7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Evaluation of a Multifluid Mesoscopic Eulerian Formalism on the Large Eddy
Simulation of an aeronauticaltype configuration
A. Vi6*, M. Sanjos6 S. Jay*, C. Angelberger*, B. Cuenott and M. Massott
IFP, 1&4 av. de Bois Pr6au, 92852 RueilMalmaison Cedex, FRANCE
SCERFACS, 42 av. G. Coriolis, 31057 Toulouse Cedex 1, FRANCE
SEM2C, CNRS UPR288, Ecole Centrale de Paris, 92295 ChAtenayMalabry, FRANCE
aymeric.vie@ifp.fr
Keywords: Large Eddy Simulation, Twophase flow, Eulerian Mesoscopic, Multifluid, aeronautical swirl injector
Abstract
This work describes the evaluation of a fully eulerian method to simulate polydisperse turbulent two phase flows:
the Multifluid Mesoscopic Eulerian Formalism (MMEF). This approach is derived from the coupling between the
Mesoscopic Eulerian Formalism (MEF) of F6vrier et al. (2005), and the Multifluid Approach (MA) of Laurent and
Massot (2001). The MA handles polydispersion by using a discretization of the droplet size space, whereas the
MEF describes the velocity dispersion of monodisperse turbulent particle flows. The coupling of the two methods is
expected to capture both behaviors. A first step toward this kind of approach was initially studied in Massot (2007),
and the differences with the present work lie in the closure and modelling assumptions.
MMEF is integrated into the AVBP code from CERFACS/IFP, which solves the compressible NavierStokes equations
for reactive flows with low dissipation schemes adapted to Large Eddy Simulation. These schemes encounter some
difficulty for the Eulerian description of the liquid dispersed spray, highly compressible and showing stiff gradients
and vacuum areas of droplet density. Specific numerical procedures are used to stabilize locally the solution, with
limited effect on the accuracy.
The model is applied to the MERCATO aeronauticaltype configuration. An analysis of characteristic time scales
allows to evaluate the importance of polydispersion in this test rig. A preliminary test case consists in the evaporation
of a chosen distribution in one computational cell. Physical conditions are the same as in MERCATO. Results are
compared to the evaporation strategy used with MEF, and show the necessity to account for polydispersion description
in order to capture sizevelocity correlations.
A second test case is a twodimensional vortex entraining droplets, in which droplets are injected uniformly. Due to
the polydispersion of the liquid and its resulting centrifugal force, a spatial distribution of droplet mean diameter is
observed, and confirms again the necessity of polydispersion.
Finally, the MMEF is applied to the MERCATO test rig, for which both experimental and numerical data are available.
Comparisons of velocity profiles at selected downstream positions show a good agreement with measurements of
MEF and MMEF in the central zone, whereas only the MMEF is able to capture the external zone. Comparisons
of Droplet Number Density function at selected volumes inside the chamber also show a good agreement between
MMEF and experiments.
Nomenclature
Roman symbols
c, droplet velocity (m.s1)
fp Number Density Function
N number of sections
S droplet surface (mn2)
D deformation tensor (s 1)
T droplet temperature (K)
u phase velocity
Greek symbols
p kinetic viscosity (nm2.s 1)
p density (kg.m 3)
T, relaxation time (s)
87 Random Uncorrelated Energy (nm2.s 2)
Subscripts
9 gas
1 liquid
p particle
Superscripts
(p) section p
Others
6cp
6R1
6Si
uncorrelated droplet velocity (m.s 1)
2nd order velocity correlations tensor (mi2 .s 2)
3rd order velocity correlations tensor (mn3.s 3)
Favre averaged quantity
Reynolds averaged quantity
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
sion is a difficult question. The Mesoscopic Eulerian
Formalism (MEF)(F6vrier et al. (2005)) is based on a
locally monodisperse assumption, so it has no particle
size distribution. To tackle this problem, the Multifluid
Approach (MA) of Laurent and Massot (2001) has been
coupled with the MEF. This formalism, called Multifluid
Mesoscopic Eulerian Formalism (MMEF), is integrated
in the AVBP code. After presenting the model equa
tions and preliminary test cases, MMEF is applied to
an aeronauticaltype configuration. This application was
simulated with lagrangian and eulerian methods (San
jos6 et al. (2008)), but never with a polydisperse eulerian
formalism.
Model equations
The MEF is a statistical approach based on the droplet
number density function (NDF) fp:
where t, x, Cp, S, T are respectively the time, droplet
position, velocity, surface and temperature. The evo
lution of this NDF is driven by a WilliamsBoltzmann
equation (WBE) (Williams (1958)):
afp+ pfp Fdfp aRsfp aETf
Ot Ox 9 cpj S OT
0 (2)
Introduction
Large Eddy Simulation (LES) is a promising tool to
improve the prediction of the mixture inside aeronau
tical combustion chambers. Indeed, its ability to solve a
wide range of the turbulence spectrum, modelling only
the smallest scales, allows to reproduce more accurately
interactions between the turbulence and the liquid fuel
droplets.
This work aims at evaluating performance for liquid
injection inside an aeronautical combustor. Lagrangian
particle tracking methods are classically used but they
suffer from numerical issues (i.iI.illlis.iiiI i statisti
cal convergence, high number of particles) and special
care must be taken for the liquid/gas coupling problems
(interpolation of gas velocity, distribution of liquid/gas
source terms)(Garcia 2,li "i,'. An alternative is eule
rian methods, which use the same numerical methods
for the gas and the liquid phases, and can be efficiently
parallelized. So it seems well suited for complex indus
trial applications. However, it raises a number of mod
elling issues. In particular, the modelling of polydisper
where Fd, Rs and ET correspond to drag force, mass
and heat transfer. The drag force Fd is written as :
Cp Ugli
where ugl is the gas velocity at droplet position, and
T = plS/187rp, is the particle relaxation time. Rs and
Ee are defined as:
dS
dt
cEdT
dt
4w4
mrmp
plS1/2
Qp
mpCpl
where Tip is the mass rate, and Qp is the heating flux.
Those source terms are modeled using a finite liquid
conductivity with a gas spherical symmetry diffusion
model (Kuo (2005), Sirignano (1999)).
Using < 4 > 1/ni fR 4fpdcpdT as an averaging
fp fp (X, t, CpS, T)
operator, some characteristic quantities are defined:
ni(x, t, S)
ulTi(x,t, S)
Ti(x, t, S)
6c, i(x, t, S)
6R1,ij (x, t, S)
601(x,t, S)
8Sl~ijk (x, t,S)
R fpdcpdT
< Cp,i >
Cp,i Ul,i
< 6cp, *.. i >
6Ri(zx, t, S)/2
< SCp, '. i6cp,k >
where ul, is the mesoscopic eulerian velocity field,
shared by all particles of size S at position x and time
t and Scp is a residual velocity, proper to each particle.
Ti, 6R ,ij, 5Slijk and 608 are respectively the liquid
mean temperature, the second and third order velocity
correlation tensors, and the Random Uncorrelated En
ergy (RUE), corresponding to the residual velocity.
Multiplying the WBE (Eq. 2) by any function I, and
integrating over particles velocities and temperatures,
one can obtain the equation of Enskog:
at
it ** ()+ n{(Rs)
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
but differences appear when the uncorrelated terms are
closed.
To obtain the multifluid system, the semikinetic sys
tem (Eq. 710) is integrated over particule size, assuming
a discretization in N intervals [Sp, S,+1] called section
from Greenberg et al. (1993): superscript (p) designates
massweighted averaged quantities in interval p. Sec
tionnal quantities are then defined:
ni(x, t, S) = m (x, tt, (S) (11)
(P) ,t. f S3/2
m (x, t) p'i: i n,dS (12)
sh p 6oV7
where j '(S) is the shape of the distribution function
over each section. Multiplying the kinetic system (Eq. 7
S3/2
10) by pi 6 and integrating over sections gives:
atm() amm (m E(P)(1) (13)
a a(13)
0 m )u) (P)(;) a *(p)()
am 1xm n l1 = E(P) (ul
T@
a m m(P)~ p) (14)
+ ni Fd,j ) + niRs as ) + niET a
acpj as O
+ i( t ) + nl (cp' Oxf )+ x n I)
(6)
Replacing I by target quantities, the semikinetic sys
tem (Laurent et al. (2004)) is obtained:
9 (p)60(p) I (p)60(p) (p)
atm 1 1 xm "1 1 1'm
(p 1()
2 (p) + ()
m9 (J)H(;) + 9x (m)H) (PU)
at m 1 m
+ m C1 ,iElT
(15)
ECP) (Hi 6)
(16)
a a a
9 + 9x lm HS
at aOxmiai aSfliis
ntUl,i + in Ul,inUl,am = SnpuaiRs
a a
nia; + nnilui,m
at Oxm
2 ni
Tp
a niS6Ri,im
9Xm
na OnRs
9S
nS6R,ij a 6S
j, 9xj
(7) where E(P)() is the evaporation source term defined
as:
(P)((P)) = ) E(P) m P)j(p)
(8) + E l) l)(+l) (17)
and (p) is the uncorrelated flux of 60p) defined as:
=(p) a(p)6R(P) (p)
 m1 1,ij Uli
09 i(P)S(P)
OXm 1 1,iim
nHj + niHiuai,m = ni RsHi + nCp,1Er
at axm as
(10)
where H = CpTi is the liquid enthalpy. The approach
described here is similar to the one used in Reveillon
et al. (2002) and Massot (2004), as they start from a
decomposition into mean and uncorrelated velocities,
In Massot (2004), bS1( is neglected, whereas only
the isotropic part of 56R( is considered in Eq. 14 and
Eq. 18. For the isotropic test cases studied in Massot
(2004), these assumptions have been verified by eule
rian filtering of lagrangian computations. In the final
application studied here, it is not possible to use these
assumptions, due to the anisotropy of shear flows. So
ni
77l (
the uncorrelated flux 8R(f is decomposed into spheri
cal and deviatoric parts:
6R(P) ) .. + (p)
5lij RUM6^ + 61,ij
whr 2/3(ip)
where (P) 236 is the pressure due to Ran
dom Uncorrelated Motion. 8E6Y and 6S( are closed
using respectively a viscous assumption and a diffusion
term similar to Fick's law (Kaufmann et al. (2008)):
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
volume occupied by the liquid is neglected:
a a
at dOXm
p
p=1
9xi 9xm p,
N rnp) u()
p= (P
a1
,. ,'W )
6Y(P)
1,ij
6S(p)
1,iim
K' ij
'Mi OX
where Dli t + 'i' ~ 6ij
w xj Ox i x xk 3
VPM = 7P) 6 /3, and ,. 2 (p/3. Those
closures are derived from the assumption of a velocity
distribution close to the maxwellian equilibrium one,
which has been validated on Homogeneous Isotropic
Turbulence for low Stokes Number (Vi6 et al. 12' ri 1.
But its validity for high Stokes numbers or shear flows
is quite questionable, because the nearequilibrium
assumption would not be verified. This issue is investi
gated in Masi (2010). It is shown that the anisotropy of
the shear stress has to be accounted for.
As proposed in de Chaisemartin (2009) and advised in
Laurent (2006), a distribution constant in radius is cho
sen. The function '(S) is then:
(S) 12
2 *'2) S)
(S C
Evaporation terms are derived by an integration by
part fs1 piS3/2/(6 )RsnidS (de ( II.IIni. iiii
(2009)):
,3/2
p, p Rn 7
^6^ ^
JSp+1 ds(S3 /2)
P 1 P6 RsnidS
3 P(p)
Sp + Sp+1S
a a
tpE9 +x (p,E, + P
+ P9v9D,,imm + a Qg,i
N mE T1P m Op,t
P (p) (p) (p) .(p
P) 1,i)
(27)
where E, is the total energy and Q,,ij is the heat flux.
In the LES approach, liquid and gas equations are fil
tered through a Favre averaging procedure (Moreau et
al. (2010)). The resulting set of equations for the liquid
phase is:
a )+ a (1) (28)
at 1 xm "Im
a TP)T(P ) a (.p) (P)(U(P)
at +a a' ltmx '"ii
,TTJ(P)
S__(p) (p) (p)() .
2 +1 6z +
TP
at aX '
(p) C, V(p)
+ 1 1 IT
Smp)Cp,tE p
(29)
(30)
P(P)(H 31))
(31)
(24) where Q() and ^(p) are defined as:
where Ep) is the mass exchange between sections, and
Ep) is the mass exchange with the gas phase.
Gas phase equations are the classical NavierStokes
equations, with the coupling terms between liquid and
gas phases due to drag and evaporation. As the liquid
volume fraction is very low in the final application, the
Q(P) a J ^(P)6 (P) 0 )Ti
i OXZ, O ^m J
aP) a
1 ( 1 ,ir O2m 1,i
e(p) _+I() a () ( O 9 ( )
Sxm _1 l1,iim
07f rh 5 ,
+ )+^
2S' RP) (23)
Sp+ S
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
where Ti' is the subgrid scale tensor modeled usi
SmagorinskiYoshizawa model (Moreau et al. (20
The resulting set of equations for the gas phase is:
a a
a a
at' OX am'
N
p=l
9P 8 89
x+ pvD,,im + x gij
a ( a) a1 ,
ax, mp
)) (P
1,i
atPgE9 + (p E, + P,,
8xm 
a a
oi~ '~
I
ng a
10)).
The MERCATO test rig
The test rig is an experimental swirl combustor (Fig. 1).
Air is injected inside the plenum and enters the combus
tion chamber through the swirler. The liquid injection is
(34) located at the end of the swirler stage, and at the center
of the swirled gaseous jet. The case studied is a two
phase evaporating flow without combustion. Pressure
and temperature of the gas phase are Ibar and 463K. Ex
perimental data obtained by PDA and LDA is available
(GarciaRosa (2008)).
The injection system is a pressureswirl atomizer. It
S generates a hollow cone spray with an half angle of 40
and orthoradial motion, which produces a polydisperse
S cloud of droplets. The NDF at 13mm downstream the
(35) injector is shown in Fig. 2. The injected fuel is a sur
rogate for kerosene JP10 for which thermodynamic and
transport properties are taken from Boileau (2007).
combustion
chamber
,h,
plenum
L
(36)
air net
channel
where T9,j is the subgrid scale tensor modeled using the
WALE model (Nicoud et Ducros (1999)).
Numerical Context in the AVBP
Code
The AVBP code solves compressible NavierStokes
equation for reactive flows on unstructured grids in a
cellvertex formulation. To perform Large Eddy Sim
ulations, a 3rd order in time and space scheme is used
(TTGC)(Colin et al. (2000)).
The eulerian liquid phase is similar to the eulerian
gas phase in terms of equations, but has different be
haviours. It is highly compressible, so that vacuum
and strong gradients occurs. As the TTGC scheme
doesn't preserve positivity, it encounters difficulties in
such zones. A classical way to handle such numerical
problems is to use artificial viscosity. This methodol
ogy has been applied to the simulation of a decaying
Homogeneous Isotropic Turbulence (Vi6 et al. ,2,1 ,' I,
and seems to be very accurate. However more adapted
schemes are needed, like kinetic schemes in de Chaise
martin (2009). But their adaptation to unstructured cell
vertex 3dimensional formulation is not straightforward
and requires important developments.
 h
swirler holder
fcl yc neton
Figure 1: Scheme of the MERCATO test rig.
0.025
)4 06 08 1 12 14 1.6 1.8 2
Diameter [m] x10"4
Figure 2: MERCATO: Number Density Function at
13mm downstream the injector location (his
togram) and Multifluid representation with 10
sections (solid line).
Preliminary test cases
As evaporation modifies droplets size, it is a good test
for a multifluid approach. Such test case was already
_I
Ilr.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
studied in Laurent (2006), Dufour et al. (2005), or de
(Ii.iisL'ii.liiin (2009). The same evaporation conditions
as in the MERCATO are used. Because of heating, no
analytical solution is available but MEF and MMEF are
compared. For MEF, the droplet distribution is reduced
to a Dirac distribution around the mean diameter which
evolves in time. For MMEF, the initial droplet distribu
tion is discretized in sections, as shown on Fig. 2. The
initial mean diameter (d10) is 55pm. Figure 3 shows
the time evolution of the liquid phase with MEF, which
gives a total evaporation time of 50ms, with a heating
time of 25ms. As MEF does not take into account the
droplet number flux at size zero (as done for DQMOM
in Fox et al. (2008)), the droplet number density is con
stant. Figure 4 shows the same quantities for MMEF, giv
ing a total evaporation time of 400ms and a heating time
of 150ms, respectively eight times and six times higher
than from MEF. This is due to the presence of large
droplets which heat and evaporate very slowly compared
to smaller ones. As MMEF takes into account droplet
number flux at size zero, the total droplet number de
creases. Figure 4 also shows that the mean diameter
does not become zero at the end of evaporation. This
is due to the 1st order of the multilfluid approach used
here, which causes diffusion in phase space. This draw
back was already shown in Laurent (2006), and can be
tackle using higher order method with linear (Laurent
(2006)) or exponential (Dufour et al. (2005)) reconstruc
tion. This last one is a special case of a more general
setting of Maximization of Entropy developed in Mas
sot et al. (2010) and Kah et al. (2009). But this is not
the scope of this work. This test case shows clearly that
accounting for polydispersion is first order for the final
application, as a monodisperse formalism highly under
estimates heating and evaporation times.
Another interesting test case is related to drag, which
also depend on droplet size and impacts the spatial dis
persion. It is proposed here to analyse the spatial dis
persion effects of a vortex having the same caracteris
tics than the swirl motion in MERCATO. The test case
is then a twodimensional frozen vortex, with a spa
tially homogeneous initial droplet distribution having
the same velocity as the gas phase. Ten sections are
used, with the same droplet number in each section. Ra
dial velocity profile for the test case and in MERCATO
at first position (13mm) are shown in Fig. 5. The com
putation is run for 5ms, which is the time needed by the
liquid phase to impact the wall.
Figure 6 shows the number density and mean diame
ter radial profiles for both MEF and MMEF. After 5ms,
droplet number density is widely distributed spatially
in MMEF, while concentrated around 0.038m in MEF.
Bigger droplets, due do the weaker influence of the gas
phase, keep their initial velocity out of the vortex, and so
Time [' 06
"lme Ii ]
z
15
0.5 L  L
0 0.01 0.02 0.03 0.04 0 05 006
Time [s]
SeS 
4c 05
2c05 
Time [s]
Figure 3: MEF Evaporation test case in MERCATO
conditions. Evolution of total mass (upper
left), droplet number density (upper right),
liquid temperature (lower left) and mean di
ameter (lower right).
Figure 4: MMEF Evaporation test case in MERCATO
conditions. Evolution of total mass (upper
left), droplet number density (upper right),
liquid temperature (lower left) and mean di
ameter (lower right).
their initial radial deviation. Small droplets are in equi
librium with the gas phase, so they reach a small velocity
out of the vortex. The mean diameters are affected, with
small diameters near the vortex center and big diameters
far from. Two drawbacks are shown in Fig.6. The first
drawback is linked to the mean diameter. In the center
of the vortex, it is supposed to have no droplet, and so a
mean diameter equal to zero. But the numerical scheme
used here imposes to keep a low droplet number density,
imposing a nonzero mean diameter. The second draw
0
o
20 
20 
48. 0.03 0.02 0.01 0 0.01 0.02 0.03 0.04
Radial Position [m]
Figure 5: Gas velocity profile for the 2D vortex (solid
line) and in MERCATO test case (13mm
downstream the injection location)(squares).
back concerns the spatial dispersion. Considering drag
force, each section behaves like its mean surface, and so
follow only one trajectory. This is resulting in discrete
accumulations for each class. In such a canonical test
case, it seems to be relevant to tackle this issue. But in
practical cases, considering turbulent mixing and species
diffusion it has no effect on the fuel spatial dispersion.
03.5 
3 II
i 2
S 
5 a 1.
0 001 002 003 004 0.05 0.06 ** 06
Radial position [m] Radial position [ml
Radial Position [in]
Figure 6: Radial distribution of droplet number density
for MEF (top left) and MMEF (top right), and
arithmetic mean diameter (bottom) for MEF
(dashed line) and MMEF (solid line).
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Simulation of the MERCATO test
rig
Injecton is a keypoint in the simulation of twophase
flow burners. Then the injection system is modeled by
the FIMUR model (Senoner et al. i2', i",,, based on
the model of Cossali (2000), and using autosimilarity as
sumptions for the velocity profiles. This model was ini
tially designed for monodisperse injection, and the ex
tension to multifluid injection is made by distributing the
liquid volume fraction and droplet number density at all
nodes over all diameters, following the size distribution
of Fig. 2.
The computational domain is discretized with
1351767 tetrahedrons (Fig. 7). Averages are calculated
during 40ms, which ensures statistical convergence.
13 mm 33mm 63mm
Figure 7: Mesh and velocity profiles position.
Figure 8 shows the total liquid volume fraction for
MEF and MMEF computations. Firstly, due to its life
time in a polydisperse framework (as shown in the evap
oration test case), the liquid phase exists in the whole
the combustion chamber for MMEF, whereas evaporates
faster with the MEF Secondly, due do the bigger inertia
of large droplets, the liquid phase impacts the wall with
MMEF, but not with the MEF.
Figure 9 shows the fuel mass fraction for MEF and
MMEF computations, which is an important quantity for
combustion. The two formalism lead to very different
fuel repartitions. Due to the higher life time of droplet,
the fuel mass fraction near the injection location is lower
for the MMEF than for the MEF. More precisely, the
high mass fraction zone around the injection location
showed with MEF does not exist with the MMEF.
Velocity profiles at three downstream positions
(13mm, 33mm and 63mm, Fig. 7) are compared to ex
perimental data (GarciaRosa (2008)) for both MEF and
MMEF results in Fig. 1012. The axial velocity cal
culated with MEF and MMEF are in good agreement
with experiments for the two first positions, and slightly
underestimated inside the spray at the last position, al
though the outer zone is well captured by MMEF. Good
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
S B 0.02
4 0.04
6i 0.06
5 0 5 10 15 20
0.02
0 .04
0.06
0 10 20 30 5 0 5 10 15 20
Figure 10: Axial velocity at 13mm (right), 33mm
(center) and 63mm (left) for experiments
(squares), MEF (full line) and MMEF
(dashed line).
Figure 8: Instantaneous field of liquid volume fraction
for MEF (up) and MMEF (down).
Figure 9: Instantaneous field of fuel mass fraction
(JPJP) for MEF (up) and MMEF (down).
0.07
0.04
0.02
0.00
Figure 9: Instantaneous field of fuel mass fraction
(JP10) for MEF (up) and MMEF (down).
agreement is also obtained for the radial velocity at the
first position, but some discrepancies appear at the two
other positions. MEF is not able to capture any be
haviours, while MMEF reproduces the velocity in the
outer zone. Finally, concerning the orthoradial velocity,
it appears that MMEF reproduces the velocity profile ac
curately, while MEF catpures tendencies, but with less
accurate results. Note that the good results for all veloc
ity profiles at the first measurement location validate the
injection methodology for both MEF and MMEF.
The Number Density Function (NDF) is compared
to experimental data (GarciaRosa (2008)) at different
points inside the burner (Fig. 13). Fig.14 shows that the
NDF inside the spray (point 2) is well captured, whereas
0.06 0.06
0.04 0.04
0.02 0.02
0 n 0
0.02 0.02
0.04 0.04
0.06 0.06
20 10 0 10 20 20 10 0 10
0.04
0.02
0 .a
0.02
0.04
0.06
20 5 0
Figure 11: Radial velocity at 13mm (right), 33mm
(center) and 63mm (left) for experiments
(squares), MEF (full line) and MMEF
(dashed line).
0.06 0.06
0.04 0.04
0.02 0.02
0 0
0.02 0.02
0.04 0.04
0.06 0.06
20 10 0 10 20 20 10 0 10 20
5 0 5
Figure 12: Orthoradial velocity at 13mm (right), 33mm
(center) and 63mm (left) for experiments
(squares), MEF (full line) and MMEF
(dashed line).
the NDF in the center of the hollow cone (point 1) is
shifted to big diameters. In Fig.15, the NDF in the cen
ter of the hollow cone (point 3) is not reproduced, but
there the droplet number density is low both in measure
ments and computation. The NDF inside (point 4) and
outside (point 5) the spray are well captured, with a good
position of the maximum. In Fig. 16, the NDF is well re
5.e05,
5.0e05,
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
produced at point 6 and point 8, especially considering
the maximum. The NDF at point 7 is relatively well
captured, but the position of the maximum is shifted to
small diameters.
S'
r: r4 2.  ..
7 
. , , . ', I "
r::'. .; ". . '  /, ',1
Figure 13: Mesh and NDF measurement location (cir
cles)
0.02 0.03
SExperimental
AVBP 0.025
0.015
0.02
0.01 0.015
0.01
0.005 0.0
0.005
0 50 100 150 0 50 100 150
diameter [pm] diameter [gm]
Figure 14: NDF at 13mm at point 1 (left) and point 2
(right) for MMEF (solid line) and experi
ment histogramm).
0.04 0.035 0.025
0.03
0.02
0.03 0.025
0.02 0.015
0.02
0.015 0.01
0.01 0.01 0.01
0.005
0.005
0 50 100 150 0 50 100 150 0 50 100 150
diameter [pm] diameter [pm] diameter [pim]
Figure 15: NDF at 33mm at point 3 (left), point 4 (cen
ter) and point 5 (right) for MMEF (solid line)
and experiment histogramm).
Conclusions
Starting from the coupling of MEF and MA, the Multi
fluid Mesoscopic Eulerian Formalism (MMEF) has been
0 50 100 150 0 50 100 150 0 50 100 150
diameter [pm] diameter [pm] diameter [gm]
Figure 16: NDF at 63mm at point 6 (left), point 7 (cen
ter) and point 8 (right) for MMEF (solid line)
and experiment histogramm).
derived. Preliminary test cases, designed to be repre
sentative of the final application, highlight the necessity
to take into account polydispersion. However, they also
show the limitation of numerical schemes used here to
simulate a dispersed phase, especially in vaccum zones.
Available numerical procedures allow to limitate the in
fluence of this weakness.
Application of MMEF to the MERCATO test rig
shows a good agreement between MMEF, MEF and ex
periments for velocity profiles at 9mm, making a good
validation of the FIMUR injection strategy. For farther
profiles (33mm and 63mm), MEF and MMEF show sim
ilar results in the central zone, but exhibit some discrep
ancies for the radial velocity. Only MMEF is able to
capture the external zone, where mainly big droplets are
present.
MMEF is also able to capture NDF profiles in the
external zone (points 5 and 8). Results at other points
show some discrepancies, although MMEF captures the
global behaviour.
Such results confirm the necessity to take into ac
count polydispersion in eulerian simulation of such
aeronauticaltype application. The main effect is the
spatial segregation due to size/velocity correlations,
which drive the drag force.
The next step of this work is to apply higher order
multifluid approaches to the simulation of the MER
CATO configuration. These methods use more than one
moment for the liquid phase. As MEF is a twomoment
method, the first step is to implement 2 d order multi
fluid methods of Laurent (2006), Dufour et al. (2005).
But the increase of order is coupled with the problem
of the advection of moment sets, which needs to be
addressed (Kah et al. (2009), Wright (2007)).
An other interesting and promising approach for such
applications is the Direct Quadrature Method Of Mo
ment Fox et al. (2008). In this approach, the pdf is not
discretized into fixed sections, but is defined through a
Gauss quadrature, and can be seen as a set of Diracs,
linked by a linear system. This approach has proven
its ability to simulate evaporation and coalescence with
very low diffusion in phase space, allowing to limit the
number of Diracs compared to the number of sections
needed by the classical multifluid approach. The origi
nal formalism has been extended by taking into account
a velocity dispersion in Belt et al. (2009), and applied to
the simulation of coalescence in Homogeneous Isotropic
Turbulence in Wunsch et al. (2009).
Considering the evaporation, the strong influence of
the droplet dissappearence with high order moment
method or DQMOM is demonstrated in Fox et al.
(2008). Modeled by a flux evaluated at zero size, a sta
ble approach is shown in Fox et al. (2008), but its lack of
precision needs further studies. A solution is proposed in
Massot et al. (2010) with a high order moment method.
In this approach, the pdf is defined by its four first mo
ments. A reconstruction by Entropy Maximization al
lows to compute the evaporation flux at zero size. The
pdf is then evaporated using an integrated formulation
of the DQMOM approach. The robustness and the pre
cision of this method has been proven in Massot et al.
(2010), and make it a good candidate for final applica
tions since it only involves one section, as long as the
problem of moments set advection in complex industrial
applications is addressed.
Acknowledgements
This research was supported by ANRT/IFP Ph.D. grant
for A. Vi6, DGA Ph.D. grant for M. Sanjos6.
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