A COMPRESSIBLE MODEL FOR
LOW MACH TWO-PHASE FLOW WITH
HEAT AND MASS EXCHANGES
N. GRENIER, J.P. VILA & Ph. VILLEDIEU
CONTEXT AND MOTIVATION
Context : COMPERE program from CNES &
DLR
External Heat flux
Gas phase
Evaporation
Research partners : ONERA, ZARM, CNRS,
Erlangen university, Air LIquide (Grenoble),
Astrium ST (Bremen)
Capillary raise
Liquid phase
Objectives : development of numerical tools for
simulating complex fluid behavior inside space
Separated two-phase launcher tanks:
flow with a free
• dynamical behavior  sloshing
moving interface
• thermal effects  heat and mass exchanges
• low gravity effects  capillary effects,
Marangoni convection …
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MULTIMAT 2011 - 5-9 septembre 2011
OUTLINE OF THE PRESENTATION
• 1. Presentation of the model
• 2. Numerical method
• 3. Numerical test cases
• 4. Conclusion
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MULTIMAT 2011 - 5-9 septembre 2011
OUTLINE OF THE PRESENTATION
• 1. Presentation of the model
• 2. Numerical method
• 3. Numerical test cases
• 4. Conclusion
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MULTIMAT 2011 - 5-9 septembre 2011
1. Presentation of the model
Modeling choices
• Two fluid model  diffuse interface model
• Advantage : not necessary to localize (level set method) or reconstruct (VOF
method) the interface between the two fluids  easy to implement
• Drawback : interface diffusion  necessary to define a “mixture” physical model
and to use low diffusive numerical scheme
• Compressible model
• Advantage : more general, easier to implement into a gas dynamics code (ONERA
context)
• Drawback : ill conditioned for low Mach number flows  low Mach Scheme
• Same velocity field for both fluids
• Advantage : hyperbolic model, no closure assumption needed
• Drawback : impossible to deal with subscale phenomena (subgrid bubbles or
droplets …)
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MULTIMAT 2011 - 5-9 septembre 2011
1. Presentation of the model
Inviscid two-fluid Model
  g
    g v   0


t

 
Liquid bulk density
 t      v   0
(1) 
  v      v  v  pI    g
 t
  E

     Ev  pv    g.v
 t
Gas bulk density
with  E   e 
  g  l
1 2
 v being the mixture total energy per unit volume and
2
the mixture bulk density.
To get a close model, it is now necessary to give a relation between the
“mixture” pressure p, the bulk densities  g ,l and the mixture specific
internal energy e.
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1. Presentation of the model
Extension to non isothermal flows
Let T denote the mixture temperature and  the gas volume fraction. 
and T are assumed to be the unique solution of the following system :
(2)
  g 
 

p
,
T

p
,
T
 g



1


 




g
l


e


e
(
,
T
)


e
(
,T )
g g
l l


1
Local mechanical equilibrium
Local thermal equilibrium
where p = pg(g,T), p = pl(l,T) denote the gas and liquid EOS and
e=eg(g,T), e=el(l,T) denote the gas and liquid colorific laws.
The mixture EOS is then (implicitly) defined by :
 ρg 
 ρ

(3) p(ρ g ,ρ ,  e)  pg  , T   p 
,T 
 1 


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MULTIMAT 2011 - 5-9 septembre 2011
1. Presentation of the model
Other interpretation of closure equations (5)-(6)
Closure relations (2)-(3) can also be interpreted as a direct consequence of the
following modeling assumption for the mixture Gibbs potential :
g mix ( p, T )  yg g g ( p, T )  yl gl ( p, T )
with
yg 
g

, yl  l


Ideal mixture assumption
Important consequence : System (1) with pressure law given by (2)-(3) is
thermodynamically consistent in the sense that it has a convex entropy in the
sense of Lax defined as :

 (  g , l ,  E ,  v)    g s g (  g ,  g eg )  l sl ( l , l el )


Φ(  g , l ,  E ,  v)   (  g , l ,  E )V
where eg, el are the gas and liquid specific internal energies (implicitly
defined by the solution of (2)), sg and sl are specific entropies, and


are the real densities.
 g = g , l = l

1
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1. Presentation of the model
Inclusion of diffusion and capillary effects.
  g
    g v   0


t

 
 t      v   0
(4) 
  v      v  v  pI    g  div( τ  τ )
v
c
 t
  E

     Ev  pv    g.v  div(φc )  div( τ v : v) + div( τ c : v)
 t
with :
τv   div(v) I + (v  t v)

   
τ c   (T )   I 

2





φc   T
Viscous stress tensor
Capillary stress tensor
(body force formulation)
Heat flux
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1. Presentation of the model
Approximate enthalpy equation for low Mach flows
Neglecting viscous and capillary effects, the energy equation is equivalent
to :
 h
p
     hv      T  +
 v.p
t
t
Heat flux
presssure contribution
which is the Eulerian formulation of the well-known thermodynamic relation :
dh   q 
1

dp
For low Mach number flow, with imposed pressure on one of the boundaries,
one generally has : 1/ dp << q, and therefore the energy equation can be
replaced by the heat equation :
 h
     hv      T   0
t
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1. Presentation of the model
Phase change modeling
Phase change phenomena can be included in model (7) by just adding
a relaxation source term in the r.h.s. :
U
 (U )  U
   F(U , U ) 
 S (U )
t

where (U) is the thermodynamic equilibrium state corresponding to U,
defined as the state which maximizes the mixture entropy under the
constraints of imposed total volume, total mass and total energy :
Max yg sg (eg , vg )  yl sl (el , vl )
yg  yl 1
yg eg  yl el  e
yg vg  yl vl  v
1
, e  e(U )
where v 
 (U )
In practice, the thermodynamic equilibrium time scale is assumed to be
infinitely small compared to the macroscopic time scale  local thermodynamic
equilibrium assumption.
This idea was first proposed in : HELLUY P., SEGUIN N., “Relaxation model of phase transition flows”, M2AN, Math. Model. Numer. Anal., vol. 40,
num. 2, 2006, p. 331–352.
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OUTLINE OF THE PRESENTATION
• 1. Presentation of the model
• 2. Numerical method
• 3. Numerical test cases
• 4. Conclusion
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MULTIMAT 2011 - 5-9 septembre 2011
2. Numerical scheme
A finite volume relaxation scheme
Each time step is divided in two stages :
Transport step  Eulerian finite volume scheme
U U
*
K
n
K
t

mK

eK
Gen1/ 2 me  t S Kn1/ 2
K
e
Numerical flux on edge e
ne,K
Ke
Relaxation step  local thermodynamic equilibrium
U Kn1  (U K* )
Note that, by construction, the second step is entropy diminishing.
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2. Numerical scheme
Expression of the hyperbolic numerical flux
(isothermal case  no energy equation)
0



G(U L ,U R , ne )  ve+U L  ve-U R   0

pn 
advection term
 e e
Low Mach Scheme
Which expression choosing for pe and ve ?
pe =
1
2
1
ve =
2
 pL  pR 
 v L  v R  .n e
e
ne
UL
UR
pressure term
Centered scheme for pressure (see Dellacherie
(2011) recent work on low Mach number schemes)
-  e  pR  pL 
Centered expression + stabilizing
pressure term. Expression of the
positive parameter e will be
given later.
Remark : a similar idea has been proposed by Liou (AUSM+up scheme, JCP 2006) and by Li & Gu (all Mach Roe type
scheme, JCP 2008) for the compressible gas dynamics system.
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2. Numerical scheme
Semi-implicit version of the scheme
(isothermal case)
To avoid a restrictive stability condition based on the sound celerity, mass
conservation equations are solved with a implicit scheme.
  g* , K 
 * 
 
 l ,K 
  gn, K  t
 n 
   mK
 l ,K 

+

v

e

eK

  g* , K  -   g* , K 
 *   ve  * 
 
 
 l ,K 
 l ,K 

 me

Newton
algorithm
An explicit scheme is used to compute the new velocity from the
momentum equation :
 K* v*K  Kn v nK 
with
ve =
1
2
t
mK
 v
eK

e
*
K* v nK + ve Ke
v nKe  pene,K  me
 v nK  v nKe  .ne -  e  pK*  pKe* 
and
pe =
1
2
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p
*
K
*
 pKe

2. Numerical scheme
Formal justification of the stabilizing role of “-  (pR - pL) “
Let us consider the following modified system for isothermal flows
Modified convective velocity
 ρ
 t     ( v   v)ρ   0
 g 
(1') 
where ρ   
 l 
  v     ( v   v)  (  v)  pI   0
 t
Proposition : the term  v   h p has a stabilizing effect in the sense for that any
smooth solution of (1’) one has the following free energy balance equation :

1

1


2
2

F


v

di
v

F


v

p
(
v


v
)



  p. v

t 
2
2




Dissipative source term if v
is proportional to – grad(p)




g
l
 + l fl 
 denotes the free energy
where  F (  g , l )   g f g 

(

,

)
1


(

,

)
g
l 
g
l 


of the mixture.
Remark : the same property holds for the non isothermal case but with the entropy
instead of the free energy.
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2. Numerical scheme
How to choose the value of e ?
Stability theoretical result : Under the two following conditions
(i)
 t mK
t mK 
 e  Max 
,
*
* 
2
m

2
m

K
K
K
K 


e
e
(ii)
2t v K mK
1
Sup m
K
K
e
with
vK 
CFL like condition
1
 K* mK

*
Ke
ve me
e
mean cell velocity
the semi-implicit scheme is entropic (in the sense of Lax).
In practice, we take :
 t mK t mKe
1
 e  Max 
,
n
n
2
m

 K K mKe  Ke
 t ( vnK  cKn )mK 

  cfl
 and Sup 
m



K
with cfl much larger than 1 for low mach number flows.
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2. Numerical scheme
Discretization of the enthalpy equation
To respect the maximum principle on the temperature, we use the following
upwind scheme based on the sign of the mass fluxes :
 K* hK*   Kn hKn 
t
mK
 (h
*
g ,K
max( g* ,e , 0)  hg* , Ke min( g* ,e , 0))me 
e
t
mK
 (h
*
l ,K
max(l*,e , 0)  hl*, Ke min(l*,e , 0))me
e
*
*


Where g ,K , respectively l ,K , denote the gas, respectively the
liquid, mass numerical flux.
In practice, two variants of the scheme can be used :
• an explicit scheme with respect to the fluid temperature,
• a fully implicit scheme with respect to all thermodynamic
variables  g , l , T
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2. Numerical scheme
Relaxation step U*  Un+1
, v and h are left unchanged during this step. We thus have :
n 1
n 1
*
*
*










l
g
l
 g
 * *
n 1
n 1
n 1

h


h
(
p
,
T
)



g
g
l hl ( p, T )

If both phases can coexist (gas – liquid thermodynamic equilibrium)
p(ln1,gn1,T n1 )=psat (T n1 )
System of 3 equations
and 3 unknowns
else only one phase can be present in the cell at the end of the time step
 gn1  * , ln1  0 or ln1  * , gn1  0
Remark : in practice, for numerical purpose, a minimal lower value is imposed for
gas and liquid mass fractions.
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OUTLINE OF THE PRESENTATION
• 1. Presentation of the model
• 2. Numerical method
• 3. Numerical test cases
• 4. Conclusion
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3. Numerical test cases
Linear oscillations in a 2D rectangular tank
•ρ1=1 kg.m-3 ; c1=300 m.s-1
•ρ2=1000 kg.m-3 ; c2=1200 m.s-1
•Transverse acceleration : a0 = 0.01 g
•Coarse cartesian grid : 40 X 20
•Ma = 2 10-5
2
Possibility to compute an analytical solution
as a série expansion by potential flow theory.
(see for example Landau & Lifschitz T6, fluid
Mechanics)
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3. Numerical test cases
Linear oscillations in a 2D rectangular tank
Exact solution
Numerical
Scheme
Godunov
scheme
Low Mach
scheme
Time step
0.0005
0.05
Total CPU
time
Second order low Mach scheme
Second order Godunov type scheme
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MULTIMAT 2011 - 5-9 septembre 2011
20
1
3. Numerical test cases
Dynamical test case :
bubble rise inside a liquid : Sussman et al test case
(Sussman, M. and Smereka, P. and Osher, S., A Level Set approach for computing solutions to
incompressible two-phase flow, Journal of Computational Physics, 114, 146-159, 1994)
Explicit
Godunov type
scheme with
real EOS
Explicit
Godunov type
scheme with
modified EOS
Semi-implicit
low Mach
scheme with
real EOS
Cartesian mesh
Cartesian mesh
140 X 233
Cartesian mesh
140 X 233
140 X 233
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3. Numerical test cases
Bubble rise inside a liquid : Sussman et al test case
(Sussman, M. and Smereka, P. and Osher, S., A Level Set approach for computing solutions to
incompressible two-phase flow, Journal of Computational Physics, 114, 146-159, 1994)
Semi-implicit
low Mach scheme
with real EOS
Sussman et al
Solution with
Level Set
method and
incompressible
model
Usual Godunov
type scheme with
real EOS
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3. Numerical test cases
Rayleigh-Bénard instability
Wall with
imposed
temperature
Periodic
boundary
conditions
Gas phase
g
Liquid phase
Ra  g 
TH 3

with

1   
  T  p
Critical Rayleigh number for instability : Rac  1708
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3. Numerical test cases
Rayleigh-Bénard instability
Stable
Unstable
Unstable
Stable
Stable
Unstable
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MULTIMAT 2011 - 5-9 septembre 2011
3. Numerical test cases
Marangoni convection test case
Adiabatic Wall
Gas phase
Wall with
imposed
temperature
T = T0
Liquid phase
Wall with
imposed
temperature
T = T1<T0
Adiabatic Wall
No gravity. Static contact angle : q = 90°
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3. Numerical test cases
Marangoni convection test case
Coarse grid
Medium grid
Fine grid
Temperature
field
Volume fraction
field
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3. Numerical test cases
1D Evaporation test case
Evaporation front
Outlet with
imposed
Liquid phase
pressure :
p = p0
=l , p = p0 , T = Tsat(p0),
u = uI
Gas phase
=v , p= p0 , u= 0,
T = f(x)
Approximate theoretical solution
qw
m
Lv
uI  
m
v
 v 
ul  u I  1  
 l 
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Wall with
imposed
heat flux
qw
3. Numerical test cases
1D Evaporation test case
l  1000kg m3
v  1 kg m3
Interface position vs time
for several values of Lv and qw.
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3. Numerical test cases
1D Evaporation test case
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OUTLINE OF THE PRESENTATION
• 1. Presentation of the model
• 2. Numerical method
• 3. Numerical test cases
• 4. Conclusion
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CONCLUSIONS AND FUTURE PROSPECTS
• An Eulerian two-fluid model with diffuse interface has been applied to the
simulation of low Mach separated two-phase flows with heat and mass
transfers.
•Using formal arguments, a simple semi-implicit low Mach scheme has been
proposed for this model. For isothermal flows, this scheme has been proved to
be entropy diminishing under a CFL condition which do not depend on the
sound celerity.
• This methodology can be very easily implemented in existing industrial
compressible CFD codes for multi-physics applications (work in progress at
ONERA). It is a very interesting alternative to classical approaches based on
one-fluid incompressible model with VOF or Level Set methods.
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CONCLUSIONS AND FUTURE PROSPECTS
•This two-fluid approach has been successfully applied to several academic
problems for low Mach two-phase flows.
• Future works will be devoted to the
• assessment of the method for more complex phase change problems.
• extension of the model to more complex physical problems : multicomponent gas phase with an incondensable specie, 3 phases
problems …
• parallelization of the code for 3D applications
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Thank you for your attention …..
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Back – up
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1. Presentation of the model
Purely Dynamical model
(inviscid Isothermal flow)
  g
    g v   0

 t
 
(1) 
    v  0
 t
  v
 t      v  v  pI    g

with
  g  l the mixture bulk density.
To get a close model, it is necessary to give a relation between the
“mixture” pressure p and the bulk gas density  g and the bulk liquid
density l .
Remark: The gas volume fraction  is not explicitly transported in this model.
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1. Presentation of the model
Purely dynamical model
• Let  denote the gas volume fraction : g  g ; l  (1   )l
 is defined as the solution of
(2)
 g
pg 


  

p



1





Local pressure equilibrium between
the two non miscible fluids
where p = pg(g) and p = pl(l) denote the gas and liquid equation of state.
Mixture EOS
 ρg
(3) p(ρ g , ρ )  pg 


 ρ 

p



1





Remark : if the expressions of pg and pl are complex, p is only implicitly defined in
function of the bulk densities.
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MULTIMAT 2011 - 5-9 septembre 2011
1. Presentation of the model
Example : stiffened gas model for both fluids.
Expression of the Gibbs potential for each fluid :
gi ( p, T )   i cvi T 1  ln(T )  ( i 1)cvi T ln( p  i )  ui0  Tsi0
Fluid i Equation of state
1
T
 vi ( p, T )  ( i  1)cvi
 i ( p, T )
p  i
Fluid i calorific law
p   i i
ei ( p, T )  ei0  cvi T
p  i
With these notations, system (5)-(6) is equivalent to :
Mixture specific
volume
g

l
1
v


v
(
p
,
T
)

vl ( p, T )
g

 g  l  g  l
 g  l

(7) 
 e   g e ( p , T )   l e ( p, T )

 g  l g
 g  l l

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MULTIMAT 2011 - 5-9 septembre 2011
1. Presentation of the model
Other interpretation of closure equations (5)-(6)
Closure relations (5)-(6) can also be interpreted as a direct consequence of the
Ideal mixture assumption
following modeling assumption for the mixture Gibbs potential :
g mix ( p, T )  yg g g ( p, T )  yl gl ( p, T ) - T ( yg , yl )
with
yg 
g

, yl  l


Important property : System (4) with pressure law given by (5)-(6) is
thermodynamically consistent in the sense that it has a infinite set of convex
entropies in the sense of Lax defined as :
 g l


(

,

,

E
,

v
)



s
(

,
e
)


s
(

,
e
)


(
, )

g
l
g
g
g
g
l l
l
l



Φ(  ,  ,  E ,  v)   (  ,  ,  E )V
g
l
g
l

where  is an arbitrary concave function, eg, el are the specific internal
energies, implicitly defined by the solution of (5), sg and sl are the specific


entropies, and
are the real fluid densities.
 g = g , l = l

1
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MULTIMAT 2011 - 5-9 septembre 2011
1. Presentation of the model
Purely dynamical model (3/3)
Proposition : If pg and pl are strictly non decreasing functions, model
(1)-(2)-(3) is hyperbolic and has a convex entropy in the sense of Lax
defined as :

 g 
1
 l 
2
 (  g , l ,  V)   V   g f g   + l fl 

2

1





 
Φ(  ,  ,  V )   (  ,  ,  V )V  pV
g
l
g
l

Lax entropy (convex function
of the conservative variables)
Entropy flux
where fg and fl are the free energy of the gas and liquid phases and are
defined as :
pi (  )
pi
fi ( ) 
d
dfi   pi d i  2 d i
2
i


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MULTIMAT 2011 - 5-9 septembre 2011
4. APPLICATIONS
Linear oscillations in a 2D rectangular tank
•ρ1=1 kg.m-3 ; c1=300 m.s-1
•ρ2=1000 kg.m-3 ; c2=1200 m.s-1
•Transverse acceleration : a0 = 0.01 g
•Coarse cartesian grid : 40 X 20
•Ma = 2 10-5
2
Possibility to compute an analytical solution
as a série expansion by potential flow theory.
(see for example Landau & Lifschitz T6, fluid
Mechanics)
42
MULTIMAT 2011 - 5-9 septembre 2011
4. APPLICATIONS
Linear oscillations in a 2D rectangular tank
Exact solution
Numerical
Scheme
Godunov
scheme
Low Mach
scheme
Time step
0.0005
0.05
Total CPU
time
Second order low Mach scheme
Second order Godunov type scheme
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MULTIMAT 2011 - 5-9 septembre 2011
20
1
1. Presentation of the model
References
• R. ABGRALL, R. SAUREL. A simple method for compressible multifuid flows, SIAM J. Sci.
Comput. 21 (3) : 1115-1145, (1999). 66
• G. ALLAIRE, G. FACCANONI et S. KOKH, A strictly hyperbolic equilibrium phase transition model.
C. R. Acad. Sci. Paris Sér. I, 344 pp. 135–140, 2007.
• CARO F., COQUEL F., JAMET D., KOKH S., “A Simple Finite-Volume Method for
Compressible Isothermal Two-Phase Flows Simulation”, Int. J. on Finite Volumes, vol. 3,
num. 1, 2006, p. 1–37.
• HELLUY P., SEGUIN N., “Relaxation model of phase transition flows”, M2AN, Math. Model. Numer.
Anal., vol. 40, num. 2, 2006, p. 331–352.
• LE METAYER O., MASSONI J., SAUREL R., “Elaborating equations of state of a liquid
and its vapor for two-phase flow models”, Int. J. of Th. Sci., vol. 43, num. 3, 2004, p. 265–276.
• G. CHANTEPERDRIX, JP VILA, P. VILLEDIEU, A compressible model for separated two-phase
flow computations, FEDSM02, 14-18 July, Montreal, Quebec, Canada, 2002
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MULTIMAT 2011 - 5-9 septembre 2011