Numerical algorithm for two-phase flows drift-flux modelin a porous media on a staggered grid

(1) CEA PARIS-SACLAY

(2) PARIS 13 UNIVERSITY, SUPGALILÉE

Anouar MEKKAS 1

Sami KOURAICHI 2

anouar.mekkas@cea.frsami.kouraichi@math.univ-paris13.fr

Context

FLICA4 is a 3D compressible code specially devoted to reactor core analysis which solves a com-pressible drift-flux model for two-phase flows in a porous medium [1]. To define convective fluxes,FLICA4 uses a specific finite volume numerical method based on an extension of the Roe’s approx-imate Riemann colocated solver [2]. Nevertheless, an analysis of this type of method shows that inlow-Mach number, it is necessary to apply modifications to the 2D or 3D geometries on a cartesianmesh otherwise this method does not converge to the right solution when the mach number tendsto zero [3]. For this reason, we apply a so-called “pressure correction“. Although this correction isnecessary to reach the required precision, it may produces some checkerboard oscillations in space,especially in the 1D case.

Since these checkerboard oscillations are sometimes critical and may lead to unstable resolutionsor even divergence in some cases, we also investigate another numerical algorithm to solve this com-pressible drift-flux model in the low Mach regim. The key point is to develope a compressible solveron staggered grid since checkerboard oscillations cannot exist on this type of discretisation. The aimof this work is to present such a compressible scheme and to validate it in low Mach regime withtest cases describing a simplified nuclear core.

Four equation Drift-flux model in porous medium

The four equations model (M) is established from the six equations equations model in porousmedium [1]. A Drift-flux model is used to take into account the slip between the vapor and the liquidphase.

(M)

φ∂ρ

∂t+∇ · (φρ

−→V ) = 0

φ∂(ρC)

∂t+∇ · (φρC

−→V ) +∇ · (φρC(1− C)

−→V r)−∇ · (φKcv∇C) = φ(Γlv + Γwv)

φρ

(∂−→V

∂t+−→V ∇−→V

)+∇ ·

(φρC(1− C)

−→Vr ⊗

−→Vr

)+ φ∇P = ∇ · (¯̄σ) + φ(τs + τw) + φρ−→g

φ∂(ρe)

∂t+∇ ·

(φρe−→V)

+ P∇ ·(φ−→V)

+∇ ·(φρC(1− C) (Hv −Hl)

−→Vr

)= φQ +∇ · (q)

â Unknows:

•−→V : mixture velocity• C = αvρv

ρ : concentration• P : mixture pressure• e or h: internal mixture energy or specific

mixture enthalpy

â Given quantities:

• φ: porosity• −→g : gravity• Q: power density•K (τs): singular pressure loss coefficient

â Simplifying hypothesis (neglected terms):• Γwv: vapor production at the heating surfaces,• τw: friction between the wall and the fluid,•Kcv: phase mass diffusion term due to two-phase flow turbulence,• ¯̄σ: viscous stress tensor and turbulence effects modeling,• q: fluid heat conduction and energy turbulence diffusion terms.

â Closure models:• Equation of state: Stiffened gas EOS [5] ⇒ E(P, ρ, h ou e) = 0,• Non-equilibrium thermodynamics: the vapor phase is assumed to be saturated ⇒ hv(ρv, P ) =

hsatv (P ),• Non-equilibrium Kinematics: the relative velocity between liquid and vapor phases is taken into

account by a kinetic constitutive equation ⇒−→Vr =

−→V v −

−→V l =

−→Vr(−→V , h, C, P ). Here, we choose Ishii

drift-model [4].• Interphase mass exchange Γlv: we choose F3 model [1].

â Boundary conditions:• Inlet: concentration, velocity and specific enthalpy,• Outlet: pressure.

Time-Space discretization

STAGGERED GRID

A principal grid is used for scalar transport equations (mass, internal energy, vapor mass). Theso-called scalar variables are defined at the center of every mesh cell: pressure, internal energies,concentration. Three other grids are used, one for each velocity component. Velocity nodes arelocated on the center of the faces normal to the velocity component.

X ∈ {x, y, z}VMk

X= (φ∆x∆y∆z)Mk

Xσ: common interface betweentwo neighboring cellsSσ: surface of σ

In the equations of mixture momentum, we use the centered value for the evaluation of scalar vari-ables to the faces. To ensure the stability of the numerical scheme, we use a donor (upwind) schemeapproximation in each convection terms for each balance equations.

â MIXTURE MASS F 1:ρn+1

MKc−ρn

MKc

∆t + 1VMKc

∑σ∈εK F n+1

σ = 0

F n+1σ = Sσρnσ~V n+1

σ .~nσ

â VAPOR MASS F 2:(ρC)n+1

MKc−(ρC)n

MKc

∆t + 1VMKc

∑σ∈εK

[F n+1

σ + G n+1σ

]− (Γlv)

n+1MKc

= 0

F n+1σ = Sσ[ρC]nσ

~V n+1σ .~nσ G n+1

σ = Sσ[ρrCr(1− Cr)]nσ(~Vr)n+1σ .~nσ

â MIXTURE INTERNAL ENERGY F 3:(ρe)n+1

MKc−(ρe)n

MKc

∆t + 1VMKc

∑σ∈εK

[F n+1

σ + P n+1MKc

G n+1σ + H n+1

σ

]−Qn

MKc

= 0

F n+1σ = Sσ[ρe]nσ~V

n+1σ .~nσ G n+1

σ = Sσ~V n+1σ .~nσ H n+1

σ = Sσ[ρrCr(1− Cr)(Hv −Hl)]nσ(~Vr)

n+1σ .~nσ

â MIXTURE MOMENTUM F 4, F 5 and F 6: ρnMKX

(V X )n+1

MKX−(V X )n

MKX

∆t + 1VMKX

∑σ∈εK

[F n+1

σ − (V X )n+1MKXG n+1σ + H n+1

σ

]+[

SM+LxcP n+1

M+LXc− SMK

cP n+1MKc

]− (ρgX )n

MKX

+ KX

2 ρnMKX

(V X )n+1MKX|(V X )n+1

MKX| = 0

F n+1σ = Sσρnσ(V X )n+1

σ~V n+1σ .~nσ G n+1

σ = Sσρnσ~V n+1σ .~nσ H n+1

σ = Sσ(ρrCr(1− Cr))nσ(V Xr )n+1σ (~Vr)

n+1σ .~nσ

Solution algorithm

Let (S) denote the non linear system we ought to solve at each physical time step. This system islinearized using a Newton-Raphson iterative method. This method gives a linear system of equationsfor the increments of the principal variables U = (P, h, C, ~V )t:

(S)

F 1(P, h, ~V ) = 0

F 3(P, h, ~V ) = 0

F 2(P, h, C, ~V ) = 0

F 4(P, ~V ) = 0

F 5(P, ~V ) = 0

F 6(P, ~V ) = 0

⇒

∂F 1

∂P

∂F 1

∂h

∂F 1

∂C

∂F 1

∂V x

∂F 1

∂V y

∂F 1

∂V z

∂F 2

∂P

∂F 2

∂h

∂F 2

∂C

∂F 2

∂V x

∂F 2

∂V y

∂F 2

∂V z

∂F 3

∂P

∂F 3

∂h

∂F 3

∂C

∂F 3

∂V x

∂F 3

∂V y

∂F 3

∂V z

∂F 4

∂P

∂F 4

∂h

∂F 4

∂C

∂F 4

∂V x

∂F 4

∂V y

∂F 4

∂V z

∂F 5

∂P

∂F 5

∂h

∂F 5

∂C

∂F 5

∂V x

∂F 5

∂V y

∂F 5

∂V z

∂F 6

∂P

∂F 6

∂h

∂F 6

∂C

∂F 6

∂V x

∂F 6

∂V y

∂F 6

∂V z

︸ ︷︷ ︸A(Uk) : jacobian of nonlinear system (S)

∆U1

∆U2

∆U3

∆U4

∆U5

∆U6

︸ ︷︷ ︸

∆Uk+1 : increments

=

S1 = −F 1

S2 = −F 2

S3 = −F 3

S4 = −F 4

S5 = −F 5

S6 = −F 6

︸ ︷︷ ︸S(Un,Uk) : residuals

.

• Step 1: Eliminating the velocity incrementsWe consider vector (momentum) balance equations of system (S):

â F 4(P n+1MK

c, P n+1

M +Lxc

, (V x)n+1MK

x) = 0

â F 5(P n+1MK

c, P n+1

M +Lyc

, (V y)n+1MK

y) = 0

â F 6(P n+1MK

c, P n+1

M +Lzc

, (V z)n+1MK

z) = 0

⇒

∂F 4

∂P 0 0 ∂F 4

∂V x 0 0

∂F 5

∂P 0 0 0 ∂F 5

∂V y 0

∂F 6

∂P 0 0 0 0 ∂F 6

∂V z

∆U1

∆U2

∆U3

∆U4

∆U5

∆U6

=

S4

S5

S6

⇒

(∆V x)MKx

= 1∂F4MKx

∂V xMKx

[S4MKx−

∂F 4MKx

∂PMKc

(∆P )MKc−

∂F 4MKx

∂PM+Lxc

(∆P )M+Lxc

]

(∆V y)MKy

= 1∂F5MKy

∂Vy

MKy

[S5MKy−

∂F 5MKy

∂PMKc

(∆P )MKc−

∂F 5MKy

∂PM

+Lyc

(∆P )M

+Lyc

]

(∆V z)MKy

= 1∂F6MKy

∂V zMKy

[S6MKy−

∂F 6MKy

∂PMKc

(∆P )MKc−

∂F 6MKy

∂PM+Lzc

(∆P )M+Lzc

]

(1)

By analogy, we obtain (∆V x)M−Lxx, (∆V y)

M−Lyy

et (∆V z)M−Lzz.

• Step 2: Pressure solverWe consider scalar balance equations of system (S):

â F 1(P n+1MK

c, hn+1

MKc, ~V n+1

σ ) = 0

â F 2(P n+1MK

c, hn+1

MKc, Cn+1

MKc, ~V n+1

σ ) = 0

â F 3(P n+1MK

c, hn+1

MKc, ~V n+1

σ ) = 0

⇒

∂F 1

∂P∂F 1

∂h 0 ∂F 1

∂V x∂F 1

∂V y∂F 1

∂V z

∂F 2

∂P∂F 2

∂h∂F 2

∂C∂F 2

∂V x∂F 2

∂V y∂F 2

∂V z

∂F 3

∂P∂F 3

∂h 0 ∂F 3

∂V x∂F 3

∂V y∂F 3

∂V z

∆U1

∆U2

∆U3

∆U4

∆U5

∆U6

=

S1

S2

S3

(2)

At this stage applying the operations (L1)←− (L1)× ∂F 2

∂h − (L3)× ∂F 1

∂h and then (L2)↔ (L3) on the system(2) results in: J1,1 0 0 J1,4 J1,5 J1,6

∂F 3

∂P∂F 3

∂h 0 ∂F 3

∂V x∂F 3

∂V y∂F 3

∂V z∂F 2

∂P∂F 2

∂h∂F 2

∂C∂F 2

∂V x∂F 2

∂V y∂F 2

∂V z

∆U1

∆U2

∆U3

∆U4

∆U5

∆U6

=

D1 = S1∂F 2

∂h − S2∂F 1

∂hS3

S2

(3)

We develop the first row of the system (3), we get the following equation:

(J1,1)MKc

(∆P )MKc

+ (J1,4)MKx

(∆V x)MKx

+ (J1,4)M−Lxx(∆V x)M−Lxx

+ (J1,5)MKy

(∆V y)MKy

+ (J1,5)M−Lyy

(∆V y)M−Lyy

+(J1,6)MKz

(∆V z)MKz

+ (J1,6)M−Lzz(∆V z)M−Lzz

= D1MKc

(4)

We make use of the velocity increments calculated in the last step. Their integration in the equation(4) gives us:

A(∆P )MKc

+ B(∆P )M+Lxc

+ C(∆P )M−Lxc+ D(∆P )

M+Lyc

+ E(∆P )M−Lyc

+F(∆P )M+Lzc

+ G(∆P )M−Lzc= S (5)

• Step 3: Other incrementsâ Velocity increments As a solution of the pressure equation (5), the pressure increments will be

used to compute the velocity increments (step 1).

â Enthalpy increments To compute the enthalpy increments we need is to develop the second rowof the system (3).

â Concentration increments The same method as above is applied to calculate the concentrationincrements that can be computed using the third row of the system (3).

Numerical tests

The physical quantities that we use in these tests match the functioning of the Pressurized WaterReactors. We consider a 4.2 meter long channel heated by a uniform thermal flux Q = 1.E8 w/m3 onwhich we impose the following conditions: inlet concentration Ci = 0, inlet enthalpies hl = 1.3E6 J/kgand hv = 2.6E6 J/kg, inlet velocities ul = uv = 1.0 m/s and outlet pressure Ps = 155 bar. In the figurebelow, we compare the results of homogeneous and Ishii models to an analytical solutions obtainedwith low Mach mixture model [5].

References

[1] A. Bergeron and Ph. Fillion and D. Gallo and E. Royer, FLICA4 v1.8. Modèles physiques, Note Technique CEA 2005, SFME/LETR/RT/02-005/A.

[2] S. Clerc and Ph. Fillion, FLICA4 v1.8. Méthode numérique, Note Technique CEA 2002, SFME/LETR/RT/02-005/A.

[3] Stéphane Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, J. Comp, Phys.,4(229):978-1016, 2010.

[4] ISHII M., One-dimensionnal Drift-Flux Model and constitutive Equations for Relative Motion between Phases in Various Two-Phase Flow Regimes,Technical Report Argonne National Laboratory, ANL-77-47, 1977.

[5] BERNARD M., DELLACHERIE S., FACCANONI G., GREC B. ET PENEL Y., Study of a low Mach nuclear core model for two-phase flows with phase

transition I: stiffened gaz law, M2AN, 48(6), pp. 1639-1679, 2014.

Low velocity flows: applications to low Mach and low Froude regimes – Paris Descartes University , 5-6 november 2015, FRANCE