X-Mesh: A new approach for the simulation of two-phase flow with sharp interface

TL;DR

The paper tackles accurate sharp-interface simulation of two-phase flows without remeshing by introducing X-Mesh, a framework in which the mesh locally deforms to conform to the moving interface while preserving fixed topology. Interface tracking is achieved with a level-set function, and surface tension is incorporated via a direct pressure jump at interface nodes, yielding an exactly sharp interface (the static Laplace case gives $p_{bubble}=\sigma/R$). A sequential coupling of a stabilized Navier–Stokes solver, level-set advection, and front-relaying mesh deformation enables robust handling of large topological changes. The approach is validated on sloshing, dambreak, Rayleigh–Taylor, single-bubble rise, and bubble merging, demonstrating sharp interfaces, reduced parasitic currents to machine precision, and good agreement with analytical and benchmark results, albeit with mass-variation challenges inherent to level-set representations and PSPG stabilization.

Abstract

Accurate modeling of moving boundaries and interfaces is a difficulty present in many situations of computational mechanics. We use the eXtreme Mesh deformation approach (X-Mesh) to simulate the interaction between two immiscible flows using the finite element method, while maintaining an accurate and sharp description of the interface without remeshing. In this new approach, the mesh is locally deformed to conform to the interface at all times, which can result in degenerated elements. The surface tension between the two fluids is added by imposing the pressure jump condition at the interface, which, when combined with the X-Mesh framework, allows us to have an exactly sharp interface. If a numerical scheme fails to properly balance surface tension and pressure gradients, it leads to numerical artefacts called spurious or parasitic currents. The method presented here is well balanced and reduces such currents down to the level of machine precision.

Figures (13)

• Figure 1: Two-phase flow sketch.
• Figure 2: A triangle with its three internal angles $\theta_i$, $\theta_j$ and $\theta_k$.
• Figure 3: Moving the nodes along their velocity (top) or along the edges (bottom).
• Figure 4: Mesh deformation algorithm.
• Figure 5: Static bubble.