A hierarchical level-set numerical approach for immiscible incompressible n-phase flows (n≥3)

In this article, we propose an approach for interface-resolved simulations of immiscible incompressible n-phase flows ($n\ge 3$). The standard level set (LS) method can accurately simulate the motion of two phases. However, when more than two phases are considered, the interfaces cannot intersect and numerical overlaps occur, leading to unphysical spurious velocities and unrealistic distortions of the interfaces. The proposed approach organizes a group of $n-1$ LS functions into a hierarchy to resolve the issue of numerical overlaps. The n-phase flow field is discretized using the extended finite element method (XFEM), which adequately addresses the discontinuities occurring in the pressure and velocity fields due to distinct fluid properties and surface tension effects at the interface between the fluids. This approach is applied to three-phase and four-phase fluid dynamics, involving gas and liquids under various conditions of surface tension. The $2D$ numerical tests prove that the proposed numerical methods can effectively model interactions between multi-fluid interfaces, avoiding numerical overlaps.