Parabolic interface reconstruction for 2D volume of fluid methods

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. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

The interface curvature is crucial for the modelling of surface tension in capillary driven flow. Traditional geometric volume of fluid (VOF) methods based on a piecewise linear interface approximation fail to converge under mesh refinement in time-dependent problems. Instead, the proposed piecewise parabolic interface calculation (PPIC) methods, specifically the parabolic LVIRA and MOF methods (PLVIRA and PMOF), accurately capture the interface dynamics. Numerical experiments demonstrate improved reconstruction accuracy, convergence of interface curvature, and Weber number independent convergence in droplet translation problems when using the PPIC methods coupled with a two-phase Navier-Stokes solver. The PLVIRA method is successfully applied to the simulation of a 2D rising bubble, showing good agreement with a reference solution.