A hybrid Eulerian–Lagrangian method to simulate the dispersed phase in turbulent gas-particle flows

This paper presents a methodology to combine stochastic Lagrangian approach and continuum model to simulate the dispersed phase in gas-particle turbulent flows using that both approaches are based on the same Boltzmann-like kinetic equation governing the joint fluid-particle probability density function (pdf). The proposed hybrid method is based on the separate application of each approach in two adjacent domains and their coupling at the interface via flux boundary conditions. Validation of the method is carried out for non-colliding solid particles suspended in homogeneous turbulent shear flow without two-way coupling.     

Two-phase turbulence models for simulating dense gas-particle flows

The two-fluid model is widely adopted in simulations of dense gas-particle flows in engineering facili- ties. Present two-phase turbulence models for two-fluid modeling are isotropic. However, turbulence in actual gas-particle flows is not isotropic. Moreover, in these models the two-phase velocity correlation is closed using dimensional analysis, leading to discrepancies between the numerical results, theoretical analysis and experiments. To rectify this problem, some two-phase turbulence models were proposed by the authors and are applied to simulate dense gas-particle flows in downers, risers, and horizontal channels; Experimental results validate the simulation results. Among these models the USM-O and the two-scale USM models are shown to give a better account of both anisotropic particle turbulence and particle-particle collision using the transport equation model for the two-phase velocity correlation.     

Diffusion-based coarse graining in hybrid continuum–discrete solvers: Theoretical formulation and a priori tests

Coarse graining is an important ingredient in many multi-scale continuum–discrete solvers such as CFD–DEM (computational fluid dynamics–discrete element method) solvers for dense particle-laden flows. Although CFD–DEM solvers have become a mature technique that is widely used in multiphase flow research and industrial flow simulations, a flexible and easy-to-implement coarse graining algorithm that can work with CFD solvers of arbitrary meshes is still lacking. In this work, we proposed a new coarse graining algorithm for continuum–discrete solvers for dense particle-laden flows based on solving a transient diffusion equation. Via theoretical analysis we demonstrated that the proposed method is equivalent to the statistical kernel method with a Gaussian kernel, but the current method is much more straightforward to implement in CFD–DEM solvers. A priori numerical tests were performed to obtain the solid volume fraction fields based on given particle distributions, the results obtained by using the proposed algorithm were compared with those from other coarse graining methods in the literature (e.g., the particle centroid method, the divided particle volume method, and the two-grid formulation). The numerical tests demonstrated that the proposed coarse graining procedure based on solving diffusion equations is theoretically sound, easy to implement and parallelize in general CFD solvers, and has improved mesh-convergence characteristics compared with existing coarse graining methods. The diffusion-based coarse graining method has been implemented into a CFD–DEM solver, the results of which are presented in a separate work.     

转动非平衡玻尔兹曼模型方程统一算法与全流域绕流计算应用

基于过去开展稀薄自由分子流到连续流气体运动论统一算法框架,采用转动惯量描述气体分子自旋运动,确立含转动非平衡效应各流域统一玻尔兹曼模型方程.基于转动能量对分布函数守恒积分,得到计及转动非平衡效应气体分子速度分布函数方程组,使用离散速度坐标法对分布函数方程所依赖速度空间离散降维;应用拓展计算流体力学有限差分方法,构造直接求解分子速度分布函数的气体动理论数值格式;基于物面质量流量通量守恒与能量平衡关系,发展计及转动非平衡气体动理论边界条件数学模型及数值处理方法,提出模拟各流域转动非平衡效应玻尔兹曼模型方程统一算法.通过高、低不同马赫数1:5～25氮气激波结构与自由分子流到连续流全飞行流域不同克努森数（9×10-4～10）Ramp制动器、圆球、尖双锥飞行器、飞船返回舱外形体再入跨流域绕流模拟研究,将计算结果与有关实验数据、稀薄流DSMC模拟值等结果对比分析,验证统一算法模拟自由分子流到连续流再入过程高超声速绕流问题的可靠性与精度.      

GAS-KINETIC UNIFIED ALGORITHM FOR BOLTZMANN MODEL EQUATION IN ROTATIONAL NONEQUILIBRIUM AND ITS APPLICATION TO THE WHOLE RANGE FLOW REGIMES

基于过去开展稀薄自由分子流到连续流气体运动论统一算法框架,采用转动惯量描述气体分子自旋运动,确立含转动非平衡效应各流域统一玻尔兹曼模型方程.基于转动能量对分布函数守恒积分,得到计及转动非平衡效应气体分子速度分布函数方程组,使用离散速度坐标法对分布函数方程所依赖速度空间离散降维;应用拓展计算流体力学有限差分方法,构造直接求解分子速度分布函数的气体动理论数值格式;基于物面质量流量通量守恒与能量平衡关系,发展计及转动非平衡气体动理论边界条件数学模型及数值处理方法,提出模拟各流域转动非平衡效应玻尔兹曼模型方程统一算法.通过高、低不同马赫数1：5～25氮气激波结构与自由分子流到连续流全飞行流域不同克努森数（9×10^-4～10）Ramp制动器、圆球、尖双锥飞行器、飞船返回舱外形体再入跨流域绕流模拟研究,将计算结果与有关实验数据、稀薄流DSMC模拟值等结果对比分析,验证统一算法模拟自由分子流到连续流再入过程高超声速绕流问题的可靠性与精度.     

Simple lattice Boltzmann model for traffic flows

A lattice Boltzmann model with 5-bit lattice for traffic flows is proposed. Using the Chapman-Enskog expansion and multi-scale technique, we obtain the higher-order moments of equilibrium distribution function. A simple traffic light problem is simulated by using the present lattie Boltzmann model, and the result agrees well with analytical solution.     

微通道人工泡状流的Lattice Boltzmann数值模拟

采用高频电控热激发汽泡的方式构造微通道人工泡状流,可以有效抑制微通道沸腾流动的不稳定性和强化传热。本文基于Lattice Boltzmann大密度比多相流复合模型,数值研究了通道内人工泡状流的流动和传热,通过比较分析不同发泡频率的泡状流,量化分析了汽泡运动和增长对微通道流动与传热的相互影响。一方面着重分析了汽泡运动对微通道运动边界层以及汽泡相变增长对热边界层的影响,另一方面也研究了边界层对汽泡动力行为的影响,所得结论对研究抑制微通道沸腾流动不稳定性和强化传热有参考意义。     

Studies on gas turbulence and particle fluctuation in dense gas-particle flows

Particle fluctuation and gas turbulence in dense gas-particle flows are less studied due to complexity of the phenomena. In the present study, simulations of gas turbulent flows passing over a single particle are carried out first by using RANS modeling with a Reynolds stress equation turbulence model and sufficiently fine grids, and then by using LES. The turbulence enhancement by the particle wake effect is studied under various particle sizes and relative gas velocities, and the turbulence enhancement is found proportional to the particle diameter and the square of velocity. Based on the above results, a turbulence enhancement model for the particle-wake effect is proposed and is incorporated as a sub-model into a comprehensive two-phase flow model, which is then used to simulate dilute gas-particle flows in a horizontal channel. The simulation results show that the predicted gas turbulence by using the present model accounting for the particle wake effect is obviously in better agreement with the experimental results than the prediction given by the model not accounting for the wake effect. Finally, the proposed model is incorporated into another two-phase flow model to simulate dense gasparticle flows in a downer. The results show that the particle wake effect not only enhances the gas turbulence, but also amplifies the particle fluctuation.     

A discrete contact model for complex arbitrary-shaped convex geometries

The shape of particles has a significant influence on the behavior of suspensions, as the particle-fluid, particle-particle, and particle-wall interactions depend on it. However, the simultaneous consideration of complex particle shapes and four-way coupling remains a major challenge. This is mainly due to a lack of suitable contact models. Contact models for complex shapes have been proposed in literature, and most limit the accuracy of the particle-fluid interaction. For this reason, this paper presents a novel contact model for complex convex particle shapes for use with partially saturated methods, in which we propose to obtain necessary contact properties, such as the indentation depth, by a discretization of the contact area. The goal of the proposed model is to enable comprehensive and accurate studies of particulate flows, especially with high volume fractions, that lead to new insights and contribute to the improvement of existing industrial processes. To ensure correctness and sustainability, we validate the model extensively by studying cases with and without fluid. In the latter case, we use the homogenized lattice Boltzmann method. The provided investigations show a great agreement of the proposed discrete contact model with analytical solutions and the literature.     

Development of LBGK and incompressible LBGK‐based lattice Boltzmann flux solvers for simulation of incompressible flows

This paper presents lattice Boltzmann Bhatnagar–Gross–Krook (LBGK) model and incompressible LBGK model‐based lattice Boltzmann flux solvers (LBFS) for simulation of incompressible flows. LBFS applies the finite volume method to directly discretize the governing differential equations recovered by lattice Boltzmann equations. The fluxes of LBFS at each cell interface are evaluated by local reconstruction of lattice Boltzmann solution. Because LBFS is applied locally at each cell interface independently, it removes the major drawbacks of conventional lattice Boltzmann method such as lattice uniformity, coupling between mesh spacing, and time interval. With LBGK and incompressible LBGK models, LBFS are examined by simulating decaying vortex flow, polar cavity flow, plane Poiseuille flow, Womersley flow, and double shear flows. The obtained numerical results show that both the LBGK and incompressible LBGK‐based LBFS have the second order of accuracy and high computational efficiency on nonuniform grids. Furthermore, LBFS with both LBGK models are also stable for the double shear flows at a high Reynolds number of 10⁵. However, for the pressure‐driven plane Poiseuille flow, when the pressure gradient is increased, the relative error associated with LBGK model grows faster than that associated with incompressible LBGK model. It seems that the incompressible LBGK‐based LBFS is more suitable for simulating incompressible flows with large pressure gradients. Copyright © 2014 John Wiley & Sons, Ltd.     

Accelerating fully resolved simulation of particle-laden flows on heterogeneous computer architectures

An efficient computing framework, namely PFlows, for fully resolved-direct numerical simulations of particle-laden flows was accelerated on NVIDIA General Processing Units (GPUs) and GPU-like accelerator (DCU) cards. The framework is featured as coupling the lattice Boltzmann method for fluid flow with the immersed boundary method for fluid-particle interaction, and the discrete element method for particle collision, using two fixed Eulerian meshes and one moved Lagrangian point mesh, respectively. All the parts are accelerated by a fine-grained parallelism technique using CUDA on GPUs, and further using HIP on DCU cards, i.e., the calculation on each fluid grid, each immersed boundary point, each particle motion, and each pair-particle collision is responsible by one computer thread, respectively. Coalesced memory accesses to LBM distribution functions with the data layout of Structure of Arrays are used to maximize utilization of hardware bandwidth. Parallel reduction with shared memory for data of immersed boundary points is adopted for the sake of reducing access to global memory when integrate particle hydrodynamic force. MPI computing is further used for computing on heterogeneous architectures with multiple CPUs-GPUs/DCUs. The communications between adjacent processors are hidden by overlapping with calculations. Two benchmark cases were conducted for code validation, including a pure fluid flow and a particle-laden flow. The performances on a single accelerator show that a GPU V100 can achieve 7.1–11.1 times speed up, while a single DCU can achieve 5.6–8.8 times speed up compared to a single Xeon CPU chip (32 cores). The performances on multi-accelerators show that parallel efficiency is 0.5–0.8 for weak scaling and 0.68–0.9 for strong scaling on up to 64 DCU cards even for the dense flow (φ = 20%). The peak performance reaches 179 giga lattice updates per second (GLUPS) on 256 DCU cards by using 1 billion grids and 1 million particles. At last, a large-scale simulation of a gas-solid flow with 1.6 billion grids and 1.6 million particles was conducted using only 32 DCU cards. This simulation shows that the present framework is prospective for simulations of large-scale particle-laden flows in the upcoming exascale computing era.     

The setting up of the equations for two-phase flows by the kinetic theory method

The fundamental equations for two-phase flows are deduced from the Boltzmann's equation. The collision terms are treated with a method similar to what is used in the classical kinetic theory for handling the transport properties of dense gases. It is shown that collision pressure and collision thermal flux exist in gas-particle flows in addition to the general partial pressure and partial thermal flux. Their physical natures are quite different from those of the general partial pressure and partial thermal flux. The applicability of the binary collision assumption and the molecular chaos assumption to gas-particle flows is also discussed. Finally, the equations for two-phase flows obtained by the method of the kinetic theory are compared with those obtained by average continuum models and by the model of particle clouds. The results from the kinetic theory show clearly the physical significance of various parameters and clarify some confusing concepts. Institute of Mechanics, Academia Sinica     

颗粒流体系统的格子Boltzmann数值方法研究进展

介绍Euler-Lagrange框架下基于格子Boltzmann方法 LBM(Lattice Boltzmann Method)发展的两种不同层次（即不同时-空尺度和精度）的颗粒流体系统离散模拟方法,即格子Boltzmann颗粒解析直接数值模拟（LB-based PR-DNS）方法和格子Boltzmann离散颗粒模拟（LB-based DPS）方法,总结了Euler-Euler框架下基于格子Boltzmann双流体模型（LB-based TFM）方面的探索研究。LB-based PR-DNS方法中颗粒尺寸远大于格子步长,能够直接解析出流体在颗粒表面的流动以及颗粒所受完整的动力学信息;LB-based DPS方法中格子步长远大于颗粒直径,其在计算精度、时间耗费和计算效率之间能达到很好的平衡,可获得流体的宏观平均流动及颗粒的运动轨迹信息。LB-based DNS和DPS是探索颗粒流体系统的有力手段,但LB-based TFM应用于模拟颗粒流体系统仍需进一步探索。     

Unstructured lattice Boltzmann method in three dimensions

Over the last decade, the lattice Boltzmann method (LBM) has evolved into a valuable alternative to continuum computational fluid dynamics (CFD) methods for the numerical simulation of several complex fluid‐dynamic problems. Recent advances in lattice Boltzmann research have considerably extended the capability of LBM to handle complex geometries. Among these, a particularly remarkable option is represented by cell‐vertex finite‐volume formulations which permit LBM to operate on fully unstructured grids. The two‐dimensional implementation of unstructured LBM, based on the use of triangular elements, has shown capability of tolerating significant grid distortions without suffering any appreciable numerical viscosity effects, to second‐order in the mesh size. In this work, we present the first three‐dimensional generalization of the unstructured lattice Boltzmann technique (ULBE as unstructured lattice Boltzmann equation), in which geometrical flexibility is achieved by coarse‐graining the lattice Boltzmann equation in differential form, using tetrahedrical grids. This 3D extension is demonstrated for the case of 3D pipe flow and moderate Reynolds numbers flow past a sphere. The results provide evidence that the ULBE has significant potential for the accurate calculation of flows in complex 3D geometries. Copyright © 2005 John Wiley & Sons, Ltd.     

An integrated smoothed particle hydrodynamics model for complex interfacial flows with large density ratios

In this paper, an integrated smoothed particle hydrodynamics (SPH) model for complex interfacial flows with large density ratios is developed. The discrete continuity equation and acceleration equation are obtained by considering the time derivative of the volume of particle and Eckart's continuum Lagrangian equation. A continuum surface force model is used to meet the fact that surface force may not be distributed uniformly on each side of the interface. An improved boundary condition is imposed to model wall free-slip and no-slip condition for interfacial flows with large density ratios. Particle shifting algorithm (PSA) is added for interfacial flows by imposing the normal correction near the interface, called as Interface-PSA. Then four representative numerical examples, including droplet deformation, Rayleigh-Taylor instability, dam breaking, and bubble rising, are presented and compared well with reference data. It is demonstrated that inherent interfacial flow physics can be well captured, including surface tension and the dynamic evolution of the complex interfaces.     

Simulation of three-dimensional incompressible flows in generalized curvilinear coordinates using a high-order compact finite-difference lattice Boltzmann method

In the present study, a high-order compact finite-difference lattice Boltzmann method is applied for accurately computing 3-D incompressible flows in the generalized curvilinear coordinates to handle practical and realistic geometries with curved boundaries and nonuniform grids. The incompressible form of the 3-D nineteen discrete velocity lattice Boltzmann method is transformed into the generalized curvilinear coordinates. Herein, a fourth-order compact finite-difference scheme and a fourth-order Runge-Kutta scheme are used for the discretization of the spatial derivatives and the temporal term, respectively, in the resulting 3-D nineteen discrete velocity lattice Boltzmann equation to provide an accurate 3-D incompressible flow solver. A high-order spectral-type low-pass compact filtering technique is applied to have a stable solution. All boundary conditions are implemented based on the solution of the governing equations in the 3-D generalized curvilinear coordinates. Numerical solutions of different 3-D benchmark and practical incompressible flow problems are performed to demonstrate the accuracy and performance of the solution methodology presented. Herein, the 2-D cylindrical Couette flow, the decay of a 3-D double shear wave, the cubic lid-driven cavity flow with nonuniform grids, the flow through a square duct with 90° bend and the flow past a sphere at different flow conditions are considered for validating the present computations. Numerical results obtained show the accuracy and robustness of the present solution methodology based on the implementation of the high-order compact finite-difference lattice Boltzman method in the generalized curvilinear coordinates for solving 3-D incompressible flows over practical and realistic geometries.     

基于Boltzmann模型方程的气体运动论统一算法研究

模型方程出发,研究确立含流态控制参数可描述不同流域气体流动特征的气体分子速度分布函数方程; 研究发展气体运动论离散速度坐标法, 借助非定常时间分裂数值计算方法和NND差分格式, 结合DSMC方法关于分子运动与碰撞去耦技术, 发展直接求解速度分布函数的气体运动论耦合迭代数值格式; 研制可用于物理空间各点宏观流动取矩的离散速度数值积分方法, 由此提出一套能有效模拟稀薄流到连续流不同流域气体流动问题统一算法. 通过对不同Knudsen数下一维激波内流动、二维圆柱、三维球体绕流数值计算表明, 计算结果与有关实验数据及其它途径研究结果(如DSMC模拟值、N-S数值解)吻合较好, 证实气体运动论统一算法求解各流域气体流动问题的可行性. 尝试将统一算法进行HPF并行化程序设计, 基于对球体绕流及类``神舟'返回舱外形绕流问题进行HPF初步并行试算, 显示出统一算法具有很好的并行可扩展性, 可望建立起新型的能有效模拟各流域飞行器绕流HPF并行算法研究方向. 通过将气体运动论统一算法推广应用于微槽道流动计算研究, 已初步发展起可靠模拟二维短微槽道流动数值算法; 通过对Couette流、Poiseuille流、压力驱动的二维短槽道流数值模拟, 证实该算法对微槽道气体流动问题具有较强的模拟能力, 可望发展起基于Boltzmann模型方程能可靠模拟MEMS微流动问题气体运动论数值计算方法研究途径.      

利用格子Boltzmann方法模拟单个气泡在复杂流道内的运动特性

利用格子Boltzmann方法模拟了单个气泡在具有三个半圆形喉部的复杂流道内的上升过程.通过分析气泡运动过程中的形态、运动轨迹及运动速度的变化,研究复杂流道对气泡运动特性的影响.研究结果表明,在上升过程中,由于壁面的影响,气泡的形状发生严重的变形,运动轨迹也发生相应的偏转.通过实验结果的对比,验证了模拟结果的正确性.结果表明格子Boltzmann方法可以用于模拟具有复杂边界的两相流问题.     

CFD development for macro particle simulations

Numerous industrial operations involve fluid-particle systems, in which both phases display very complex behaviour. Some examples include fluidisation technology in catalytic reactors, pneumatic transportation of grain or powder materials, carbon nanotube alignment in the nano-devices and circuit integration and so on. In this paper, a macro particle method is developed to model the fluid-particle flows. The macro particle is formed by a collection of micro-sized particles so that the number of macro particles to be tracked is much less than the number of smaller particles. Unlike the calculations of instantaneous point variables of fluid phase with moving discrete boundaries of the smaller particles with direct numerical simulation, the boundary of each macro particle is just dealt with the blocked-off approach. On the other hand, the flow fields based on the present method is solved by original Navier–Stokes, rather than the modified ones based on the locally averaged theorem. The flow fields are solved on the length scale of computational cells, while the resolutions of solid particles are the size of macro particle, which is determined as needed in specific applications. The macro particle method is validated by several selected cases, which demonstrate that the macro particle method could accurately resolve fluid-particle systems in an efficient, robust and flexible fashion.     

Systems analysis of hybrid, multi-scale complex flow simulations using Newton-GMRES

We present a methodology to analyze the stationary states and stability of complex fluid flows by using hybrid, discrete, and/or continuum multi-scale simulations. Building on existing theories, our scheme extracts dynamical and equilibrium characteristics from carefully chosen time integrations of these multi-scale evolution equations. Two canonical problems are presented to demonstrate the ability and accuracy of the formalism. The first is an investigation of flow-induced transitions seen in homogeneous, hard- rod liquid crystal suspensions subjected to a linear shear flow. In the second problem, we study the phenomenon of draw resonance, a dynamical instability in an isothermal fiber-spinning process, by using a multi-scale hybrid simulation that incorporates both stochastic and continuum models.