a relaxation scheme for the numerical modelling of phase transition
DESCRIPTION
International Workshop on Multiphase and Complex Flow simulation for Industry, Cargese, October 20-24, 2003. A relaxation scheme for the numerical modelling of phase transition. Philippe Helluy , Université de Toulon , Projet SMASH, INRIA Sophia Antipolis. boiling. Introduction. Cavitation. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/1.jpg)
A relaxation scheme for the numerical modelling of phase
transition.
Philippe Helluy,
Université de Toulon,
Projet SMASH, INRIA Sophia Antipolis.
International Workshop on Multiphase and Complex Flow simulation for Industry, Cargese, October 20-24, 2003.
![Page 2: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/2.jpg)
Cavitation
boiling
Introduction
![Page 3: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/3.jpg)
Demonstration
Introduction
![Page 4: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/4.jpg)
Plan
• Modelling of cavitation
• Non-uniqueness of the Riemann problem
• Relaxation and projection finite volume scheme
• Numerical results
![Page 5: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/5.jpg)
Entropy and state law
: density : internal energy
But it is an incomplete law for thermal modelling (Menikoff, Plohr, 1989)
T : temperature
The Euler compressible model needs a pressure law of the form ( , )p p
The complete state law : s is the specific entropy (concave)1/ ( , )s s
Caloric law1s
T
Tds d pd
sp T
Pressure law
Modelling
![Page 6: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/6.jpg)
Mixtures
ii i 1 2
energy fractionsiiz
ii iy
Entropy is an additive quantity : 1 2(1 )s ys y s
1 2
1 1( , , ) ( , ) (1 ) ( , )
1 1
z zs Y ys y s
y y y y
( , , )Y y z
1 2V V V
1 1 1y y z z
volume fractionsii
V
V
We consider 2 phases (with entropy functions s1 and s2) of a same simple body (liquid water and its vapor) mixed at a macroscopic scale.
mass fractionsiiy
1 2
1 2 1 2 1 21 1 1y y z z
Modelling
![Page 7: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/7.jpg)
Equilibrium lawMass and energy must be conserved. The equilibrium is thus determined by
0 1( , , ) max ( , , )eq
Ys Y s Y
If the maximum is attained for 0<Y<1, we obtain
1 2
1 2
1 1 2 2
( ) 0
( ) 0
( / / ) 0
sp p
sT T
zs
T Ty
p Ts
Generally, the maximum is attained for Y=0 or Y=1. If 0<Yeq<1, we are on the saturation curve.
(chemical potential)
Modelling
![Page 8: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/8.jpg)
Mixture law out of equilibrium
1 2
1 2
( (1 ) )p p
p TT T
Mixture pressure
1 2
1 1z z
T T T
Mixture temperature
If T1=T2, the mixture pressure law becomes
1 2(1 )p p p
(Chanteperdrix, Villedieu, Vila, 2000)
Modelling
![Page 9: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/9.jpg)
Simple model (perfect gas laws)The entropy reads
1 2(1 ) ,
ln , 1.ii ii
s ys y s
s
Temperature equilibrium
1 2 1 2( (1 ) ).T y y
Pressure equilibrium:1 21 1 2
1 2
,
1, .
1
p T T
y y
The fractions and z can be eliminated
1 1
2 2
1 2
ln ln ln
(1 ) ln ln ,
(1 ) .
s y
y
y y
Riemann
![Page 10: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/10.jpg)
Saturation curveOut of equilibrium, we have a perfect gas law
,
( 1) .
s pp
Tp
On the other side,
1 2
1 1 2 2
( ) ln
ln 1 ln 1 .
s
y
The saturation curve is thus a line in the (T,p) plane.
Riemann
![Page 11: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/11.jpg)
Optimization with constraints
Phase 2 is the most stable Phase 1 is the most stable
Phases 1 and 2 are at equilibrium
Riemann
![Page 12: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/12.jpg)
Equilibrium pressure law
Let
1 1 2
2
1
1
2
1 1 2 2
exp( 1) ,
/ , / .
A
A A
We suppose 1 2.
(fluid (2) is heavier than fluid (1))
2 2
2 1
1 1
if ,
( , ) if ,
if .
p A
Riemann
![Page 13: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/13.jpg)
Shock curves
Shock:
( )j u Shock lagrangian velocity
wL is linked to wR by a 3-shock if there is a j>0 such that:
(Hugoniot curve)
if / ,( , )
if / .L
R
w x tw t x
w x t
2 ,
,
1( ) 0.
2 L R
pj
pj
u
p p
Riemann
![Page 14: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/14.jpg)
Two entropy solutions
On the Hugoniot curve: 2 21.
2Tds d j
Menikof & Plohr, 1989 ; Jaouen 2001; …
Riemann
![Page 15: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/15.jpg)
A relaxation model for the cavitation
2
2 2
( ) 0,
( ) ( ) 0,
( / 2) ( / 2 ) 0,
( ), .
t x
t x
t x
t x eq
u
u u p
u u p u
Y uY Y Y
The last equation is compatible with the second principle because, by the concavity of s
( )
( )
( ( ) ( ))
0.
t x Y t x
Y eq
eq
s us s Y uY
s Y Y
s Y s Y
(Coquel, Perthame 1998)
Scheme
![Page 16: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/16.jpg)
Relaxation-projection schemeWhen =0, the previous system can be written in the classical form
2
2 2
( ) 0,
( , , ( / 2), ) ,
( ) ( , , ( ( / 2) ) , )
t x
T T
T T
w f w
w u u Y
f w u u p u p u uY
Finite volumes scheme (relaxation of the pressure law)
1/ 21/ 2 1/ 2
1/ 2 1
( , ),
0,
( , ) Godunov flux (computable)
ni
n n n ni i i i
n n ni i i
w w n t i x
w w F F
t x
F F w w
Projection on the equilibrium pressure law1 1/ 2 1 1/ 2 1 1/ 2, ,n n n n n n
i i i i i iu u 1 1 1 1 1
0 1( , , ) max ( , , )n n n n n
i i i i iY
s Y s Y
Scheme
![Page 17: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/17.jpg)
Numerical resultsScheme
![Page 18: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/18.jpg)
Numerical resultsScheme
![Page 19: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/19.jpg)
Numerical resultsScheme
![Page 20: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/20.jpg)
Mixture of stiffened gases
1 0ln(( )i
ii i i i i is C Q s
Caloric and pressure laws( 1) ( )
i ii i i i
iii i i i i
C T Q
p Q
( 1) ii i i i ip C T
Setting
1 2
1 2
1 2
1 1 2 2
1 2
(1 )
(1 )
(1 )
(1 )
(1 )
C yC y C
Q yQ y Q
y C y C
yC y C
The mixture still satisfies a stiffened gas law
( 1)p CT
CT Q
Scheme
Barberon, 2002
![Page 21: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/21.jpg)
Convergence and CFL Tests
0,08 mm
wall
0 mm
0,06 mm 0,015 mm
Ambient pressure (105 Pa)
High pressure(5.109 Pa)
0,005 mm
Ambient pressure (105 Pa)
200, 800, 1600, 3200 cells
Liquid
Scheme
![Page 22: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/22.jpg)
Convergence Tests
• 200, 800, 1600, 3200 cells
• convergence of the scheme
Pressure Mass Fraction
Mixture density
Scheme
![Page 23: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/23.jpg)
CFL Tests
• Jaouen (2001)
• CFL = 0.5, 0.7, 0.95
• No difference observed
Mass Fraction Pressure
Scheme
![Page 24: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/24.jpg)
45 cells
12 mm
0.2 mm
10 cells35 cells
• Liquid area heated at the center by a laser pulse (Andreae, Ballmann, Müller, Voss, 2002).
• The laser pulse (10 MJ) increases the internal energy.
• Because of the growth of the internal energy, the phase transition from liquid into a vapor – liquid mixture occurs.
• Phase transition induces growth of pressure
• After a few nanoseconds,
the bubble collapses.
IV.1 Bubble appearance
Ambient liquid (1atm)
Heated liquid (1500 atm)
Results
![Page 25: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/25.jpg)
Mixture Pressure (from 0 to 1ns)
IV.1 Bubble appearance : PressureResults
![Page 26: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/26.jpg)
Volume Fraction of Vapor (from 0 to 60ns)
IV.1 Bubble appearance : Volume FractionResults
![Page 27: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/27.jpg)
• Same example as previous test, with a rigid wall• Liquid area heated at the center by a laser pulse
IV.2 Bubble collapse near a rigid wall
Ambient liquid (1atm)
Heated liquid (1500 atm)
2.0 mm, 70 cells
2.4 mm, 70 cells
1.4 mm
0.15 mm 0.45 mm
Wall
Results
![Page 28: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/28.jpg)
Mixture pressure (from 0 to 2ns)
IV.2 Bubble close to a rigid wallResults
![Page 29: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/29.jpg)
Volume Fraction of Vapor (from 0 to 66ns)
IV.2 Bubble close to a rigid wallResults
![Page 30: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/30.jpg)
Cavitation flow in 2DFast projectile (1000m/s) in water (Saurel,Cocchi, Butler, 1999)
p<0
3 cm
2 cm45°
15 cm, 90 cells
4 cm, 24 cells
Projectile
Pressure (pa)
final time :225 s
Results
![Page 31: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/31.jpg)
Cavitation flow in 2DFast projectile (1000m/s) in water ; final time 225 s
p>0
Results
![Page 32: A relaxation scheme for the numerical modelling of phase transition](https://reader035.vdocuments.net/reader035/viewer/2022062718/56812b8b550346895d8fa69a/html5/thumbnails/32.jpg)
Conclusion
• Simple method based on physics• Entropic scheme by construction• Possible extensions : reacting flows, n phases, finite reaction rate, …
Perspectives
• More realistic laws• Critical point
Conclusion