Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 6.6.1 - Towards Large Eddy Simulation of two-phase flow with phase-change: Direct Numerical Simulation of a pseudo-turbulent two-phase condensing flow
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Permanent Link: http://ufdc.ufl.edu/UF00102023/00163
 Material Information
Title: 6.6.1 - Towards Large Eddy Simulation of two-phase flow with phase-change: Direct Numerical Simulation of a pseudo-turbulent two-phase condensing flow Multiphase Flows with Heat and Mass Transfer
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Bois, G.
Jamet, D.
Lebaigue, O.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: two-phase flow
phase change
DNS
LES
ISS
scale similarity
pseudo-turbulence
condensation sink term
Nusselt correlation
a priori test
 Notes
Abstract: In the quest of a two-phase 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 phase-change. Because of the importance of phase-change phenomena, this paper is the first step to extend their analysis to non-material interfaces. The difficulty comes from the velocity and the temperature gradient discontinuity at the interface due to phase-change. 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 pseudoturbulent subcooled liquid provides reference data on heat and mass transfers. Turbulence and interface interactions are explicitly filtered to establish a hierarchy of the subgrid-scale terms to be modeled. Then, we propose to model dominating terms using the Bardina et al. (1983) scale-similarity hypothesis. 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 two-phase flow and heat transfer for non-material interfaces up to the smallest scales, is further analyzed at the macroscopic scale to improve the understanding of the strong two-way coupling between the flow dynamics and the interfacial heat transfer. Space- and time-averaged quantities computed from the DNS (e.g., phase-change rate) are compared with standard correlations. Discrepancies are observed between classical models for the condensation sink term in the two-fluid model. We discuss the effect of void fraction showing a strong phase-change enhancement due to pseudo-turbulence not always considered in averaged models. The proposed LES-like model opens new perspectives towards complex simulations of liquid/vapor flow with explicit interface tracking at reasonable computational cost.
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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Bibliographic ID: UF00102023
Volume ID: VID00163
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 661-Bois-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Towards Large Eddy Simulation of two-phase flow with phase-change:
Direct Numerical Simulation of a pseudo-turbulent two-phase 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: Two-phase flow, phase-change, DNS, LES, ISS, scale similarity, pseudo-turbulence,
condensation sink term, Nusselt correlation, a priori test




Abstract

In the quest of a two-phase 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 phase-change. Because of the importance of phase-change phenomena, this paper is the first step to
extend their analysis to non-material interfaces. The difficulty comes from the velocity and the temperature gradient
discontinuity at the interface due to phase-change. 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 subgrid-scale terms to be modeled. Then, we propose to model
dominating terms using the Bardina et al. (1983) scale-similarity 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 two-phase flow and heat transfer for non-material interfaces up
to the smallest scales, is further analyzed at the macroscopic scale to improve the understanding of the strong two-way
coupling between the flow dynamics and the interfacial heat transfer. Space- and time-averaged quantities computed
from the DNS (e.g., phase-change rate) are compared with standard correlations. Discrepancies are observed between
classical models for the condensation sink term in the two-fluid model. We discuss the effect of void fraction showing
a strong phase-change enhancement due to pseudo-turbulence not always considered in averaged models.
The proposed LES-like model opens new perspectives towards complex simulations of liquid/vapor flow with explicit
interface tracking at reasonable computational cost.


Nomenclature

Roman symbols
Cp specific heat-capacity (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 two-phase flows
equivalent to the well-known Large Eddy Simulation
(LES). Toutant et al. (2009) developed this Interfaces
and Subgrid-Scales (ISS) model to consider momentum
turbulent transfer at an under-resolved material inter-
face. Because phase-change is a key process in many
industrial applications, (e.g., heat exchangers, nuclear
safety...), we want to extend this model to non-material
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 phase-change.
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 pseudo-turbulent two-phase 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 up-scaling step. Finally, the up-
scaling towards averaged two-fluid 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 up-scaling procedure


The two-fluid 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 3D-averaged 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 two-fluid 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 Subgrid-Scales (ISS) concept aims
at generalizing the single-phase Large Eddy Simulation
(LES) to two-phase flows, not only modeling the effect
of the smallest turbulent scales, but also modeling the
smallest interface deformations. The detailed explana-
tion of the up-scaling 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 up-scaling 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 heat-capacity,
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)= V-D
at Re


Micro Meso Macro

S Filter MAE
0.5

-1-0.5 0 0.5 1


Figure 1: Sketch of the two-step up-scaling
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 two-phase system with phase-
change, at the mesoscopic scale of Fig. 1.
Local instantaneous conservation laws for two-phase
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 one-fluid 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 one-fluid 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 one-fluid 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 two-phase problem with
phase-change 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 area-weighted 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 time-derivatives. The strong
advantage of unconditional filtering over phase-filtering
is that subgrid-scale 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 one-fluid
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 phase-change 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 pseudo-turbulent 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 phase-change 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 phase-change 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 phase-change 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 non-material 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 subgrid-scale modeling In
the one-fluid 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 non-material 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 by-pass
this drawback, we seek for a continuous velocity field
containing all the fluctuations of the one-fluid velocity v
apart from the discontinuity. The main idea illustrated
by Fig. 3 is to extend the phase-change 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
non-material 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*^v-r*n* v


where v = 1p is the one-fluid 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 phase-change. 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 subgrid-scale
fluctuations of the phase-change 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 phase-change 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 phase-change 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 scale-similarity 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 up-scaling 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 pseudo-turbulent two-phase
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 two-phase flow applications, fluctua-
tions are the combination of single-phase turbulence
and pseudo-turbulence induced by bubble wakes. In
this preliminary work, we simulate the pseudo-turbulent
interactions between deformable buoyant bubbles con-
densing in a initially quiescent liquid. Fully-developed
single-phase turbulence is not considered to limit com-
putational cost and should motivate another specific
study.


Figure 4: DNS of a pseudo-turbulent two-phase
condensing flow: multibubbles condensation in a
subcooled initially-quiescent 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 10-3 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 space-averaging on this
reduced sample. We need space- and time-averaging.
Hence, statistical steady-state has to be achieved. The
energy released by condensation warms-up the liquid
and the averaged bulk temperature tends to the satura-
tion temperature, then decreasing phase-change to zero.
In order to get a steady-state flow with phase-change,
we must artificially maintain the liquid bulk temperature
constant by adding a small time-varying 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.m-3] 574.3 113.6
p [Pa.s] 65.16x10 6 23.73 x106
c, [J.kg-1.K-1] 10.11x103 17.15x 103
A [J.s-1.m-l.K-1] 44.7x 102 13.8x10 2
C [J.kg-1] 895.7 x 103




We used an explicit Front-Tracking method with
sharp interfaces implemented in the Trio_U code (Math-
ieu 2004). The fine mesh used for the DNS calculation
had 41.5 M-nodes 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 multi-bubble 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 under-resolved data obtained on coarser meshes (see
Fig. 10-15). 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 pseudo-turbulence induced by bubble wakes.
Bubble wakes influence both the bubble shape and the
rate of phase-change.

In the following sections, these DNS results are used
(i) to validate the Bardina et al. (1983) scale-similarity
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: Pseudo-turbulent simulation: Effect of
bubble wakes on the temperature (sliced field) and the
rate of phase-change over the interfaces.


A priori tests: subgrid-scale 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 under-resolved 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 time-averaged space-maximum 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
T|nv 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 time-averaged space-maximum 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 time-derivative 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
case-dependant. 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
under-prediction (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, under-prediction of the scale-similarity models is
well reported in single-phase turbulence (Sarghini et al.
1999). Scale-similar 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 single-phase turbulence is
to use mixed models that include an eddy-viscosity part
(functional modeling) as well as a scale-similar contri-
bution (structural modeling). In this first approach of
two-phase 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 phase-change 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 scale-similarity 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 3D-averaged mod-
gure 6: Modeling of the saturation temperature at the as Ranz and Marshall 1952) used in 3D-aeraged mod-
interface: Scatter of the variable 6 versus the els e.g., Mimouni et al. (2009); Morel et al. (2010). We
mesoscopic rate of phase-change first present some correlations from the literature. Then,
mesoscopic rate of phase-change m .
we compare their predictions to the results of our DNS
in the case of a single bubble and for the multi-bubble
simulation.
For this first example, we have chosen to illustrate the
multiscale approach on the closure of the condensation
sink term in two-fluid models. This closure is essential
since it determines the void fraction of collapsing bub-
bles in a sub-cooled 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 heat-transfer-controlled 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 scale-similarity 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.
Two-fluid 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
multi-bubble 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
1-0.876

-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:
Middle-slice of the temperature field and distribution of
the rate of phase-change 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


Pseudo-turbulent 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
steady-state and we assume invariance in the cross-flow
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 phase-change with increasing bubble Reynolds
number. Zeitoun et al. (1995) correlation over-predicts
heat-transfer at the interface whereas other correlations
tend to under-predict 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: Pseudo-turbulent 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
pseudo-turbulence. 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 two-fluid
models. We have been able to attribute the 20% increase
of phase-change 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 non-material interfaces, thus extending the ISS con-
cept to two-phase flows with phase-change. 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 phase-change tends to zero. Temperature gradi-
ent and velocity discontinuities are accounted for accu-
rately because they are linked to the mesoscopic rate
of phase-change. The scale-similarity 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 phase-change.
A priori tests are conducted on a configuration where
condensing bubbles interact together thus generating a
pseudo-turbulent flow. First validating results for the
closure of the two-phase LES-like 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
pseudo-turbulence. 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 two-fluid 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 two-phase
flows with phase-change (see Fig. 16). Besides valida-
tion, work in progress concerns the second up-scaling
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
GENCI-CCRT (Grant 2009-c2009026200).


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