A consistent and conservative volume distribution algorithm and its applications to multiphase flows using Phase-Field models

TL;DR

This paper introduces a novel, consistent, and conservative volume distribution algorithm for multiphase flows, applies it to develop improved Phase-Field models, and validates their accuracy and stability in complex multiphase dynamics.

Contribution

It presents a new volume distribution algorithm satisfying key constraints and develops a multiphase conservative Allen-Cahn model with a boundedness mapping, enhancing multiphase flow simulations.

Findings

The multiphase conservative Allen-Cahn model better preserves structures than the Cahn-Hilliard model.

The boundedness mapping ensures physical bounds and properties of order parameters.

The models accurately capture complex multiphase dynamics with large density/viscosity ratios.

Abstract

In the present study, the multiphase volume distribution problem, where there can be an arbitrary number of phases, is addressed using a consistent and conservative volume distribution algorithm. The proposed algorithm satisfies the summation constraint, the conservation constraint, and the \textit{consistency of reduction}. The first application of the volume distribution algorithm is to determine the Lagrange multipliers in multiphase Phase-Field models that enforce the mass conservation, and a multiphase conservative Allen-Cahn model that satisfies the \textit{consistency of reduction} is developed. A corresponding consistent and conservative numerical scheme is developed for the model. The multiphase conservative Allen-Cahn model has a better ability than the multiphase Cahn-Hilliard model to preserve under-resolved structures. The second application is to develop a numerical procedure, called the boundedness mapping, to map the order parameters, obtained numerically from a multiphase model, into their physical interval, and at the same time to preserve the physical properties of the order parameters. Along with the consistent and conservative schemes for the multiphase Phase-Field models, the numerical solutions of the order parameters are reduction consistent, conservative, and bounded, which are theoretically analyzed and numerically validated. Then, the multiphase Phase-Field models are coupled with the momentum equation by satisfying the \textit{consistency of mass conservation} and the \textit{consistency of mass and momentum transport}, thanks to the consistent formulation. It is demonstrated that the proposed model and scheme converge to the sharp-interface solution and are capable of capturing the complicated multiphase dynamics even when there is a large density and/or viscosity ratio.