A combined method using Lattice Boltzmann Method(LBM)and Finite Volume Method(FVM)to simulate geothermal reservoirs in Enhanced Geothermal System(EGS)

With the development of industrial activities,global warming has accelerated due to excessive emission of CO_(2).Enhanced Geothermal System(EGS)utilizes deep geothermal heat for power generation.Although porous medium theory is commonly employed to model geothermal reservoirs in EGS,Hot Dry Rock(HDR)presents a challenge as it consists of impermeable granite with zero porosity,potentially distorting the physical interpretation.To address this,the Lattice Boltzmann Method(LBM)is employed to simulate CO_(2)flow within geothermal reservoirs and the Finite Volume Method(FVM)to solve the energy conservation equation for temperature distribution.This combined method of LBM and FVM is imple-mented using MATLAB.The results showed that the Reynolds numbers(Re)of 3,000 and 8,000 lead to higher heat extraction rates from geothermal reservoirs.However,higher Re values may accelerate thermal breakthrough,posing challenges to EGS operation.Meanwhile,non-equilibrium of density in fractures becomes more pronounced during the system's life cycle,with non-Darcy's law becoming significant at Re values of 3,000 and 8,000.Density stratification due to buoyancy effects significantly impacts temperature distribution within geothermal reservoirs,with buoyancy effects at Re=100 under gravitational influence being noteworthy.Larger Re values(3,000 and 8,000)induce stronger forced convection,leading to more uniform density distribution.The addition of proppant negatively affects heat transfer performance in geothermal reservoirs,especially in single fractures.Practical engineering considerations should determine the quantity of proppant through detailed numerical simulations.

Simulation of fluid-structure interaction in a microchannel using the lattice Boltzmann method and size-dependent beam element on a graphics processing unit

Fluid-structure interaction （FSI） problems in microchannels play a prominent role in many engineering applications. The present study is an effort toward the simulation of flow in microchannel considering FSI. The bottom boundary of the microchannel is simulated by size-dependent beam elements for the finite element method （FEM） based on a modified cou- ple stress theory. The lattice Boltzmann method （LBM） using the D2Q13 LB model is coupled to the FEM in order to solve the fluid part of the FSI problem. Because of the fact that the LBM generally needs only nearest neighbor information, the algorithm is an ideal candidate for parallel computing. The simulations are carried out on graphics processing units （GPUs） using computed unified device architecture （CUDA）. In the present study, the governing equations are non-dimensionalized and the set of dimensionless groups is exhibited to show their effects on micro-beam displacement. The numerical results show that the displacements of the micro-beam predicted by the size-dependent beam element are smaller than those by the classical beam element.

Application of shifted lattice model to 3D compressible lattice Boltzmann method

An additional potential energy distribution function is introduced on the basis of previous D3Q25 model,and the equilibrium distribution function of D3Q25 is obtained by spherical function.A novel three-dimensional(3D)shifted lattice model is proposed,therefore a shifted lattice model is introduced into D3Q25.Under the finite volume scheme,several typical compressible calculation examples are used to verify whether the numerical stability of the D3Q25 model can be improved by adding the shifted lattice model.The simulation results show that the numerical stability is indeed improved after adding the shifted lattice model.

A multiple-relaxation-time lattice Boltzmann method for high-speed compressible flows

This paper presents a coupling compressible model of the lattice Boltzmann method. In this model, the multiplerelaxation-time lattice Boltzmann scheme is used for the evolution of density distribution functions, whereas the modified single-relaxation-time （SRT） lattice Boltzmann scheme is applied for the evolution of potential energy distribution functions. The governing equations are discretized with the third-order Monotone Upwind Schemes for scalar conservation laws finite volume scheme. The choice of relaxation coefficients is discussed simply. Through the numerical simulations, it is found that compressible flows with strong shocks can be well simulated by present model. The numerical results agree well with the reference results and are better than that of the SRT version.

Numerical study of convective heat transfer in static arrangements of particles with arbitrary shapes:A monolithic hybrid lattice Boltzmann-finite difference-phase field solver

A compressible lattice Boltzmann-finite difference method is extended by the phase-field approach into a monolithic scheme to study fluid flow and heat transfer through regular arrangements of solid bodies of circular,elliptical and irregular shapes.The advantage of using the phase-field method is demon-strated both in its simplicity of accounting for flow and thermal boundary conditions at solid surfaces with irregular shapes and in the capability of generating such complex-shaped objects.For an array of discs,numerical results for the overall solid-to-gas heat transfer rate are validated via experiments on flow through arrays of hot cylinders.The thus validated compressible LB-FD-PF hybrid scheme is used to study the dependence of heat transfer on flow and thermal boundary conditions(Reynolds number,temperature difference between the hot solid bodies and the inlet gas),porosity as well as on the shape of solid objects.Results are rationalized in terms of the residence time of the gas close to the solid body and downstream variations of gas velocity and temperature.Perspective for further applications of the proposed methodology are also discussed.

Lattice Boltzmann Flux Solver:An Efficient Approach for Numerical Simulation of Fluid Flows

A lattice Boltzmann flux solver(LBFS)is presented for simulation of fluid flows.Like the conventional computational fluid dynamics(CFD)solvers,the new solver also applies the finite volume method to discretize the governing differential equations,but the numerical flux at the cell interface is not evaluated by the smooth function approximation or Riemann solvers.Instead,it is evaluated from local solution of lattice Boltzmann equation(LBE)at cell interface.Two versions of LBFS are presented in this paper.One is to locally apply one-dimensional compressible lattice Boltzmann(LB)model along the normal direction to the cell interface for simulation of compressible inviscid flows with shock waves.The other is to locally apply multi-dimensional LB model at cell interface for simulation of incompressible viscous and inviscid flows.The present solver removes the drawbacks of conventional lattice Boltzmann method(LBM)such as limitation to uniform mesh,tie-up of mesh spacing and time interval,limitation to viscous flows.Numerical examples show that the present solver can be well applied to simulate fluid flows with non-uniform mesh and curved boundary.

Size effect of lattice material and minimum weight design

The size effects of microstructure of lattice materials on structural analysis and minimum weight design are studied with extented multiscale finite element method（EMsFEM） in the paper. With the same volume of base material and configuration, the structural displacement and maximum axial stress of micro-rod of lattice structures with different sizes of microstructure are analyzed and compared.It is pointed out that different from the traditional mathematical homogenization method, EMsFEM is suitable for analyzing the structures which is constituted with lattice materials and composed of quantities of finite-sized micro-rods.The minimum weight design of structures composed of lattice material is studied with downscaling calculation of EMsFEM under stress constraints of micro-rods. The optimal design results show that the weight of the structure increases with the decrease of the size of basic sub-unit cells. The paper presents a new approach for analysis and optimization of lattice materials in complex engineering constructions.

Fluid-Structure Interaction in Microchannel Using Lattice Boltzmann Method and Size-Dependent Beam Element

Fluid-structure interaction(FSI)problems in microchannels play prominent roles in many engineering applications.The present study is an effort towards the simulation of flow in microchannel considering FSI.Top boundary of the microchannel is assumed to be rigid and the bottom boundary,which is modeled as a Bernoulli-Euler beam,is simulated by size-dependent beam elements for finite element method(FEM)based on a modified couple stress theory.The lattice Boltzmann method(LBM)using D2Q13 LB model is coupled to the FEM in order to solve fluid part of FSI problem.In the present study,the governing equations are non-dimensionalized and the set of dimensionless groups is exhibited to show their effects on micro-beam displacement.The numerical results show that the displacements of the micro-beam predicted by the size-dependent beam element are smaller than those by the classical beam element.

波浪在平缓岸滩上爬坡的有限差分格子Boltzmann模拟

应用有限差分格子Boltzmann方法模拟平缓岸滩上波浪爬坡过程,计算结果和半解析解以及其它数值模型模拟结果吻合。通过和标准格子Boltzmann模型计算结果比较,发现在格子Boltzmann方法的基础上采用高精度有限差分格式离散方程取较大的空间和时间步长就可以得到稳定的计算结果,节省计算时间,可以用于模拟大范围波浪爬坡现象。

基于格子Boltzmann方法和有限体积法的方柱绕流特性对比分析

基于黏性流体理论,分别采用格子Boltzmann方法(LBM)和有限体积法(FVM)建立了黏性流场中方柱绕流模型。探究LBM在非光滑曲面钝体绕流方面的应用;并结合FVM进行对比分析。在FVM模型中,采用局部加密的方法对钝体边界进行处理,而在LBM模型中,除了传统的Half-way边界处理方法,还结合了拐角边界处理方法。为获得较好的可对比数据,根据已发表文献中的理论及UDF编译码技术,分别对两模型的进出口边界条件进行了讨论和设置。对比分析了两模型下的速度云图以及获得的升、阻力系数,Strouhal数。结果发现方柱上游压力不受涡脱落影响,雷诺数对其影响也较小;两种方法下的速度、无量纲参数吻合较好;但两者最适进出口边界不同,且相同条件下,LBM比FVM数值模拟能更快达到稳定状态。

A particle-resolved heat-particle-fluid coupling model by DEM-IMB-LBM

Multifield coupling is frequently encountered and also an active area of research in geotechnical engineering.In this work,a particle-resolved direct numerical simulation(PR-DNS)technique is extended to simulate particle-fluid interaction problems involving heat transfer at the grain level.In this extended technique,an immersed moving boundary(IMB)scheme is used to couple the discrete element method(DEM)and lattice Boltzmann method(LBM),while a recently proposed Dirichlet-type thermal boundary condition is also adapted to account for heat transfer between fluid phase and solid particles.The resulting DEM-IBM-LBM model is robust to simulate moving curved boundaries with constant temperature in thermal flows.To facilitate the understanding and implementation of this coupled model for non-isothermal problems,a complete list is given for the conversion of relevant physical variables to lattice units.Then,benchmark tests,including a single-particle sedimentation and a two-particle drafting-kissing-tumbling(DKT)simulation with heat transfer,are carried out to validate the accuracy of our coupled technique.To further investigate the role of heat transfer in particle-laden flows,two multiple-particle problems with heat transfer are performed.Numerical examples demonstrate that the proposed coupling model is a promising high-resolution approach for simulating the heat-particle-fluid coupling at the grain level.

An MPI parallel DEM-IMB-LBM framework for simulating fluid-solid interaction problems

The high-resolution DEM-IMB-LBM model can accurately describe pore-scale fluid-solid interactions,but its potential for use in geotechnical engineering analysis has not been fully unleashed due to its prohibitive computational costs.To overcome this limitation,a message passing interface(MPI)parallel DEM-IMB-LBM framework is proposed aimed at enhancing computation efficiency.This framework utilises a static domain decomposition scheme,with the entire computation domain being decomposed into multiple subdomains according to predefined processors.A detailed parallel strategy is employed for both contact detection and hydrodynamic force calculation.In particular,a particle ID re-numbering scheme is proposed to handle particle transitions across sub-domain interfaces.Two benchmarks are conducted to validate the accuracy and overall performance of the proposed framework.Subsequently,the framework is applied to simulate scenarios involving multi-particle sedimentation and submarine landslides.The numerical examples effectively demonstrate the robustness and applicability of the MPI parallel DEM-IMB-LBM framework.

Lattice Boltzmann Simulations of Two Linear Microswimmers Using the Immersed Boundary Method

The performance of a single or the collection of microswimmers strongly depends on the hydrodynamic coupling among their constituents and themselves.We present a numerical study for a single and a pair of microswimmers based on lattice Boltzmann method(LBM)simulations.Our numerical algorithm consists of two separable parts.Lagrange polynomials provide a discretization of the microswimmers and the lattice Boltzmann method captures the dynamics of the surrounding fluid.The two components couple via an immersed boundary method.We present data for a single swimmer system and our data also show the onset of collective effects and,in particular,an overall velocity increment of clusters of swimmers.

线性Poisson-Boltzmann方程的Mortar有限元方法的数值计算

本文对分子生物物理学中产生的线性Poisson-Boltzmann方程（PBE）,给出了Mortar有限元方法的计算过程,数值计算例子表明,与一般的协调有限元方法相比,用Mortar元方法求解此类有间断系数的问题是非常有效的.

隧道突泥破坏的耦合格子Boltzmann-离散元法模拟

为了探究隧道突泥灾害的灾变机制,开发了耦合格子Boltzmann–离散元法(LBM-DEM)数值计算平台并对隧道突泥破坏的演化过程进行了模拟,分析了致灾介质颗粒黏结强度、水压力及突泥口尺寸等因素对隧道突泥破坏特征的影响。结果表明:基于LBM-DEM模拟能再现隧道突泥破坏"启动、加速、缓慢和稳定"等连续4个阶段的演化过程;无黏结的致灾介质突泥破坏形态近似为直线,有一定黏结强度的致灾介质突泥破坏形态总体呈圆弧或抛物线状,突泥破坏区扩展范围和稳定后的突泥量随着颗粒间黏结强度的增大而逐渐减小;水压力越大,突泥灾害发生后突泥量增长越快,最终的突泥量也越大,且颗粒间黏结强度较大时水压力的这种影响越显著;当致灾介质颗粒间无黏结时,不同突泥口尺寸的模型在稳定后突泥量和破坏区范围基本相同,而当颗粒间形成一定强度的黏结后,突泥口尺寸越大,突泥灾害发生后突泥量增长越快,稳定后的突泥量也越多;隧道突泥破坏是致灾岩土介质、水压和开挖三者综合作用的结果。

格子Boltzmann方法中不同边界处理及不同体力项表达式的对比分析

简要介绍了格子Boltzm ann方法以及耦合温度的格子Boltzm ann方法,就LBM采用不同边界处理分别对二维方腔自然对流进行了模拟以及对比。同时,推导出LBM中更为完整的体力项的表达式,并采用简化的体力项与较完整体力项分别进行数值模拟,将结果进行了对比分析。模拟的结果与前人的结果吻合良好。

基于非结构化网格的高可扩展并行有限体积格子Boltzmann方法

均匀网格格子Boltzmann方法虽然有其优势,但是在模拟大规模流场信息以及复杂几何边界时仍然存在困难。为此,文中给出了非结构化网格下的有限体积格子Boltzmann方法。该方法采用cell-centered方案,使用low-diffusion Roe方案计算对流通量密度,通过最小二乘方法计算粒子分布函数的梯度。为了能够模拟大规模复杂流场情况,文中给出了非结构化网格有限体积格子Boltzmann方法的并行方法。该法通过ParMETIS划分流场的非结构化网格,将网格近似平均地发送给MPI进程,比较了两种不同规模的网格单元的并行性能。文中通过以下两点验证了并行算法的正确性:1)顶盖方腔驱动流,Re=400,1 000,3 200,5 000;2)圆柱绕流,Re=10,20,40。并行数值实验的结果表明所提并行算法在1 920核上仍然拥有良好的可扩展性,在1 920个核上的并行效率可以达到在240核上效率的78.42%。

EXTERNAL BODY FORCEIN FINITE DIFFERENCE LATTICE BOLTZMANN METHOD

A new finite difference lattice Boltzmann scheme is developed. Because of analyzing the influence of external body force roundly, the correct Navier-Stokes equations with the external body force are recovered, without any additional unphysical terms. And some numerical results are presented. The result which close agreement with analytical data shows the good performance of the model.

A new cross-scaling method to deal with the porous flow problem

The finite volume method （FVM） and the lattice Boltzmann method （LBM） are coupled with each other to construct a new cross-scaling method to deal with the porous flow problem. To check the effectiveness of our developed cross-scaling LBM-FVM, the above mentioned problem is also solved by the well known LBM-LBM. Based on the data checking of the published data and the results of LBM-FVM and LBM-LBM, good agreement is observed.