  Simulation of multiphase flows require coupled capturing or tracking of the
interfaces in conjunction with the solution of fluid motion often occurring at
multiple scales. We will present unified cascaded LB methods based on central
moments for the solution of the incompressible two-phase flows at high density
ratios and for capturing of the interfacial dynamics. Based on a modified
continuous Boltzmann equation (MCBE) for two-phase flows, where a kinetic
transformation to the distribution function involving the pressure field is
introduced to reduce the associated numerical stiffness at high density
gradients, a central moment cascaded LB formulation for computing the fluid
motion will be constructed. In this LB scheme, the collision step is prescribed
by the relaxation of various central moments to their equilibria that are
reformulated in terms of the pressure field obtained via matching to the
continuous equilibria based on the transformed Maxwell distribution.
Furthermore, the differential treatments for the effects of the source term
representing the change due to the pressure field and of the source term due to
the interfacial tension force and body forces appearing in the MCBE on
different moments are consistently accounted for in this cascaded LB solver
that computes the pressure and velocity fields. In addition, another cascaded
LB scheme via modified equilibria will be developed to solve for the
interfacial dynamics represented by a phase field model based on the
conservative Allen-Cahn equation. Based on numerical simulations of a variety
of two-phase flow benchmark problems at high density ratios and involving the
effects of surface tension and its tangential gradients (Marangoni stresses),
we will validate our unified cascaded LB approach and also demonstrate
improvements in numerical stability.
