Color-gradient lattice Boltzmann model for 3D multiphase flows with density ratios

Abstract

In recent years, the interest on lattice Boltzmann (LB) methods for simulation of fluid flows is increasing. The reason for this increasing interest is the capability of LB methods to simulate flows in complex geometries such as porous media. The explicitness of the governing equations is also one of the advantages of this method that facilitates parallelization to reduce computational time. With the success of single-phase LB methods, some researchers developed models that can handle multiphase flows. One of them is the color-gradient model. The color-gradient model can successfully simulate multiphase flows with various viscosity ratios. For systems with density ratios not equal to or near unity, however, it contains errors that were reported in several studies. Ba et al. (2016) presented a color-gradient model that can simulate systems with density ratios for D2Q9 (two dimensions and nine velocities) LB model by modifying the equilibrium distribution function and introducing source terms into multi-relaxation collision to rigorously recover Navier-Stokes Equations (NSEs). In this study, I expanded the 2D procedure from Ba et al. (2016) to 3D systems that use the D3Q19 (three dimensions nineteen velocities) LB model. A new equilibrium distribution function and additional source terms in multi-relaxation collision were derived. This new model was validated with three cases with known analytical solutions, namely two-phase Poiseuille flow between two parallel plates, a static droplet, and deformation of a single droplet in Couette flows. Results from the first case showed clear improvements in the velocity profile. Results from the second case demonstrated the model’s capability to recover Young-Laplace equations. Results from the third case proved the accuracy of the model in a flowing system with a steadily deformed interface; the deformation is correlated to the interfacial tension between the two phases and the viscosities. With this validated model, a simulation in a small porous medium was run to prove the capability of this new model in a complex geometry. This model captured several phenomena inside the porous medium such as the effect of wettability on the shape of droplets and interfaces, pressure distribution, formation of segmented flows at Y-junctions, gas channeling, and Plateau-Rayleigh instability.