A numerical scheme for condensation and flash vaporization V. - PowerPoint PPT Presentation

A numerical scheme for condensation and flash vaporization V. Perrier, R. Abgrall, L. Hallo perrier@math.u-bordeaux1.fr Math´ ematiques Appliqu´ ees de Bordeaux CEntre des Lasers Intenses et Applications Universit´ e de Bordeaux 1 351 Cours de la Lib´ eration, 33 405 Talence Cedex V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 1

Overview 1. Thermodynamic of phase transition 2. The Riemann Problem with equilibrium E.O.S 3. The Riemann Problem out of equilibrium 4. Numerical scheme 5. Numerical results V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 2

Thermodynamic model Two equations of state: ε 1 ( P, T ) and ε 2 ( P, T ) V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 3

Thermodynamic model Two equations of state: ε 1 ( P, T ) and ε 2 ( P, T ) Mixture zone Suppose that fluids are locally non miscible V 1 + V 2 = V tot Optimization of mixture entropy ⇒ When the mixture is stable = µ 1 = µ 2 P 1 = P 2 T 1 = T 2 V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 3

Thermodynamic model Two equations of state: ε 1 ( P, T ) and ε 2 ( P, T ) Mixture zone P Liquid Mixture Gas τ 3 convex E.O.S. V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 3

Overview 1. Thermodynamic of phase transition 2. The Riemann Problem with equilibrium E.O.S 3. The Riemann Problem out of equilibrium 4. Numerical scheme 5. Numerical results V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 4

Equilibrium EOS (1/4) Look for simple waves for the Euler system V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 5

Equilibrium EOS (1/4) Look for simple waves for the Euler system  ∂ t ρ + ∂ x ( ρu ) = 0   ∂ t ( ρu ) + ∂ x ( ρu 2 + P ) = 0  ∂ t ( ρE ) + ∂ x (( ρE + P ) u ) = 0  with E = ε + 1 2 u 2 ε, P, ρ are linked with an E.O.S. V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 5

Equilibrium EOS (1/4) Look for simple waves for the Euler system Look for self similar solutions + Entropy criterion If P decreases, isentropic regular wave S = cste V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 5

Equilibrium EOS (1/4) Look for simple waves for the Euler system Look for self similar solutions + Entropy criterion If P decreases, isentropic regular wave If P increases, shock: Rankine–Hugoniot relations M = u 2 − u 1   τ 2 − τ 1    M 2 = − p 2 − p 1  τ 2 − τ 1  ε 2 − ε 1 + 1   2( p 2 + p 1 )( τ 2 − τ 1 ) = 0   V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 5

Equilibrium EOS (1/4) Look for simple waves for the Euler system Look for self similar solutions + Entropy criterion If P decreases, isentropic regular wave If P increases, shock: Rankine–Hugoniot relations if the E.O.S if globally convex, existence and uniqueness of a solution for the Riemann Problem V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 5

Equilibrium EOS (2/4) Consequences of the phase transition for Hugoniot Curves V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 6

Equilibrium EOS (2/4) Consequences of the phase transition for Hugoniot Curves P t σ 1 σ 2 x τ V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 6

Equilibrium EOS (2/4) Consequences of the phase transition for Hugoniot Curves P Liquid Mixture Gas B A τ V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 6

Equilibrium EOS (2/4) Consequences of the phase transition for Hugoniot Curves P Liquid Mixture Gas B A τ V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 6

Equilibrium EOS (2/4) Consequences of the phase transition for Hugoniot Curves Lost the uniqueness of the entropic solution V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 6

Equilibrium EOS (2/4) Consequences of the phase transition for Hugoniot Curves Lost the uniqueness of the entropic solution Liu (1975) The “ physical” solution is the one with a wave splitting in B V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 6

Equilibrium EOS (3/4) Consequences for isentropic waves V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 7

Equilibrium EOS (3/4) Consequences for isentropic waves P A B τ V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 7

Equilibrium EOS (3/4) Consequences for isentropic waves Characteristic curves in point A liquid mixture A V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 7

Equilibrium EOS (3/4) Consequences for isentropic waves Characteristic curves in point A ⇒ OK = Characteristic curves in point B B mixture gas V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 7

Equilibrium EOS (3/4) Consequences for isentropic waves Characteristic curves in point A ⇒ OK = Characteristic curves in point B ⇒ non regular wave ??? = V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 7

Equilibrium EOS (4/4) Non uniqueness for compressive waves ⇒ difficulties to compute the right solution with = approximate solvers (Jaouen Phd Thesis) No solution for undercompressive waves ⇒ Trash = V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 8

Overview 1. Thermodynamic of phase transition 2. The Riemann Problem with equilibrium E.O.S 3. The Riemann Problem out of equilibrium 4. Numerical scheme 5. Numerical results V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 9

Out of equilibrium Riemann problem (1/3) metastable states P Liquid metastable state Mixture τ ⇒ need for a multiphase code = V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 10

Out of equilibrium Riemann problem (1/3) metastable states A phase transition wave is a self–similar discontinuity ⇒ Rankine–Hugoniot relations hold = M = u 2 − u 1   τ 2 − τ 1    M 2 = − p 2 − p 1  τ 2 − τ 1  ε 2 − ε 1 + 1   2( p 2 + p 1 )( τ 2 − τ 1 ) = 0   V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 10

Out of equilibrium Riemann problem (1/3) metastable states A phase transition wave is a self–similar discontinuity ⇒ Rankine–Hugoniot relations hold = M = u 2 − u 1   τ 2 − τ 1    M 2 = − p 2 − p 1  τ 2 − τ 1  ε 2 − ε 1 + 1   2( p 2 + p 1 )( τ 2 − τ 1 ) = 0   beware! ε 1 == E.O.S of the liquid ε 2 == E.O.S of the mixture or the gas V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 10

Out of equilibrium Riemann problem (2/3) upstream state / ∈ the set of the downstream states P detonations P 0 deflagrations τ τ 0 V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 11

Out of equilibrium Riemann problem (2/3) upstream state / ∈ the set of the downstream states τ increases = ⇒ deflagration P P 0 weak deflagrations CJ strong deflagrations τ τ 0 V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 11

Out of equilibrium Riemann problem (2/3) upstream state / ∈ the set of the downstream states τ increases = ⇒ deflagration No strong deflagrations (Lax characteristic condition) ⇒ subsonic wave = V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 11

Out of equilibrium Riemann problem (2/3) upstream state / ∈ the set of the downstream states τ increases = ⇒ deflagration No strong deflagrations (Lax characteristic condition) ⇒ subsonic wave = contact surface vaporization wave (subsonic) sonic wave (rarefaction/shock) V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 11

Out of equilibrium Riemann problem (2/3) upstream state / ∈ the set of the downstream states τ increases = ⇒ deflagration No strong deflagrations (Lax characteristic condition) ⇒ subsonic wave = entropy growth is ensured for all the downstream states of weak deflagration V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 11

Out of equilibrium Riemann problem (3/3) one indeterminate contact surface vaporization state ⋆ state 0 ⋆ sonic wave state 0 V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 12

Out of equilibrium Riemann problem (3/3) one indeterminate A “physical” closure (Lemétayer et al, JCP 2005) P P ⋆ 0 M increases τ τ ⋆ 0 V. Perrier / Workshop on Numerical methods for multi-material fluid flows 2005 – p. 12