A pressure approach of cumulant phase-field lattice Boltzmann method for simulating multiphase flows

Abstract

The cumulant lattice Boltzmann method (LBM) has been recently used to simulate multiphase-multicomponent flows by applying an external force. Furthermore, the mass and momentum are not conserved when an external force is used. In the classical approach, the third-order derivatives in density necessitate information from a large stencil of neighbors, which affects parallelization and is computationally expensive. In this paper, we propose an equilibrium distribution function in the moment space, which includes diagonal and off diagonal elements of the pressure tensor. Consequently, the interfacial tension effect can be exerted into this equilibrium function, circumventing the need for an external force. The Cahn–Hilliard equation can be coupled to the method to track the interface at multiphase-multicomponent flows. This function is applied on the moment, central, and cumulant LBM and transferred back to the discrete space to be used in Bhatnagar–Gross–Krook LBM. These key advantages include simplicity, easy-to-implement, and high parallelization capability due to removing high-order derivatives. An immiscible two-component flow between two parallel plates is simulated and compared with the analytical solution at different viscosities for the moment LBM and the cumulant LBM. Numerical results are in good agreement with analytical solutions. Moreover, a dispersed droplet in a continuous phase under shear flow is simulated to show the capability of the proposed method in the breaking-up process modeling. It is demonstrated that spurious velocities are less affected by decreasing the viscosity and cumulant LBM with the proposed function, while the interfacial tension is calculated accurately. Finally, the method has been extended for three dimensions, and two cases of a three-dimensional breakup of a liquid thread and collision of two equal droplets are studied to show the ability of this method to simulate the coalescence and breakup process.