.
In addition to the statistical error due to the finite number of particles in the
averaging cell, Boivin et al. [17] showed that such projection procedures are
equivalent to a filtering of the length scales smaller than the control volume
size which limit high gradient values. This spatial error can be limited by using
smaller control volume, but this increases however the statistical error due to
20
Fig. 6. Dependence of the projected correlated energy spectra using the gaussian
projector on the mean number of particles per cell. 1D synthetic case. The line is
the correlated energy spectra.
particle number. A different alternative would be to use a projection kernel
with a smaller characteristic length scale. It can be noticed that volumic-type
projectors returns better results than box-type projectors. A compromise be-
tween small systematic error and accuracy for small particle number density
is to use the gaussian projector on a large control volume (see Table 1) but
having the box projection characteristic length scale. The Gaussian projector
error is comparable to the box projection error for high particle numbers and
to the large box for small particle number.
To analyse more in details the projection error, the energy spectra of up, up,
δv, and up are presented on fig. 6-8 with the gaussian projector for 2, 20 and
200 mean particle number per cell of the projection grid. For all the cases the
projected correlated velocity energy (fig. 6) spectra follow the one of the cor-
related velocity at large scales. The projected correlated velocity (up) energy
spectra for the 200 mean particle number per cell case underestimates the
21
Fig. 7. Dependence of the projected noise energy spectra using the gaussian pro-
jector on the mean number of particles per cell. 1D synthetic case. The continuous
line is the correlated energy spectra. The horizontal dashed lines correspond to the
noise spectrum models (Eq. 35).
energy at small scales comparing to the up energy spectra. This behaviour is
the mark of the spatial error detailled before. In the case of 2 particles per cell
case, the spectrum presents an unphysical overestimation of the small scale
energy. This effect is avoided by increasing the mean particle number density,
np so that the spectra for 20 and 200 particles per cell are nearly identical for
all waves numbers. For equally spaced particles, the error is removed, even for
the case with the lowest particle number per cell (not presented here) proving
that the overestimation of the velocity spectrum induced by the projection
procedure is due to the random repartition of the particles.
Fig. 7 presents the energy spectra of the projected noise (δv). These spectra
are mainly flat and their level quasi-linearly decrease by increasing the parti-
cle number density. The noise spectra level are comparable to the correlated
energy ones. More precisely, this level can be evaluated. The energy spectrum
22
Fig. 8. Dependence of the projected correlated energy spectra with added noise
using the gaussian projector on the mean number of particles per cell. 1D synthetic
case. The continuous line is the correlated energy spectra. The horizontal dashed
lines correspond to the noise spectrum models (Eq. 35).
of a discret white noise on Np equaly spaced particles is constant up to the
Nyquist frequency (Np/2 = 〈np〉N∆x/2). Considering that the spectrum in-
tegral gives the Random Uncorrelated Energy, the energy spectrum of the
discret noise is :
Eδv(k) =N∆x
2π(Np
2− 1)−1δq2
p (34)
Eδv(k)≈ 1
π〈np〉δq2
p (35)
For non homogeneous particle distribution, eq. 35 can be used as a noise
spectrum level estimation. Comparing the projected noise spectra level with
the non projected noise model level, the particle density effect is recovered and
the mean level too (about 2% error). The projection procedure is just filtering
the noise and cut off all the components with wave numbers bigger than N/2.
23
We now consider the projection of correlated velocity with added noise (fig. 8).
At small scales the energy spectrum of up obtained with 2 or 20 mean particle
number per cell follows the one of the noise. A statistical error occurs and
the projection is not able to eliminate noise of the discret velocity field. By
increasing the number of particle the energy of the projected noise becomes
negligeable against the correlated energy at all length scales. An a-posteriory
validity criteria in the computation of the correlated energy spectra with added
noise is introduced :
Eup(k) >1
π〈np〉δq2
p for all k (36)
To resume, when obtaining continuous eulerian fields from discret Lagrangian
quantities by projection, errors are mainly due to a statistical error and to
an intrisic filtering (or spatial error) at small scales. By using a well chosen
projector (e.g. gaussian projection) with enough of particles (more than 10
particles per cell), it is possible to circumvent the problem in a satisfactory
manner.
3.2 Validation on DPS results
Finally to validate the gaussian projection, two DPS have been performed with
10 and 80 millions Lagrangian particles. Particles are randomly placed in the
computational domain of length 2π with the fluid velocity at their position.
Initial Lagrangian quantities are then projected with a gaussian filter on a 643
and 1283 node grids. For the coarser case, 10 millions of particles projected on
a 643 grid cell (about 38 particles by cell) with a ∆x equal to the computing
mesh size, the projected spectrum is identical to the fluid one for k smaller than
24
Fig. 9. Comparison of correlated energy spectra of DPS results at the initialisation
with V(k)(t = 0) = u(X(k)(t = 0), (t = 0)). DPS with 107 or 8. 107 particles
projected on a 643 or 1283 grid with the gaussian projector.
18 (fig. 9). For larger value of k, the energy spectrum of the projected velocity
shows an unphysical increase but remains 104 smaller than the effective values
measured in the energetic region. As expected, fig. 9 shows that this effect can
be diminished by increasing the particle number and decreasing the projection
cell size but leading to very expansive simulation costs.
Both DPS are performed and the same analyse is realised on a case with RUE
(≈ 16% of the correlated energy is measured) and particle segregation. The
energy spectra of mesoscopic velocities obtained by projection on a 643 (fig. 10)
are nearly identical for the two DPS results, no statistical bias is observed
so that 10 million particles (≈ 38 particles per cell) is sufficient to analyse
mesoscopic velocity fields in term of energy spectra. Indeed, these spectra
follow the one obtained by projection on a finer grid (1283) up to the higher
wave number (fig. 10). The one-dimensional noise spectrum model (eq. 35) is
extended for the three-dimensional spectrum by replacing the total number of
25
Fig. 10. Comparison of correlated energy spectra of DPS results. DPS with 107 or
8. 107 particles projected on a 643 or 1283 grid with the gaussian projector. Case
St = 0.53 at time t = 10.8. Horizontal dashed lines are the lower limit of spectrum
validity given by Eq. 37 (upper line is for cases 107 particles projected on 643 and
8. 107 particles projected on 1283 grid; lower line is for case 8. 107 particles projected
on 643 grid).
particle by the number of particles in only one direction (〈np〉∆x3N) :
Eup(k) >1
π〈np〉∆x2δq2
p for all k (37)
We notice on fig. 10 that the correlated energy spectrum validity criterion
(eq. 37) is satisfied. This criterion is of course questionable because particle
distribution is far from homogeneous and RUE is not uniform, but it could be
considered as a spectrum validity indicator. The gaussian projector is really
able to limit the intrinsic error of the projection procedure and is used to
obtain mesoscopic fields from DPS results with 10 million particles for 643
projection cells (about 38 particles per cell) in the rest of the paper.
26
4 Homogeneous isotropic decaying turbulence test case
Homogeneous isotropic turbulence is one of the classical cases where dynam-
ics and dispersion of particle laden flows can be studied. This has been done
extensively using the DPS formalism and encouraging results and insight are
obtained by such methods. Comparison of DPS in decreasing homogeneous
isotropic turbulence [20] with experimental measurements of particle disper-
sion in grid generated turbulence [21] shows that essential features of the
particle dynamics can be captured. Preliminary computations with a simpli-
fied Eulerian formalism of this test case gave encouraging results [22]. In the
case of tracer particles (small Stokes number limit) Eulerian methods are well
suited to describe the dynamics [3,4]. With increasing inertia, the particle
velocities become de-correlated from the gaseous carrier phase velocity and
segregation occurs for particle relaxation times comparable or larger than the
Kolmogorov time scale [23].
For the sake of simplicity, the case of decaying homogeneous isotropic tur-
bulence is studied. The carrier phase is initially supposed to have uniform
density, the velocity field to be divergence free and the kinetic energy to fol-
low a Passot-Pouquet spectrum [24]. After roughly one turn over time of the
energy containing eddies, the velocity field is supposed to represent a realis-
tic turbulent flow and the dispersed phase is added. At the particle injection
time, the Reynolds number is ReL = 13.6. For DPS initial conditions, par-
ticles are randomly and homogeneously distributed in space and the initial
particle velocities are given equal to the carrier phase velocity at the location
of the particle. For the Eulerian computation, these conditions correspond
to a uniform particle number density fieldand a mesocopic particle velocity
27
field identical to the carrier phase velocity field and zero uncorrelated velocity
variances. So the initial uncorrelated kinetic energy (RUE) is zero at initial-
ization and should develop during the simulation. The spatial resolution of the
gaseous phase computation is 643 and a total of 10 million individual particles
are traced in the computational domain. This corresponds to 38.1 particles
per gaseous node.
4.1 Numerical Method
The Eulerian simulation is performed using a different code (AVBP [25]) then
the DPS reference solution (NTMIX [18]). AVBP offers several spatial and
temporal schemes. In the present study a central second order spatial scheme
with a second order temporal correction (Lax Wendroff) was used. Comparison
to third order Runge Kutta time stepping showed no significant improvement
of the numerical accuracy. NTMIX uses a high-order spectral like scheme [26]
on cartesian grids and Runge Kutta time stepping. Both numerical tools use
domain decomposition and MPI for parallel computation. For the test cases
carrier phase solutions are identical and velocity spectra superpose.
To ensure the accuracy of the DNS of the carrier phase a necessary condition
is to satisfy the balance of the volume averaged kinetic energy (q2f ) with the
dissipation rate (ε) computed directly form the viscous stress tensor (τij) on
the computational grid.
∂
∂tq2f = −ε (38)
This verification that dissipation of carrier phase kinetic energy results from
laminar viscous effects and not from the numerical diffusion in AVBP code
28
Fig. 11. Time derivative of the carrier phase turbulent kinetic energy dq2f/dt com-
puted from AVBP numerical simulation results compared to the dissipation rate
computed directly from the viscous terms.
Fig. 12. Comparison of the carrier phase turbulent kinetic energy q2f computed
from AVBP simulations (line) using a second order difference scheme with the one
computed from NTMIX simulations (triangles) using a spectral like scheme [26].
is shown by fig. 11. The carrier phase kinetic energy of the two codes are
identical (fig. 12). Furthermore, the kinetic energy spectra at a adimentional
time of 10.8 are very similar until the Kolmogorov wave number (1/ηK) which
29
Fig. 13. Kinetic energy spectra of the carrier phase from simulations with the two
numerical codes used in this study at time t = 10.8.
is 7.7 in non-dimensional units.
In the Eulerian simulation, the dispersed phase is computed using the same
numerical method as the carrier phase, imposing an additional limit on the
time step due to particle relaxation time (τp).
The Eulerian results are compared to DPS results at different levels : average
quantities over the compational domain, kinetic energy spectra and snapshot
of instantaneous local fields.
4.2 Integral Properties
The integral properties of particle kinetic energy q2p and fluid-particle correla-
tion qfp should be identical in the two simulations and therefore constitute a
first step of comparison. In the present test case of decreasing homogeneous
isotropic turbulence those quantities vary in time. The properties are averaged
over the computational volume and defined for the Eulerian simulation by
30
q2f =
1
2{uiui} (39)
qfp = {uiup,i}p (40)
q2p =
1
2{up,iup,i}p (41)
and for the DPS by
qfp =1
N
∑k
V(k)i ui(X
(k)) (42)
q2p =
1
N
∑k
1
2V
(k)i V
(k)i (43)
The volume averaged uncorrelated kinetic energy (RUE) from DPS is given
by
δq2p =
{δθp
}p
(44)
with the local RUE δθp measured by eq. 27.
4.3 Spectral treatment
Using the Fourier transformed velocities of the carrier phase ui(k) = F(ui(x))
and dispersed phase up,i(k) = F(up,i(x)) one can construct three dimensional
fluid and particle energy spectra and a fluid-particle correlation spectrum.
Ef (k) =1
2ui(k)ui(k) (45)
Ep(k) =1
2up,i(k)up,i(k) (46)
Efp(k) = ui(k)up,i(k) (47)
For established turbulent flow the undisturbed carrier phase kinetic energy fol-
lows the standard Kolmogorov spectrum. Here the interest lies on the behavior
of the spectrum of the correlated particle kinetic energy and the fluid-particle
31
correlation. Whereas the carrier phase is considered incompressible, segrega-
tion effects measured in DPS show that there must be a compressible part
in the correlated dispersed phase velocity spectrum. With the definition of
Kraichnan [27] operators in the spectral velocity can be divided into a com-
pressible and an incompressible (solenoidal) component.
ucp,i =
κiκj
κ2up,j (48)
usp,i =
(1− κiκj
κ2
)up,j (49)
This orthogonal decomposition allows to construct a compressible and a solenoidal
spectral energy such that the sum equals to the total spectral energy.
Ecp(k) =
1
2uc
p,i(k)ucp,i(k) (50)
Esp(k) =
1
2us
p,i(k)usp,i(k) (51)
5 Simulations results and discussion
Eulerian methods are well suited to describe the particle dynamics [3] for
particle relaxation times small compared to the Kolmogorov time scale. It
is therefore interesting to study how the Eulerian description behaves out-
side this range. Preliminary tests with the decaying homogeneous turbulence
described above, showed that the numerical simulation with any of the two
models proposed for the RUE is possible up to a Stokes number (St = τp/τ+)
of St = 0.042 based on the macroscopic dissipative time scale τ+(= k/ε) at
the particle injection time. This corresponds to a Stokes number of StK = 0.17
based on the Kolmogorov time scale. In contrast, mesoscopic Eulerian Simu-
lations with larger Stokes numbers failed. Detailed analysis revealed that, due
32
Fig. 14. Comparison of correlated particle kinetic energy and fluid particle correla-
tion from DPS and Eulerian simulation for the test case of St = 0.042.
to segregation effects, the particle number density field has very stiff local gra-
dients that caused dispersion errors in the numerical scheme. Increasing grid
resolution for the Eulerian simulation up to (1283, 1923, 2563) only allowed a
small increase in attainable Stokes number. Therefore in a first step the results
of DPS and the Eulerian model are compared for the limiting Stokes number
of St = 0.042. In a second step an heuristic extension of the proposed model
overcoming the numerical difficulties encountered due to massive segregation
is presented and results are compared for a Stokes number based on the macro-
scopic dissipative time scale of St = 0.53 according to a Stokes number value,
based on the Kolmogorov time scale, StK = 2.2. Eulerian results presented
here are obtained with the RUE transport equation (eq. 21).
33
Fig. 15. Comparison of volume filtered DPS and Eulerian number density PDF for
the test case with St = 0.042 at t=10.8.
5.1 Comparison at St = 0.042
At Stokes numbers as small as St = 0.042 based on the dissipative time scale
τ+(= k/ε) the particles are expected to follow closely the carrier phase veloc-
ity and the uncorrelated velocity contribution to the particle dynamics is very
small. In fig. 14 the integral quantities of correlated particle kinetic energy q2p,
fluid-particle correlation qfp of the Eulerian simulation are compared to the
DPS results. Temporal evolution of particle kinetic energy and fluid-particle
correlation are well predicted by the Eulerian Simulation.
RUE measured in DPS (≈ 10−2q2p) is of the same order of numerical error.
Then the predicted RUE is of two orders of magnitude lower than the corre-
lated kinetic energy, the contribution of RUE can be neglected in this case.
Fig. 15 shows the probability to find a computational cell with a given value
for the number density in the Eulerian prediction and the DPS reference re-
sult at t=10.8. The distribution of the Eulerian number density is less wide
than the DPS distribution. If the correlated velocity is well predicted in the
34
Eulerian simulation, a possible cause of the non-matching number density
distributions may be the numerical scheme dispersion leading to a more ho-
mogeneous droplet number density.
35
Fig. 16. Comparison of the normalized droplet number n/〈n〉 from the Lagrangian
simulation (upper graph, resolution 643) with the one from the Eulerian simulation
(lower graph, resolution 643) at the non-dimensional time t=10.8, case St = 0.042.
The drawing plane is parallel to the x-y reference plane and crosses the computa-
tional domain center.
36
Fig. 17. Comparison of total and compressible kinetic energy spectra from DPS and
Eulerian Simulation for the test case with St = 0.042 at t=10.8.
As an instantaneous local quantity the normalized droplet number density is
shown in fig. 16 for the x-y plane crossing the computational domain center at
the simulation time t=10.8. This corresponds to roughly 18 particle relaxation
times (τp) and two dissipative times scales (τ+) of the carrier phase. It shows
that the Eulerian number density prediction is quantitatively close to the DPS
result. Regions with high and low particle number densities are well correlated
in the two approaches.
Fig. 17 shows the total particle kinetic energy spectra and the compressible
kinetic energy spectra computed from Lagrangian and Eulerian simulation
results. The correlated particle kinetic energy spectrum follows closely the
spectrum of the carrier phase at large scales. Up to the kink in the DPS
energy spectra, the spectra computed from the Eulerian Simulation match well
those from the DPS. As shown in section 3, this kink is probably unphysical
and is due to a numerical error induced by the non-homogeneous particle
distribution.
37
In case of small Stokes number, the mesoscopic simulation is able to predict
integral quantities such as particle energy and also local quantities such as
segragation. Results with Eulerian mesoscopic approach are very similar to
the ones obtained using the Eulerian equilibrium approach propose by Rani
and Balachandar [4,28].
5.2 Comparison at St = 0.53
Preliminary tests with Stokes numbers larger than St = 0.042 failed since
large segregation effects imply shock like in the number density distribution
that could not be handled by the numeric scheme [15]. Supposing that the
numerical resolution of the model is insufficient, several possibilities exist to
circumvent this difficulty : a different numerical scheme using up-winding or
flux limiters are clearly able to capture those strong gradients but imply some
type of numerical diffusion. For the computation of the number density in
the equilibrium Eulerian approach with a limited Stokes number range Rani
and Balachandar [4,28] use a spectral viscosity to overcome this difficulty.
Increasing spatial resolution increases strongly numerical cost. The origin of
the preferential concentration is due to the compressibility of the particle
number density field. A pure diffusivity added to the evolution of the number
density field would bias the conservative transport of the particle velocity. The
origin of the compressibility is the compressible component of the mesoscopic
particle velocity field.
At this point it was found preferable to act on the compressible component of
the correlated velocity field to circumvent the measured difficulties associated
with steep gradients. This leads to compute for a modified correlated velocity
38
˜up,i transport equation and consequently a modified droplet number density
field ¯np. The compressible component of the correlated velocity is modified by
introducing a subgrid bulk viscosity, ξsgs.
∂
∂t¯np
˜up,i +∂
∂xj
¯np˜up,i
˜up,j =− ∂
∂xi
PQB +∂
∂xj
τij −¯np
τp
(˜up,i − uf,i
)(52)
+∂
∂xi
ξsgs∂
∂xk
˜up,k
The subgrid model has the form of a bulk viscous term ξsgs∂ ˜uk/∂xkδij which
is added to the shear viscosity term δτp,ij in eq. 17. The subgrid bulk viscosity
is mesh-size dependent (ξsgs ∝ ¯np(∆x)2|∂ ˜uk/∂xk|). The proportionality coeffi-
cient for the bulk viscosity used for all simulations with St > 0.042 is 50. The
RUE transport equation (eq. 21) is not modified.
In homogeneous turbulence, the spatial average of this bulk viscous term is
zero, still it acts locally and leads to a more homogeneous number density
field. Simulations have been carried out for several particle relaxation times.
The following computation with a Stokes number of St = 0.53 is performed
with this heuristically introduced bulk viscosity and compared to the DPS
reference results.
5.2.1 Integral quantities
Figure 18 shows the temporal evolution of carrier phase kinetic energy, par-
ticle kinetic energy and fluid-particle correlation. The carrier phase kinetic
energy decreases due to viscous dissipation. Particle kinetic energy follows the
carrier phase kinetic energy with a delay of the order of the particle relaxation
time. Due to particle inertia, the particle velocities become partially uncorre-
lated in space and the RUE begins to increase. The behavior of the integral
39
Fig. 18. Temporal evolution of the fluid, particle and fluid-particle velocity corre-
lations for DPS (symbols) and Eulerian simulations (lines). For Eulerian approach
the total particle kinetic energy q2p is computed as the sum of the kinetic energy
due to correlated motion q2p and the random uncorrelated energy δq2
p.
quantities of correlated and uncorrelated particle kinetic energy as well as the
fluid-particle correlation are well predicted by the Eulerian simulation using
the transport equation for RUE. Simulations using the isentropic approxima-
tion show results of similar quality.
Comparison of particle number density PDF is provided on fig. 19. As ex-
pected, the Eulerian approach underestimates particle segregation.
5.2.2 Instantaneous local fields
Fig 20 shows a snapshot of number density in DPS and the Eulerian simulation
in the upper and the lower graph. The number density field shows the same
kind of structures for both the Lagrangian and Eulerian simulations. But the
instantaneous distribution is less heterogeneouss for the Eulerian simulation
due to the heuristic bulk viscosity which reduces compressibility effects.
40
Fig. 19. Comparison of DPS and Eulerian number density PDF for the test case St
=0.53 at t=10.8.
The heuristically introduced bulk viscous term tends to render the spatial par-
ticle number density more uniform. Without this bulk viscous term, Eulerian
simulations can currently not be carried out : the physical particle segregation
is too large as it could be resolved by the numerical scheme. Since the spatial
average of the volume viscosity is however zero, it does not effect the tempo-
ral evolution of the averaged kinetic energy of Random Uncorrelated Motion
of the particles δq2p. In contrast, local instantaneous values of δθp may differ
notably from the values obtained in the DPS (fig. 21).
41
Fig. 20. Comparison of the normalized droplet number np/〈np〉 from the Lagrangian
simulation (upper graph, resolution 643) with the one from the Eulerian simulation
(lower graph, resolution 1283) (St = 0.53) at time (t=10.8). Same cut-plane as
fig. 16.
42
Fig. 21. Comparison of RUE (δθp) in the DPS (upper graph,resolution 643) and the
Eulerian computation (lower graph,resolution 1283) after one particle relaxation (St
= 0.53) time (t=10.8) . Same cut-plane as fig. 16.
43
5.3 Transition filter between DPS and Eulerian simulation results
The number density of the Eulerian simulation with subgrid bulk viscosity
can be interpreted as a filtered Eulerian number density. Formally the filtered
number density field is obtained by the following convolution with the filter
kernel :
¯np(x) =∫
np(x′)F (x′ − x) dx’ (53)
The actually predicted Eulerian number density without bulk viscosity is not
known. Since the DPS number density should be equal to the Eulerian number
density field in question, the particle number density from DPS np is used in
its place. Using the Fourier transform property the convolution becomes a
product in spectral space :
ˆnp(κ) = ˆnp(κ)F (κ) (54)
This allows to obtain the filtering kernel associated to the difference between
the number density field predicted in the Eulerian computation with bulk
viscosity and the DPS result that should be achieved. Finally, the space filter
kernel can be obtained by backward Fourier transform :
F (x) = F−1
ˆnp(κ)ˆnp(κ)
(55)
The filtering kernel allows a qualitative comparison of the particle number den-
sity field obtained by the grid filtering of the DPS and the Eulerian prediction
of the particle number density field. Using the number density weighted filter
it is possible to obtain from DPS the filtered particle number density that cor-
44
respond to the Eulerian prediction with subgrid bulk viscosity operator (fig.
22). These two snapshot are very comparable.
5.4 Filtering Kernel
The filter kernel is displayed in fig. 23. Since the filter is considered isotropic in
space, averaging over the different directions was performed and only the one
dimensional kernel is retained. The figure shows that the convolution kernel
averages the DPS number density field a little more than the neighboring grid
cells. When interpreting this graph, one has to keep in mind that the DPS
number density was already volume filtered to obtain a continuous field.
5.5 Spectral kinetic energies
Fig. 24 shows spectra of the total kinetic energies of the dispersed phase as
well as the compressible kinetic energies measured from the DPS and Eule-
rian simulation. First, one remarks the high compressible component of the
kinetic energy compared to the gaseous carrier phase kinetic energy. This
causes structures similar to those known as eddy shocklets in compressible
turbulence [29]. In addition, the compressible part of the energy spectrum is
of the same order than the solenoidal part at small scales or large wavenum-
ber values. The kinetic energy spectrum of the Eulerian simulation does not
reflect this behaviour to the same extend at small scales in contrast with the
one measured at large scales or small wave number values. This discrepancy
between both approaches is probably due to the bulk viscosity operator which
reduces drastically the compressible effects at small scales.
45
Fig. 22. Comparison of the normalized droplet number np/〈np〉 with filtered DPS
(upper graph, resolution 643) and with the Eulerian computation (lower graph,
resolution 1283) after one particle relaxation (St = 0.53) at time (t=10.8). Same
cut-plane as fig. 16.
46
Fig. 23. Filtering Kernel obtained by backward convolution between the DPS num-
ber density field and the Eulerian number density field. Case St = 0.53 at time
t = 10.8. The upper graph shows the spatial filter and the lower graph shows the
spectral filter.
47
Fig. 24. Comparison of DPS and Eulerian kinetic energy spectra for the Stokes
number St = 0.53 at t = 10.8.
48
6 Conclusion and perspectives
This study presents the potentials of a new Eulerian approach for particle
ensemble in gas turbulent flows that differs from standard approaches by it’s
ability to account for instantaneous local phenomena. The proposed meso-
scopic Eulerian approach allows to simulate the dynamics of particles sus-
pended in homogeneous isotropic decaying turbulence at large scales even in
the vicinity of unity Stokes numbers based on the turbulent time macroscale.
The role of the uncorrelated velocity variances in the momentum equation
is identified and a modeling approach in terms of pressure and viscosity is
presented and evaluated. Simulations were performed at very small turbulent
Reynolds numbers because simulations
with higher Reynolds numbers of the carrier phase show deficiencies for the
spatial accuracy of the dispersed numerical prediction. Therefore tests have
to be extended to higher Reynolds numbers and for colliding particles. Con-
currently such an approach is being adapted to a LES formalism [30], which
are very interesting for the unsteady computations of industrial applications
with a very large number of particles or droplets.
Eventually it is necessary to quantify the capacity of such an approach in real
geometries other than the synthetic case of boxes with periodic boundary con-
ditions. Possible configurations are particle laden jets or particle laden channel
flow with and without collisions.
49
Acknowledgements
Numerical computation of the Eulerian simulations were performed on the
COMPAQ supercomputers of CEA and CERFACS. Numerical solutions on
large grids (1923, 2563) were performed on SGI ORIGIN 3800 at CINES in
the framework of the Extreme Computing for Turbulent Combustion program
using up to 128 processors. The DPS reference solution was obtained with
numerical simulations performed at computing center IDRIS using the DPS
version of NTMIX in the Ecoulements Reactifs Diphasiques : Simulations Di-
rectes et aux Grandes Echelles project.
Financial support for this work was received from the European Community
via the STOPP research training network.
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