Sharp Interfaces

Accurately capturing sharp interfaces on a fixed Eulerian grid is a major challenge for computational fluid dynamics. Attempts to minimise phase boundaries in multiphase LBMs cause them to become fixed to the underlying computational grid instead of advecting with the fluid velocity. Paul Dellar and I proposed lattice Boltzmann implementations of conservation equations with stiff source terms to better understand lattice pinning. By incorporating stochastic sharping we were able to eliminating pinning a facetting and correctly compute very narrow interfaces.

An overview of this work is given in the following presentation from the 2010 UK Lattice Boltzmann Workshop.

Plot of the vortex reversal benchmark test at times t=1,2,3, and 6 produced by the LBM Interface Sharpening algorithm

Our first model offered a theoretical explanation of lattice pinning, and some ways to eliminate (or at least reduce ) it, but it model does not readily lend itself to flows with significant local changes in interface topology. This is because of the variable, and re-calculated, sharpening threshold (but on the plus side, no numerical differentiation in needed so the sharpening is a purely local and algebraic source term!). I proposed a alternative and perhaps more usable (in terms of engineering applications) Conservative Interface Sharpening Lattice Boltzmann Model  (first presented in the 2010 UK LBM workshop (see the slides above)). The price to pay is some (nonlocal) numerical differentiation on the lattice, but this is required in all multiphase LBMs. The model was originally presented at the 2010 UK Lattice Boltzmann workshop in Oxford, and this presentation is shown below.

Plot of the phase field against its radius after one full reversal of the vortex reversal problem, showing a narrow circular interface. 

The resulting numerical scheme is nevertheless simple, accurate, does not leak mass, and can capture significant interface deformations, as can be seen in these two plots of the stringent Vortex Reversal benchmark test. This work was inspired by the Artificial Compression Model of Olsson, Kreiss,and Zahedi.

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