Phase-field-based lattice Boltzmann method for containerless freezing.

In this paper we first propose a phase-field model for the containerless freezing problems, in which the volume expansion or shrinkage of the liquid caused by the density change during the phase change process is considered by adding a mass source term to the continuum equation. Then a phase-field-based lattice Boltzmann (LB) method is further developed to simulate solid-liquid phase change phenomena in multiphase systems. We test the developed LB method by the problem of conduction-induced freezing in a semi-infinite space, the three-phase Stefan problem, and the droplet solidification on a cold surface, and the numerical results are in agreement with the analytical and experimental solutions.

Phase-field-based lattice Boltzmann method for two-phase flows with interfacial mass or heat transfer.

In this work, we develop a phase-field-based lattice Boltzmann (LB) method for a two-scalar model of the two-phase flows with interfacial mass or heat transfer. Through the Chapman-Enskog analysis, we show that the present LB method can correctly recover the governing equations for phase field, flow field, and concentration or temperature field. In particular, to derive the two-scalar equations for the mass or heat transfer, we propose a new LB model with an auxiliary source distribution function to describe the extra flux terms, and the discretizations of some derivative terms can be avoided.

Macroscopic finite-difference scheme and modified equations of the general propagation multiple-relaxation-time lattice Boltzmann model.

In this paper we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations (MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation (NACDE) and Navier-Stokes equations (NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme.

Auto-ejection of liquid from a nozzle.

Auto-ejection of liquid is an important process in engineering applications, and is also very complicated since it involves interface moving, deforming, and jet breaking up. In this work, a theoretical velocity of meniscus at nozzle exit is first derived, which can be used to analyze the critical condition for auto-ejection of liquid. Then a consistent and conservative axisymmetric lattice Boltzmann (LB) method is proposed to study the auto-ejection process of liquid jet from a nozzle.

Macroscopic finite-difference scheme based on the mesoscopic regularized lattice-Boltzmann method.

In this paper, we develop a macroscopic finite-difference scheme from the mesoscopic regularized lattice Boltzmann (RLB) method to solve the Navier-Stokes equations (NSEs) and convection-diffusion equation (CDE). Unlike the commonly used RLB method based on the evolution of a set of distribution functions, this macroscopic finite-difference scheme is constructed based on the hydrodynamic variables of NSEs (density, momentum, and strain rate tensor) or macroscopic variables of CDE (concentration and flux), and thus shares low memory requirement and high computational efficiency. Based on an accuracy analysis, it is shown that, the same as the mesoscopic RLB method, the macroscopic finite-difference scheme also has a second-order accuracy in space.

Rectangular multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: General equilibrium and some important issues.

In this paper, we develop a general rectangular multiple-relaxation-time lattice Boltzmann (RMRT-LB) method for the Navier-Stokes equations (NSEs) and nonlinear convection-diffusion equation (NCDE) by extending our recent unified framework of the multiple-relaxation-time lattice Boltzmann (MRT-LB) method [Chai and Shi, Phys. Rev. E 102, 023306 (2020)10.

Fourth-order multiple-relaxation-time lattice Boltzmann model and equivalent finite-difference scheme for one-dimensional convection-diffusion equations.

In this paper, we first develop a fourth-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the one-dimensional convection-diffusion equation (CDE) with the constant velocity and diffusion coefficient, where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is used. We also perform the Chapman-Enskog analysis to recover the CDE from the MRT-LB model. Then an explicit four-level finite-difference (FLFD) scheme is derived from the developed MRT-LB model for the CDE.

Improved hybrid Allen-Cahn phase-field-based lattice Boltzmann method for incompressible two-phase flows.

In this work we develop an improved phase-field based lattice Boltzmann (LB) method where a hybrid Allen-Cahn equation (ACE) with a flexible weight instead of a global weight is used to suppress the numerical dispersion and eliminate the coarsening phenomenon. Then two LB models are adopted to solve the hybrid ACE and the Navier-Stokes equations, respectively. Through the Chapman-Enskog analysis, the present LB model can correctly recover the hybrid ACE, and the macroscopic order parameter used to label different phases can be calculated explicitly.

Multiple-distribution-function lattice Boltzmann method for convection-diffusion-system-based incompressible Navier-Stokes equations.

In this paper, a multiple-distribution-function lattice Boltzmann method (MDF-LBM) with a multiple-relaxation-time model is proposed for incompressible Navier-Stokes equations which are considered as coupled convection-diffusion equations. Through direct Taylor expansion analysis, we show that the Navier-Stokes equations can be recovered correctly from the present MDF-LBM, and additionally, it is also found that the velocity and pressure can be directly computed through the zero and first-order moments of the distribution function. Then in the framework of the present MDF-LBM, we develop a locally computational scheme for the velocity gradient in which the first-order moment of the nonequilibrium distribution is used; this scheme is also extended to calculate the velocity divergence, strain rate tensor, shear stress, and vorticity.

Consistent and conservative phase-field-based lattice Boltzmann method for incompressible two-phase flows.

In this work, we consider a general consistent and conservative phase-field model for the incompressible two-phase flows. In this model, not only the Cahn-Hilliard or Allen-Cahn equation can be adopted, but also the mass and the momentum fluxes in the Navier-Stokes equations are reformulated such that the consistency of reduction, consistency of mass and momentum transport, and the consistency of mass conservation are satisfied. We further develop a lattice Boltzmann (LB) method, and show that through the direct Taylor expansion, the present LB method can correctly recover the consistent and conservative phase-field model.

Diffuse-interface lattice Boltzmann modeling of charged particle transport in Poiseuille flow.

In this paper, we developed a coupled diffuse-interface lattice Boltzmann method (DI-LBM) to study the transport of a charged particle in the Poiseuille flow, which is governed by the Navier-Stokes equations for fluid field and the Poisson-Boltzmann equation for electric potential field. We first validated the present DI-LBM through some classical benchmark problems, and then investigated the effect of electric field on the lateral migration of the particle in the Poiseuille flow. The numerical results show that the electric field has a significant influence on the particle migration.

Improved phase-field-based lattice Boltzmann method for thermocapillary flow.

In this paper, we present an improved phase-field-based lattice Boltzmann (LB) method for thermocapillary flows with large density, viscosity, and thermal conductivity ratios. The present method uses three LB models to solve the conservative Allen-Cahn equation, the incompressible Navier-Stokes equations, and the temperature equation. To overcome the difficulty caused by the convection term in solving the convection-diffusion equation for the temperature field, we first rewrite the temperature equation as a diffuse equation where the convection term is regarded as the source term and then construct an improved LB model for the diffusion equation.

Multiple-relaxation-time finite-difference lattice Boltzmann model for the nonlinear convection-diffusion equation.

In this paper, a multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) is developed for the nonlinear convection-diffusion equation (NCDE). Through designing the equilibrium distribution function and the source term properly, the NCDE can be recovered exactly from MRT-FDLBM. We also conduct the von Neumann stability analysis on the present MRT-FDLBM and its special case, i.

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.

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.

Phase-field-based lattice Boltzmann model for immiscible incompressible N-phase flows.

In this paper, we develop an efficient and alternative lattice Boltzmann (LB) model for simulating immiscible incompressible N-phase flows (N≥2) based on the Cahn-Hilliard phase field theory. In order to facilitate the design of LB model and reduce the calculation of the gradient term, the governing equations of the N-phase system are reformulated, and they satisfy the conservation of mass, momentum and the second law of thermodynamics. In the present model, (N-1) LB equations are employed to capture the interface, and another LB equation is used to solve the Navier-Stokes (N-S) equations, where a new distribution function for the total force is delicately designed to reduce the calculation of the gradient term.

Discrete unified gas kinetic scheme for nonlinear convection-diffusion equations.

In this paper, we develop a discrete unified gas kinetic scheme (DUGKS) for a general nonlinear convection-diffusion equation (NCDE) and show that the NCDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the present DUGKS through some classic convection-diffusion equations, and we find that the numerical results are in good agreement with analytical solutions and that the DUGKS model has a second-order convergence rate. Finally, as a finite-volume method, the DUGKS can also adopt the nonuniform mesh.

Lattice Boltzmann method for contact-line motion of binary fluids with high density ratio.

Within the phase-field framework, we present an accurate and robust lattice Boltzmann (LB) method for simulating contact-line motion of immiscible binary fluids on the solid substrate. The most striking advantage of this method lies in that it enables us to handle two-phase flows with mass conservation and a high density contrast of 1000, which is often unavailable in the existing multiphase LB models. To simulate binary fluid flows, the present method utilizes two LB evolution equations, which are respectively used to solve the conservative Allen-Cahn equation for interface capturing, and the incompressible Navier-Stokes equations for hydrodynamic properties.

Mixed bounce-back boundary scheme of the general propagation lattice Boltzmann method for advection-diffusion equations.

In this work, a mixed bounce-back boundary scheme of general propagation lattice Boltzmann (GPLB) model is proposed for isotropic advection-diffusion equations (ADEs) with Robin boundary condition, and a detailed asymptotic analysis is also conducted to show that the present boundary scheme for the straight walls has a second-order accuracy in space. In addition, several numerical examples, including the Helmholtz equation in a square domain, the diffusion equation with sinusoidal concentration gradient, one-dimensional transient ADE with Robin boundary and an ADE with a source term, are also considered. The results indicate that the numerical solutions agree well with the analytical ones, and the convergence rate is close to 2.

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.

Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows.

In this paper, we present a simple and accurate lattice Boltzmann (LB) model for immiscible two-phase flows, which is able to deal with large density contrasts. This model utilizes two LB equations, one of which is used to solve the conservative Allen-Cahn equation, and the other is adopted to solve the incompressible Navier-Stokes equations. A forcing distribution function is elaborately designed in the LB equation for the Navier-Stokes equations, which make it much simpler than the existing LB models.

General propagation lattice Boltzmann model for nonlinear advection-diffusion equations.

In this paper, a general propagation lattice Boltzmann model is proposed for nonlinear advection-diffusion equations (NADEs), and the Chapman-Enskog analysis shows that the NADEs with variable coefficients can be recovered correctly from the present model. We also perform some simulations of the linear advection-diffusion equation, nonlinear heat conduction equation, NADEs with anisotropic diffusion, and variable coefficients to test the present model, and find that the numerical results agree well with the corresponding analytical solutions. Moreover, it is also shown that by properly adjusting the two free parameters introduced into the propagation step, the present model could be more stable and more accurate than the standard lattice Bhatnagar-Gross-Krook model.

Lattice Boltzmann model for high-order nonlinear partial differential equations.

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved.