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

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