On a relation between the volume of fluid, level-set and phase field interface models

TL;DR

This paper explores the relationship between fluid volume, level-set, and phase field interface models, introducing a statistical physical model for non-flat interfaces and new numerical techniques for improved accuracy and stability.

Contribution

It establishes a relation between level-set re-initialization and phase equilibrium conditions, and proposes novel numerical schemes for accurate interface advection and re-initialization.

Findings

Enhanced re-initialization accuracy using constrained interpolation.

Second-order accurate semi-analytical Lagrangian scheme for advection.

Achieved complete second-order convergence in interface simulations.

Abstract

This paper discusses a relation between the re-initialization equation of the level-set functions derived by Wac{\l}awczyk [J.Comp.Phys., 299, (2015)] and the condition for the phase equilibrium provided by the stationary solution to the modified Allen-Cahn equation [Acta Metall., 27, (1979)]. As a consequence, the statistical model of the non-flat interface in the state of phase equilibrium is postulated. This new physical model of the non-flat interface is introduced based on the statistical picture of the sharp interface disturbed by the field of stochastic forces, it yields the relation between the sharp and diffusive interface models. Furthermore, the new techniques required for the accurate solution of the model equations are proposed. First it is shown, the constrained interpolation improves re-initialization of the level-set functions as it avoids oscillatory numerical errors typical for the second-order accurate interpolation schemes. Next, the new semi-analytical, second order accurate Lagrangian scheme is put forward to integrate the advection equation in time avoiding interface curvature oscillations introduced by the second-order accurate flux limiters. These techniques provide means to obtain complete, second-order convergence during advection and re-initialization of the interface in the state of phase equilibrium.