Diffuse interfaces with compressible fluids, phase transition and capillarity

Richard SaurelAix Marseille University, M2P2, CNRS 7340

RS2N SAS, Saint-Zacharie

Aims of the “diffuse interface” approach

- Solve liquid and gas flows with real density jumps at arbitrary speeds, separated by interfaces.

- Consider compressibility of both phases.

- Consider surface tension, heat diffusion and phase transition.

- Address waves propagation (possibly shocks).

Petitpas, Massoni, Saurel and Lapébie, 2009, Int. J. Mult. Flows

Example: Underwater supercavitating missile (400 km/h)2 interfaces are present: liquid-vapor and gas-vapor

• The ‘diffuse interfaces’ under consideration come from numerical

diffusion (not capillary nor viscous regularization),

• These interfaces are captured with an hyperbolic flow solver.

• The numerical diffusion can be reduced (1-2 points – work in progress).

• All fluids are considered compressible with a convex EOS.

• The ‘Noble-Abel-stiffened-gas’ EOS is a good prototype example:

(Le Metayer-Saurel, POF, 2016)

It is both an improved SG formulation and simplified van der

Waals formulation

� well defined sound speed

• The liquid and vapour constants are coupled through the phase diagram

(and saturation curves).

( ) k k kk 1 k k

k k

(e q )p 1 p

1 b ∞ρ −= γ − − γ

− ρk =1,2

Preliminary remarks

k k(p cst. instead of p a ²)∞ ∞= = ρ

How to compute the pressure in this artificial mixture zone?

The EOS are discontinuous with restricted domain of validity.

G

L

L

G

�Mixture cell

Diffuse interfaces of simple contact (with compressible fluids)

Starting point of the diffuse interfaces method:Consider mixture cells through a total non-equilibrium two-phase model with 2 velocities, 2 temperatures, 2 pressures

)pp(x

ut

211

i1 −µ=

∂∂α+

∂∂α

0x

u

t11111 =

∂ρ∂α

+∂

ρ∂α

)uu(x

px

)p²u(

t

u12

1i

11111111 −λ+∂

∂α=∂

α+ρα∂+∂ρ∂α

)uu(u)pp(p x

upx

)pE(u

t

E12

'

i21'i

kii

111111111 −λ+−µ−∂

∂α=∂

α+ρα∂+∂ρ∂α

Each phase evolves in its own sub-volume.

The pressure equilibrium condition is replaced by a PDE with relaxation � hyperbolicity is preserved.

3 symmetric equations are used for the second phase.

Fulfill interface conditions of equal pressures and equal velocities � stiff relaxation

ε−<α<ε 1k

‘bubbles’ expansion or contraction forces mechanical equilibrium locally:

)pp(x

ut

211

i1 −µ=

∂∂α+

∂∂α

Similar relaxation is done for velocities (Saurel and Abgrall, JCP, 1999)

This trick enables fulfilment of interface conditions.

µ tends to infinity

In other words…

Solve this ODE system in the limit

This ODE system is replaced by two algebraic systems.

)pp(t

211 −µ=

∂∂α

0t

11 =∂

ρ∂α

)uu(t

u12

111 −λ=∂ρ∂α

)uu(u)pp(p t

E12

'

i21

'

i111 −λ+−µ−=

∂ρ∂α

This method deals with 2 steps:

-Solve the 7 equations model without relaxation terms during a

time step � 2 velocities and 2 pressures are computed in the mixture zone.

-Relax (reset) the 2 velocities and pressures to their equilibrium values:

+∞→µ+∞→λ

Example with velocity relaxation

**2

*1 uuu ==

We have to solve:

At relaxed state

0t

22 =∂

ρ∂α

0t

11 =∂

ρ∂α

)uu(t

u12

111 −λ=∂ρ∂α

)uu(t

u12

222 −λ−=∂ρ∂α

Summing the two momentum equations results in:

( )0

t

uu 222111 =∂

ρα+ρα∂

Integrating this equation between the end of the propagation step and the relaxed step results in:

( ) ( )0

222111

*

222111 uuuu ρα+ρα=ρα+ρα

Thus: ( )( )

( )( )0

2211

0

222111*

2211

0

222111* uuuuu

ρα+ραρα+ρα=

ρα+ραρα+ρα=

The same type of procedure is done for pressure relaxation.

t

u

Computational example: Impact of copper projectile at 5 km/s on a copper tank filled with water, surrounded by air.

Very robust but quite complicated if interface only are under consideration.

Helpful when velocity disequilibrium effects are under consideration.

Reduced model with single pressure and single velocity but two temperatures

+∞→ε=µλ 1,

• Faster computations

• Add physical effects easily (surface tension, phase transition)…

1o fff ε+=

Relaxation coefficients are assumed stiff.

Each flow variable evolves with small perturbations around an equilibrium state.

( )x

u

cc

cc)pp(

2

222

1

211

2

11

2

2221 ∂

∂

αρ

+α

ρρ−ρ

→−µ

Consequence

The pressure relaxation term becomes a differential one:

)pp(x

ut

211

i1 −µ=

∂∂α+

∂∂α

Kapila et al., POF, 2001

2211 ρα+ρα=ρ

( ) 1 1 1 1 1 2 2 2 2 21 1 2 2

1 21 1

1 1 1 2 2 2

1 2

(1 b ) (1 b )e (Y q Y q )

,Y(1 b ) (1 b )

p p1 1

p p( ,e, )

1 1

∞ ∞

−ρ −ρ− +

−ρ −ρ

α γ α γρ − +γ − γ −= ρ α = α α+

γ − γ −

x

u

cc

)cc(

xu

t

2

2

22

1

2

11

2

11

2

2211

∂∂

αρ+

αρ

ρ−ρ=

∂∂α

+∂

∂α

0x

u

t1111 =

∂ρ∂α

+∂

ρ∂α

0x

)p²u(tu =∂

+ρ∂+∂∂ρ

0x

)pE(utE =∂

+ρ∂+∂∂ρ

0x

u

t2222 =

∂ρ∂α

+∂

ρ∂α

A mixture EOS is obtained from the mixture energy definition and pressure equilibrium:

Two-phase shock relations are given in Saurel et al., Shock Waves, 2007.

1 1 1 2 2 2E E Eρ = α ρ + α ρ

1 22 2

1 1 2 2

1c² c c

+ρ ρ ρ

α α=

2 22 2 1 1

2 21 1 2 2

1 2

(ρ c ρ c )αu. α div(u) 0

ρ c ρ ctα α

ρYdiv( Yu) 0

tρ

div(ρu) 0t

ρu Y Ydiv PI ρu u σ Y I 0

t Y

σ Y 1ρ(e u²)

σ Y 1 Y Yρ 2div (ρ(e u²) P)u σ Y I

t ρ 2 Y

−∂ + ∇ − =∂ +

∂ + ρ =∂

∂ + =∂

∂ ∇ ⊗∇ + + ⊗ − ∇ − = ∂ ∇

∇∂ + + ∇ ∇ ⊗∇

+ + + + − ∇ −∂ ∇

��

�

�

� ��

�� �

�

�

��

� �

�� �

� .u 0

=

�

Capillary effects can be added (Perigaud and Saurel, JCP, 2005)

- Compressibilty is considered as well as surface tension.

- Thermodynamics and capillarity are decoupled: no need to enlarge the interface.

- Two Gibbs identities are used: and

- There is no capillary lenght to resolve: Jump conditions deal with capillarity as they deal with shocks (genuine property of the conservative formulation),

k k k kde T ds pdv= − de dSσ =σ

Falling droplet from the roof

(computations done in 2004)

Qualitative comparisons (2D computations) Quantitative comparisons (2D axi)

Phase transition

Examination of the mixture entropy production suggests (Saurel et al., JFM, 2008)

( ) ( )

( ) ( )122222

121111

ggudivt

ggudivt

−ρν−=ρα+∂

ρα∂

−ρν=ρα+∂

ρα∂

�

�

2

2

22

1

2

11

2

2

1

1

12

2

2

22

1

2

11

2

22

1

21

12

2

2

22

1

2

11

2

11

2

221

1

cc)TT(H

cc

cc

)gg()u(divcc

)cc()(grad.u

t

αρ+

αρ

αΓ

+αΓ

−+

αρ+

αρ

α+

α−ρν+

αρ+

αρ

ρ−ρ=α+

∂∂α ��

Pressure relaxation Phase change Heat exchange

( )

( ) 0)upE(divt

E

0)p(graduudivt

u

=+ρ+∂ρ∂

=+⊗ρ+∂ρ∂

Phase transition is modeled as a Gibbs free energy relaxation process.

But two kinetic parameters are now present.

The model involves two entropies

02 <c

Phase transition connects the two isentropes through a kinetic path (mass source terms).Sound speed is defined everywhere:

• Very different of the van der Waals (and variants) approaches

With VdW phase transition is modeled as a kinetic path …. and sound speed is undefined

� Ill posed model.

Mixture

Mixture

cstes =1

cstes =2

2 2

s

pc v 0

v

∂ =− <∂

liquid

liquid

vapor

vapor

1 22 2

1 1 2 2

1c² c c

+ρ ρ ρ

α α=

Thermodynamic closureEach fluid is governed by the NASG EOS (Le Metayer and Saurel, Phys.

Fluids, 2016)

With the help of Gibbs identity and Maxwell relations:

Vh(P,T) C T bP q=γ + +

v V 1

Tg(P,T) ( C q')T C Tln bP q

(p p )

γ

γ−∞

= γ − − + ++

V

P pe(P,T) C T q

P p∞

∞

+ γ= ++

( )�

agitation

k k kk k k k k k

k k attraction

shortdistancesrepulsion

(e q )p ( ,e ) 1 p

1 b ∞ρ −ρ = γ − − γ

− ρ

�����

����

V( 1)C Tv(P,T) b

P P∞

γ −= ++

Each fluid EOS requires 6 parameters:

V,P ,C ,b,q,q'∞γ

EOS parameters

k,VkCγ

k,Vk, C,P∞

)T(h\\ sat,k

)T(v\\ sat,k

kh (P,T)

The liquid and gas saturation curves are used

v(P,T)

VL (T) )T(L\\ sat,V lg q,q

lglglg gg,pp,TT ===,l

Bln(p) A ClnT Dln(p p )

T ∞= + + + + )T(P\\ sat lg 'q,'q

Psat

hl

vl

Lv

hg

vg

Data for water

Kinetic parameters ?

The model requires two relaxation parameters H and ν (heat exchange and evaporation kinetics).

Evaporation front observations in expansion tubes (Simoes Moreira and

Shepherd, JFM, 1999)

Existence of 4 waves: left expansion, evaporation front, contact discontinuity, right shock wave.

Local thermodynamic equilibrium is assumed at interfaces

� Thermodynamic equilibrium is forced locally.

� Waves propagate on both sides of the front and produce metastable states that relax to equilibrium at interfaces only.

� When heat diffusion is present metastable states result of heat transfer as well.

ε−<α<ε∞+

=ν otherwise 0

1 ifH,

k

εαε −<< 1k

Mixture cellSame idea as with contact interfaces:

Stiff thermodynamic relaxation

g g g Le Y e (T,P) (1 Y )e (T,P)= + −

The asymptotic solution when H and ν tend to infinity corresponds to the solution of the algebraic system:

g g g Lv Y v (T,P) (1 Y )v (T,P)= + −

g LT T T= =

g LP P P= =

g L satg (T,P) g (T,P) T T (P)= ⇔ =

Unknowns (Psat,Yg) � equilibriumgY

Non-linear system quite delicate to solve in particular when the states range from two-phase to single phase : Allaire, Faccanoni and Kokh, CRAS 2007

Le Metayer, Massoni and Saurel, ESAIM, 2013

g g sat g L sate Y e (P ) (1 Y )e (P )= + −

g g sat g L satv Y v (P ) (1 Y )v (P )= + −

Simple and fast equilibrium solver (Chiapolino, Boivin, Saurel, IJNMF, 2016)

eql sat lIf (Y and T T )then Y (overheated vapor)≤ ε > = ε

Single phase bounds:

eql sat lIf (Y 1 and T T )then Y 1 (subcooled liquid)≥ − ε < = − ε

Otherwise,

Compute:

eqlY 1ε < < − ε

from the mixture mass definition,

from the mixture energy definition.

These two estimates are equal only when p=psatwhich is unknown.

Minmod

Then compute the product

If ( ) then,Take the minimum variation

Otherwise,Set the variation to zero

Update the mass fraction at equilibrium:

m ml l lY Y Yδ = −

e el l lY Y Yδ = −

compute the variations

m el lY . Yδ δ

m el lY . Y 0δ δ >

m el l lY Min( Y , Y )δ = δ δ

lY 0δ =

equl l lY Y Y= + δ

Example: Shock tube computations with two liquid-gas mixtures and phase change

Symbols: Equilibrium computed with the Newton method.Lines: Equilibrium computed with MinmodDotted lines: Solution in temperature and pressure equilibrium without phase transition.

Liquid

ρliquid = 500 kg/m3

pliquid = 1000 Bar

Vapor

ρvapor = 2 kg/m3

pvapor = 1 Barx

Contact discontinuity

shock

Evaporation front

expansion

29

Back to the Simoes-Moreira Shepherd experiment

4 waves appear

Flashing front in an expansion tube (Moreira and Shepherd, JFM, 1999)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

180 200 220 240 260 280 300Température de surchauffe (°C)

Pression derrière le front (Bar)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

180 200 220 240 260 280 300Température de surchauffe (°C)

Vitesse du front de transition (m/s)Phase transition front speed (m/s) Pressure jump across the front (atm)

Experiments: blue

Computations: redSimilar results were obtained by Zein, Hantke and Warnecke, JCP, 2010Saurel et al., JFM, 2008

Reduced model when heat diffusion is

considered � single temperature

31

==

21

21

TT

gg

Tqcond ∇λ−=�

�

q

g

σ

Example: boiling flows

32

Reduced model in both mechanical and thermal

equilibrium

( )

( )

2 111

(g g )Ydiv Y u

0t

div u 0t

ρν −∂ρ + ρ = ∂

∂ρ + ρ =∂

�

�

( )

( )( )

udiv u u PI 0

tE

div E P u 0t

∂ρ + ρ ⊗ + =∂

∂ρ + ρ + =∂

�

� �

�

The phases evolve with same pressure and same temperature but have

different chemical potentials and evolve in their own subvolumes (very

different of the Dalton law).

Conservative system ( ), comfortable for numerical

resolution.

1 2

1 2

1 2

1 2

u u

p p

T T

g g

===≠

33

U F0

t x

∂ ∂+ =∂ ∂

The model is reminiscent of the reactive Euler equations

but the EOS is very different

( )( ) ( )( )2

1 2 ,1 ,2 2 1 ,2 ,1 1 2

1 1P A A P P A A P P A A

2 4∞ ∞ ∞ ∞= + − + + − − − +

( )( ) ( ) ( )( )k k vk

k ,k1 1 v1 2 2 v,2

Y 1 CA e q P

Y 1 C Y 1 C ∞

γ −= ρ − −

γ − + γ −

34

1 2

1 1 2 2e Y e (T,p) Y e (T,p)= +

1 1 2 2v Y v (T,p) Y v (T,p)= +

1 2

1 2

p p p

T T T

= == =

The sound speed in lower than the previous one

T P W o o dc c≤

35

∑ ρα

=ρ k

2kk

k2w cc

1

Inserting heat diffusion and surface tension

( )11 2 1

Ydiv Y u (g g )

t

∂ρ + ρ = ρν −∂

�

( ) 1 1 2 2

u m mdiv u u PI m I g

t m

E m m mdiv E P m u m I u q q g u

t m

∂ρ ⊗+ ρ ⊗ + − σ − = ρ ∂

∂ρ + σ ⊗+ ρ + + σ − σ − • + α + α = ρ • ∂

��

� �

�� � �

( )div u 0t

∂ρ + ρ =∂

�

36

1m Y= ∇�

Le Martelot, Saurel, Nkonga, IJMF, 2014

11vap

11liq

2

1

K.m.W025.0

K.m.W68.0

s.m81.9g

m.mN73

−−

−−

−

−

=λ

=λ==σ

37

Computations done with an implicit low Mach hyperbolic solver.

Liquid initially at saturation temperature.

uniform heat flux

11vap

11liq

2

1

K.m.W025.0

K.m.W68.0

s.m81.9g

m.mN73

−−

−−

−

−

=λ

=λ==σ

38

Same computations without nucleation sites.

Another example: Atomization of O2 liquid jet surrounded by H2 vapour

40

Thank you!

Ongoing work:- Extend the Minmod method for mass transfer to the presence of non condensable

gas (liquid suspended in air for example): Chiapolino, Boivin, Saurel, Comp & Fluids, submitted

- Sharpen diffuse interfaces on unstructured meshes (Chiapolino and Saurel, SIAM, in preparation)

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