A COMPRESSIBLE MODEL FOR LOW MACH TWO-PHASE FLOW WITH HEAT AND MASS EXCHANGES N. GRENIER, J.P. VILA...
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Transcript of A COMPRESSIBLE MODEL FOR LOW MACH TWO-PHASE FLOW WITH HEAT AND MASS EXCHANGES N. GRENIER, J.P. VILA...
A COMPRESSIBLE MODEL FORLOW MACH TWO-PHASE FLOW WITH
HEAT AND MASS EXCHANGES
N. GRENIER, J.P. VILA & Ph. VILLEDIEU
MULTIMAT 2011 - 5-9 septembre 20112
CONTEXT AND MOTIVATION
Context : COMPERE program from CNES & DLR
Research partners : ONERA, ZARM, CNRS, Erlangen university, Air LIquide (Grenoble), Astrium ST (Bremen)
Objectives : development of numerical tools for simulating complex fluid behavior inside space launcher tanks:• dynamical behavior sloshing • thermal effects heat and mass exchanges• low gravity effects capillary effects, Marangoni convection …
Separated two-phase flow with a free
moving interface
Gas phase
Liquid phase
External Heat flux
Evaporation Capillary raise
MULTIMAT 2011 - 5-9 septembre 20113
OUTLINE OF THE PRESENTATION
• 1. Presentation of the model
• 2. Numerical method
• 3. Numerical test cases
• 4. Conclusion
MULTIMAT 2011 - 5-9 septembre 20114
OUTLINE OF THE PRESENTATION
• 1. Presentation of the model
• 2. Numerical method
• 3. Numerical test cases
• 4. Conclusion
MULTIMAT 2011 - 5-9 septembre 20115
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 …)
MULTIMAT 2011 - 5-9 septembre 20116
To get a close model, it is now necessary to give a relation between the “mixture” pressure p, the bulk densities , and the mixture specific internal energy e.
1. Presentation of the model
0
0 (1)
.
ggt
t
pt
EE p
t
v
v
vv v g
v v g v
I
g
Inviscid two-fluid Model
l
with being the mixture total energy per unit volume and
21
2eE v
the mixture bulk density. g l
Gas bulk density
Liquid bulk density
MULTIMAT 2011 - 5-9 septembre 20117
1. Presentation of the model
, ,1
(2)
( , ) ( , )1
gg
g lg g l l
T T
T T
p p
e e e
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 :
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 :
(3) (ρ
,ρρ
,ρ ) ,,1
gggp T p Te p
MULTIMAT 2011 - 5-9 septembre 20118
1. Presentation of the model
Other interpretation of closure equations (5)-(6)
,
( , , , ) s ( ,
( , , ) ( ,
s (
,
, ))
)
l l l lg g g g g g
g
l l
l g l E
eE
E
e
V
v
Φ v
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 :
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.
= , =1
g lg l
Closure relations (2)-(3) can also be interpreted as a direct consequence of the following modeling assumption for the mixture Gibbs potential :
( , ) ( , ) ( , ) with , mix
g lg g l l g lg p T y g p T y g p T y y
Ideal mixture assumption
MULTIMAT 2011 - 5-9 septembre 20119
1. Presentation of the model
0
0 (4)
( )
. ( ) ( : ) + ( : )
v
v
g
c
c
gt
t
p divt
EE p div div div
t
c
v
v
vv v g
v v g v v
τ
vτφ
τ
τ
I
v div( ) + ( )t τ v I v v
Inclusion of diffusion and capillary effects.
with :
c 2( )T I
τ
Viscous stress tensor
Capillary stress tensor (body force formulation)
T cφ Heat flux
MULTIMAT 2011 - 5-9 septembre 201110
1. Presentation of the model
1 dh q dp
0h
h Tt
v
Approximate enthalpy equation for low Mach flows
Neglecting viscous and capillary effects, the energy equation is equivalent to :
which is the Eulerian formulation of the well-known thermodynamic relation :
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 :
Heat flux
presssure contribution
+ .h
ht
Tp
pt
vv
MULTIMAT 2011 - 5-9 septembre 201111
1. Presentation of the model
( , )(
))
(U
U St
UU U
U
F
Phase change modeling Phase change phenomena can be included in model (7) by just adding a relaxation source term in the r.h.s. :
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 :
1( , ) ( , )
g l
g g l l
g g l l
g g g g l l l ly yy e y ey v
evy v
Max y s e v y s e v
where
1 , ( )
( )e Uv e
U
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.
In practice, the thermodynamic equilibrium time scale is assumed to be infinitely small compared to the macroscopic time scale local
thermodynamic equilibrium assumption.
MULTIMAT 2011 - 5-9 septembre 201112
OUTLINE OF THE PRESENTATION
• 1. Presentation of the model
• 2. Numerical method
• 3. Numerical test cases
• 4. Conclusion
MULTIMAT 2011 - 5-9 septembre 201113
2. Numerical scheme
A finite volume relaxation scheme
Each time step is divided in two stages :
* 1/ 2 1/ 2 n n nK K e e K
e KK
tU U G m t S
m
Transport step Eulerian finite volume scheme
Numerical flux on edge e
K
Ke
ne,Ke
1 * ( ) nK KU U
Relaxation step local thermodynamic equilibrium
Note that, by construction, the second step is entropy diminishing.
MULTIMAT 2011 - 5-9 septembre 201114
2. Numerical scheme
+ -e e
advection ter
pressure term
m
v v ( , , )
0
0 L
e e
RL R eG U U
p
U U
n
n
Expression of the hyperbolic numerical flux (isothermal case no energy equation)
Which expression choosing for pe and ve ?
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.
Low Mach Scheme
1=
2 e L Rp p p Centered scheme for pressure (see Dellacherie
(2011) recent work on low Mach number schemes)
e e
1v = . -
2 R Le L R p p v v nCentered expression + stabilizing pressure term. Expression of the positive parameter e will be given later.
ne
UL
UR
e
MULTIMAT 2011 - 5-9 septembre 201115
2. Numerical scheme
* * *, , , ,+ -
e e* * *, , , ,
v v n
g K g K g K g K
ene KKl K l K l K l K
tm
m
* * * *, v + v n n n n
K K K K e K K e Ke Ke e e K ee KK
tp m
m
v v v v n
* *e e
1 v = . -
2n n
e K Ke K Kep p v v n
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.
An explicit scheme is used to compute the new velocity from the momentum equation :
with and * *1 =
2e K Kep p p
Newton algorithm
MULTIMAT 2011 - 5-9 septembre 201116
2. Numerical scheme
(
( ) 0(1') where
) 0 ( )
g
l
t
pt
v v
v
ρρ
ρv
vv I
Formal justification of the stabilizing role of “- (pR - pL) “
Let us consider the following modified system for isothermal flowsModified convective velocity
Remark : the same property holds for the non isothermal case but with the entropy instead of the free energy.
2 2 )1 1
(2 2
.F di pt
pv F
vv v vv
Proposition : the term has a stabilizing effect in the sense for that any smooth solution of (1’) one has the following free energy balance equation :
h p v
where denotes the free energy of the mixture.
( , ) + ( , ) 1 ( , )
g lg l g g l l
g l g l
F f f
Dissipative source term if v
is proportional to – grad(p)
MULTIMAT 2011 - 5-9 septembre 201117
2. Numerical scheme
the semi-implicit scheme is entropic (in the sense of Lax).
* *,
2 2e
e e
KKe
K K K K
t mt mMax
m m
**
2 v 11 v v
e
K KK K e e
eK K K
mean cell velocity
K
CFL like condition
t mwith m
m mSup
Stability theoretical result : Under the two following conditions
(i)
(ii)
In practice, we take : )1
,2
e
e e
nK KKK
e n nK K K K K
t c mt mt mMax Sup cfl
m m m
nK ( v
and
How to choose the value of e ?
with cfl much larger than 1 for low mach number flows.
MULTIMAT 2011 - 5-9 septembre 201118
Where , respectively , 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
2. Numerical scheme
* * * * * * * * * *, , , , , , , ,( max( ,0) min( ,0)) ( max( ,0) min( ,0))n n
K K K K g K g e g Ke g e e l K l e l Ke l e ee eK K
t th h h h m h h m
m m
*,Kl
, ,g l T
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 :
*,Kg
MULTIMAT 2011 - 5-9 septembre 201119
2. Numerical scheme
*1 1
1 1 1
* *
* * ( , ) ( , )
n ng l
n n ng g l
g l
lh h p hT p T
1 1 1 1, , ( )= ( )n nsg a
n ntl Tp p T
Relaxation step U* Un+1
If both phases can coexist (gas – liquid thermodynamic equilibrium)
, v and h are left unchanged during this step. We thus have :
System of 3 equations and 3 unknowns
else only one phase can be present in the cell at the end of the time step 1 * *1 1 1, 0 or , 0n n n n
g l l g
Remark : in practice, for numerical purpose, a minimal lower value is imposed for
gas and liquid mass fractions.
MULTIMAT 2011 - 5-9 septembre 201120
OUTLINE OF THE PRESENTATION
• 1. Presentation of the model
• 2. Numerical method
• 3. Numerical test cases
• 4. Conclusion
MULTIMAT 2011 - 5-9 septembre 201121
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
Possibility to compute an analytical solution as a série expansion by potential flow theory. (see for example Landau & Lifschitz T6, fluid Mechanics)
2
MULTIMAT 2011 - 5-9 septembre 201122
3. Numerical test cases
Linear oscillations in a 2D rectangular tank
Second order low Mach scheme Second order Godunov type scheme
Exact solution
Numerical Scheme
Godunov scheme
Low Mach scheme
Time step 0.0005 0.05
Total CPU time
20 1
MULTIMAT 2011 - 5-9 septembre 201123
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
Cartesian mesh 140 X 233
Semi-implicit low Mach
scheme with real EOS
Cartesian mesh 140 X 233
Explicit Godunov type scheme with
modified EOS
Cartesian mesh 140 X 233
MULTIMAT 2011 - 5-9 septembre 201124
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)
Sussman et alSolution with Level Set method and incompressible model
Usual Godunov type scheme with real EOS
Semi-implicit low Mach scheme with real EOS
MULTIMAT 2011 - 5-9 septembre 201125
3. Numerical test cases
3
TH
Ra g
1708cRa
Rayleigh-Bénard instability
Critical Rayleigh number for instability :
Wall with imposed temperature
Liquid phase
Gas phase
Periodic boundary conditions
g
with 1
pT
MULTIMAT 2011 - 5-9 septembre 201126
3. Numerical test cases
Rayleigh-Bénard instability
Stable
Stable
Stable
Unstable Unstable
Unstable
MULTIMAT 2011 - 5-9 septembre 201127
3. Numerical test cases
Marangoni convection test case
No gravity. Static contact angle : = 90°
Liquid phase
Adiabatic Wall
Wall with imposed temperature T = T0
Adiabatic Wall
Gas phase
Wall with imposed temperature T = T1<T0
MULTIMAT 2011 - 5-9 septembre 201128
3. Numerical test cases
Marangoni convection test case
Volume fraction field
Temperature field
Coarse grid Medium grid Fine grid
MULTIMAT 2011 - 5-9 septembre 201129
3. Numerical test cases
1D Evaporation test case
Outlet with imposed pressure : p = p0
Gas phaseWall with imposed heat flux
qw
Evaporation front
=l , p = p0 , T = Tsat(p0), u = uI
=v , p= p0 , u= 0,
T = f(x)
Approximate theoretical solution
v
wqm
L 1 v
l Il
u u
I
v
mu
Liquid phase
MULTIMAT 2011 - 5-9 septembre 201130
3. Numerical test cases
1D Evaporation test case
3 1000 mkgl 31 v kg m
Interface position vs timefor several values of Lv and qw.
MULTIMAT 2011 - 5-9 septembre 201132
OUTLINE OF THE PRESENTATION
• 1. Presentation of the model
• 2. Numerical method
• 3. Numerical test cases
• 4. Conclusion
MULTIMAT 2011 - 5-9 septembre 201133
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.
MULTIMAT 2011 - 5-9 septembre 201134
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 : multi-component gas phase with an incondensable specie, 3 phases problems …• parallelization of the code for 3D applications
MULTIMAT 2011 - 5-9 septembre 201137
with the mixture bulk density.
1. Presentation of the model
0
(1) 0
ggt
t
pt
v
v
vv v gI
g l
Purely Dynamical model (inviscid Isothermal flow)
To get a close model, it is necessary to give a relation between the “mixture” pressure p and the bulk gas density and the bulk liquid density .
g
Remark: The gas volume fraction is not explicitly transported in this model. l
MULTIMAT 2011 - 5-9 septembre 201138
1. Presentation of the model
ρ
1 (3) (ρ , )
ρρ g
g g pp p
Purely dynamical model
(2) 1
ggp p
• Let denote the gas volume fraction :
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.
; (1 )lg g l
Mixture EOS
Remark : if the expressions of pg and pl are complex, p is only implicitly defined in
function of the bulk densities.
is defined as the solution of
MULTIMAT 2011 - 5-9 septembre 201139
1. Presentation of the model
0( , )i
i ii i v
i
pe p T e c T
p
Example : stiffened gas model for both fluids.
Expression of the Gibbs potential for each fluid :
1( , ) ( 1)
( , ) ii i vi i
Tv p T c
p T p
0 0( , ) 1 ln( ) ( 1) ln( )i ii i v i v i i ig p T c T T c T p u Ts
Fluid i Equation of state Fluid i calorific law
With these notations, system (5)-(6) is equivalent to :
1( , ) ( , )
(7)
( , ) ( , )
g lg l
g l g l g l
g lg l
g l g l
v v v
e
p p
p
T T
Te e p T
Mixture specific
volume
MULTIMAT 2011 - 5-9 septembre 201140
1. Presentation of the model
Other interpretation of closure equations (5)-(6)
,
( , , , ) ( , )
( , , ) (
s
, , )
s ( , ( , ))g g gl l lg g
g
g
l g
l
l
lleE
E E
e
V
v
Φ v
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 :
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. = , =
1g l
g l
Closure relations (5)-(6) can also be interpreted as a direct consequence of the following modeling assumption for the mixture Gibbs potential :
( , ) ( , ) ( , ) - with ,( , )mix gg l
g g lll l gg p T y g p T y g p T y yT y y
Ideal mixture assumption
MULTIMAT 2011 - 5-9 septembre 201141
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 :
1. Presentation of the model
21( , , ) +
2
( , , , ,
) )
1
(lg
lgg g g
g
l l l
l p
ff
V V
V V V VΦ
d
pf i
i 2
)()(
where fg and fl are the free energy of the gas and liquid phases and are defined as :
Lax entropy (convex function of the conservative variables)
Entropy flux
2i
i i i ii
pdf p d d
Purely dynamical model (3/3)
MULTIMAT 2011 - 5-9 septembre 201142
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
Possibility to compute an analytical solution as a série expansion by potential flow theory. (see for example Landau & Lifschitz T6, fluid Mechanics)
2
MULTIMAT 2011 - 5-9 septembre 201143
4. APPLICATIONS
Linear oscillations in a 2D rectangular tank
Second order low Mach scheme Second order Godunov type scheme
Exact solution
Numerical Scheme
Godunov scheme
Low Mach scheme
Time step 0.0005 0.05
Total CPU time
20 1
MULTIMAT 2011 - 5-9 septembre 201144
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 forCompressible 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 liquidand 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-phaseflow computations, FEDSM02, 14-18 July, Montreal, Quebec, Canada, 2002