A numerical method based on quasi-Lagrangian Voronoi cells for two-phase flows with large density contrast

TL;DR

This work develops a quasi-Lagrangian moving Voronoi framework (SILVA) for two-phase flows with large density contrasts, achieved through a robust remapping stage using a smoothed color function to stabilize sharp interfaces and enable a semi-implicit, conservative solver that accommodates both shocks and surface tension. The method combines a first-order Voronoi gradient operator, a sharp-interface two-phase formulation with surface tension, and an operator-splitting time integration into reversible and irreversible steps, with optional implicit treatment of viscosity. Six 2D benchmarks, including circular patches, dam breaks, shock–bubble interactions, rotating squares, and rising bubbles, demonstrate good agreement with analytical solutions, experiments, and SPH references, validating stability and robustness at density ratios up to about $10^3$. The authors also implement a Lloyd-based remapping to prevent mesh degeneration and report a publicly available code, highlighting practical impact for simulating complex multi-phase flows in engineering and physics contexts; extensions to 3D and higher-order accuracy are identified as future directions.

Abstract

In this work, we use a moving Voronoi and sharp interface approach for simulating two-phase flows. At every time step, the mesh is generated anew from Voronoi seeds that behave as material points. The paper is a continuation of our previous works on moving Voronoi meshes where we have considered single phase incompressible and compressible flows. In the context of quasi-Lagrangian Voronoi simulations, problems with large density contrasts (such as water and air interface) are being treated here for the first time to the best of our knowledge. This is made possible through a remapping stage, which relies on a filtering of a color function. The resulting semi-implicit scheme is conservative and robust, allowing us to simulate both compressible and incompressible flows, including shock waves and surface tension.

Figures (11)

• Figure 1: Two Voronoi cells in two-dimensional grid. Note that the midpoint between the generating seeds $\overline{\bm{x}}_{ij} = \frac{1}{2}(\bm{x}_i + \bm{x}_j)$ is generally distinct from the edge midpoint $\bm{m}_{ij}$.
• Figure 2: Flowchart explaining the time-marching scheme of SILVA.
• Figure 3: Circular path test results.
• Figure 4: Selected frames of the dam break test. The color map indicates magnitude of velocity. Air cells are hidden from view.
• Figure 5: Comparing our numerical simulation in dam break test with an SPH result violeau and an experiment by Koshizuka and Oka koshizuka1996moving. We observe excellent agreement with the particle simulation and small discrepancy with the experiment, pointing to the possibility that some physical aspect is missing in both simulations.