A mass-conserving and volume-preserving lattice Boltzmann method with dynamic grid refinement for immiscible ternary flows

In this paper, a lattice Boltzmann model with dynamic grid refinement is proposed for immiscible ternary flows, which is capable of conserving the total mass and preserving the volume of each phase. The application of interpolation schemes in adaptive mesh refinement (AMR) techniques results in the violation of the total mass of the fluids system within the lattice Boltzmann method (LBM) framework. In the present model, a source term with two free parameters is introduced into the interface capturing equation, which can be determined by the mass conservation and the volume preservation properties. A piecewise constant function has been deliberately incorporated into the source term in order to avoid the appearance of an unphysical fluid at the interface of other two fluids. Based on a block-structured AMR method, the governing equations for phase-field variables and flow hydrodynamic properties are solved by the finite-difference multiple-relaxation-time (MRT) LBM. Simulations of several typical problems are performed in order to evaluate the accuracy and applicability of the proposed model. The numerical results demonstrate that the present model can conserve both mass and volume at the same time as well as reduce numerical dispersion in the bulk region.