Phase-field Model For Two-phase Flows Research Articles

A ternary phase-field model for two-phase flows in complex geometries

In this work, a ternary phase-field model for two-phase flows in complex geometries is proposed. In this model, one of the three components in the classical ternary Cahn–Hilliard model is considered as the solid phase, and only one Cahn–Hilliard equation with degenerate mobility needs to be solved due to the condition of volume conservation, which is consistent with the standard phase-field model with a single-scalar variable for two-phase flows. To depict different wetting properties at the complex fluid–solid boundaries, the spreading parameters in ternary phase-field model are determined based on the Young’s law, in which the liquid–solid surface tension coefficient is assumed to be a linear function of gas–liquid surface tension coefficient and related to the contact angle and the minimum curvature of the solid surface. In addition, to achieve a high viscosity in the solid phase and preserve the velocity boundary conditions on the solid surface, the phase-field variable of the solid phase is also used to derive the modified Navier–Stokes equations. To test the present model, we further develop a consistent and conservative Hermite-moment based lattice Boltzmann method where an adjustable scale factor is introduced to improve the numerical stability, and conduct the numerical simulations of several benchmark problems. The results illustrate that present model has the good capability in the study of the two-phase flows in complex geometries.

Efficient decoupled second-order numerical scheme for the flow-coupled Cahn–Hilliard phase-field model of two-phase flows

We construct in this paper a fully-decoupled and second-order accurate numerical scheme for solving the Cahn–Hilliard–Navier–Stokes phase-field model of two-phase incompressible flows. A full decoupling method is used by introducing several nonlocal variables and their ordinary differential equation to deal with the nonlinear and coupling terms. By combining with some effective methods to handle the Navier–Stokes equation, we obtain an efficient and easy-to-implement numerical scheme in which one only needs to solve several fully-decoupled linear elliptic equations with constant coefficients at each time step. We further prove the unconditional energy stability and solvability rigorously, and present various numerical simulations in 2D and 3D to demonstrate the efficiency and stability of the proposed scheme, numerically.

Energy stable and mass conservative numerical method for a generalized hydrodynamic phase-field model with different densities

A general hydrodynamic phase-field model for two-phase flows with general average velocity and variable densities is rigorously derived using thermodynamic laws and Onsager’s variational principle. The pressure is naturally involved in both the Cahn–Hilliard equation and Navier–Stokes equation. The proposed model includes two famous phase-field models as its special cases. The model admits a natural energy dissipation law. A semi-implicit, totally linear, and energy stable numerical scheme is proposed for the model, which uses an intermediate velocity involving all driving forces, including surface tension, pressure, and gravity. The tight coupling relationship between pressure and velocity is decoupled. Another advantage of the proposed scheme is that the intermediate velocity allows us to preserve the mass conservation, and consequently, there is no need to impose any mass balance equation in the Navier–Stokes equation as usual. The discrete energy dissipation law is proved rigorously. Several numerical examples are simulated to demonstrate that the proposed method can preserve the energy stability and total mass conservation for complex hydrodynamical flow problems with large density contrasts and gravity.

A diffuse interface model for two-phase flows with phase transition

A thermodynamically consistent diffuse interface model is developed for two-phase flows with phase transition. Inside the liquid-vapor mixing layer, the Cahn-Hilliard equation governing the evolution of liquid/vapor species explicitly includes the phase transition between the two species. In the present model, the specific Helmholtz free energy of the binary mixing fluid as well as other thermodynamic functions consists of a classical part (ideal mixing free energy plus a double-well potential of the mass fraction) and a nonclassical part consisting of the gradient of mass fraction. The nonclassical chemical potential is defined in a manner similar to the classical chemical potential. According to the present model, there exists a (conserved) nonclassical stress tensor in the mixing fluid, which is a distributed form of the surface tension across the mixing layer. Using the principle of entropy increase, it is observed that the rate of local phase transition is proportional to the local generalized chemical potential, and a system of transport equations of the mixing fluid is derived. This system of equations can be reduced to that of Guo and Lin [“A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects,” J. Fluid Mech. 766, 226 (2015)] except for the formula on chemical potential. The present diffuse-interface model will also be reduced to the general classical sharp interface model when the mixing layer collapses to a sharp surface.

A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations

We develop a fractional extension of a mass-conserving Allen–Cahn phase field model that describes the mixture of two incompressible fluids. The fractional order controls the sharpness of the interface, which is typically diffusive in integer-order phase-field models. The model is derived based on an energy variational formulation. An additional constraint is employed to make the Allen–Cahn formulation mass-conserving and comparable to the Cahn–Hilliard formulation but at reduced cost. The spatial discretization is based on a Petrov–Galerkin spectral method whereas the temporal discretization is based on a stabilized ADI scheme both for the phase-field equation and for the Navier–Stokes equation. We demonstrate the spectral accuracy of the method with fabricated smooth solutions and also the ability to control the interface thickness between two fluids with different viscosity and density in simulations of two-phase flow in a pipe and of a rising bubble. We also demonstrate that using a formulation with variable fractional order we can deal simultaneously with both erroneous boundary effects and sharpening of the interface at no extra resolution.

A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects

AbstractIn this paper, we develop a phase-field model for binary incompressible (quasi-incompressible) fluid with thermocapillary effects, which allows for the different properties (densities, viscosities and heat conductivities) of each component while maintaining thermodynamic consistency. The governing equations of the model including the Navier–Stokes equations with additional stress term, Cahn–Hilliard equations and energy balance equation are derived within a thermodynamic framework based on entropy generation, which guarantees thermodynamic consistency. A sharp-interface limit analysis is carried out to show that the interfacial conditions of the classical sharp-interface models can be recovered from our phase-field model. Moreover, some numerical examples including thermocapillary convections in a two-layer fluid system and thermocapillary migration of a drop are computed using a continuous finite element method. The results are compared with the corresponding analytical solutions and the existing numerical results as validations for our model.

An efficient moving mesh spectral method for the phase-field model of two-phase flows

We develop in this paper a moving mesh spectral method for the phase-field model of two-phase flows with non-periodic boundary conditions. The method is based on a variational moving mesh PDE for the phase function, coupled with efficient semi-implicit treatments for advancing the mesh function, the phase function and the velocity and pressure in a decoupled manner. Ample numerical results are presented to demonstrate the accuracy and effectiveness of the moving mesh spectral method.