Fokker-Planck Central Moment Lattice Boltzmann Methods with a Renormalization Principle for Interfacial Tracking and Simulation of Two-Phase Flows

The Fokker-Planck (FP) equation represents drift and diffusion in stochastic processes in various statistical physics applications but is also a well-defined model for the collision integral of the continuous Boltzmann equation for hydrodynamical phenomena. The lattice Boltzmann method (LBM) is a dramatic simplification of the Boltzmann equation and is commonly used for the numerical simulation of fluid dynamics, including multiphase flows. Extending our recent work, we construct novel collision operators by taking central moments of the respective FP models to recover the conservative Allen-Cahn equation for interface tracking and a pressure-based kinetic formulation for two-phase flows with surface tension effects in both 2D and 3D. They effectively involve relaxations to the so-called Markovian central moment attractors that depend on products of lower order central moments and diffusion tensor parameters in a recurrent manner. When the different central moments are relaxed at different rates for improved numerical stability, hyperviscosity effects can arise when simulating flows at relatively very high Reynolds numbers with using other existing collision models based on the Maxwellians; such artifacts are naturally eliminated in our FP-based formulation via determining the diffusion tensor parameters based on a renormalization principle involving the second order central moments as their fixed points under collisional relaxations. We validate the resulting LBMs for a variety of interface tracking and two-phase flow benchmarks at high density ratios in 2D and 3D and show significant numerical stability advantages of our formulation when compared to other existing models used in the LBM for multiphase flows.