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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.

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a relaxation scheme for the numerical modelling of phase transition

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.

cavitation

boiling

Introduction

Cavitation
slide4
Plan
  • Modelling of cavitation
  • Non-uniqueness of the Riemann problem
  • Relaxation and projection finite volume scheme
  • Numerical results
entropy and state law

The Euler compressible model needs a pressure law of the form

The complete state law : s is the specific entropy (concave)

Pressure law

Caloric law

Modelling

Entropy and state law

r : density

e : internal energy

But it is an incomplete law for thermal modelling (Menikoff, Plohr, 1989)

T : temperature

mixtures

Modelling

Mixtures

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.

Entropy is an additive quantity :

equilibrium law

Modelling

Equilibrium law

Mass and energy must be conserved. The equilibrium is thus determined by

If the maximum is attained for 0<Y<1, we obtain

(chemical potential)

Generally, the maximum is attained for Y=0 or Y=1. If 0<Yeq<1, we are on the saturation curve.

mixture law out of equilibrium

Modelling

Mixture law out of equilibrium

Mixture pressure

Mixture temperature

If T1=T2, the mixture pressure law becomes

(Chanteperdrix, Villedieu, Vila, 2000)

simpl e model perfect gas laws

Riemann

Simple model (perfect gas laws)

The entropy reads

Temperature equilibrium

Pressure equilibrium:

The fractions a and z can be eliminated

saturation curve

Riemann

Saturation curve

Out of equilibrium, we have a perfect gas law

On the other side,

The saturation curve is thus a line in the (T,p) plane.

optimization with constraints

Riemann

Optimization with constraints

Phase 2 is the most stable

Phase 1 is the most stable

Phases 1 and 2 are at equilibrium

equilibrium p ressure law

Riemann

Equilibrium pressure law

Let

We suppose

(fluid (2) is heavier than fluid (1))

shock curves

Riemann

Shock curves

Shock:

Shock lagrangian velocity

wL is linked to wR by a 3-shock if there is a j>0 such that:

(Hugoniot curve)

two entropy solutions

Riemann

Two entropy solutions

On the Hugoniot curve:

Menikof & Plohr, 1989 ; Jaouen 2001; …

a relaxation model for the cavitation

Scheme

A relaxation model for the cavitation

The last equation is compatible with the second principle because, by the concavity of s

(Coquel, Perthame 1998)

relaxation projection scheme

Scheme

Relaxation-projection scheme

When l=0, the previous system can be written in the classical form

Finite volumes scheme (relaxation of the pressure law)

Projection on the equilibrium pressure law

mixture of stiffened gases

Scheme

Mixture of stiffened gases

Barberon, 2002

Caloric and pressure laws

The mixture still satisfies a stiffened gas law

Setting

slide21

Scheme

Liquid

High pressure

(5.109Pa)

Ambient pressure (105 Pa)

Ambient pressure (105 Pa)

0 mm

0,08 mm

wall

0,06 mm

0,015 mm

0,005 mm

200, 800, 1600, 3200 cells

Convergence and CFL Tests

slide22

Scheme

Convergence Tests

  • 200, 800, 1600, 3200 cells
  • convergence of the scheme

Mixture density

Pressure

Mass Fraction

slide23

Scheme

CFL Tests

  • Jaouen (2001)
  • CFL = 0.5, 0.7, 0.95
  • No difference observed

Mass Fraction

Pressure

slide24

Results

45 cells

35 cells

10 cells

0.2 mm

12 mm

IV.1 Bubble appearance

  • 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.

Heated liquid (1500 atm)

Ambient liquid (1atm)

slide25

Results

IV.1 Bubble appearance : Pressure

Mixture Pressure (from 0 to 1ns)

slide26

Results

IV.1 Bubble appearance : Volume Fraction

Volume Fraction of Vapor (from 0 to 60ns)

slide27

Results

IV.2 Bubble collapse near a rigid wall

  • Same example as previous test, with a rigid wall
  • Liquid area heated at the center by a laser pulse

Wall

2.0 mm, 70 cells

Heated liquid (1500 atm)

1.4 mm

0.15 mm

0.45

mm

Ambient liquid (1atm)

2.4 mm, 70 cells

slide28

Results

IV.2 Bubble close to a rigid wall

Mixture pressure (from 0 to 2ns)

slide29

Results

IV.2 Bubble close to a rigid wall

Volume Fraction of Vapor (from 0 to 66ns)

slide30

Results

Cavitation flowin 2D

Fast projectile (1000m/s) in water (Saurel,Cocchi, Butler, 1999)

final time :

225 s

15 cm, 90 cells

4 cm, 24 cells

p<0

45°

2 cm

Projectile

3 cm

Pressure (pa)

slide31

Results

Cavitation flow in 2D

Fast projectile (1000m/s) in water ; final time 225 s

p>0

conclusion

Conclusion

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