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Fine-Scale Modeling Approaches for Two-Phase Flow

Sanjoy BanerjeeCUNY Distinguished Professor of Chemical

EngineeringDirector, The CUNY Energy Institute

City College New York,The City University of New York

email: banerjee@che.ccny.cuny.eduCollaborators: Frederic Gibou, Jean-Christophe

Nave, Vittorio Badalassi, Djamel Lakehal

Keynote Lecture, Grenoble, France

September 11, 2008

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Main Topics

1. Approaches to fine-scale modeling

2. Interface tracking methods

3. Phase field simulations

4. Level-set and ghost fluid

5. Boundary fitting DNS: physical insights

6. Conclusions

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Interface Resolving Methods

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Interface Tracking Approaches

1. Phase Field

2. Boundary fitting: low steepness waves

3. Implicit interface tracking:

4. Level set/ghost fluid

5. VOF/MARS

6. Explicit tracking/Lagrangian

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Why a phase-field approach ?

Diffuse interface approach facilitates numerical convergence

Equations of motion for material rather than boundary

Simulations to realistic length and time scales

Simple – order parameter distinguishes between fluid material

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Semi-Implicit Procedure (Majorization)

Semi-implicit method using a majorization technique (example for Euler method):

If the method is unconditionally stable. For the nonlinear term in we write

and let

Result: we remove the fourth and second order time step restrictions!

12 1 21n n

n n n n n nC Cv C a a

t Peµ χ µ µ

++−

+ ⋅∇ = ∇ + ∇ ⋅ ∇ − ∇ ∆r

1 max2

a χ≥

( ) ( )2 C Cψ ψ′ ′′∇ = ∇ ⋅ ∇( )' Cψ

( )( ) ( )1 1max 1 12 2

a Cψ ψ′′ ′′= = ± =( ) 2' 0C Cµ βψ α= − ∇ =

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Model Structure

( ) ( )( ) 1TvRe v v p v v C

t Caϑ µ∂ + ⋅∇ = −∇ + ∇ ⋅ ∇ + ∇ + ∇ ∂

rr r r r r rr r r r

1Cv C

t Peλ µ∂ + ⋅∇ = ∇ ⋅ ∇

∂r

3 2C C Cµ = − − ∇

2

2, ,

3

c c c c

c

U U U URe Pe Ca

M

ρ ξ ξ αη ηη β β ξ σ

= = = =

is the chemical potential

Order parameter C (conserved form): Model H

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Free Energy: Landau-Ginzburg Form

Free energy A[C]:

At equilibrium the functional A[C] will be a minimum with respect to variations of the function C:

Surface tension , Interface thickness

is the homogeneous free energy (e.g. double well pot.)

is the gradient free energy

[ ] ( ) 21

2A C C Cβψ α

Ω

= + ∇∫

( ) 2' 0A

C CC

δµ βψ αδ

= = − ∇ =

ψσ αβ∝ αξ β∝

2C∇

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Double-well Potential and Interface Thickness

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Falling Drop on Free Surface

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Rayleigh Taylor Instability

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Rayleigh Taylor Instability (Cont’d)

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Drop Coalescence

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Phase Separation: Density Mismatch

Critical (50/50) Not Critical (20/80)

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Self Assembled Nanostructure

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Phase Diagram/Structure

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Model Structure: Level Set

( ) ( )( ) ( )

( ) ε

εεε

επε

µµηηηµρρλλλρ

>Φ<Φ<−

<ΦΦ+Φ+

=

=Φ−+=Φ=Φ−+=Φ

sin1

1

21

0

e wher)1(

where)1(

21

21

H

H

H

( ) ( ) ( ) ( ) ( )

( ) Φ∇Φ∇=⋅∇=Φ

∂Φ∂ΦΦ−Φ+

∂∂

+∂∂−=Φ

∂∂+Φ

∂∂

nnK

xKg

xx

puu

xu

t i

i

i

ij

i

ji

j

i

rr and

δσρτρρ

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Model Structure: Level Set

( ) ( )( ) ( )

( ) ε

εεε

επε

µµηηηµρρλλλρ

>Φ<Φ<−

<ΦΦ+Φ+

=

=Φ−+=Φ=Φ−+=Φ

sin1

1

21

0

e wher)1(

where)1(

21

21

H

H

H

( ) ( ) ( ) ( ) ( )

( ) Φ∇Φ∇=⋅∇=Φ

∂Φ∂ΦΦ−Φ+

∂∂

+∂∂−=Φ

∂∂+Φ

∂∂

nnK

xKg

xx

puu

xu

t i

i

i

ij

i

ji

j

i

rr and

δσρτρρ

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The Level Set Approach

Forcing Φ to be a distance function (reinitialization)

The Level set equation

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Applications: Bubble Mergers

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Applications: Bubble Interface Interaction

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Ghost Fluid Method

Sharp treatment of interfacial changes

N – local interface normal

T1/T2 – local interface tangents

P – pressure tensor

tau – viscous tress tensor

Sigma – surface tension

Kappa – local interface curvature

[.] denotes the local interface jump condition

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2D Vertical Falling Liquid Films: Analysis of a Wave: Streamlines

Re= 69

We= 0.2785

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Wave Effects on Interfacial Scalar Transfer

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Turbulent Film Flow (Vortices Scaled Up)

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Coherent Structures: Turbulent Film Flow

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Gas-driven Two-phase Flow

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Vortices in gas-driven flow

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Drop Vaporisation

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Film Boiling

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Temperture Contours

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Convergence on Mesh Refinement

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g Steam & water cocurrent or countercurrentg Low steepness wave field (ak = 0.01-0.17,

capillary/gravity ripples)g Variable Pr/Sc, shear velocities u*, and

subcooling/superheating

Condensation

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Comparison of RELAP5 predicted values for the interfacial heat transfer coefficient in subcooled boiling vs. McMasters University data

Scalar transfer: sheared interfaces

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Canonical Problem

Physical Analog

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DNS Problem Schematic

Fractional time step taken in gas and liquid domains

Finer grid on liquid side for mass transfer calculations

Mass TransferMass Transfer

TTbulk,Lbulk,L

TTbulk,Gbulk,G

TTIntInt

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DNS method

• Pseudo-spectral DNS solver (Fourier-Fourier-Chebyshev).

• Uses mapping in the gravity direction (De Angelis, 1998).

• Based on a projection method.

• Grid resolution: typically 128X128X129 (each domain)

• Alternate solution of gas and liquid domains

• Fractional steps in each domain: interfacial stress from gas

• Interfacial velocity from liquid to gas

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Non-Orthogonal Mapping

The equations are expressed in new coordinate system.

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Boundary Fitting Method

2

2

PrRe

1

ij

jx

T

x

Tu

t

T

∂∂=

∂∂+

∂∂

j

j

i

i

ij

i

j

i

x

u

x

u

x

p

x

uu

t

u

∂∂

∂∂+

∂∂−=

∂∂+

∂∂

;Re

12

2

0=∂∂+

∂∂

j

jx

fu

t

fInterface advection

( )[ ]( )[ ]

=

=

=⋅⋅−

=+⋅∇−−+⋅⋅−

LG

LG

iGL

LGGL

TT

uu

tn

fFr

nWe

ppnn

0

011

Re

1

ττ

ττ

Interfacial jump conditions

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Interfacial Motion

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Interfacial heat transfer

τρ uTTC

qK

bulkp)(

int

int

−=+

Interfacial shear

HTC (flat & wavy interface); Pr=1

Heat Transfer at the Interface

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Gas Side Heat Flux vs. Shear Stress(DeAngelis et al 1997, DeAngelis 1998, DeAngelis et al 2000)

Heat Flux

Shear Stress

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Liquid Side Heat Flux vs. Shear Stress(DeAngelis et al 1997, DeAngelis 1998, DeAngelis et al 2000)

Heat Flux

Shear Stress

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Shear stress distribution by quadrant: gas & liquid sides

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Surface Renewal Prediction(Banerjee 1990, DeAngelis et al 1997, DeAngelis 1998, DeAngelis et al 2000)

Experiment (Rashidi and Banerjee Phys. Fluids A2 1827 (1990)) + DNS (Lombardi, De Angelis and Banerjee Phys. Of Fluids 8 1643 (1996)) suggest:

Time between burst

Lines between ejections and sweeps:

From surface renewal theory:

Liquid side: mobile boundary

D ~ molecular diffusivity; ~ time between renewals.

Liquid side:

Gas side:

35~

2

*

νuT

tE

E=+

LiiL

iL

LL

uuScK ρτ=*

*

5.0

; 0.11 ~ DNS ; 0.15 to0.1~

90~

2

*

νuT

tB

B=+

tK RLD=

GiiG

iG

GG

uuScK ρτ=*

*

3/1

; 0.075 ~ DNS ; 0.09 to0.07~

t R

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Prediction versus experiment(DeAngelis et al 1997, DeAngelis 1998, DeAngelis et al 2000)

Comparison with the data by Wanninkhof and Bliven (1994). The outlying points are large amplitude waves that were breaking.

Comparison of Eq. 1 with wind-wave tank gas transfer data from (Ocampo-Torres et al.) Assuming Sc =660.

Comparison of Eq. 2 for moisture flux coefficient with data from Ocampo-Torres et al. (1994)

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Bubble column mass transfer(Cockx et al 1995)

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( )[ ] ( )( )[ ]

==

=⋅

−

−Γ=⋅

−

=⋅⋅−

=−Γ++⋅∇−−+⋅⋅−

GLLG

iLG

LG

iGL

LGGL

RTT

tuR

u

R

Rnu

Ru

tn

RfFr

nWe

ppnn

ρρ

ττ

ττ

/;

01

11

0

0111

Re

1

2

22

nTJa

nTJa

G

G

GL

L

L ⋅∇−⋅∇=ΓPrRePrRe

2- Apply Energy jump conditions:

1- Solve Navier Stokes Eqs. in each domain:

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DNS vs. model

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Conclusions

Interface capturing models provide a framework that can elucidate many supergird features of two-phase flow structures.

Fine-scale modeling such as DNS can clarify closure relationships, e.g. for condensation.

Implicit interface capturing on an Euleriangrid using GFM or phase field is attractive and perhaps can be combined with LBM for bulk fluid computations in future for parralelization.