The lattice Boltzmann equation is often advocated as a simulation tool that is particularly effective for complex fluids such as multiphase and multicomponent flows through porous media. We construct a three-dimensional 19 velocity lattice Boltzmann model for immiscible binary fluids with variable viscosities and density ratio based on the model proposed by Gunstensen. The model is tested for the following binary fluid flow problems: a stationary planar interface among two fluids; channel flow of immiscible binary fluids; the Laplace problem; and a rising bubble. The results agree well with semi-analytic results in a range of the Eötvös, Morton and Reynolds number. We also present preliminary simulation results for two large-scale realistic applications: the flow of an air-water mixture in a waste-water batch reactor and the saturation hysteresis effect in soil flow. We discuss some limitations of the lattice Boltzmann method in the simulation of realistic and difficult multiphase problems.
After a short discussion of recent discretization techniques for the lattice-Boltzmann equations we motivate and discuss some alternative approaches using implicit, nonuniform FD discretization and mesh refinement techniques. After presenting results of a stability analysis we use an implicit approach to simulate a boundary layer test problem. The numerical results compare well to the reference solution when using strongly refined meshes. Some basic ideas for a nonuniform mesh refinement (with non-cartesian mesh topology) are introduced using the standard discretization procedure of alternating collision and propagation.
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