modeling transport phenomena on fluid interfaces in combination … · 2011. 7. 5. · conclusion...

CHAIR FOR

COMPUTATIONAL

ANALYSIS OF

TECHNICAL

SYSTEMS

Modeling transport phenomena on fluid

interfaces in combination with the XFEM

H. Sauerland and T.-P. Fries

March 23, 2011

16. Internat. Conference on Finite Elements in Flow ProblemsMunich, Germany

Motivation

Topic of increasing interest in the scientific community.

Transport of quantities on moving surfaces is not intuitive.

Various applications

Surfactant transport (fluid dynamics).

Texture synthesis (computer graphics).

Species diffusion along grain boundaries (material science).

Pattern formation on growing organisms (biology).

[1] [2] [3]

[1] Bertalmio, JCP 2001; [2] Brown University, Gao Group; [3] Leung, JCP 2003

H. Sauerland and T.-P. Fries Modeling transport phenomena on fluid interfaces 2

Motivation

Here: Two-phase/free-surface flow problems.

Flow field

Advection/deformationof the interface

Variable surface tension

Scalar transporton the interface


Outline

Governing equationsInterface description

Challenges in two-phase flows

XFEM

Modeling transport phenomena on hypersurfacesImplicit approach

Numerical examples

Conclusion




XFEM


Numerical examples

Conclusion


Governing equations

Incompressible and isothermal Navier-Stokes equations foreach phase j = 1, 2 in Ω:

ρj

(∂u

∂t+ u · ∇u− f

)

−∇ · σ = 0 in Ωj × [0,T ],

∇ · u = 0 in Ωj × [0,T ].

Constitutive equation:

σ(u, p) = −pI+ µj

[

∇u+ (∇u)T]

.

Boundary conditions:

u = u on Γu × [0,T ],

n · σ = h on Γh × [0,T ].

Interface conditions:

[u]Γd = 0 on Γd (t) × [0,T ],

[n · σ]Γd = γκn on Γd (t) × [0,T ].

Ω1

Ω2

Ω2

n

n

n

Γd

Γ

Initial condition:

u(x, 0) = u0(x) in Ω.


Interface description

The Level-set method offers a flexible representation of themoving interface.

Zero-level of the scalar level-set function φ implicitly describesthe interface:

φ(x) = ± minx⋆∈Γd

‖x− x⋆‖ , ∀x ∈ Ω.

Motion of the interface Γd is covered by the level-settransport equation:

∂φ

∂t+ u(x, t) · ∇φ = 0 in Ω× [0,T ].

→ Strongly coupled problem of the fluid and level-set field.




XFEM


Numerical examples

Conclusion



Discontinuous field variables:

Density differences

→ Kink in the velocity/pressure field.Viscosity differences

Surface tension → Jump in the pressure field.

→ XFEM

up

ρ2, µ2

ρ1, µ1

u

p

ρ2, µ2

ρ1, µ1

γ




XFEM


Numerical examples

Conclusion


XFEM

Discontinuities inside elements can be accounted for.

Here: Pressure approximation space is locally enriched:

ph(x) =∑

i∈I

Ni (x)pi

︸︷︷︸

strd. FE approx.

+∑

i∈I⋆

Ni (x) · [ψ(x, t)− ψ(xi , t)] ai

︸︷︷︸

enrichment

.

Sign-enrichment:

ψ(x, t) = sign (φ(x, t)) =

−1 : φ < 0,

0 : φ = 0,

1 : φ > 0.

I ⋆

φ(x, t) = 0


XFEM

Topics worth talking about:

Numerical integration of enriched elements.

Choice of enrichment functions[Sauerland and Fries, JCP 2011].

Time-integration[Fries and Zilian, IJNME 2009].

Adaptive mesh refinement at the interface[Fries et al., IJNME 2010].




XFEM


Numerical examples

Conclusion


Modeling transport phenomena on hypersurfaces

Physical description of transport processes is simple.

Solving PDEs on arbitrary, moving manifolds is difficult.

Explicit approach:

Moving surface mesh.

Generalized differentialoperators required(cf. Laplace-Beltramioperator).

Usual disadvantages ofexplicit descriptions.

Implicit approach:

Level-set method.

Quantities on the surfaceare extended to the wholedomain, Rn → R

n+1.

Transport equations aresolved in a fixed volume.


Implicit approach

Transport of a scalar quantity G on an implicit hypersurface:

Prerequisite: Scalar G is known in the whole domain.

Situation: Only GΓ on the interface is available.

Extend GΓ off the interface, orthogonal to φ

sign(φ)∇G · ∇φ = 0 in Ω,

G = GΓ on Γd .Ω1

Ω2

Γd

Adalsteinsson, JCP 1999; Chessa, IJNME 2002


Implicit approach

Application of the boundary condition G = GΓ on the implicit Γd :

Find intersections xintj of Γd with the element edges.

For each cut element node i ∈ I ⋆ find x⋆ = minj

‖xi − xintj ‖.

Prescribe Gi = GΓ(x⋆).

I ⋆

Γd


Implicit approach

Conservation equation for G in Ω (for brevity in 2D)

∂G

∂t=− u · ∇G

︸︷︷︸

advection

−

[

n2y∂u

∂x− nxny

(∂u

∂y+∂v

∂x

)

+ n2x∂v

∂y

]

G

︸︷︷︸

compression/expansion

+ σ

[∂2G

∂x2n2y − 2

∂2G

∂x∂ynxny +

∂2G

∂y2n2x − κ(∇G · n)

]

︸︷︷︸

diffusion

,

Adalsteinsson, JCP 2003

velocity: u = (u, v)T , diffusion coefficient: σ,

normal vector: n = (nx , ny )T =

∇φ

|∇φ|, curvature: κ.


Implicit approachAdvection in a constant flow field:

u = (−y , x , 0)T , σ = 0,GΓ = x + y + z .


Implicit approachDiffusion on a fixed surface:

u = 0, σ = 1.0,GΓ = z .

Diffusion on moving interfaces not considered here.


Implicit approach

Integration in the two-phase flow solver:

Navier-Stokes Level-set

f (G )

Hypersurface transport

u(x, t)

φ(x, t)

φ(x, t)u(x, t)

G (xΓ, t)

γ(xΓ, t)




XFEM


Numerical examples

Conclusion


Numerical examples

Importance of the compression/expansion term for generalcases.

Tank Sloshing

0 0.2 0.4 0.6 0.8 1

0.1

0.3

0.5

0.7

x

GΓ

t=20s

pure advectionadv.+compr.initial


Numerical examples3D tank sloshing (40 × 40× 60 elements)

t = [0, 40s], σ = 0,GΓ = 0.5.


Numerical examples3D rising drop (40 × 40× 80 elements)

t = [0, 0.02s], σ = 0,GΓ = 0.5,Eo = 7.64,Mo = 1.22 · 10−6.




XFEM


Numerical examples

Conclusion


Conclusion

Results of this case study:

Implict approach for hypersurface transport is easilyimplemented in an existing level-set framework.

A smooth and very accurate level-set field has to bemaintained.

Frequent “re-extension” of the scalar quantity from theinterface is required (cf. level-set reinitialization).

Outlook:

Consideration of diffusion on moving interfaces.

Variable surface tension coefficient depending on the scalarconcentration.


Thank you for your attention.

www.xfem.rwth-aachen.de


modeling transport phenomena on fluid interfaces in combination … · 2011. 7. 5. · conclusion...

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