A Path-Conservative Method for a Weakly Compressible Two-Phase Model with Surface Tension

TL;DR

This work develops a novel HLLC-type path-conservative scheme for a weakly compressible two-phase model with surface tension, enabling a hyperbolic treatment of non-conservative terms and seamless inclusion of capillary effects. The method combines a first-order flux with a high-order solution reconstruction to achieve robust accuracy on unstructured meshes, and it enforces physically correct jump conditions via generalized Riemann invariants that incorporate the Young-Laplace pressure jump. Through benchmark problems (static drop, linear sloshing, capillary waves, and Rayleigh–Taylor instability), the scheme demonstrates well-balanced behavior, proper interface dynamics, and improved capillary-dominated accuracy compared to approaches that decouple surface tension as a source term. The results underscore the importance of embedding surface-tension effects directly in the hyperbolic solver for reliable simulations of incompressible two-phase flows.

Abstract

When extended to two-phase flows, weakly compressible models lead to a non-conservative system, which precludes its treatment using standard finite volume techniques. In this paper, a novel HLLC-type path-conservative scheme is formulated for the weakly compressible two-phase model. Furthermore, capillary effects are included in the proposed path-conservative scheme, eliminating the need to discretize surface tension terms separately. The first-order path-conservative formulation is combined with a local solution reconstruction technique to obtain high-order spatial accuracy. The solver is tested on several benchmark two-phase flow problems to demonstrate its efficacy.

Figures (14)

• Figure 1: The wave structure of the HLLC Riemann solver.
• Figure 2: The $i^\text{th}$ finite volume cell. Here $\Omega_i$ denotes the area of the cell and $j$ is one of the von Neumann neighbours of the $i^\text{th}$ cell. $\mathbf{n}^{ij}$ is unit normal vector to the interface between cells $i$ and $j$, whose edge length is $\Gamma_{ij}$.
• Figure 3: Static drop pressure profiles along: (a) horizontal centreline from various meshes, (b) horizontal centreline from $128 \times 128$ mesh compared against exact profile, and (c) diagonal direction from $128 \times 128$ mesh compared against exact profile.
• Figure 4: Temporal variation of spurious velocity in the static drop simulation on a $32 \times 32$ mesh, with the exact interface curvature prescribed.
• Figure 5: Schematic of the linear sloshing in a rectangular tank ($L = 1$ m, $H = 2.25$ m, and $h = 1$ m).

Theorems & Definitions (2)

• Remark
• Remark