Post on 05-Jun-2018
Level-Set and Phase-Field Methods: Application to Moving Interfaces and
Two-Phase Fluid Flows
Maged IsmailClaremont Graduate University
& Keck Graduate Institute
HMC Math 164, Scientific Computing, Spring 2007
Outline
• Background & Motivation• Level Set Method• Sharp-Interface Phase-Field Method• Conservative Level Set Method• Mathematical Formulation for Two-phase Flows• Conclusions & Future Work• Acknowledgments• References
Background & Motivation
• Tracking of moving interfaces- Wide range of scientific and engineering applications(e.g. two-phase fluid flows, melting and solidification, computer graphics, image segmentation- Typically two types of approaches to simulate moving interfaces:▪ Particle Methods (Lagrangian, explicit)
can’t handle topological changes, sharp corners…▪ Level Set Methods (Eulerian, implicit)
today’s topic
Level Set Method
• An implicit method for capturing the evolution of an interface.
• History: Devised by Sethian and Osher (J. Computational Physics,1988) as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions.
• Based upon representing an interface as the zero level set of some higher dimensional function.
Level Set Method
• Basic ideaTo track the evolution of a moving interface:▪ Recast problem with one additional dimension – the distance from the interface.
( , ) ( , ) 0
( , )
d x t xx t x
d x t xφ
+ ∉Ω⎧⎪= ∈∂Ω⎨⎪− ∈Ω⎩
0φ <0φ >
outside
inside
interface0φ =
Ω
∂Ω
Level Set Method
• The interface always lies at the zeroth level set of the function
• i.e., the interface is defined by the implicit equation
• To advance the interface:Given a velocity field, uThe evolution equation becomes:
• Interfacial geometric quantities can be easily calculated using
φ
( ), 0t x yφ =
φ| |,φ φ κ= ∇ ∇ =∇⋅n n
0tφ φ∂+ ⋅∇ =
∂u
Level Set Method
• Advantages:- capable of capturing topological changes- intrinsic geometric properties are easy to determine- relatively easy to implement- accurate high order computational schemes exist
• Difficulties:- computationally expensive- re-initialization is needed to maintain the signed distance function- not conservativeloss or gain of mass (area) due to numerical diffusion
Sharp-Interface Phase-Field Method
• Based on: Sun, Y., and Beckermann, C., "Sharp Interface Tracking Using the Phase-Field Equation," J. Computational Physics, 2007.
• Instead of signed distance function, the phase field function is assumed to be
• Using same interface advection equation
tanh2xW
φ ⎛ ⎞= − ⎜ ⎟⎝ ⎠
1 insidesmooth near thetransition interface
1 outside
φ
+⎧⎪⎪→ ⎨⎪−⎪⎩
0tφ φ∂+ ⋅∇ =
∂u
Sharp-Interface Phase-Field Method
• However, using
and
• The following evolution equation can be derived
• Descritized using finite difference method• Simple forward Euler for time discretization (explicit)• spatial discretization…
( )e a bκ= + −u u n( )21 φ φφκ φ
φ φ φ⎛ ⎞ ⎡ ⎤∇ ⋅∇ ∇∇
= ∇⋅ = ∇ ⋅ = ∇ −⎜ ⎟ ⎢ ⎥⎜ ⎟∇ ∇ ∇⎝ ⎠ ⎣ ⎦n
( )22
2
1ea b
t Wφ φφ φφ φ φ φ
φ
⎡ ⎤− ⎛ ⎞∂ ∇⎢ ⎥+ ∇ + ⋅∇ = ∇ + − ∇ ∇⋅⎜ ⎟⎜ ⎟∂ ∇⎢ ⎥⎝ ⎠⎣ ⎦
u
Sharp-Interface Phase-Field Method• Laplacian (9-point finite-difference stencil):
• Norm of the gradient (central difference):
• Curvature (???):
( ) ( )1, , 1 1, , 1 , 1, 1 1, 1 1, 1 1, 1 ,2, 2
2 4 0.5 43
i j i j i j i j i j i j i j i j i j i ji j x
φ φ φ φ φ φ φ φ φ φφ + + − − + + + − + − − −+ + + − + + + + −
∇ =Δ
( ) ( )2 2
1, 1, , 1 , 1
,
14 4
i j i j i j i j
i j xφ φ φ φ
φ + − + −− −∇ = +
Δ
( ) ( ) ( ) ( )
( ) ( )
1, , , 1,
2 2 2 2
1, , 1, 1 , 1 1, 1 , 1 , 1, 1, 1 , 1 1, 1 , 1
, 1 ,,2 2
, 1 , 1, 1 1, 1, 1 1,
/16 /161
/16
i j i j i j i j
i j i j i j i j i j i j i j i j i j i j i j i j
i j i ji j
i j i j i j i j i j i j
x
φ φ φ φ
φ φ φ φ φ φ φ φ φ φ φ φφφ φφ
φ φ φ φ φ φ
+ −
+ + + + + − − − − + + − − −
+
+ + + + − + −
− −−
− + + − − − + + − −⎛ ⎞∇∇⋅ =⎜ ⎟⎜ ⎟ −∇ Δ⎝ ⎠ +
− + + − − ( ) ( ), , 1
2 2
, , 1 1, 1 1, 1, 1 1, /16
i j i j
i j i j i j i j i j i j
φ φ
φ φ φ φ φ φ
−
− + − + − − −
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟
−⎜ ⎟−⎜ ⎟
⎜ ⎟− + + − −⎝ ⎠
ii-1 i+1 i+2
Need 4 points to discretize with third order accuracy φ∇
This often leads to oscillations at the interface
Fix: pick the best four points out of a larger set of grid points to avoid/minimize oscillations (“essentially-non-oscillatory”)
i-3 i-2 i+3 i+4
Set 1 Set 2 Set 3
Sharp-Interface Phase-Field Method• Hyperbolic term
(3rd order Essentially-Non-Oscillatory scheme)e φ⋅∇u
Sharp-Interface Phase-Field Method• Results
- Interface motion with a constant normal speedperiodic cosine curve propagating with normal speed of unity
( ) ( )0 1 , 1 cos 2 / 4 , 0 1s s sγ π= − + ≤ ≤⎡ ⎤⎣ ⎦
Swallowtail solution Weak solution (meaningful)
Analytical solutions Numerical solution
Sharp-Interface Phase-Field Method• Results
- Interface motion due to external flow fields(Diagonal translation of a circle)
Conservative Level Set Method• Based on :Olsson, E., Kreiss, G., A conservative level set method
for two phase flow, Journal of Computational Physics, 2005.• Level set function
smeared out Heaviside instead of signed distance function
where
• Interface represented by 0.5φ =
( )
0, ,1 1 sin , ,2 2 2
1, ,
sd
sd sdsm sd sd
sd
H
φ εφ πφφ φ ε φ εε π ε
φ ε
< −⎧⎪⎪ ⎛ ⎞= = + + − ≤ ≤⎨ ⎜ ⎟
⎝ ⎠⎪⎪ >⎩
φ
( ) ( ) ( )minsd dφΓ
Γ∈Γ= = −
xx x x x
Conservative Level Set Method
• Evolution equation of- Standard level set method
▪ loss or gain of mass (area) due to numerical diffusion▪ interface shape is not preserved
- Modified equation (non-conservative form): add▪ shape preserving artificial compression ▪ artificial diffusion to smear the profile (avoid discontinuities)
φ
0tφ φ∂+ ⋅∇ =
∂u
( )1tφ φφ γ ε φ φ φ
φ
⎡ ⎤⎛ ⎞∂ ∇+∇ ⋅ = ∇ ⋅ ∇ − −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∇⎢ ⎥⎝ ⎠⎣ ⎦
u
Conservative Level Set Method- Modified equation (conservative form)▪ divergence free velocity field
▪ shape of the level set function stabilized across the interface▪ exact numerical conservation of the integral of ▪ the above equation can be split into two steps:
Advection + Reinitialization
,
( ) ( )1tφ φφ γ ε φ φ φ
φ
⎡ ⎤⎛ ⎞∂ ∇+∇⋅ = ∇ ⋅ ∇ − −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∇⎢ ⎥⎝ ⎠⎣ ⎦
u
φ
( ) 0tφ φ∂+∇ ⋅ =
∂u ( )1φ φγ ε φ φ φ
τ φ
⎡ ⎤⎛ ⎞∂ ∇= ∇⋅ ∇ − −⎢ ⎥⎜ ⎟⎜ ⎟∂ ∇⎢ ⎥⎝ ⎠⎣ ⎦
Mathematical Modeling of Two-phase Flows
• Incompressible Navier-Stokes equations ($10^6 prize)Conservation of mass (Continuity equation):
Conservation of linear momentum (Newton’s 2nd law):
density and dynamic viscosity depend on
0∇⋅ =u
( )( )( )Tpρ μ ρ σκδ∂⎛ ⎞+ ⋅∇ = −∇ +∇⋅ ∇ + ∇ + + +⎜ ⎟∂⎝ ⎠u u u u u g n Ft
φ
( )1 2 1ρ ρ ρ ρ φ= + −
( )1 2 1μ μ μ μ φ= + −
Numerical Simulation• COMSOL Multiphysics 3.3a (formerly FEMLAB)
• solving the NS eqns coupled with the modified conservative / non-conservative level-set equation using finite-element method
• computational time for each case ~ 1-6 hours on Pentium 4, 3GHz processor with 2GB of RAM
• special treatment of moving three-phase contact-line
wetted wall boundary condition:
▪ enforces slip
▪ adds a frictional force
▪ adds the weak boundary term
wall 0⋅ =u n
frηβ
= −F u
( ) ( )( )walltest cos dSσ θ δ∂Ω
⎡ ⎤⋅ −⎣ ⎦∫ u n n
Results• Rising bubble
oil bubble rising up through water
2D case 2D axisymmetric (revolved to 3D)
Oil, ρ=880 kg/m^3Water, ρ=997 kg/m^3
Conclusions
• Several variants of the standard level-set method exist.• Sharp-interface phase-field method implemented in
Matlab using FD discretization and used for tracking propagating interfaces.
• Conservative level-set method tested on several two-phase flow benchmark cases.
• Problems with moving contact-line also considered.
Future Work
• Validation with published experimental/analytical results• Implementing the sharp-interface phase-field method in
COMSOL Multiphysics• Application to droplet-microfluidics• Extension to three-dimensional cases (parallel-computing!)• Funding…proposals
Acknowledgements• Dr. Ali Nadim (KGI/CGU)• Dr. Darryl Yong (HMC)• Support from KGI/CGU
References
• Osher, S., Fedkiw, R. ,2003, Level Set Methods and Dynamic Implicit Surfaces, Springer.
• Olsson, E., Kreiss, G., 2005, A conservative level set method for two phase flow, Journal of Computational Physics, Vol. 210, pp. 225-246.
• Sun, Y., Beckermann, C., 2007, Sharp Interface Tracking Using the Phase-Field Equation, Journal of Computational Physics, Vol. 220, pp. 626-653.