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.

Cavitation

boiling

Introduction

Demonstration

Introduction

Plan

• Modelling of cavitation

• Non-uniqueness of the Riemann problem

• Relaxation and projection finite volume scheme

• Numerical results

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

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

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

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

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

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

Optimization with constraints

Phase 2 is the most stable Phase 1 is the most stable

Phases 1 and 2 are at equilibrium

Riemann

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

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

Two entropy solutions

On the Hugoniot curve: 2 21.

2Tds d j

Menikof & Plohr, 1989 ; Jaouen 2001; …

Riemann

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

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

Numerical resultsScheme

Numerical resultsScheme

Numerical resultsScheme

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

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

Convergence Tests

• 200, 800, 1600, 3200 cells

• convergence of the scheme

Pressure Mass Fraction

Mixture density

Scheme

CFL Tests

• Jaouen (2001)

• CFL = 0.5, 0.7, 0.95

• No difference observed

Mass Fraction Pressure

Scheme

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

Mixture Pressure (from 0 to 1ns)

IV.1 Bubble appearance : PressureResults

Volume Fraction of Vapor (from 0 to 60ns)

IV.1 Bubble appearance : Volume FractionResults

• 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

Mixture pressure (from 0 to 2ns)

IV.2 Bubble close to a rigid wallResults

Volume Fraction of Vapor (from 0 to 66ns)

IV.2 Bubble close to a rigid wallResults

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

Cavitation flow in 2DFast projectile (1000m/s) in water ; final time 225 s

p>0

Results

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