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Dynamic relaxation processes in compressible
multiphase flows
Application to evaporation phenomena
Olivier Le Métayer
Aix-Marseille University, Polytech Marseille
UMR CNRS 7343
Some industrial applications where evaporation is crucial
Fuel injectors at
high pressure
BLEVE phenomena (Security) : rapid
depressurization of liquefied gas + combustion
Cryogenic injection
device in space launcher
Typical physical processes we are interested in
Low pressure
chamber
Liquid/vapor mixture (drops, bubbles,
pockets, ,…)
Liquid flow from a
high pressure chamber
Drops evaporation and jet explosion
« Flashing » (rapid depressurization)
generated by the initial pressure ratio
Valve/diaphragm
Cavitation created by geometric
singularities (strong rarefaction waves)
Interface problems
Two-velocities flow
(inter-penetration)
How can be solved the multiphase flow when the topology strongly varies ?
A way is to consider relaxation effects
Plan
Presentation of the relaxation effects
Some multidimensional numerical results
Description of the non-equilibrium multiphase flow model
Non-equilibrium
multiphase compressible
flow model
Multiphase compressible flow model
0)u(.t
)(k
k
kIkk αPIPuuρ.
t
)u(
kIIkk α.uPuPρE.
t
)E(
0α.ut
αkI
k
Each phase has its own set of equations and associated variables
(velocity, pressure temperature, entropy,…) :
+ Equation of state : ),P(e kkk
Solved by the Discrete Equations Method (Abgrall & Saurel, JCP, 2003 ; Saurel & al.,
JFM, 2003 ; Chinnayya & al., JCP, 2004 ; Le Métayer & al., JCP, 2005 ; Saurel & al.,
IJNMF, 2007 ; Le Métayer & al., JCP, 2011)
Initially proposed by (Baer & Nunziato,IJMF,1986) for two phases
Fixed control volume = sum of the sub-volumes filled with the fluids
Each phase has a time-dependent sub-volume (volume fraction)
Discrete Equations Method :
Godunov’s concept applied to multiphase cells
•DEM=averaging procedure by solving Riemann problems between pure fluids at each cells interface
Cell i Cell (i-1) Cell (i+1)
•Godunov’s method=averaging procedure by solving a single Riemann problem at each cells interface
Cell i Cell (i-1) Cell (i+1)
Fixed control volume = volume filled with the fluid
Major contributions of DEM (1)
Discrete equations traducing the time evolution of phases and mixture variables are
obtained = Numerical scheme
When dealing with a two-phases bubbly flow, a continuous model has been obtained :
All interface variables are determined !
x
αP
x
Pρu
t
)u( 1I
1
2
1
12 uuλ
x
αu
t
α 1I
1 21 PPμ
x
αuP
x
uPρE
t
)E( 1II
11
21I PPμP 12I uuλu
21
I
ZZ
A
21
21I
ZZ
ZZA
21
2211I
ZZ
uZuZu
21
2
2
1
1
I
Z
1
Z
1
Z
P
Z
P
P
Interfacial area
between phases
µ and λ express the rates at which pressure and velocity equilibrium are reached respectively
Major contributions of DEM (2)
This non-equilibrium model is able to treat simultaneously :
- mixtures with several velocities, temperatures,…
- material interfaces (contact)
- permeable interfaces (interfaces separating a cloud of drops and a gas for example,…)
Evaporation effects
This model is completed by heat and mass transfer source terms (evaporation,
condensation) at infinite rates (relaxations)
They are particularly relevant when :
- the interfacial area and the flow topology are unknown,
- the Direct Numerical Simulation of interface problems is considered.
Relaxation processes (at infinite rate) allow the obtention of solutions in limit cases
corresponding to reduced models.
Two-phase compressible flow model + relaxation terms
m
IH~
m
Iu~m
Pressure
relaxation Velocity
relaxation Mass
transfer
Heat
transfer Hyperbolic
1
1 )u.(t
)(
12 uuλ
1II11 α.uPuPρE.
t
)E(
21I PPPμ 12I uu.uλ
12T TTH
21I PPPμ IH~
m 12T TTH
)gg(~m 21I
12 uuλ
12I uu.uλ
2
2 )u.(t
)(
1I22 αPIPuuρ.
t
)u(
1II22 α.uPuPρE.
t
)E(
1I11 αPIPuuρ.
t
)u(
m
21 PPμ
Iu~m
I~m
1I
1 α.ut
Relaxation coefficients
12 uuλ
12T TTH
)gg(~m 21I
21 PPμ
µ and λ express the rate at which pressure and velocity equilibria are
reached respectively :
HT expresses the rate at which temperature equilibrium is reached :
ν expresses the rate at which Gibbs free energy equilibrium is reached :
Heat exchange between phases
Mass transfer (evaporation/condensation process)
Mechanical interactions (drags, compressibility ratio) between phases
μ , HT , ν → ∞ Instantaneous exchanges : relaxation effects
Relaxation effects
at infinite rates
Hierarchy of compressible flow models : reduced models
Multi-velocity flow model
Non-equilibrium flows (u1,u2,p1,p2,T1,T2,g1,g2)
Single-velocity flow model
Interface problems (u,p,T1,T2,g1,g2)
Homogeneous Euler model
Thermodynamic equilibrium flows (u,p,T,g)
Thermal
relaxation
(temperature)
Mechanical
relaxation
(velocity and
pressure)
Chemical
relaxation
(Gibbs free
energy)
Multi-species Euler mixture model
Thermal equilibrium flows
(u,p,T, g1,g2)
,
TH
Velocity relaxation procedure : λ → ∞
0t
)( 1
t
)u( 1
12 uuλ
0t
α1
12I uu.uλ
0t
)( 2
cst)(
Y 11
cst)(
Y 22
2211
* uYuYu
)uu).(uu(2
1ee 1
*
1
*
1
*
1
EOS of both phases are not used explicitly
)uu).(uu(2
1ee 2
*
2
*
2
*
2
cst1
12 uuλ-
t
)u( 2
12I uu.uλ-
t
)E( 1
t
)E( 2
Momentum and energy
conservation
Pressure relaxation procedure : μ → ∞
0t
)( 1
0t
)u( 1
t
α1
cst)(
Y 22
21I PPPμ
cstu1
cstu2
1
*
1
*
1
*
1
11pee
Use of EOS
),p(e *
1
**
1
21I PPPμ
0t
)( 2
21 PPμ
t
)E( 1
0t
)u( 2
t
)E( 2
cst)(
Y 11
2
*
2
*
2
*
2
11pee
),p(e *
2
**
2
)p( **
1
)p( **
2
)p(
Y
)p(
Y1**
2
2
**
1
1
Function of the final
relaxed pressure only
When considering ideal gas or ‘Stiffened Gas’ EOS an
analytical relation is available for the pressure
Mass and energy
conservation
Single-velocity flow model : Interface problems
Each phase has its own temperature and Gibbs free energy
(Kapila & al., 2001; Saurel & al., 2008)
Mechanical equilibrium
Mixture variables :
Asymptotic analysis of the full non-equilibrium model
Wood sound speed :
10
100
1000
10000
0 0.2 0.4 0.6 0.8 1
Fraction volumique eau
Vitesse du son de wood dans un melange eau/air (m/s)
21
21
)E()E(E
)()(
ppp
uuu
21
21
0u)pE(.t
)E(
0Ipuu.t
)u(
0u)(.t
)(
0u)(.t
)(
u.cc
)cc(.u
t
22
11
2
2
22
1
2
11
2
11
2
221
1
2
22
2
2
11
1
2
wood ccc
1
,
Pressure and temperature relaxation procedure : μ , HT → ∞
0t
)( 1
0t
)u( 1
t
)E( 1
t
α1
0t
)( 2
0t
)u( 2
t
)E( 2
cst)(
Y 11
cst)(
Y 22
21I PPPμ
21 PPμ
21I PPPμ
cstu1
cstu2
cteYY1
*
2
2
*
1
1
Mass and energy
conservation
12T TTH
12T TTH
cteeYeYe *
22
*
11
Use of EOS
)T,p( ***
k
)T,p(e ***
k
)T,p(
Y
)T,p(
Y1***
2
2
***
1
1
When considering ideal gas or ‘Stiffened Gas’ EOS analytical
relations are available for pressure and temperature
)T,p(eY)T,p(eYe ***
22
***
11
Two-species Euler model
0u)pE(.t
E
0Ipuu.t
u
0u)(.t
)(
0u)(.t
)(
22
11
Mixture variables :
u.u2/1eE
0u)pE(.t
E
0Ipuu.t
u
0)uY.(t
Y
0)uY.(t
Y
22
11
Mechanical and thermal
equilibrium
Asymptotic analysis of the full non-equilibrium model
TTT
ppp
uuu
21
21
21
TH,,
Each phase has its own Gibbs free energy
2
2
1
1 YY1
2211eYeYe
Pressure, temperature and Gibbs free energy relaxation
procedure : μ , HT , ν→ ∞
I~m
m
IH~
m
Iu~m
t
)( 1
t
)u( 1
t
)E( 1
t
α1
t
)( 2
t
)u( 2
t
)E( 2
21I PPPμ
21 PPμ
21I PPPμ
Mass and energy
conservation
12T TTH
12T TTH
Use of EOS
)T,p( ***
k
)T,p(e ***
k
m
Iu~m
IH~
m
)gg(~m 21I
*
2
*
2
*
1
*
1 eYeYe + Gibbs free
energy equality
)p(TT *
sat
*
*
2
*
2
*
1
*
1 YY1
)p(h)p(h
pe)p(h
Y**
1
**
2
***
2
*
1
)p(
1
)p(
1
1
)p(
1
Y
**
1
**
2
**
2*
1
Latent heat of
vaporization
)p(L *
V
Homogeneous Euler model
0u)pE(.t
E
0Ipuu.t
u
0)u.(t
ggg
TTT
ppp
uuu
21
21
21
21
Mechanical and thermodynamic
equilibrium
Closure relations :
u.u2/1eE
Equilibrium sound speed :
2
2
2,p
2
2
1
1,p
1
2
wood
2
eq dp
ds
C
Y
dp
ds
C
YT
c
1
c
1
woodeq cc
Asymptotic analysis of the full non-equilibrium model
,H,, T
)p(TT sat
12
21 11
11
Y
2211eYeYe
For an arbitrary
number of phases…
Multiphase compressible flow model
Pressure
relaxation
Velocity
relaxation
Temperature
relaxation Hyperbolic
k
k )u(.t
)(
kIkk αPIPuuρ.
t
)u(
kIIkk α.uPuPρE.
t
)E(
kI
k α.ut
α
N,1l
klkl )uu(
N,1l
lkkl )PP(
N,1l
klklI )uu(.u
N,1l
lkklI )PP(P
N,1l
klkl,T )TT(H
Pressure, temperature relaxation and Gibbs free energy
relaxation (liquid/vapor) : μkl → ∞, HT,kl → ∞, ν → ∞
mt
)( 1
I
1 u~
mt
)u(
IN,1l
l1l1 ~m
)PP(
t
α1
N,1l
l1l1I )PP(P
t
)E( 1
I
N,1l
1ll1,T H~
m)TT(H
mt
)( 2
I
2 u~
mt
)u(
IN,1l
l2l2 ~m
)PP(
t
α2
N,1l
l2l2I )PP(P
t
)E( 2
I
N,1l
2ll2,T H~
m)TT(H
0t
)( k
0
t
)u( k
N,1l
lkkl )PP(
t
αk
N,1l
lkklI )PP(P
t
)E( k
N,1l
klkl,T )TT(H
Phase k (inert) : Phase 1 (liquid) :
Phase 2 (vapor) :
-pressure
-temperature
Relaxation : Relaxation :
-Gibbs free energy
-pressure
-temperature
)gg(~m 21I
-pressure
-temperature
Relaxation :
3k
Pressure, temperature and Gibbs free energy (liquid/vapor)
relaxation for an arbitrary number of phases
Mass and energy
conservation
Use of EOS
)T,p( ***
k
)T,p(e ***
k
N,3k
*
k
0
k
*
2
*
2
*
1
*
1 eYeYeYe
+ Gibbs free energy
equality
)p(TT *
sat
*
N,3k
*
k
0
k
*
2
*
2
*
1
*
1 YYY1
)p(h)p(h
)p(hYp
e)p(hYY
Y**
1
**
2
N,3k
**
k
0
k
***
2
0
2
0
1
*
1
)p(
1
)p(
1
)p(
Y1
)p(
YY
Y
**
1
**
2
N,3k**
k
0
k
**
2
0
2
0
1
*
1
1YYYN,3k
0
k
*
2
*
1
)N,3k(cstYY 0
k
*
k
Modification of the final
thermodynamic state between the
liquid and its vapor
One-dimensional
examples
Shock tube (water, steam, air)
BarP 1BarP 10
K467)P(TT sat
4
Liq 10.5
Cloud of droplets in a gaz (air/steam) mixture
K373)P(TT sat
2.0Air
Mechanical equilibrium :
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Mass fraction
Liq
Vap
Air
1
2
3
4
5
6
7
8
9
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Pressure(Bar)
Liq
Vap
Air
0
50
100
150
200
250
300
350
400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Velocity(m/s)
Liq
Vap
Air
300
350
400
450
500
550
600
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Temperature(K)
Liq
Vap
Air
,
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Mass fraction
Liq
Vap
Air
1
2
3
4
5
6
7
8
9
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Pressure(Bar)
Liq
Vap
Air
0
50
100
150
200
250
300
350
400
450
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Velocity(m/s)
Liq
Vap
Air
No interaction between phases
300
350
400
450
500
550
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Temperature(K)
Liq
Vap
Air
Mechanical equilibrium
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Mass fraction
Liq
Vap
Air
1
2
3
4
5
6
7
8
9
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Pressure(Bar)
Liq
Vap
Air
0
50
100
150
200
250
300
350
400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Velocity(m/s)
Liq
Vap
Air
300
350
400
450
500
550
600
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Temperature(K)
Liq
Vap
Air
,Mechanical/thermal equilibrium
TH,,
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Mass fraction
Liq
Vap
Air
1
2
3
4
5
6
7
8
9
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Pressure(Bar)
Liq
Vap
Air
0
50
100
150
200
250
300
350
400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Velocity(m/s)
Liq
Vap
Air
360
380
400
420
440
460
480
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Temperature(K)
Liq
Vap
Air
Mechanical/thermodynamical
equilibrium ,H,, T
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Mass fraction
Liq
Vap
Air
1
2
3
4
5
6
7
8
9
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Pressure(Bar)
Liq
Vap
Air
0
50
100
150
200
250
300
350
400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Velocity(m/s)
Liq
Vap
Air
380
390
400
410
420
430
440
450
460
470
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Temperature(K)
Liq
Vap
Air
Another possibility : heat exchanges between inert phases and
liquid/vapor are absent
mt
)( 1
I
1 u~
mt
)u(
IN,1l
l1l1 ~m
)PP(
t
α1
N,1l
l1l1I )PP(P
t
)E( 1
I
N,1l
1ll1,T H~
m)TT(H
mt
)( 2
I
2 u~
mt
)u(
IN,1l
l2l2 ~m
)PP(
t
α2
N,1l
l2l2I )PP(P
t
)E( 2
I
N,1l
2ll2,T H~
m)TT(H
0t
)( k
0
t
)u( k
N,1l
lkkl )PP(
t
αk
N,1l
lkklI )PP(P
t
)E( k
N,1l
klkl,T )TT(H
Phase k (inert) : Phase 1 (liquid) :
Phase 2 (vapor) :
-pressure
-temperature
Relaxation : Relaxation :
-Gibbs free energy
-pressure
-temperature
)gg(~m 21I
-pressure
-temperature
Relaxation :
3k
Mechanical/thermodynamical
equilibrium ,H,, T
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Mass fraction
Liq
Vap
Air
1
2
3
4
5
6
7
8
9
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Pressure(Bar)
Liq
Vap
Air
0
50
100
150
200
250
300
350
400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Velocity(m/s)
Liq
Vap
Air
380
390
400
410
420
430
440
450
460
470
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Temperature(K)
Liq
Vap
Air
Thermodynamical equilibrium
between liquid and vapor only
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Mass fraction
Liq
Vap
Air
1
2
3
4
5
6
7
8
9
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Pressure(Bar)
Liq
Vap
Air
0
50
100
150
200
250
300
350
400
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Velocity(m/s)
Liq
Vap
Air
300
350
400
450
500
550
600
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X(m)
Temperature(K)
Liq
Vap
Air
Multi-dimensional
examples
Cryogenic liquid and inert gas injections
Inert gas
injection
Tank Bar1P
Tank filled
with cryogenic
liquid
~50000
tetrahedrons
Inert gas
Bar1P
bar20p
Pressure and temperature equilibrium flow
Pressure, temperature and Gibbs free energy equilibrium flow
Vapor volume fraction evolution
Filling of cavity under gravity effect
Walls
Connection to
atmosphere
Hot
steam
Bar1P
Air Bar1P
Cold
water Gravity
Liquid volume fraction evolution : pressure relaxation
Pressure and temperature relaxation
Pressure, temperature and Gibbs free energy relaxation
Perspectives
Exact steady solutions through converging-diverging nozzles should be
investigated under mechanical, thermal and Gibbs free energy equilibria for
an arbitrary number of phases
Reference solutions
The knowledge of the interfacial area at each point of the flow must be strongly
improved and particularly the change of the multiphase flow topology.
Additional equations are thus required (number of particles per unit volume,
geometrical equations,…)
Big challenge !!!
Thanks for your attention…
