An inverse analysis of fluid flow through granular media using differentiable lattice Boltzmann method

This study presents a method for the inverse analysis of fluid flow problems.The focus is put on accurately determining boundary conditions and characterizing the physical properties of granular media,such as permeability,and fluid components,like viscosity.The primary aim is to deduce either constant pressure head or pressure profiles,given the known velocity field at a steady-state flow through a conduit containing obstacles,including walls,spheres,and grains.The lattice Boltzmann method(LBM)combined with automatic differentiation(AD)(AD-LBM)is employed,with the help of the GPU-capable Taichi programming language.A lightweight tape is used to generate gradients for the entire LBM simulation,enabling end-to-end backpropagation.Our AD-LBM approach accurately estimates the boundary conditions for complex flow paths in porous media,leading to observed steady-state velocity fields and deriving macro-scale permeability and fluid viscosity.The method demonstrates significant advantages in terms of prediction accuracy and computational efficiency,making it a powerful tool for solving inverse fluid flow problems in various applications.

Volumetric lattice Boltzmann method for pore-scale mass diffusionadvection process in geopolymer porous structures

Porous materials present significant advantages for absorbing radioactive isotopes in nuclear waste streams.To improve absorption efficiency in nuclear waste treatment,a thorough understanding of the diffusion-advection process within porous structures is essential for material design.In this study,we present advancements in the volumetric lattice Boltzmann method(VLBM)for modeling and simulating pore-scale diffusion-advection of radioactive isotopes within geopolymer porous structures.These structures are created using the phase field method(PFM)to precisely control pore architectures.In our VLBM approach,we introduce a concentration field of an isotope seamlessly coupled with the velocity field and solve it by the time evolution of its particle population function.To address the computational intensity inherent in the coupled lattice Boltzmann equations for velocity and concentration fields,we implement graphics processing unit(GPU)parallelization.Validation of the developed model involves examining the flow and diffusion fields in porous structures.Remarkably,good agreement is observed for both the velocity field from VLBM and multiphysics object-oriented simulation environment(MOOSE),and the concentration field from VLBM and the finite difference method(FDM).Furthermore,we investigate the effects of background flow,species diffusivity,and porosity on the diffusion-advection behavior by varying the background flow velocity,diffusion coefficient,and pore volume fraction,respectively.Notably,all three parameters exert an influence on the diffusion-advection process.Increased background flow and diffusivity markedly accelerate the process due to increased advection intensity and enhanced diffusion capability,respectively.Conversely,increasing the porosity has a less significant effect,causing a slight slowdown of the diffusion-advection process due to the expanded pore volume.This comprehensive parametric study provides valuable insights into the kinetics of isotope uptake in porous structures,facilitating the development of porous materials for nuclear waste treatment applications.

On the spreading behavior of a droplet on a circular cylinder using the lattice Boltzmann method

The study of a droplet spreading on a circular cylinder under gravity was carried out using the pseudo-potential lattice Boltzmann high-density ratios multiphase model with a non-ideal Peng–Robinson equation of state. The calculation results indicate that the motion of the droplet on the cylinder can be divided into three stages: spreading, sliding, and aggregating.The contact length and contact time of a droplet on a cylindrical surface can be affected by factors such as the wettability gradient of the cylindrical wall, the Bond number, and droplet size. Furthermore, phase diagrams showing the relationship between Bond number, cylinder wall wettability gradient, and contact time as well as maximum contact length for three different droplet sizes are given. A theoretical foundation for additional research into the heat and mass transfer process between the droplet and the cylinder can be established by comprehending the variable rules of maximum contact length and contact time.

A High-Accuracy Curve Boundary Recognition Method Based on the Lattice Boltzmann Method and Immersed Moving Boundary Method

Applying numerical simulation technology to investigate fluid-solid interaction involving complex curved bound-aries is vital in aircraft design,ocean,and construction engineering.However,current methods such as Lattice Boltzmann(LBM)and the immersion boundary method based on solid ratio(IMB)have limitations in identifying custom curved boundaries.Meanwhile,IBM based on velocity correction(IBM-VC)suffers from inaccuracies and numerical instability.Therefore,this study introduces a high-accuracy curve boundary recognition method(IMB-CB),which identifies boundary nodes by moving the search box,and corrects the weighting function in LBM by calculating the solid ratio of the boundary nodes,achieving accurate recognition of custom curve boundaries.In addition,curve boundary image and dot methods are utilized to verify IMB-CB.The findings revealed that IMB-CB can accurately identify the boundary,showing an error of less than 1.8%with 500 lattices.Also,the flow in the custom curve boundary and aerodynamic characteristics of the NACA0012 airfoil are calculated and compared to IBM-VC.Results showed that IMB-CB yields lower lift and drag coefficient errors than IBM-VC,with a 1.45%drag coefficient error.In addition,the characteristic curve of IMB-CB is very stable,whereas that of IBM-VC is not.For the moving boundary problem,LBM-IMB-CB with discrete element method(DEM)is capable of accurately simulating the physical phenomena of multi-moving particle flow in complex curved pipelines.This research proposes a new curve boundary recognition method,which can significantly promote the stability and accuracy of fluid-solid interaction simulations and thus has huge applications in engineering.

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.

Experimental Study of Heat Transfer in an Insulated Local Heated fromBelow and Comparison with Simulation by Lattice Boltzmann Method

In this paper,experimental and numerical studies of heat transfer in a test local of side H=0.8 m heated from below are presented and compared.All the walls,the rest of the floor and the ceiling are made from plywood and polystyrene in sandwich form(3 mmplywood-3 cm polystyrene-3 mmplywood)just on one of the vertical walls contained a glazed door(2 H/3×0.15 m).This local is heated during two heating cycles by a square plate of iron the width L=0.6 H,which represents the heat source,its temperature Th is controlled.The plate is heated for two cycles by an adjustable set-point heat source placed just down the center of it.For each cycle,the heat source is switched“on”for 6 h and switched“off”for 6 h.The outdoor air temperature is kept constant at a low temperature Tc<Th.All measurements are carried out with k-type thermocouples and with flux meters.Results will be qualitatively presented for two cycles of heating in terms of temperatures and heat flux densitiesϕfor various positions of the test local.The temperature evolution of the center and the profile of the temperature along the vertical centerline are compared by two dimensions simulation using the lattice Boltzmann method.The comparison shows a good agreement with a difference that does not exceed±1℃.

FLOW FIELD ANALYSES OF PLANE JET AT LOW REYNOLDS NUMBERS USING LATTICE BOLTZMANN METHOD

A two-dimensional（2-D） incompressible plane jet is investigated using the lattice Boltzmann method（LBM） for low Reynolds numbers of 42 and 65 based on the jet-exit-width and the maximum jet-exit-velocity. The results show that the mean centerline velocity decays as x-1/3 and the jet spreads as x2/3 in the self-similar region, which are consistent with the theoretical predictions and the experimental data. The time histories and PSD analyses of the instantaneous centerline velocities indicate the periodic behavior and the interaction between periodic components of velocities should not be neglected in the far field region, although it is invisible in the near field region.

NOVEL IMMERSED BOUNDARY-LATTICE BOLTZMANN METHOD BASED ON FEEDBACK LAW

The lattice Boltzmann method （LBM） and the immersed boundary method （IBM） are alternative, com- putational techniques for solving complex fluid dynamics systems, and can take the place of the Navier-Stokes（N- S） equation. This paper proposes a novel immersed boundary-lattice Boltzmann method （IB-LBM） based on the feedback law. The method uses the immersed boundary concept in the LBM framework to capture the coupling between a body with complex geometry and a uniform fluid, Then, the flows around a stationary circular cylinder and two circular cylinders in a side by side arrangement are simulated by using the method. Results are agreed well with the benchmark data, so, the capability of the method for complex geometry is demonstrated. Different from the conventional IB-LBM, which uses the Hook＇s law or the direct forcing method to compute the interae- tion force, the method uses the feedback law--the feedback of velocity field and displacement information to calculate the force, thus ensuring the method has advantages of easy implementation and full parallelism.

COMPRESSIBLE FLOW SIMULATION AROUND AIRFOIL BASED ON LATTICE BOLTZMANN METHOD

The flow around airfoil NACA0012 enwrapped by the body-fitted grid is simulated by a coupled doubledistribution-function （DDF） lattice Boltzmann method （LBM） for the compressible Navier-Stokes equations. Firstly, the method is tested by simulating the low Reynolds number flow at Ma =0. 5,a=0. 0, Re=5 000. Then the simulation of flow around the airfoil is carried out at Ma：0. 5, 0. 85, 1.2; a=-0.05, 1.0, 0.0, respectively. And a better result is obtained by using a local refined grid. It reduces the error produced by the grid at Ma=0. 85. Though the inviscid boundary condition is used to avoid the problem of flow transition to turbulence at high Reynolds numbers, the pressure distribution obtained by the simulation agrees well with that of the experimental results. Thus, it proves the reliability of the method and shows its potential for the compressible flow simulation. The suecessful application to the flow around airfoil lays a foundation of the numerical simulation of turbulence.

Zakharov-Rubenchik方程组的格子Boltzmann方法

Zakharov-Rubenchik方程组常用于描述非线性介质中高、低频波间相互作用的波耦合现象。本文针对该方程组的数值求解问题,构建了一种格子Boltzmann方法的D1Q3演化模型,并利用Chapman-Enskog展开和多尺度分析技术,推导出了各个方向的平衡态分布函数和修正函数的具体表达式,从而将所建的演化模型准确恢复到宏观方程组。最后,通过数值算例证明了该方法的有效性。

浸入运动边界-格子Boltzmann方法4种固含率计算方法对比研究

为了达到流固耦合,格子Boltzmann方法(LBM)可采用浸入运动边界法(IMB)实现移动颗粒边界上的无滑移条件.该耦合方式(IMB-LBM)中固含率计算方法对流固耦合计算精度和效率有影响.对常用的固含率4种计算方法,即蒙特卡洛法(MCM)、单元分解法(UDM)、近似多边形法(APM)和闭合边界法(CBM),分别阐述其具体算法,对比了它们的计算精度和计算效率;最后通过圆盘颗粒非连续变形分析方法(DDDA)与IMB-LBM耦合模型下的一个多颗粒沉降流固耦合算例,对比分析了它们在流固耦合计算过程中的耗时.结果表明:1)CBM无误差,MCM和UDM在随机点数取1000,子单元数取100时误差稳定在1%以下,APM在颗粒直径大于格子长度10倍时,误差小于0.44%;2)MCM和UDM的计算精度及耗时分别与随机点数和子单元数相关,它们的计算耗时大于APM和CBM;3)计算效率上,APM>CBM>UDM>MCM,其中CBM计算耗时略微大于APM,APM和UDM计算耗时分别比MCM少2个和1个数量级.该结果可为IMB-LBM耦合模型中固含率计算方法优选提供借鉴.

Free convection of nanofluid filled enclosure using lattice Boltzmann method (LBM)

The lattice Boltzmann method （LBM） is used to examine free convection of nanofluids. The space between the cold outer square and heated inner circular cylinders is filled with water including various kinds of nanoparticles： TiO2, Ag, Cu, and A1203. The Brinkman and Maxwell-Garnetts models are used to simulate the viscosity and the effective thermal conductivity of nanofluids, respectively. Results from the performed numerical analysis show good agreement with those obtained from other numerical meth- ods. A variety of the Rayleigh number, the nanoparticle volume fraction, and the aspect ratio are examined. According to the results, choosing copper as the nanoparticle leads to obtaining the highest enhancement for this problem. The results also indicate that the maximum value of enhancement occurs at λ =2.5 when Ra = 106 while at A = 1.5 for other Rayleigh numbers.

On the interaction between bubbles and the free surface with high density ratio 3D lattice Boltzmann method

The bubbles rise up and burst at the free surface is a complex two-phase process.A free energy lattice Boltzmann method(LBM)model is adopted in this paper to study this phenomenon.The interface capturing technique[Zheng et al.,2006]is used to deal with the high density ratio problem.The Laplace law and the air-water interface capturing ability are validated for the multiphase model.The interaction between the single bubble or multiple bubbles and the free surface are studied by the multiphase model.The force acting on the bubble and the evolution of the free surface is studied.Meanwhile,effect of the initial distance between two adjacent bubbles on interaction effects of multiple bubbles is investigated as well.

Lattice Boltzmann method for simulation of time-dependent neutral particle transport

In this paper, a novel model is proposed to investigate the neutron transport in scattering and absorbing medium. This solution to the linear Boltzmann equation is expanded from the idea of lattice Boltzmann method(LBM) with the collision and streaming process. The theoretical derivation of lattice Boltzmann model for transient neutron transport problem is proposed for the first time.The fully implicit backward difference scheme is used to ensure the numerical stability, and relaxation time and equilibrium particle distribution function are obtained. To validate the new lattice Boltzmann model, the LBM formulation is tested for a homogenous media with different sources, and both transient and steady-state LBM results get a good agreement with the benchmark solutions.

非线性时间分数阶空气动力学方程的格子Boltzmann研究

针对非线性时间分数阶空气动力学方程,提出了一种新的格子Boltzmann模型.通过使用Chapman-Enskog展开和时间多尺度展开技术,得到了一系列不同时间尺度上的系列偏微分方程.选择合适的平衡态分布函数的矩,恢复出宏观方程,数值模拟出非线性时间分数阶空气动力学方程的解.数值实验表明,格子Boltzmann方法是研究非线性时间分数阶空气动力学方程的有效工具.

基于格子Boltzmann方法的地埋管管群换热的模拟研究

土壤结构和地下水通常会影响地埋管换热器的传热,本文将利用随机四参数生成法建立双分布函数格子Boltzmann模型,对有渗流条件下地埋管管群换热器与饱和土壤多孔介质的换热情况进行了模拟,分析了影响埋管外土壤平均温度的主要因素。研究表明,土壤热传导特性主要受土壤孔隙度、地下水渗流速度和土壤骨架导热系数k s的影响。本文的研究结论对考虑土壤渗流的地下换热器的设计具有一定的指导意义。

Simulating high Reynolds number flow in two-dimensional lid-driven cavity by multi-relaxation-time lattice Boltzmann method

By coupling the non-equilibrium extrapolation scheme for boundary condition with the multi-relaxation-time lattice Boltzmann method, this paper finds that the stability of the multi-relaxation-time model can be improved greatly, especially on simulating high Reynolds number （Re） flow. As a discovery, the super-stability analysed by Lallemand and Luo is verified and the complex structure of the cavity flow is also exhibited in our numerical simulation when Re is high enough. To the best knowledge of the authors, the maximum of Re which has been investigated by direct numerical simulation is only around 50 000 in the literature; however, this paper can readily extend the maximum to 1000 000 with the above combination.

New Boundary Treatment Methods for Lattice Boltzmann Method

In practical fluid dynamic simulations, the bou n dary condition should be treated carefully because it always has crucial influen ce on the numerical accuracy, stability and efficiency. Two types of boundary tr eatment methods for lattice Boltzmann method (LBM) are proposed. One is for the treatment of boundaries situated at lattice nodes, and the other is for the appr oximation of boundaries that are not located at the regular lattice nodes. The f irst type of boundary treatment method can deal with various dynamic boundaries on complex geometries by using a general set of formulas, which can maintain sec ond\|order accuracy. Based on the fact that the fluid flows simulated by LBM are not far from equilibrium, the unknown distributions at a boundary node are expr essed as the analogous forms of their corresponding equilibrium distributions. T herefore, the number of unknowns can be reduced and an always\|closed set of equ ations can be obtained for the solutions to pressure, velocity and special bound ary conditions on various geometries. The second type of boundary treatment is a complete interpolation scheme to treat curved boundaries. It comes from careful analysis of the relations between distribution functions at boundary nodes and their neighboring lattice nodes. It is stable for all situations and of second\| order accuracy. Basic ideas, implementation procedures and verifications with ty pical examples for the both treatments are presented. Numerical simulations and analyses show that they are accurate, stable, general and efficient for practica l simulations.

Numerical modeling of condensate droplet on superhydrophobic nanoarrays using the lattice Boltzmann method

In the present study,the process of droplet condensation on superhydrophobic nanoarrays is simulated using a multicomponent multi-phase lattice Boltzmann model.The results indicate that three typical nucleation modes of condensate droplets are produced by changing the geometrical parameters of nanoarrays.Droplets nucleated at the top（top-nucleation mode）,or in the upside interpillar space of nanoarrays（side-nucleation mode）,generate the non-wetting Cassie state,whereas the ones nucleated at the bottom corners between the nanoarrays（bottom-nucleation mode） present the wetting Wenzel state.Time evolutions of droplet pressures at the upside and downside of the liquid phase are analyzed to understand the wetting behaviors of the droplets condensed from different nucleation modes.The phenomena of droplet condensation on nanoarrays patterned with different hydrophilic and hydrophobic regions are simulated,indicating that the nucleation mode of condensate droplets can also be manipulated by modifying the local intrinsic wettability of nanoarray surface.The simulation results are compared well with the experimental observations reported in the literature.

A study of the upper limit of solid scatters density for gray Lattice Boltzmann Method

The upper limit of the solid scatters density ns （x）, a key parameter for the simulation of flows in porous media with a gray Lattice Boltzmann Method, is studied by an analytical way for the infiltration Poiseuille flow between two infinite parallel plates. Analyses of three different gray Lattice Boltzmann schemes, separately proposed by Gao and Sharma et al., Dardis and McCloskey, and Thorne and Sukop, indicate that the effective domain of Gao and Sharma＇s scheme is restricted to ns 〈 1/2√3≈0.289, Dardis and McCloskey＇s scheme is restricted to ns 〈 （√57-1）/28≈0.234, and that there is no extra restriction on ns（x） with Thorne and Sukop＇s scheme. These results are obtained for the dimensionless relaxation time τ= 1. The above analytical results are verified by our numerical simulations. The use of a gray LBM is further illustrated by simulating the flow at the interface of a porous medium. Simulation results yield velocity profiles which agree very well with Brinkman＇s prediction.