1. A SUBCELL BASED INTERFACE RECONSTRUCTION METHOD TO DEAL WITH FILAMENTS
ECCOMAS 2012 | Christophe Fochesato1, Raphaël Loubère2, Renaud Motte1, Jean Ovadia3
1CEA, DAM, DIF, F Arpajon, France
2Institut de Mathématiques de Toulouse, CNRS, Université de Toulouse
3Retired fellow from CEA, CESTA, F Le Barp, France

2. II Presentation of the method
Outline
I Introduction
Context / Problem / Proposed solution / Example
II Presentation of the method
Detection / Centroids / Sub-zones / Volume distribution / Reconstruction
III Test cases
Infinite vertical filament / isolated fragment / finite vertical filament
IV Conclusions / Perspectives

3. I - Context Bi-fluids hydrodynamical flows with interfaces
VoF interface reconstruction
Maximum possible resolution is not always sufficient
structure size < mesh cell size

4. I - Problem
Usual VoF methods use only one interface per mixed cell, typically a straight line (PLIC)
numerical surface tension
generation of flotsam reduces robustness
Youngs’ normal can be irrelevant : ~0 because of almost symmetrical volume fractions on the stencil
Reconstruction is coarse whereas there is enough information to do better with the same 9-cells stencil

5. I – Proposed solution : a subgradients method
The idea is to compute subgradients from a local stencil associated with each corner (2x2 cells for instance) in order to reconstruct one interface per subzone, with normals given by these subgradients
Method does not contain more physics
Another geometrical choice
with less numerical surface tension
avoiding to use irrelevant normal information
Localized algorithm in the code : no change in the numerical scheme

6. A filament with our method VoF / PLIC with Youngs’ normal
I - Example
A filament with our method
VoF / PLIC with Youngs’ normal

7. II – Method summary
Detection of marked cells with subgradients computation
Estimation of centroid if the fluid is just entering
Determination of subzones with centroid as center of subdivision
Distribution of fluid per subzone with priority
Youngs/PLIC reconstruction : one straight line per subzone
Computation of centroids
Advection of centroids

8. II – Detection of concerned cells
Computation of a volume fraction gradient per cell
The cell is marked
if normal is potentially not relevant : or
if the fluid potentially passes through the cell f7 f8 f9 f4 f5 f6 f1 f2 f3
If marked cell, compute a gradient associated with each corner on a 2x2 stencil
Confirmation that cell is marked if gradient is less relevant than subgradients

9. II – Estimation of centroids
If the fluid is coming in the cell no centroid information
compute Youngs/PLIC interface,
compute the volume centroid
else
use the advected centroids from the previous step or init

10. II – Determination of subzones
keep only relevant subgradients
detect particular configurations : vertical filament subzones, …
with relevant subgradients, we associate subzones
4 relevant subgradients
3 relevant subgradients
2 relevant subgradients
1 relevant subgradient

11. II – Distribution of the volume of fluid
In priority, fill in subzones next to non empty cells
uniform distribution in volume
If it remains some fluid to distribute, fill in other subzones
Correction if more volume of fluid than volume of the subzone
uniform distribution on other non full subzones

12. II – Distribution of the volume of fluid
In priority, fill in subzones next to non empty cells
uniform distribution in volume
If it remains some fluid to distribute, fill in other subzones
Correction if more volume of fluid than volume of the subzone
uniform distribution on other non full subzones

13. II – Reconstruction of interfaces
PLIC per subzone
Youngs’ algorithm

14. II – Computation of centroid and advection
Compute the centroid for reconstructed cell
for now, mean of centroid of each subzone
to be improved because different from real centroid
Advect the centroid

15. III – Code used for the tests
Cartesian grid
Gradients computation with Youngs’ Finite Difference formula
Subgradients computation with Finite Difference formula
Lagrange + remap scheme with direction splitted remapping
interface reconstruction on 1D-stretched cells for each direction
exact intersection of transfer volume of the cell with reconstructed interface gives transfer volume for the fluid

16. Advection of an infinite vertical filamentu u
Good advection of filament with our method
Youngs’ VoF reconstruction does not move !

17. Advection of a fragment smaller than the cellu u
Advection of the fragment with our method
too slow through mesh interfaces
Youngs’ VoF reconstruction does not move !

18. Advection of a finite vertical filament
Better advection speed
Filament ends go slower than the middle u
Youngs’ VoF reconstruction moves !
Surprisingly well, but advection speed too large u

19. Advection of a filament with (u,v)
v = u/2
v = u/2
Filament is nearly preserved with our method
Advection speed is quite good
Youngs’ VoF reconstruction early leads to blobby flows !

20. Conclusions, perspectives
summary
A subgradient method to compute VoF/PLIC interfaces in subzones
At given mesh, better representation of thin structures of fluid
Not a subgrid physical model : method as geometrical as original VoF with the same volume fraction field information
Ability to advect filaments and fragments proven on simple cases
fluid distribution into subzones
centroid estimation
to be tested on more and more complex cases
… extension to 3D
… extension to unstructured mesh
… extension to more than 2 fluids
in progress
perspectives

21. Commissariat à l’énergie atomique et aux énergies alternatives
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