a relaxation scheme for the numerical modelling of phase transition. philippe helluy, université de...
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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.
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CavitationIntroduction
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DemonstrationIntroduction
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PlanModelling of cavitationNon-uniqueness of the Riemann problem
Relaxation and projection finite volume schemeNumerical results
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Entropy and state lawr : densitye : internal energyBut it is an incomplete law for thermal modelling (Menikoff, Plohr, 1989)T : temperatureModelling
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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
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Mixture law out of equilibriumMixture pressureMixture temperatureIf T1=T2, the mixture pressure law becomes(Chanteperdrix, Villedieu, Vila, 2000)Modelling
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Simple model (perfect gas laws)The entropy readsTemperature equilibriumPressure equilibrium:The fractions a and z can be eliminatedRiemann
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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
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Optimization with constraintsPhase 2 is the most stablePhase 1 is the most stablePhases 1 and 2 are at equilibriumRiemann
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Equilibrium pressure lawLetWe suppose(fluid (2) is heavier than fluid (1))Riemann
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Shock curvesShock:Shock lagrangian velocitywL is linked to wR by a 3-shock if there is a j>0 such that:(Hugoniot curve)Riemann
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Two entropy solutionsOn the Hugoniot curve:Menikof & Plohr, 1989 ; Jaouen 2001; Riemann
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A relaxation model for the cavitationThe last equation is compatible with the second principle because, by the concavity of s(Coquel, Perthame 1998)Scheme
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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
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Numerical resultsScheme
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Numerical resultsScheme
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Numerical resultsScheme
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Mixture of stiffened gasesCaloric and pressure lawsSettingThe mixture still satisfies a stiffened gas lawSchemeBarberon, 2002
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Convergence and CFL TestsScheme
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Convergence Tests 200, 800, 1600, 3200 cells convergence of the schemePressureMass FractionMixture densityScheme
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CFL Tests Jaouen (2001) CFL = 0.5, 0.7, 0.95 No difference observedMass Fraction PressureScheme
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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
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Mixture Pressure (from 0 to 1ns)IV.1 Bubble appearance : PressureResults
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Volume Fraction of Vapor (from 0 to 60ns)IV.1 Bubble appearance : Volume FractionResults
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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
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Mixture pressure (from 0 to 2ns)IV.2 Bubble close to a rigid wallResults
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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
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Cavitation flow in 2DFast projectile (1000m/s) in water ; final time 225 sp>0Results
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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