An adaptive scheme using hierarchical grids for lattice Boltzmann multi-phase flow simulations

The lattice Boltzmann (LB) method is extended and adapted to simulate multi-phase flows on non-uniform tree-type grids. Our model is an extension of the model developed by Gunstensen [Gunstensen AK, Rothman D. Lattice Boltzmann model of immiscible fluids. Phys Rev A 1991;43(8):4320–4327], which is based on the Rothman–Keller model [Rothman DH, Keller JM. Immiscible cellular automaton fluids. J Stat Phys 1988;52:1119–1127]. A first approach we use an a priori grid refinement. We find that the maximum number of possible grid levels for problems with dominant capillary forces is very restricted, if the physical interface is allowed to pass over grid interfaces. Thus a second approach based on adaptive grids was developed, where the physical interface is always discretized on the finest grid level. Efficient and flexible data structures have been developed to manage the remeshing. The application of the scheme for a rising bubble in three dimensions shows very good agreement with the semi-analytical solution and demonstrates the efficiency of our approach.