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.

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.

Exploration of the coupled lattice Boltzmann model based on a multiphase field model:A study of the solid-liquid-gas interaction mechanism in the solidification process

A multiphase field model coupled with a lattice Boltzmann(PF-LBM)model is proposed to simulate the distribution mechanism of bubbles and solutes at the solid-liquid interface,the interaction between dendrites and bubbles,and the effects of different temperatures,anisotropic strengths and tilting angles on the solidified organization of the SCN-0.24wt.%butanedinitrile alloy during the solidification process.The model adopts a multiphase field model to simulate the growth of dendrites,calculates the growth motions of dendrites based on the interfacial solute equilibrium;and adopts a lattice Boltzmann model(LBM)based on the Shan-Chen multiphase flow to simulate the growth and motions of bubbles in the liquid phase,which includes the interaction between solid-liquid-gas phases.The simulation results show that during the directional growth of columnar dendrites,bubbles first precipitate out slowly at the very bottom of the dendrites,and then rise up due to the different solid-liquid densities and pressure differences.The bubbles will interact with the dendrite in the process of flow migration,such as extrusion,overflow,fusion and disappearance.In the case of wide gaps in the dendrite channels,bubbles will fuse to form larger irregular bubbles,and in the case of dense channels,bubbles will deform due to the extrusion of dendrites.In the simulated region,as the dendrites converge and diverge,the bubbles precipitate out of the dendrites by compression and diffusion,which also causes physical phenomena such as fusion and spillage of the bubbles.These results reveal the physical mechanisms of bubble nucleation,growth and kinematic evolution during solidification and interaction with dendrite growth.

Lattice Boltzmann simulation study of anode degradation in solid oxide fuel cells during the initial aging process

For present solid oxide fuel cells(SOFCs),rapid performance degradation is observed in the initial aging process,and the dis-cussion of the degradation mechanism necessitates quantitative analysis.Herein,focused ion beam-scanning electron microscopy was em-ployed to characterize and reconstruct the ceramic microstructures of SOFC anodes.The lattice Boltzmann method(LBM)simulation of multiphysical and electrochemical processes in the reconstructed models was performed.Two samples collected from industrial-size cells were characterized,including a reduced reference cell and a cell with an initial aging process.Statistical parameters of the reconstructed microstructures revealed a significant decrease in the active triple-phase boundary and Ni connectivity in the aged cell compared with the reference cell.The LBM simulation revealed that activity degradation is dominant compared with microstructural degradation during the initial aging process,and the electrochemical reactions spread to the support layer in the aged cell.The microstructural and activity de-gradations are attributed to Ni migration and coarsening.

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.

Data-driven optimization study of the multi-relaxation-time lattice Boltzmann method for solid-liquid phase change

Sharp phase interfaces and accurate temperature distributions are important criteria in the simulation of solid-liquid phase changes.The multi-relaxation-time lattice Boltzmann method(MRT-LBM)shows great numerical performance during simulation;however,the value method of the relaxation parameters needs to be specified.Therefore,in this study,a random forest(RF)model is used to discriminate the importance of different relaxation parameters to the convergence,and a support vector machine(SVM)is used to explore the decision boundary of the convergent samples in each dimensional model.The results show that the convergence of the samples is consistent with the sign of the decision number,and two types of the numerical deviations appear,i.e.,the phase mushy zone and the non-physical heat transfer.The relaxation parameters chosen on the decision boundary can further suppress the numerical bias and improve numerical accuracy.

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.

Simulation of gas–liquid two-phase flow in a flow-focusing microchannel with the lattice Boltzmann method

A lattice Boltzmann method for gas–liquid two-phase flow involving non-Newtonian fluids is developed. Bubble formation in a flow-focusing microchannel is simulated by the method. The influences of flow rate ratio, surface tension,wetting properties, and rheological characteristics of the fluid on the two-phase flow are analyzed. The results indicate that the flow pattern transfers from slug flow to dry-plug flow with a sufficiently small capillary number. Due to the presence of three-phase contact lines, the contact angle has a more significant effect on the dry-plug flow pattern than on the slug flow pattern. The deformation of the front and rear meniscus of a bubble in the shear-thinning fluid can be explained by the variation of the capillary number. The reduced viscosity and increased contact angle are beneficial for the drag reduction in a microchannel. It also demonstrates the effectiveness of the current method to simulate the gas–liquid two-phase flow in a microchannel.

A joint absorbing boundary for the multiple-relaxation-time lattice Boltzmann method in seismic acoustic wavefield modeling

Conventional seismic wave forward simulation generally uses mathematical means to solve the macroscopic wave equation,and then obtains the corresponding seismic wavefield.Usually,when the subsurface structure is finely constructed and the continuity of media is poor,this strategy is difficult to meet the requirements of accurate wavefield calculation.This paper uses the multiple-relaxation-time lattice Boltzmann method(MRT-LBM)to conduct the seismic acoustic wavefield simulation and verify its computational accuracy.To cope with the problem of severe reflections at the truncated boundaries,we analogize the viscous absorbing boundary and perfectly matched layer(PML)absorbing boundary based on the single-relaxation-time lattice Boltzmann(SRT-LB)equation to the MRT-LB equation,and further,propose a joint absorbing boundary through comparative analysis.We give the specific forms of the modified MRT-LB equation loaded with the joint absorbing boundary in the two-dimensional(2D)and three-dimensional(3D)cases,respectively.Then,we verify the effects of this absorbing boundary scheme on a 2D homogeneous model,2D modified British Petroleum(BP)gas-cloud model,and 3D homogeneous model,respectively.The results reveal that by comparing with the viscous absorbing boundary and PML absorbing boundary,the joint absorbing boundary has the best absorption performance,although it is a little bit complicated.Therefore,this joint absorbing boundary better solves the problem of truncated boundary reflections of MRT-LBM in simulating seismic acoustic wavefields,which is pivotal to its wide application in the field of exploration seismology.

A rescaling algorithm for multi-relaxation-time lattice Boltzmann method towards turbulent flows with complex configurations

Understanding and modeling flows over porous layers are of great industrial significance.To accurately solve the turbulent multi-scale flows on complex configurations,a rescaling algorithm designed for turbulent flows with the Chapman-Enskog analysis is proposed.The mesh layout and the detailed rescaling procedure are also introduced.Direct numerical simulations(DNSs)for a turbulent channel flow and a porous walled turbulent channel flow are performed with the three-dimensional nineteen-velocity(D3Q19)multiple-relaxation-time(MRT)lattice Boltzmann method(LBM)to validate the accuracy,adaptability,and computational performance of the present rescaling algorithm.The results,which are consistent with the previous DNS studies based on the finite difference method and the LBM,demonstrate that the present method can maintain the continuity of the macro values across the grid interface and is able to adapt to complex geometries.The reasonable time consumption of the rescaling procedure shows that the present method can accurately calculate various turbulent flows with multi-scale and complex configurations while maintaining high computational efficiency.

Study on Free Surface and Channel Flow Induced by Low Temperature Plasma via Lattice Boltzmann Method

Active boundary layer flow control and boundary layer manipulation in the channel flow that was based on low temperature plasma were studied by means of a lattice Boltzmann method. Two plasma actuators were placed in a row to obtain the influence rule of their separation distance on the velocity profile at three locations and maximum velocity in the flow field. Two plasma actuators were placed symmetrically inside a channel to examine the effect of channel height and voltage on the velocity profile and flow rate. It was found that the channel height controls the distribution of flow velocity, which affected the flow rate and its direction. Increasing plasma voltage had a negative effect on the flow rate due to the generation of a larger and stronger flow vortex.

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 dis- tribution 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 par- ticularly 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.

High velocity flow simulation using lattice Boltzmann method with no-free-parameter dissipation scheme

A finite difference lattice Boltzmann method of second-order accuracy in time is developed based on non-oscillatory scheme with no-free-parameter dissipation （NND） difference scheme in this paper. The NND lattice Boltzmann method is used to simulate high-speed flows by constructing a new equilibrium distribution function of the lattice Boltzmann method. Compared with a variation of lattice Boltzmann method developed by Qu, et al., the present method can capture shock waves and handle oscillations of high velocity flows accurately in larger time steps and in shorter computing time. Numerical results indicate the correctness and capability of simulating shock wave interactions of the NND lattice Boltzmann method.

Lattice Boltzmann simulation of the effects of cavity structures and heater thermal conductivity on nucleate boiling heat transfer

The boiling heat transfer technology with cavity surfaces can provide higher heat flux under lower wall superheat,which is of great significance for the cooling of electronic chips and microelectromechanical devices.In this paper,the boiling characteristics of the cavity surfaces are investigated based on the lattice Boltzmann(LB)method,focusing on the effects of cavity shapes,sizes,and heater thermal conductivity on the heat transfer performance.The results show that the triangular cavity has the best boiling performance since it has less residual vapor and higher bubble departure frequency than those of the trapezoidal and rectangular cavities.As the cavity size increases,the enhancement of heat transfer by the cavity mouth is suppressed by the heat accumulation effect at the heater bottom.The liquid rewetting process during bubble departure is the reason for the fluctuation of the space-averaged heat flux,and the heater thermal conductivity determines the fluctuation amplitude.The evaporation of liquid in the cavity with high thermal conductivity walls is more intense,resulting in shorter waiting time and higher bubble departure frequency.

Improved contact angle measurement in multiphase lattice Boltzmann

Contact angle is an essential parameter to characterize substrate wettability.The measurement of contact angle in experiment and simulation is a complex and time-consuming task.In this paper,an improved method of measuring contact angle in multiphase lattice Boltzmann simulations is proposed,which can accurately obtain the real-time contact angle at a low temperature and larger density ratio.The three-phase contact point is determined by an extrapolation,and its position is not affected by the local deformation of flow field in the three-phase contact region.A series of simulations confirms that the present method has high accuracy and gird-independence.The contact angle keeps an excellent linear relationship with the chemical potential of the surface,so that it is very convenient to specify the wettability of a surface.The real-time contact angle measurement enables us to obtain the dynamic contact angle hysteresis on chemically heterogeneous surface,while the mechanical analyses can be effectively implemented at the moving contact line.

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.

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.

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.

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.