A Two Fluid Model for Two-Phase A Two Fluid Model for Two-Phase - - PowerPoint PPT Presentation

Workshop “Numerical Methods for Multimaterial Compressible Fluid Flows”, Paris, September 23-25, 2002

A Two Fluid Model for Two-Phase A Two Fluid Model for Two-Phase Flows with Free Interface Flows with Free Interface

• G. Chanteperdrix
• P. Villedieu

J.P. Vila

ONERA/CNES ONERA/MIP MIP/ONERA

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WNMMCFF, Paris, September 23-25, 2002

Context and Applications

Fluids behaviour in tanks :

• Sloshing effects due to

transverse accelerations

• Surface tension effects due

to possible gravity reduction and interface curvature

• Thermal effects due to

external heat fluxes

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Main Features of our approach

• Eulerian method on a fixed grid, with no interface reconstruction or

interface tracking, existence of an artificial mixture zone → need of a mixture closure law leading to a well-posed problem (hyperbolic system, thermodynamic consistency) → need of a low diffusive numerical scheme

• able to deal with a wide range of applications relative to fluid

behaviour in a launcher tank (including coupling between thermal and hydrodynamic effects) → compressible two fluid model, eventually with “pseudo” sound speeds, to overcome low Mach number difficulties

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Modeling Hypothesis

• Gas of density ρg , liquid of density ρ
• , supposed to be

compressible and inviscid

• no thermal effects : linearized EOS
• the two fluid “fictitiously” coexist everywhere, presence

characterised by volume fraction αg , α

: αg+α

=1

• only one velocity field

) ( ) ( , ) ( ) (

2 2

ρ ρ ρ ρ ρ ρ − + = − + = c p p c p p

g g g g g

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WNMMCFF, Paris, September 23-25, 2002

The Equilibrium Model

( )

( ) ( )

    

• 

      = + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂

y x g g g

P v y uv x v t uv y P u x u t v y u x t v y u x t E

2 2

~ ~ ~ ~ ~ ~ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρ α ρ αρ ρ ~ ~ , ) 1 ( ~ , ~ : with + = − = =

g g g

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WNMMCFF, Paris, September 23-25, 2002

The Equilibrium Model : closure laws

mixture pressure

Mixture pressure P :

( ) ( )

      − − +         = 1 ~ 1 ~ ~ , ~

* * * *

α ρ α α ρ α ρ ρ

p p P

g g g

( )

ρ ρ α α ~ , ~

* * g

=

      − =         1 ~ ~

* *

α ρ α ρ

p p

g g

where : is the equilibrium gas volume fraction insuring :

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WNMMCFF, Paris, September 23-25, 2002

The Equilibrium Model : closure laws

source terms

Source terms :

• acceleration in the local reference frame

(gravity + transverse acceleration)

• surface tension, CSF way :

(Brackbill et al, JCP 1992) σ the surface tension coefficient κ the interface curvature with

α σκ ∇ = c F ∇ ∇ ∇ ⋅ −∇ = ∇

       

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WNMMCFF, Paris, September 23-25, 2002

The Equilibrium Model : closure laws “conservative” capillary force

joint work with D. Jamet, CEA

( )

c

∇ ∇ + ∇ ⊗ ∇ ∇ ⋅ ∇

       

− = σ σ F

c

∇ ∇ ∇ ⋅ ∇

       

− = σ F

The capillary force : also reads :

→ yields a completely conservative formulation of momentum equation → numerically, ∇α needs only to be computed at the faces center

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• The mixture pressure law leads to an hyperbolic system

with mixture sound speed c* such that :

• Existence of a Lax entropy
• Difficulties with the explicit solution of the Riemann problem

due to the complexity of the mixture pressure law → Relaxation model (in the spirit of Coquel & Perthame, SIAM 1998)

        ∂ ∂ + ∂ ∂ − =       − − +       + =

ρ α ρ ρ α ρ α ρ α ρ ρ ~ ~ ~ ~ with 1 1 ~ 1 ~

* * * 2 * 2 2 * g g g g

K K c K c c

( )

energy kinetic energy free

2 2 2

2 1 ~ ~ , ~ , ~ , u u

d p g d g g p g g

ρ ρ ρ ρ ρ ρ α

ρ ρ ρ ρ

+ + =

∫ ∫ F

The Equilibrium Model :

mathematical properties (without surface tension) ( ) ( )

: with , ~ , ~ , , ~ , ~

*

u u

g g

ρ ρ ρ α ρ ρ ρ

F F *

=

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WNMMCFF, Paris, September 23-25, 2002

The Relaxation Model

      − ) − (1 +         = ,  

• 

   = ∂ ∂ + ∂ ∂ + ∂ ∂ << < − = ∂ ∂ + ∂ ∂ + ∂ ∂ α α α α ε ε α α α

ε

1 ~ ~ ) ~ , ~ ( with ) 1 (0 ) (

,

p p P y x t w p p y v x u t R

g g g g

Remark : the relaxation source term is linked to F by :

        ∂ ∂ − ∂ ∂ − = ∂ ∂ − = −

α α ε α ε ε

F F F

g g

p p 2 1 1

thermodynamic potential

similar models in :

• Saurel & Abgrall (JCP 1999), Saurel & Lemetayer (JFM 2001)
• Dellacherie & Rency (preprint 2001)

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WNMMCFF, Paris, September 23-25, 2002

2 2 2

~ ~

c c c

g g

ρ ρ ρ + =

• Also hyperbolic with mixture sound speed c such that :
• F is an entropy, compatible with the relaxation source term :

the relaxation source term contributes to the decrease in entropy and verifying the sub-characteristic condition : c*

2 ≤ c2

→ good mathematical frame for relaxation

a Chapman-Enskog like expansion of the relaxation model can formally be derived (as in Coquel & Perthame, SIAM 1998)

The Relaxation Model :

mathematical properties (without surface tension)

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Numerical Method

• Finite Volume method :

– RK2 in time (2nd order) – Godunov-MUSCL in space (2nd order)

• Each RK2 stage divided in two parts

– hyperbolic step (exact Riemann solver) – pressure relaxation step

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WNMMCFF, Paris, September 23-25, 2002

Numerical Method :

hyperbolic step

 

• 

   = ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ y x t w y v x u t ρα ρα ρα Given αn and wn, we solve : by to obtain intermediate state ∆t is fixed by a CFL condition

( )

*

,

n

w α

ij j j i i n ij n ij

t y g g x f f t w w

2 / 1 2 / 1 2 / 1 2 / 1 *

∆ +         ∆ − + ∆ − ∆ −         =        

− + − +

ρα ρα

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WNMMCFF, Paris, September 23-25, 2002

Numerical Method : hyperbolic step

numerical scheme

Numerical flux functions f and g based on exact resolution of the associated Riemann problem : Godunov Scheme

u-c u u+c Nature shock

• r

rarefaction contact discontinuity shock

• r

rarefaction

u+c u-c u

x t

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WNMMCFF, Paris, September 23-25, 2002

Numerical Method :

pressure relaxation step

Given , we solve with ε→0 : to obtain αn+1 and wn+1 :

 

• 

   = ∂ ∂ − = ∂ ∂ τ ε τ α w p pg

• nly α change

( )



• 

  = =

+ + * 1 * 2 * 1 * 1

~ , ~

n n n n n

w w ρ ρ α α

no numerical scheme is needed for the volume fraction with this approach

Remark : α* can be explicitly computed for linearized EOS

( )

*

,

n

w α

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Numerical Results : Sloshing in a 2D wedge

high Froude number, high Bond number

mesh size : 150 × 75

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WNMMCFF, Paris, September 23-25, 2002

Numerical Results : Sloshing in a 2D wedge

high Froude number, high Bond number

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WNMMCFF, Paris, September 23-25, 2002

Numerical Results : Sloshing in a 2D wedge

high Froude number, high Bond number

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WNMMCFF, Paris, September 23-25, 2002

Numerical Results : Sloshing in a 2D wedge

high Froude number, high Bond number

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WNMMCFF, Paris, September 23-25, 2002

Physically Relevant Quantities

• Total gas volume :
• Energies :

∫

Ω

= V

V

d

g

α

) (F energy capillary : energy free the

• f

part remaining ) (F energy kinetic ) (F energy free the

• f

part pressure tot

ts ec p

∫ ∫

Ω + Ω

∇ + = V V

F F

d d α σ

This total energy Ftot is decreasing for the relaxation model (up to a slight modification of the pressure closure law, current work with D. Jamet, CEA)

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Numerical Results : square to circular bubble

mesh size : 80×80

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WNMMCFF, Paris, September 23-25, 2002

Numerical Results : square to circular bubble

energy balance

Bubble oscillations in a inviscid fluid (Lamb, 1932), deformation modes frequencies :

3

) 2 )( 1 )( 1 ( 1 R n n n fn

ρ σ π + − + =

n=2 n=4

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Conclusions and Future Developments

• Model and numerical method efficient and accurate for sloshing

effects at high Bond number : – no numerical scheme required for the gas volume fraction – numerical scheme with very low diffusivity compared to similar approaches ⇒ low damping of the interface oscillations – possible extension of the Riemann solver to more complex EOS (local linearization)

• Capillary forces : validation in progress
• Free energy balance with surface tension effect
• Introduction of viscous and thermal effects

( p(ρ,T) , conduction, cavitation, evaporation)