Parabolic interface reconstruction for 2D volume of fluid methods

TL;DR

This paper introduces parabolic interface reconstruction methods for 2D volume of fluid simulations, significantly improving curvature accuracy and convergence in time-dependent capillary flow problems.

Contribution

It proposes a novel piecewise parabolic interface approximation (PPIC) framework, including the PLVIRA and PMOF methods, enhancing accuracy and convergence over traditional linear approaches.

Findings

Increased reconstruction accuracy by one order

Convergence of interface curvature with mesh refinement

Weber number independent droplet translation results

Abstract

For capillary driven flow the interface curvature is essential in the modelling of surface tension via the imposition of the Young--Laplace jump condition. We show that traditional geometric volume of fluid (VOF) methods, that are based on a piecewise linear approximation of the interface, do not lead to an interface curvature which is convergent under mesh refinement in time-dependent problems. Instead, we propose to use a piecewise parabolic approximation of the interface, resulting in a class of piecewise parabolic interface calculation (PPIC) methods. In particular, we introduce the parabolic LVIRA and MOF methods, PLVIRA and PMOF, respectively. We show that a Lagrangian remapping method is sufficiently accurate for the advection of such a parabolic interface. It is numerically demonstrated that the newly proposed PPIC methods result in an increase of reconstruction accuracy by one order, convergence of the interface curvature in time-dependent advection problems and Weber number independent convergence of a droplet translation problem, where the advection method is coupled to a two-phase Navier--Stokes solver. The PLVIRA method is applied to the simulation of a 2D rising bubble, which shows good agreement to a reference solution.