7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
A DNS study ranging from dense to dilute turbulent twophase flows
G. Lurett, T. Menardt, A. Berlemontt, J. Reveillont and F.X. Demoulint
CNRS CORIA UMR 6614, University of Rouen, Technopl6e du Madrillet, BP12, 76801 SaintEtienneduRouvray Cedex,
France
gautier.luret@coria.fr and demoulin@coria.fr
Keywords: Atomization, Collision, Modeling, Direct numerical simulation
Abstract
The atomization of a highvelocity Diesel jet is a difficult process to apprehend. The presence of a very dense zone in the
vicinity of the injector exit contributes extensively to the primary atomization and requires particular attention. The modelled
spray behaviour depends strongly on the description of this region. Improvements of atomization predictions have been
obtained thanks to the EulerianLagrangian Spray Atomization (ELSA) model that takes into account explicitly the dense
part of the spray (Lebas, Menard et al. 2009). A particular feature of this approach is to use a transport equation of the
liquid/gas interface density I to generalize the notion of droplet diameter in the dense region of the spray. Several source
terms related to physical phenomena acting on the interface topology are not elucidated yet. The present work focuses on
collision processes that become dominant for dense spray.
Recently, we have used Lagrangian Simulation of a twophase flow to scrutinize the collision/coalescence phenomena in a
moderately dense spray (Luret, Menard et al. 2010). It happens that most of collisions occur between nonspherical droplets.
We have shown that DNS is a possible way to study this phenomenon. Therefore, based on the Archer's code (Menard,
Tanguy et al. 2007), a numerical configuration has been carried out that allows the effect of successive collisions to be
studied statistically. A twophase flow is simulated in a cubic computational domain with periodic boundary conditions at a
prescribed turbulence level, see figure 1. From that study, a relevant parameter has been put forward: the equilibrium Weber
number We'. Recently, the simulations have been extended to a large range of conditions: liquid volume fraction varying
from 1% to 99%, various turbulent kinetic energy and density ratio. The numerical results underline the turbulence effects on
the liquid/gas interface topology.
Introduction
For many years research on atomization has been carried on
to improve the characteristic and the control of sprays. This
is particularly true as far as fuel injection is considered. In
the context of the atomisation of Diesel jet several works
have been proposed to improve the reliability of the
modelling approach (Lebas, Menard 2009, Vallet and
Borghi 1999, Vallet, Burluka et al. 2001, Demoulin, Beau et
al. 2007). The modelling proposal, the socalled ELSA
model is based on a realistic description of the dense zone
of the spray. While most of two phase flow approaches used
for atomisation consider that the liquid phase is dispersed
and composed of liquid parcels mainly isolated, in the
ELSA approach no assumption is done on the topology of
the interface. Instead the liquidgas mixture is considered in
the whole. Within the mixture, the liquid and the gas may
evolve like standard species. The interfacial phenomena are
then characterized mainly by the knowledge of an additional
variable, the surface density. This model has been
developed for flows with high value of the Reynolds
number. Accordingly, corresponding velocity fields fell in
the category of turbulent flows. However, this turbulence
may differ from the classical turbulence encountered in
single phase flows. These two points: surface density and
turbulence in twophase flows are the main subject of the
present work. Because experiments are difficult for dense
twophase flows, they are completed here by a numerical
approach. It is based on a DNS of well controlled twophase
flow. The classical homogeneous and isotropic turbulence
configuration initially proposed by (Eswaran and Pope
1988) to study scalar mixing is extended to twophase flows.
The DNS solver used in this study is presented first, and
then the main features of the ELSA model are recalled. This
show the importance for the surface density equation of
terms like the equilibrium Weber number We Finally
first results are extracted from DNS test case.
Nomenclature
Kinetic turbulent energy
Normal to the interface
Velocity vector (ms1)
Species mass fraction
Schmidt number
Source term for surface density
Weber number
Greek letters
0 Liquid volume fraction
K Main curvature (m 1)
A Level set function (m)
U Dynamic viscosity (kgm s ')
Q Surface density per unit of mass (m2kg')
T Dense function indicator
P Density (kgm3)
I Surface density per unit of volume (m1)
o Surface tension coefficient
Subscripts
G Gas
L Liquid
T Turbulence
Upper scripts
Equilibrium
DNS simulation of the atomisation
The proposed work concerns mainly the atomisation process
that is relevant for Diesel spray, but it has certainly
application for other injection devices. The main drawback
for this kind of atomisation is the lack of experimental data
in the vicinity of the injector tip. The high velocity and high
density variation in this zone prevent to use classical
measurement apparatus. In particular, the diffraction effect
is the main reason of failure for optical diagnostic. Even if
new measurement techniques have been developed (Leick,
Riedel et al. 2007, Blaisot and Yon 2005, Linne, Paciaroni
et al. 2006, Chaves, Kirmse et al. 2004), DNS simulation is
still a very interesting tools to explore the vicinity of the
liquid jet exit.
We use for this work a DNS code "ARCHER" developed at
the CORIA laboratory (Menard, Tanguy 2007, Tanguy and
Berlemont 2005). It has been used already to collect
statistical information in the dense zone of the spray where
nearly no experimental data are available. These simulations
are sufficiently predictive and quantitative to be used for
validation of modelling proposals (Lebas, Menard 2009).
The numerical method describes the interface motion
precisely, handles jump conditions at the interface without
artificial smoothing, and respect mass conservation.
Accordingly, the interface tracking is performed by a Level
Set method. The Ghost Fluid Method is used to capture
accurately sharp discontinuities. The Level Set and VOF
methods are coupled to ensure mass conservation. A
projection method is used to solve the incompressible
NavierStokes equations that are coupled to a transport
equation for level set and VOF functions.
Level Set methods are based on the transport of a
continuous function A which describes the interface
between two phases (Sussman, Fatemi et al. 1998, Sethian
1999). This function is defined by the algebraic distance
between any point of the domain and the interface. The
interface is thus described by the 0 level of the Level Set
function. Solving a convection equation allows to determine
the evolution of the interface in a given velocity field V
(Sethian 1999):
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
 + V.V = (1)
at
Particular attention must be paid to this transport equation.
Problems may arise when the level set method is developed:
a high velocity gradient can produce wide spreading and
stretching of the level sets, such that A no longer remains
a distance function. Thus, a redistancing algorithm
(Sussman, Fatemi 1998) is applied to keep A as the
algebraic distance to the interface.
To avoid singularities in the distance function field, a 5th
order WENO scheme has been used for convective terms
(Jiang and Shu 1996). Temporal derivatives are computed
with a third order Runge Kutta scheme.
One advantage of the Level Set method is its ability to
represent topological changes both in 2D or 3D geometry
quite naturally. Moreover, geometrical information on the
interface, such as normal vector n or curvature K are
easily obtained through:
VA
n K(A) = V n (2)
VAI
It is well known that numerical computation of equation (1)
and a resistance algorithm can generate mass loss in
underresolved regions. This is the main drawback of Level
Set methods. However, to improve mass conservation a
coupling between VOF and Level Set (Sussman and Puckett
2000) method has been performed.
vi
.6.
Figure 1: Instantaneous snapshot of the surface for the
reference case
This DNS approach is applied in a cubic domain with
periodic conditions. The amount of liquid can be prescribed
and the total turbulence energy is kept constant thanks to a
linear forcing (Rosales and Meneveau 2005). The figure 1
present a snapshot of the liquid interface after that a
constant state has been achieved in average. This
configuration is of upmost interest to get data for modelling
in a relatively dense zone of the two phase flow. The mean
liquid volume fraction is for this case: =5 = 5%. It is clear
that a direct application of an optic diagnostic is such a
media will suffer multiple scattering of the light. Here, the
DNS approach is clearly a very efficient tool, though the
11A
numerical convergence is still an issue. While for single
phase flows the smallest length scales can be estimated
(Batchelor for the scalar and Kolmogorov for the velocity)
this is not anymore the case as far as liquid gas flows are
concerned. The diffusive effect that limits the size of the
smallest inclusion for classical scalar field does not act on
this immiscible mixture. The diffusion is replaced by the
surface tension force. But as long as the main curvature
remains small the surface tension force does not act. This is
the case for liquid sheets than can become very thin.
Moreover due to the important density ratio between gas
and liquid, the smallest scale of the liquid scalar field should
modify also the smallest scale of the velocity field.
Despite these drawbacks, we consider that this DNS
configuration is able to give interesting information to
improve modelling in dense two phase flow. The next
section overviews the ELSA model that contains a
modelling description of dense two phase flows.
Description of the ELSA model
The goal of the ELSA model is to describe realistically the
dense zone of the spray. Based on the assumption of a high
Reynolds and injection Weber number values, the ELSA
model is naturally well adapted to Diesel Direct Injection
conditions. This assumption corresponds to an initial
atomization dominated by aerodynamic forces. The global
behaviour of the model and its ability to describe Diesel
injection have been checked out by Lebas et al. (Lebas,
Menard 2009).
A liquidgas flow is considered as a unique flow with a
highly variable density p which can be determined thanks to
the following equation:
1 Y 1Y,
+ (3)
P P1 Pg
Yl corresponds to the mean liquid mass fraction. While,
pg and p, are respectively the gas and the liquid densities.
Considering the twophase flow as a unique mixture flow
with a highly variable density implies that the transport
equation for the mean velocity does not contain any
momentum exchange terms between the liquid and the gas
phases. This "mixture" approach has to be combined with a
turbulence model. The (kc) model is generally used even if
other models have been tested (Demoulin, Beau 2007). A
regular transport equation for the mean liquid mass fraction
Y1 can be written in the complete case with a source term
representing the effect of vaporization:
a+ Y + a F P= .vELSAQ (4)
at ax, ax, Sc,, axJ
is the liquidgas interface density per unit of mass and
mvELSA represents the vaporization rate per unit of mass
(Abramzon and Sirignano 1989, Sirignano 1999).
To determine the amount of surface between the two phases,
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
classical approaches consist in considering spherical liquid
drops and using the diameter as geometrical parameter. But
a more general parameter has to be used where a diameter
of droplet cannot be defined: the liquidgas interface density,
noted I when expressed per unit of volume or Q when
given per unit of mass. The following equation relates both
definitions:
pQ =Y (5)
The transport equation for this variable is postulated. In the
latest version of the ELSA model, it takes the following
form (Lebas, Menard 2009):
at ax, &x Sc,Q ax, 9
+ T(S + Sturb ) (6)
+ (I T^coill /coal + S2ndBU + Svapo)
This equation must be applicable from the dense zone up to
the dispersed spray where droplets are eventually formed. In
this latter case, an equivalent diameter of Sauter can be
defined using the liquidgas interface density and the mean
liquid mass fraction :
6Y
D32 (7)
pQ
Each source term S, (equation 6) models a specific
physical phenomenon encountered by the liquid blobs or
droplets.
Snt is an initialization term, taking high values near the
injector nozzle, where the mass fraction gradients take its
highest values. It corresponds to the minimum production of
liquidgas interface density necessarily induced by the
mixing between the liquid and gas phases (Beau and
Demoulin 2004).
turb t
Sturb corresponds to the production/destruction of liquid
gas interface density due to the turbulent flow stretching and
the effects of collision and coalescence in the dense part of
the spray. It is supposed to be driven by a kind of turbulent
time scale Tt. This production/destruction term is defined
to reach an equilibrium liquidgas interface density 2 It
corresponds to the quantity of surface obtained at
equilibrium under given flow conditions. Several
formulations can be proposed. Without any additional
information the equilibrium Weber number has been taken
as (Lebas, Menard 2009):
Wed Ik Y, 1
Wedense dene
Pl"I^ dense
Once the equilibrium Weber number is given the
corresponding equilibrium surface density is given by:
= _
dense T *
PIOtWedense
(10
Where oC is the surface tension of the liquid phase.
Scoll /coal models the production/destruction of liquidgas
interface density due to the effects of collision and
coalescence in the dilute spray region. Various forms of this
term have been proposed (Luret, Menard 2010). One is
compatible to the production term used in the dense zone of
the spray (equation 8). But the others are based on binary
collisions between droplets and take forms that are not
similar to the one propose in equation (8). However, they all
predict an equilibrium state if the liquid volume fraction
remain constant in a flow with a constant turbulent agitation.
This equilibrium state can be characterized by a Weber
number. The proposed equilibrium Weber numbers have
different values depending on the model. Weber numbers
have many definitions depending on the phenomenon under
study. In the context of collision, it takes the following
form:
We = 4 = 12 15
UoQ
This definition of Weber number will be kept in the
following.
S2ndBU deals with the production of liquidgas interface
density due to the effects of secondary breakup in the dilute
spray region. This source term is derived from the work of
Pilch and Erdman (Pilch and Erdman 1987). It enables the
estimation of the breakup time scaleT2ndB accordingly to
the Weber number of the gas phase We thanks to
empirical correlations.
Vaporisation is characterized thanks to po It comes from
a classical adaptation of the "D2" law of vaporization
models for droplets and deals with the effects of destruction
of liquidgas interface density due to vaporization.
The transport equation of 2 takes into account several
physical phenomena encountered by the liquid phase. Some
of them are specifically observed in the dense zone of the
spray and other are dedicated to dispersed spray regions. A
function TI has been introduced to switch from the dense
formulation to the dispersed formulation continuously and
linearly in term of liquid volume fraction (Lebas, Menard
2009).
This description of the ELSA model show how an Eulerian
method can be derived to deal with atomisation. Though it
has been shown (Lebas, Menard 2009) that the presented
form of the model is able to capture the global features of
the atomisation of a Diesel jet, more detailed studies are still
required. In the following the DNS configuration previously
described will be used to scrutinize the flows for various
liquid volume fractions varying from dilute to very dense
spray.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Behaviour liquidgas flows in statistically constant
environment
The motivation of this study is to scrutinize a part of a liquid
jet during its atomization. The test case is based on previous
calculation of Diesel jet (Lebas, Menard 2009). This is why
the following test conditions have been retained:
p (kg/m3) 25.0
tg (kg/m/s) 1.879 105
p (kg/m3) 753.6
pt (kg/m/s) 1.337 103
o (N/m) 2.222 102
Pi Pg 30.14
Table 1: Fluids properties
They correspond to an injection of Diesel like liquid in air at
a pressure of about 25 bars. Concerning the resolution there
is mainly two opposite constraints: in one hand the smallest
scales of the flow have to be resolved and in other hand the
total size of the domain must be big enough to prevent any
bias due to forced periodic conditions.
The smallest sizes of the flow could be the Kolmogorov
scales, however the viscosity is different between the gas
and the liquid. Moreover, as outlined before the
Kolmogorov scales should be perturbed by the scalar field.
Because of the inertia small liquid structures may produce
in the gas smaller scales. On the contrary surface tension
force tends to promote a coherent motion inside a liquid
parcel or gas parcel. Clearly, all requirements cannot be
satisfied at the same time. Thus a kind of arbitrary choice
has prevailed when determining the following reference
case:
Volume (m3) 0.01
Mesh () 128
S(%)_ 5
k (m2/s2) 0.08
Table 2: Reference test case
To maintain a constant turbulence level a linear forcing
procedure has been used. A snapshot of the liquid surface is
presented once the flow has been established in figure 1.
The figure 2 presents the turbulent kinetic energy obtained
for the mixture, the liquid and the gas phase.
Turbulent kinetic energies presented on figure 2 are average
spatially over the whole computational domain. They are
drawn as functions of time (dashed line). Clearly, the
mixture and liquid kinetic energy are not constant because
there is not enough statistical convergence using only spatial
averaging. Consequently we compute also the mean value in
time of these spatial averaging (solid line).
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Figure 2: Turbulent kinetic energy over the time for the
reference test case: dashed line spatial average solid line
spatial and time average; : complete mixture, l:liquid, g:gas.
Additionally Reynolds or Favre averaging can be used.
They are equivalent when considering only one phase.
Indeed, for this conditioned mean the density is constant.
Presented results are Favre averaging, this choice is
important for the mixture: by using Favre averaging, the
average is mass weighted. Thus, the Favre averaging is
mainly driven by the liquid average. On the contrary
Reynolds average is volume weighted, hence the mixture
Reynolds average kinetic energy (not shown but equal to
0.08 by forcing) his very close to the gas kinetic energy.
From this simulation it appears than the liquid kinetic
energy experience bigger fluctuations than the gas liquid
kinetic energy. This happens because the liquid phase is the
minor phase ( 4 = 5% ). Thus even with a special
averaging there is no sufficient statistical convergence.
Additionally, the velocity of a liquid parcel can be very
different than the surrounding gas because of their inertia.
Due to linear forcing a liquid parcel velocity can increase
rapidly. The redistribution happen suddenly when collisions
occurs with another liquid parcels. This phenomenon
explains the important fluctuations of liquid kinetic energy.
It is confirmed by the figure 3 that shows the surface density
fluctuations. They are clearly correlated to the sudden
decrease of liquid kinetic energy. Liquid collisions ensure
the redistribution of the liquid velocity in all direction and at
smaller length scale. This increases the velocity fluctuation
dissipation that reduces the liquid kinetic energy.
Additionally liquid kinetic energy is transferred to the
surface energy by increasing the liquid surface.
Figure 2 and 3 show also that the liquid kinetic energy is
not equal to the gas kinetic energy. There is a little bit more
kinetic energy in the liquid than in the gas. This may be due
to the linear forcing and collision phenomena as described
above but also to a difference of kinematic viscosity
between both phases.
0 0,1 0,2 0,3 0,4 0,5 0.6 0,7 0,8
t
Figure 3: In black: spatial average of surface density; in
red ratio of liquid kinetic energy over the gas kinetic energy.
From this analysis it is possible to compute the equilibrium
Weber number We* defined according to equation (11).
Figure 4 presents the evolution of the Weber number as a
function of time.
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
t
Figure 4: Temporal variation of the Weber number;
dashed line: spatial average; solid line: spatial and temporal
average.
Not surprisingly this Weber number show strong
oscillations that correspond to liquid surface oscillations.
The result value is about 3.6 for this. This is below the
values proposed on the previous models (equation 11). It is
important to recall that previous model consider only
collision between spherical droplets. Figure 1 shows that
most of the droplets are not spherical. Due to their internal
agitation before a collision they are more affected by the
collision than spherical droplet. Consequently the
equilibrium is achieved for a certain level of turbulence with
a surface density. This trend induces a decrease of the
equilibrium Weber number.
For really diluted mixture, it is expected that droplets may
recover their spherical shapes between two successive
collisions.
1=5%
P/Pg 30
0,1 0,2 0,3 0,4 0,5 0,6 ,7 0,8
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8
s15 200
o 150
1 0
100
50
0 . . I ... I . . I . . I ,11 1 1 1 0
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
01
Figure 5: Equilibrium Weber number We for different
liquid volume fractions.
Figure 5 shows the effect of the volume fraction on the
equilibrium Weber number. The range of variation for
extends from 4, = 0.01 to ( = 0.99 due to the size of
the computational domain. More diluted cases would suffer
of bias due to periodic boundaries.
For flows with a major quantity of gas (sprays), the
equilibrium Weber number does not take the values
proposed in previous modelling approaches, though it is in
the same range. For flows with a major quantity of liquid
the definition of the Weber number We is not suitable
because it diverges toward the infinity.
Notice that small values of the liquid volume fraction
correspond to a liquid discrete phase (droplets), while values
close to the unity referred to a gaseous discrete phase
(bubbles). Intermediate cases, where no discrete phase can
be defined, are also covered by this study. For this later case
a snapshot of the liquidgas interface is presented in figure
6.
Figure 6: Liquidgas interface for 4, = 0.50 Blue:
surface seen from the gas; Red: surface seen from the liquid.
The twophase flow in this case in continuous for both
phases, though one bubble can be seen in the down left
corner. This kind of turbulent flows should be difficult to
explore with classical measurement techniques. The DNS in
this case appears as a powerful tool.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Conclusions
In the context of the collision/coalescence modelling the
importance of the dense zone of a spray has been outlined.
To characterise interactions between turbulence motion and
the state of the liquidgas interface an equilibrium Weber
number has been used. To find its value a well defined DNS
configuration has been set up. It has been demonstrated that
present numerical techniques allow such a Weber number to
be determined for a wide range of liquid volume fractions.
From this study it was possible to follow continuously with
the same numerical tools a situation ranging from bubble
flows to droplets flows. However the current definition of
the Weber number is not valid for the whole range of liquid
volume fractions. A more general definition should be
proposed. Cases with about the same level of liquid and gas
can be studied. Contrary to bubble and droplet flows no
discrete phase can be defined. This equally dense twophase
flows will be studied in further works.
Acknowledgements
Authors want to thanks PSA for supporting this work.
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