A relaxation scheme for the numerical modelling of phase transition.Philippe Helluy,Universit de Toulon,Projet SMASH, INRIA Sophia Antipolis.International Workshop on Multiphase and Complex Flow simulation for Industry, Cargese, October 20-24, 2003.

CavitationIntroduction

DemonstrationIntroduction

PlanModelling of cavitationNon-uniqueness of the Riemann problem

Relaxation and projection finite volume schemeNumerical results

Entropy and state lawr : densitye : internal energyBut it is an incomplete law for thermal modelling (Menikoff, Plohr, 1989)T : temperatureModelling

MixturesEntropy is an additive quantity :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.Modelling

Equilibrium lawMass and energy must be conserved. The equilibrium is thus determined byIf the maximum is attained for 0

Mixture law out of equilibriumMixture pressureMixture temperatureIf T1=T2, the mixture pressure law becomes(Chanteperdrix, Villedieu, Vila, 2000)Modelling

Simple model (perfect gas laws)The entropy readsTemperature equilibriumPressure equilibrium:The fractions a and z can be eliminatedRiemann

Saturation curveOut of equilibrium, we have a perfect gas lawOn the other side,The saturation curve is thus a line in the (T,p) plane.Riemann

Optimization with constraintsPhase 2 is the most stablePhase 1 is the most stablePhases 1 and 2 are at equilibriumRiemann

Equilibrium pressure lawLetWe suppose(fluid (2) is heavier than fluid (1))Riemann

Shock curvesShock:Shock lagrangian velocitywL is linked to wR by a 3-shock if there is a j>0 such that:(Hugoniot curve)Riemann

Two entropy solutionsOn the Hugoniot curve:Menikof & Plohr, 1989 ; Jaouen 2001; Riemann

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

Relaxation-projection schemeWhen l=0, the previous system can be written in the classical formFinite volumes scheme (relaxation of the pressure law)Projection on the equilibrium pressure lawScheme

Numerical resultsScheme

Numerical resultsScheme

Numerical resultsScheme

Mixture of stiffened gasesCaloric and pressure lawsSettingThe mixture still satisfies a stiffened gas lawSchemeBarberon, 2002

Convergence and CFL TestsScheme

Convergence Tests 200, 800, 1600, 3200 cells convergence of the schemePressureMass FractionMixture densityScheme

CFL Tests Jaouen (2001) CFL = 0.5, 0.7, 0.95 No difference observedMass Fraction PressureScheme

Liquid area heated at the center by a laser pulse (Andreae, Ballmann, Mller, 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 pulseIV.2 Bubble collapse near a rigid wallAmbient liquid (1atm)Heated liquid (1500 atm) 2.0 mm, 70 cells 2.4 mm, 70 cells 1.4 mm 0.15 mm0.45 mmWallResults

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

Cavitation flow in 2DFast projectile (1000m/s) in water ; final time 225 sp>0Results

Conclusion Simple method based on physics Entropic scheme by construction Possible extensions : reacting flows, n phases, finite reaction rate, Perspectives More realistic laws Critical pointConclusion