Compressible two-phase flow models: asymptotics, coupling and adaptation Nicolas Seguin Laboratoire J.-L. Lions, UPMC Paris 6, France 22 juin, AMIS 2012 Nicolas Seguin (LJLL, UPMC) 1 / 32
Context Collaboration with EDF R&D Mathematical modeling and numerical simulation of compressible two-phase flows 2009 – 2012 : PhD of Khaled Saleh, with J.-M. H´ erard and F. Coquel (X) Collaboration with the CEA Saclay theoretical and numerical coupling for nuclear thermohydraulics 2004 – 2009 : Research group with A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagouti` ere, P.-A. Raviart... 2009 – 2012 : LRC Manon (joint research department LJLL–CEA Saclay) Nicolas Seguin (LJLL, UPMC) 2 / 32
Outline of the talk Thermohydraulics in PWR 1 Overview and LOCA Modeling assumptions Models for two-phase flows 2 The “most” accurate model Hierarchy by asymptotic limits Toward a simulation of the whole domain 3 Interfacial coupling of hyperbolic systems Optimization of the localization of the coupling interfaces Conclusion et perspectives 4 Nicolas Seguin (LJLL, UPMC) 3 / 32
Thermohydraulics in PWR Outline of the talk Thermohydraulics in PWR 1 Overview and LOCA Modeling assumptions Models for two-phase flows 2 The “most” accurate model Hierarchy by asymptotic limits Toward a simulation of the whole domain 3 Interfacial coupling of hyperbolic systems Optimization of the localization of the coupling interfaces Conclusion et perspectives 4 Nicolas Seguin (LJLL, UPMC) 4 / 32
Thermohydraulics in PWR Overview and LOCA Pressurized water reactor (PWR) Enceinte de confinement Vapeur d’eau Pressuriseur Cuve Barres de contrˆ ole Turbine Vapeur G´ en´ erateur de vapeur Alternateur (´ echangeur de chaleur) Caloporteur Tour de chaud refroidissement (330 ◦ C) Pompe Condenseur ou rivi` ere ou mer Cœur Liquide Circuit de refroidissement Pompe Caloporteur froid (280 ◦ C) Pompe Simulation of water circuits Primary circuit : liquid water at 300 ◦ C and 150 × 10 5 Pa Secondary circuit : liquid water and vapor (about 270 ◦ C and 50 × 10 5 Pa ) Nicolas Seguin (LJLL, UPMC) 5 / 32
Thermohydraulics in PWR Modeling assumptions Characteristics of the models Regimes of interest Standard configuration Loss of coolant accident Fissure in the pipes of the primary circuit 1 Violent decrease of pressure and liquid to vapor phase transition 2 Entrance of air in the pipes 3 Assumptions on the models (and on the numerical schemes) Multiphase flows Phase appearance and disappearance Compressible fluids Phase transition Account for temperature All Mach number Nicolas Seguin (LJLL, UPMC) 6 / 32
Thermohydraulics in PWR Modeling assumptions Characteristics of the models Regimes of interest Standard configuration Loss of coolant accident Fissure in the pipes of the primary circuit 1 Violent decrease of pressure and liquid to vapor phase transition 2 Entrance of air in the pipes 3 Assumptions on the models (and on the numerical schemes) Multiphase flows Phase appearance and disappearance Compressible fluids Phase transition Account for temperature All Mach number Nicolas Seguin (LJLL, UPMC) 6 / 32
Thermohydraulics in PWR Modeling assumptions Characteristics of the models Regimes of interest Standard configuration Loss of coolant accident Fissure in the pipes of the primary circuit 1 Violent decrease of pressure and liquid to vapor phase transition 2 Entrance of air in the pipes 3 Assumptions on the models (and on the numerical schemes) Average models (space, time, statistics, homogenization...) compressible Euler based models Finite Volume Schemes Complex (tabulated) equations of state Nicolas Seguin (LJLL, UPMC) 6 / 32
Thermohydraulics in PWR Modeling assumptions Characteristics of the models Regimes of interest Standard configuration Loss of coolant accident Fissure in the pipes of the primary circuit 1 Violent decrease of pressure and liquid to vapor phase transition 2 Entrance of air in the pipes 3 Assumptions on the models (and on the numerical schemes) Average models (space, time, statistics, homogenization...) compressible Euler based models Finite Volume Schemes Complex (tabulated) equations of state Difficulties [Ishii ’75] [Drew, Passman ’98] Non conservative systems, non strict/conditional hyperbolicity, closure laws... Nicolas Seguin (LJLL, UPMC) 6 / 32
Models for two-phase flows Outline of the talk Thermohydraulics in PWR 1 Overview and LOCA Modeling assumptions Models for two-phase flows 2 The “most” accurate model Hierarchy by asymptotic limits Toward a simulation of the whole domain 3 Interfacial coupling of hyperbolic systems Optimization of the localization of the coupling interfaces Conclusion et perspectives 4 Nicolas Seguin (LJLL, UPMC) 7 / 32
Models for two-phase flows The “most” accurate model The Baer-Nunziato model Fine description of each phase k , k = 1 , 2 density ρ k , velocity u k and pressure p k void fraction α k (probability of presence, volume fraction...) Phasic equations of state : p k = P k ( ρ k , ρ k e k ) , E k = | u k | 2 / 2 + e k Exchanges between phases : mechanics, heat, drag, mass [Baer, Nunziato ’86] [Embid, Baer ’92] [Kapila et al. ’97] [Saurel, Abgrall ’99]... ∂ t α 1 + u I ∂ x α 1 = S p ρ 1 ρ 1 u 1 0 ρ 1 ( u 1 ) 2 + p 1 ∂ t α 1 + ∂ x α 1 − ∂ x α 1 = − S ρ 1 u 1 p I u 1 ( ρ 1 E 1 + p 1 ) ρ 1 E 1 p I u I ρ 2 ρ 2 u 2 0 ρ 2 ( u 2 ) 2 + p 2 + ∂ x α 2 − ∂ x α 2 = + S ∂ t α 2 ρ 2 u 2 p I ρ 2 E 2 u 2 ( ρ 2 E 2 + p 2 ) p I u I where α 1 + α 2 = 1 and S stands for all the exchange source terms Nicolas Seguin (LJLL, UPMC) 8 / 32
Models for two-phase flows The “most” accurate model The Baer-Nunziato model Fine description of each phase k , k = 1 , 2 density ρ k , velocity u k and pressure p k void fraction α k (probability of presence, volume fraction...) Phasic equations of state : p k = P k ( ρ k , ρ k e k ) , E k = | u k | 2 / 2 + e k Exchanges between phases : mechanics, heat, drag, mass [Baer, Nunziato ’86] [Embid, Baer ’92] [Kapila et al. ’97] [Saurel, Abgrall ’99]... ∂ t α 1 + u I ∂ x α 1 = S p ρ 1 ρ 1 u 1 0 ρ 1 ( u 1 ) 2 + p 1 ∂ t α 1 + ∂ x α 1 − ∂ x α 1 = − S ρ 1 u 1 p I u 1 ( ρ 1 E 1 + p 1 ) ρ 1 E 1 p I u I ρ 2 ρ 2 u 2 0 ρ 2 ( u 2 ) 2 + p 2 + ∂ x α 2 − ∂ x α 2 = + S ∂ t α 2 ρ 2 u 2 p I ρ 2 E 2 u 2 ( ρ 2 E 2 + p 2 ) p I u I where α 1 + α 2 = 1 and S stands for all the exchange source terms Nicolas Seguin (LJLL, UPMC) 8 / 32
Models for two-phase flows The “most” accurate model The Baer-Nunziato model ∂ t α 1 + u I ∂ x α 1 = S p ρ 1 ρ 1 u 1 0 ρ 1 ( u 1 ) 2 + p 1 ∂ t α 1 ρ 1 u 1 + ∂ x α 1 − p I ∂ x α 1 = − S u 1 ( ρ 1 E 1 + p 1 ) ρ 1 E 1 p I u I 0 ρ 2 ρ 2 u 2 ρ 2 ( u 2 ) 2 + p 2 ∂ t α 2 + ∂ x α 2 + ∂ x α 1 = + S ρ 2 u 2 p I ρ 2 E 2 u 2 ( ρ 2 E 2 + p 2 ) p I u I General properties One directly has α k ∈ [0 , 1] , ρ k > 0 et T k > 0 Global mass, momentum and total energy conservations S p = 0 ⇐ ⇒ p 1 = p 2 and S = 0 ⇐ ⇒ u 1 = u 2 , T 1 = T 2 , µ 1 = µ 2 Entropy dissipative source terms Nicolas Seguin (LJLL, UPMC) 9 / 32
Models for two-phase flows The “most” accurate model The Baer-Nunziato model ∂ t α 1 + u I ∂ x α 1 = S p ρ 1 ρ 1 u 1 0 ρ 1 ( u 1 ) 2 + p 1 ∂ t α 1 ρ 1 u 1 + ∂ x α 1 − p I ∂ x α 1 = − S u 1 ( ρ 1 E 1 + p 1 ) ρ 1 E 1 p I u I 0 ρ 2 ρ 2 u 2 ρ 2 ( u 2 ) 2 + p 2 ∂ t α 2 + ∂ x α 2 + ∂ x α 1 = + S ρ 2 u 2 p I ρ 2 E 2 u 2 ( ρ 2 E 2 + p 2 ) p I u I [Coquel, Gallou¨ et, H´ erard, Seguin ’02] [Gallou¨ et, H´ erard, Seguin ’04] If the source terms S p S are neglected : The PDE system is strictly hyperbolic iff the wave u I is not superposed with acoustic waves u k ± c k The wave u I is a contact discontinuity iff u I = u 1 , u 2 or α 1 ρ 1 u 1 + α 2 ρ 2 u 2 α 1 ρ 1 + α 2 ρ 2 One may define p I such that the BN model admits a global entropy inequality Nicolas Seguin (LJLL, UPMC) 9 / 32
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