Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 2.5.4 - Modeling of the Random Uncorrelated Velocity Stress Tensor for Unsteady Particle Eulerian Simulation in Turbulent Flows
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 Material Information
Title: 2.5.4 - Modeling of the Random Uncorrelated Velocity Stress Tensor for Unsteady Particle Eulerian Simulation in Turbulent Flows Particle-Laden Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Masi, E.
Riber, E.
Simonin, O.
Gicquel, L.Y.M.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: particle-laden flows
Eulerian modeling
one-point closure
axisymmetric state
 Notes
Abstract: In the Eulerian framework, the dispersed phase of a particle-laden turbulent flow may be modeled by using a statistical approach known as the mesoscopic Eulerian formalism (Février 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 second-order 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 particle-laden homogeneous isotropic turbulence but it fails when performing preliminary a posteriori tests (Riber 2007) in mean-sheared turbulent particle-laden 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.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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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, F-31400 Toulouse, France
CNRS; Institut de MWcanique des Fluides de Toulouse; F-31400 Toulouse, France
t CERFACS; 42, avenue Gaspard Coriolis, F-31057 Toulouse Cedex 01, France
masi@imft.fr, simonin@imft.fr
Keywords: particle-laden flows, Eulerian modeling, one-point closure, axisymmetric state




Abstract

In the Eulerian framework, the dispersed phase of a particle-laden 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 second-order
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 particle-laden homogeneous isotropic turbulence but it fails when performing preliminary a posteriori tests (Riber
2007) in mean-sheared turbulent particle-laden 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 second-rank 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 one-point con-
ditional probability distribution function (p.d.f.) asso-


ciated to one-fluid 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 particle-velocity space of
a Boltzmann-like 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 spatially-uncorrelated 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 particle-velocity 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 Navier-Stokes
(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 mean-sheared turbu-
lent particle-laden flows (Hishida 1987). Comparisons
with experiments showed a re-laminarization 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 second-order 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 so-called 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) velocity-gradient 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 second-order 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
mean-shear 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 particle-laden
turbulent planar jet is chosen. Direct numerical sim-
ulations of the turbulence, one-way coupled with La-
grangian tracking of particles, were performed. Several
simulations varying in particle inertia were carried out.
Lagrangian values were then post-processed 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 y-planes parallel to the streamwise direction
and the value associated to each simulation refers to that
found at the periphery of jet. The so-called "periphery"
of the jet is a portion of the slab of y-coordinate ~ 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 Navier-Stokes equa-
tions are integrated by using a third order Runge-Kutta
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 Runge-Kutta 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 k-particle 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 y-planes. 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 particle-density-weighted 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 exact-to-modeled 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 well-known limit of
every viscosity-like 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 re-laminarization
of the dispersed phase observed by Riber (2007) when
performing Eulerian-Eulerian 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 rate-of-
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 one-phase tur-
bulent flow analysis (e.g., Tao et al. 2002, Higgins et
al. 2003). This parameter, originally called "strain-state
parameter" and used to study the "shape" of the defor-



























"' ". /
"" *' ,
-2 -1 0 1 2 3
S10-3


Figure 2: P.d.f. of the exact (dot-dashed 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 rate-of-strain 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 rate-of-strain 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 second-rank 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 rate-of-strain 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 one-component 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 one-component
limit state of the deviatoric RUM are used in order to
suggest a new formulation of the stresses in function
of the rate-of-strain tensor. This new formulation over-
comes the problem of the tensor magnitude because it
uses only information of the rate-of-strain 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 well-known
relation may be used


R*XR* AR*XR*


6* p,ij
2.. ,


for which the eigenvalues associated to the one-
component-limit 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 one-component 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 one-component 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 viscosity-type 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 rate-of-strain tensor as
the square-root 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 rate-of-strain:
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 one-component 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 one-component 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 rate-of-strain 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 one-component 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 viscosity-like 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
S10-3

Figure 9: P.d.f. of the exact (dot-dashed 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 particle-laden 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 rate-of-strain 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 one-component limit state. Informa-
tion of the tensor structure were then used for build-
ing alternative models. The linear viscosity-type 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 TIMECOP-AE project (Project AST5-
CT-2006-030828). 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.557-577, 1950

F6vrier P, Simonin O. and Squires K.D., Partitioning
of particle velocities in gas-solid turbulent flows into
a continuous field and a spatially uncorrelated random
distribution: theoretical formalism and numerical study,
J. Fluid Mech., 533, pp. 1-46, 2005

Higgins C.W., Parlange M., and Meneveau C., Align-
ment Trends of Velocity Gradients and Subgrid-Scale
Fluxes in the Turbulent Atmospheric Boundary Layer,
Boundary-Layer Meteorology, 109, pp.5983, 2003

Hishida K., Takemoto K. and Maeda M., Turbulent
characteristics of gas-solids two-phase 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 Prandtl-Kolmogorov


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. 48-50, 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: 6448-6472, 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. 1838-1847, 1994

Maxey R. and Riley J., Equation of motion of a small
rigid sphere in a nonuniform flow, Phys. Fluids, 26 (4),
883-889, 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.inp-toulouse.fr/

Reeks M.W., On a kinetic equation for the transport
of particles in turbulent flows, Phys. Fluids A 3,
pp.446-456, 1991

Riber E., Modeling turbulent two-phase flows using
Large-Eddy 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 subgrid-scale model for finite-difference large-
eddy simulations, Phys. of Fluids, 7 (11), pp.2831-2847,
1995

Simonin O., F6vrier P. and Lavidville J., On the spatial
distribution of heavy-particles velocities in turbulent
flow: from continuous field to particulate chaos, J. of
Turb., 3, pp.l-18, 2002

Simonin O., Prediction of the dispersed phase turbu-
lence in particulate laden jet, In Proc. 4th Int. Symp. on
Gas-Solid lows, ASME-FED, 121, 197-206, 1991

Simonsen A.J. and Krogstad P.-A., Turbulent stress
invariant analysis: Clarification of existing terminology,
Phys. Fluids 17, 088103, pp.1-4, 2005

Tao B., Katz J. and Meneveau C., Statistical geometry
of subgrid-scale stresses determined from holographic
particle image velocimetry measurements, J. Fluid
Mech., 457, 35-78, 2002












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