a compressible model for low mach two-phase flow with heat and mass exchanges n. grenier, j.p. vila...

A COMPRESSIBLE MODEL FORLOW MACH TWO-PHASE FLOW WITH

HEAT AND MASS EXCHANGES

N. GRENIER, J.P. VILA & Ph. VILLEDIEU

MULTIMAT 2011 - 5-9 septembre 20112

CONTEXT AND MOTIVATION

Context : COMPERE program from CNES & DLR

Research partners : ONERA, ZARM, CNRS, Erlangen university, Air LIquide (Grenoble), Astrium ST (Bremen)

Objectives : development of numerical tools for simulating complex fluid behavior inside space launcher tanks:• dynamical behavior sloshing • thermal effects heat and mass exchanges• low gravity effects capillary effects, Marangoni convection …

Separated two-phase flow with a free

moving interface

Gas phase

Liquid phase

External Heat flux

Evaporation Capillary raise


OUTLINE OF THE PRESENTATION

• 1. Presentation of the model

• 2. Numerical method

• 3. Numerical test cases

• 4. Conclusion






• 4. Conclusion


1. Presentation of the model

Modeling choices • Two fluid model diffuse interface model • Advantage : not necessary to localize (level set method) or reconstruct (VOF method) the interface between the two fluids easy to implement• Drawback : interface diffusion necessary to define a “mixture” physical model and to use low diffusive numerical scheme

• Compressible model • Advantage : more general, easier to implement into a gas dynamics code (ONERA context) • Drawback : ill conditioned for low Mach number flows low Mach Scheme

• Same velocity field for both fluids• Advantage : hyperbolic model, no closure assumption needed• Drawback : impossible to deal with subscale phenomena (subgrid bubbles or droplets …)


To get a close model, it is now necessary to give a relation between the “mixture” pressure p, the bulk densities , and the mixture specific internal energy e.


0

0 (1)

.

ggt

t

pt

EE p

t

v

v

vv v g

v v g v

I

g

Inviscid two-fluid Model

l

with being the mixture total energy per unit volume and

21

2eE v

the mixture bulk density. g l

Gas bulk density

Liquid bulk density



, ,1

(2)

( , ) ( , )1

gg

g lg g l l

T T

T T

p p

e e e

Extension to non isothermal flows

Let T denote the mixture temperature and the gas volume fraction. and T are assumed to be the unique solution of the following system :

Local mechanical equilibrium

Local thermal equilibrium

where p = pg(g,T), p = pl(l,T) denote the gas and liquid EOS and

e=eg(g,T), e=el(l,T) denote the gas and liquid colorific laws.

The mixture EOS is then (implicitly) defined by :

(3) (ρ

,ρρ

,ρ ) ,,1

gggp T p Te p



Other interpretation of closure equations (5)-(6)

,

( , , , ) s ( ,

( , , ) ( ,

s (

,

, ))

)

l l l lg g g g g g

g

l l

l g l E

eE

E

e

V

v

Φ v

Important consequence : System (1) with pressure law given by (2)-(3) is thermodynamically consistent in the sense that it has a convex entropy in the sense of Lax defined as :

where eg, el are the gas and liquid specific internal energies (implicitly defined by the solution of (2)), sg and sl are specific entropies, and are the real densities.

= , =1

g lg l

Closure relations (2)-(3) can also be interpreted as a direct consequence of the following modeling assumption for the mixture Gibbs potential :

( , ) ( , ) ( , ) with , mix

g lg g l l g lg p T y g p T y g p T y y

Ideal mixture assumption



0

0 (4)

( )

. ( ) ( : ) + ( : )

v

v

g

c

c

gt

t

p divt

EE p div div div

t

c

v

v

vv v g

v v g v v

τ

vτφ

τ

τ

I

v div( ) + ( )t τ v I v v

Inclusion of diffusion and capillary effects.

with :

c 2( )T I

τ

Viscous stress tensor

Capillary stress tensor (body force formulation)

T cφ Heat flux



1 dh q dp

0h

h Tt

v

Approximate enthalpy equation for low Mach flows

Neglecting viscous and capillary effects, the energy equation is equivalent to :

which is the Eulerian formulation of the well-known thermodynamic relation :

For low Mach number flow, with imposed pressure on one of the boundaries, one generally has : 1/dp << q, and therefore the energy equation can be replaced by the heat equation :

Heat flux

presssure contribution

+ .h

ht

Tp

pt

vv



( , )(

))

(U

U St

UU U

U

F

Phase change modeling Phase change phenomena can be included in model (7) by just adding a relaxation source term in the r.h.s. :

where (U) is the thermodynamic equilibrium state corresponding to U, defined as the state which maximizes the mixture entropy under the constraints of imposed total volume, total mass and total energy :

1( , ) ( , )

g l

g g l l

g g l l

g g g g l l l ly yy e y ey v

evy v

Max y s e v y s e v

where

1 , ( )

( )e Uv e

U

This idea was first proposed in : HELLUY P., SEGUIN N., “Relaxation model of phase transition flows”, M2AN, Math. Model. Numer. Anal., vol. 40, num. 2, 2006, p. 331–352.

In practice, the thermodynamic equilibrium time scale is assumed to be infinitely small compared to the macroscopic time scale local

thermodynamic equilibrium assumption.






• 4. Conclusion


2. Numerical scheme

A finite volume relaxation scheme

Each time step is divided in two stages :

* 1/ 2 1/ 2 n n nK K e e K

e KK

tU U G m t S

m

Transport step Eulerian finite volume scheme

Numerical flux on edge e

K

Ke

ne,Ke

1 * ( ) nK KU U

Relaxation step local thermodynamic equilibrium

Note that, by construction, the second step is entropy diminishing.


2. Numerical scheme

+ -e e

advection ter

pressure term

m

v v ( , , )

0

0 L

e e

RL R eG U U

p

U U

n

n

Expression of the hyperbolic numerical flux (isothermal case no energy equation)

Which expression choosing for pe and ve ?

Remark : a similar idea has been proposed by Liou (AUSM+up scheme, JCP 2006) and by Li & Gu (all Mach Roe type scheme, JCP 2008) for the compressible gas dynamics system.

Low Mach Scheme

1=

2 e L Rp p p Centered scheme for pressure (see Dellacherie

(2011) recent work on low Mach number schemes)

e e

1v = . -

2 R Le L R p p v v nCentered expression + stabilizing pressure term. Expression of the positive parameter e will be given later.

ne

UL

UR

e


2. Numerical scheme

* * *, , , ,+ -

e e* * *, , , ,

v v n

g K g K g K g K

ene KKl K l K l K l K

tm

m

* * * *, v + v n n n n

K K K K e K K e Ke Ke e e K ee KK

tp m

m

v v v v n

* *e e

1 v = . -

2n n

e K Ke K Kep p v v n

Semi-implicit version of the scheme (isothermal case)

To avoid a restrictive stability condition based on the sound celerity, mass

conservation equations are solved with a implicit scheme.

An explicit scheme is used to compute the new velocity from the momentum equation :

with and * *1 =

2e K Kep p p

Newton algorithm


2. Numerical scheme

(

( ) 0(1') where

) 0 ( )

g

l

t

pt

v v

v

ρρ

ρv

vv I

Formal justification of the stabilizing role of “- (pR - pL) “

Let us consider the following modified system for isothermal flowsModified convective velocity

Remark : the same property holds for the non isothermal case but with the entropy instead of the free energy.

2 2 )1 1

(2 2

.F di pt

pv F

vv v vv

Proposition : the term has a stabilizing effect in the sense for that any smooth solution of (1’) one has the following free energy balance equation :

h p v

where denotes the free energy of the mixture.

( , ) + ( , ) 1 ( , )

g lg l g g l l

g l g l

F f f

Dissipative source term if v

is proportional to – grad(p)


2. Numerical scheme

the semi-implicit scheme is entropic (in the sense of Lax).

* *,

2 2e

e e

KKe

K K K K

t mt mMax

m m

**

2 v 11 v v

e

K KK K e e

eK K K

mean cell velocity

K

CFL like condition

t mwith m

m mSup

Stability theoretical result : Under the two following conditions

(i)

(ii)

In practice, we take : )1

,2

e

e e

nK KKK

e n nK K K K K

t c mt mt mMax Sup cfl

m m m

nK ( v

and

How to choose the value of e ?

with cfl much larger than 1 for low mach number flows.


Where , respectively , denote the gas, respectively the liquid, mass numerical flux.

In practice, two variants of the scheme can be used :• an explicit scheme with respect to the fluid temperature, • a fully implicit scheme with respect to all thermodynamic variables

2. Numerical scheme

* * * * * * * * * *, , , , , , , ,( max( ,0) min( ,0)) ( max( ,0) min( ,0))n n

K K K K g K g e g Ke g e e l K l e l Ke l e ee eK K

t th h h h m h h m

m m

*,Kl

, ,g l T

Discretization of the enthalpy equation

To respect the maximum principle on the temperature, we use the

following upwind scheme based on the sign of the mass fluxes :

*,Kg


2. Numerical scheme

*1 1

1 1 1

* *

* * ( , ) ( , )

n ng l

n n ng g l

g l

lh h p hT p T

1 1 1 1, , ( )= ( )n nsg a

n ntl Tp p T

Relaxation step U* Un+1

If both phases can coexist (gas – liquid thermodynamic equilibrium)

, v and h are left unchanged during this step. We thus have :

System of 3 equations and 3 unknowns

else only one phase can be present in the cell at the end of the time step 1 * *1 1 1, 0 or , 0n n n n

g l l g

Remark : in practice, for numerical purpose, a minimal lower value is imposed for

gas and liquid mass fractions.






• 4. Conclusion


3. Numerical test cases

Linear oscillations in a 2D rectangular tank

•ρ1=1 kg.m-3 ; c1=300 m.s-1

•ρ2=1000 kg.m-3 ; c2=1200 m.s-1

•Transverse acceleration : a0 = 0.01 g•Coarse cartesian grid : 40 X 20•Ma = 2 10-5

Possibility to compute an analytical solution as a série expansion by potential flow theory. (see for example Landau & Lifschitz T6, fluid Mechanics)

2




Second order low Mach scheme Second order Godunov type scheme

Exact solution

Numerical Scheme

Godunov scheme

Low Mach scheme

Time step 0.0005 0.05

Total CPU time

20 1



Dynamical test case : bubble rise inside a liquid : Sussman et al test case

(Sussman, M. and Smereka, P. and Osher, S., A Level Set approach for computing solutions to incompressible two-phase flow, Journal of Computational Physics, 114, 146-159, 1994)

Explicit Godunov type scheme with

real EOS

Cartesian mesh 140 X 233

Semi-implicit low Mach

scheme with real EOS


Explicit Godunov type scheme with

modified EOS




Bubble rise inside a liquid : Sussman et al test case(Sussman, M. and Smereka, P. and Osher, S., A Level Set approach for computing solutions to

incompressible two-phase flow, Journal of Computational Physics, 114, 146-159, 1994)

Sussman et alSolution with Level Set method and incompressible model

Usual Godunov type scheme with real EOS

Semi-implicit low Mach scheme with real EOS



3

TH

Ra g

1708cRa

Rayleigh-Bénard instability

Critical Rayleigh number for instability :

Wall with imposed temperature

Liquid phase

Gas phase

Periodic boundary conditions

g

with 1

pT



Rayleigh-Bénard instability

Stable

Stable

Stable

Unstable Unstable

Unstable



Marangoni convection test case

No gravity. Static contact angle : = 90°

Liquid phase

Adiabatic Wall

Wall with imposed temperature T = T0

Adiabatic Wall

Gas phase

Wall with imposed temperature T = T1<T0



Marangoni convection test case

Volume fraction field

Temperature field

Coarse grid Medium grid Fine grid



1D Evaporation test case

Outlet with imposed pressure : p = p0

Gas phaseWall with imposed heat flux

qw

Evaporation front

=l , p = p0 , T = Tsat(p0), u = uI

=v , p= p0 , u= 0,

T = f(x)

Approximate theoretical solution

v

wqm

L 1 v

l Il

u u

I

v

mu

Liquid phase




3 1000 mkgl 31 v kg m

Interface position vs timefor several values of Lv and qw.






• 4. Conclusion


CONCLUSIONS AND FUTURE PROSPECTS

• An Eulerian two-fluid model with diffuse interface has been applied to the simulation of low Mach separated two-phase flows with heat and mass transfers.

•Using formal arguments, a simple semi-implicit low Mach scheme has been proposed for this model. For isothermal flows, this scheme has been proved to be entropy diminishing under a CFL condition which do not depend on the sound celerity.

• This methodology can be very easily implemented in existing industrial compressible CFD codes for multi-physics applications (work in progress at ONERA). It is a very interesting alternative to classical approaches based on one-fluid incompressible model with VOF or Level Set methods.


CONCLUSIONS AND FUTURE PROSPECTS

•This two-fluid approach has been successfully applied to several academic problems for low Mach two-phase flows.

• Future works will be devoted to the • assessment of the method for more complex phase change problems.• extension of the model to more complex physical problems : multi-component gas phase with an incondensable specie, 3 phases problems …• parallelization of the code for 3D applications


Thank you for your attention …..


Back – up


with the mixture bulk density.


0

(1) 0

ggt

t

pt

v

v

vv v gI

g l

Purely Dynamical model (inviscid Isothermal flow)

To get a close model, it is necessary to give a relation between the “mixture” pressure p and the bulk gas density and the bulk liquid density .

g

Remark: The gas volume fraction is not explicitly transported in this model. l



ρ

1 (3) (ρ , )

ρρ g

g g pp p

Purely dynamical model

(2) 1

ggp p

• Let denote the gas volume fraction :

Local pressure equilibrium between the two non miscible fluids

where p = pg(g) and p = pl(l) denote the gas and liquid equation of state.

; (1 )lg g l

Mixture EOS

Remark : if the expressions of pg and pl are complex, p is only implicitly defined in

function of the bulk densities.

is defined as the solution of



0( , )i

i ii i v

i

pe p T e c T

p

Example : stiffened gas model for both fluids.

Expression of the Gibbs potential for each fluid :

1( , ) ( 1)

( , ) ii i vi i

Tv p T c

p T p

0 0( , ) 1 ln( ) ( 1) ln( )i ii i v i v i i ig p T c T T c T p u Ts

Fluid i Equation of state Fluid i calorific law

With these notations, system (5)-(6) is equivalent to :

1( , ) ( , )

(7)

( , ) ( , )

g lg l

g l g l g l

g lg l

g l g l

v v v

e

p p

p

T T

Te e p T

Mixture specific

volume



Other interpretation of closure equations (5)-(6)

,

( , , , ) ( , )

( , , ) (

s

, , )

s ( , ( , ))g g gl l lg g

g

g

l g

l

l

lleE

E E

e

V

v

Φ v

Important property : System (4) with pressure law given by (5)-(6) is thermodynamically consistent in the sense that it has a infinite set of convex entropies in the sense of Lax defined as :

where is an arbitrary concave function, eg, el are the specific internal energies, implicitly defined by the solution of (5), sg and sl are the specific entropies, and are the real fluid densities. = , =

1g l

g l

Closure relations (5)-(6) can also be interpreted as a direct consequence of the following modeling assumption for the mixture Gibbs potential :

( , ) ( , ) ( , ) - with ,( , )mix gg l

g g lll l gg p T y g p T y g p T y yT y y

Ideal mixture assumption


Proposition : If pg and pl are strictly non decreasing functions, model (1)-(2)-(3) is hyperbolic and has a convex entropy in the sense of Lax defined as :


21( , , ) +

2

( , , , ,

) )

1

(lg

lgg g g

g

l l l

l p

ff

V V

V V V VΦ

d

pf i

i 2

)()(

where fg and fl are the free energy of the gas and liquid phases and are defined as :

Lax entropy (convex function of the conservative variables)

Entropy flux

2i

i i i ii

pdf p d d

Purely dynamical model (3/3)


4. APPLICATIONS


•ρ1=1 kg.m-3 ; c1=300 m.s-1

•ρ2=1000 kg.m-3 ; c2=1200 m.s-1

•Transverse acceleration : a0 = 0.01 g•Coarse cartesian grid : 40 X 20•Ma = 2 10-5

Possibility to compute an analytical solution as a série expansion by potential flow theory. (see for example Landau & Lifschitz T6, fluid Mechanics)

2


4. APPLICATIONS


Second order low Mach scheme Second order Godunov type scheme

Exact solution

Numerical Scheme

Godunov scheme

Low Mach scheme

Time step 0.0005 0.05

Total CPU time

20 1



References • R. ABGRALL, R. SAUREL. A simple method for compressible multifuid flows, SIAM J. Sci. Comput. 21 (3) : 1115-1145, (1999). 66

• G. ALLAIRE, G. FACCANONI et S. KOKH, A strictly hyperbolic equilibrium phase transition model.

C. R. Acad. Sci. Paris Sér. I, 344 pp. 135–140, 2007.

• CARO F., COQUEL F., JAMET D., KOKH S., “A Simple Finite-Volume Method forCompressible Isothermal Two-Phase Flows Simulation”, Int. J. on Finite Volumes, vol. 3,num. 1, 2006, p. 1–37.

• HELLUY P., SEGUIN N., “Relaxation model of phase transition flows”, M2AN, Math. Model. Numer. Anal., vol. 40, num. 2, 2006, p. 331–352.

• LE METAYER O., MASSONI J., SAUREL R., “Elaborating equations of state of a liquidand its vapor for two-phase flow models”, Int. J. of Th. Sci., vol. 43, num. 3, 2004, p. 265–276.

• G. CHANTEPERDRIX, JP VILA, P. VILLEDIEU, A compressible model for separated two-phaseflow computations, FEDSM02, 14-18 July, Montreal, Quebec, Canada, 2002

a compressible model for low mach two-phase flow with heat and mass exchanges n. grenier, j.p. vila...

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