In this paper, a phase-field method under the framework of discrete unified gas-kinetic scheme (DUGKS) for incompressible multiphase fluid flows is proposed. Two kinetic models are constructed to solve the conservative Allen-Cahn equation that accounts for the interface behavior and the incompressible hydrodynamic equations that govern the flow field, respectively. With a truncated equilibrium distribution function as well as a temporal derivative added to the source term, the macroscopic governing equations can be exactly recovered from the kinetic models through the Chapman-Enskog analysis. Calculation of source terms involving high-order derivatives existed in the quasi-incompressible model is simplified. A series of benchmark cases including four interface-capturing tests and four binary flow tests are carried out. Results compared to that of the lattice Boltzmann method (LBM) have been obtained. A convergence rate of second order can be guaranteed in the test of interface diagonal translation. The capability of the present method to track the interface that undergoes a severe deformation has been verified. Stationary bubble and spinodal decomposition problems, both with a density ratio as high as 1000, are conducted and reliable solutions have been provided. The layered Poiseuille flow with a large viscosity ratio is simulated and numerical results agree well with the analytical solutions. Variation of positions of the bubble front and spike tip during the evolution of Rayleigh-Taylor instability has been predicted precisely. However, the detailed depiction of complicated interface patterns appearing during the evolution process is failed, which is mainly caused by the relatively large numerical dissipation of DUGKS compared to that of LBM. A high-order DUGKS is needed to overcome this problem.

The discrete unified gas kinetic scheme (DUGKS) is a new finite volume (FV) scheme for continuum and rarefied flows, which combines the benefits of both the lattice Boltzmann method and UGKS. By the reconstruction of the gas distribution function using particle velocity characteristic lines, the flux contains more detailed information of fluid flow and more concrete physical nature. In this work, a simplified DUGKS is proposed with the reconstruction stage on a whole time step instead of a half time step in the original DUGKS. Using the temporal/spatial integral Boltzmann Bhatnagar–Gross–Krook equation, the auxiliary distribution function with the inclusion of the collision effect is adopted. The macroscopic and mesoscopic fluxes of the cell on the next time step are predicted by the reconstruction of the auxiliary distribution function at interfaces along particle velocity characteristic lines. According to the conservation law, the macroscopic variables of the cell on the next time step can be updated through its flux, which is a moment of the predicted mesoscopic flux at cell interfaces. The equilibrium distribution function on the next time step can also be updated. The gas distribution function is updated by the FV scheme through its predicted mesoscopic flux in a time step. Compared with the original DUGKS, the computational process of the proposed method is more concise because of the omission of half time step flux calculation. The numerical time step is only limited by the Courant–Friedrichs–Lewy condition, and a relatively good stability has been preserved. Several test cases, including the Couette flow, lid-driven cavity flow, laminar flows over a flat plate, a circular cylinder, and an airfoil, and microcavity flow cases, are conducted to validate the present scheme. The observed numerical simulation results reasonably agree with the reported results.

In this paper, the simplified discrete unified gas-kinetic scheme presented in the former paper is extended from incompressible flow to compressible flow at a high Mach number. In our earlier work, a simplified discrete unified gas–kinetic scheme was developed for low-speed flow in which the Mach number is small for keeping the incompressible property. To simulate compressible flow, the governing equation of the internal energy distribution function presented as potential energy including the Prandtl number effect is introduced to the present method. The velocity field is coupled with density and internal energy by the evolution of distribution functions related to mass, momentum, and temperature. For simplification and computational efficiency, the D2Q13 circular distribution function is applied as the equilibrium model. Compared to our earlier work, higher Mach number flows can be simulated by the proposed method, which is of the ability to simulate compressible flow. A number of numerical test cases from incompressible to compressible flows have been conducted, including incompressible lid-driven cavity flow, Taylor vortex flow, transonic flow past NACA (National Advisory Committee for Aeronautics) 0012 airfoil, Sod shock tube, supersonic flow past a circular cylinder, and isentropic vortex convection. All simulation results agree well with the reference data.

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