Simple extended lattice Boltzmann methods for incompressible viscous single-phase and two-phase fluid flows

The lattice Boltzmann method (LBM) is a numerical method with second-order spatial accuracy for incompressible viscous fluid flows based on an analogy with the kinetic theory of gases. Recently, the collision model of the LBM has become more and more complicated as its numerical stability has been enhanced. In this paper, conversely, we propose simple extended LBMs having good numerical stability with simple collision models by using the lattice kinetic scheme (LKS), which is an extended LBM. First, several schemes for single-phase flows based on the LKS are presented. The LKS is a simple stable scheme but has higher-order dissipation errors in the calculation of high-Reynolds-number flows. To solve this problem, the LKS is improved by using the linkwise artificial compressibility method (LWACM). Then, the numerical stability and accuracy of the LBM, the LKS, the LWACM, and the improved LKS are compared in a simulation of a doubly periodic shear layer at high Reynolds number with low-resolution grids. Next, a simple scheme for two-phase flows with large density ratio is constructed on the basis of the LKS and the improved LKS. The scheme is stable even for two-phase flows with large density ratio. The validation of the scheme is studied in a simulation of a binary droplet collision.