Abstract
Two well-established classes of the interface capturing models are the level-set and phase-field models. Level-set formulations satisfy the maximum principle for the density but are not energy-stable. On the other hand, phase-field models do satisfy the second law of thermodynamics but lack the maximum principle for the density. In this paper we derive a novel model for incompressible immiscible two-phase flow with non-matching densities and surface tension that is both energetically-stable and satisfies the maximum principle for the density. The model finds its place at the intersection of level-set and phase-field models. Its derivation is based on a diffusification of the incompressible two-phase Navier–Stokes equations with non-matching densities and surface tension and involves functional entropy variables. Additionally, we present an associated fully-discrete energy-stable method. Isogeometric analysis is used for the spatial discretization and the temporal-integration is performed with a new time-integration scheme that is a perturbation of the second-order midpoint scheme. The fully-discrete scheme is unconditionally energy-dissipative, pointwise divergence-free and satisfies the maximum principle for the density. Numerical examples in two and three dimensions verify the energetic-stability of the methodology.
Highlights
Free-surface flows appear in a large class of applications ranging from marine and offshore engineering, e.g. violent sloshing of LNG in tanks or wave impacts, to bubble dynamics
Weber number (We) derive the variable v via the concept of functional entropy variables, proposed by Liu and coworkers [27], which is basically the same as the introduction of the chemical potential known in phase-field literature
In this work we have proposed a novel diffuse-interface model for immiscible two-phase flow with non-matching densities and surface tension
Summary
Free-surface flows appear in a large class of applications ranging from marine and offshore engineering, e.g. violent sloshing of LNG in tanks or wave impacts, to bubble dynamics. Classical numerical methods for such free-surface flow problems follow the moving free-surfaces with mesh-motion. These so-called interfacetracking methods become very complex in case of topological changes (e.g. break-up or coalescence). A good alternative is formed by the interface-capturing methods which introduce an extra field that naturally captures the topological changes. This strategy transforms the moving boundary problem into a system of PDEs on a fixed domain. This significantly simplifies the complexity of associated numerical solution strategies, it introduces new challenges
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