Lattice Boltzmann equation for microscale gas flows of binary mixtures

Discrete unified gas kinetic scheme for all Knudsen number flows. III. Binary gas mixtures of Maxwell molecules

Recently a discrete unified gas kinetic scheme (DUGKS) in a finite-volume formulation based on the Boltzmann model equation has been developed for gas flows in all flow regimes. The original DUGKS is designed for flows of single-species gases. In this work, we extend the DUGKS to flows of binary gas mixtures of Maxwell molecules based on the Andries-Aoki-Perthame kinetic model [P. Andries et al., J. Stat. Phys. 106, 993 (2002)JSTPBS0022-471510.1023/A:1014033703134. A particular feature of the method is that the flux at each cell interface is evaluated based on the characteristic solution of the kinetic equation itself; thus the numerical dissipation is low in comparison with that using direct reconstruction. Furthermore, the implicit treatment of the collision term enables the time step to be free from the restriction of the relaxation time. Unlike the DUGKS for single-species flows, a nonlinear system must be solved to determine the interaction parameters appearing in the equilibrium distribution function, which can be obtained analytically for Maxwell molecules. Several tests are performed to validate the scheme, including the shock structure problem under different Mach numbers and molar concentrations, the channel flow driven by a small gradient of pressure, temperature, or concentration, the plane Couette flow, and the shear driven cavity flow under different mass ratios and molar concentrations. The results are compared with those from other reliable numerical methods. The results show that the proposed scheme is an effective and reliable method for binary gas mixtures in all flow regimes.

Discrete unified gas kinetic scheme for all Knudsen number flows: low-speed isothermal case

Based on the Boltzmann-BGK (Bhatnagar-Gross-Krook) equation, in this paper a discrete unified gas kinetic scheme (DUGKS) is developed for low-speed isothermal flows. The DUGKS is a finite-volume scheme with the discretization of particle velocity space. After the introduction of two auxiliary distribution functions with the inclusion of collision effect, the DUGKS becomes a fully explicit scheme for the update of distribution function. Furthermore, the scheme is an asymptotic preserving method, where the time step is only determined by the Courant-Friedricks-Lewy condition in the continuum limit. Numerical results demonstrate that accurate solutions in both continuum and rarefied flow regimes can be obtained from the current DUGKS. The comparison between the DUGKS and the well-defined lattice Boltzmann equation method (D2Q9) is presented as well.

Lattice-type-dependent momentum-exchange method for moving boundaries

The conventional momentum-exchange method (CME) is verified to be accurate for the stationary boundary by Mei et al. [Phys. Rev. E 65, 041203 (2002)], but it might be inaccurate when the boundary is moving in the lattice Boltzmann simulations. A lattice-type-dependent momentum-exchange method (LME) is presented to evaluate the hydrodynamic force on moving boundaries, in which the additional momenta induced by the type-changing lattices are well considered. LME preserves the superior features of CME, such as reliability, simplicity, and parallelism. Without any interpolation and integration, the algorithm is independent of boundary geometries, and therefore, efficient in computation and easy to be implemented in both two and three dimensions. A series of cylinder sedimentations are simulated to illustrate the accuracy and robustness of LME, and the results are in excellent agreement with those by the arbitrary Lagrangian-Eulerian technique (ALE). The lateral migrations of a particle are also investigated in the simulations of a neutrally buoyant cylinder in a Poiseuille flow, and consistent with the Segré-Silberberg effect.

Multicomponent lattice Boltzmann model from continuum kinetic theory

We derive from the continuum kinetic theory a multicomponent lattice Boltzmann model with intermolecular interaction. The resulting model is found to be consistent with the model previously derived from a lattice-gas cellular automaton [X. Shan and H. Chen, Phys. Rev. E 47, 1815 (1993)] but applies in a much broader domain. A number of important insights are gained from the kinetic theory perspective. First, it is shown that even in the isothermal case, the energy equipartition principle dictates the form of the equilibrium distribution function. Second, thermal diffusion is shown to exist and the corresponding diffusivities are given in terms of macroscopic parameters. Third, the ordinary diffusion is shown to satisfy the Maxwell-Stefan equation at the ideal-gas limit.