Physical formulation and numerical algorithm for simulating N immiscible incompressible fluids involving general order parameters

• Published in: Journal of computational physics Vol. 283; pp. 98 - 128
• Main Author: Dong, S.
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 15.02.2015
• Subjects: ALGORITHMS; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPARATIVE EVALUATIONS; Computational fluid dynamics; Computer simulation; COMPUTERIZED SIMULATION; DENSITY; ENERGY DENSITY; FIELD EQUATIONS; Fluid flow; Fluids; General order parameters; Large density contrast; LIQUIDS; Mathematical analysis; Mathematical models; MIXTURES; MULTIPHASE FLOW; N-phase flow; Numerical analysis; ORDER PARAMETERS; Pairwise surface tensions; Phase field; SURFACE TENSION; THERMODYNAMICS; VISCOSITY; Large density contrast; Multiphase flow; Pairwise surface tensions; General order parameters; Phase field; N-phase flow
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2014.11.039

We present a family of physical formulations, and a numerical algorithm, based on a class of general order parameters for simulating the motion of a mixture of N (N⩾2) immiscible incompressible fluids with given densities, dynamic viscosities, and pairwise surface tensions. The N-phase formulations stem from a phase field model we developed in a recent work based on the conservations of mass/momentum, and the second law of thermodynamics. The introduction of general order parameters leads to an extremely strongly-coupled system of (N−1) phase field equations. On the other hand, the general form enables one to compute the N-phase mixing energy density coefficients in an explicit fashion in terms of the pairwise surface tensions. We show that the increased complexity in the form of the phase field equations associated with general order parameters in actuality does not cause essential computational difficulties. Our numerical algorithm reformulates the (N−1) strongly-coupled phase field equations for general order parameters into 2(N−1) Helmholtz-type equations that are completely de-coupled from one another. This leads to a computational complexity comparable to that for the simplified phase field equations associated with certain special choice of the order parameters. We demonstrate the capabilities of the method developed herein using several test problems involving multiple fluid phases and large contrasts in densities and viscosities among the multitude of fluids. In particular, by comparing simulation results with the Langmuir–de Gennes theory of floating liquid lenses we show that the method using general order parameters produces physically accurate results for multiple fluid phases.