Phase-field-theory-based lattice Boltzmann equation method for N immiscible incompressible fluids

Phase-field-based lattice Boltzmann model for immiscible incompressible N-phase flows

In this paper, we develop an efficient and alternative lattice Boltzmann (LB) model for simulating immiscible incompressible N-phase flows (N≥2) based on the Cahn-Hilliard phase field theory. In order to facilitate the design of LB model and reduce the calculation of the gradient term, the governing equations of the N-phase system are reformulated, and they satisfy the conservation of mass, momentum and the second law of thermodynamics. In the present model, (N-1) LB equations are employed to capture the interface, and another LB equation is used to solve the Navier-Stokes (N-S) equations, where a new distribution function for the total force is delicately designed to reduce the calculation of the gradient term. The developed model is first validated by two classical benchmark problems, including the tests of static droplets and the spreading of a liquid lens, the simulation results show that the current LB model is accurate and efficient for simulating incompressible N-phase fluid flows. To further demonstrate the capability of the LB model, two numerical simulations, including dynamics of droplet collision for four fluid phases and dynamics of droplets and interfaces for five fluid phases, are performed to test the developed model. The results show that the present model can successfully handle complex interactions among N (N≥2) immiscible incompressible flows.

Reduction-consistent phase-field lattice Boltzmann equation for N immiscible incompressible fluids

In this paper, we develop a reduction-consistent conservative phase-field method for interface-capturing among N (N≥ 2) immiscible fluids, which is governed by conservative Allen-Cahn equation (CACE); here the reduction-consistent property is that if only M (1≤M≤N-1) immiscible fluids are present in a N-phase system, the governing equations for N immiscible fluids must reduce to the corresponding M immiscible fluids system. Then we propose a reduction-consistent lattice Boltzmann equation (LBE) method for solving N immiscible incompressible fluids with high density and viscosity contrasts. Some numerical simulations are carried out to validate the present LBE such as stationary droplets, spreading of a liquid lens, and spinodal decomposition together with the reduction-consistent property, and the numerical results predicted by present LBE are in good agreement with the analytical solutions/other numerical results, which also demonstrate the reduction-consistent property by present LBE.

Multiphase flows of N immiscible incompressible fluids: Conservative Allen-Cahn equation and lattice Boltzmann equation method

In this paper, we develop a conservative phase-field method for interface-capturing among N (N≥2) immiscible fluids, the evolution of the fluid-fluid interface is captured by conservative Allen-Cahn equation (CACE), and the interface force of N immiscible fluids is incorporated to Navier-Stokes equation (NSE) by chemical potential form. Accordingly, we propose a lattice Boltzmann equation (LBE) method for solving N (N≥2) immiscible incompressible NSE and CACE at high density and viscosity contrasts. Numerical simulations including stationary droplets, Rayleigh-Taylor instability, spreading of liquid lenses, and spinodal decompositions are carried out to show the accuracy and capability of present LBE, and the results show that the predictions by use of the present LBE agree well with the analytical solutions and/or other numerical results.