An evaluation of lattice Boltzmann schemes for porous medium flow simulation

We quantitatively evaluate the capability and accuracy of the lattice Boltzmann equation (LBE) for modeling flow through porous media. In particular, we conduct a comparative study of the LBE models with the multiple-relaxation-time (MRT) and the Bhatnagar–Gross–Krook (BGK) single-relaxation-time (SRT) collision operators. We also investigate several fluid–solid boundary conditions including: (1) the standard bounce-back (SBB) scheme, (2) the linearly interpolated bounce-back (LIBB) scheme, (3) the quadratically interpolated bounce-back (QIBB) scheme, and (4) the multi-reflection (MR) scheme. Three-dimensional flow through two porous media—a body-centered cubic (BCC) array of spheres and a random-sized sphere-pack—are examined in this study. For flow past a BCC array of spheres, we validate the linear LBE model by comparing its results with the nonlinear LBE model. We investigate systematically the viscosity-dependence of the computed permeability, the discretization error, and effects due to the choice of relaxation parameters with the MRT and BGK schemes. Our results show unequivocally that the MRT–LBE model is superior to the BGK–LBE model, and interpolation significantly improves the accuracy of the fluid–solid boundary conditions.     

非均匀分布颗粒群中的曳力分布

本文采用格子Boltzmann方法(LBM)在图形处理器(GPU)上计算了由静止圆柱阵列组成的团聚物周期单元内的不可压缩流体流动,流固交界面处采用直接反弹以实现无滑移边界,每个圆柱上的曳力通过统计动量交换直接求得。根据LBM求得的流体速度,对于团聚物中的单圆柱按能量最小多尺度(EMMS)模型计算平均曳力系数,并考察了将聚团近似为均匀悬浮的临界条件。对颗粒雷诺数Re_p在0～10之间的80种固相份额的模拟结果表明,密相空隙率可以表征这种临界条件。当固相份额恒定时,该临界空隙率随着Re_p的增加而降低;当Re_p恒定时,该临界空隙率随着固相份额的增加而降低。     

A viscosity counteracting approach in the lattice Boltzmann BGK model for low viscosity flow: Preliminary verification

Due to numerical instability, the lattice Boltzmann model (LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator has some limitations in the simulation of low viscosity flows. In this paper, we propose a viscosity counteracting approach for simulating a moderate viscosity flow. An extra negative viscosity term is introduced to counteract part of the moderate viscosity by using the lattice Boltzmann equation with a source term. The counteracting viscosity term is treated as a non-uniform unsteady source. The stability is enhanced; thus small viscosity flows can be simulated. Model verification consists of benchmark cases such as those of Poiseuille flow, Couette flow, waterhammer waves, Taylor–Green vortex flow, and lid-driven cavity flow. The flow patterns, error characteristics, and representative parameters are carefully analyzed. It is shown that this approach can simulate flows with lower viscosities than may be simulated using the normal LBGK model; the second-order accuracy of the LBGK model is definitely retained, although a little dissipation is added. These preliminary studies prove the effectiveness and accuracy of the model. Sophisticated analysis and further verification of the stability mechanism will be done in the near future.     

Three-dimensional discrete-velocity BGK model for the incompressible Navier–Stokes equations

The lattice Boltzmann method (LBM) has been widely used for the simulations of the incompressible Navier–Stokes (NS) equations. The finite difference Boltzmann method (FDBM) in which the discrete-velocity Boltzmann equation is solved instead of the lattice Boltzmann equation has also been applied as an alternative method for simulating the incompressible flows. The particle velocities of the FDBM can be selected independently from the lattice configuration. In this paper, taking account of this advantage, we present the discrete velocity Boltzmann equation that has a minimum set of the particle velocities with the lattice Bharnagar–Gross–Krook (BGK) model for the three-dimensional incompressible NS equations. To recover incompressible NS equations, tensors of the particle velocities have to be isotropic up to the fifth rank. Thus, we propose to apply the icosahedral vectors that have 13 degrees of freedom to the particle velocity distributions. Validity of the proposed model (D3Q13BGK) is confirmed by numerical simulations of the shear-wave decay problem and the Taylor–Green vortex problem. With respect to numerical accuracy, computational efficiency and numerical stability, we compare the proposed model with the conventional lattice BGK models (D3Q15, D3Q19 and D3Q27) and the multiple-relaxation-time (MRT) model (D3Q13MRT) that has the same degrees of freedom as our proposal. The comparisons show that the compressibility error of the proposed model is approximately double that of the conventional lattice BGK models, but the computational efficiency of the proposed model is superior to that of the others. The linear stability of the proposed model is also superior to that of the lattice BGK models. However, in non-linear simulations, the proposed model tends to be less stable than the others.     

Adjoint parameter sensitivity analysis for the hydrodynamic lattice Boltzmann method with applications to design optimization

We present an adjoint parameter sensitivity analysis formulation and solution strategy for the lattice Boltzmann method (LBM). The focus is on design optimization applications, in particular topology optimization. The lattice Boltzmann method is briefly described with an in-depth discussion of solid boundary conditions. We show that a porosity model is ideally suited for topology optimization purposes and models no-slip boundary conditions with sufficient accuracy when compared to interpolation bounce-back conditions. Augmenting the porous boundary condition with a shaping factor, we define a generalized geometry optimization formulation and derive the corresponding sensitivity analysis for the single relaxation LBM for both topology and shape optimization applications. Using numerical examples, we verify the accuracy of the analytical sensitivity analysis through a comparison with finite differences. In addition, we show that for fluidic topology optimization a scaled volume constraint should be used to obtain the desired “0-1” optimal solutions.     

Non-body-fitted Cartesian-mesh simulation of highly turbulent flows using multi-relaxation-time lattice Boltzmann method

This paper presents a lattice Boltzmann method (LBM) based study aimed at numerical simulation of highly turbulent and largely inclined flow around obstacles of curved geometry using non-body-fitted Cartesian meshes. The approach features (1) combining the interpolated bounce-back scheme with the LBM of multi-relaxation-time (MRT) type to enable the use of simple Cartesian mesh for the flow cases even with complex geometries; and (2) incorporating the Spalart–Allmaras (SA) turbulence model into LBM in order to represent the turbulent flow effect. The numerical experiments are performed corresponding to flows around an NACA0012 airfoil at Re=5×10⁵ and around a flat plate at Re=2×10⁴, respectively. The agreement between all simulation results obtained from this study and the data provided by other literature demonstrates the reliability of the enhanced LBM proposed in this paper for simulating, simply on Cartesian meshes, complex flows that may involve bodies of curved boundary, high Reynolds number, and large angle of attack.     

Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods—Part I: Formulation and application to the 2-D Burgers’ equation

The lattice Boltzmann method is being increasingly employed in the field of computational fluid dynamics due to its computational efficiency. Floating-point operations in the lattice Boltzmann method involve local data and therefore allow easy cache optimization and parallelization. Due to this, the cache-optimized lattice Boltzmann method has superior computational performance over traditional finite difference methods for solving unsteady flow problems. When solving steady flow problems, the explicit nature of the lattice Boltzmann discretization limits the time step size and therefore the efficiency of the lattice Boltzmann method for steady flows. To quantify the computational performance of the lattice Boltzmann method for steady flows, a comparison study between the lattice Boltzmann method (LBM) and the alternating direction implicit (ADI) method was performed using the 2-D steady Burgers’ equation. The comparison study showed that the LBM performs comparatively poor on high-resolution meshes due to smaller time step sizes, while on coarser meshes where the time step size is similar for both methods, the cache-optimized LBM performance is superior. Because flow domains can be discretized with multiblock grids consisting of coarse and fine grid blocks, the cache-optimized LBM can be applied on the coarse grid block while the traditional implicit methods are applied on the fine grid blocks. This paper finds the coupled cache-optimized lattice Boltzmann-ADI method to be faster by a factor of 4.5 over the traditional methods while maintaining similar accuracy.     

Numerical prediction of three-dimensional time-dependent viscoelastic extrudate swell using differential and algebraic models

The dynamic behavior of a droplet on a solid surface is simulated by the lattice Boltzmann method (LBM) for two-phase fluids with large density differences; the wetting boundary condition on solid walls is incorporated in this simulation. By using the method, the dynamic behavior of a droplet impinging on a horizontal wall is investigated in terms of various Weber numbers. The dynamic contact angle, the contact line velocity, and the wet length are calculated, and found to be in good agreement with available experimental data. In addition, the method is applied to simulations of the collision of a falling droplet with a stationary droplet on a solid surface. The behavior of the droplets and the mixing process during their collision are simulated in terms of various impact velocities and several static contact angles on the solid surface. It is seen that mixing occurs around the rim of the coalescent droplet due to the circular flows. Also, the relationship between the mixing rate of the primary coalescent droplet and Weber number is investigated.     

Optimization Strategies for the Entropic Lattice Boltzmann Method

The entropic formulation of the lattice Boltzmann method (LBM) features enhanced numerical stability due to its compliance with the Boltzmann H-theorem. This stability comes at the price of some computational overhead, associated with the need of adjusting the local relaxation time of the standard LBM in such a way as to secure compliance with the H-theorem. In this paper, we discuss a number of possible optimization strategies to reduce the computational overhead of entropic LBMs.     

A scalable interface-resolved simulation of particle-laden flow using the lattice Boltzmann method

We examine the scalable implementation of the lattice Boltzmann method (LBM) in the context of interface-resolved simulation of wall-bounded particle-laden flows. Three distinct aspects relevant to performance optimization of our lattice Boltzmann simulation are studied. First, we optimize the core sub-steps of LBM, the collision and the propagation (or streaming) sub-steps, by reviewing and implementing five different published algorithms to reduce memory loading and storing requirements to boost performance. For each, two different array storage formats are benchmarked to test effective cache utilization. Second, the vectorization of the multiple-relaxation-time collision model is discussed and our vectorized collision and propagation algorithm is presented. We find that careful use of Intel’s Advance Vector Extensions and appropriate array storage formats can significantly enhance performance. Third, in the presence of many finite-size, moving solid particles within the flow field, three different communication schemes are proposed and compared in order to optimize the treatment of fluid-solid interactions. These efforts together lead to a very efficient LBM simulation code for interface-resolved simulation of particle-laden flows. Overall, the optimized scalable code of particle-laden flow is a factor of 4.0-to-8.5 times faster than our previous implementation.     

Application of lattice Boltzmann method in free surface flow simulation of micro injection molding

The filling flow in micro injection molding was simulated by using the lattice Boltzmann method (LBM). A tracking algorithm for free surface to handle the complex interaction between gas and liquid phases in LBM was used for the free surface advancement. The temperature field in the filling flow is also analyzed by combining the thermal lattice Boltzmann model and the free surface method. To simulate the fluid flow of polymer melt with a high Prandtl number and high viscosity, a modified lattice Boltzmann scheme was adopted by introducing a free parameter in the thermal diffusion equation to overcome the restriction of the thermal relaxation time. The filling flow simulation of micro injection molding was successfully performed in the study.     

Adjoint-based fluid flow control and optimisation with lattice Boltzmann methods

A lattice Boltzmann (LB) framework to solve fluid flow control and optimisation problems numerically is presented. Problems are formulated on a mesoscopic basis. In a side condition, the dynamics of a Newtonian fluid is described by a family of simplified Boltzmann-like equations, namely BGK–Boltzmann equations, which are linked to an incompressible Navier–Stokes equation. It is proposed to solve the non-linear optimisation problem by a line search algorithm. The needed derivatives are obtained by deriving the adjoint equations, referred to as adjoint BGK–Boltzmann equations. The primal equations are discretised by standard lattice Boltzmann methods (LBM) while for the adjoint equations a novel discretisation strategy is introduced. The approach follows the main ideas behind LBM and is therefore referred to as adjoint lattice Boltzmann methods (ALBM). The corresponding algorithm retains most of the basic features of LB algorithms. In particular, it enables a highly-efficient parallel implementation and thus solving large-scale fluid flow control and optimisation problems. The overall solution strategy, the derivation of a prototype adjoint BGK–Boltzmann equation, the novel ALBM and its parallel realisation as well as its validation are discussed in detail in this article. Numerical and performance results are presented for a series of steady-state distributed control problems with up to approximately 1.6 million unknown control parameters obtained on a high performance computer with up to 256 processing units.     

Lattice Boltzmann modeling of dendritic growth in forced and natural convection

A two-dimensional (2D) coupled model is developed for the simulation of dendritic growth during alloy solidification in the presence of forced and natural convection. Instead of conventional continuum-based Navier–Stokes (NS) solvers, the present model adopts a kinetic-based lattice Boltzmann method (LBM), which describes flow dynamics by the evolution of distribution functions of moving pseudo-particles, for the numerical computations of flow dynamics as well as thermal and solutal transport. The dendritic growth is modeled using a solutal equilibrium approach previously proposed by Zhu and Stefanescu (ZS), in which the evolution of the solid/liquid interface is driven by the difference between the local equilibrium composition and the local actual liquid composition. The local equilibrium composition is calculated from the local temperature and curvature. The local temperature and actual liquid composition, controlled by both diffusion and convection, are obtained by solving the LB equations using the lattice Bhatnagar–Gross–Krook (LBGK) scheme. Detailed model validation is performed by comparing the simulations with analytical predictions, which demonstrates the quantitative capability of the proposed model. Furthermore, the convective dendritic growth features predicted by the present model are compared with those obtained from the Zhu–Stefanescu and Navier–Stokes (ZS–NS) model, in which the fluid flow is calculated using an NS solver. It is found that the evolution of the solid fraction of dendritic growth calculated by both models coincides well. However, the present model has the significant advantages of numerical stability and computational efficiency for the simulation of dendritic growth with melt convection.     

Direct numerical and large eddy simulation of longitudinal flow along triangular array of rods using the lattice Boltzmann method

Results of direct numerical (DNS) and large eddy simulation (LES) of turbulent longitudinal flow in rod bundles are presented using the lattice Boltzmann method with the Bhatnagar–Gross–Krook collision operator [P.L. Bhatnagar, E.P. Gross, M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94 (1954) 511; Y.H. Qian, d’Humiéres, P. Lallemand, Lattice BGK models for Navier-Stokes equation, Europhys. Lett. 17 (1992) 479] as a computational framework. The problem requires the accurate modeling of curved walls, to which the method proposed by Yu et al. [D. Yu, M.R. Luo, W. Shyy, Viscous flow computations with the method of lattice Boltzmann equation, Prog. Aerospace Sci. 39 (2003) 329] has been applied. The computational domain is a regular hexagonal prism around the rod. Opposite sides of the prism are coupled periodically. In the longitudinal direction periodical boundary conditions are applied and the flow is driven by a body force. Simulations were carried out using two three-dimensional lattices. It has been found that the application of the model with 19 velocities (D3Q19) gives qualitatively false result. However, we have found that the application of the model with 27 links (D3Q27) can provide the proper mean axial velocity profile, and it also predicts the secondary flow patterns deduced from measurements [A.C. Trupp, R.S. Azad, The structure of turbulent flow in triangular array rod bundles, Nucl. Eng. Des. 32 (1975) 47]. Flow pulsation phenomenon is also observed in our simulations just like in some recent measurements of Krauss and Meyer [T. Krauss, L. Meyer, Experimental investigation of turbulent transport of momentum and energy in heated rod bundle, Nucl. Eng. Des. 180 (1998) 185].     

A parallel workload balanced and memory efficient lattice-Boltzmann algorithm with single unit BGK relaxation time for laminar Newtonian flows

A parallel workload balanced and memory efficient lattice-Boltzmann algorithm for laminar Newtonian fluid flow through large porous media is investigated. It relies on a simplified LBM scheme using a single unit BGK relaxation time, which is implemented by means of a shift algorithm and comprises an even fluid node partitioning domain decomposition strategy based on a vector data structure. It provides perfect parallel workload balance, and its two-nearest-neighbour communication pattern combined with a simple data transfer layout results in 20-55% lower communication cost, 25-60% higher computational parallel performance and 40-90% lower memory usage than previously reported LBM algorithms. Performance tests carried out using scale-up and speed-up case studies of laminar Newtonian fluid flow through hexagonal packings of cylinders and a random packing of polydisperse spheres on two different computer architectures reveal parallel efficiencies with 128 processors as high as 75% for domain sizes comprising more than 5 billion fluid nodes.     

Sailfish: A flexible multi-GPU implementation of the lattice Boltzmann method

We present Sailfish, an open source fluid simulation package implementing the lattice Boltzmann method (LBM) on modern Graphics Processing Units (GPUs) using CUDA/OpenCL. We take a novel approach to GPU code implementation and use run-time code generation techniques and a high level programming language (Python) to achieve state of the art performance, while allowing easy experimentation with different LBM models and tuning for various types of hardware. We discuss the general design principles of the code, scaling to multiple GPUs in a distributed environment, as well as the GPU implementation and optimization of many different LBM models, both single component (BGK, MRT, ELBM) and multicomponent (Shan–Chen, free energy). The paper also presents results of performance benchmarks spanning the last three NVIDIA GPU generations (Tesla, Fermi, Kepler), which we hope will be useful for researchers working with this type of hardware and similar codes.     

Efficient LBM Visual Simulation on Face-Centered Cubic Lattices

The Lattice Boltzmann method (LBM) for visual simulation of fluid flow generally employs cubic Cartesian (CC) lattices such as the D3Q13 and D3Q19 lattices for the particle transport. However, the CC lattices lead to suboptimal representation of the simulation space. We introduce the face-centered cubic (FCC) lattice, fD3Q13, for LBM simulations. Compared to the CC lattices, the fD3Q13 lattice creates a more isotropic sampling of the simulation domain and its single lattice speed (i.e., link length) simplifies the computations and data storage. Furthermore, the fD3Q13 lattice can be decomposed into two independent interleaved lattices, one of which can be discarded, which doubles the simulation speed. The resulting LBM simulation can be efficiently mapped to the GPU, further increasing the computational performance. We show the numerical advantages of the FCC lattice on channeled flow in 2D and the flow-past-a-sphere benchmark in 3D. In both cases, the comparison is against the corresponding CC lattices using the analytical solutions for the systems as well as velocity field visualizations. We also demonstrate the performance advantages of the fD3Q13 lattice for interactive simulation and rendering of hot smoke in an urban environment using thermal LBM.     

基于Python的大规模高性能LBM多相流模拟

Python由于具有丰富的第三方库、开发高效等优点,已成为数据科学、智能科学等应用领域最流行的编程语言之一。Python强调了对科学与工程计算的支持,目前已积累了丰富的科学与工程计算库和工具。例如,SciPy和NumPy等数学库提供了高效的多维数组操作及丰富的数值计算功能。以往,Python主要作为脚本语言,起到连接数值模拟前处理、求解器和后处理的“胶水”功能,以提升数值模拟的自动化处理水平。近年来,国外已有学者尝试采用Python代码实现求解计算功能,并在高性能计算机上开展了超大规模并行计算研究,取得了不错的效果。由于自身特点,高效大规模Python数值模拟的实现和性能优化与传统基于C/C++和Fortran的数值模拟等具有很大的不同。文中实现了国际上首个完全基于Python的大规模并行三维格子玻尔兹曼多相流模拟代码PyLBMFlow,探索了Python大规模高性能计算和性能优化方法。首先,利用NumPy多维数组和通用函数设计实现了LBM流场数据结构和典型计算内核,通过一系列性能优化并对LBM边界处理算法进行重构,大幅提升了Python的计算效率,相对于基准实现,优化后的串行性能提升了两个量级。在此基础上,采用三维流场区域分解方法,基于mpi4py和Cython实现了MPI+OpenMP混合并行;在天河二号超级计算机上成功模拟了基于D3Q19离散方法和Shan-Chen BGK碰撞模型的气液两相流,算例规模达百亿网格,并行规模达1024个结点,并行效率超过90%。     

An algorithm based on collision theory for the lattice Boltzmann simulation of isothermal mass diffusion with chemical reaction

The lattice Boltzmann method (LBM) provides a framework for the simulation of mass transport and chemical reaction in complex geometries when accurate pointwise mesoscopic-scale solutions are sought. Herein, an algorithm based on collision theory to determine the rate of chemical reaction during the collision step in LBM is proposed. The model is validated against three isothermal problems with simple analytical solutions: a batch reactor with homogeneous chemical reactions of first and second-order and a cylindrical pore with one-dimensional mass diffusion and surface (heterogeneous) chemical reaction. The results of the LBM simulations agree to within 1% when compared to the analytical solutions, presenting a promising opportunity for the simulation of detailed chemical reaction mechanisms.     

An analysis of natural convection using the thermal finite element discrete Boltzmann equation

The lattice Boltzmann method (LBM) is the simple numerical simulator for fluids because it consists of linear equations. Excluding the higher differential term, the LBM for a temperature field is also achieved as an easy numerical simulation method. However, the LBM is hardly applied to body fitted coordinates for its formulation. It is then difficult to calculate complex lattices using the LBM. In this paper, the finite element discrete Boltzmann equation (FEDBE) is introduced to deal with this weakness of the LBM. The finite element method is applied to the discrete Boltzmann equation (DBE) of the basic equation of the LBM. For FEDBE, the simulation using complex lattices is achieved, and it will be applicable for the development in engineering fields. The natural convection in a square cavity and the Rayleigh–Bernard convection are chosen as the test problem. Each simulation model is accurate enough for the flow patterns, the temperature distribution and the Nusselt number. This method is now considered good for the flow and temperature field, and is expected to be introduced for complex lattices using the DBE.