Direct assessment of lattice Boltzmann hydrodynamics and boundary conditions for recirculating flows

A hydrodynamic boundary condition is developed for lattice Boltzmann hydrodynamics using a square, orthogonal grid. A constraint based on energy considerations is developed to provide closure for the equations which govern the particle distribution at the boundaries. This boundary condition is applied to the two-dimensional, steady flow of an incompressible fluid behind a grid, known as Kovasznay flow. The results are compared to those using alternate boundary conditions using the known exact solution. The hydrodynamic boundary condition produces quadratic spatial convergence, while alternate techniques fail to maintain this second-order accuracy.     

A time-reversal lattice Boltzmann method

In this paper we address the time-reversed simulation of viscous flows by the lattice Boltzmann method (LB). The theoretical derivation of the reversed LB from the Boltzmann equation is detailed, and the method implemented for weakly compressible flows using the D2Q9 scheme. The implementation of boundary conditions is also discussed. The accuracy and stability are illustrated by four test cases, namely the propagation of an acoustic wave in a medium at rest and in an uniform mean flow, the Taylor–Green vortex decay and the vortex pair–wall collision.     

Local second-order boundary methods for lattice Boltzmann models

A new way to implement solid obstacles in lattice Boltzmann models is presented. The unknown populations at the boundary nodes are derived from the locally known populations with the help of a second-order Chapman-Enskog expansion and Dirichlet boundary conditions with a given momentum. Steady flows near a flat wall, arbitrarily inclined with respect to the lattice links, are then obtained with a third-order error. In particular, Couette and Poiseuille flows are exactly recovered without the Knudsen layers produced for inclined walls by the bounce back condition.     

Non—equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method

In this paper, we propose a new approach to implementing boundary conditions in the lattice Boltzmann method (LBM). The basic idea is to decompose the distribution function at the boundary node into its equilibrium and non-equilibrium parts, and then to approximate the non-equilibrium part with a first-order extrapolation of the non-equilibrium part of the distribution at the neighbouring fluid node. Schemes for velocity and pressure boundary conditions are constructed based on this method. The resulting schemes are of second-order accuracy. Numerical tests show that the numerical solutions of the LBM together with the present boundary schemes are in excellent agreement with the analytical solutions. Second-order convergence is also verified from the results. It is also found that the numerical stability of the present schemes is much better than that of the original extrapolation schemes proposed by Chen et al. (1996 Phys. Fluids 8 2527).     

一种滑移区气体流动的格子Boltzmann曲边界处理新格式

针对滑移区复杂气-固边界存在速度滑移现象,提出了一种基于格子Boltzmann方法的非平衡态外推与有限差分相结合的曲边界处理新格式.该格式具有可考虑实际物理边界与网格线偏移量的优势,较传统half-way DBB(diffusive bounce-back)格式更能准确反映实际边界情况,同时还可获取壁面处气体宏观量及其法向梯度等信息.采用本文所提曲边界处理格式模拟分析了滑移区气体平直/倾斜微通道Poiseuille流、微圆柱绕流和同心微圆柱面旋转Couette流问题.研究结果表明,采用曲边界处理新格式所得结果与理论值以及文献结果符合良好,适用于滑移区气体流动的复杂边界处理,且比half-way DBB格式具有更高的精度,较修正DBB格式具有更好的适应性.     

Boundary conditions for lattice Boltzmann simulations

A heuristic interpretation of no-slip boundary conditions for lattice Boltzmann and lattice gas simulations is developed. An improvement is suggested which consists of including the wall nodes in the collision operation.     

Factorization symmetry in the lattice Boltzmann method

A non-perturbative algebraic theory of the lattice Boltzmann method is developed based on the symmetry of a product. It involves three steps: (i) Derivation of admissible lattices in one spatial dimension through a matching condition which imposes restricted extension of higher-order Gaussian moments, (ii) A special quasi-equilibrium distribution function found analytically in closed form on the product-lattice in two and three spatial dimensions, and which proves the factorization of quasi-equilibrium moments, and (iii) An algebraic method of pruning based on a one-into-one relation between groups of discrete velocities and moments. Two routes of constructing lattice Boltzmann equilibria are distinguished. The present theory includes previously known limiting and special cases of lattices, and enables automated derivation of lattice Boltzmann models from two-dimensional tables, by finding the roots of one polynomial and solving a few linear systems.     

On the lattice Boltzmann method for phonon transport

The lattice Boltzmann method is a discrete representation of the Boltzmann transport equation that has been employed for modeling transport of particles of different nature. In the present work, we describe the lattice Boltzmann methodology and implementation techniques for the phonon transport modeling in crystalline materials. We show that some phonon physical properties, e.g., mean free path and group velocity, should be corrected to their effective values for one- and two-dimensional simulations, if one uses the isotropic approximation. We find that use of the D2Q9 lattice for phonon transport leads to erroneous results in transient ballistic simulations, and the D2Q7 lattice should be employed for two-dimensional simulations. Furthermore, we show that at the ballistic regime, the effect of direction discretization becomes apparent in two dimensions, regardless of the lattice used. Numerical methodology, lattice structure, and implementation of initial and different boundary conditions for the D2Q7 lattice are discussed in detail.     

Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model

In this paper we analytically solve the velocity of the lattice Boltzmann BGK equation (LBGK) for several simple flows. The analysis provides a framework to theoretically analyze various boundary conditions. In particular, the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition. We find that the slip velocity is zero as long as fe=0 at boundaries, no matter what combination of distributions is chosen. The schemes proposed by Nobleet al. and by Inamuroet al. yield the correct zeroslip velocity, while some other schemes, such as the bounce-back scheme and the equilibrium distribution scheme, would inevitably generate a nonzero slip velocity. The bounce-back scheme with the wall located halfway between a flow node and a bounce-back node is also studied for the simple flows considered and is shown to produce results of second-order accuracy. The momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries. To be specific, the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids.     

基于浸入边界-多松弛时间格子玻尔兹曼通量求解法的流固耦合算法研究

针对流固耦合问题,发展了基于浸入边界-多松弛时间格子玻尔兹曼通量求解法(immersed boundary method multi-relaxation-time lattice Boltzmann flux solver,IB-MRT-LBFS)的弱耦合算法.依据多尺度Chapman-Enskog展开,建立不可压宏观方程状态变量和通量与格子玻尔兹曼方程中粒子密度分布函数之间的关系;采用强制浸入边界法处理流固界面使固壁表面满足无滑移边界条件,根据修正的速度求解动量方程力源项;结构运动方程采用四阶龙格-库塔法求解.格子模型与浸入边界法的引入使流固耦合计算可以在笛卡尔网格下进行,无需生成贴体网格及运用动网格技术,简化了计算过程.数值模拟了单圆柱横向涡激振动、单圆柱及串列双圆柱双自由度涡激振动问题.结果表明,IB-MRT-LBFS能够准确预测圆柱涡激振动的锁定区间、振动响应、受力情况以及捕捉尾流场结构形态,验证了该算法在求解流固耦合问题的有效性和可行性.     

A lattice Boltzmann model for the generalized Burgers-Huxley equation

In this paper we develop a lattice Boltzmann model for the generalized Burgers-Huxley equation (GBHE). By choosing the proper time and space scales and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation, and the local equilibrium distribution functions are obtained. Excellent agreement with the exact solution is observed, and better numerical accuracy is obtained than the available numerical result. The results indicate the present model is satisfactory and efficient. The method can also be applied to the generalized Burgers-Fisher equation and be extended to multidimensional cases.     

A lattice Boltzmann model for coupled diffusion

Diffusion coupling between different chemical components can have significant effects on the distribution of chemical species and can affect the physico-chemical properties of their supporting medium. The coupling can arise from local electric charge conservation for ions or from bound components forming compounds. We present a new lattice Boltzmann model to account for the diffusive coupling between different chemical species. In this model each coupling is added as an extra relaxation term in the collision operator. The model is tested on a simple diffusion problem with two coupled components and is in excellent agreement with the results obtained through a finite difference method. Our model is observed to be numerically very stable and unconditional stability is shown for a class of diffusion matrices. We further develop the model to account for advection and show an example of application to flow in porous media in two dimensions and an example of convection due to salinity differences. We show that our model with advection loses the unconditional stability, but offers a straight-forward approach to complicated two-dimensional advection and coupled diffusion problems.     

任意复杂流-固边界的格子Boltzmann处理方法

本文提出了一种适用于流固耦合领域中任意复杂边界条件的lattice Boltzmann处理方法.该方法基于half-way反弹模型,在流固耦合处构建了一层虚拟边界,并结合有限差分的方法,获取虚拟边界上的变量值.改进后的方法确保了粒子反弹位置与宏观速度采集点的位置相同,计入了实际物理边界与网格线不重合时,偏移量对计算结果的准确影响,而且其适用范围被扩展到了任意静止或运动、平直或弯曲的复杂边界.文中研究了该方法在Poiseuille流、圆柱绕流和Couette流等经典条件下的边界处理能力,结果表明该方法与理论值符合良好,且当实际物理边界与网格线不重合时,与已发表文献中的结果相比,具有更高的精度.     

A improved incompressible lattice Boltzmann model for time-independent flows

It is well known that the lattice Boltzmann equation method (LBE) can model the incompressible Navier-Stokes (NS) equations in the limit where density goes to a constant. In a LBE simulation, however, the density cannot be constant because pressure is equal to density times the square of sound speed, hence a compressibility error seems inevitable for the LBE to model incompressible flows. This work uses a modified equilibrium distribution and a modified velocity to construct an LBE which models time-independent (steady) incompressible flows with significantly reduced compressibility error. Computational results in 2D cavity flow and in a 2D flow with an exact solution are reported.     

Advanced lattice Boltzmann scheme for high-Reynolds-number magneto-hydrodynamic flows

Is the lattice Boltzmann method suitable to investigate numerically high-Reynolds-number magneto-hydrodynamic (MHD) flows? It is shown that a standard approach based on the Bhatnagar–Gross–Krook (BGK) collision operator rapidly yields unstable simulations as the Reynolds number increases. In order to circumvent this limitation, it is here suggested to address the collision procedure in the space of central moments for the fluid dynamics. Therefore, an hybrid lattice Boltzmann scheme is introduced, which couples a central-moment scheme for the velocity with a BGK scheme for the space-and-time evolution of the magnetic field. This method outperforms the standard approach in terms of stability, allowing us to simulate high-Reynolds-number MHD flows with non-unitary Prandtl number while maintaining accuracy and physical consistency.     

A fluctuating lattice Boltzmann scheme for the one-dimensional KPZ equation

Based on the well-known mapping between the Burgers equation with noise and the Kardar–Parisi–Zhang (KPZ) equation for fluctuating interfaces, we develop a fluctuating lattice Boltzmann (LB) scheme for growth phenomena, as described by the KPZ formalism. A very simple LB-KPZ scheme is demonstrated in 1+1 spacetime dimensions, and is shown to reproduce the scaling exponents characterizing the growth of one-dimensional fluctuating interfaces.     

Rotational invariance in the three-dimensional lattice Boltzmann method is dependent on the choice of lattice

The lattice Boltzmann method has recently gained popularity as a tool for simulating complex fluid flows. It uses discrete sets of velocity vectors, or lattices, to create a reduced model of the molecular dynamics of a continuum fluid. While several lattices are believed to behave isotropically, there are reports of qualitatively incorrect results. However, thus far, the reason as to why a lack of isotropy occurs is not known. Based on the hypothesis that lower order lattices may not display rotational invariance, this study tests the isotropy of the D3Q15, D3Q19 and D3Q27 lattices by performing simulations at intermediate Reynolds numbers (50–500) and low Knudsen number (<0.0005) in an axisymmetrical geometry with a nozzle leading to a throat followed by a sudden expansion. The symmetry properties of the results were examined. It was found that at Re ? 250 the D3Q15 and D3Q19 lattices produced different results depending on the plane of the lattice with which the flow was aligned. Lattice planes with fewer than six velocity vectors consistently produced results which were qualitatively different from the planes with six or more velocity vectors. These errors were not observed at Re = 50 or when a D3Q27 lattice was used. They appeared to be independent of grid density, collision operator and Ma. This suggests that the lattices which contain these planes are not fully isotropic and therefore do not properly replicate the behavior of a real fluid in this particular situation, notably downstream from the expansion. Predictions made using these models in more complex geometries may therefore be affected by the orientation of the lattice. When using LBM in CFD simulation (including validation) this study highlights the need for caution to ensure that the solution obtained is independent of the lattice orientation throughout the domain.     

Multiple-relaxation-time lattice Boltzmann model for compressible fluids

We present an energy-conserving multiple-relaxation-time finite difference lattice Boltzmann model for compressible flows. The collision step is first calculated in the moment space and then mapped back to the velocity space. The moment space and corresponding transformation matrix are constructed according to the group representation theory. Equilibria of the nonconserved moments are chosen according to the need of recovering compressible Navier-Stokes equations through the Chapman-Enskog expansion. Numerical experiments showed that compressible flows with strong shocks can be well simulated by the present model. The new model works for both low and high speeds compressible flows. It contains more physical information and has better numerical stability and accuracy than its single-relaxation-time version.     

A large eddy lattice Boltzmann simulation of magnetohydrodynamic turbulence

Large eddy simulations (LES) of a lattice Boltzmann magnetohydrodynamic (LB-MHD) model are performed for the unstable magnetized Kelvin–Helmholtz jet instability. This algorithm is an extension of Ansumali et al. [1] to MHD in which one performs first an expansion in the filter width on the kinetic equations followed by the usual low Knudsen number expansion. These two perturbation operations do not commute. Closure is achieved by invoking the physical constraint that subgrid effects occur at transport time scales. The simulations are in very good agreement with direct numerical simulations.     

Analytical solutions of the lattice Boltzmann BGK model

Analytical solutions of the two-dimensional triangular and square lattice Boltzmann BGK models have been obtained for the plane Poiseuille flow and the plane Couette flow. The analytical solutions are written in terms of the characteristic velocity of the flow, the single relaxation time , and the lattice spacing. The analytic solutions are the exact representation of these two flows without any approximation. Using the analytical solution, it is shown that in Poiseuille flow the bounce-back boundary condition introduces an error of first order in the lattice spacing. The boundary condition used by Kadanoffet al. in lattice gas automata to simulate Poiseuille flow is also considered for the triangular lattice Boltzmann BGK model. An analytical solution is obtained and used to show that the boundary condition introduces an error of second order in the lattice spacing.