7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Towards Large Eddy Simulation of twophase flow with phasechange:
Direct Numerical Simulation of a pseudoturbulent twophase condensing flow
G. Bois, D. Jamet and O. Lebaigue
CEA, DEN, D6partement d'Etudes des Reacteurs
CEA/Grenoble, DEN/DER/SSTH/LDAL, 17 rue des martyrs, 38054 Grenoble Cedex 9, FRANCE
didier.jamet@cea.fr
Keywords: Twophase flow, phasechange, DNS, LES, ISS, scale similarity, pseudoturbulence,
condensation sink term, Nusselt correlation, a priori test
Abstract
In the quest of a twophase equivalent of Large Eddy Simulation (LES), which is attractive to tackle simulations
with enough bubbles to extract statistics needed in averaged models, our goal is to model subgrid turbulence and
interface transfers. Toutant et al. (2009) propose an Interfaces and Subgrid Scales (ISS) model for material interfaces,
i.e., without phasechange. Because of the importance of phasechange phenomena, this paper is the first step to
extend their analysis to nonmaterial interfaces. The difficulty comes from the velocity and the temperature gradient
discontinuity at the interface due to phasechange. The saturation condition of the interface is also a difficult task to
tackle.
Theoretical analysis and first validating results are presented in this paper. A DNS on condensing bubbles in a pseudo
turbulent subcooled liquid provides reference data on heat and mass transfers. Turbulence and interface interactions
are explicitly filtered to establish a hierarchy of the subgridscale terms to be modeled. Then, we propose to model
dominating terms using the Bardina et al. (1983) scalesimilarity ii'.lpsc'i, Some modifications are discussed in
order to consider the velocity and the temperature gradient discontinuity at the interface. The saturation condition of
the interface is closed using a deconvolution procedure.
In a multiscale approach, our original DNS, resolving twophase flow and heat transfer for nonmaterial interfaces up
to the smallest scales, is further analyzed at the macroscopic scale to improve the understanding of the strong twoway
coupling between the flow dynamics and the interfacial heat transfer. Space and timeaveraged quantities computed
from the DNS (e.g., phasechange rate) are compared with standard correlations. Discrepancies are observed between
classical models for the condensation sink term in the twofluid model. We discuss the effect of void fraction showing
a strong phasechange enhancement due to pseudoturbulence not always considered in averaged models.
The proposed LESlike model opens new perspectives towards complex simulations of liquid/vapor flow with explicit
interface tracking at reasonable computational cost.
Nomenclature
Roman symbols
Cp specific heatcapacity (J.kg 1.K 1)
g acceleration of gravity (m.s 2)
k thermal conductivity (W.m 1.K 1)
latent heat (J.kg 1)
mr condensation rate (kg.m 2.s 1)
p pressure (N.m 2)
T temperature (K)
v velocity (m.s 1)
Greek symbols
a thermal diffusivity (m2.s 1) or void fraction ()
3 inverse of the thermal diffusivity (s.m 2)
X phase indicator ()
6 interface volume area (m 1)
K curvature (m 1)
S dynamic viscosity (kg.m .s 1)
p density (kg.rn3)
a surface tension (N.m 1)
H product of conductivity and temperature (W.m 1)
Dimensionless numbers
At Atwood ()
Fr Froude ()
Fo Fourier ()
Ja Jacob ()
Nu Nusselt ()
Pe P6clet ()
Pr Prandtl ()
Re Reynolds ()
We Weber ()
Operators
[I' jump
i  norm
(.) averaging
S volume filter
surface filter
Subscripts
b bubble
bk bulk
i interface
(o mesoscopic interface
1 liquid
m multibubble
o initial
s single bubble
v vapor
conv convection
diff diffusion
interf interface
superf superficial
temp temporal
Superscripts
+ dimensional variable
* extended variable
i interfacial
m model
sat saturation
T temperature
vap vaporisation
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Reynolds number. The accuracy of numerical simula
tions depends on the computational cost and does not
rely on technological breakthrough. Even with the rapid
increase of computing power, industrial flows remain
out of reach of Direct Numerical Simulation (DNS).
Following the path of Toutant et al. (2009), we aim at
developing a new simulation tool for twophase flows
equivalent to the wellknown Large Eddy Simulation
(LES). Toutant et al. (2009) developed this Interfaces
and SubgridScales (ISS) model to consider momentum
turbulent transfer at an underresolved material inter
face. Because phasechange is a key process in many
industrial applications, (e.g., heat exchangers, nuclear
safety...), we want to extend this model to nonmaterial
interfaces.
In this paper, we first give an overview of the up
scaling procedure followed by a detailed explanation of
the first filtering step leading to the continuous LES.
The main difficulty is to correctly consider the jump of
velocity and temperature gradient due to phasechange.
An equivalent of the saturation condition of the inter
face must also be expressed in terms of mesoscopic vari
ables. In the second part of the present contribution,
we present the Direct Numerical Simulation (DNS) of
a pseudoturbulent twophase condensing flow. A large
number of bubbles generates velocity and temperature
fluctuations. The detailed description given by the DNS
calculation is then explicitly filtered in order to evaluate
the models proposed for the continuous LES. This pro
cedure, known as < a priori tests , gives first validating
results to close the first upscaling step. Finally, the up
scaling towards averaged twofluid models is achieved.
In this prospect, the sink terms and Reynolds numbers
are estimated on horizontal slices of the flow. Then, they
are compared to the prediction of standard condensation
Nusselt number using correlations from the literature.
Introduction Overview of the upscaling procedure
The twofluid model is considered as the most accurate
formulation able to deal with industrial flows. Still, it
requires an accurate prediction of the interfacial trans
port and transfers. Classically, interfacial information
is closed by experimental correlations and accuracy is
limited by technical abilities. Extending operating con
ditions of PWR towards improved efficiency requires
more accurate closures. For 3Daveraged models, re
cent needs for more local closures combined with the
advances in computing power leads to new perspectives:
the application of local simulations to estimate closure
laws for averaged twofluid models. This reasoning is
called << multiscale approach .
DNS calculations have to entail a number of de
grees of freedom proportional to the third power of the
The Interfaces and SubgridScales (ISS) concept aims
at generalizing the singlephase Large Eddy Simulation
(LES) to twophase flows, not only modeling the effect
of the smallest turbulent scales, but also modeling the
smallest interface deformations. The detailed explana
tion of the upscaling procedure is far too long to be in
cluded in this paper. It can be found in Toutant et al.
(2009) for adiabatic flows. In this paper, only the main
guidelines of the approach are reminded. Modeling is
achieved by two steps:
In the first step, an unconditional filter is applied
to both phases resulting in a smearing of the in
terface (see Fig. 1). Choosing a space and time
independent filter simplifies the derivation of some
parts of the model because the filtering operation
commutes with derivation. Compared to condi
tional filtering, the subgrid terms are a combination
of classical phase turbulence and interface subgrid
deformation. A continuous LES description is ob
tained at the mesoscopic scale and closures are re
quired to model turbulent and interfacial transfers.
*In the second step, we seek for an equivalent dis
continuous formulation, referred to as macroscopic
scale. Interfacial subgrid models from step one are
transferred onto the discontinuous interfaces under
the form of specific source terms in the expression
of velocity discontinuity and jump conditions.
In the next section, we focus on the first upscaling step,
namely the filtering operation.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Its time evolution is given by the interface velocity vi:
vi n, 6' = vi VX
Let us note t the time, v the velocity, p the pressure, T
the temperature, p the density, p the dynamic viscosity, k
the thermal conductivity, Cp the specific heatcapacity,
a k/(pCp) the thermal diffusivity, g the acceleration
of gravity, K the local curvature of the interface, a the
surface tension, Cvap the latent heat of vaporization and
Tr the condensation rate defined as:
mIp (vv vi) *n = pi(vi vi) nv (3)
The following set of mass, momentum and energy bal
ance equations is valid on the whole domain in the sense
of distributions:
V v yin 6l /p
2/
+pvV 1V
 +V(pvv)= VD
at Re
Micro Meso Macro
S Filter MAE
0.5
10.5 0 0.5 1
Figure 1: Sketch of the twostep upscaling
methodology: centered filtering and matched
asymptotic expansions (MAE).
Step 1: Continuous LES
In this section, we establish a closed form of the equa
tions that describe a twophase system with phase
change, at the mesoscopic scale of Fig. 1.
Local instantaneous conservation laws for twophase
flows can be described by two equivalent ways: (i) either
conservation laws are written in each phase and coupled
by jump conditions at the interfaces (Delhaye 1974); (ii)
or conservation laws are interpreted in the sense of dis
tributions in the whole domain (Kataoka 1986). This last
formulation, called the onefluid formulation, is compact
and efficient to apply unconditional averaging. One
fluid variables 4 = s Xkek are based on the phase
indicator function Xk (Xk 1 in phase k and 0 else
where). The vapor phase indicator X, is linked to the
unit normal towards the liquid n, and to the Dirac delta
function 56 indicating the interface by:
VX. = n, 6' (1)
tKnl6 pez
Vp+ + F
We Fr
T+V (pvCpT) = V (kVT) + mTb
at Pe Ja
where the jump of any variable 4 is defined by []1
01 P,, and the diffusion stress tensor is given by:
p D^ (Vv + VT 2ri [1/pl n () nu ,6) (5)
This tensor is naturally defined by the product of the dy
namic viscosity and the velocity gradient in each phase.
Using onefluid variables, the last term in the brackets
appears to eliminate the velocity jump at the interface
that would lead to infinite viscous stress. Any dimen
sionless physical property is defined by 4 = +/1,
where the superscript + denotes dimensional variable.
Main variables are put in dimensionless form by: v
v+/V, p p+/ (pV2), r = m+/ (pV) and T
(T+ Tsat) / (Tbk Tsat), where Tbk is the liquid bulk
temperature and Tsat is the saturation temperature (at
the system pressure). Dimensionless Reynolds, Weber,
Froude, Peclet, Jacob and Atwood numbers are defined
from the reference bulk temperature Tbk, bubble diame
ter Db and velocity V by:
m plVDb
plV2Db
gDb
piCplVDb
Pe I
ki
P1 Cp1 (Tbk 
Pv Cvap
A P1 P
RePr = Re ml
Tsat)
0%t
In the set of equations (4) we assume that the surface ten
sion and the physical properties in each phase are con
stant and that the viscous dissipation is negligible.
At a liquid/vapor interface the local thermodynamic
equilibrium condition leads to the continuity of the tem
perature, of the tangential velocity and of the Gibbs
free energy under some assumptions (see Delhaye et al.
1981). The continuity of the tangential velocity and of
the temperature is already included in the diffusion D,
and conduction kVT terms of the onefluid system (4).
In order to fully describe the system, the condition of
continuity of the Gibbs free energy must be added to the
system (4). This condition can be formulated in terms of
the interface temperature (Delhaye et al. 1981):
rT 2acTsat(p l)
pllCvapAT
where Tsat(pl) is the equilibrium saturation tempera
ture at the liquid pressure pi. In this work, we neglect
the effect of curvature on the interface temperature as it
quickly becomes negligible for sufficiently large radii.
The interface saturation condition thus reads:
T 6 = 0 (7)
At the microscopic scale, the main variables
(Xv, v, p, T and m) of a twophase problem with
phasechange are fully described by the local instanta
neous equations (2), (4) and (7).
Filtering Let us note the volume filtering operation
defined by
(8
(,xo) = G(xo x)y(x)dx (8)
where G is the convolution kernel of the filtering opera
tion. We also introduce a filtering operation for interfa
cial variables that is defined by:
s(xo) i(xo) = 6i(xo)
The above relation is an areaweighted average of any
interfacial variable 4. In the neighborhood of the inter
face, the resulting volume field q is almost constant in
the normal direction.
When G is independent of space and time, the filter
is called unconditional and the filtering operation com
mutes with the space and timederivatives. The strong
advantage of unconditional filtering over phasefiltering
is that subgridscale fluctuations are not only turbulent,
but also linked to the smallest scales of the interface de
formations. Hence, interfacial deformations do not have
to be fully resolved. They can be included in the model.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Applying the filtering operation (8) to the onefluid
formulation (2), (4) and (7), one gets:
t + vi VX 0
iv Ip
V ;v = d... [1
9 Vp _v v
 + V pv v
tt
1V
ReV
8pCpTT
S+ V pCpv T
at
{Vxu pez
We F+
We Fr
1 1
V k VT+ 
Pe Ja
T6 = 0
Interfacial mass transfer adds three specific modeling
issues that are absent in the adiabatic study of Toutant
et al. (2009): (i) the rate of phasechange Tr is an addi
tional variable associated to the saturation condition (7);
(ii) the temperature gradient and (iii) the velocity are dis
continuous across the interface. In the following subsec
tions, we deal with these issues. Suggested solutions are
validated in the next section, evaluating models by a pri
ori tests on a pseudoturbulent configuration.
Specific treatment of the saturation tempera
ture at the interface To account for the local ther
modynamic equilibrium (7) at the mesoscopic scale, one
must find an equivalent boundary condition at the inter
face expressed in terms of main variables of the meso
scopic problem, formally:
f (x, v p, T, ) 0
Fig. 2 gives the microscopic profile of a continuous vari
able with discontinuous gradient and the corresponding
continuous profile at the mesoscopic scale. The satura
tion condition is not satisfied by the mesoscopic variable
at the interface. We aim at determining the difference
represented by the double arrow in Fig. 2. Inverting
the filtering operation in order to reach the local (mi
croscopic) temperature T is known as a deconvolution
procedure. It can be achieved by expending variables in
Taylor series in the neighborhood of the interface. Let
us define a new variable = kT. In view of the sat
uration condition (7), this variable is continuous across
the interface and its gradient jump is related to the rate
of phasechange by the microscopic heat flux jump con
dition
Pe
[VO n, kVT] n, = (11)
Ja
because the thermal conductivity is assumed constant in
each phase and the interface temperature is null in view
of the saturation condition (7). Unlike the jump of the
temperature gradient [VT], the jump [VO] is given in
terms of the main variables only. Applying the filter to
the Taylor series of phase variables 01 and 0, in the
normal direction, one gets at main order
e(xa) = e(x) + C' [V] n,
where the constant C' characterizes the filter. In view of
equations (7) and (11), we have:
Pe
e(xi) = C'tn(xi)
Ja
Finally, assuming that the mesoscopic variable m' is a
good approximation of the local phasechange rate Tr,
we propose to express the formal relation (10) as
Pe
6, C = 6, C (12)
Ja
where the Dirac delta function 6, corresponds to the
mesoscopic interface and is defined by the levelset
x 1/2. The main advantage in using e instead of
the temperature T is that its jump is directly related to
the rate of phasechange by the heat flux condition (11).
Relation (12) gives a closure to the last equation of the
system (9). The accuracy of the closure relation (12)
is estimated in the second part of this work. We show a
strong correlation between the mesoscopic rate of phase
change m*S and the value of the mesoscopic variable e
at the interface.
S= kT
2 ~ kT 
1.5
1
0.5
0  H
0.5 I
1 0.5 0 0.5 x
Figure 2: Profile of e across a nonmaterial interface.
Subgrid heat flux modeling The second advan
tage in taking e as a main mesoscopic variable is that
we no longer need a model for heat flux at the interface.
Indeed, we can apply the filter to the energy equation of
system (4) formulated in terms of e to get
at
1 V2 1 +
Pe V a +
where 3 1/a is the inverse of the diffusivity, be
cause the thermal conductivity is constant in each phase
and the dimensionless temperature is null at the inter
face. The conduction term of equation (13) is defined
with the main variable e and does not have to be mod
eled. Only the time derivative and the convection sub
grid terms must be modeled (terms of the LHS of equa
tion (13)). Because e is continuous across the interface,
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
we can use the scale similarity hi\ldicsis, of Toutant
et al. (2009) to close the first term of equation (13):
S= temp
MPQ~Q
(14a)
(14b)
Throughout the article, T is used to define a subgrid
term and the superscript m indicates the corresponding
model. The convection term is a triple product of the
inverse of the microscopic thermal diffusivity 3, the ve
locity v and the variable e. The product of a physical
property and the velocity requires a specific treatment
that is presented in the following subsection.
Velocity induced subgridscale modeling In
the onefluid formulation, filtering interfaces and dis
continuities is transparent. Toutant et al. (2009) show
that scale similarity models predict accurately correla
tions between two variables (e.g., density and velocity)
even if one of them is discontinuous at the interface
(e.g., density). The problem of nonmaterial interfaces
is more complex than adiabatic flows studied by Toutant
et al. (2009) because both physical properties and veloc
ity are discontinuous at the interface. We noticed under
prediction of subgrid fluctuations when one applies this
class of models to two discontinuous fields. To bypass
this drawback, we seek for a continuous velocity field
containing all the fluctuations of the onefluid velocity v
apart from the discontinuity. The main idea illustrated
by Fig. 3 is to extend the phasechange rate m and nor
mal n, fields in the vicinity of the interface and combine
them to the fluid velocity to build an extension of the in
terface velocity.
Figure 3: Normal velocity profile through a
nonmaterial interface.
According to the Helmoltz's theorem (Arfken 1985),
we define r* n* as the unique vector field satisfying the
following conditions in the whole domain 2:
Vx E Q2,
Vx e 2,
V (5* n) = Ti (xi) K (xi) (15a)
Vx (rm* n) 0 (15b)
Vx E d m* n* m nv
(15c)
xi is the orthogonal projection of x over the interface
80. The extended vector field r* n* is continuous at
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
the interface. It can be used to extend the interface ve
locity to the whole domain:
VxeQ, v*^vr*n* v
where v = 1p is the onefluid specific volume. The
vector field v* is continuous across the interface and
still includes local velocity fluctuations. v is the classi
cal convection velocity whereas ri* n* v represents the
fluid velocity induced by phasechange. Thus, we have
in the whole domain:
vi VX v* VX (17a)
pv = pv* + n* n (17b)
pv v = pv ( v* + v ( m* n (17c)
V = V [p (Vv* + VTv*)] (17d)
pCpvT = pCp T v* + CpT T n (17e)
The volume field v* can be filtered to obtain at the meso
scopic scale:
v ^f r* n v (18)
This velocity is not defined in terms of the main vari
ables of the mesoscopic scale. We aim at closing the def
inition (18) with an appropriate modeling. In the neigh
borhood of the interface, we assume that subgridscale
fluctuations of the phasechange rate and the normal are
negligible. Hence, the velocity v* is approximated by:
v* wv ^ v n V (19)
The mesoscopic velocity v, is not a filtered quantity.
The notation is used abusively here to indicate a meso
scopic variable. In view of the approximation (19), the
convolution of equations (17) with the filter kernel leads
to the following definitions of subgrid terms T:
Vi VXv = V Vv X, Tinterf
pv = pvy + ar n+v Ttemp
p v v = Pv g) + v ~ Va V n + Tconv
V T , V [ (VW + VTV,) + Tdiff]
pCpvT= fv3v + vf m8 n + rTcn
(20a)
(20b)
(20c)
(20d)
(20e)
sures:
Tierf^ V Xv, va Vx,
temp^pVJ P,
ToQnp v, O v p V O V
Tff^ (V, + VT~) 
Tcon" v, 13 Ov,
(Vv, + V V)
(21a)
(21b)
(21c)
(21d)
(21e)
Let us underline here the importance of the velocity v
that ensures an accurate prediction of the subgrid terms
unlike the fluid velocity v. In the momentum equation,
we assume that subgrid terms for diffusion and capillary
forces are negligible according to the work of Toutant
et al. (2009). This assumption will be corroborated by
subsequent tests in the following section.
Because of the approximation (19), the models pro
posed here assume that the subgrid correlations between
the rate of phasechange and the normal to the interface
are negligible in the reconstruction process. The ma
jor contribution of v, to the models is due to the fluid
velocity v. This I\11' ,llc'is indicates that the interface
geometric fluctuations have to be limited. It should be
noted that consistent models proposed here naturally de
generate to the proposition of Toutant et al. (2009) for
adiabatic flows. Their models could be valid under the
assumption that the rate of phasechange and the density
jump remains small compared to the fluid velocity:
m n)V <
Nonetheless, our proposition is more general as it en
sures an accurate time evolution of macroscopic quanti
ties such as the void fraction even for high rates of phase
change and high density jumps.
Summarizing the closures given by equations (12),
(14b) and (21), we propose the following description for
the continuous LES (mesoscopic scale of Fig. 1):
a . V VX Tin7erf 0
V V = 1m si 1/p (23)
p + V f v, a ni + Tmn V
at V Tcov
1
+ [_V (V;, + VT,) + Tdm
Re
VIvX pe
We Fr
In order to close the set of equations (9), we aim
at modeling the subgrid terms T in terms of meso
scopic variables. Toutant et al. (2009) have adapted
the scalesimilarity Ihp,1,liLcis to discontinuous vari
ables (e.g., density). Combining their approach with the
approximation (19), we can propose the following clo
aCpOP T
at +C V (pC Tv + Cp T m v + T8n nv)
1 V 1 8
Pe Ja
e6, = 6, Pe
Ja
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
where:
pv+pv + n + Teip
pCp T^3 0 + Ttin,
(24a)
(24b)
The system (23) ends the first step of our upscaling pro
cedure (see Fig. 1). Indeed, we are now able to fully de
scribe the evolution of mesoscopic variables. We have
dealt with the temperature gradient and velocity discon
tinuities accurately and we have applied a deconvolution
procedure to the product of conductivity and tempera
ture so as to consider the saturation condition at the mi
croscopic interface. This system is closed at the meso
scopic scale.
First validating results are given in the following sec
tion. The DNS calculation is explicitly filtered to evalu
ate subgrid terms and corresponding models and to rank
them according to their importance. This methodology
is called a priori test, as the effective efficiency of the
model is not tested. Actually, we do not have the appro
priate numerical method to fully evaluate the continuous
LES model.
DNS of a pseudoturbulent twophase
condensing flow
We specified the DNS configuration presented here to
evaluate subgrid terms and models on a complex flow
configuration by an explicit filtering of the DNS results.
Thus, choosing the configuration is a subtle balance be
tween complexity and realizability. Analysis of the a
priori test is given in the subsequent section. At the end
of this section, we will go beyond the frame of the ISS
modeling and validation presenting a statistical analysis
of the flow. This application aims at illustrating the fea
sibility of the multiscale approach to extract information
from DNS data.
In industrial twophase flow applications, fluctua
tions are the combination of singlephase turbulence
and pseudoturbulence induced by bubble wakes. In
this preliminary work, we simulate the pseudoturbulent
interactions between deformable buoyant bubbles con
densing in a initially quiescent liquid. Fullydeveloped
singlephase turbulence is not considered to limit com
putational cost and should motivate another specific
study.
Figure 4: DNS of a pseudoturbulent twophase
condensing flow: multibubbles condensation in a
subcooled initiallyquiescent liquid.
Bubbles are injected at the bottom of a rectangular
column of initially quiescent liquid. They rise under
the effect of gravity and are destroyed when they reach
the top of the domain (see Fig. 4). Monodispersed bub
bles are injected at an arbitrary horizontal position and
the injection frequency (4Hz) is adapted to maintain a
low void fraction I'. < a < 7.5'., thus limiting coa
lescence. The column size is limited by computational
cost to 5Db width and 15Db height. Symmetry and adi
abatic conditions are applied on lateral borders to rep
resent core flow condensation. At the top, free outlet
allows fluid recirculation. The corresponding incoming
fluid temperature is assumed to be equal to the averaged
temperature in the column T Tat 1.7524K. At
the bottom, the saturation temperature is imposed. We
limit the Reynolds number to Re z 50 by reducing the
gravity constant to g = 5 x 103 m.s 2, so as to resolve
the thermal boundary layer at the interfaces. Surface ten
sion has been decreased compared to standard pressur
ized water (p 15.5 MPa) to compensate the small rel
ative velocity: o 6 x 10 6 N.m 1. Computational
cost limits the domain size and therefore the number of
bubbles in the simulation domain around 40. Converged
statistics cannot be obtained by spaceaveraging on this
reduced sample. We need space and timeaveraging.
Hence, statistical steadystate has to be achieved. The
energy released by condensation warmsup the liquid
and the averaged bulk temperature tends to the satura
tion temperature, then decreasing phasechange to zero.
In order to get a steadystate flow with phasechange,
we must artificially maintain the liquid bulk temperature
constant by adding a small timevarying energy source
term in the liquid phase. This source compensates the
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
latent heat released by condensation and is proportional
to the local subcooling.
Physical properties of pressurized water used for
this simulation are given in Table 1. Dimensionless
Reynolds, Weber, Froude, Prandtl and Jacob numbers
are:
50, We
1iCpi/ki :
1.6, Fr 0.92, At 0.80,
1.474 and Ja 0.1
Table 1: Physical properties of saturated water
(Tsat 618 K, Psat 15.5 MPa).
Liquid Vapor
p [kg.m3] 574.3 113.6
p [Pa.s] 65.16x10 6 23.73 x106
c, [J.kg1.K1] 10.11x103 17.15x 103
A [J.s1.ml.K1] 44.7x 102 13.8x10 2
C [J.kg1] 895.7 x 103
We used an explicit FrontTracking method with
sharp interfaces implemented in the Trio_U code (Math
ieu 2004). The fine mesh used for the DNS calculation
had 41.5 Mnodes spread on 1536 processors. The sim
ulation has been validated by mesh convergence study
on a single bubble. We used three meshes with a re
finement factor of 2. We determined a minimum of 38
nodes per diameter for converged results. We extrapo
lated this result to the multibubble case choosing a mesh
fine enough to ensure that the local curvature does not
exceed this criterion. Mesh convergence was also ver
ified a posteriori, comparing DNS averaged quantities
to underresolved data obtained on coarser meshes (see
Fig. 1015). The convergence is reasonable. Evaluating
the rate of convergence from the three meshes used, we
can evaluate the uncertainty on the Nusselt number to
:'. Fig. 5 gives the instantaneous solution obtained on
the finest mesh. A slice of the temperature field illus
trates the pseudoturbulence induced by bubble wakes.
Bubble wakes influence both the bubble shape and the
rate of phasechange.
In the following sections, these DNS results are used
(i) to validate the Bardina et al. (1983) scalesimilarity
lIpi ,ilI'hei and (ii) to demonstrate the potential informa
tion transfer from a fine solution of a complex flow to
averaged models.
Pseudocolor
Var: Tem erature
0.9204
1.841
2.761
3.681
Max 0.6328
Min: 4.096
Pseudocolor
Var: Phase change
I 0.01527
0.01145
0.007636
0.003818
Max: 0.01527
Min: 0.000
Figure 5: Pseudoturbulent simulation: Effect of
bubble wakes on the temperature (sliced field) and the
rate of phasechange over the interfaces.
A priori tests: subgridscale evaluation The
DNS data grant us access to the detailed solution (see
microscopic scale of Fig. 1). Defining an explicit fil
ter, we can evaluate the subgrid terms and their model.
Then, ranking and comparing them we estimate the
quality of the proposed modeling. A representative sam
ple of 25 timesteps is taken throughout the simulation.
Subgrid terms and models are evaluated for every mesh
point using a simple space filtering, averaging quanti
ties on a control volume of 9x9x9 cubes. The magni
tudes given in tables 2 and 3 correspond to the time
average of the space absolute maximum. We check that
time fluctuations of extrema are actually small. Vectors
are compared in norms and the three invariant of ten
sors (trace, tensor contraction and determinant) are eval
uated. In table 2, percentages refer to the open meso
scopic term (e.g., the percentage for momentum is given
by 100 x Ttemp/TP). It gives valuable information on the
relevance of the definition of the subgrid term. One has
to bear in mind that the intensity of the subgrid terms
is related to the difference of scales between the filter
size and the Kolmogorov scale. Noting that the partic
ular filter considered here is rather large compared to
the Kolmogorov scale, table 2 indicates the pertinence
of subgrid terms because the intensity does not exceed
one third. Comparing orders of magnitude, we can ne
glect subgrid terms due to diffusion Tdiff and surface ten
sion Tsuperf in the momentum equation. This result, in
agreement with previous work of Toutant et al. (2008)
for adiabatic flows, indicates that the underresolved so
lution of the mesoscopic system is sufficiently fine to ac
curately describe the boundary layer at the interface and
the geometry of the interface at the dominating order.
The magnitude of other subgrid terms is large enough to
justify the need for a model.
Table 2: Results of the a priori test. Magnitude com
pares the timeaveraged spacemaximum of each sub
grid term or model to the corresponding open meso
scopic quantity. The slope of the line of best fit and
the correlation coefficient characterize the quality of the
models.
Term Magnitude (%) Slope Correlation
T Tm a coeff. (r)
Tinterf 33.10 25.66 0.898 0.997
IIkteimp 16.93 12.72 0.848 0.995
tr (Tonv) 23.53 13.51 0.696 0.977
con (Tonv) 4.30 1.41 0.411 0.966
det (Tconv) 0.00 0.00 0.279 0.742
tr (Tdiff) 0.89 N/A N/A N/A
con (Tdiff) 3.04 N/A N/A N/A
det (Tdiff) 0.04 N/A N/A N/A
Tsuperf 8.89 N/A N/A N/A
Nrmp 9.43 7.06 0.771 0.984
Tnv I 15.44 11.32 0.827 0.984
Ci N/A N/A 0.155 0.995
Table 3: Results of the a priori test. Magnitude com
pares the timeaveraged spacemaximum of the contri
bution to the balance equations of each subgrid term or
model to the corresponding open mesoscopic contribu
tion. The slope of the line of best fit and the correlation
coefficient characterize the quality of the models.
Contribution Magnitude (%) Slope Correlation
Contribution
7T T a coeff. (r)
Tinterf 33.10 25.66 0.898 0.997
IIdtTtemp  10.82 8.14 0.841 0.996
V Tconv 25.71 17.33 0.806 0.980
V"rde 9.16 N/A N/A N/A
Re
Tsuperf/We 33.89 N/A N/A N/A
Petermp 40.31 30.16 0.759 0.986
PeV 7nv 103.25 55.48 0.784 0.954
In table 3, a different point of view is adopted. A
single reference for each equation is considered, namely
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
vi VXv, V pv 0 v and V2 The contribution of each
term evaluated by space or timederivative is then com
pared to this reference. The main advantage of table 3
is that it measures the true effect of the model on the
conservation equation. The major drawback is that it is
casedependant. Indeed, the subgrid hierarchy mainly
depends on the particular dimensionless numbers taken
for the simulation. This hierarchy also depends on the
particular frame of reference of the study as both Ttemp
and Tconv contribute to convection and still, their contri
bution is considered separately. Table 3 roughly con
firms the previous analysis.
Now that the need for subgrid models has been
asserted, let us qualify the closures given by equa
tions (12), (14b) and (21) on this particular flow. In
view of turbulent structures and interface deformations
observed on Fig. 5 compared to the viscous flow around
a single bubble (see Fig. 9), we believe that this example
is complex enough to establish the quality of the mod
eling. First of all, qualitative agreement between sub
grid terms and models is verified on flow slices. Results
for the momentum are illustrated on Fig. 7 where good
qualitative agreement is found. The model predicts high
subgrid terms in the region where friction is high as ex
pected. For a quantitative comparison, we estimate the
scattering of the model versus the corresponding subgrid
term to be modeled (see for example momentum on Fig.
8). Tables 2 and 3 summarize the slope a and the cor
relation coefficient r of the line of best fit T = aT.
The slope is smaller than unity. This indicates a slight
underprediction (from 10 to : '.) of the model that
is coherent with the previous records of Toutant et al.
(2008); Magdeleine (2009) for adiabatic flows. Actu
ally, underprediction of the scalesimilarity models is
well reported in singlephase turbulence (Sarghini et al.
1999). Scalesimilar models provide backscatter and
predict subgrid stresses that are well correlated with the
actual Reynolds stress but they are not enough dissipa
tive. A solution proposed in singlephase turbulence is
to use mixed models that include an eddyviscosity part
(functional modeling) as well as a scalesimilar contri
bution (structural modeling). In this first approach of
twophase LES, our suggestion is to calibrate a model
ing constant (noted a in tables 2 and 3) from a priori
tests.
Strong correlations between the subgrid terms and the
corresponding models (see Fig. 8 and tables 2 and 3) as
sert the quality of the modeling. Finally, we evaluate the
quality of the closure relation (12) estimating the corre
lation between the mesoscopic rate of phasechange m
and the mesoscopic variable over the interface. Strong
correlation is again observed (see Fig. 6). The slope of
the line of best fit is related to the interfacial constant
(for the chosen filter): C' 0.155. In the end, the com
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
plex microscopic solution of the bubbly flow illustrated
on Fig. 5 enables the qualification of the model proposed
in this work. Good modeling properties are obtained.
eo (x)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.4
0.
Fi
Figure 8: Validation of the scalesimilarity II\ 1llc'i
Scatter of the model Ttmp versus the subgrid term Ttemp.
Ideal correlation ( ) and line of best fit ( ).
5 Statistical analysis for averaged models The
5 = 0.107. Pem (p)objective of this section is to evaluate the validity of cor
4.5 4 3.5 3 2.5 2 1.5 1 0.5 Ja relations between Nusselt and Reynolds numbers such
as Ranz and Marshall (1952) used in 3Daveraged mod
gure 6: Modeling of the saturation temperature at the as Ranz and Marshall 1952) used in 3Daeraged mod
interface: Scatter of the variable 6 versus the els e.g., Mimouni et al. (2009); Morel et al. (2010). We
mesoscopic rate of phasechange first present some correlations from the literature. Then,
mesoscopic rate of phasechange m .
we compare their predictions to the results of our DNS
in the case of a single bubble and for the multibubble
simulation.
For this first example, we have chosen to illustrate the
multiscale approach on the closure of the condensation
sink term in twofluid models. This closure is essential
since it determines the void fraction of collapsing bub
bles in a subcooled liquid core. Chen and Mayinger
(1992) showed that for a single bubble, the condensa
tion is completely controlled by interfacial heat transfer
S for Jacob numbers Ja < 80, which is already well be
0..,.. yond realistic conditions for water in PWR. The present
..66o work focuses on heattransfercontrolled bubble conden
sation. Let us note (.) an averaging operation. The aver
aged condensation rate (4) is represented in dimension
less parameters by the Nusselt number:
Nu (VT) D32
(Tbk Tsat)
(T) CvapD3
kl(Tbk Tsat)
2.0 1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Figure 7: Validation of the scalesimilarity hl' 11 lllc i,
Qualitative comparison of the subgrid term Ttemp to be
modeled (upper slice) versus its model rTmp (bottom
slice)
where Tbk (xiT) is the bulk temperature and D32
6a/ai is the mean Sauter diameter defined by the ratio
of the void fraction a = (X) and the volume interfacial
area a, = (6i). In the previous expression, we have
assumed that the steam has reached thermal equilibrium
leading to a uniform steam temperature: T, Tsat.
Twofluid models are based on the assumption that
the averaged condensation rate (rT) (represented by the
Nusselt number) is correlated to other averaged quanti
ties, e.g., Reynolds and Jacob number. Sometimes, the
void fraction is added as a parameter to account for
multibubble effect. In the literature, two paths (some
time coupled) are followed to determine these correla
tions: Warrier et al. (2002) or Park et al. (2007) de
velop mechanistic models from experimental observa
tions whereas Ruckenstein (1959) or Dimic (1977) base
their analysis on theoretical considerations only, look
ing for the flow characteristics under some assumptions,
e.g., potential or laminar flows around a solid sphere
moving in a stagnant liquid with uniform subcooling.
Limitations of the first class of work are linked to the
accuracy of experimental data and the capacity to get lo
cal interfacial measurement whereas analytical solutions
have a limited representation of reality embodied by the
lni 'liLc'c, used. With DNS, one can have access to lo
cal and complete flow representation in rather complex
configurations. In the following, we show an example
on how to follow this path to assess the existing correla
tions.
First of all, let us present some of the literature corre
lations valid in the range of our application. They will
be used as reference data for comparison with our DNS
results. Many theoretical investigations (e.g., Rucken
stein 1959) have been carried out to study the thermal
boundary layer around a solid sphere in an irrotational
flow. Experiments have been carried out on single bub
bles. The various models and correlations proposed dis
agree on the effect of the Jacob number on the conden
sation heat transfer coefficient. Whereas first results of
Ruckenstein (1959); Isenberg and Sideman (1970) and
Akiyama (1973) show an independence of the Nusselt
number to the subcooling, DimiC (1977) find an increase
of the Nusselt number with increasing liquid subcooling
and the model of Chen and Mayinger (1992); Zeitoun
et al. (1995) show an opposite trend.
Ruckenstein (1959) firstly reported the requirement of
accounting for multibubble effects. Zeitoun et al. (1995)
included the effect of the void fraction on the Nusselt
number in their correlation when Warrier et al. (2002)
found the void fraction effect negligible (for low void
fraction a < '. .) in front of the residence time (ac
counted for by the Fourier number Fo alt/D o where
Dbo is the initial bubble diameter or the detachment di
ameter). Available models and correlations for conden
sation Nusselt number are listed in Table 4.
To sum up, numerous difficulties are encountered by
experimentalists to understand the detailed mechanisms
that prevail at an interface in condensing flows. Most re
sults are given within an uncertainty range of +'i' r., as
a result of the many experimental difficulties. Numerical
simulation is a valuable tool because it gives an insight
into local transfers.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Table 4: Bubble condensation models for the condensa
tion Nusselt number.
Author Nusselt number
Ranz et al.
Ruckenstein
Isenberg et al.
Akiyama
DimiC
with
Chen et al.
Zeitoun et al.
Warrier et al.
Pseudocolor
Var: Temperature
0.438
10.876
1.7S
Max: 0.03670
Min: 1.752
peudoncolov
0.00523
0.00349
0.00174
0.00
Max: 0.006979
Min: 0.0003025
2 + 0.6 Re1/2Pr1/3
2/ / (RePr)1/2
S1/ vRe/2Pr1/3
0.37 Re06Pr1/3
= 2/ (RePr)l/2 1/2
[1 6/FJaRe/2 Pr1/2 Foo]2/3
=0.185Re.7Pr1/2
2.04 Re0.61 328Ja 0.308
=0.6Re/2Pr/3 [1 1.2Jao.9Foo]2/3
Figure 9: Laminar flow around an ellipsoidal bubble:
Middleslice of the temperature field and distribution of
the rate of phasechange over the interface.
Single bubble simulation First of all, let us ana
lyze the case of a single deformable bubble rising freely
in uniformly subcooled liquid (see Fig. 9). The bubble
shape is close to an ellipsoid. The ratio of major and
minor diameters is 1.7. Fig. 10 compares the time evo
lution of the Nusselt number to predictions of Table 4.
Predictions are estimated using the actual Reynolds, Ja
cob and Fourier numbers of the DNS simulation. The
beginning of the simulation (until t = 0.7s) should not
be considered as it corresponds to the establishment of
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
translational regime and of the thermal boundary layer.
Zeitoun et al. (1995) is not represented on Fig. 10 as
the void fraction tends to zero for a single bubble. We
believe that the dependency on the void fraction is not
expressed >.'lIsiclnil\ in this correlation as it does not
degenerate to standard single bubble correlations when
the void fraction goes to zero. We found a good agree
ment with Ruckenstein (1959); DimiC (1977) whereas
the other correlations underestimate the heat transfer.
For these correlations, our results are not within the un
certainty range of  '' given by most authors. From
numerical data, time evolution of Nusselt and Reynolds
numbers can be correlated as follow:
Nu= ( 0 I1 (26)
where the subscript s indicates that this correlation is
valid for a single bubble only. We are really closed to
the theoretical solution proposed by Ruckenstein (1959).
Slight discrepancy can be attributed to the bubble shape.
In this first step, we have established an accurate cor
relation giving the evolution of the Nusselt number of a
single bubble as a function of the Reynolds number only.
In the short range of validity of this correlation, the influ
ence of the Fourier number (representing dimensionless
time) does not seem significant. Therefore, our results
tend to indicate that the decrease in heat transfer asso
ciated to thickening of the thermal layer suggested by
Warrier et al. (2002) does not seem to be the dominating
term.
Predicted Nu
18
16
14
12
Pseudoturbulent simulation The great advan
tage of numerical simulations is that we can easily in
crease the size of the domain and the number of bubbles
in order to get a complex flow. This was achieved in
the DNS presented above. The flow is in a statistical
steadystate and we assume invariance in the crossflow
directions because of the lateral symmetry conditions.
Converged statistics are obtained averaging the local in
stantaneous field over a period of 40s and over five sub
domains corresponding to five axial slices of approxi
mately 2Db width (see Fig. 4). The averaging period is
150 times higher than the period at which bubbles are
injected. Sliding averages of the Nusselt number (resp.
Reynolds number) over the 3rd slice are given on Fig. 11
(resp. Fig. 12) for several averaging periods Tm. We de
termine a minimum period of 40s for converged statis
tics within .'. Convergence is corroborated on the
other slides by Fig. 13 (resp. Fig. 14).
Nu
12
11.75
11.5
11.25
11
10.75
10.5
10.25
10
Tm = 5s ..................
T 10s 
Tm 20s
T, = 40s 
__
1 I I I
:
0 20 40 60 80 100 120
Figure 11: Sliding average of the Nusselt number for
the 3rd slice for different averaging periods Tm.
8 
6 L
4
S....
S
0_____
53
52
t 51
50
49
48
47
0 0.5 1 1.5 2 2.5 3 3.5 4
Coarse mesh Chen *
Medium mesh Akiyama a
DNS Ruckenstien
Ranz E Warrier v
Isenberg Dimic
TRACE Single bubble Eq. (26)
Figure 10: Single bubble laminar flow Correlation
between the Nusselt and Reynolds numbers. Mesh
convergence and comparison to correlations from the
literature.
T,
T
S'Tm
Tm
>t ;ii ;:
 5 s ..................
= 10s 
 20s
= 40s
irLi /
46 t
0 20 40 60 80 100 120
Figure 12: Sliding average of the Nusselt number for
the 3rd slice for different averaging periods Tm.
!'V
Nu
12.25
mean
12 1
11.75 2
11.5 8... .. s3 .. ........
11.25" s 
11 i...... '::................. .... .
10.75
10.5
10.25
10
9.75 ... 
9.5
9.25 t
20 30 40 50 60 70 80 90 100
Figure 13: Sliding average of the Nusselt number
(Slices are numbered from bottom (sl) to top (ss)).
Re
57
56
55
54
53
52
51
50
49
46
45
44
43
2
41
40
39
38
37
20
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
eter reduction. Meanwhile the thermal boundary layer
thickens. Both these effects are particularly well consid
ered by DimiC (1977) correlation as we can see on Fig.
15 that it goes from the Ruckenstein (1959) solution for
high Reynolds number (i.e., low residence time) to our
DNS results for low Reynolds numbers (i.e., for longer
residence time).
Predicted Nu
 . . . . . ...
, 0 i
t
30 40 50 60 70 80 90 100
Figure 14: Sliding average of the Reynolds number
(Slices are numbered from bottom (si) to top (ss)).
The calculated condensation Nusselt number versus
the bubble Reynolds number is shown in Fig. 15. Results
are compared to the literature and to the single bubble
correlation (26). The Jacob, Fourier and Reynolds num
bers and the void fraction are evaluated for each slice.
The time in the Fourier number is an estimation of the
residence time of the bubbles since the injection. The
strong dispersion on Fig. 15 illustrates the difficulties
of getting accurate and representative experimental mea
surements. DNS results and correlations agree on the in
crease of phasechange with increasing bubble Reynolds
number. Zeitoun et al. (1995) correlation overpredicts
heattransfer at the interface whereas other correlations
tend to underpredict its intensity. Our results are in
good agreement with the theoretical work of Rucken
stein (1959); DimiC (1977). The void fraction a and the
Sauter mean diameter decrease from slice si to slice s5
while the residence time increases. On this single flow
example, it is thus difficult to distinguish the effects of
the void fraction a from the effects of the Fourier num
ber Fo. In the region of interest, the Reynolds number
decreases with residence time as a result of the diam
35 40 45 50 55 60
Coarse mesh Chen
Medium mesh Akiyama *
DNS Ruckenstien
Zeitoun E Warrier
Ranz Dimic
Isenberg Single bubble Eq. (26) .
TRACE *
Figure 15: Pseudoturbulent bubbly flow Correlation
between the Nusselt and Reynolds numbers. Mesh
convergence and comparison to correlations from the
literature.
The single bubble correlation (26) gives valuable in
formation as it has been established in a very simi
lar configuration, except for the collective effects and
pseudoturbulence. Hence, the difference between the
Nusselt number predicted by correlation (26) and the ac
tual measurement is to be directly attributed to pseudo
turbulence. Following Zeitoun et al. (1995) sugges
tion, we can estimate the trend. Because of pseudo
turbulence, the Nusselt number increases with increas
ing void fraction as a result of both turbulent mixing
and larger temperature gradients due to a smaller min
imum temperature (see Fig. 5). To overcome Zeitoun
et al. (1995) issues with low void fractions, we propose a
correlation for bubbly flows Num of the following form:
Num = Nu (1 + f(a))
where the subscript m stands for multibubbles. Unfor
tunately, due to the lack of sample we could not express
the dependency f. For the range I'. < a < 7.5 '., the
ilniiliili.i.ii,n factor f is almost constant and equal to
0.2 .,'. Obviously, one has to bear in mind that com
plementary work is required to extend the validity range
of the correlation (27) and to differentiate the effect of
the void fraction (a) from the influence of the residence
time (Fo). Nonetheless, our work illustrates the poten
tial contribution of DNS to the closure issue of twofluid
models. We have been able to attribute the 20% increase
of phasechange rate to global effects. Accessing full lo
cal data is thus valuable to understand and propose new
physics modeling.
Conclusions and Perspectives
This paper is the first step to define filtered discontinu
ous nonmaterial interfaces, thus extending the ISS con
cept to twophase flows with phasechange. We pro
pose consistent closures to describe a continuous LES
at the mesoscopic scale (see first step of Fig. 1) that
naturally degenerate to Toutant et al. (2009) modeling
when phasechange tends to zero. Temperature gradi
ent and velocity discontinuities are accounted for accu
rately because they are linked to the mesoscopic rate
of phasechange. The scalesimilarity ll\p,,lllci re
mains accurate to predict interface/velocity and inter
face/temperature correlations provided that a continuous
velocity based on the interfacial velocity is used in the
neighborhood of the interface. An equivalent to the lo
cal thermodynamic equilibrium condition is obtained at
the mesoscopic scale. Using a deconvolution procedure,
we relate interfacial value of the mesoscopic variable e
to the mesoscopic rate of phasechange.
A priori tests are conducted on a configuration where
condensing bubbles interact together thus generating a
pseudoturbulent flow. First validating results for the
closure of the twophase LESlike model show a good
predictivity of the subgrid scale transfers and interface
transport. The mesoscopic condition for the temperature
of the interface turns out to be accurate on the particular
complex flow tested.
Even though these results need further investigation,
they are promising. This system could be solved with
appropriate numerical methods dealing with diffuse in
terfaces. However for numerical reasons, following the
path of Toutant et al. (2009), we aim at stiffening the in
terface and transforming this formulation into an equiv
alent discontinuous problem. One perspective of this
work is to evaluate the difference between the meso
scopic and the macroscopic formulations in order to
determine the excess quantities that should be applied
through specific jump conditions (see 2nd step in Fig. 1).
Moreover, valuable data has been extracted from
a complete DNS of deformable buoyant bubbles to
demonstrate the potentialities of the multiscale ap
proach. We have been able to validate our numerical
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
method against the literature and to demonstrate and
quantify the enhancement of interfacial mass transfer by
pseudoturbulence. Hence, this work aims at improving
the understanding of heat and mass transfers at a liq
uid/vapor interface, giving an insight into local transfer.
This can be particularly helpful to help and close local
closure relations required by twofluid averaged models.
However, computational cost to get converged aver
aged quantities is still high and the bubble Reynolds
number is limited. Thus, the ISS model is required to
be able to extend Magdeleine (2009) work to twophase
flows with phasechange (see Fig. 16). Besides valida
tion, work in progress concerns the second upscaling
step with the application of matched asymptotic expan
sions to the continuous LES model.
Figure 16: ISS of an intermittent bubbly flow (from
Magdeleine 2009).
Acknowledgements
This work was performed using HPC resources from
GENCICCRT (Grant 2009c2009026200).
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