7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Modeling the Random Uncorrelated Velocity Stress Tensor for Unsteady Particle
Eulerian Simulation in Turbulent Flows
E. Masi* E. Ribert, P. Sierra, O. Simonin* L.YM. Gicquelt
University de Toulouse; INPT, UPS; IMFT; All6e Camille Soula, F31400 Toulouse, France
CNRS; Institut de MWcanique des Fluides de Toulouse; F31400 Toulouse, France
t CERFACS; 42, avenue Gaspard Coriolis, F31057 Toulouse Cedex 01, France
masi@imft.fr, simonin@imft.fr
Keywords: particleladen flows, Eulerian modeling, onepoint closure, axisymmetric state
Abstract
In the Eulerian framework, the dispersed phase of a particleladen turbulent flow may be modeled by using a statistical
approach known as the mesoscopic Eulerian formalism (F6vrier et al., 2005). At the first order, the evolution
of the first moments of the conditional probability distribution function, namely the mesoscopic particle number
density and the mesoscopic particle velocity, supply the description of the dispersed phase, and the secondorder
moment appearing in the momentum equation, accounting for the momentum transport due to the particle random
uncorrelated velocity, needs to be modeled. In literature, in order to close this moment, referred to as RUM stress
tensor, an additional transport equation for the spherical part of the tensor, including the RUM kinetic energy, has
been used and a viscosity model for the deviatoric RUM has been suggested (Simonin et al. 2002). The latter when
a priori (Moreau 2006) and a posteriori (Kaufmann et al. 2008, Riber 2007) tested gives quite satisfactory results
in particleladen homogeneous isotropic turbulence but it fails when performing preliminary a posteriori tests (Riber
2007) in meansheared turbulent particleladen flows (Hishida 1987).
In this paper, the concern of the adequacy of such a model for predicting uncorrelated stresses is addressed. An
analysis of the tensor structure will make it possible to supply an alternative modeling less sensitive to the change
of the particle inertia. The new models are a priori checked over several simulations varying in Stokes number; a
posteriori tests are ongoing.
Nomenclature
Symbols
g mesocopic quantities
6g RUM quantities
G secondrank tensors
G* Traceless tensor
Introduction
FRvrier, Simonin & Squires (2005) suggested a statis
tical approach, known as the mesoscopic Eulerian for
malism (MEF), able to describe the local and instan
taneous behavior of particles interacting with unsteady
and inhomogeneous turbulent flows. Eulerian quanti
ties are defined as the moments of the onepoint con
ditional probability distribution function (p.d.f.) asso
ciated to onefluid flow realization. The Eulerian local
and instantaneous transport equations are then obtained
in the general frame of the kinetic theory of dilute gases,
namely by integration over the particlevelocity space of
a Boltzmannlike equation (Reeks 1991). Such an ap
proach consists in partitioning the particle velocity into
two contributions: i) an Eulerian particle velocity field
referred to as "mesoscopic" which is a continuous field
shared by all the particles and accounting for correla
tions between particles and between particles and fluid;
ii) a random spatiallyuncorrelated contribution, asso
ciated with each particle and satisfying the molecular
chaos assumption. It is referred to as "RUM" (Random
Uncorrelated Motion) and it is characterized in terms of
Eulerian fields of particlevelocity moments. The origi
nality of this formalism is to make it possible to separate
contributions which are intrinsically different, which do
not interact in the same way with the different scales of
the turbulence and thus should be modeled in their own
way.
At the first order, in isothermal condition, the dispersed
phase is fully described by the evolution of the meso
scopic particle number density and the mesoscopic par
ticle velocity as follows (F6vrier et al. 2005):
9, p npup
at xi
OnpUip,i 9pippil,
at 9Oxj
" (tp,i
T7P
x(2)
The first term on the right hand side (r.h.s.) of Eq. (2)
represents the momentum transfer through the drag force
while the second term is the momentum transport due
to the particle uncorrelated velocity contribution. The
RUM tensor 6Rp,ij may be considered as equivalent
to the stress tensor accounting for in the NavierStokes
(NS) equations, similarly derived by using the Boltz
mann kinetic theory. The RUM is composed of a spher
ical part including the RUM kinetic energy 60p which
corresponds to the translation temperature in kinetic the
ory of dilute gases, and a deviatoric part 6SR*j which is
analogous to the viscous contribution due to the thermal
agitation in the NS equations:
6Rp,ij = 6Rij + 6Rp,kk6iJ = 6Rpij + ., .2
S3.
(3)
The RUM kinetic energy is generally obtained by an ad
ditional closed transport equation (e.g., see Kaufmann
et al. 2008) and only the deviatoric RUM needs to be
modeled. The latter is the topic of the present study.
State of the art
In literature, only one model has been suggested in or
der to close the unknown deviatoric RUM (Simonin et
al. 2002). This model was a priori (Moreau 2006) and
a posteriori (Kaufmann et al. 2008, Riber 2007) tested
in homogeneous isotropic turbulence showing quite sat
isfactory results for moderate Stokes numbers. Instead,
such a model strongly fails (Riber 2007) when perform
ing preliminary a posteriori tests in meansheared turbu
lent particleladen flows (Hishida 1987). Comparisons
with experiments showed a relaminarization of the dis
persed phase: the particle mesoscopic fluctuations were
considerably damped and only the RUM contributed to
the particle agitation.
In this work the concern of the adequacy of such a model
to predict the secondorder moment is addressed.
The viscosity model is here recalled:
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
where vt = 60p is the socalled mesoscopic viscosity,
T is the mesoscopic particle relaxation time and
sTij 1 ai0, p i 2 u,~,m
S 2 Oxjy Oxi 3 *m
is the deviatoric symmetric part of the mesoscopic
(1) velocitygradient tensor accounting for shearing or dis
tortion of any element of the dispersed phase. The vis
cosity model, hereinafter referred to as "VISCO", may
be obtained from the secondorder moment transport
pnnntinn"
D
np Dt 8Rp1ij
npRp,k j aO
aXk
pStRp,i, i
~X k
2 6Rpij a pQp,ijk (6)
assuming equilibrium, i.e. neglecting all the transport
terms, and light anisotropy.
The assumptions of both equilibrium and light
anisotropy revealed wrong especially in presence of a
meanshear and when increasing inertia. Results of a
priori testing will confirm the inadequacy of such a
model to predict deviatoric RUM stresses.
Numerical simulations
In order to a priori test the model in presence of a mean
shear, a configuration corresponding to a particleladen
turbulent planar jet is chosen. Direct numerical sim
ulations of the turbulence, oneway coupled with La
grangian tracking of particles, were performed. Several
simulations varying in particle inertia were carried out.
Lagrangian values were then postprocessed and e\
act" Eulerian mesoscopic fields were computed by us
ing a projection algorithm (Kaufmann et al. 2008). The
number of Stokes, ranging from 0.03 to 10, was com
puted over a "large" timescale of the turbulence seen
by the particles, estimated by evoking the Tchen equi
librium in the spanwise direction which remains mean
flow free (Simonin 1991); the Stokes numbers were esti
mated over yplanes parallel to the streamwise direction
and the value associated to each simulation refers to that
found at the periphery of jet. The socalled "periphery"
of the jet is a portion of the slab of ycoordinate ~ 0.7
corresponding to the periphery at the initial time.
The slab corresponds to the dispersion of a particle
laden turbulent planar jet into a homogenous isotropic
decaying turbulence. The simulation domain is a cube
of length size 2w7, with a mesh grid of 1283 cells and
periodic in boundary conditions. Within the slab, of
d/Lbox 0.25 of width, particles are randomly embed
ded. Mean velocity of particles is imposed equal to that
erruation:
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
of the gas at the initial time. The NavierStokes equa
tions are integrated by using a third order RungeKutta
time stepping and a sixth order compact finite difference
scheme on a cartesian grid. Also the advancement in
time of the Lagrangian tracking is ensured by a third or
der RungeKutta scheme. The interpolation of the turbu
lent fields at the particle location is performed by a third
order Lagrange polynomial algorithm. Without gravity,
for small heavy particles, Lagrangian equations govern
ing the motion of each kparticle are (Maxey & Riley
1983):
dx(
dt
( k
(k) dv()
p dt
1 (V()
r,
_ (k)
U1 v)
where uft is the undisturbed velocity of the gas at the
particle centre location. The particle relaxation time Tp
is defined in the Stokes regime as:
Ppd2
P p (8)
18Pl
where pp is the particle density, dp the particle diameter
and pf the dynamic viscosity of the gas.
A priori testing
"Exact" Eulerian mesoscopic fields, obtained by La
grangian simulations, are then used to a priori check
models. The model accuracy is evaluated by means of
correlation coefficients at scalar level. Correlation co
efficients are computed as follows (Salvetti & Banerjee
1995):
< AB > < A >< B >
c =(9)
(< A2 > < A >2)(< B2 > < B >2)
where brackets denote averages over yplanes. The eval
uated scalar quantity represents the dissipation, by shear,
of the mean mesoscopic energy into the mean RUM ki
netic energy and it is written, accounting for the sign, as
follows:
9u' i, 9du' ,
< np*RI;j O > np < 6R aij >p (10)
where np is the mean particle number density and the
operator < >p denotes particledensityweighted aver
age (over planes); if a mean mesoscopic velocity is de
fined Up, < up,i >p, then the fluctuating component
is the residual ui,i Ui Up,i. Moreover, the p.d.f. of
the scalar quantity is also evaluated. The p.d.f. is plotted
including the magnitude ratio between exact and mod
eled averaged quantities; in this manner only the shape
2 4 6 8
Figure 1: "VISCO" model (Eq. 4): correlation coeffi
cients evaluated at scalar level over the mean
dissipation (Eq. 10), at the periphery of jet.
is evaluated. The exacttomodeled magnitude ratio rep
resents a multiplicative coefficient which should be ac
counted for in order to predict the exact "mean" magni
tude. This quantity is very important and it can seriously
affect the success of the model if its value is far from one
and it is not taken into account. Figure 1 shows corre
lation coefficients computed at the periphery of the jet,
far from the start, for the model "VISCO". The model
gives quite good correlations for small Stokes numbers
while it fails when inertia increases. Looking at the
p.d.f. of the mean dissipation (Figure 2), results show
that the model is no able to reproduce positive values,
meaning the local inverse transfer of energy from RUM
to mesoscopic motion. This is a wellknown limit of
every viscositylike model. Concerning the magnitude
of modeled against exact dissipations, Figure 3 shows
that such a model strongly overestimates the predictions;
this behaviour is consistent with the relaminarization
of the dispersed phase observed by Riber (2007) when
performing EulerianEulerian simulations of the Hishida
(1987) jet configuration. The overestimation increases
with the particle inertia.
In order to supply an alternative to the model "VISCO",
the structure of the deviatoric RUM (R*) and rateof
strain (S*) tensors is analysed and information are used
to construct new models.
The structure of the tensors
In order to study the structure of the tensors R* and
S*, since they are locally defined, we use a local dimen
sionless parameter proposed by Lund & Rogers (1994)
and used by several authors involved in onephase tur
bulent flow analysis (e.g., Tao et al. 2002, Higgins et
al. 2003). This parameter, originally called "strainstate
parameter" and used to study the "shape" of the defor
"' ". /
"" *' ,
2 1 0 1 2 3
S103
Figure 2: P.d.f. of the exact (dotdashed line) against
modeled by "VISCO" (solid line) local dissi
pation for simulation corresponding to St ~
1, at the periphery of jet.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
is small, while 1 means one large eigenvalue and the
other two, identically small. For the sake of simplicity,
we will use the same vocabulary than that used in turbu
lence, missing out, for the moment, any physical mean
ing. Figures 4 and 5 show the p.d.f. of s* evaluated for
both the deviatoric RUM and rateofstrain tensors, for
three different Stokes numbers and far from the initial
time. Results show that the tensors R* and S* behave as
in axisymmetric contraction and expansion respectively,
independently from the inertia. Same results are found
over all the planes of the jet. However, for large inertia
and only at the periphery of the jet, the tensor S* tends
toward a more random distribution. Moreover, the
........ S t
St
..... St
0 ...... ..
1 0.5 0 0.5 1
Figure 4: P.d.f.s of the parameter s* evaluated for R*,
at the periphery of jet.
Figure 3: Profiles of exact (line) and modeled by
"VISCO" (symbols) mean dissipations, for
simulation corresponding to St ~ 1.
nations caused by the rateofstrain tensor, may be used
to investigate the structure of a traceless tensor giving lo
cal information about the relative magnitude of the ten
sor eigenvalues and reproducing information similarly to
that found in the invariant Lumley's map. Hereinafter,
the bold writing denotes secondrank tensors, brackets
{.} represents the trace and the asterisk means traceless
when associated with a tensor. The dimensionless pa
rameter is then defined as follows:
3/ 6A1 A2 A3
s = (11)
where A1, A2 and A3 are the tensor eigenvalues. This
parameter is bounded between +1 and 1 where +1
corresponds to an axisymmetric expansion and 1 is an
axisymmetric contraction, in classical turbulence way of
speaking. From a mathematical point of view +1 means
that two identical eigenvalues dominate and that the third
*....... St = 0.03
St= 1
.. St= 3
o fluid
0.5 0 0.5 1
Figure 5: P.d.f.s of the parameter s* evaluated for S*, at
the periphery of jet. The rateofstrain of the
fluid is also evaluated.
analysis shows that the tensor R* behaves as in one
component limit state, which means that the smallest
eigenvalues of RUM tend to zero. This behaviour is de
velopped immediately, in terms of time, in all the zones
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
of the jet, and it persists up to the end of each simula
tion for every Stokes number. In this case, theoretically,
the eigenvalues are known. Indeed, following Lumley
(1978), the eigenvalues of an averaged anisotropy tensor
g*
g G 1
{G} 3'
G*
{G}'
corresponding to the onecomponent limit, are A1 = ,
A2 A3 S in descending order. Consequently, the
eigenvalues of the averaged tensor G or that of its devi
atoric part G* will be deducted (Simonsen & Krogstad
2005) via the relation
AG, 1
) {G 3
{G} 3
Gc*
{G}
In the same way, but locally, we define the RUM
anisotropy stress tensor as follows:
Modeling by using eigenvalues
In this section, information about the onecomponent
limit state of the deviatoric RUM are used in order to
suggest a new formulation of the stresses in function
of the rateofstrain tensor. This new formulation over
comes the problem of the tensor magnitude because it
uses only information of the rateofstrain principal di
rections assuming alignment between tensors.
As R* and S* are real and symmetric, then they always
have an orthonormal basis composed by their real eigen
vectors. Thus, in the principal axes, the assumption of
the relative alignment of the two tensors may be traduced
by the equality between their eigenvectors, being the
magnitude taken into account by the respective eigenval
ues. By using the linear transformation, the wellknown
relation may be used
R*XR* AR*XR*
6* p,ij
2.. ,
for which the eigenvalues associated to the one
componentlimit are known. Figure 6 shows the p.d.f.
of the exact eigenvalues of b* measured at the periphery
of jet for a Stokes number close to one. The analysis
shows that over all planes, for every Stokes number,
the onecomponent limit state is reproduced. Hence,
) where XR* is a right column eigenvector of R* and AR*
(14)
its correspondent eigenvalue. This relation may also be
written in a matrix form as follows:
i*XR,* XR* AR*
where XR* is the matrix of the tensor eigenvectors and
AR* is the diagonal matrix of the eigenvalues. As R*
and AR* are similar matrix, XR* is nonsingular, we
may also write
R* XR*AR*XR!.
Then, using the assumption of alignment, which leads to
XR* = Xs*, the deviatoric RUM tensor can be straight
forwardly obtained as follows:
R* Xs*AR*XS* 1
0.2 0 0.2 0.4 0.6
Figure 6: P.d.f.s of the eigenvalues of b*, at the periph
ery of the jet, for a simulation corresponding
to St ~ 1.
the eigenvalues of the deviatoric tensor R*, sorted in
descending order, are:
A R (
+1,,,
0
0
0 0
0
_ .o o
.5 ^
where AR* includes the known onecomponent limit
eigenvalues. This model, referred to as "EIGEN", here
evaluated by the a priori analysis, is instead less easy
to handle when performing a posteriori Eulerian simu
lations because of the operation of matrix inversion and
since it requires to initially check the right position of
the deviatoric RUM eigenvalues into the matrix. In the
next section another approach will be used leading to an
equivalent formulation in Cartesian frame of reference.
A viscositytype model for axisymmetric
dispersed phases
In this section, the same procedure than in Jovanovid et
al. (2000) is applied. Following their idea for turbulent
flows, tensors are assumed, as observed, axisymmetric
i
I
~.L:....
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
with the respect to a preferred direction and written in a
bilinear form (Chandrasekhar 1950) leading to:
Sij =.'I + BAiAj (19)
Rij = C6ij + DAiAj. (20)
As the tensors are traceless, Eq. (19) and (20) can be
rewritten as follows:
S* =j A.' 3AAiAy (21)
86Rij Cij 3CAiAj. (22)
Defining the magnitude of the rateofstrain tensor as
the squareroot of its second invariant S II /2
{S*2}1/2, Eq. (21), if contracted, makes it possible to
obtain A as a function of S:
A =sign(IIIs) (2 S (23)
3( S
where Ills = {S*3} is the third invariant of the
sor, and its sign accounts for the possibility to hav
axisymmetric contraction or expansion. Eq. (21)
becomes:
S = sign(IIIs)
sp*,j2
3
3 2)
2 3
ten
e an
then
1/2
S
(24)
From Eq. (24), the product AiAj may be expressed in
function of the rateofstrain:
2 / \ 1/2 Sg* +lj 1
AiA, sign(IIIs) ()+ 6 (25)
3 2 S 3
and injected into Eq. (22) which takes the form:
2 /(3 /2 5*
6R =j sign(IIIs)3C 3 '. (26)
The same relation may be provided for the anisotropy
tensor
bt* sign(IIIs) (3) 2 (27)
which contracted leads to the second invariant of the
anisotropy lib, from which we can obtain C as follows
C2 IIb6Op2. (28)
3
As well as for A, the sign of C is established by the sign
of the third invariant IIIb accounting for axisymmetric
contraction or expansion
C = sign(IIIb) (2)1/2 20
3"
0.1
0.1
5 0
{R*S*}/ < {R*S*}
5 10
Figure 7: Conditional average of the normalized IIIs
over the normalized contracted product
R* ijS ,i, for a simulation corresponding to
St ~ 1, at the periphery of jet.
Including C into Eq. (26), we finally obtain
S*
6R,ij = sign(IIIs)sign(IIIb)II 2..,.
(30)
1The product of the invariant signs accounts for the fact
\,, that the tensors may be both in the same configuration
of "contraction" or c\plnSion" (positive sign) or in the
opposite configuration (negative sign). Numerical simu
lations showed that the local sign of IIIb is always posi
tive (b* is strictly in axisymmetric contraction) and only
that of Ills changes. Moreover, numerical simulations
pointed out the local onecomponent limit state of b*
which gives IIb So, the model may be rewritten
as:
6R,= sign(IIIs) 2., 5.
/ S
The latter represents a model where the axisymmetry of
tensors, their alignment, and the onecomponent limit
state are assumed. Concerning the sign of the model, the
sign of Ils is used to reproduce both positive and neg
ative viscosities, i.e. negative and positive values of the
contracted product R P* SP,*j, theoretically in that spe
cial case in which the axisymmetric rateofstrain ten
sor moves from a configuration of expansion to contrac
tion and vice versa. However, this assumption may be
relaxed in practical applications. Figure 7 shows the
conditional average of Ills over the contracted product
R* iSi* for a simulation corresponding to a number
of Stokes St ~ 1. Results support such an assumption.
Hereinafter, the model corresponding to Eq. (31) will
refer to as "AXISY" model.
The relation between the model "EIGEN" and the
model "AXISY" should be understood. The model "AX
0 
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
ISY" rewritten in terms of anisotropy leads to:
1/2 2 S*
=siqgn(IIIs) 3 (32)
As b* and S* are real and symmetric, they may be writ
ten in the principal axes. Then, the assumption of the
alignment, which leads to impose Xb*=Xs*, involves
that also the eigenvalues are the same, of course account
ing for every scalar included in the model. So, the ex
pression for the eigenvalues becomes:
x* sign(IIIs) ( A. (33)
Contracting the axisymmetric tensor S*, for instance
when written in the configuration of expansion as fol
lows
2S 0
0 S
0 0
2S, 0
0 0
0 0
leads to write S = 6SA where SA is the largest eigen
value of S*. Including this relation into Eq. (33) and the
right sign, gives:
A 1 A* (34)
3 SA
which is fulfilled in our onecomponent limit state, giv
ing evidence of the same model assumptions. Hence,
theoretically, accounting for all the assumptions, using
the model "EIGEN" is like to use the model "AXISY"
in the principal directions. In practice, the two mod
els can give different results. The models "EIGEN" and
"AXISY" are viscositylike models accounting for pos
itive or negative local viscosity and using the timescale
J(S 1) at the place of the particle relaxation time fp
of the model "VISCO". Figure 8 shows correlation co
efficients for all models evaluated over several simula
tions varying in Stokes number. Results for the mod
els "EIGEN" and "AXISY' show a strongly improve
ment of predictions when compared with results of the
model "VISCO". The ability of the new models to pre
dict the reverse of sign is given by Figure 9 which shows
the p.d.f. of the local dissipation evaluated by using the
model "AXISY". Moreover, predictions of the mean dis
sipation magnitude are also improved, as shown by Fig
ure 10 where the exact mean dissipation is compared to
that modeled by using "VISCO" and "AXISY'.
Conclusions and perspectives
In this paper the concern of the modeling of the second
order tensor accounting for in the momentum equation
Figure 8: "VISCO" (triangles), "AXISY" (circles) and
"EIGEN" (stars) models: correlation coeffi
cients evaluated at scalar level for the mean
dissipation (Eq. 10), at the periphery of jet.
103
102
100
3 2 1 0 1 2 3
S103
Figure 9: P.d.f. of the exact (dotdashed line) against
modeled by "AXISY' (solid line) local dissi
pation for simulation corresponding to St ~
1, at the periphery of jet.
of the dispersed phase was addressed. An a priori study
on a particleladen turbulent planar jet, pointed out the
inability of the viscosity model, commonly used, to pre
dict such a tensor when the particle inertia increases.
This model showed poor correlations at scalar level also
overestimating the magnitude of the mean mesoscopic
dissipation. An analysis of the tensor structure, includ
ing that of the rateofstrain tensor, made it possible
to characterize the "shape" of both the tensors. Ten
sors were found axisymmetric and the deviatoric RUM
behaving as in a onecomponent limit state. Informa
tion of the tensor structure were then used for build
ing alternative models. The linear viscositytype models
suggested in this paper strongly improve predictions of
RUM stresses. A further study about nonlinear models
is ongoing as well as preliminary a posteriori tests.
Figure 10: Profiles of exact (line) and modeled by
"VISCO" (triangles) and "AXISY" (circles)
mean dissipation, for simulation correspond
ing to St ~ 1.
Acknowledgements
This work received funding from the European Commu
nity through the TIMECOPAE project (Project AST5
CT2006030828). It reflects only the author's views
and the Community is not liable for any use that may
be made of the information contained therein. Numer
ical simulations were performed by the IBM Power6
machine; support of Institut de Ddveloppment et des
Ressources en Informatique Scientifique (IDRIS) is
gratefully acknowledged.
References
Chandrasekhar S., The Theory of Axisymmetric Turbu
lence, Philosophical Transactions of the Royal Society
of London, Series A, Mathematical and Physical Sci
ences, 242, n 855, pp.557577, 1950
F6vrier P, Simonin O. and Squires K.D., Partitioning
of particle velocities in gassolid turbulent flows into
a continuous field and a spatially uncorrelated random
distribution: theoretical formalism and numerical study,
J. Fluid Mech., 533, pp. 146, 2005
Higgins C.W., Parlange M., and Meneveau C., Align
ment Trends of Velocity Gradients and SubgridScale
Fluxes in the Turbulent Atmospheric Boundary Layer,
BoundaryLayer Meteorology, 109, pp.5983, 2003
Hishida K., Takemoto K. and Maeda M., Turbulent
characteristics of gassolids twophase confined jet,
Japanese Journal of Multiphase Flow 1, 1, pp. 5669,
1987
JovanoviC J. and Otic I., On the Constitutive Relation
for the Reynolds Stresses and the PrandtlKolmogorov
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Hypothesis of Effective Viscosity in Axisymmetric
Strained Turbulence, Journal of fluids engineering, 122,
pp. 4850, 2000
Kaufmann A., Moreau M., Simonin O. and Helie
J., Comparison between Lagrangian and Mesoscopic
Eulerian Modeling Approaches for Inertial Particles
Suspended in Decaying Isotropic Turbulence, J. Comp.
Physics, 227, 13: 64486472, 2008
Lumley J.L., Computational Modeling of Turbulent
Flows, Advances in applied mechanics, 18, pp. 123176,
1978
Lund, T and Rogers M., An improved measure of strain
state probability in turbulent flows, Phys. Fluids, 6 (5),
pp. 18381847, 1994
Maxey R. and Riley J., Equation of motion of a small
rigid sphere in a nonuniform flow, Phys. Fluids, 26 (4),
883889, 1983
Moreau M., Mod6lisation numdrique directed et des
grandes 6chelles des 6coulements turbulents gaz
particules dans le formalisme eul6rien mesoscopique,
PhD thesis, INP Toulouse 2006, avalaible on the web
site http://ethesis.inptoulouse.fr/
Reeks M.W., On a kinetic equation for the transport
of particles in turbulent flows, Phys. Fluids A 3,
pp.446456, 1991
Riber E., Modeling turbulent twophase flows using
LargeEddy Simulation, PhD thesis, INP Toulouse
2007, avalaible on the web site http://ethesis.inp
toulouse.fr/
Salvetti M.V and Banerjee S., A priori tests of a new
dynamic subgridscale model for finitedifference large
eddy simulations, Phys. of Fluids, 7 (11), pp.28312847,
1995
Simonin O., F6vrier P. and Lavidville J., On the spatial
distribution of heavyparticles velocities in turbulent
flow: from continuous field to particulate chaos, J. of
Turb., 3, pp.l18, 2002
Simonin O., Prediction of the dispersed phase turbu
lence in particulate laden jet, In Proc. 4th Int. Symp. on
GasSolid lows, ASMEFED, 121, 197206, 1991
Simonsen A.J. and Krogstad P.A., Turbulent stress
invariant analysis: Clarification of existing terminology,
Phys. Fluids 17, 088103, pp.14, 2005
Tao B., Katz J. and Meneveau C., Statistical geometry
of subgridscale stresses determined from holographic
particle image velocimetry measurements, J. Fluid
Mech., 457, 3578, 2002
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
