Compressible two-phase flow models: asymptotics, coupling and - - PowerPoint PPT Presentation

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

• f 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

1

Thermohydraulics in PWR Overview and LOCA Modeling assumptions

2

Models for two-phase flows The “most” accurate model Hierarchy by asymptotic limits

3

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems Optimization of the localization of the coupling interfaces

4

Conclusion et perspectives

Nicolas Seguin (LJLL, UPMC) 3 / 32

Thermohydraulics in PWR

Outline of the talk

1

Thermohydraulics in PWR Overview and LOCA Modeling assumptions

2

Models for two-phase flows The “most” accurate model Hierarchy by asymptotic limits

3

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems Optimization of the localization of the coupling interfaces

4

Conclusion et perspectives

Nicolas Seguin (LJLL, UPMC) 4 / 32

Thermohydraulics in PWR Overview and LOCA

Pressurized water reactor (PWR)

Enceinte de confinement Liquide Vapeur Pompe G´ en´ erateur de vapeur (´ echangeur de chaleur) Turbine Alternateur Tour de refroidissement

• u rivi`

ere ou mer Circuit de refroidissement Condenseur Pompe Pompe Cœur Cuve Barres de contrˆ

• le

Pressuriseur Caloporteur chaud (330 ◦C) Caloporteur froid (280 ◦C) Vapeur d’eau

Simulation of water circuits Primary circuit : liquid water at 300◦C and 150 × 105Pa Secondary circuit : liquid water and vapor (about 270◦C and 50 × 105Pa)

Nicolas Seguin (LJLL, UPMC) 5 / 32

Thermohydraulics in PWR Modeling assumptions

Characteristics of the models

Regimes of interest Standard configuration Loss of coolant accident

1

Fissure in the pipes of the primary circuit

2

Violent decrease of pressure and liquid to vapor phase transition

3

Entrance of air in the pipes

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

1

Fissure in the pipes of the primary circuit

2

Violent decrease of pressure and liquid to vapor phase transition

3

Entrance of air in the pipes

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

1

Fissure in the pipes of the primary circuit

2

Violent decrease of pressure and liquid to vapor phase transition

3

Entrance of air in the pipes

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

1

Fissure in the pipes of the primary circuit

2

Violent decrease of pressure and liquid to vapor phase transition

3

Entrance of air in the pipes

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

1

Thermohydraulics in PWR Overview and LOCA Modeling assumptions

2

Models for two-phase flows The “most” accurate model Hierarchy by asymptotic limits

3

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems Optimization of the localization of the coupling interfaces

4

Conclusion et perspectives

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 ρk, velocity uk and pressure pk void fraction αk (probability of presence, volume fraction...) Phasic equations of state : pk = Pk(ρk, ρkek), Ek = |uk|2/2 + ek Exchanges between phases : mechanics, heat, drag, mass [Baer, Nunziato ’86] [Embid, Baer ’92] [Kapila et al. ’97] [Saurel, Abgrall ’99]...                     

∂tα1 + uI∂xα1 = Sp ∂t α1    ρ1 ρ1u1 ρ1E1    + ∂x α1    ρ1u1 ρ1(u1)2 + p1 u1(ρ1E1 + p1)    −    pI pIuI    ∂xα1 = − S ∂t α2    ρ2 ρ2u2 ρ2E2    + ∂x α2    ρ2u2 ρ2(u2)2 + p2 u2(ρ2E2 + p2)    −    pI pIuI    ∂xα2 = + S

where α1 + α2 = 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 ρk, velocity uk and pressure pk void fraction αk (probability of presence, volume fraction...) Phasic equations of state : pk = Pk(ρk, ρkek), Ek = |uk|2/2 + ek Exchanges between phases : mechanics, heat, drag, mass [Baer, Nunziato ’86] [Embid, Baer ’92] [Kapila et al. ’97] [Saurel, Abgrall ’99]...                     

∂tα1 + uI∂xα1 = Sp ∂t α1    ρ1 ρ1u1 ρ1E1    + ∂x α1    ρ1u1 ρ1(u1)2 + p1 u1(ρ1E1 + p1)    −    pI pIuI    ∂xα1 = − S ∂t α2    ρ2 ρ2u2 ρ2E2    + ∂x α2    ρ2u2 ρ2(u2)2 + p2 u2(ρ2E2 + p2)    −    pI pIuI    ∂xα2 = + S

where α1 + α2 = 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

                    

∂tα1 + uI∂xα1 = Sp ∂t α1    ρ1 ρ1u1 ρ1E1    + ∂x α1    ρ1u1 ρ1(u1)2 + p1 u1(ρ1E1 + p1)    −    pI pIuI    ∂xα1 = − S ∂t α2    ρ2 ρ2u2 ρ2E2    + ∂x α2    ρ2u2 ρ2(u2)2 + p2 u2(ρ2E2 + p2)    +    pI pIuI    ∂xα1 = + S

General properties One directly has αk ∈ [0, 1], ρk > 0 et Tk > 0 Global mass, momentum and total energy conservations Sp = 0 ⇐ ⇒ p1 = p2 and S = 0 ⇐ ⇒ u1 = u2, T1 = T2, µ1 = µ2 Entropy dissipative source terms

Nicolas Seguin (LJLL, UPMC) 9 / 32

Models for two-phase flows The “most” accurate model

The Baer-Nunziato model

                    

∂tα1 + uI∂xα1 = Sp ∂t α1    ρ1 ρ1u1 ρ1E1    + ∂x α1    ρ1u1 ρ1(u1)2 + p1 u1(ρ1E1 + p1)    −    pI pIuI    ∂xα1 = − S ∂t α2    ρ2 ρ2u2 ρ2E2    + ∂x α2    ρ2u2 ρ2(u2)2 + p2 u2(ρ2E2 + p2)    +    pI pIuI    ∂xα1 = + S

[Coquel, Gallou¨ et, H´ erard, Seguin ’02] [Gallou¨ et, H´ erard, Seguin ’04] If the source terms Sp S are neglected : The PDE system is strictly hyperbolic iff the wave uI is not superposed with acoustic waves uk ± ck The wave uI is a contact discontinuity iff uI = u1, u2 or α1ρ1u1+α2ρ2u2

α1ρ1+α2ρ2

One may define pI such that the BN model admits a global entropy inequality

Nicolas Seguin (LJLL, UPMC) 9 / 32

Models for two-phase flows The “most” accurate model

The Baer-Nunziato model

                    

∂tα1 + uI∂xα1 = Sp ∂t α1    ρ1 ρ1u1 ρ1E1    + ∂x α1    ρ1u1 ρ1(u1)2 + p1 u1(ρ1E1 + p1)    −    pI pIuI    ∂xα1 = − S ∂t α2    ρ2 ρ2u2 ρ2E2    + ∂x α2    ρ2u2 ρ2(u2)2 + p2 u2(ρ2E2 + p2)    +    pI pIuI    ∂xα1 = + S

[Coquel, Gallou¨ et, H´ erard, Seguin ’02] [Gallou¨ et, H´ erard, Seguin ’04] If the source terms Sp S are neglected : Theorem [Isaacson, Temple ’95] [Goatin, LeFloch ’04] The Riemann problem may admit up to three distinct solutions

Nicolas Seguin (LJLL, UPMC) 9 / 32

Models for two-phase flows The “most” accurate model

The Baer-Nunziato model

                    

∂tα1 + uI∂xα1 = Sp ∂t α1    ρ1 ρ1u1 ρ1E1    + ∂x α1    ρ1u1 ρ1(u1)2 + p1 u1(ρ1E1 + p1)    −    pI pIuI    ∂xα1 = − S ∂t α2    ρ2 ρ2u2 ρ2E2    + ∂x α2    ρ2u2 ρ2(u2)2 + p2 u2(ρ2E2 + p2)    +    pI pIuI    ∂xα1 = + S

[Coquel, H´ erard, Saleh, Seguin] The system is symmetrisable (out of resonance) The global entropy is a (non strictly) convex function Construction of positive preserving finite volume schemes

Nicolas Seguin (LJLL, UPMC) 9 / 32

Models for two-phase flows Hierarchy by asymptotic limits

The drift-flux model

Barotropic Baer-Nunziato model + external forces fk               

∂tα1 + uI∂xα1 = Sp ∂t α1

• ρ1

ρ1u1

• + ∂x α1
• ρ1u1

ρ1(u1)2 + p1

• −
• pI
• ∂xα1 = − S +
• α1ρ1f1
• ∂t α2
• ρ2

ρ2u2

• + ∂x α2
• ρ2u2

ρ2(u2)2 + p2

• +
• pI
• ∂xα1 = + S +
• α2ρ2f2
• Define mean and

relative variables ρ = α1ρ1 + α2ρ2 ρY = α2ρ2 p = α1p1 + α2p2 ρv = α1ρ1u1 + α2ρ2u2 vr = u2 − u1 pr = p2 − p1      ∂tρ + ∂x(ρv) = 0 ∂t(ρY ) + ∂x

• ρvY + ρY (1 − Y )vr
• = 0

∂t(ρv) + ∂x

• ρv2 + p + ρY (1 − Y )(vr)2

= ρ(1 − Y )f1 + ρY f2 + PDE’s for ur and pr

Nicolas Seguin (LJLL, UPMC) 10 / 32

Models for two-phase flows Hierarchy by asymptotic limits

The drift-flux model

One pressure p, mean velocity v and relative velocity vr [Zuber, Findlay ’65]      ∂tρ + ∂x(ρv) = 0 ∂t(ρY ) + ∂x

• ρvY + ρY (1 − Y )vr
• = Γ

∂t(ρv) + ∂x

• ρv2 + p + ρY (1 − Y )(vr)2

= ρ(1 − Y )f1 + ρY f2 p(ρ, Y ) = P1

• ρ 1 − Y

1 − α2

• = P2
• ρ Y

α1

• et

vr = Vr(ρ, ρY, ρv) with ρY = α2ρ2 and external forces fk From Baer-Nunziato to drift-flux model [Ambroso, Chalons, Coquel, Gali´ e, Godlewski, Raviart, Seguin ’08] Asymptotic study with pr = O(ε2) and vr = O(ε) Chapman-Enskog expansion → parabolic model [Guillard, Duval 07] vr = Vr(ρ, ρY, ρv, ∂xp, f1, f2) Permanent flow → drift-flux model ∂xp = ρ(1 − Y )f1 + ρY f2

Nicolas Seguin (LJLL, UPMC) 11 / 32

Models for two-phase flows Hierarchy by asymptotic limits

The drift-flux model

One pressure p, mean velocity v and relative velocity vr [Zuber, Findlay ’65]      ∂tρ + ∂x(ρv) = 0 ∂t(ρY ) + ∂x

• ρvY + ρY (1 − Y )vr
• = Γ

∂t(ρv) + ∂x

• ρv2 + p + ρY (1 − Y )(vr)2

= ρ(1 − Y )f1 + ρY f2 p(ρ, Y ) = P1

• ρ 1 − Y

1 − α2

• = P2
• ρ Y

α1

• et

vr = Vr(ρ, ρY, ρv) with ρY = α2ρ2 and external forces fk From Baer-Nunziato to drift-flux model [Ambroso, Chalons, Coquel, Gali´ e, Godlewski, Raviart, Seguin ’08] Asymptotic study with pr = O(ε2) and vr = O(ε) Chapman-Enskog expansion → parabolic model [Guillard, Duval 07] vr = Vr(ρ, ρY, ρv, ∂xp, f1, f2) Permanent flow → drift-flux model ∂xp = ρ(1 − Y )f1 + ρY f2

Nicolas Seguin (LJLL, UPMC) 11 / 32

Models for two-phase flows Hierarchy by asymptotic limits

Homogeneous models

two-phase flow ≡ one fluid (vr = 0) with mass transfer Homogeneous relaxation model (HRM)          ∂t(ρY ) + ∂x(ρY v) = λ(µ2 − µ1) ∂tρ + ∂x(ρv) = 0 ∂t(ρv) + ∂x(ρv2 + p) = 0 ∂t(ρE) + ∂x(v(ρE + p)) = 0 with p = Pr(ρ, ρe, ρY ) Homogeneous equilibrium model (HEM)      ∂tρ + ∂x(ρv) = 0 ∂t(ρv) + ∂x(ρv2 + p) = 0 ∂t(ρE) + ∂x(v(ρE + p)) = 0 with p = Pe(ρ, ρe) satisfying the compatibility relation (λ → +∞) Pr(ρ, ρe, ρY ) = Pe(ρ, ρe) ⇐ ⇒ µ1 = µ2

Nicolas Seguin (LJLL, UPMC) 12 / 32

Models for two-phase flows Hierarchy by asymptotic limits

Homogeneous models

two-phase flow ≡ one fluid (vr = 0) with mass transfer Homogeneous relaxation model (HRM)          ∂t(ρY ) + ∂x(ρY v) = λ(µ2 − µ1) ∂tρ + ∂x(ρv) = 0 ∂t(ρv) + ∂x(ρv2 + p) = 0 ∂t(ρE) + ∂x(v(ρE + p)) = 0 with p = Pr(ρ, ρe, ρY ) Homogeneous equilibrium model (HEM)      ∂tρ + ∂x(ρv) = 0 ∂t(ρv) + ∂x(ρv2 + p) = 0 ∂t(ρE) + ∂x(v(ρE + p)) = 0 with p = Pe(ρ, ρe) satisfying the compatibility relation (λ → +∞) Pr(ρ, ρe, ρY ) = Pe(ρ, ρe) ⇐ ⇒ µ1 = µ2

Nicolas Seguin (LJLL, UPMC) 12 / 32

Models for two-phase flows Hierarchy by asymptotic limits

Asymptotic limits

Baer-Nunziato model → Drift-flux → HRM → HEM Hyperbolic/hyperbolic relaxation (simplified...)

• ∂tu + ∂xf(u, v) = 0

∂tv + ∂xg(u, v) = 1

εr(u, v)

→

• ∂tu + ∂xf(u, ve(u)) = 0

v = ve(u) o` u r(u, ve(u)) = 0 Hyperbolic/parabolic relaxation (simplified...)

• ε∂tu + ∂xvf(u) = 0

ε∂tv + ∂xg(u, v) = 1

εr(u)v

→

• ∂tu + ∂xf(u, ve(u, ∂xu)) = 0

v = εve(u, ∂xu) := ε r(u)−1∂xg(u, 0) Theoretical studies : [Chen, Levermore, Liu ’95], but also Yong, Serre, Hanouzet, Natalini, Coulombel, Tzavaras... + kinetic theory...

Nicolas Seguin (LJLL, UPMC) 13 / 32

Toward a simulation of the whole domain

Outline of the talk

1

Thermohydraulics in PWR Overview and LOCA Modeling assumptions

2

Models for two-phase flows The “most” accurate model Hierarchy by asymptotic limits

3

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems Optimization of the localization of the coupling interfaces

4

Conclusion et perspectives

Nicolas Seguin (LJLL, UPMC) 14 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Interfacial coupling

Each model is dedicated to a particular part of the domain Interfacial coupling for a global simulation Non intrusive interfacial coupling by boundary conditions Coupling problem Two different systems to couple through an interface x = 0 CODE [L] CODE [R] ∂tUL + ∂xFL(UL) = SL(UL) ∂tUR + ∂xFR(UR) = SR(UR) x < 0 x > 0 x = 0 Compatibility relations between UL(t, 0−) and UR(t, 0+) are needed The interfacial coupling is ARTIFICIAL

Nicolas Seguin (LJLL, UPMC) 15 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Interfacial coupling

Each model is dedicated to a particular part of the domain Interfacial coupling for a global simulation Non intrusive interfacial coupling by boundary conditions Coupling problem Two different systems to couple through an interface x = 0 CODE [L] CODE [R] ∂tUL + ∂xFL(UL) = SL(UL) ∂tUR + ∂xFR(UR) = SR(UR) x < 0 x > 0 x = 0 Compatibility relations between UL(t, 0−) and UR(t, 0+) are needed The interfacial coupling is ARTIFICIAL

Nicolas Seguin (LJLL, UPMC) 15 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Interfacial coupling

Each model is dedicated to a particular part of the domain Interfacial coupling for a global simulation Non intrusive interfacial coupling by boundary conditions Coupling problem Two different systems to couple through an interface x = 0 CODE [L] CODE [R] ∂tUL + ∂xFL(UL) = SL(UL) ∂tUR + ∂xFR(UR) = SR(UR) x < 0 x > 0 x = 0 Compatibility relations between UL(t, 0−) and UR(t, 0+) are needed The interfacial coupling is ARTIFICIAL

Nicolas Seguin (LJLL, UPMC) 15 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Interfacial coupling

One may hope that [R] is the asymptotic limit of [L] physical arguments may provide the compatibility relations between UL(t, 0−) and UR(t, 0+) the coupling interface is located where the models are close [L] ≈ [R] BUT... the codes are developed independently the models are not fully compatible all the physical principles (continuity, conservation) cannot be fulfilled at the same time in extreme cases, the coupling interface may be ill located Simplest example : 2 Euler systems with different equations of state

Nicolas Seguin (LJLL, UPMC) 16 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Interfacial coupling

One may hope that [R] is the asymptotic limit of [L] physical arguments may provide the compatibility relations between UL(t, 0−) and UR(t, 0+) the coupling interface is located where the models are close [L] ≈ [R] BUT... the codes are developed independently the models are not fully compatible all the physical principles (continuity, conservation) cannot be fulfilled at the same time in extreme cases, the coupling interface may be ill located Simplest example : 2 Euler systems with different equations of state

Nicolas Seguin (LJLL, UPMC) 16 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Interfacial coupling

One may hope that [R] is the asymptotic limit of [L] physical arguments may provide the compatibility relations between UL(t, 0−) and UR(t, 0+) the coupling interface is located where the models are close [L] ≈ [R] BUT... the codes are developed independently the models are not fully compatible all the physical principles (continuity, conservation) cannot be fulfilled at the same time in extreme cases, the coupling interface may be ill located Simplest example : 2 Euler systems with different equations of state

Nicolas Seguin (LJLL, UPMC) 16 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Interfacial coupling

Several works : A. Ambroso, C. Chalons, B. Boutin, F. Coquel, T. Gali´ e,

• E. Godlewski, F. Lagouti`

ere, P.-A. Raviart, N. Seguin Simplest example 2 Euler systems with different equations of state : pL(.) ≡ pR(.)      ∂tρ + ∂xρu = 0 ∂tρu + ∂x(ρu2 + pL) = 0 ∂tρE + ∂x(u(ρE + pL)) = 0      ∂tρ + ∂xρu = 0 ∂tρu + ∂x(ρu2 + pR) = 0 ∂tρE + ∂x(u(ρE + pR)) = 0 x < 0 x > 0 x = 0

Nicolas Seguin (LJLL, UPMC) 17 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Interfacial coupling

Several works : A. Ambroso, C. Chalons, B. Boutin, F. Coquel, T. Gali´ e,

• E. Godlewski, F. Lagouti`

ere, P.-A. Raviart, N. Seguin Simplest example 2 Euler systems with different equations of state : pL(.) ≡ pR(.)      ∂tρ + ∂xρu = 0 ∂tρu + ∂x(ρu2 + pL) = 0 ∂tρE + ∂x(u(ρE + pL)) = 0      ∂tρ + ∂xρu = 0 ∂tρu + ∂x(ρu2 + pR) = 0 ∂tρE + ∂x(u(ρE + pR)) = 0 x < 0 x > 0 x = 0

Nicolas Seguin (LJLL, UPMC) 17 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Interfacial coupling

Compatibility relations to prescribe at the interface : ϕL(UL)(t, 0) = ϕR(UR)(t, 0)          ∂tρ + ∂xρu = 0 ∂tρu + ∂x(ρu2 + pL) = 0 ∂tρE + ∂x(u(ρE + pL)) = 0 ϕL(UL)(t, 0) = ϕR(UR)(t, 0)          ∂tρ + ∂xρu = 0 ∂tρu + ∂x(ρu2 + pR) = 0 ∂tρE + ∂x(u(ρE + pR)) = 0 ϕR(UR)(t, 0) = ϕL(UL)(t, 0)

Nicolas Seguin (LJLL, UPMC) 18 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Interfacial coupling

Compatibility relations to prescribe at the interface : ϕL(UL)(t, 0) = ϕR(UR)(t, 0)          ∂tρ + ∂xρu = 0 ∂tρu + ∂x(ρu2 + pL) = 0 ∂tρE + ∂x(u(ρE + pL)) = 0 ϕL(UL)(t, 0) = ϕR(UR)(t, 0)          ∂tρ + ∂xρu = 0 ∂tρu + ∂x(ρu2 + pR) = 0 ∂tρE + ∂x(u(ρE + pR)) = 0 ϕR(UR)(t, 0) = ϕL(UL)(t, 0) (weak) boundary conditions ` a la [Dubois, LeFloch ’88] : = ⇒ ϕL(UL)(t, 0) = ϕR(UR)(t, 0) not always satisfied ! Fictitious state technique for the numerical coupling

Nicolas Seguin (LJLL, UPMC) 18 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Interfacial coupling

Compatibility relations to prescribe at the interface : ϕL(UL)(t, 0) = ϕR(UR)(t, 0)          ∂tρ + ∂xρu = 0 ∂tρu + ∂x(ρu2 + pL) = 0 ∂tρE + ∂x(u(ρE + pL)) = 0 ϕL(UL)(t, 0) = ϕR(UR)(t, 0)          ∂tρ + ∂xρu = 0 ∂tρu + ∂x(ρu2 + pR) = 0 ∂tρE + ∂x(u(ρE + pR)) = 0 ϕR(UR)(t, 0) = ϕL(UL)(t, 0) Theorem (Riemann problem) If |u| ≪ c, there exists a unique continuous solution. If |u| ≃ c, it may exists a one-parameter family of continuous solutions. If |u| ≃ c, there exists at most one discontinuous solution.

Nicolas Seguin (LJLL, UPMC) 18 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Simulation in the case of multiple solutions

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

• 0.4
• 0.2

0.2 0.4 Velocity Rusanov I Rusanov II Relaxation 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4

• 0.4
• 0.2

0.2 0.4 Density Rusanov I Rusanov II Relaxation 2 4 6 8 10 12

• 0.4
• 0.2

0.2 0.4 Pressure Rusanov I Rusanov II Relaxation 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

• 0.4
• 0.2

0.2 0.4 Mach number Rusanov I Rusanov II Relaxation

Nicolas Seguin (LJLL, UPMC) 19 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Simulation in the case of multiple solutions

0.0001 0.001 0.01 0.1 100 1000 10000 Difference of the velocities at x=0+ Number of cells Comparison between Rusanov I (CFL=0,45), Rusanov II (CFL=0,05) and the relaxation method Rusanov I vs Rusanov II Relaxation vs Rusanov II Relaxation vs Rusanov I

Nicolas Seguin (LJLL, UPMC) 20 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Intermediate conclusion on the interfacial coupling

Coupling of 2 Euler systems Coupling relations have to be chosen In subsonic flows, everything goes well... The numerics well reproduce the theoretical results How to overcome the problems of the interfacial coupling Locate the coupling interfaces where “nothing” happens Propose an indicator of the “nothing” Use the simplest model as much as possible = ⇒ Adaptive algorithm to select the “good” model in the “good” domain

Nicolas Seguin (LJLL, UPMC) 21 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Intermediate conclusion on the interfacial coupling

Coupling of 2 Euler systems Coupling relations have to be chosen In subsonic flows, everything goes well... The numerics well reproduce the theoretical results How to overcome the problems of the interfacial coupling Locate the coupling interfaces where “nothing” happens Propose an indicator of the “nothing” Use the simplest model as much as possible = ⇒ Adaptive algorithm to select the “good” model in the “good” domain

Nicolas Seguin (LJLL, UPMC) 21 / 32

Toward a simulation of the whole domain Interfacial coupling of hyperbolic systems

Intermediate conclusion on the interfacial coupling

Coupling of 2 Euler systems Coupling relations have to be chosen In subsonic flows, everything goes well... The numerics well reproduce the theoretical results How to overcome the problems of the interfacial coupling Locate the coupling interfaces where “nothing” happens Propose an indicator of the “nothing” Use the simplest model as much as possible = ⇒ Adaptive algorithm to select the “good” model in the “good” domain

Nicolas Seguin (LJLL, UPMC) 21 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Local model adaptation

Algorithm for a dynamical mode selection Choice between a fine model and a coarse model Indicator of error between the two models Use the coarse model as much as possible Automatic model adaptation Fine model Mf, coarse model Mc (with solutions uf and uc) Iteration tn → tn+1 :

1

Decomposition of the computational domain R in Dn

f and Dn g :

Computation of the local indicator E (x, tn) ∼ uf − uc(x, tn+1) Dn

f := {x | E (x, tn) > Θ}

Dn

c := {x | E (x, tn) < Θ}

2

Computation of the coupling problem between tn and tn+1 :

Model Mf in Dn

f

Model Mc in Dn

c

Coupling conditions at ¯ Dn

f ∩ ¯

Dn

c

Nicolas Seguin (LJLL, UPMC) 22 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Local model adaptation

Algorithm for a dynamical mode selection Choice between a fine model and a coarse model Indicator of error between the two models Use the coarse model as much as possible Automatic model adaptation Fine model Mf, coarse model Mc (with solutions uf and uc) Iteration tn → tn+1 :

1

Decomposition of the computational domain R in Dn

f and Dn g :

Computation of the local indicator E (x, tn) ∼ uf − uc(x, tn+1) Dn

f := {x | E (x, tn) > Θ}

Dn

c := {x | E (x, tn) < Θ}

2

Computation of the coupling problem between tn and tn+1 :

Model Mf in Dn

f

Model Mc in Dn

c

Coupling conditions at ¯ Dn

f ∩ ¯

Dn

c

Nicolas Seguin (LJLL, UPMC) 22 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Local model adaptation

Applications Hyperbolic/hyperbolic relaxation

• C. Canc`

es, F. Coquel, E. Godlewski, H. Mathis, N. Seguin Hyperbolic/parabolic relaxation A.-C. Boulanger, C. Canc` es, H. Mathis, K. Saleh, N. Seguin (Cemracs’11) Local indicators for the models and the numerics E (x, tn) ∼ uf − uc(x, tn+1) A posteriori estimates [Eymard-Gallou¨ et-Ghilani-Herbin, Ohlberger, Bouchut-Perthame, Mercier...] first-order term in Chapman-Enskog expansions [Chen-Levermore-Liu, Yong...] & kinetic theory... Relative entropy for hyperbolic systems [Lax-Wendroff, DiPerna, Dafermos, Chainais, Tzavaras...]

Nicolas Seguin (LJLL, UPMC) 23 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Local model adaptation

Applications Hyperbolic/hyperbolic relaxation

• C. Canc`

es, F. Coquel, E. Godlewski, H. Mathis, N. Seguin Hyperbolic/parabolic relaxation A.-C. Boulanger, C. Canc` es, H. Mathis, K. Saleh, N. Seguin (Cemracs’11) Local indicators for the models and the numerics E (x, tn) ∼ uf − uc(x, tn+1) A posteriori estimates [Eymard-Gallou¨ et-Ghilani-Herbin, Ohlberger, Bouchut-Perthame, Mercier...] first-order term in Chapman-Enskog expansions [Chen-Levermore-Liu, Yong...] & kinetic theory... Relative entropy for hyperbolic systems [Lax-Wendroff, DiPerna, Dafermos, Chainais, Tzavaras...]

Nicolas Seguin (LJLL, UPMC) 23 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Hyperbolic models with relaxation

Simplified framework for relaxation Relaxation model (R)

• ∂tU + ∂xf(U, v)

= 0, ∂tv + ∂xg(U, v) = 1

ε(h(U) − v),

where ε is a small parameter. Equilibrium states : v = h(U). Associated equilibrium model (E) ∂tU + ∂xf(U, h(U)) = 0 smaller system of PDE (R) : Accurate model with small time scale (E) : Coarse model, simpler to solve

Nicolas Seguin (LJLL, UPMC) 24 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Chapman-Enskog expansion

Take a solution of (R) near equilibrium v = h(U) + εv1 + O(ε2) Plug this expansion in (R)... Deduce the first order correction v1 = ∇h(U)T ∂x (f(U, h(U))) − ∂xg(U, h(U)) + O(ε) Solutions of (R) near equilibrium satisfy the intermediate model ∂tU + ∂xf(U, h(U)) = −ε∂x

• ∇2f(U, h(U))v1
• + O(ε2)

Nicolas Seguin (LJLL, UPMC) 25 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Chapman-Enskog expansion

Take a solution of (R) near equilibrium v = h(U) + εv1 + O(ε2) Plug this expansion in (R)... Deduce the first order correction v1 = ∇h(U)T ∂x (f(U, h(U))) − ∂xg(U, h(U)) + O(ε) Solutions of (R) near equilibrium satisfy the intermediate model ∂tU + ∂xf(U, h(U)) = −ε∂x

• ∇2f(U, h(U))v1
• + O(ε2)

Nicolas Seguin (LJLL, UPMC) 25 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Chapman-Enskog expansion

Take a solution of (R) near equilibrium v = h(U) + εv1 + O(ε2) Plug this expansion in (R)... Deduce the first order correction v1 = ∇h(U)T ∂x (f(U, h(U))) − ∂xg(U, h(U)) + O(ε) Solutions of (R) near equilibrium satisfy the intermediate model ∂tU + ∂xf(U, h(U)) = −ε∂x

• ∇2f(U, h(U))v1
• + O(ε2)

Nicolas Seguin (LJLL, UPMC) 25 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Chapman-Enskog expansion

Take a solution of (R) near equilibrium v = h(U) + εv1 + O(ε2) Plug this expansion in (R)... Deduce the first order correction v1 = ∇h(U)T ∂x (f(U, h(U))) − ∂xg(U, h(U)) + O(ε) Solutions of (R) near equilibrium satisfy the intermediate model ∂tU + ∂xf(U, h(U)) = −ε∂x

• ∇2f(U, h(U))v1
• + O(ε2)

Nicolas Seguin (LJLL, UPMC) 25 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Derivation of the indicator

We have supposed that v = h(U) + εv1 + O(ε2) and then v1 = ∇h(U)T ∂x (f(U, h(U))) − ∂xg(U, h(U)) + O(ε) ∂tU + ∂xf(U, h(U)) = −ε∂x[∇2f(U, h(U)) v1] + O(ε2) Indicator : εv1 Everything can be done identically at the numerical point of view ! From (U n

i )i one can compute εvn+1 1,i

without solving (R)

Nicolas Seguin (LJLL, UPMC) 26 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Derivation of the indicator

We have supposed that v = h(U) + εv1 + O(ε2) and then v1 = ∇h(U)T ∂x (f(U, h(U))) − ∂xg(U, h(U)) + O(ε) ∂tU + ∂xf(U, h(U)) = −ε∂x[∇2f(U, h(U)) v1] + O(ε2) Indicator : εv1 Everything can be done identically at the numerical point of view ! From (U n

i )i one can compute εvn+1 1,i

without solving (R)

Nicolas Seguin (LJLL, UPMC) 26 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Algorithm for adaptation

1

Knowing (U n

i )i and given a threshold θ

2

One each cell i compute E n

i := εvn+1 1,i

3

If [|E n

i | > θ]

(R) must be applied in cell i Else (E) must be applied in cell i

4

Compute the coupling problem : → (U, v)n+1

i

in the domain where (R) is used → U n+1

i

in the domain where (E) is used Idem in multi-D on unstructured meshes

Nicolas Seguin (LJLL, UPMC) 27 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Algorithm for adaptation

1

Knowing (U n

i )i and given a threshold θ

2

One each cell i compute E n

i := εvn+1 1,i

3

If [|E n

i | > θ]

(R) must be applied in cell i Else (E) must be applied in cell i

4

Compute the coupling problem : → (U, v)n+1

i

in the domain where (R) is used → U n+1

i

in the domain where (E) is used Idem in multi-D on unstructured meshes

Nicolas Seguin (LJLL, UPMC) 27 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Algorithm for adaptation

1

Knowing (U n

i )i and given a threshold θ

2

One each cell i compute E n

i := εvn+1 1,i

3

If [|E n

i | > θ]

(R) must be applied in cell i Else (E) must be applied in cell i

4

Compute the coupling problem : → (U, v)n+1

i

in the domain where (R) is used → U n+1

i

in the domain where (E) is used Idem in multi-D on unstructured meshes

Nicolas Seguin (LJLL, UPMC) 27 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Algorithm for adaptation

1

Knowing (U n

i )i and given a threshold θ

2

One each cell i compute E n

i := εvn+1 1,i

3

If [|E n

i | > θ]

(R) must be applied in cell i Else (E) must be applied in cell i

4

Compute the coupling problem : → (U, v)n+1

i

in the domain where (R) is used → U n+1

i

in the domain where (E) is used Idem in multi-D on unstructured meshes

Nicolas Seguin (LJLL, UPMC) 27 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Algorithm for adaptation

1

Knowing (U n

i )i and given a threshold θ

2

One each cell i compute E n

i := εvn+1 1,i

3

If [|E n

i | > θ]

(R) must be applied in cell i Else (E) must be applied in cell i

4

Compute the coupling problem : → (U, v)n+1

i

in the domain where (R) is used → U n+1

i

in the domain where (E) is used Idem in multi-D on unstructured meshes

Nicolas Seguin (LJLL, UPMC) 27 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

R´ esultats num´ eriques 1D

Coupling between HEM and HRM (phase change) Domain : [−0.5, 0.5]. Perfect gas : γ1 = 1.6, γ2 = 1.5. Left state Right state ρL = 1 ρR = 1 uL = −0.5 uR = 0.5 pL = 1 pR = 1 At t = 0, one sets ϕ0(x) = ϕeq(ρ0(x)). Parameters : ε = 10−2, Dx = 0.001, θ = 500.

Nicolas Seguin (LJLL, UPMC) 28 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Numerical results in 1D

Figure: Evolution of ρ and indicator : t = 0.2s.

Nicolas Seguin (LJLL, UPMC) 29 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Numerical results in 2D

Mass fraction at t = 0.2s : coarse (top), fine (middle), adaptation (bottom) liquid in blue (ϕ = 0), vapor in red (ϕ = 1)

Nicolas Seguin (LJLL, UPMC) 30 / 32

Toward a simulation of the whole domain Optimization of the localization of the coupling interfaces

Numerical results in 2D

Figure: Indicator at t = 0.2s (red = coarse, blue = fine).

Nicolas Seguin (LJLL, UPMC) 31 / 32

Conclusion and ongoing works

Conclusion

Models and numerical methods for two-phase flows

Understand the models (derivation and analysis) Construction of robust numerical methods Study of the asymptotic limits Construction of numerical methods compatible with these asymptotics

Coupling of models

Proposition of theoretical and numerical methods of coupling Coupling of models of a same hierarchy

Dynamical adaptation of model

Hyperbolic/hyperbolic relaxation and hyperbolic/parabolic relaxation Academic test cases and more industrial tests Other asymptotics (1D/2D...)

Nicolas Seguin (LJLL, UPMC) 32 / 32