bogner jens

Transcription

bogner jens

Surface Tension and Wetting
with the
Free Surface Lattice Boltzmann Method
(FSLBM)
■
■
Simon Bogner, Jens Harting1 and Ulrich Rüde
Chair for System Simulation (LSS),
University of Erlangen-Nürnberg, Germany
1
Technical University Eindhoven, Netherlands
Simon Bogner, Ulrich Rüde (Universität Erlangen-Nürnberg), Jens Harting (TU Eindhoven)
1
Outline
■
Free Surface Lattice Boltzmann Method


■
LBM and Free Boundaries
Interface Tracking (Volume of Fluid)
Comparison of 3 surface tension models:
a) FD: Finite Differences (Brackbill et al. 1992)
b) TRI: Triangulation (Pohl 2007, Donath 2010)
c) LSQR: Least Squares Reconstruction (Renardy 2002,
Popinet 2009)
■
Numerical Experiments


■
Equilibrium Bubble
Sessile Drop (Wetting)
Conclusion
2
Section I
The Free Surface
Lattice Boltzmann Method
(FSLBM)
3
Ingredients of the FSLBM
■
3 Ingredients:
1) Lattice Boltzmann Equation

Solve incompressible Navier-Stokes
in the fluid subdomain
2) Interface Tracking

VOF indicator function
to represent and advect the
interface (sharp interface; thickness
)
3) Coupling of 1) and 2)


Free surface boundary conditions at the free boundary
Advection of according to LBE (streaming step)
4
Ingredients of the FSLBM
■
3 Ingredients:
1) Lattice Boltzmann Equation

Solve incompressible Navier-Stokes
in the fluid subdomain
2) Interface Tracking

VOF indicator function
to represent and advect
the interface (sharp interface; thickness
)
3) Coupling of 1) and 2)


Free surface boundary conditions at the free boundary
Advection of
according to LBE (stream step)
5
Interface Tracking
■
Volume of Fluid (VOF) indicator function




Hirt, Nichols (1981)
is the volume fraction of fluid within the
control volume of cell
fluid cells:
Cell types: interface cells:
gas cells (inactive):
Advection of according to flow
6
Coupling
■
Coupling of VOF with LBM

Free surface boundary condition

Most important is the boundary value for the pressure
➔
➔
■
static gas pressure
Laplace pressure due to surface tension
Laplace Pressure

Requires numerical estimate of interface curvature
surface tension
curvature
7
Section II
Curvature Computation
(for surface tension and wetting)
8
Method a)
■
a) Finite Differences (FD)




Brackbill et al. (1992)
Interface normal:
Interface curvature:
Boundary conditions (wetting)
Fluid-solid interface
➔ At contact line, the interface normal must satisfy
➔
Normal of solid wall
Normal to contact line, tangential to solid wall
Ideal equilibrium contact angle
(controls wettability of solid wall)

Warning: Involves finite differences of non-smooth
!!
9
Method a)
■
a) Finite Differences (FD)




Brackbill et al. (1992)
Interface normal:
Interface curvature:
finite differences
for
2nd order
derivative
Boundary conditions (wetting)
Fluid-solid interface
➔ At contact line, the interface normal must satisfy
➔
Normal of solid wall
Normal to contact line, tangential to solid wall
Ideal equilibrium contact angle
(controls wettability of solid wall)

Warning: Involves finite differences of non-smooth
!!
10
Method b)
■
b) Triangulation (TRI)


Pohl (2007)
Two steps:
Piecewise Linear Interface (Re-) Construction (PLIC)
➔ Curvature from triangulation
➔
■
PLIC: Piecewise Linear Interface Construction

For each interface cell
with


define
satisfying
I.e., the plane cuts off exactly
of the unit volume
centered around
Scardovelli, Zaleski (1999)
.
11
Method b)
■


Pohl (2007)
Two steps:
➔
■
Curvature from triangulation

For each interface cell:
Triangulate surface points in 3x3x3 neighborhood
➔ Obtain “triangle fan”
➔

Estimate curvature of triangle fan
➔
Taubin (1995) 's algorithm for curvature
of polyhedral meshes!
12
Method b)
■


Pohl (2007)
Two steps:
➔
■
Curvature from triangulation

For each interface cell:
Triangulate surface points in 3x3x3 neighborhood
➔ Obtain “triangle fan”
➔

Estimate curvature of triangle fan
➔

Taubin (1995) 's algorithm for curvature
of polyhedral meshes!
Boundary Conditions (wetting):
Rather complex, cf. Donath (2011)
➔ Involves modification of triangle fans at solid walls.
➔
13
Method c)
■
c) Least Squares Approximation (LSQR)

Two Steps:
PLIC: obtain surface points
➔ LSQR: Fit quadratic function
through surface points.
➔
■
LSQR ansatz (in 3x3x3 neighborhood)

Model function

Minimize:

Curvature:

Cf. Renardy et al. (2002), Popinet (2009)
14
Method c)
■
Boundary conditions (wetting):

Model function

Minimize:

Boundary conditions:

■
➔
Interface normal in contact point is (as before!)
➔
Constrain the optimization problem in contact point
Solve constrained LSQR problem.
Contact points:

Intersect PLIC - segment with obstacle
15
Section III
Numerical Experiment
16
Equilibrium Bubble
■
Spherical bubble test case:


Expected: Laplace pressure in balance → vanishing
velocity field
Setup for
Bubble radius
➔ Fluid density
➔ Fluid viscosity
➔ Surface tension
➔ Time step dt=1e-4
➔ Space step 1/96, 1/128, 1/160, 1/192
➔ Top/bottom boundaries: noslip
➔ Other boundaries: periodic
➔

Evaluation: “spurious” velocity field
Average of velocity magnitude
➔ Maximum of velocity magnitude
➔
17
Spurious Currents
■
Errors in the curvature computation...
Spurious velocities of FD-approach
18
Evaluation of Equilibrium Bubble
■
Spherical Bubble



Comparison of different methods
The accuracy of TRI and LSQR documents the
importance of surface reconstruction
For LSQR,
: Spurious currents are of the
order of machine precision.
19
Sessile Droplets
■
Boundary Conditions of Wetting


Prescribe ideal contact angles
Setup:
Droplet radius: R = 5 l.u.
➔ At t=0, drop is slightly touching solid wall
➔ For t >> 0, drop will assume equilibrium shape
➔ Equilibrium contact angle wall: 30° … 150°
➔

Evaluate:
➔
Equilibrium height of drop
20
Evaluation of Droplet Height
24
Evaluation of Droplet Height (cont.)
■
■
■
■
First: Horizontal plane
Same test-case with 45°
inclined plane
(step towards arbitrary
geometries)
Extreme
→ max. Error
LSQR overestimates
wettability
25
Section IV
Conclusion
26
Conclusion
■
Surface tension is critical



■
Curvature estimation best with “ geometric
reconstruction” methods (TRI, LSQR)
Non-smooth indicator → Problems with finite differences
→ not accurate enough for capillary flows
LSQR: Function fitting / optimization approach
Wetting ~ Boundary condition interface

LSQR: Optimization with constraint (contact normal)
■
■
TRI: fast (performant) but difficult for wetting
LSQR: slow (involves linear system)
■
Outlook: Dynamic Behavior
27
Literature
(1) Chen, Doolen. 1998. Lattice Boltzmann Method for Fluid Flows. Annu. Rev. Fluid Mech. 30/329-364
(2) Benzi, Succi, Vergassola. 1992. The Lattice Boltzmann Equation: Theory and Applications. Physical Reports 222(3) / 145-197
(3) Körner, Thies, Hofmann, Thürey, Rüde. 2005. Lattice Boltzmann Model for Free Surface Flow for Modeling Foaming. J. Stat. Phys. 121(1) /
179-196
(4) Hirt, Nichols. 1981. Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J. Comp. Phys. 39 / 201-225
(5) Ginzburg, Steiner. 2003. Lattice Boltzmann model for free-surface flow and its application to filling process in casting. J. Comp. Phys. 185 /
61-99
(6) Brackbill, Kothe, and Zemach. 1992. A continuum method for modeling surface tension. Journal of Computational Physics, 100 / 335–354.
(7) Harvie, Davidson, Rudman. 2006. An analysis of parasitic current generation in volume of fluid simulations. Applied Mathematical
Modelling,30(10):1056 – 1066
(8) Pohl. 2007. High Performance Simulation of Free Surface Flows Using the Lattice Boltzmann Method. PhD thesis, University of ErlangenNuremberg
(9) Renardy and M. Renardy. 2002. Prost: A parabolic reconstruction of surface tension for the volume-of-fluid method. Journal of Computational
Physics, 183:400–421
(10) Scardovelli, Zaleski. 1999. DIRECT NUMERICAL SIMULATION OF FREE-SURFACE AND INTERFACIAL FLOW. Annu. Rev. Fluid Mech.
1999. 31:567–603
(11) Taubin. 1995. Estimating the Tensor of Curvature of a Surface from a Polyhedral Approximation. In: Proceedings of the Fifth International
Conference on Computer Vision, S. 902–905
(12) Donath. 2011. Wetting Models for a Parallel High-Performance Free Surface Lattice Boltzmann Method. PhD thesis, University of ErlangenNuremberg
(13) Popinet. 2009. An accurate adaptive solver for surface-tension-driven interfacial flows. Journal of Computational Physics 228 / 5838-5866
28

bogner jens

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