A two-dimensional lattice Boltzmann model for uniform channel flows

In this paper a novel two-dimensional lattice Boltzmann model (LBM) is developed for uniform channel flows. The axial velocity is solved from a momentum diffusion equation over the cross-sectional plane. An extrapolation boundary condition is also introduced to enhance the no-slip boundary in the momentum equation. This boundary treatment can also be applied to LBM simulations of other diffusion processes. The algorithm and boundary treatment are validated by simulations of steady Poiseuille and pulsatile Womersley flows in circular pipes. The numerical convergence and accuracy are comparable to those of existing models. Moreover, comparison with general three-dimensional lattice Boltzmann simulations demonstrates the advantages of our two-dimensional model, including lower computational resource requirements (memory and time), easier boundary treatment for arbitrary cross-sectional shapes, and no velocity constraint. These features are attractive for practical applications with uniform channel flows.     

LES of turbulent square jet flow using an MRT lattice Boltzmann model

In this paper we consider the application of multiple-relaxation-time (MRT) lattice Boltzmann equation (LBE) for large-eddy simulation (LES) of turbulent flows. The implementation is discussed in the context of 19-velocity (D3Q19) MRT-LBE model in conjunction with the Smagorinsky subgrid closure model. The MRT-LBE-LES is then tested in the turbulent square jet flow case. We compare MRT-LBE-LES results with (a) single-relaxation-time (SRT) or BGK LBE results and (b) experimental data. Computed results include the distribution of centerline mean streamwise velocity, jet spread, and spanwise profiles of mean streamwise velocity in the near-field region. The phenomenon of axis switching is investigated. The advantages of MRT over SRT are demonstrated. Reasonable agreement between our numerical results and experimental data demonstrate that the MRT-LBE is a potentially viable tool for LES of turbulence.     

A thermal lattice Boltzmann model with diffuse scattering boundary condition for micro thermal flows

A lattice Boltzmann model for simulating isothermal micro flows has been proposed by us recently [Niu XD, Chew YT, Shu C. A lattice Boltzmann BGK model for simulation of micro flows. Europhys Lett 2004;67(4):600]. In this paper, we extend the model to simulate the micro thermal flows. In particular, the thermal lattice Boltzmann equation (TLBE) [He X, Chen S, Doolen GD. A novel thermal model for the lattice Boltzmann method in incompressible limit. J Comput Phys 1998;146:282] is used with modification of the relaxation times linking to the Knudsen number. The diffuse scattering boundary condition (DSBC) derived in our early model is extended to consider temperature jump at wall boundaries. Simple theoretical analyses of the DSBC are presented and the results are found to be consistent with the conventional velocity slip and temperature jump boundary conditions. Numerical validations are carried out by simulating two-dimensional thermal Couette flows and developing thermal flows in a microchannel, and the obtained results are found to be in good agreement with those given from the direct simulation Monte Carlo (DSMC), the molecular dynamics (MD) approaches and the Maxwell theoretical prediction.     

General regularized boundary condition for multi-speed lattice Boltzmann models

The lattice Boltzmann method is nowadays a common tool for solving computational fluid dynamics problems. One of the difficulties of this numerical approach is the treatment of the boundaries, because of the lack of physical intuition for the behavior of the density distribution functions close to the walls. A massive effort has been made by the scientific community to find appropriate solutions for boundaries. In this paper we present a completely generic way of treating a Dirichlet boundary for two- and three-dimensional flat walls, edges or corners, for weakly compressible flows, applicable for any lattice topology. The proposed algorithm is shown to be second-order accurate and could also be extended for compressible and thermal flows.     

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.     

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.     

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.     

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 parallel lattice Boltzmann method for large eddy simulation on multiple GPUs

To improve the simulation efficiency of turbulent fluid flows at high Reynolds numbers with large eddy dynamics, a CUDA-based simulation solution of lattice Boltzmann method for large eddy simulation (LES) using multiple graphics processing units (GPUs) is proposed. Our solution adopts the “collision after propagation” lattice evolution way and puts the misaligned propagation phase at global memory read process. The latest GPU platform allows a single CPU thread to control up to four GPUs that run in parallel. In order to make use of multiple GPUs, the whole working set is evenly partitioned into sub-domains. We implement Smagorinsky model and Vreman model respectively to verify our multi-GPU solution. These two LES models have different relaxation time calculation behavior and lead to different CUDA implementation characteristics. The implementation based on Smagorinsky model achieves 190 times speedup over the sequential implementation on CPU, while the implementation based on Vreman model archives more than 90 times speedup. The experimental results show that the parallel performance of our multi-GPU solution scales very well on multiple GPUs. Therefore large-scale (up to 10,240 $\times $ 10,240 lattices) LES–LBM simulation becomes possible at a low cost, even using double-precision floating point calculation.     

Lattice BBGKY scheme for two-phase flows: One-dimensional case

A novel lattice Boltzmann model for two-phase fluids is presented. We begin with the two-body BBGKY equation, and perform a coordinate transformation to split it into a Boltzmann equation for the one-body distribution, coupled to a kinetic equation for the correlation function. The coupling is accomplished by a self-consistent force. The resulting lattice Boltzmann model for nonideal fluids is grounded in the physics of the two-body distribution function. The discrete velocity model is described in detail, and numerical results are given for phase separation in one dimension.     

A direct-forcing immersed boundary method for the thermal lattice Boltzmann method

In this study, a direct-forcing immersed boundary method (IBM) for thermal lattice Boltzmann method (TLBM) is proposed to simulate the non-isothermal flows. The direct-forcing IBM formulas for thermal equations are derived based on two TLBM models: a double-population model with a simplified thermal lattice Boltzmann equation (Model 1) and a hybrid model with an advection–diffusion equation of temperature (Model 2). As an interface scheme, which is required due to a mismatch between boundary and computational grids in the IBM, the sharp interface scheme based on second-order bilinear and linear interpolations (instead of the diffuse interface scheme, which uses discrete delta functions) is adopted to obtain the more accurate results. The proposed methods are validated through convective heat transfer problems with not only stationary but also moving boundaries – the natural convection in a square cavity with an eccentrically located cylinder and a cold particle sedimentation in an infinite channel. In terms of accuracy, the results from the IBM based on both models are comparable and show a good agreement with those from other numerical methods. In contrast, the IBM based on Model 2 is more numerically efficient than the IBM based on Model 1.     

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.     

An analysis of natural convection using the thermal finite element discrete Boltzmann equation

The lattice Boltzmann method (LBM) is the simple numerical simulator for fluids because it consists of linear equations. Excluding the higher differential term, the LBM for a temperature field is also achieved as an easy numerical simulation method. However, the LBM is hardly applied to body fitted coordinates for its formulation. It is then difficult to calculate complex lattices using the LBM. In this paper, the finite element discrete Boltzmann equation (FEDBE) is introduced to deal with this weakness of the LBM. The finite element method is applied to the discrete Boltzmann equation (DBE) of the basic equation of the LBM. For FEDBE, the simulation using complex lattices is achieved, and it will be applicable for the development in engineering fields. The natural convection in a square cavity and the Rayleigh–Bernard convection are chosen as the test problem. Each simulation model is accurate enough for the flow patterns, the temperature distribution and the Nusselt number. This method is now considered good for the flow and temperature field, and is expected to be introduced for complex lattices using the DBE.     

五阶精度WENO差分型格子波尔兹曼算法求解单守恒方程

给出了五阶精度WENO差分型格子波尔兹曼算法求解单守恒模型方程的计算方法.根据WENO差分格式的特点,定义了广义格子波尔兹曼分布函数,将守恒型方程的求解问题,转化成用WENO格式的差分算法对该分布函数进行求解.该方法的意义在于,将高精度高分辨率的WENO格式差分方法与近几十年发展起来的格子波尔兹曼方法相结合,从而很方便地构造出可以用于求解守恒型方程的格子波尔兹曼模型,使格子波尔兹曼方法在可压缩流领域的使用更简单.利用该方法分别构造了不同初值条件下的一维Burgers守恒型方程的求解模型,求出结果,并分析了模型的精度和稳定性.最后总结了方法的优点和不足,以及有待进一步研究解决的问题.     

Simulation of incompressible two-phase flows with large density differences employing lattice Boltzmann and level set methods

A hybrid lattice Boltzmann and level set method (LBLSM) for two-phase immiscible fluids with large density differences is proposed. The lattice Boltzmann method is used for calculating the velocities, the interface is captured by the level set function and the surface tension force is replaced by an equivalent force field. The method can be applied to simulate two-phase fluid flows with the density ratio up to 1000. In case of zero or known pressure gradient the method is completely explicit. In order to validate the method, several examples are solved and the results are in agreement with analytical or experimental results.     

Three-dimensional discrete-velocity BGK model for the incompressible Navier–Stokes equations

The lattice Boltzmann method (LBM) has been widely used for the simulations of the incompressible Navier–Stokes (NS) equations. The finite difference Boltzmann method (FDBM) in which the discrete-velocity Boltzmann equation is solved instead of the lattice Boltzmann equation has also been applied as an alternative method for simulating the incompressible flows. The particle velocities of the FDBM can be selected independently from the lattice configuration. In this paper, taking account of this advantage, we present the discrete velocity Boltzmann equation that has a minimum set of the particle velocities with the lattice Bharnagar–Gross–Krook (BGK) model for the three-dimensional incompressible NS equations. To recover incompressible NS equations, tensors of the particle velocities have to be isotropic up to the fifth rank. Thus, we propose to apply the icosahedral vectors that have 13 degrees of freedom to the particle velocity distributions. Validity of the proposed model (D3Q13BGK) is confirmed by numerical simulations of the shear-wave decay problem and the Taylor–Green vortex problem. With respect to numerical accuracy, computational efficiency and numerical stability, we compare the proposed model with the conventional lattice BGK models (D3Q15, D3Q19 and D3Q27) and the multiple-relaxation-time (MRT) model (D3Q13MRT) that has the same degrees of freedom as our proposal. The comparisons show that the compressibility error of the proposed model is approximately double that of the conventional lattice BGK models, but the computational efficiency of the proposed model is superior to that of the others. The linear stability of the proposed model is also superior to that of the lattice BGK models. However, in non-linear simulations, the proposed model tends to be less stable than the others.     

Velocity inversion of micro cylindrical Couette flow: A lattice Boltzmann study

In this work the micro gas flow between two concentric cylinders is investigated by a lattice Boltzmann equation (LBE) model with multiple relaxation times. A local kinetic boundary condition is proposed for the LBE to model the gas–wall interaction. Numerical simulations are conducted to examine the tangential velocity distribution under different flow conditions. It is shown that the proposed LBE can capture the velocity inversion phenomenon successfully. Comparisons with the Navier–Stokes solutions and DSMC results are also made and it is shown that the LBE yields better predictions.     

三分子自催化反应扩散系统中化学波的二维格子Boltzmann模拟

建立二维区域带平方衰减反应的三分子自催化反应扩散系统的数学模型,利用浓度分布的Chapmann-Enskogz展开及多尺度技术,给出基于格子Boltzmann模型的二维三分子自催化反应扩散系统的数值求解法,得到半开放自催化反应扩散系统在反应与扩散机制同时作用所产生化学波的过程和浓度空间分布值.数值结果表明本文提供的求解化学波现象的方法是有效的.     

An upwind discretization scheme for the finite volume lattice Boltzmann method

The fact that the classic lattice Boltzmann method is restricted to Cartesian Grids has inspired several researchers to apply Finite Volume [Nannelli F, Succi S. The lattice Boltzmann equation on irregular lattices. J Stat Phys 1992;68:401–7; Peng G, Xi H, Duncan C, Chou SH. Finite volume scheme for the lattice Boltzmann method on unstructured meshes. Phys Rev E 1999;59:4675–82; Chen H. Volumetric formulation of the lattice Boltzmann method for fluid dynamics: basic concept. Phys Rev E 1998;58:3955–63] or Finite Element [Lee T, Lin CL. A characteristic Galerkin method for discrete Boltzmann equation. J Comp Phys 2001;171:336–56; Shi X, Lin J, Yu Z. Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element. Int J Numer Methods Fluids 2003;42:1249–61] methods to the Discrete Boltzmann equation. The finite volume method proposed by Peng et al. works on unstructured grids, thus allowing an increased geometrical flexibility. However, the method suffers from substantial numerical instability compared to the standard LBE models. The computational efficiency of the scheme is not competitive with standard methods.We propose an alternative way of discretizing the convection operator using an upwind scheme, as opposed to the central scheme described by Peng et al. We apply our method to some test problems in two spatial dimensions to demonstrate the improved stability of the new scheme and the significant improvement in computational efficiency. Comparisons with a lattice Boltzmann solver working on a hierarchical grid were done and we found that currently finite volume methods for the discrete Boltzmann equation are not yet competitive as stand alone fluid solvers.     

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.