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 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.

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.

Improving the Stability of theMultiple-Relaxation-Time Lattice Boltzmann Method by a Viscosity Counteracting Approach

Numerical instability may occur when simulating high Reynolds number flows by the lattice Boltzmann method(LBM).The multiple-relaxation-time(MRT)model of the LBM can improve the accuracy and stability,but is still subject to numerical instability when simulating flows with large single-grid Reynolds number(Reynolds number/grid number).The viscosity counteracting approach proposed recently is a method of enhancing the stability of the LBM.However,its effectiveness was only verified in the single-relaxation-time model of the LBM(SRT-LBM).This paper aims to propose the viscosity counteracting approach for the multiple-relaxationtime model(MRT-LBM)and analyze its numerical characteristics.The verification is conducted by simulating some benchmark cases:the two-dimensional(2D)lid-driven cavity flow,Poiseuille flow,Taylor-Green vortex flow and Couette flow,and threedimensional(3D)rectangular jet.Qualitative and Quantitative comparisons show that the viscosity counteracting approach for the MRT-LBMhas better accuracy and stability than that for the SRT-LBM.

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.

Combined immersed boundary method and multiple-relaxation-time lattice Boltzmann flux solver for numerical simulations of incompressible flows

A method combining the immersed boundary technique and a multi- relaxation-time （MRT） lattice Boltzmann flux solver （LBFS） is presented for numerical simulation of incompressible flows over circular and elliptic cylinders and NACA 0012 Airfoil. The method uses a simple Cartesian mesh to simulate flows past immersed complicated bodies. With the Chapman-Enskog expansion analysis, a transform is performed between the Navier-Stokes and lattice Boltzmann equations （LBEs）. The LBFS is used to discretize the macroscopic differential equations with a finite volume method and evaluate the interface fluxes through local reconstruction of the lattice Boltzmann solution. The immersed boundary technique is used to correct the intermediate velocity around the solid boundary to satisfy the no-slip boundary condition. Agreement of simulation results with the data found in the literature shows reliability of the proposed method in simulating laminar flows on a Cartesian mesh.

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.

Modeling the Wettability of Microstructured Hydrophobic Surface Using Multiple-relaxation-time Lattice Boltzmann Method

Wetting properties are significant for a hydrophobic surface and normally characterized by the equilibrium contact angle.In this manuscript,a mesoscopic method based on multiphase multiple-relaxation-time Lattice Boltzmann method has been presented and applied to simulate the contact angle at three-phase interfaces of a solid surface with micro-pillar structure.The influence of different parameters including pillar height,pillar width,inter-pillar spacing,intrinsic contact angle and the volume of the liquid drop on the equilibrium contact angle has been comprehensively investigated.The effect of geometry parameters of the micro-pillar structure on the wetting transition from Cassie–Baxter state to Wenzel state has also been studied.The results indicate that when the inter-pillar spacing is less than a certain value or the pillar height is greater than a certain value,the contact form between the droplet and the surface satisfies the Cassie–Baxter state.When the contact form satisfies the Cassie–Baxter state,the contact angle gradually increases with the increase of the inter-pillar spacing;the contact angle does not change significantly with the pillar height;the contact angle gradually decreases and approaches the intrinsic contact angle with the pillar width increases.Moreover,the contact angle increases with the increase of the intrinsic contact angle,and the contact angle is not sensitive to the change of droplet volume when the droplet volume is between 0.5 and10μl.

Effect of viscosity on stability and accuracy of the two-component lattice Boltzmann method with a multiple-relaxation-time collision operator investigated by the acoustic attenuation model

A two-component lattice Boltzmann method(LBM) with a multiple-relaxation-time(MRT) collision operator is presented to improve the numerical stability of the single relaxation time(SRT) model. The macroscopic and the momentum conservation equations can be retrieved through the Chapman–Enskog(C-E) expansion analysis. The equilibrium moment with the diffusion term is calculated, a diffusion phenomenon is simulated by utilizing the developed model, and the numerical stability is verified. Furthermore, the binary mixture channel model is designed to simulate the sound attenuation phenomenon, and the obtained simulation results are found to be consistent with the analytical solutions. The sound attenuation model is used to study the numerical stability and calculation accuracy of the LBM model. The simulation results show the stability and accuracy of the MRT model and the SRT model under different viscosity conditions. Finally,we study the influence of the error between the macroscopic equation of the MRT model and the standard incompressible Navier–Stokes equation on the calculation accuracy of the model to demonstrate the general applicability of the conclusions drawn by the sound attenuation model in the present study.

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.

Multiple-Relaxation-Time Lattice Boltzmann Approach to Richtmyer-Meshkov Instability

The aims of the present paper are twofold. At first, we further study the Multiple-Relaxation-Time （MRT） Lattice Boltzmann （LB） model proposed in [Europhys. Lett. 90 （2010） 54003]. We discuss the reason why the Gram Schmidt orthogonalization procedure is not needed in the construction of transformation matrix M; point out a reason why the Kataoka-Tsutahara model [Phys. Rev. E 69 （2004） 035701（R）] is only valid in subsonic flows. The yon Neumann stability analysis is performed. Secondly, we carry out a preliminary quantitative study on the Richtmyer- Meshkov instability using the proposed MRT LB model. When a shock wave travels from a light medium to a heavy one, the simulated growth rate is in qualitative agreement with the perturbation model by Zhang-Sohn. It is about half of the predicted value by the impulsive model and is closer to the experimental result. When the shock wave travels from a heavy medium to a light one, our simulation results are also consistent with physical analysis.

Cavitation Bubble Collapse near a Curved Wall by the Multiple-Relaxation-Time Shan-Chen Lattice Boltzmann Model

The cavitation bubble collapse near a cell can cause damage to the cell wall. This effect has received increasing attention in biomedical supersonics. Based on the lattice Boltzmann method, a multiple-relaxation-time Shan–Chen model is built to study the cavitation bubble collapse. Using this model, the cavitation phenomena induced by density perturbation are simulated to obtain the coexistence densities at certain temperature and to demonstrate the Young–Laplace equation. Then, the cavitation bubble collapse near a curved rigid wall and the consequent high-speed jet towards the wall are simulated. Moreover, the influences of initial pressure difference and bubble-wall distance on the cavitation bubble collapse are investigated.

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.