Diffuse-interface two-phase flow models with different densities: A new quasi-incompressible form and a linear energy-stable method

Abstract

While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable schemes for their numerical simulation are lacking. In this paper, we derive a new form of thermodynamically-consistent quasi-incompressible diffuse-interface Navier-Stokes Cahn-Hilliard model for a two-phase flow of incompressible fluids with different densities. The derivation is based on mixture theory by invoking the second law of thermodynamics and Coleman-Noll procedure. We also demonstrate that our model and some of the existing models are equivalent and we provide a unification between them. In addition, we develop a linear and energy-stable time-integration scheme for the derived model. Such a linearly-implicit scheme is nontrivial, because it has to suitably deal with all nonlinear terms, in particular those involving the density. Our proposed scheme is the first linear method for quasi-incompressible two-phase flows with nonsolenoidal velocity that satisfies discrete energy dissipation independent of the time-step size, provided that the mixture density remains positive. The scheme also preserves mass. Numerical experiments verify the suitability of the scheme for two-phase flow applications with high density ratios using large time steps by considering the coalescence and break-up dynamics of droplets including pinching due to gravity.

A diffuse domain method for two-phase flows with large density ratio in complex geometries

“…2018; Shokrpour Roudbari et al. 2018). Some of these models, however, do not satisfy Galilean invariance, local mass conservation or are not thermodynamically consistent.…”

“…2017; Shokrpour Roudbari et al. 2018) were developed to study binary fluids with variable densities, where the volume fraction was employed as the phase variable and a different free energy was used for the model derivations.…”

An unconditionally energy-stable scheme based on an implicit auxiliary energy variable for incompressible two-phase flows with different densities involving only precomputable coefficient matrices

“…Several thermodynamically consistent phase field models are already available in the literature for two-phase and multiphase flows, see e.g. [29,25,1,42,2,11,28,16,14,34], with various degrees of sophistication or the observance/violation of other physical principles such as Galilean invariance and reduction consistency. The mass conservation of the individual fluid components in the system, and the choice of an appropriate form for the free energy density function, naturally give rise to the Cahn-Hilliard equation in two phases or a system of coupled Cahn-Hilliard type equations (see e.g.…”

“…On the other hand, energy-stable schemes (see e.g. [40,35,21,18,41,20,47,34], among others) can potentially allow the use of much larger time step sizes in dynamic simulations. The downside lies in that, the computational cost per time step of these schemes can be very high.…”

“…The schemes can preserve the mass conservation on the discrete level, and they lead to coupled highly-nonlinear algebraic systems after discretization. Numerical schemes for related quasi-incompressible hydrodynamic phase field models have also been proposed in [16,34,17]. In particular, in [17] the so-called invariant energy quadratization method has been used to reformulate the phase field equation, and the resultant numerical schemes are second-order in time, and involve the solution of linear algebraic systems that couple together the different flow variables.…”