A lattice Boltzmann model for blood flows

A lattice Boltzmann model for blood flows is proposed. The lattice Boltzmann Bi-viscosity constitutive relations and control dynamics equations of blood flow are presented. A non-equilibrium phase is added to the equilibrium distribution function in order to adjust the viscosity coefficient. By comparison with the rheology models, we find that the lattice Boltzmann Bi-viscosity model is more suitable to study blood flow problems. To demonstrate the potential of this approach and its suitability for the application, based on this validate model, as examples, the blood flow inside the stenotic artery is investigated.     

A new lattice Boltzmann model for the laplace equation

In this paper, a new lattice Boltzmann equation which is independent of time is proposed. Based on the new lattice Boltzmann equation, some steady problems can be modeled by the lattice Boltzmann method. In the further study, the Laplace equation is investigated with the method of the higher-order moment of equilibrium distribution functions and a series of partial differential equations in different space scales. The numerical results show that the new method is effective.     

Lattice Boltzmann simulation for high-speed compressible viscous flows with a boundary layer

In this study, the lattice Boltzmann method is employed for simulating high-speed compressible viscous flows with a boundary layer. The coupled double-distribution-function lattice Boltzmann method proposed by Li et al. (2007) is employed because of its good numerical stability and non-free-parameter feature. The non-uniform mesh construction near the wall boundary in fine grids is combined with an appropriate wall boundary treatment for the finite difference method in order to obtain accurate spatial resolution in the boundary layer problem. Three typical problems in high-speed viscous flows are solved in the lattice Boltzmann simulation, i.e., the compressible boundary layer problem, shock wave problem, and shock boundary layer interaction problem. In addition, in-depth comparisons are made with the non-oscillatory and non-free-parameter dissipation (NND) scheme and second order upwind scheme in the present lattice Boltzmann model. Our simulation results indicate the great potential of the lattice Boltzmann method for simulating high-speed compressible viscous flows with a boundary layer. Further research is needed (e.g., better numerical models and appropriate finite difference schemes) because the lattice Boltzmann method is still immature for high-speed compressible viscous flow applications.     

Analysis of lattice Boltzmann boundary conditions

The correct implementation of Navier‐Stokes boundary conditions in the framework of lattice Boltzmann schemes is complicated by the non‐availability of analytical methods to assess the consistency of such discretizations. To close this gap, we propose a simple direct asymptotic analysis which is readily applicable to finite difference discretizations of initial boundary value problems in general and to lattice Boltzmann methods in particular. Results of the analysis applied to the classical lattice Boltzmann scheme with bounce back boundary condition are reported.     

Lattice Boltzmann model for two-dimensional unsteady Burgers’ equation

In this paper, a special lattice Boltzmann model is proposed to simulate two-dimensional unsteady Burgers’ equation. The maximum principle and the stability are proved. The model has been verified by several test examples. Excellent agreement is obtained between numerical predictions and exact solutions. The cases of steep oblique shock waves are solved and compared with the two-point compact scheme results. The study indicates that lattice Boltzmann model is highly stable and efficient even for the problems with severe gradients.     

Lattice Boltzmann模型在CFD中应用

近年来,格子Boltzmann方法(LBM)已发展为一种模拟流体和物理问题的新颖的、有前景的数值方法,在许多领域的各种数值问题求解上取得很大的成功.文章介绍了一种模拟复杂流动的高效建模数值算法Lattice Boltzmann方法,和它的基本原理及其应用.并通过两个实例数值模拟计算,说明Lattice Boltzmann方法正确、有效,并展示了广阔的应用前景,为今后更深入的研究和广泛应用打下基础.     

Lattice Boltzmann model for two-dimensional generalized sine-Gordon equation

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical model based on lattice Boltmann method to obtain the numerical solutions of two-dimensional generalized sine-Gordon equation, including damped and undamped sine-Gordon equation. By choosing properly the conservation condition between the macroscopic quantity $u_t$ and the distribution functions and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation. Moreover, the local equilibrium distribution function is obtained. The numerical results of the first three examples agree well with the analytic solutions, which indicates the lattice Boltzmann model is satisfactory and efficient. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given. Numerical experiments also show that the present scheme has a good long-time numerical behavior for the generalized sine-Gordon equation. Moreover, the model can also be applied to other two-dimensional nonlinear wave equations, such as nonlinear hyperbolic telegraph equation and Klein-Gordon equation.     

Application of lattice Boltzmann method for incompressible viscous flows

Because of the presence of corner eddies that change in number and pattern the lid-driven cavity problem has been found suitable to study various aspects of the performance of solution algorithms for incompressible viscous flows. It retains all the difficult flow physics and is characterized by a large primary eddy at the centre and secondary eddies located near the cavity corners. In this work, lid-driven cavity flow is simulated by lattice Boltzmann method with single-relaxation-time and it is compared with those by lattice Boltzmann method with multi-relaxation-time and finite difference method. The effects of the Reynolds number on the size, centre position and number of vortices are studied in detail together with the flow pattern in the cavity. The close agreement of the results bears testimony to the validity of this relatively new approach. However lattice Boltzmann method with multi-relaxation-time model is seen to remove the difficulties faces by the lattice Boltzmann method with single-relaxation-time at higher Reynolds numbers.     

Lattice Boltzmann Constraints for Standard Optimization Problems

This paper presents a simultaneous analysis and design (SAND) approach for the control of incompressible, stationary fluid problems ruled by the lattice Boltzmann equation. Therefore the discretized lattice Boltzmann fix point problem is used as constraint for the optimization problem. This allows for the use of standard non-linear programming techniques. To validate the proposed method a topology optimization problem is solved with the NLP solver WORHP at moderate Reynolds numbers (Re < 1000) for a 2-dimensional flow domain. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lattice evolution solution for the nonlinear Poisson–Boltzmann equation in confined domains

The lattice evolution method for solving the nonlinear Poisson–Boltzmann equation in confined domain is developed by introducing the second-order accurate Dirichlet and Neumann boundary implements, which are consistent with the non-slip model in lattice Boltzmann method for fluid flows. The lattice evolution method is validated by comparing with various analytical solutions and shows superior to the classical numerical solvers of the nonlinear Poisson equations with Neumann boundary conditions. The accuracy and stability of the method are discussed. This lattice evolution nonlinear Poisson–Boltzmann solver is suitable for efficient parallel computing, complex geometry conditions, and easy extension to three-dimensional cases.     

Lattice Boltzmann model for the perfect gas flows with near-vacuum region

It is known that the standard lattice Boltzmann method has near-vacuum limit, i. e., when the density is near zero, this method is invalid. In this letter, we propose a simple lattice Boltzmann model for one-dimensional flows. It possesses the ability of simulating near-vacuum area by setting a limitation of the relaxation factor. Thus, the model overcomes the disadvantage of non-physical pressure and the density. The numerical examples show these results are satisfactory.     

Lattice Boltzmann solver of Rossler equation

We proposed a lattice Boltzmann model for the Rossler equation. Using a method of multiscales in the lattice Boltzmann model, we get the diffusion reaction as a special case. If the diffusion effect disappeared, we can obtain the lattice Boltzmann solution of the Rossler equation on the mesescopic scale. The numerical results show the method can be used to simulate Rossler equation.     

A large-eddy-based lattice Boltzmann model for turbulent flow simulation

In this paper a novel and simple large-eddy-based lattice Boltzmann model is proposed to simulate two-dimensional turbulence. Unlike existing lattice Boltzmann models for turbulent flow simulation, which were based on primitive-variables Navier–Stokes equations, the target macroscopic equations of the present model are vorticity-streamfunction equations. Thanks to the intrinsic features of vorticity-streamfunction equations, the present model is efficient, stable and simple for two-dimensional turbulence simulation. The advantages of the present model are validated by numerical experiments.     

一类非线性偏微分方程的四阶格子Boltzmann模型

通过Chapman-Enskog展开技术和多尺度分析,建立了一种新的D1Q4带修正项的四阶格子Boltzmann模型,一类非线性偏微分方程从连续的Boltzmann方程得到正确恢复．统一了KdV和Burgers等已知方程类型的格子BGK模型,还首次给出了组合KdV-Burgers,广义Burgers—Huxley等方程...     

Lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation

A lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation is proposed. By using multi-scale technique and the Chapman–Enskog expansion on complex lattice Boltzmann equation, we obtain a series of complex partial differential equations, complex equilibrium distribution function and its complex moments. Then, the complex reaction–diffusion equation is recovered with higher-order accuracy of the truncation error. This equation can be used to describe the bimolecular autocatalytic reaction–diffusion systems, in which a rich variety of behaviors have been observed. Based on this model, the Fitzhugh–Nagumo model and the Gray–Scott model are simulated. The comparisons between the LBM results and the Alternative Direction Implicit results are given in detail. The numerical examples show that assumptions of source term can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the complex reaction–diffusion equation.     

General propagation lattice Boltzmann model for variable-coefficient non-isospectral KdV equation

In this paper, a general propagation lattice Boltzmann model for variable-coefficient non-isospectral Korteweg–de Vries (vc-nKdV) equation, which can describe the interfacial waves in a two layer liquid and Alfvén waves in a collisionless plasma, is proposed by selecting appropriate equilibrium distribution function and adding the compensate function. The Chapman–Enskog analysis shows that the vc-nKdV equation can be recovered correctly from the present model. Numerical simulation for the non-propagating one soliton of this equation in different situations is conducted as validation. It is found that the numerical results match well with the analytical solutions, which demonstrates that the current general propagation lattice Boltzmann model is a satisfactory and efficient method, and could be more stable and accurate than the standard lattice Bhatnagar–Gross–Krook model.     

均匀格子气Boltzmann方程的稳定性研究

１．引言格子气的基本方程是在几何空间、速度空间和时间上都是离散的Ｂｏｌｔｚｍａｎｎ方程（Ｂ方程）．这是一个有限差分方程．在离散速度气体运动论中［１］,Ｂ方程在速度空间上是离散的,在几何空间和时间上是连续的．这是一个偏微分方程．人们对离散速度气体Ｂ方程的稳定性和渐近特性的研究已经取得了很多结果．Ｍａａｓｓ~［２］通过构造Ｌｙaｐｕｎｏｖ函数族,在分布函数在空间上均匀的条件下,证明了平衡分布的渐近稳定性．信息函数Ｈ是该函数族的一员．Ｂｅｌｌｏｍｏｅｔａｌ~［３］．采用小扰动线性化方法在初值距离平衡解足…     

Lattice Boltzmann model for a generalized Gardner equation with time-dependent variable coefficients

In this paper, lattice Boltzmann model for a generalized Gardner equation with time-dependent variable coefficients, which can provide some more realistic models than their constant-coefficient counterparts, is derived through selecting equilibrium distribution function and adding the compensate function, appropriately. Effects and approximate value range of the free parameters, which are introduced to adjust the single relaxation time and equilibrium distribution function, are discussed in detail, as well as the impact of the lattice space step and velocity. Numerical simulations in different situations of this equation are conducted, including the propagation and interaction of the solitons, the evolution of the non-propagating soliton and the propagation of the double-pole solutions. It is found that the numerical results match well with the analytical solutions, which demonstrates that the current lattice Boltzmann model is a satisfactory and efficient algorithm.     

Convergence of lattice Boltzmann methods for Navier–Stokes flows in periodic and bounded domains

Combining an asymptotic analysis of the lattice Boltzmann method with a stability estimate, we are able to prove some convergence results which establish a strict relation to the incompressible Navier–Stokes equation. The proof applies to the lattice Boltzmann method in the case of periodic domains and for specific bounded domains if the Dirichlet boundary condition is realized with the bounce back rule.     

Lattice Boltzmann 3D flow simulations on a metallic foam

In this paper a lattice Boltzmann method (LBM) is used to simulate isothermal incompressible flow in a RCM-NCX-1116 metallic foam. The computational technique is a multiple relaxation time (MRT) lattice Boltzmann equation model. Computer aided X-ray micro-tomography is used to obtain 3D images of the metallic foam, providing the geometry and information required for LB simulations of a single phase flow.Pressure drops are computed and successfully compared to experimental measures and correlated with Ergun’s equation. Invariance of Ergun’s parameters A and B with the sampling rate of the images is observed.