S. Dong

Z. Yang, S. Dong

We present an energy-stable scheme for numerically approximating the governing equations for incompressible two-phase flows with different densities and dynamic viscosities for the two fluids. The proposed scheme employs a scalar-valued auxiliary energy variable in its formulation, and it satisfies a discrete energy stability property. More importantly, the scheme is computationally efficient. Within each time step, it computes two copies of the flow variables (velocity, pressure, phase field function) by solving individually a linear algebraic system involving a constant and time-independent coefficient matrix for each of these field variables. The coefficient matrices involved in these linear systems only need to be computed once and can be pre-computed. Additionally, within each time step the scheme requires the solution of a nonlinear algebraic equation about a {\em scalar-valued number} using the Newton's method. The cost for this nonlinear solver is very low, accounting for only a few percent of the total computation time per time step, because this nonlinear equation is about a scalar number, not a field function. Extensive numerical experiments have been presented for several two-phase flow problems involving large density ratios and large viscosity ratios. Comparisons with theory show that the proposed method produces physically accurate results. Simulations with large time step sizes demonstrate the stability of computations and verify the robustness of the proposed method. An implication of this work is that energy-stable schemes for two-phase problems can also become computationally efficient and competitive, eliminating the need for expensive re-computations of coefficient matrices, even at large density ratios and viscosity ratios.

S. Dong

We present an effective method for simulating wall-bounded multiphase flows consisting of $N$ ($N\geqslant 2$) immiscible incompressible fluids with different densities, viscosities and pairwise surface tensions. The N-phase physical formulation is based on a modified thermodynamically consistent phase field model that is more general than in a previous work, and it is developed by considering the reduction consistency if some of the fluid components were absent from the system. We propose an N-phase contact-angle boundary condition that is reduction consistent between $N$ phases and $M$ phases ($2\leqslant M\leqslant N-1$). We also present a numerical algorithm for solving the N-phase governing equations together with the contact-angle boundary conditions developed herein. Extensive numerical experiments are presented for several flow problems involving multiple fluid components and solid-wall boundaries to investigate the wettability effects with multiple types of contact angles. In particular, we compare simulation results with the de Gennes theory for the contact-angle effects on the liquid drop spreading on wall surfaces, and demonstrate that our method produces physically accurate results.

S. H. Challa, S. Dong, L. D. Zhu

We present a hybrid spectral element-Fourier spectral method for solving the coupled system of Navier-Stokes and Cahn-Hilliard equations to simulate wall-bounded two-phase flows in a three-dimensional domain which is homogeneous in at least one direction. Fourier spectral expansions are employed along the homogeneous direction and $C^0$ high-order spectral element expansions are employed in the other directions. A critical component of the method is a strategy we developed in a previous work for dealing with the variable density/viscosity of the two-phase mixture, which makes the efficient use of Fourier expansions in the current work possible for two-phase flows with different densities and viscosities for the two fluids. The attractive feature of the presented method lies in that the two-phase computations in the three-dimensional space are transformed into a set of de-coupled two-dimensional computations in the planes of the non-homogeneous directions. The overall scheme consists of solving a set of de-coupled two-dimensional equations for the flow and phase-field variables in these planes. The linear algebraic systems for these two-dimensional equations have constant coefficient matrices that need to be computed only once and can be pre-computed. We present ample numerical simulations for different cases to demonstrate the accuracy and capability of the presented method in simulating the class of two-phase problems involving solid walls and moving contact lines.

S. Dong, X. Wang

Two-phase outflows refer to situations where the interface formed between two immiscible incompressible fluids passes through open portions of the domain boundary. We present in this paper several new forms of open boundary conditions for two-phase outflow simulations within the phase field framework. In addition, we also present a rotational pressure-correction based algorithm for numerically treating these open boundary conditions. Our algorithm gives rise to linear algebraic systems for the velocity and the pressure that involve only constant and time-independent coefficient matrices after discretization, despite the variable density and variable viscosity of the two-phase mixture. By comparing simulation results with the theory and the experimental data, we show that the method developed herein produces physically accurate results. We also present numerical experiments to demonstrate the long-term stability of the method in situations where large density contrast, large viscosity contrast, and backflows are present at the two-phase outflow/open boundaries.