A Coupled Lattice Boltzmann Method to Solve Nernst–Planck Model for Simulating Electro-osmotic Flows

In this paper, we focus on the nonlinear coupling mechanism of the Nernst–Planck model and propose a coupled lattice Boltzmann method (LBM) to solve it. In this method, a new LBM for the Nernst–Planck equation is developed, a multi-relaxation-time (MRT)-LBM for flow field and an LBM for the Poisson equation are used. And then, we discuss the choice of the model and found that the MRT-LBM is much more stable and accurate than the LBGK model. A reasonable iterative sequence and evolution number for each LBM are proposed by considering the properties of the coupled LBM. The accuracy and stability of the presented coupled LBM are also discussed through simulating electro-osmotic flows (EOF) in micro-channels. Furthermore, to test the applicability of it, the EOF with non-uniform surface potential in micro-channels based on the Nernst–Planck model is simulated. And we investigate the effects of non-uniform surface potential on the pattern of the EOF at different external applied electric fields. Finally, a comparison of the difference between the Nernst–Planck model and the Poisson–Boltzmann model is presented.     

New model and scheme for compressible fluids of the finite difference lattice Boltzmann method and direct simulations of aerodynamic sound

A new scheme for the finite difference lattice Boltzmann method is proposed, in which negative viscosity term is introduced to reduce the viscosity and the calculation time can be remarkably reduced for high Reynolds number flows. A model with additional internal degree of freedom is also presented for diatomic gases such as air, in which an additional distribution function is introduced. Direct simulations of aero-acoustics by using the proposed model and scheme are presented. Speed of sound is correctly recovered. As typical examples, the Aeolian tone emitted by a circular cylinder is successfully simulated even very low Mach number flow. Full three-dimensional sound emission is also given.     

Lattice Boltzmann modeling of microchannel flows in the transition flow regime

Owing to its kinetic nature and distinctive computational features, the lattice Boltzmann method for simulating rarefied gas flows has attracted significant research interest in recent years. In this article, a lattice Boltzmann (LB) model is presented to study microchannel flows in the transition flow regime, which have gained much attention because of fundamental scientific issues and technological applications in various micro-electro-mechanical system (MEMS) devices. In the model, a Bosanquet-type effective viscosity is used to account for the rarefaction effect on gas viscosity. To match the introduced effective viscosity and to gain an accurate simulation, a modified second-order slip boundary condition with a new set of slip coefficients is proposed. Numerical investigations demonstrate that the results, including the velocity profile, the non-linear pressure distribution along the channel, and the mass flow rate, are in good agreement with the solution of the linearized Boltzmann equation, the direct simulation Monte Carlo (DSMC) results, and the experimental results over a broad range of Knudsen numbers. It is shown that taking the rarefaction effect on gas viscosity into consideration and employing an appropriate slip boundary condition can lead to a significant improvement in the modeling of rarefied gas flows with moderate Knudsen numbers in the transition flow regime.     

Simulation of floating bodies with the lattice Boltzmann method

This paper is devoted to the simulation of floating rigid bodies in free surface flows. For that, a lattice Boltzmann based model for liquid–gas–solid flows is presented. The approach is built upon previous work for the simulation of liquid–solid particle suspensions on the one hand, and on an interface-capturing technique for liquid–gas free surface flows on the other. The incompressible liquid flow is approximated by a lattice Boltzmann scheme, while the dynamics of the compressible gas are neglected. We show how the particle model and the interface capturing technique can be combined by a novel set of dynamic cell conversion rules. We also evaluate the behaviour of the free surface–particle interaction in simulations. One test case is the rotational stability of non-spherical rigid bodies floating on a plane water surface–a classical hydrostatic problem known from naval architecture. We show the consistency of our method in this kind of flows and obtain convergence towards the ideal solution for the heeling stability of a floating box.     

A Lattice BGK Scheme with General Propagation

The standard Lattice BGK (LBGK) scheme often encounters numerical instability in simulation of fluid flow with small kinematic viscosity or as the nondimensional relaxation time is close to 0.5. In this paper, based on a time-splitting scheme for the Boltzmann equation with discrete velocities, a new LBGK scheme with general propagation step is proposed to address this problem. In this model, two free parameters are introduced into the propagation step, which can be adjusted to obtain a small kinematic viscosity and improved numerical stability as well. Numerical simulations of the two-dimensional Taylor vortex and the unsteady Womersley flow are carried out to test the kinematic viscosity, numerical diffusion, and numerical stability of the proposed scheme.     

A higher-order moment method of the lattice Boltzmann model for the Korteweg–de Vries equation

In this paper, a lattice Boltzmann model for the Korteweg–de Vries (KdV) equation with higher-order accuracy of truncation error is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model, our method has a wide flexibility to select equilibrium distribution function. The higher-order moment method bases on so-called a series of lattice Boltzmann equation obtained by using multi-scale technique and Chapman–Enskog expansion. We can also control the stability of the scheme by modulating some special moments to design the dispersion term and the dissipation term. The numerical example shows the higher-order moment method can be used to raise the accuracy of truncation error of the lattice Boltzmann scheme.     

Modelling solute transport in shallow water with the lattice Boltzmann method

The lattice Boltzmann method is used to investigate the solute transport in shallow water flows. Shallow water equations are solved using the lattice Boltzmann equation on a D2Q9 lattice with multiple-relaxation-time (MRT-LBM) and Bhatnagar–Gross–Krook (BGK-LBM) terms separately, and the advection–diffusion equation is also solved with a LBM-BGK on a D2Q5 lattice. Three cases: open channel flow with side discharge, shallow recirculation flow and flow in a harbour are simulated to verify the described methods. Agreements between predictions and experiments are satisfactory. In side discharge flow, the reattachment length for different ratios of side discharge velocity to main channel velocity has been studied in detail. Furthermore, the performance of MRT-LBM and BGK-LBM for these three cases has been investigated. It is found that LBM-MRT has better stability and is able to satisfactorily simulate flows with higher Reynolds number. The study shows that the lattice Boltzmann method is simple and accurate for simulating solute transport in shallow water flows, and hence it can be applied to a wide range of environmental flow problems.     

Evaluation of three lattice Boltzmann models for multiphase flows in porous media

A free energy (FE) model, the Shan–Chen (S–C) model, and the Rothman and Keller (R–K) model are studied numerically to evaluate their performance in modeling two-dimensional (2D) immiscible two-phase flow in porous media on the pore scale. The FE model is proved to satisfy the Galilean invariance through a numerical test and the mass conservation of each component in the simulations is exact. Two-phase layered flow in a channel with different viscosity ratios was simulated. Comparing with analytical solutions, we see that the FE model and the R–K model can give very accurate results for flows with large viscosity ratios. In terms of accuracy and stability, the FE model and the R–K model are much better than the S–C model. Co-current and countercurrent two-phase flows in complex homogeneous media were simulated and the relative permeabilities were obtained. Again, it is found that the FE model is as good as the R–K model in terms of accuracy and efficiency. The FE model is shown to be a good tool for the study of two-phase flows with high viscosity ratios in porous media.     

A fractional step lattice Boltzmann method for simulating high Reynolds number flows

A fractional step lattice Boltzmann scheme is presented to greatly improve the stability of the lattice Boltzmann method (LBM) in modelling incompressible flows at high Reynolds number. This method combines the good features of the conventional LBM and the fractional step technique. Through the fractional step, the flow at an extreme case of infinite Reynolds number (inviscid flow) can be effectively simulated. In addition, the non-slip boundary condition can be directly implemented.     

Optimal relaxation collisions for lattice Boltzmann methods

The optimal relaxation time of about 0.8090 has been proposed to balance the efficiency, stability, and accuracy at a given lattice size of numerical simulations with lattice Boltzmann methods. The optimal lattice size for a desired Reynolds number can be refined by reducing the Mach number for incompressible flows. The functioned polylogarithm polynomials are defined and used to express the lattice Boltzmann equations at different time scales and to analyze the impact of relaxation times and lattice sizes on truncation errors. Smaller truncation errors can be achieved when relaxation times are greater than 0.5 and less than 1.0. The steady-state lid-driven cavity flow was chosen to validate the code of lattice Boltzmann procedures. The applications of the optimal relaxation parameters numerically balance the stability, efficiency, and accuracy through Hartmann flow. The optimal relaxation time can also be used to select the initial lattice size for the channel flow over a square cylinder with a given Mach number.     

A lattice Boltzmann approach for the non-Newtonian effect in the blood flow

A new lattice Boltzmann approach within the framework of D2Q9 lattice for simulating shear-thinning non-Newtonian blood flows described by the power-law, Carreau-Yasuda and Casson rheology models is proposed in this study. The essence of this method lies in splitting the complete non-Newtonian effect up into two portions: one as the Newtonian result and the other as an effective external source. This arrangement takes the advantage in remaining fixed relaxation time during the whole course of numerical simulation that can avoid the potential numerical instability caused by the relaxation time approaches to 1/2, an inherent difficulty in the conventional lattice Boltzmann methods using varying relaxation times for the non-Newtonian effect. Macroscopically, consistency of the proposed model with the equations of motion for the three target non-Newtonian models is demonstrated through the technique of Chapman-Enskog multi-scale expansion. The feasibility and accuracy of the method are examined by comparing with the analytical solutions of the two-dimensional Poiseuille flows based on the power-law and Casson models. The results show that the velocity profiles agree very well with those of analytical solutions and the error analyses demonstrate that the proposed scheme is with second-order accuracy. The present approach also demonstrates its superiority over the conventional lattice Boltzmann method in the extent of numerical stability for simulating the power-law-based shear-thinning flows. The straightforwardness in scheme derivation and implementation renders the present approach as a potential method for the complex non-Newtonian flows.     

A New Multiple-relaxation-time Lattice Boltzmann Method for Natural Convection

This article is devoted to the study of multiple-relaxation-time (MRT) lattice Boltzmann method with eight-by-eight collision matrix for natural convection flow. In the velocity space, eight speed directions are used and the corresponding incompressible multiple-relaxation-time model with force term is presented. D2Q4 model is for temperature field. The coupled double distribution functions (DDF) overcome artificial compressible effect corresponding to the standard MRT model. The simulations of natural convection flows with Pr=0.71 for air and Ra=10³–10⁹ are carried out and excellent agreements are obtained to demonstrate the numerical accuracy and stability of the proposed model.     

Application of lattice Boltzmann method in free surface flow simulation of micro injection molding

The filling flow in micro injection molding was simulated by using the lattice Boltzmann method (LBM). A tracking algorithm for free surface to handle the complex interaction between gas and liquid phases in LBM was used for the free surface advancement. The temperature field in the filling flow is also analyzed by combining the thermal lattice Boltzmann model and the free surface method. To simulate the fluid flow of polymer melt with a high Prandtl number and high viscosity, a modified lattice Boltzmann scheme was adopted by introducing a free parameter in the thermal diffusion equation to overcome the restriction of the thermal relaxation time. The filling flow simulation of micro injection molding was successfully performed in the study.     

Lattice Boltzmann modeling of dendritic growth in forced and natural convection

A two-dimensional (2D) coupled model is developed for the simulation of dendritic growth during alloy solidification in the presence of forced and natural convection. Instead of conventional continuum-based Navier–Stokes (NS) solvers, the present model adopts a kinetic-based lattice Boltzmann method (LBM), which describes flow dynamics by the evolution of distribution functions of moving pseudo-particles, for the numerical computations of flow dynamics as well as thermal and solutal transport. The dendritic growth is modeled using a solutal equilibrium approach previously proposed by Zhu and Stefanescu (ZS), in which the evolution of the solid/liquid interface is driven by the difference between the local equilibrium composition and the local actual liquid composition. The local equilibrium composition is calculated from the local temperature and curvature. The local temperature and actual liquid composition, controlled by both diffusion and convection, are obtained by solving the LB equations using the lattice Bhatnagar–Gross–Krook (LBGK) scheme. Detailed model validation is performed by comparing the simulations with analytical predictions, which demonstrates the quantitative capability of the proposed model. Furthermore, the convective dendritic growth features predicted by the present model are compared with those obtained from the Zhu–Stefanescu and Navier–Stokes (ZS–NS) model, in which the fluid flow is calculated using an NS solver. It is found that the evolution of the solid fraction of dendritic growth calculated by both models coincides well. However, the present model has the significant advantages of numerical stability and computational efficiency for the simulation of dendritic growth with melt convection.     

Artificial compressibility method and lattice Boltzmann method: Similarities and differences

The artificial compressibility method and the lattice Boltzmann method yield the solutions of the incompressible Navier–Stokes equations in the limit of the vanishing Mach number. The inclusion of the bulk viscosity is considered to be one of the reasons for the success of the lattice Boltzmann method at least in the 2D case. In the present paper, the robustness of the artificial compressibility method is enhanced by introducing a new dissipation term, which makes high cell-Reynolds number computation possible. The increase of the stability is also confirmed in the linear stability analysis; the magnitude of the eigenvalues are drastically reduced for low resolution. Comparisons are made with the lattice Boltzmann method. It is confirmed that the fortified ACM is more robust as well as more accurate than the lattice Boltzmann method.     

Hydrodynamics in Porous Media: A Finite Volume Lattice Boltzmann Study

Fluid flow through porous media is of great importance for many natural systems, such as transport of groundwater flow, pollution transport and mineral processing. In this paper, we propose and validate a novel finite volume formulation of the lattice Boltzmann method for porous flows, based on the Brinkman–Forchheimer equation. The porous media effect is incorporated as a force term in the lattice Boltzmann equation, which is numerically solved through a cell-centered finite volume scheme. Correction factors are introduced to improve the numerical stability. The method is tested against fully porous Poiseuille, Couette and lid-driven cavity flows. Upon comparing the results with well-documented data available in literature, a satisfactory agreement is observed. The method is then applied to simulate the flow in partially porous channels, in order to verify its potential application to fractured porous conduits, and assess the influence of the main porous media parameters, such as Darcy number, porosity and porous media thickness.     

Numerical Method Based on the Lattice Boltzmann Model for the Kuramoto-Sivashinsky Equation

In this paper, we proposed a lattice Boltzmann model based on the higher-order moment method for the Kuramoto-Sivashinsky equation. A series of partial differential equations obtained by using multi-scale technique and Chapman-Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the fifth-order dispersion term and the sixth-order dissipation term. As results, the Kuramoto-Sivashinsky equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the Kuramoto-Sivashinsky equation.     

Multi-block lattice Boltzmann simulations of subcritical flow in open channel junctions

A two-dimensional lattice Boltzmann model (LBM) for subcritical flows in open channel junctions is developed. Shallow water equations coupled with the large eddy simulation model is numerically simulated by the lattice Boltzmann method, so that the turbulence, caused by the combination of the main channel and tributary flows, can be taken into account and modeled efficiently. In order to obtain more detailed and accurate results, a multi-block lattice scheme is designed and applied at the area of combining flows. The model is first verified by experimental data for a 90° junction flow, then is used to investigate the effect of the junction angle on flow characteristics, such as velocity field, water depth and separation zone. The objectives of this study are to validate the two-dimensional LBM in junction flow simulation and compare the results with available experimental data and classical analytical solutions in the separation zone.     

Optimal design for non-Newtonian flows using a topology optimization approach

We study non-Newtonian effects on the layout and geometry of flow channels using a material distribution based topology optimization approach. The flow is modeled with the single-relaxation hydrodynamic lattice Boltzmann method, and the shear dependence of viscosity is included through the Carreau–Yasuda model for non-Newtonian fluids. To represent the viscosity of blood in this model, we use non-Newtonian similarity. Further, we introduce a scaling to decrease the effects of the non-Newtonian model in porous regions in order to stabilize the coupling of the LBM porosity and non-Newtonian flow models. For the resulting flow model, we derive the non-Newtonian sensitivity analysis for steady-state conditions and illustrate the non-Newtonian effect on channel layouts for a 2D dual-pipe design problem at different Reynolds numbers.     

Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods—Part I: Formulation and application to the 2-D Burgers’ equation

The lattice Boltzmann method is being increasingly employed in the field of computational fluid dynamics due to its computational efficiency. Floating-point operations in the lattice Boltzmann method involve local data and therefore allow easy cache optimization and parallelization. Due to this, the cache-optimized lattice Boltzmann method has superior computational performance over traditional finite difference methods for solving unsteady flow problems. When solving steady flow problems, the explicit nature of the lattice Boltzmann discretization limits the time step size and therefore the efficiency of the lattice Boltzmann method for steady flows. To quantify the computational performance of the lattice Boltzmann method for steady flows, a comparison study between the lattice Boltzmann method (LBM) and the alternating direction implicit (ADI) method was performed using the 2-D steady Burgers’ equation. The comparison study showed that the LBM performs comparatively poor on high-resolution meshes due to smaller time step sizes, while on coarser meshes where the time step size is similar for both methods, the cache-optimized LBM performance is superior. Because flow domains can be discretized with multiblock grids consisting of coarse and fine grid blocks, the cache-optimized LBM can be applied on the coarse grid block while the traditional implicit methods are applied on the fine grid blocks. This paper finds the coupled cache-optimized lattice Boltzmann-ADI method to be faster by a factor of 4.5 over the traditional methods while maintaining similar accuracy.