Development and Comparative Studies of Three Non-Free Parameter Lattice Boltzmann Models for Simulation of Compressible Flows

This paper at first shows the details of finite volume-based lattice Boltzmann method (FV-LBM) for simulation of compressible flows with shock waves. In the FV-LBM, the normal convective flux at the interface of a cell is evaluated by using one-dimensional compressible lattice Boltzmann model, while the tangential flux is calculated using the same way as used in the conventional Euler solvers. The paper then presents a platform to construct one-dimensional compressible lattice Boltzmann model for its use in FV-LBM. The platform is formed from the conservation forms of moments. Under the platform, both the equilibrium distribution functions and lattice velocities can be determined, and therefore, non-free parameter model can be developed. The paper particularly presents three typical non-free parameter models, D1Q3, D1Q4 and D1Q5. The performances of these three models for simulation of compressible flows are investigated by a brief analysis and their application to solve some one-dimensional and two-dimensional test problems. Numerical results showed that D1Q3 model costs the least computation time and D1Q4 and D1Q5 models have the wider application range of Mach number. From the results, it seems that D1Q4 model could be the best choice for the FV-LBM simulation of hypersonic flows.     

Relaxation and transport in FCHC lattice gases

FCHC lattice gases are the basic models for studying flow problems in three-dimensional systems. This paper presents a self-contained theoretical analysis and some computer simulations of such lattice gases, extended to include an arbitrary number of rest particles, with special emphasis on non-semi-detailed balance (NSDB) models. The special FCHC lattice symmetry guarantees isotropy of the Navier-Stokes equations, and enumerates the 12 spurious conservation laws (staggered momenta). The kinetic theory is based on the mean field approximation or the nonlinear Boltzmann equation. It is shown how calculation of the eigenvalues of the linearized Boltzmann equation offers a simple alternative to the Chapman-Enskog method or the multi-time-scale methods for calculating transport coefficients and relaxation rates. The simulated values for the speed of sound in NSDB models slightly disagree with the Boltzmann prediction.     

Lattice Boltzmann computational fluid dynamics in three dimensions

The recent development of the lattice gas method and its extension to the lattice Boltzmann method have provided new computational schemes for fluid dynamics. Both methods are fully paralleled and can easily model many different physical problems, including flows with complicated boundary conditions. In this paper, basic principles of a lattice Boltzmann computational method are described and applied to several three-dimensional benchmark problems. In most previous lattice gas and lattice Boltzmann methods, a face-centered-hyper-cubic lattice in four-dimensional space was used to obtain an isotropic stress tensor. To conserve computer memory, we develop a model which requires 14 moving directions instead of the usual 24 directions. Lattice Boltzmann models, describing two-phase fluid flows and magnetohydrodynamics, can be developed based on this simpler 14-directional lattice. Comparisons between three-dimensional spectral code results and results using our method are given for simple periodic geometries. An important property of the lattice Boltzmann method is that simulations for flow in simple and complex geometries have the same speed and efficiency, while all other methods, including the spectral method, are unable to model complicated geometries efficiently.     

Asymptotic Analysis of Lattice Boltzmann Boundary Conditions

In this article, we use a general method for the analysis of finite difference schemes to investigate lattice Boltzmann algorithms for Navier–Stokes problems with Dirichlet boundary conditions. Several link based boundary conditions for commonly used lattice Boltzmann BGK models are considered. With our method, the accuracy of the algorithms can be exactly predicted. Moreover, the analytical results can be used to construct new algorithms which is demonstrated with a corrected bounce back rule that requires only local evaluations but still yields second order accuracy for the velocity. The analysis is applicable to general geometries and instationary flows     

Comparison between lattice Boltzmann method and Navier–Stokes high order schemes for computational aeroacoustics

Computational aeroacoustic (CAA) simulation requires accurate schemes to capture the dynamics of acoustic fluctuations, which are weak compared with aerodynamic ones. In this paper, two kinds of schemes are studied and compared: the classical approach based on high order schemes for Navier–Stokes-like equations and the lattice Boltzmann method. The reference macroscopic equations are the 3D isothermal and compressible Navier–Stokes equations. A Von Neumann analysis of these linearized equations is carried out to obtain exact plane wave solutions. Three physical modes are recovered and the corresponding theoretical dispersion relations are obtained. Then the same analysis is made on the space and time discretization of the Navier–Stokes equations with the classical high order schemes to quantify the influence of both space and time discretization on the exact solutions. Different orders of discretization are considered, with and without a uniform mean flow. Three different lattice Boltzmann models are then presented and studied with the Von Neumann analysis. The theoretical dispersion relations of these models are obtained and the error terms of the model are identified and studied. It is shown that the dispersion error in the lattice Boltzmann models is only due to the space and time discretization and that the continuous discrete velocity Boltzmann equation yield the same exact dispersion as the Navier–Stokes equations. Finally, dispersion and dissipation errors of the different kind of schemes are quantitatively compared. It is found that the lattice Boltzmann method is less dissipative than high order schemes and less dispersive than a second order scheme in space with a 3-step Runge–Kutta scheme in time. The number of floating point operations at a given error level associated with these two kinds of schemes are then compared.     

Numerical method based on the lattice Boltzmann model for the Fisher equation

In this paper, a lattice Boltzmann model for the Fisher equation is proposed. First, the Chapman-Enskog expansion and the multiscale time expansion are used to describe higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. Second, the modified partial differential equation of the Fisher equation with the higher-order truncation error is obtained. Third, comparison between numerical results of the lattice Boltzmann models and exact solution is given. The numerical results agree well with the classical ones.     

Consistent lattice Boltzmann method

Lack of energy conservation in lattice Boltzmann models leads to unrealistically high values of the bulk viscosity. For this reason, the lattice Boltzmann method remains a computational tool rather than a model of a fluid. A novel lattice Boltzmann model with energy conservation is derived from Boltzmann's kinetic theory. Simulations demonstrate that the new lattice Boltzmann model is the valid approximation of the Boltzmann equation for weakly compressible flows and microflows.     

Analytical solution of axi-symmetrical lattice Boltzmann model for cylindrical Couette flows

Analytical solution for the axi-symmetrical lattice Boltzmann model is obtained for the low-Mach number cylindrical Couette flows. In the hydrodynamic limit, the present solution is in excellent agreement with the result of the Navier–Stokes equation. Since the kinetic boundary condition is used, the present analytical solution using nine discrete velocities can describe flows with the Knudsen number up to 0.1. Meanwhile, the comparison with the simulation data obtained by the direct simulation Monte Carlo method shows that higher-order lattice Boltzmann models with more discrete velocities are needed for highly rarefied flows.     

基于总能形式的耦合的双分布函数热晶格玻尔兹曼数值方法

双分布函数热晶格玻尔兹曼数值方法在微尺度热流动系统中得到广泛的应用. 本文基于晶格玻尔兹曼平衡分布函数低阶Hermite展开式, 创新性地提出了包含黏性热耗散和压缩功的耦合的双分布函数热晶格玻尔兹曼数值方法, 将能量场内温度的变化以动量源的形式引入晶格波尔兹曼动量演化方程, 实现了能量场与动量场之间的耦合. 研究了考虑黏性热耗散和压缩功的和不考虑的两种热自然对流模型, 重点分析了不同瑞利数和普朗特数下流场内的流动情况以及温度、速度和平均努赛尔数的变化趋势. 本文实验结果与文献结果一致, 验证了本文数值方法的可行性和准确性. 研究结果表明: 随着瑞利数和普朗特数的增大, 方腔内对流传热作用逐渐增强, 边界处形成明显的边界层; 考虑黏性热耗散和压缩功的模型对流作用相对增强, 黏性热耗散和压缩功对自然对流的影响在微尺度流动过程中不能忽略.     

Thermodynamic Foundations of Kinetic Theory and Lattice Boltzmann Models for Multiphase Flows

This paper demonstrates that thermodynamically consistent lattice Boltzmann models for single-component multiphase flows can be derived from a kinetic equation using both Enskog's theory for dense fluids and mean-field theory for long-range molecular interaction. The lattice Boltzmann models derived this way satisfy the correct mass, momentum, and energy conservation equations. All the thermodynamic variables in these LBM models are consistent. The strengths and weaknesses of previous lattice Boltzmann multiphase models are analyzed.     

Nonexistence of H Theorem for Some Lattice Boltzmann Models

In this paper, we provide a set of sufficient conditions under which a lattice Boltzmann model does not admit an H theorem. By verifying the conditions, we prove that a number of existing lattice Boltzmann models does not admit an H theorem. These models include D2Q6, D2Q9 and D3Q15 athermal models, and D2Q16 and D3Q40 thermal (energy-conserving) models. The proof does not require the equilibria to be polynomials.     

Lattice Boltzmann Study of Mixed Convection in aCubic Cavity

The problem of the mixed convection in a cubic cavity is studied with lattice Boltzmann method. A multiple-relaxation-time lattice Boltzmann model for incompressible flow in the cubic cavity and another thermal lattice Boltzmann model for solving energy/temperature equation are proposed. The present models are first validated through a comparison with some available results, and then, we present a detailed parameter study on the mixed
convection in the cubic cavity. The numerical results show that the flow and temperature patterns change greatly with variations of the Reynolds and Richardson numbers.     

Investigation of Stability and Hydrodynamics of Different Lattice Boltzmann Models

Stability and hydrodynamic behaviors of different lattice Boltzmann models including the lattice Boltzmann equation (LBE), the differential lattice Boltzmann equation (DLBE), the interpolation-supplemented lattice Boltzmann method (ISLBM) and the Taylor series expansion- and least square-based lattice Boltzmann method (TLLBM) are studied in detail. Our work is based on the von Neumann linearized stability analysis under a uniform flow condition. The local stability and hydrodynamic (dissipation) behaviors are studied by solving the evolution operator of the linearized lattice Boltzmann equations numerically. Our investigation shows that the LBE schemes with interpolations, such as DLBE, ISLBM and TLLBM, improve the numerical stability by increasing hyper-viscosities at large wave numbers (small scales). It was found that these interpolated LBE schemes with the upwind interpolations are more stable than those with central interpolations because of much larger hyper-viscosities.     

生物过滤器中非均匀性流动的数值研究

生物过滤技术因其具有有效性、低成本和环境友好等优点引起了人们的广泛关注.该技术主要通过生物过滤器去除含有H2S等废气的有毒有害气体.运用格子Boltzmann方法对三种生物过滤器模型中多孔介质的非均匀性流动进行了数值模拟.数值模拟结果表明,多孔介质的性质和进口流动条件对临界Rayleigh数有显著影响,临界Rayleigh数随着多孔介质的孔隙度和Darcy数的增大而逐渐变小,并随着进口Reynolds数的增大而逐渐变大.所得结果可望为生物过滤器的优化设计提供一个合理的理论依据.     

Star-triangle relation for a three-dimensional model

The solvablesl(n)-chiral Potts model can be interpreted as a three-dimensional lattice model with local interactions. To within a minor modification of the boundary conditions it is an Ising-type model on the body-centered cubic lattice with two- and three-spin interactions. The corresponding local Boltzmann weights obey a number of simple relations, including a restricted star-triangle relation, which is a modified version of the well-known star-triangle relation appearing in two-dimensional models. We show that these relations lead to remarkable symmetry properties of the Boltzmann weight function of an elementary cube of the lattice, related to the spatial symmetry group of the cubic lattice. These symmetry properties allow one to prove the commutativity of the row-to-row transfer matrices, bypassing the tetrahedron relation. The partition function per site for the infinite lattice is calculated exactly.On leave of absence from the Institute for High Energy Physics, Protvino, Moscow Region, 142284, Russia.     

Note on Invariance of One-Dimensional Lattice-Boltzmann Equation

Invariance of the one-dimensional lattice Boltzmann model is proposed together with its rigorous theoretical background. It is demonstrated that the symmetry inherent in Navier-Stokes equations is not really recovered in the one-dimensional lattice Boltzmann equation （LBE）, especially for shock calculation. Symmetry breaking may be the inherent cause for the non-physical oscillations in the vicinity of the shock for LBE calculation.     

Extended Lattice Boltzmann Model

Conventional lattice Boltzmann models for the simulation of fluid dynamics are restricted by an error in the stress tensor that is negligible only for small flow velocity and at a singular value of the temperature. To that end, we propose a unified formulation that restores Galilean invariance and the isotropy of the stress tensor by introducing an extended equilibrium. This modification extends lattice Boltzmann models to simulations with higher values of the flow velocity and can be used at temperatures that are higher than the lattice reference temperature, which enhances computational efficiency by decreasing the number of required time steps. Furthermore, the extended model also remains valid for stretched lattices, which are useful when flow gradients are predominant in one direction. The model is validated by simulations of two- and three-dimensional benchmark problems, including the double shear layer flow, the decay of homogeneous isotropic turbulence, the laminar boundary layer over a flat plate and the turbulent channel flow.     

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.     

Steady-state analytical solutions for the lattice boltzmann equation

A general class of analytical solutions of the lattice Boltzmann equation is derived for two-dimensional, steady-state unidirectional flows. A subset of the solutions that verifies the corresponding Navier-Stokes equations is given. It is pointed out that this class includes, e.g., the Couette and the Poiseuille flow but not, e.g., the basic Kolmogorov flow. For steady-state non-unidirectional flows, first and second order solutions of the lattice Boltzmann equation are derived. Practical consequences of the analysis are mentioned. Differences between the technique applied here and those used in some earlier works are emphasized.