Comparative study of lattice‐Boltzmann and finite volume methods for the simulation of laminar flow through a 4:1 planar contraction

In the present paper, a comparative study of numerical solutions for Newtonian fluids based on the lattice‐Boltzmann method (LBM) and the classical finite volume method (FVM) is presented for the laminar flow through a 4:1 planar contraction at a Reynolds number of value one, Re=1. In this study, the stress field for LBM is directly obtained from the distribution function. The calculations of the stress based on the FVM‐data use the evaluations of velocity gradients with finite differences. The stress field for both LBM and FVM is expressed in the present study in terms of the shear stress and the first normal stress difference. The lateral and axial profiles of the velocity, the shear stress and the first normal stress difference for both methods are investigated. It is shown that the LBM results for the velocity and the stresses are in excellent agreement with the FVM results. Copyright © 2004 John Wiley & Sons, Ltd.     

A hybrid FVM–LBM method for single and multi‐fluid compressible flow problems

The lattice Boltzmann method (LBM) has established itself as an alternative approach to solve the fluid flow equations. In this work we combine LBM with the conventional finite volume method (FVM), and propose a non‐iterative hybrid method for the simulation of compressible flows. LBM is used to calculate the inter‐cell face fluxes and FVM is used to calculate the node parameters. The hybrid method is benchmarked for several one‐dimensional and two‐dimensional test cases. The results obtained by the hybrid method show a steeper and more accurate shock profile as compared with the results obtained by the widely used Godunov scheme or by a representative flux vector splitting scheme. Additional features of the proposed scheme are that it can be implemented on a non‐uniform grid, study of multi‐fluid problems is possible, and it is easily extendable to multi‐dimensions. These features have been demonstrated in this work. The proposed method is therefore robust and can possibly be applied to a variety of compressible flow situations. Copyright © 2009 John Wiley & Sons, Ltd.     

A LBM–DEM solver for fast discrete particle simulation of particle–fluid flows

The lattice Boltzmann method (LBM) for simulating fluid phases was coupled with the discrete element method (DEM) for studying solid phases to formulate a novel solver for fast discrete particle simulation (DPS) of particle–fluid flows. The fluid hydrodynamics was obtained by solving LBM equations instead of solving the Navier–Stokes equation by the finite volume method (FVM). Interparticle and particle–wall collisions were determined by DEM. The new DPS solver was validated by simulating a three-dimensional gas–solid bubbling fluidized bed. The new solver was found to yield results faster than its FVM–DEM counterpart, with the increase in the domain-averaged gas volume fraction. Additionally, the scalability of the LBM–DEM DPS solver was superior to that of the FVM–DEM DPS solver in parallel computing. Thus, the LBM–DEM DPS solver is highly suitable for use in simulating dilute and large-scale particle–fluid flows.     

Lattice Boltzmann simulation of natural convection in open end cavity with inclined hot wall

Natural convection in an open end cavity with a hot inclined wall is simulated based on the lattice Boltzmann method (LBM). The physics of flow and energy transfer in open end cavities are addressed when the hot wall is inclined. The combination of the two topics (open cavity and inclined walls) is the main novelty of the present study. The effects of the angle of the hot inclined wall on the flow field and heat transfer are thoroughly investigated. The Prandtl number is fixed to 0.71 (air). The Rayleigh number and the angle of the hot inclined wall are varied in the range of 10⁴ to 10⁶ and 60? to 85?, respectively. The results are presented for two different aspect ratios, i.e., A = 1 and 2. The results obtained with the LBM are also compared with those of the finite volume method (FVM). The predicted results of the LBM conform to those of the FVM. The results show that by increasing the angle of the hot inclined wall and the aspect ratio of the cavity, the average Nusselt number decreases. The trend of the local Nusselt number on the inclined wall is also discussed.     

求解变分型积分方程的一种新型数值方法——有限变分法

将作者提出的多虚拟裂纹扩展法(MVCE法)拓展为求解变分型积分方程问题的一种新型数值方法——有限变分法(FVM)。它的基本思想是,给定有限个(N个)局部变分模式,将所求解的未知量用适当的方法离散化,针对这N个局部变分模式列出N个方程,求解N个未知系数,从而求得未知量。单一未知变量FVM的最终方程组的系数矩阵通常是一个对称的窄带矩阵,对角元是大数,有很好的数值计算性能。用FVM求解了三维I型裂纹前缘的应力强度因子(SIF)分布。利用基于FVM的通用权函数法计算程序,可以高精度和高效率地求解表面力、体积力和温度载荷共同作用情况下三维裂纹前缘SIF的分布及其时间历程。FVM可以被推广到更广泛的领域,是一个求解变分型积分方程问题的普遍适用的新型数值方法。     

Moving‐boundary analysis of net vapor generation point in a steam generator

This paper deals with the study of the dynamics of net vapor generation point in the boiling channel of the steam generator of Kaiga‐1 nuclear power plant. The dynamics has been studied by perturbing liquid velocity at the inlet of boiling channel with a step function and heating rate with a ramp function. Both finite volume method (FVM) and finite difference method (FDM) have been applied to solve the model equations that have been developed to predict boiling boundary. The effect of thermal non‐equilibrium conditions on subcooled boiling has been taken into consideration. A comparative study of the two methods has been carried out based on the analytical solution of the equations. The study shows that at higher system frequency, increasing number of computational grids increase the accuracy of numerical solutions using FVM while FDM fails to achieve the same. The superiority of FVM over FDM for the problem has also been confirmed by grid convergence analysis. An attempt has also been made to find the analytical solution of the effect of change of heat input on boiling boundary, which is an essential part of computations for the simulation of startup and shutdown of the steam generator. Copyright © 2008 John Wiley & Sons, Ltd.     

Airfoil design optimization based on lattice Boltzmann method and adjoint approach

We present a new aerodynamic design method based on the lattice Boltzmann method (LBM) and the adjoint approach. The flow field and the adjoint equation are numerically simulated by the GILBM (generalized form of interpolation supplemented LBM) on non-uniform meshes. The first-order approximation for the equilibrium distribution function on the boundary is proposed to diminish the singularity of boundary conditions. Further, a new treatment of the solid boundary in the LBM is described particularly for the airfoil optimization design problem. For a given objective function, the adjoint equation and its boundary conditions are derived analytically. The feasibility and accuracy of the new approach have been perfectly validated by the design optimization of NACA0012 airfoil.     

Fully resolved simulations of viscoelastic suspensions by an efficient immersed boundary-lattice Boltzmann method

An efficient immersed boundary-lattice Boltzmann method (IB-LBM) is proposed for fully resolved simulations of suspended solid particles in viscoelastic flows. Stress LBM based on Giesekus and Oldroyd-B constitutive equation are used to model the viscoelastic stress tensor. A boundary thickening-based direct forcing IB method is adopted to solve the particle–fluid interactions with high accuracy for non-slip boundary conditions. A universal law is proposed to determine the diffusivity constant in a viscoelastic LBM model to balance the numerical accuracy and stability over a wide range of computational parameters. An asynchronous calculation strategy is adopted to further improve the computing efficiency. The method was firstly applicated to the simulation of sedimentation of a single particle and a pair of particles after good validations in cases of the flow past a fixed cylinder and particle migration in a Couette flow against FEM and FVM methods. The determination of the asynchronous calculation strategy and the effect of viscoelastic stress distribution on the settling behaviors of one and two particles are revealed. Subsequently, 504 particles settling in a closed cavity was simulated and the phenomenon that the viscoelastic stress stabilizing the Rayleigh–Taylor instabilities was observed. At last, simulations of a dense flow involving 11001 particles, the largest number of particles to date, were performed to investigate the instability behavior induced by elastic effect under hydrodynamic interactions in a viscoelastic fluid. The elasticity-induced ordering of the particle structures and fluid bubble structures in this dense flow is revealed for the first time. These simulations demonstrate the capability and prospects of the present method for aid in understanding the complex behaviors of viscoelastic particle suspensions.     

Simulation of three‐dimensional flows over moving objects by an improved immersed boundary–lattice Boltzmann method

An improved immersed boundary–lattice Boltzmann method (IB–LBM) developed recently [28] was applied in this work to simulate three‐dimensional (3D) flows over moving objects. By enforcing the non‐slip boundary condition, the method could avoid any flow penetration to the wall. In the developed IB–LBM solver, the flow field is obtained on the non‐uniform mesh by the efficient LBM that is based on the second‐order one‐dimensional interpolation. As a consequence, its coefficients could be computed simply. By simulating flows over a stationary sphere and torus [28] accurately and efficiently, the proposed IB–LBM showed its ability to handle 3D flow problems with curved boundaries. In this paper, we further applied this method to simulate 3D flows around moving boundaries. As a first example, the flow over a rotating sphere was simulated. The obtained results agreed very well with the previous data in the literature. Then, simulation of flow over a rotating torus was conducted. The capability of the improved IB–LBM for solving 3D flows over moving objects with complex geometries was demonstrated via the simulations of fish swimming and dragonfly flight. The numerical results displayed quantitative and qualitative agreement with the date in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

A semi‐Lagrangian multi‐moment finite volume method with fourth‐order WENO projection

Numerical oscillation has been an open problem for high‐order numerical methods with increased local degrees of freedom (DOFs). Current strategies mainly follow the limiting projections derived originally for conventional finite volume methods and thus are not able to make full use of the sub‐cell information available in the local high‐order reconstructions. This paper presents a novel algorithm that introduces a nodal value‐based weighted essentially non‐oscillatory limiter for constrained interpolation profile/multi‐moment finite volume method (CIP/MM FVM) (Ii and Xiao, J. Comput. Phys., 222 (2007), 849–871) as an effort to pursue a better suited formulation to implement the limiting projection in schemes with local DOFs. The new scheme, CIP‐CSL‐WENO4 scheme, extends the CIP/MM FVM method by limiting the slope constraint in the interpolation function using the weighted essentially non‐oscillatory (WENO) reconstruction that makes use of the sub‐cell information available from the local DOFs and is built from the point values at the solution points within three neighboring cells, thus resulting a more compact WENO stencil. The proposed WENO limiter matches well the original CIP/MM FVM, which leads to a new scheme of high accuracy, algorithmic simplicity, and computational efficiency. We present the numerical results of benchmark tests for both scalar and Euler conservation laws to manifest the fourth‐order accuracy and oscillation‐suppressing property of the proposed scheme. Copyright © 2016 John Wiley & Sons, Ltd.     

求解三维裂纹前缘SIF分布时间历程的通用权函数法和有限变分法

将三维热权函数法扩展为适用于表面力、体积力和温度载荷的通用权函数法(UWF).推导出以变分型积分方程表达的UWF法基本方程,从变分的角度,将求解三维热权函数法基本方程的多虚拟裂纹扩展法(MVCE)改造为可以适用于一般的变分型积分方程的一类新型数值方法--有限变分法(FVM).在FVM中可以引入无穷多种线性无关的局部变分模式,可以根据计算要求在求解域中插入任意多个计算节点,单一型裂纹问题FVM所得到的最终方程组的系数矩阵总是一个对称的窄带矩阵,而且对角元总是大数,具有良好的数值计算性能.FVM对于SIF沿裂纹前缘急剧变化的复杂情况具有较好的数值模拟能力和较高的计算精度,利用自身一致性,可以求得三维裂纹前缘SIF的高精度解.     

Implicit–explicit finite‐difference lattice Boltzmann method with viscid compressible model for gas oscillating patterns in a resonator

Difficulties for the conventional computational fluid dynamics and the standard lattice Boltzmann method (LBM) to study the gas oscillating patterns in a resonator have been discussed. In light of the recent progresses in the LBM world, we are now able to deal with the compressibility and non‐linear shock wave effects in the resonator. A lattice Boltzmann model for viscid compressible flows is introduced firstly. Then, the Boltzmann equation with the Bhatnagar–Gross–Krook approximation is solved by the finite‐difference method with a third‐order implicit–explicit (IMEX) Runge–Kutta scheme for time discretization, and a fifth‐order weighted essentially non‐oscillatory (WENO) scheme for space discretization. Numerical results obtained in this study agree quantitatively with both experimental data available and those using conventional numerical methods. Moreover, with the IMEX finite‐difference LBM (FDLBM), the computational convergence rate can be significantly improved compared with the previous FDLBM and standard LBM. This study can also be applied for simulating some more complex phenomena in a thermoacoustics engine. Copyright © 2008 John Wiley & Sons, Ltd.     

MEMS稀薄气体内部流动模拟中的信息保存法

首先综述了处理低速稀薄气体流动的一些方法: 线化Boltzmann方程方法、Lattice Boltzmann方法(LBM)、加滑移边界的Navier-Stokes方程、以及DSMC方法, 并讨论它们在模拟MEMS中过渡领域低速流动特别是内部流动所遇到的困难, 其中表明了LBM现有方案不适合模拟过渡领域中的MEMS流动问题. 信息保存(IP)法通过保存一个模拟分子所代表的大量分子的平均信息,克服了流速低使得信息噪声比小而引起统计模拟的困难. 本文给出了方法的一些理论证实. MEMS中内部流动的特点, 即流速低和大的长宽比的特点, 引起椭圆性问题, 即出入口边界条件相互影响需要协调的问题. 通过对(长约几千微米的)微槽道流动应用IP方法的算例,演示了采用守恒形式的质量守恒方程和超松弛法可成功地解决这一问题. 借助同样的方法,用IP方法求解了真实长度(1\,000\,$\mu$m)硬盘驱动器读写头在过渡领域的薄膜支撑问题, 压力分布与具有严格气体动理论基础的概括化Reynolds方程完全相符, 而在此之前, DSMC方法只对短的读写头(5\,$\mu$m)与Reynolds方程做了校验. 作者建议将原来用于求解读写头润滑问题的Reynolds方程退化来求解过渡领域中的微槽道流动问题, 从而提供了一个有严格气体动理论品性的检验方法来验证求解MEMS内部流动的各种方法.      

基于格子-波尔兹曼法的流体能耗求解

格子-波尔兹曼法是近年来新兴的一种计算流体力学数值方法。随着这种方法的不断发展,人们将它用于流体的仿真、优化等不同场合。与此同时,一些与流场流速和压强相关的物理量(如能耗)的求解也成为关注的焦点。本文介绍了能耗这一流体宏观量的格子-波尔兹曼法求解及其实现。与传统的有限差分法不同,本文在求解有关的速度梯度时使用了格子-波尔兹曼-矩法,这种方法不但能够避免有限差分法在边界处失效的缺点,而且计算简单,算法局部性好,适合大规模并行计算。本文在分析其数值解精度的基础上,使用这种方法进行了以能耗极小为目标的直通道内椭圆挡块的参数优化。这些分析和算例分别定量和定性地说明了本文算法的准确性。     

A direct‐forcing pressure‐based lattice Boltzmann method for solving fluid–particle interaction problems

A direct‐forcing immersed boundary‐lattice Boltzmann method (IB–LBM) is developed to simulate fluid–particle interaction problems. This method uses the pressure‐based LBM to solve the incompressible flow field and the immersed boundary method to handle the fluid–particle interactions. The pressure‐based LBM uses the pressure distribution functions instead of the density distribution functions as the independent dynamic variables. The main idea is to explicitly eliminate the compressible effect due to the density fluctuation. In the IB method, a direct‐forcing method is introduced to capture the particle motion. It directly computes an IB force density at each lattice grid from the differences between the pressure distribution functions obtained by the LBM and the equilibrium pressure distribution functions computed from the particle velocity. By applying this direct‐forcing method, the IB–LBM becomes a purely LBM version. Also, by applying the Gauss theorem, the formulas for computing the force and the torque acting on the particle from the flows are derived from the volume integrals over the particle volume instead of from the surface integrals over the particle surface. The order of accuracy of the IB–LBM is demonstrated on the errors of velocity field, wall stress, and gradients of velocity and pressure. As a demonstration of the efficiency and capabilities of the new method, sedimentation of a large number of spherical particles in an enclosure is simulated. Copyright © 2010 John Wiley & Sons, Ltd.     

Lattice Boltzmann method for simulating particle–fluid interactions

The lattice Boltzmann method （LBM） has gained increasing popularity in the last two decades as an alternative numerical approach for solving fluid flow problems. One of the most active research areas in the LBM is its application in particle-fluid systems, where the advantage of the LBM in efficiency and parallel scalability has made it superior to many other direct numerical simulation （DNS） techniques. This article intends to provide a brief review of the application of the LBM in particle-fluid systems. The numerical techniques in the LBM pertaining to simulations of particles are discussed, with emphasis on the advanced treatment for boundary conditions on the particle-fluid interface. Other numerical issues, such as the effect of the internal fluid, are also briefly described. Additionally, recent efforts in using the LBM to obtain closures for particle-fluid drag force are also reviewed.     

Numerical simulation of laminar jet-forced flow using lattice Boltzmann method

In the paper, a numerical study on symmetrical and asymmetrical laminar jet-forced flows is carried out by using a lattice Boltzmann method （LBM） with a special boundary treatment. The simulation results are in very good agreement with the available numerical prediction. It is shown that the LBM is a competitive method for the laminar jet-forced flow in terms of computational efficiency and stability.     

Lattice Boltzmann modeling of pore‐scale fluid flow through idealized porous media

In order to capture the hydro‐mechanical impacts on the solid skeleton imposed by the fluid flowing through porous media at the pore‐scale, the flow in the pore space has to be modeled at a resolution finer than the pores, and the no‐slip condition needs to be enforced at the grain–fluid interface. In this paper, the lattice Boltzmann method (LBM), a mesoscopic Navier–Stokes solver, is shown to be an appropriate pore‐scale fluid flow model. The accuracy and lattice sensitivity of LBM as a fluid dynamics solver is demonstrated in the Poiseuille channel flow problem (2‐D) and duct flow problem (3‐D). Well‐studied problems of fluid creeping through idealized 2‐D and 3‐D porous media (J. Fluid Mech. 1959; 5 (2):317–328, J. Fluid Mech. 1982; 115 :13–26, Int. J. Multiphase Flow 1982; 8 (4):343–360, Phys. Fluids A 1989; 1 (1):38–46, Int. J. Numer. Anal. Meth. Geomech. 1999; 23 :881–904, Int. J. Numer. Anal. Meth. Geomech. 2010; DOI: 10.1002/nag.898, Int. J. Multiphase Flow 1982; 8 (3):193–206) are then simulated using LBM to measure the friction coefficient for various pore throats. The simulation results agree well with the data reported in the literature. The lattice sensitivity of the frictional coefficient is also investigated. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical and experimental study on the motion characteristics of single bubble in a complex channel

This paper is an extended study from previous work. In this study, the focus is paid to the dynamics of bubble rising and deformation in a complex channel, while the previous work is in straight channel. For this purpose, a three-dimensional lattice Boltzmann method (LBM) is employed to simulate the dynamics behaviour of a bubble rising in a complex channel consisting of three half-round throats. To validate the numerical method, a visual experiment was carried out by means of a high-speed digital camera and computer image processing technology. The behaviour of the rising bubble through glycerine solution in a complex channel was recorded. Some physical parameters such as rising velocities, trajectory and shapes of the bubble were calculated and processed based on the experimental data. In the same conditions, the trajectory, shapes and rising velocities of the bubble were simulated during its rising process by the proposed LBM. The numerical results are in good agreement with the experimental results. It demonstrates that LBM used in this work is feasible for simulating two-phase flow in such a complex channel.     

THE RESEARCH PROGRESS OF LATTICE BOLTZMANN METHOD IN NON-NEWTONIAN FLUID FLOW

格子玻尔兹曼方法（lattice Boltzmann method,LBM）能够直接计算局部剪切速率并可以达到二次精度,因此在非牛顿流动数值模拟中展现出一定优势。尽管已证实LBM 对于非牛顿流动的适用性,但是LBM 需要通过即时调节BGK（Bhatnagar-Gross-Krook）碰撞项中的松弛时间来实时反映黏度改变,当松弛时间接近1/2 时,迭代会出现数值不稳定现象。该文对LBM 在非牛顿流体研究中的进展进行了总结,介绍了增加数值稳定性的方法并对结果的精度进行了比较,在此基础上对LBM 在非牛顿研究中的进一步发展进行了展望。