Generalized modification in the lattice Bhatnagar-Gross-Krook model for incompressible Navier-Stokes equations and convection-diffusion equations

Multiple-relaxation-time lattice Boltzmann model-based four-level finite-difference scheme for one-dimensional diffusion equations

In this paper, we first present a multiple-relaxation-time lattice Boltzmann (MRT-LB) model for one-dimensional diffusion equation where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is considered. Then through the theoretical analysis, we derive an explicit four-level finite-difference scheme from this MRT-LB model. The results show that the four-level finite-difference scheme is unconditionally stable, and through adjusting the weight coefficient ω_{0} and the relaxation parameters s_{1} and s_{2} corresponding to the first and second moments, it can also have a sixth-order accuracy in space. Finally, we also test the four-level finite-difference scheme through some numerical simulations and find that the numerical results are consistent with our theoretical analysis.

Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: Modeling, analysis, and elements

In this paper, we first present a unified framework of multiple-relaxation-time lattice Boltzmann (MRT-LB) method for the Navier-Stokes and nonlinear convection-diffusion equations where a block-lower-triangular-relaxation matrix and an auxiliary source distribution function are introduced. We then conduct a comparison of the four popular analysis methods (Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion, and recurrence equations approaches) that have been used to obtain the macroscopic Navier-Stokes and nonlinear convection-diffusion equations from the MRT-LB method and show that from mathematical point of view, these four analysis methods can give the same equations at the second-order of expansion parameters. Finally, we give some elements that are needed in the implementation of the MRT-LB method and also find that some available LB models can be obtained from this MRT-LB method.

Discrete effects on boundary conditions of the lattice Boltzmann method for fluid flows with curved no-slip walls

The lattice Boltzmann method (LBM) has been formulated as a powerful numerical tool to simulate incompressible fluid flows. However, it is still a critical issue for the LBM to overcome the discrete effects on boundary conditions successfully for curved no-slip walls. In this paper, we focus on the discrete effects of curved boundary conditions within the framework of the multiple-relaxation-time (MRT) model. We analyze in detail a single-node curved boundary condition [Zhao et al., Multiscale Model. Simul. 17, 854 (2019)10.1137/18M1201986] for predicting the Poiseuille flow and derive the numerical slip at the boundary dependent on a free parameter as well as the distance ratio and the relaxation times. An approach by virtue of the free parameter is then proposed to eliminate the slip velocity while with uniform relaxation parameters. The theoretical analysis also indicates that for previous curved boundary schemes only with the distance ratio and the halfway bounce-back (HBB) boundary scheme, the numerical slip cannot be removed with uniform relaxation times virtually. We further carried out some simulations to validate our theoretical derivations, and the numerical results for the case of straight and curved boundaries confirm our theoretical analysis. Finally, for fluid flows with curved boundary geometries, resorting to more degrees of freedom from the boundary scheme may have more potential to eliminate the discrete effect at the boundary with uniform relaxation times.

Stability of the lattice kinetic scheme and choice of the free relaxation parameter

The lattice kinetic scheme (LKS), a modified version of the classical single relaxation time (SRT) lattice Boltzmann method, was initially developed as a suitable numerical approach for non-Newtonian flow simulations and a way to reduce memory consumption of the original SRT approach. The better performances observed for non-Newtonian flows are mainly due to the additional degree of freedom allowing an independent control over the relaxation of higher-order moments, independently from the fluid viscosity. Although widely applied to fluid flow simulations, no theoretical analysis of LKS has been performed. The present work focuses on a systematic von Neumann analysis of the linearized collision operator. Thanks to this analysis, the effects of the modified collision operator on the stability domain and spectral behavior of the scheme are clarified. Results obtained in this study show that correct choices of the "second relaxation coefficient" lead, to a certain extent, to a more consistent dispersion and dissipation for large values of the first relaxation coefficient. Furthermore, appropriate values of this parameter can lead to a larger linear stability domain. At moderate and low values of viscosity, larger values of the free parameter are observed to increase dissipation of kinetic modes, while leaving the acoustic modes untouched and having a less pronounced effect on the convective mode. This increased dissipation leads in general to less pronounced sources of nonlinear instability, thus improving the stability of the LKS.

Theoretical and numerical analysis of the lattice kinetic scheme for complex-flow simulations

The lattice kinetic scheme (LKS) is a modified version of the classical single relaxation time lattice Boltzmann method. Although used for many applications, especially when large variations in viscosity are involved, a thorough analysis of the scheme has not been provided yet. In the context of this work, the macroscopic behavior of this scheme is evaluated through the Chapman-Enskog analysis. It is shown that the additional degree of freedom provided in the scheme allows for an independent control of higher-order moments. These results are further corroborated by numerical simulations. The behavior of this numerical scheme is studied for selected external and internal flows to clarify the effect of the free parameter on the different moments of the distribution function. It is shown that it is more stable than SRT (single relaxation time) when confronted to fully periodic under-resolved simulations (especially for λ≈1). It can also help minimize the error coming from the viscosity-dependence of the wall position when combined with the bounce-back approach; although still present, viscosity-dependence of the wall position is reduced. Furthermore, as shown through the multiscale analysis, specific choices of the free parameter can cancel out the leading-order error. Overall, the LKS is shown to be a useful and efficient alternative to the SRT method for simulating numerically complex flows.

Maxwell-Stefan-theory-based lattice Boltzmann model for diffusion in multicomponent mixtures

The phenomena of diffusion in multicomponent (more than two components) mixtures are universal in both science and engineering, and from the mathematical point of view, they are usually described by the Maxwell-Stefan (MS)-theory-based diffusion equations where the molar average velocity is assumed to be zero. In this paper, we propose a multiple-relaxation-time lattice Boltzmann (LB) model for the mass diffusion in multicomponent mixtures and also perform a Chapman-Enskog analysis to show that the MS continuum equations can be correctly recovered from the developed LB model. In addition, considering the fact that the MS-theory-based diffusion equations are just a diffusion type of partial differential equations, we can also adopt much simpler lattice structures to reduce the computational cost of present LB model. We then conduct some simulations to test this model and find that the results are in good agreement with the previous work. Besides, the reverse diffusion, osmotic diffusion, and diffusion barrier phenomena are also captured. Finally, compared to the kinetic-theory-based LB models for multicomponent gas diffusion, the present model does not include any complicated interpolations, and its collision process can still be implemented locally. Therefore, the advantages of single-component LB method can also be preserved in present LB model.

Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn-Hilliard equations

In this paper, a comparative study of the lattice Boltzmann (LB) models for the Allen-Cahn (A-C) and Cahn-Hilliard (C-H) equations is conducted. To this end, a new LB model for the A-C equation is first proposed, where the equilibrium distribution function and the source term distribution function are delicately designed to recover the A-C equation correctly. The gradient term in this model can be computed by the nonequilibrium part of the distribution function such that the collision process can be implemented locally. Then a detailed numerical study on several classical problems is performed to give a comparison between the present model for the A-C equation and the recently developed LB model [H. Liang et al., Phys. Rev. E 89, 053320 (2014)PLEEE81539-375510.1103/PhysRevE.89.053320] for the C-H equation in terms of tracking the interface of two-phase flow. The results show that the present LB model for the A-C equation is more accurate and more stable, and also has a second-order convergence rate in space, while the convergence rate of the previous LB model for the C-H equation is only about 1.5.

Multiple-relaxation-time lattice Boltzmann model for incompressible miscible flow with large viscosity ratio and high Péclet number

A lattice Boltzmann model with a multiple-relaxation-time (MRT) collision operator is proposed for incompressible miscible flow with a large viscosity ratio as well as a high Péclet number in this paper. The equilibria in the present model are motivated by the lattice kinetic scheme previously developed by Inamuro et al. [Philos. Trans. R. Soc. London, Ser. A 360, 477 (2002)]. The fluid viscosity and diffusion coefficient depend on both the corresponding relaxation times and additional adjustable parameters in this model. As a result, the corresponding relaxation times can be adjusted in proper ranges to enhance the performance of the model. Numerical validations of the Poiseuille flow and a diffusion-reaction problem demonstrate that the proposed model has second-order accuracy in space. Thereafter, the model is used to simulate flow through a porous medium, and the results show that the proposed model has the advantage to obtain a viscosity-independent permeability, which makes it a robust method for simulating flow in porous media. Finally, a set of simulations are conducted on the viscous miscible displacement between two parallel plates. The results reveal that the present model can be used to simulate, to a high level of accuracy, flows with large viscosity ratios and/or high Péclet numbers. Moreover, the present model is shown to provide superior stability in the limit of high kinematic viscosity. In summary, the numerical results indicate that the present lattice Boltzmann model is an ideal numerical tool for simulating flow with a large viscosity ratio and/or a high Péclet number.