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

Abstract

This paper discusses a relation between the re-initialization equation of the level-set functions derived by Wac lawczyk [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 reinitialization of the interface in the state of phase equilibrium.

Topological changes due to non-equilibrium effects by means of the statistical model of two-phase flow

“…The answers to some of these questions are given in [21]. Unlike described by the sharp and diffusive interface models the mesoscopic interface Γ between two phases is rough and/or turbulent.…”

“…is the absolute temperature and σ [J/m 2 ] is the surface tension coefficient [16,17]. The discrepancy between true nature of the interface and its smooth, deterministic models has led the present author to development of the description that is alternative for the sharp and diffusive interface models and yet it shows they can be considered as complementary components of the same statistical description [22,21,23].…”

“…In the present work, recently derived [22] equation describing evolution of the intermittency region is coupled with the Navier-Stokes incompressible flow solver and used to simulate flow of the rising, two-dimensional gas bubble. In addition, we use non-equilibrium solution of the modified Ginzburg-Landau functional [21,23] derived by the present author to introduce the new mechanism of the topological changes in two-phase flow. To this goal, the two cases are considered: first, when characteristic length scale field ε h (x,t) governing the local intermittency region width is constant, and second, when ε h (x,t) is given by the prescribed analytical function.…”

“…This paper is organized as follows. In the next section, derivation of the intermittency region evolution equation and its relation with the modified Ginzburg-Landau free energy functional is recalled [21,23]. Next, we present the numerical method and set-up of the two-dimensional numerical experiment.…”

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This paper presents the first results of the two-phase flow obtained using recently introduced physical, mathematical and numerical model of the intermittency region between two-phases (Wacławczyk 2017(Wacławczyk , 2021. The statistical interpretation of the intermittency region evolution equations allows to account for the non-equilibrium effects in the domain separating two phases.

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