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

Mixtures Entropy is an additive quantity : We consider 2 phases (with entropy functions s 1 and s 2 ) of a same simple body (liquid water and its vapor) mixed at a macroscopic scale. 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 Generally, the maximum is attained for Y=0 or Y=1. If 0<Y eq <1, we are on the saturation curve. (chemical potential) Modelling

Mixture law out of equilibrium Mixture pressure Mixture temperature If T 1 =T 2, the mixture pressure law becomes (Chanteperdrix, Villedieu, Vila, 2000) Modelling

Simple model (perfect gas laws) The entropy reads Temperature equilibrium Pressure equilibrium: The fractions  and z can be eliminated 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. Riemann

Optimization with constraints Phase 2 is the most stablePhase 1 is the most stable Phases 1 and 2 are at equilibrium Riemann

Equilibrium pressure law Let We suppose (fluid (2) is heavier than fluid (1)) Riemann

Shock curves Shock: Shock lagrangian velocity w L is linked to w R by a 3- shock if there is a j>0 such that: (Hugoniot curve) Riemann

Two entropy solutions On the Hugoniot curve: Menikof & Plohr, 1989 ; Jaouen 2001; … Riemann

A relaxation model for the cavitation The last equation is compatible with the second principle because, by the concavity of s (Coquel, Perthame 1998) Scheme

Relaxation-projection scheme When =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 Scheme

Numerical results Scheme

Numerical results Scheme

Numerical results Scheme

Mixture of stiffened gases Caloric and pressure laws Setting The mixture still satisfies a stiffened gas law Scheme Barberon, 2002

Convergence and CFL Tests 0,08 mm wall 0 mm 0,06 mm0,015 mm Ambient pressure (10 5 Pa) High pressure ( Pa) 0,005 mm Ambient pressure (10 5 Pa) 200, 800, 1600, 3200 cells Liquid Scheme

Convergence Tests 200, 800, 1600, 3200 cells convergence of the scheme PressureMass Fraction Mixture density Scheme

CFL Tests Jaouen (2001) CFL = 0.5, 0.7, 0.95 No difference observed Mass FractionPressure 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 : Pressure Results

Volume Fraction of Vapor (from 0 to 60ns) IV.1 Bubble appearance : Volume Fraction Results

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 mm0.45 mm Wall Results

Mixture pressure (from 0 to 2ns) IV.2 Bubble close to a rigid wall Results

Volume Fraction of Vapor (from 0 to 66ns) IV.2 Bubble close to a rigid wall Results

Cavitation flow in 2D Fast projectile (1000m/s) in water (Saurel,Cocchi, Butler, 1999) p<0 3 cm 2 cm 45° 15 cm, 90 cells 4 cm, 24 cells Projectile Pressure (pa) final time : 225  s Results

Cavitation flow in 2D Fast 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