Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling

Abstract

Phase-field models provide a way to model fluid interfaces as having finite thickness. This can allow the computation of interface movement and deformation on fixed grids. This paper applies phase-field modeling to the computation of two-phase incompressible Navier–Stokes flows. The Navier–Stokes equations are modified by the addition of the continuum forcing −C∇→φ, where C is the composition variable and φ is C's chemical potential. The equation for interface advection is replaced by a continuum advective-diffusion equation, with diffusion driven by C's chemical potential gradients. The paper discusses how solutions to these equations approach those of the original sharp-interface Navier–Stokes equations as the interface thickness ϵ and the diffusivity both go to zero. The basic flow-physics of phase-field interfaces is discussed. Straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion. On the other hand, too large a diffusion will overly damp the flow. These two constraints result in an upper bound for the diffusivity of O(ϵ) and a lower bound of O(ϵ2). Within these two bounds, the phase-field Navier–Stokes equations appear to generate an O(ϵ) error relative to the exact sharp-interface equations. An O(h2/ϵ2) numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing. An O(h4/ϵ4) compact scheme is introduced that takes advantage of the asymptotic, comparatively smooth, behavior of the chemical potential. For O(ϵ) accurate phase-field models the optimum path to convergence for this scheme appears to be ϵ∝h4/5. The asymptotic rate of convergence corresponding to this is O(h4/5) but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear. Extensive analysis and computations show that this scheme is very effective and accurate. It allows the accurate calculation of two-phase flows with interfaces only two cells wide. Computational results are given for linear capillary waves and for Rayleigh–Taylor instabilities. The first set of computations is compared to exact solutions of the diffuse-interface equations and of the original sharp-interface equations. The Rayleigh–Taylor computations test the ability of the method to compute highly deforming flows. These flows include near-singular phenomena such as interface coalescences and breakups, contact line movement, and the formation and breakup of thin wall-films. Grid-refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed, wavelength, and the general physical characteristics of the instability and mass transfer rates.