Unstructured lattice Boltzmann equation with memory

Starting from a second-order differential form of the semi-discrete Boltzmann equation, we construct a new finite-volume lattice Boltzmann equation on unstructured grids (ULBE). The new scheme (ULBE with memory) is demonstrated for the case of a Taylor-vortex flow and shown to produce stable and accurate results with time-step more than an order of magnitude above the standard LBE stability threshold.     

Adjoint lattice Boltzmann equation for parameter identification

The lattice Boltzmann equation is briefly introduced using moments to clearly separate the propagation and collision steps in the dynamics. In order to identify unknown parameters we introduce a cost function and adapt control theory to the lattice Boltzmann equation to get expressions for the derivatives of the cost function vs. parameters. This leads to an equivalent of the adjoint method with the definition of an adjoint lattice Boltzmann equation.To verify the general expressions for the derivatives, we consider two elementary situations: a linearized Poiseuille flow to show that the method can be used to optimize parameters, and a nonlinear situation in which a transverse shear wave is advected by a mean uniform flow. We indicate in the conclusion how the method can be used for more realistic situations.     

Progress in the understanding of the fluctuating lattice Boltzmann equation

We give a brief account of the development of methods to include thermal fluctuations into lattice Boltzmann algorithms. Emphasis is put on our recent work [B. Dünweg, U.D. Schiller, A.J.C. Ladd, Phys. Rev. E 76 (2007) 036704] which provides a clear understanding in terms of statistical mechanics.     

Numerical solution of the Boltzmann equation on the uniform grid

In the present paper a new numerical method for the Boltzmann equation is developed. The gain part of the collision integral is written in a form which allows its numerical computation on the uniform grid to be carried out efficiently. The amount of numerical work is shown to be of the order O(n ⁶log(n)) for the most general model of interaction and of the order O(n ⁶) for the Variable Hard Spheres (VHS) interaction model, while the formal accuracy is of the order O(n ⁻²). Here n denotes the number of discretisation points in one direction of the velocity space. Some numerical examples for Maxwell pseudo-molecules and for the hard spheres model illustrate the accuracy and the efficiency of the method in comparison with DSMC computations in the spatially homogeneous case. Received June 18, 2002; revised August 23, 2002 Published online: October 24, 2002     

WBK方程稳态解的保结构分析

水动力学系统稳定状态下的总能量与系统初始能量之差直观反映了水力系统的水头损失.本文基于保结构思想,以色散浅水波WBK模型为例,推导了其对称形式及空间辛结构等守恒性质.随后,采用Euler Box差分离散方法构造对称形式的保结构差分格式,并推导其离散空间辛结构,为数值格式保结构性能检验提供理论依据.最后,通过数值实验,考察数值格式的保结构性能,并将数值格式用于研究不同相对扩散系数条件下,WBK方程保结构稳态水质点系统的总能量,为水力系统水头损失的分析提供参考.     

基于格子Boltzmann方法的一维Burgers方程的数值模拟

基于格子Boltzmann方法(LBM)的一维Burgers方程的数值解法,已有2-bit和4-bit模型。文中通过选择合适的离散速度模型构造出恰当的平衡态分布函数; 然后, 利用单松弛的格子Bhatnagar-Gross-Krook模型、Chapman-Enskog展开和多尺度技术, 提出了用于求解一维Burgers方程的3-bit的格子Boltzmann模型,即D1Q3模型,并进行了数值实验。实验结果表明,该方法的数值解与解析解吻合的程度很好,且误差比现有文献中的误差更小,从而验证了格子Boltzamnn模型的有效性。     

A lattice Boltzmann model for the Korteweg-de Vries equation with two conservation laws

A lattice Boltzmann model for the Korteweg-de Vries (KdV) equation is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model to the KdV equation, our method has higher-order accuracy. Two key steps in the development of this model are the addition of a momentum conservation condition, and the construction of a correlation between the first conservation law and the second conservation law. The numerical example shows the higher-order moment method can be used to raise the truncation error of the lattice Boltzmann scheme.     

Bluff body flow simulation using lattice Boltzmann equation with multiple relaxation time

Numerical simulations using multiple-relaxation-time lattice Boltzmann model (MRT-LBM) are carried out for a long slender rigid circular cylinder in a cross flow to examine three-dimensional wake effect on the flow-induced forces. A mesh refinement technique is applied in the MRT-LBM calculation. The aim is to assess the validity and efficiency of the MRT-LBM model in three-dimensional calculation. In order to simulate the practical situation correctly, wall boundary conditions are specified at both ends of the cylinder. The aspect ratio of the slender cylinder is 16. The calculation is compared with results obtained from a finite volume method (FVM) and a lattice BGK model [Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases. 1. Small amplitude processes in charged and neutral one-component systems. Phys Rev 1954;94:511–25] with refined grid. Good agreement is obtained. It is found that the MRT-LBM is more efficient and faster in three-dimensional calculations.     

Solving the Boltzmann equation on GPUs

We show how to accelerate the direct solution of the Boltzmann equation using Graphics Processing Units (GPUs). In order to fully exploit the computational power of the GPU, we choose a method of solution which combines a finite difference discretization of the free-streaming term with a Monte Carlo evaluation of the collision integral. The efficiency of the code is demonstrated by solving the two-dimensional driven cavity flow. Computational results show that it is possible to cut down the computing time of the sequential code of two order of magnitude. This makes the proposed method of solution a viable alternative to particle simulations for studying unsteady low Mach number flows.     

On the small-scale dynamical behavior of lattice BGK and lattice Boltzmann schemes

In this paper, we report some comparative simulations between lattice BGK and lattice Bolzmann schemes for two-dimensional fluid flows. A quantitative assessment of the validity of the lattice BGK and lattice Bolzmann schemes is presented for the two-dimensional weakly compressibleKolmogorov flow. We use this flow to study the difference of the two schemes at small scales. A lowReynolds (R _e 300) number simulation shows the almost identical energy spectra for both schemes except for the small-scale dynamics of lattice Bolzmann which is more noisy. Because of the intrinsic difficulties of nonlinear stability analysis, we use numerical simulations to investigate which scheme is more stable. It turns out the lattice BGK is more stable. It turns out the lattice BGK is more robust than lattice Bolzmann by increasing theReynolds numbers. Detailed comparison with other methods (e.g., spectral method) remains to be done in the near future.     

A cache‐efficient implementation of the lattice Boltzmann method for the two‐dimensional diffusion equation

The lattice Boltzmann method is an important technique for the numerical solution of partial differential equations because it has nearly ideal scalability on parallel computers for many applications. However, to achieve the scalability and speed potential of the lattice Boltzmann technique, the issues of data reusability in cache‐based computer architectures must be addressed. Utilizing the two‐dimensional diffusion equation, , this paper examines cache optimization for the lattice Boltzmann method in both serial and parallel implementations. In this study, speedups due to cache optimization were found to be 1.9–2.5 for the serial implementation and 3.6–3.8 for the parallel case in which the domain decomposition was optimized for stride‐one access. In the parallel non‐cached implementation, the method of domain decomposition (horizontal or vertical) used for parallelization did not significantly affect the compute time. In contrast, the cache‐based implementation of the lattice Boltzmann method was significantly faster when the domain decomposition was optimized for stride‐one access. Additionally, the cache‐optimized lattice Boltzmann method in which the domain decomposition was optimized for stride‐one access displayed superlinear scalability on all problem sizes as the number of processors was increased. Copyright © 2004 John Wiley & Sons, Ltd.     

On the stability structure for lattice Boltzmann schemes

The stability structure for lattice Boltzmann schemes has been introduced in Banda et al. (2006) [16], Junk and Yong (2007) [14] to analyze the stability of numerical algorithms. The first purpose of this paper is to discuss the stability structure from the perspective of matrix analysis. Its second goal is to illustrate and apply the results to different classes of lattice Boltzmann collision operators. In particular we formulate an equivalence condition–just recently also reported in Yong (2008) [18]–that guarantees the existence of a pre-stability structure. It is then illustrated by several examples, how this equivalence condition can be effectively employed for the systematic verification and construction of stable collision operators. Finally, we point out some shortcomings of the stability structure approach arising in certain cases.     

Multi-level lattice Boltzmann model on square lattice for compressible flows

A compressible lattice Boltzmann model is established on a square lattice. The model allows large variations in the mean velocity by introducing a large particle-velocity set. To maintain tractability, the support set of the equilibrium distribution is chosen to include only four directions and three particle-velocity levels in which the third level is introduced to improve the stability of the model. This simple structure of the equilibrium distribution makes the model efficient for the simulation of flows over a wide range of Mach numbers and gives it the capability of capturing shock jumps. Unlike the standard lattice Boltzmann model, the formulation eliminated the fourth-order velocity tensors, which were the source of concerns over the homogeneity of square lattices. A modified collision invariant eliminates the second-order discretization error of the fluctuation velocity in the macroscopic conservation equation from which the Navier–Stokes equation and energy equation are recovered. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Two-dimensional shock-wave propagations and boundary layer flows were successfully simulated. The model can be easily extended to three-dimensional cubic lattices.     

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.     

The fractional Boltzmann transport equation

The fractional Boltzmann transport equation is derived making use of the fractional Hamilton’s equations based on the fractional actionlike variational approach. By simply defining a distribution function and inspecting its time derivative, many important results in statistical physics can be derived.     

Suppressing the coalescence in the multi-component lattice Boltzmann method

This article discusses a novel phenomenological approach for suppressing the coalescence in the Gunstensen multi-component lattice Boltzmann method (LBM). The suppression of coalescence is achieved by perturbing the terminal nodes of the ambient fluid’s thin layer trapped between the approaching droplets. This additional perturbation creates a local high pressure fluid layer which eventually leads to suppressing the coalescence of the neighboring droplets while maintaining a suitable qualitative force balance representative of the physical intermolecular forces which act between them.