A numerical study of pressure calculation in a two-phase lattice kinetic scheme: A comparison between explicit and implicit methods

In this study, we numerically evaluate a characteristic explicit pressure calculation method for the improved lattice kinetic scheme (LKS) for two-phase flows. The explicit scheme utilizes a pressure evolution equation and incorporates inner iterations to increase computational stability. For the comparative analysis, we introduce an implicit approach in which the pressure is calculated by solving the Poisson equation using the simplified marker and cell (SMAC) method. The primary objective is to elucidate the relationship between the number of inner iterations and the acoustic error in the improved LKS, which is verified through a comparative analysis with the implicit method. We perform benchmark simulations under identical conditions for three canonical two-phase flow problems: (i) a stationary droplet, (ii) a single rising bubble, and (iii) Rayleigh–Taylor instability. Our results reveal that the number of inner iterations in the improved LKS affects not only numerical stability but also pressure oscillations at the acoustic scale. Specifically, the frequency of the acoustic waves depends on the number of inner iterations. Particularly under external forces such as gravity, the use of many iterations leads to a lower frequency and faster decay of acoustic waves. The implicit scheme successfully suppresses such acoustic wave scale effects, resulting in smoother pressure evolution; however, in terms of computational efficiency, the explicit method significantly outperforms the implicit method. These findings elucidate the trade-offs among numerical stability, acoustic wave errors, and computational cost in two-phase simulations.