Third-order discrete unified gas kinetic scheme for continuum and rarefied flows: Low-speed isothermal case

Improved discrete unified gas-kinetic scheme for interface capturing

In this paper, we extend the improved discrete unified gas-kinetic scheme (DUGKS) from solving the hydrodynamic equations to addressing the phase field equations, building upon our prior work [Wang et al., Phys. Fluids 35, 017106 (2023)10.1063/5.0128912]. The conservative Allen-Cahn equation and its modified form are presented first, followed by the construction of two improved DUGKS methods for interface capturing, based on the corresponding kinetic equations. The improved DUGKS for interface capturing utilizes the node distribution function instead of the interface center distribution function for evaluating the interface flux. The improved DUGKS enhances the numerical stability of the original DUGKS, and the good stability allows the calculations to be performed using large time steps, reducing the cumulative error from which more accurate predictions can be obtained. To verify the validity of the scheme, a series of numerical experiments were further carried out, including the diagonal translation, Zalesak's disk rotation, reversed single vortex, and deformation field. The comparison with the benchmark data shows that the improved DUGKS can simply and effectively capture the sharp interface and complex deformation interface of the two-phase flow interface.

Novel Schemes of No-Slip Boundary Conditions for the Discrete Unified Gas Kinetic Scheme Based on the Moment Constraints

The boundary conditions are crucial for numerical methods. This study aims to contribute to this growing area of research by exploring boundary conditions for the discrete unified gas kinetic scheme (DUGKS). The importance and originality of this study are that it assesses and validates the novel schemes of the bounce back (BB), non-equilibrium bounce back (NEBB), and Moment-based boundary conditions for the DUGKS, which translate boundary conditions into constraints on the transformed distribution functions at a half time step based on the moment constraints. A theoretical assessment shows that both present NEBB and Moment-based schemes for the DUGKS can implement a no-slip condition at the wall boundary without slip error. The present schemes are validated by numerical simulations of Couette flow, Poiseuille flow, Lid-driven cavity flow, dipole-wall collision, and Rayleigh-Taylor instability. The present schemes of second-order accuracy are more accurate than the original schemes. Both present NEBB and Moment-based schemes are more accurate than the present BB scheme in most cases and have higher computational efficiency than the present BB scheme in the simulation of Couette flow at high Re. The present Moment-based scheme is more accurate than the present BB, NEBB schemes, and reference schemes in the simulation of Poiseuille flow and dipole-wall collision, compared to the analytical solution and reference data. Good agreement with reference data in the numerical simulation of Rayleigh-Taylor instability shows that they are also of use to the multiphase flow. The present Moment-based scheme is more competitive in boundary conditions for the DUGKS.

A Strong-Form Off-Lattice Boltzmann Method for Irregular Point Clouds

Radial basis function generated finite differences (RBF-FD) represent the latest discretization approach for solving partial differential equations. Their benefits include high geometric flexibility, simple implementation, and opportunity for large-scale parallel computing. Compared to other meshfree methods, typically based upon moving least squares (MLS), the RBF-FD method is able to recover a high order of algebraic accuracy while remaining better conditioned. These features make RBF-FD a promising candidate for kinetic-based fluid simulations such as lattice Boltzmann methods (LB). Pursuant to this approach, we propose a characteristic-based off-lattice Boltzmann method (OLBM) using the strong form of the discrete Boltzmann equation and radial basis function generated finite differences (RBF-FD) for the approximation of spatial derivatives. Decoupling the discretizations of momentum and space enables the use of irregular point cloud, local refinement, and various symmetric velocity sets with higher order isotropy. The accuracy and computational efficiency of the proposed method are studied using the test cases of Taylor–Green vortex flow, lid-driven cavity, and periodic flow over a square array of cylinders. For scattered grids, we find the polyharmonic spline + poly RBF-FD method provides better accuracy compared to MLS. For Cartesian node layouts, the results are the opposite, with MLS offering better accuracy. Altogether, our results suggest that the RBF-FD paradigm can be applied successfully also for kinetic-based fluid simulation with lattice Boltzmann methods.

Analysis and assessment of the no-slip and slip boundary conditions for the discrete unified gas kinetic scheme

The discrete unified gas kinetic scheme (DUGKS) with a force term is a finite volume solver for the Boltzmann equation. Unlike the standard lattice Boltzmann method (LBM), DUGKS can be applied on nonuniform grids. For both the LBM and DUGKS, the boundary conditions need to be processed through the density distribution function. So researchers introduced the boundary conditions from the LBM frame into the DUGKS. However, the accuracy of these boundary conditions in the DUGKS has not been studied thoroughly. Through strict theoretical deduction, we find that the bounce-back (BB) scheme leads to a different dependence of the numerical error term in the DUGKS as compared to the LBM. The error term is influenced by the relaxation time and the body force. And it can be reduced by lowering the kinetic viscosity. Unlike the BB scheme, the nonequilibrium bounce-back scheme has the ability to implement real no-slip boundary condition. Furthermore, two slip boundary conditions incorporated with Navier's slip model are introduced from the LBM framework into the DUGKS. The tangential momentum change-based (TMAC) scheme can be used directly in the DUGKS because it generates no numerical error term in the DUGKS. For the combination of the bounce-back and specular reflection schemes (BSR), the relation between the slip length and the combination parameter should be modified in accordance with the numerical error term. Analysis shows that the TMAC scheme can simulate a wider range of slip length than the BSR scheme. Numerical simulations of the Couette flow and the Poiseuille flow confirm our theoretical analysis.

Arbitrary Lagrangian-Eulerian-type discrete unified gas kinetic scheme for low-speed continuum and rarefied flow simulations with moving boundaries

In this paper, the original discrete unified gas kinetic scheme (DUGKS) is extended to the arbitrary Lagrangian-Eulerian (ALE) framework to enable simulation of low-speed continuum and rarefied flows with moving boundaries. For the proposed ALE-type DUGKS, the mesh motion velocity is introduced in the Boltzmann-BGK equation and a remapping-free scheme is used to discretize the governing equation. Under this coupling framework, the complex rezoning and remapping phases implemented in the traditional ALE method are avoided. In some application areas, large discretization errors are introduced in the simulation if the geometric conservation law (GCL) is not guaranteed. Therefore, three GCL-compliant approaches are discussed, and a uniform flow test case is conducted to validate these schemes. Further, to illustrate the performance of the proposed method, four test cases are simulated, including the continuum flow around an oscillating circular cylinder, the continuum flow around a pitching NACA0012 airfoil, a moving piston driven by a rarefied gas, and the rarefied flow caused by a plate oscillating in the normal direction. Finally, an extended test case considering the rarefied flow over an oscillating circular cylinder is also studied, as this condition is not sufficiently researched. Consistent and good results obtained from the above test cases demonstrate the capability of the proposed ALE-type DUGKS to simulate moving boundary problems in different flow regimes.

Phase-field method based on discrete unified gas-kinetic scheme for large-density-ratio two-phase flows

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.