A SUBCELL BASED INTERFACE RECONSTRUCTION METHOD TO DEAL WITH FILAMENTS ECCOMAS 2012 | Christophe Fochesato1, Raphaël Loubère2, Renaud Motte1, Jean Ovadia3 1CEA, DAM, DIF, F-91297 Arpajon, France 2Institut de Mathématiques de Toulouse, CNRS, Université de Toulouse 3Retired fellow from CEA, CESTA, F-33114 Le Barp, France

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

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

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

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

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

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

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

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

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

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

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

II – Reconstruction of interfaces PLIC per subzone Youngs’ algorithm

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

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

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

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

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

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 !

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

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