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Page 1Oleksiy Varfolomiyev | Dortmund
Introduction to the Phase Field MethodAllen-Cahn vs. Cahn-Hilliard Model
Oleksiy Varfolomiyev
Supervisor: Prof. S.Turek
LSIII, TU Dortmund
Page 2Page 2Oleksiy Varfolomiyev | Dortmund
What for?
The phase-field method (PFM), as presented here, grows out of the work of Cahn, Hilliard and Allen
It is used for two general purposes:
• to model systems in which the diffuse nature of interfaces is essential to the problem, such as spinodal decomposition and solute trapping during rapid phase boundary motion;
• as a front tracking technique to model general multi-phase systems.
Page 3Page 3Oleksiy Varfolomiyev | Dortmund
PFM Applications
Multiphase Systems
Spinodal Decomposition
Page 4Page 4Oleksiy Varfolomiyev | Dortmund
PFM Applications
Atomization
Page 5Page 5Oleksiy Varfolomiyev | Dortmund
PFM Applications
Dynamics of drop formation from a capillary tube: inkjet printing
Page 6Page 6Oleksiy Varfolomiyev | Dortmund Oleksiy Varfolomiyev | Dortmund
Two types of phase field models
Cahn Hillard
Phase is uniquely determined by the value of a conserved field variable, e.g. concentrationC < C1 we are in one phaseC > C2 we are in the other
Allen-Cahn
Phase is not uniquely determined byconcentration, temperature, pressure, etc.We define the order parameter fieldvariable to determine the phase, φ
Page 7Page 7Oleksiy Varfolomiyev | Dortmund
Models
( )( )dVFE ∫Ω
+∇= ϕϕ ( )( )dVCFCE ∫Ω
+∇=
Cahn-HilliardFree Energy
Allen-Cahn
Page 8Page 8Oleksiy Varfolomiyev | Dortmund
Models
( )( )dVFE ∫Ω
+∇= ϕϕ ( )( )dVCFCE ∫Ω
+∇=
JdtdC
⋅−∇=
Cahn-HilliardFree Energy
Allen-Cahn
Because is not conservedϕ Because C is locally conserved, according to Fick‘s second law
δϕδγϕ E
dtd
−=
( ) ( )2
22
41
ηϕϕ −
=F
Double-well potential
Page 9Page 9Oleksiy Varfolomiyev | Dortmund
Models
( )( )dVFE ∫Ω
+∇= ϕϕ ( )( )dVCFCE ∫Ω
+∇=
JdtdC
⋅−∇=
( ) μ∇−= CMJ
Cahn-HilliardFree Energy
Allen-Cahn
Because is not conservedϕ Because C is locally conserved, according to Fick‘s second law
Define potentialδϕδγϕ E
dtd
−=
( )CFCCE '+Δ−==
δδμ
Constitutive equation
( ) ( )CFCf ':=( ) ( )ϕϕ ': Ff =
Denote
( ) ( )2
22
41
ηϕϕ −
=F
Double-well potential
Page 10Page 10Oleksiy Varfolomiyev | Dortmund
( ) ( )( )[ ]CfCCMCutC
−Δ∇⋅∇=∇⋅+∂∂
( ) ( )( )
0 =
+−Δ=∇⋅+∂∂
∫Ω
dxdtd
tfut
ϕ
ξϕϕγϕϕ
Allen-Cahn Equation Cahn-Hilliard Equation
( ) ( )( ) dxtxft ,1∫ΩΩ
= ϕξ
Lagrange multiplier
Page 11Page 11Oleksiy Varfolomiyev | Dortmund
( ) ( )( )[ ]CfCCMCutC
−Δ∇⋅∇=∇⋅+∂∂
( ) ( )( )
0 =
+−Δ=∇⋅+∂∂
∫Ω
dxdtd
tfut
ϕ
ξϕϕγϕϕ
Allen-Cahn Equation Cahn-Hilliard Equation
( ) ( )( ) dxtxft ,1∫ΩΩ
= ϕξ
Lagrange multiplier
( ) ( )
2
expression theExploiting0
2ϕϕϕϕϕ
ϕϕλν
∇∇+∇Δ=∇⊗∇
=⋅∇
+∇⊗∇⋅∇=∇+Δ−∇⋅+∂∂
u
gpuuutu
Momentum equation with continuity condition
Page 12Page 12Oleksiy Varfolomiyev | Dortmund
( ) ( )( )[ ]CfCCMCutC
−Δ∇⋅∇=∇⋅+∂∂
( ) ( )( )
0 =
+−Δ=∇⋅+∂∂
∫Ω
dxdtd
tfut
ϕ
ξϕϕγϕϕ
Allen-Cahn Equation Cahn-Hilliard Equation
( ) ( )( ) dxtxft ,1∫ΩΩ
= ϕξ
Lagrange multiplier
( ) ( )
2
expression theExploiting0
2ϕϕϕϕϕ
ϕϕλν
∇∇+∇Δ=∇⊗∇
=⋅∇
+∇⊗∇⋅∇=∇+Δ−∇⋅+∂∂
u
gpuuutu
Momentum equation with continuity condition
+IC & BC
Page 13Page 13Oleksiy Varfolomiyev | Dortmund
Allen-Cahn-Hilliard-Navier-Stokes Problems
( ) ( )( ) ( ) Ω∈=
Ω∈=
xxx
xxxu u ,0,
,0,
0
0
ϕϕ
( )
( )( )
( ) CF'1
22
222
Cw
wfwCutC
gCCpuuutu
Δ−=
−Δ=∇⋅+∂∂
+∇Δ−=∇+Δ−∇⋅+∂∂
εε
γ
λν( )
( ) ( )( )
( ) ( )( ) ( ) 0 , , ,1
1
11
111
=Ω
=
+−Δ=∇⋅+∂∂
+∇Δ−=∇+Δ−∇⋅+∂∂
∫∫ ΩΩdxtx
dtddxtxft
tfut
gpuuutu
ϕϕξ
ξϕϕγϕϕ
ϕϕλν
( ) ( )( ) ( ) Ω∈=
Ω∈=
xxxC
xxxu
Cu
,0,
,0,
0
0
Initial conditions
( )
( ) T
T
txn
Ttxhu
Ω∂∈=∂∂
×Ω=Ω∂∈=
, ,0
),0(:, ,ϕ
Boundary conditions
Allen-Cahn Problem Cahn-Hilliard Problem
Initial conditions
Boundary conditions( )
( )
( ) Ω∂∈
Ω∂∈Ω∂∈
=∂∂
=∂∂
=
T
T
T
, ,0
, ,0
, ,
txnw
txnC
txqu
Page 14Page 14Oleksiy Varfolomiyev | Dortmund
Solver for the CHNS Problem
{ }
( )
( ) ( )
Ω∂=
+∇Δ−∇−∇⋅−=Δ−Δ−
=
on ,0~
, ~ ~
:schemeimplicit -semi aby on ~,~~ field velocity teintermedia theCompute :) (
,, data initialGiven
u
xgpuuutuu
Ivuustepevolutionfluidthe
pu
nh
nh
nh
nh
nh
n
n
n
nnn
ϕϕλν
ϕStep 0:
Step1:
Half-staggered mesh
A projection method on a fixed half-staggered mesh
Page 15Page 15Oleksiy Varfolomiyev | Dortmund
Solver for the CHNS Problem
{ }
( )
( ) ( )
Ω∂=
+∇Δ−∇−∇⋅−=Δ−Δ−
=
on ,0~
, ~ ~
:schemeimplicit -semi aby on ~,~~ field velocity teintermedia theCompute :) (
,, data initialGiven
u
xgpuuutuu
Ivuustepevolutionfluidthe
pu
nh
nh
nh
nh
nh
n
n
n
nnn
ϕϕλν
ϕ
,on 0
,on 0
,on ~
10
1
11
Ω∂=⋅
=⋅∇
∇Δ+=
+
+
++
nu
Iu
Ituu
n
nh
nn
hnn ψ
,on ~0
11 Iupp hnnn ⋅∇+−= ++ νψ
Step2: (the projection step) Project the intermediate velocity field onto thedivergence-free vector space
Step 0:
Step1:
Half-staggered mesh
A projection method on a fixed half-staggered mesh
Update the pressure
Page 16Page 16Oleksiy Varfolomiyev | Dortmund
Solver for the CHNS Problem
{ }
( )
( ) ( )
Ω∂=
+∇Δ−∇−∇⋅−=Δ−Δ−
=
on ,0~
, ~ ~
:schemeimplicit -semi aby on ~,~~ field velocity teintermedia theCompute :) (
,, data initialGiven
u
xgpuuutuu
Ivuustepevolutionfluidthe
pu
nh
nh
nh
nh
nh
n
n
n
nnn
ϕϕλν
ϕ
,on 0
,on 0
,on ~
10
1
11
Ω∂=⋅
=⋅∇
∇Δ+=
+
+
++
nu
Iu
Ituu
n
nh
nn
hnn ψ
,on ~0
11 Iupp hnnn ⋅∇+−= ++ νψ
Step2: (the projection step) Project the intermediate velocity field onto thedivergence-free vector space
Step 0:
Step1:
Half-staggered mesh
A projection method on a fixed half-staggered mesh
0
~1
=⋅∇
⋅∇Δ
=Δ
n
ut
h
hn
h
ψ
ψ
Pressure-Poisson Equation
Update the pressure
Page 17Page 17Oleksiy Varfolomiyev | Dortmund
Step 3 (the phase evolution step): Compute the phase field by
( ) ( ) ( ) 011
1
on Itfut
nnnnh
nh
n
nn
γξϕγϕφγϕϕ+−⋅−∇=Δ−
Δ− ++
+
Page 18Page 18Oleksiy Varfolomiyev | Dortmund
Solver for the CHNS Problem
Step 3 (the phase evolution step): Compute the phase field by
( ) ( ) ( ) 011
1
on Itfut
nnnnh
nh
n
nn
γξϕγϕφγϕϕ+−⋅−∇=Δ−
Δ− ++
+
Simulation Analysis
Page 19Page 19Oleksiy Varfolomiyev | Dortmund
Simulation – Surface Tension
Page 20Page 20Oleksiy Varfolomiyev | Dortmund
Simulation – 2 Kissing Bubbles
Page 21Page 21Oleksiy Varfolomiyev | Dortmund
Adaptive vs fixed mesh method
Page 22Page 22Oleksiy Varfolomiyev | Dortmund
Inference
Conclusion
The phase-field method is a very versatile and robust method for studying interfacial motion in multi-component flows. It casts geometric evolution in Lagrangian coordinates into an Eulerianformulation, and provides a way to represent surface effects as bulk effects. The whole process allows us to use an energetic variationalformulation that makes it possible to ensure the stability of corresponding numerical algorithms. The elastic relaxation built into the phase-field dynamics prevents the interfacial mixing layer from spreading out. Moreover, being a physically motivated approximation based on the competition between different parts of the energy functionals, the phase-field model can be adapted easily to incorporate more complex physical phenomena such as Marangonieffect and non-Newtonian rheology.
Page 23Page 23Oleksiy Varfolomiyev | Dortmund
The End
Page 24Page 24Oleksiy Varfolomiyev | Dortmund
References
Literature
• 1 Adam Powell, Introduction to Phase Field Method, Group Seminar, September 5, 2002• 2 Xiaofeng Yang, James J. Feng, Chun Liu, Jue Shen, Numerical simulations of jet pinching-off
and drop formation using an energetic variational phase-field method, Journal of Computational Physics 218 (2006) pp.417-428
• 3 Chun Liu, Jie Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D 179 (2003) pp.211-228
• 4 James J. Feng, Chun Liu, Jie Shen, Pengtao Yue, An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids - advantages and challenges, In Modeling of soft matter, vol. 141 of IMA Vol. Math Appl., pp.1-26, Springer, New York, 2005
• 5 Yana Di, Ruo Li, Tao Tang, A General Moving Mesh Framework in 3D and its Application for Simulating the Mixture of Multi-Phase Flows, Communications in Computational Physics, Vol. 3, No.3, pp.582-602
• 6 C.M. Elliott, D.A. French, and F.A. Milner, A Second Order Splitting Method for the Cahn-Hillard Equation, Numer Math. 54, 575-590 (1989)
• 7 David Kay, Richard Welford, A Multigrid Finite Element Solver for the Cahn-Hilliard Equation, Journal of Computational Physics, Volume 212, Issue 1, (2006), pp.288-304
• 8 David Kay, Richard Welford, Efficient Numerical Solution of Cahn-Hillard-Navier-Stokes Fluids in 2D, SIAM J. Sci. Comput. Vol 29, No. 6, pp. 2241-2257
• 9 C.M. Elliott, The Cahn-Hillard model for the kinetics of phase separation, in Mathematical Models for Phase Problems, Internat, Ser. Numer. Math. 88, Birkhäuser-Verlag, Basel, Switzerland, 1989, pp. 35-73
• 10 Zhengru Zhang, Huazhong Tang, An adaptive phase field method fort he mixture of two incompressible fluids, Computers & Fluids 36, (2007), pp.1307-1318
Page 25Page 25Oleksiy Varfolomiyev | Dortmund
Allen-Cahn-Navier-Stokes Problem
( )( ) Ω∈=
Ω∈=
=
=
xx
xx
t
t uu ,
,
00
00
||
ϕϕ
( ) ( )
0
=⋅∇
+∇⊗∇⋅∇=∇+Δ−∇⋅+∂∂
u
gpuuutu ϕϕλν
( )
( ) T
T
txn
Ttxhu
Ω∂∈=∂∂
×Ω=Ω∂∈=
, ,0
),0(:, ,ϕ
Initial conditions
Boundary conditions
Momentum equation with continuity condition
( ) ( )( )
0 =
+−Δ=∇⋅+∂∂
∫Ω
dxdtd
tfut
ϕ
ξϕϕγϕϕ
Allen-Cahn Equation
Page 26Page 26Oleksiy Varfolomiyev | Dortmund
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