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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: [email protected]: 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
4
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
IBM has trumpeted a nanotech method for making microchip components which it says should enable electronic devices to continue to get smaller and faster. Current techniques use light to help etch tiny circuitry on a chip, but IBM is now using molecules that assemble themselves into even smaller patterns.
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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.