### Speaker

Mrs
Marie Béchereau
(Ecole normale supérieure de Cachan (Paris-Saclay))

### Description

Two-fluid extensions of Lattice Boltzmann methods with free boundaries usually consider "microscopic''
pseudopotential interface models. In this paper, we rather propose an interface-capturing Lattice
Boltzmann approach where the mass fraction variable is considered as an unknown and is advected.
Several works have reported the difficulties of LBM methods to deal with such two-fluid systems
especially for high-density ratio configurations. This is due to the mixing nature of LBM, as with Flux
vector splitting approaches for Finite Volume methods. We here give another explanation of the lack of
numerical diffusion of Lattice Boltzmann approaches to accurately capture contact discontinuities. To fix
the problem, we propose an arbitrary Lagrangian-Eulerian (ALE) formulation of Lattice-Boltzmann
methods. In the Lagrangian limit, it allows for a proper separated treatment of pressure waves and
advection phenomenon. After the ALE solution, a remapping (advection) procedure is necessary to
project the variables onto the Eulerian Lattice-Boltzmann grid.
We explain how to derive this remapping procedure in order to get second-order accuracy and achieve
sharp stable oscillation-free interfaces. It has been shown that mass fractions variables satisfy a local
discrete maximum principle and thus stay in the range $[0,1]$. The theory is supported by numerical
computations of the free fall of an initial square block
of a dense fluid surrounded by a lighter fluid into a box. Figures 1 and 2 are showing the mass fraction
field of the light fluid at two successive instants. The density ratio equal to 4 and the computational lattice
grid is 400x400. One can appreciate the thickness of the numerical diffuse interface, the capture of
complex structures and the capability to compute strong changes of free boundary topology.
Even if our methods are currently used for inviscid flows (Euler equations) by projecting the discrete
distributions onto equilibrium ones at each time step, we believe that it is possible to extend the
framework formulation for multifluid viscous problems. This will be at the aim of a next work.