Staggered schemes for compressible flows R. Herbin , with et , L. - PowerPoint PPT Presentation

Staggered schemes for compressible flows R. Herbin ⋆ , with et ⋆ , L. Gastaldo † , W. Kheriji ⋆ † , J.-C. Latch´ e † , T.T. Nguyen ⋆ † T. Gallou¨ ⋆ Universit´ e de Provence † Institut de Radioprotection et de Sˆ uret´ e Nucl´ eaire (IRSN)

Context ⊲ General context: nuclear safety Examples: Accident in a nuclear reactor, fol- lowing loss of coolant. Interaction corium–concrete → two phase flow brèche circuit primaire Numerical simulation of compressible flows – cuve numerical code: ISIS, developed at IRSN coeur brèche cuve corium béton béton 1 à 4 m phase liquide mouvement de convection 6 m

Context ⊲ General context: nuclear safety Examples: Accident in a nuclear reactor, fol- lowing loss of coolant. Interaction corium–concrete → two phase flow brèche circuit primaire Numerical simulation of compressible flows – cuve numerical code: ISIS, developed at IRSN coeur ⊲ Aim : obtain efficient schemes brèche cuve ◮ stable and precise for all Mach number corium ◮ sufficiently decoupled so that the béton numerical solution is not too difficult. � fractional time step methods. Classical for incompressible flows (pressure correction, Chorin 68, Temam 69), (Guermond béton 06) for a synthesis. 1 à 4 m phase liquide Also developed for compressible flows, mouvement with either colocated or staggered unknowns.. de convection 6 m

Context ⊲ General context: nuclear safety Examples: Accident in a nuclear reactor, fol- lowing loss of coolant. Interaction corium–concrete → two phase flow brèche circuit primaire Numerical simulation of compressible flows – cuve numerical code: ISIS, developed at IRSN coeur ⊲ Aim : obtain efficient schemes brèche cuve ◮ stable and precise for all Mach number corium ◮ sufficiently decoupled so that the béton numerical solution is not too difficult. � fractional time step methods. Classical for incompressible flows (pressure correction, Chorin 68, Temam 69), (Guermond béton 06) for a synthesis. 1 à 4 m phase liquide Also developed for compressible flows, mouvement with either colocated or staggered unknowns.. de convection ⊲ Theoretical proof of stability and consis- tency, confirmed by numerical tests. 6 m

The drift-diffusion model Drift-diffusion model for two phase flows � ∂ ρ � ∂ t + ∇ · ( ρ u ) = 0 (mass) � � � ∂ ρ y � � + ∇ · ( ρ y u ) = −∇ · ( ρ y (1 − y ) u r ) + ∇ · ( D ∇ y ) (ffl) � ∂ t � � ∂ρ u � + ∇ · ( ρ u ⊗ u ) + ∇ p − ∇ · τ ( u ) = 0 (mom) � ∂ t � � � p = Ψ( ρ, y ) Physical properties ◮ Positivity of the density ◮ Gas mass fraction between 0 and 1 ◮ Total mass conservation and fractional mass conservation ◮ Transport of an interface between phases with constant pressure and velocity ◮ Stability: control on the entropy.

Staggered discretization ◮ Positivity of density � finite volume scheme on ρ : ρ piecewise constant on the cells.

Staggered discretization ◮ Positivity of density � finite volume scheme on ρ : ρ piecewise constant on the cells. Primal mesh : M = { set of control volumes K , L , M ... } .

Staggered discretization ◮ Positivity of density � finite volume scheme on ρ : ρ piecewise constant on the cells. Primal mesh : M = { set of control volumes K , L , M ... } . Scalar variables defined at cell centers: ( p K ) K ∈M , ( ̺ K ) K ∈M ,. . . ◮ Velocity components defined at the (or some of the) edges : ( v ( i ) σ ) σ ∈F ( i ) .

Staggered discretization ◮ Positivity of density � finite volume scheme on ρ : ρ piecewise constant on the cells. Primal mesh : M = { set of control volumes K , L , M ... } . Scalar variables defined at cell centers: ( p K ) K ∈M , ( ̺ K ) K ∈M ,. . . ◮ Velocity components defined at the (or some of the) edges : ( v ( i ) σ ) σ ∈F ( i ) . Dual mesh(es) : M n = ( D ( i ) σ ) σ ∈F ( i ) .

Staggered discretization ◮ Positivity of density � finite volume scheme on ρ : ρ piecewise constant on the cells. Primal mesh : M = { set of control volumes K , L , M ... } . Scalar variables defined at cell centers: ( p K ) K ∈M , ( ̺ K ) K ∈M ,. . . ◮ Velocity components defined at the (or some of the) edges : ( v ( i ) σ ) σ ∈F ( i ) . Dual mesh(es) : M n = ( D ( i ) σ ) σ ∈F ( i ) . Normal velocity to the face σ denoted by v σ · n σ . σ | | σ D σ = L K ǫ = D σ | D σ ′ | L = K | M ′ K σ D σ ′ M Figure: Primal and dual meshes for the Rannacher-Turek and Crouzeix-Raviart elements.

The MAC mesh : D σ (or K y : D σ (or K x 2 , j ) ) i − 1 i , j − 1 2 y j + 3 2 u y i , j + 1 2 y j + 1 u x 2 i − 1 2 , j +1 y j + 1 2 u y u y u y i − 1 , j − 1 i , j − 1 i +1 , j − 1 y j − 1 2 2 2 u x u x u x i − 3 i − 1 i + 1 2 , j 2 , j 2 , j 2 y j − 1 2 u y u x i , j − 3 y j − 3 i − 1 y j − 3 2 2 , j − 1 2 2 x i − 3 x i − 1 x i + 1 x i − 3 x i − 1 x i + 1 x i + 3 2 2 2 2 2 2 2 The dual mesh for the MAC scheme, x and y -component of the velocity.

Finite volume discretization of the mass equation ∂ t ρ + div ( ρ u ) = 0 , (mass)

Finite volume discretization of the mass equation ∂ t ρ + div ( ρ u ) = 0 , (mass) � (mass) � + implicit time discretization � ◮ K ρ n +1 − ρ n � � ( ρ n +1 u n +1 · n K ) = 0 . + δ t K ∂ K

Finite volume discretization of the mass equation ∂ t ρ + div ( ρ u ) = 0 , (mass) � (mass) � + implicit time discretization � ◮ K ρ n +1 − ρ n � � ( ρ n +1 u n +1 · n K ) = 0 . + δ t K ∂ K ◮ discretization of the fluxes: | K | � ( ρ n +1 F n +1 − ρ n K ) + K ,σ = 0 , K δ t σ ∈E ( K ) ◮ F n +1 ρ n +1 u n +1 K ,σ = | σ | ˇ · n K ,σ , numerical flux through σ . σ σ upwind approximation of ρ n +1 at the face σ with respect to u n +1 ρ n +1 ◮ ˇ · n K ,σ . σ σ

Finite volume discretization of the mass equation ∂ t ρ + div ( ρ u ) = 0 , (mass) � (mass) � + implicit time discretization � ◮ K ρ n +1 − ρ n � � ( ρ n +1 u n +1 · n K ) = 0 . + δ t K ∂ K ◮ discretization of the fluxes: | K | � ( ρ n +1 F n +1 − ρ n K ) + K ,σ = 0 , K δ t σ ∈E ( K ) ◮ F n +1 ρ n +1 u n +1 K ,σ = | σ | ˇ · n K ,σ , numerical flux through σ . σ σ upwind approximation of ρ n +1 at the face σ with respect to u n +1 ρ n +1 ◮ ˇ · n K ,σ . σ σ ◮ � Positive density: ρ n +1 > 0 if ( ρ n > 0 and ρ > 0 at inflow boundary).

Discretization of the phase mass fraction balance ∂ t ρ y + div ( ρ u y ) = −∇ · ( ρ y (1 − y ) u r ) + ∇ · ( D ∇ y ) , (ff)

Discretization of the phase mass fraction balance ∂ t ρ y + div ( ρ u y ) = −∇ · ( ρ y (1 − y ) u r ) + ∇ · ( D ∇ y ) , (ff) � (mass) � + implicit time discretization � ◮ K � ρ n +1 y n +1 − ρ n y n � ( ρ n +1 y n +1 u n +1 · n K ) = + δ t K ∂ K � � − ρ y (1 − y ) u r · n K ) + D ∇ y · n K . ∂ K ∂ K

Discretization of the phase mass fraction balance ∂ t ρ y + div ( ρ u y ) = −∇ · ( ρ y (1 − y ) u r ) + ∇ · ( D ∇ y ) , (ff) � (mass) � + implicit time discretization � ◮ K � ρ n +1 y n +1 − ρ n y n � ( ρ n +1 y n +1 u n +1 · n K ) = + δ t K ∂ K � � − ρ y (1 − y ) u r · n K ) + D ∇ y · n K . ∂ K ∂ K ◮ discretization of the left hand side: | K | � ( ρ n + 1 y n +1 − ρ n K y n F n + 1 K ,σ y n +1 K ) + K σ K δ t σ ∈E ( K ) upwind approximation of y n +1 at the face σ with respect to F n +1 y n +1 K ,σ . σ � ◮ monotone numerical flux for ∂σ ρ y (1 − y ) u r · n K ) � ◮ two point flux approximation for ∂σ D ∇ y · n K .

Discretization of the phase mass fraction balance ∂ t ρ y + div ( ρ u y ) = −∇ · ( ρ y (1 − y ) u r ) + ∇ · ( D ∇ y ) , (ff) � (mass) � + implicit time discretization � ◮ K � ρ n +1 y n +1 − ρ n y n � ( ρ n +1 y n +1 u n +1 · n K ) = + δ t K ∂ K � � − ρ y (1 − y ) u r · n K ) + D ∇ y · n K . ∂ K ∂ K ◮ discretization of the left hand side: | K | � ( ρ n + 1 y n +1 − ρ n K y n F n + 1 K ,σ y n +1 K ) + K σ K δ t σ ∈E ( K ) upwind approximation of y n +1 at the face σ with respect to F n +1 y n +1 K ,σ . σ � ◮ monotone numerical flux for ∂σ ρ y (1 − y ) u r · n K ) � ◮ two point flux approximation for ∂σ D ∇ y · n K . � : 0 < y n + 1 < 1 if (0 < y n < 1). Existence and uniqueness of the solution y n +1 for a given y n (Gastaldo-H.-Latch´ e 2009)

FV-FE discretization of the momentum equation ∂ t ( ρ u ) + div ( ρ u ⊗ u ) + ∇ p − div ( τ ( u )) = 0

FV-FE discretization of the momentum equation ∂ t ( ρ u ) + div ( ρ u ⊗ u ) + ∇ p − div ( τ ( u )) = 0 � (momentum) � + implicit time discretization � ◮ D σ ρ n +1 u n +1 − ρ n u n +1 � � ( ρ n +1 u n +1 ⊗ u n +1 · n K ) + δ t D σ ∂ D σ � ( ∇ p n +1 − div ( τ ( u n +1 ))) = 0 . + D σ