Boundary Scheme Research Articles

Convective flow over heat dissipating fins for application of electronic package cooling using curved boundary scheme lattice Boltzmann method

PurposeThe purpose of this study is to address a problem in cooling of an electronic package where the dissipating fins transfer the extra heat energy from the heat source (i.e. electronic devices) to the heat sink (i.e. environment). To this end, the convective heat transfer of nanofluid flow over dissipating fins is simulated using a numerical approach, whereas the properties of nanofluid are evaluated based on the experimental measurements and used in the numerical process.Design/methodology/approachTo simulate the convective flow, the lattice Boltzmann method is used. Also, the curved boundary scheme is used to enhance the capability of lattice Boltzmann method (LBM) in the simulation of natural convection in curved boundaries. In addition, the second law analysis is used based on total and local approaches.FindingsTo improve the cooling performance of fins, a modern technique is used, which is using of nanofluid. For this purpose, samples of SiO2-liquid paraffin with mass fractions of 0.01, 0.05, 0.1, 0.5 and 1 (Wt.%) in a temperature range of 25–60 °C are provided, and the required thermal and physical properties of samples including thermal conductivity and dynamic viscosity are measured during experimental work. The extracted results are used in the numerical simulations using derived correlations.Originality/valueThe originality of the present work is using a modern numerical method in the investigation of an engineering application and combining it with experimental data.

Interface-resolved simulation of solid-liquid phase change with thermal convection using discrete unified gas kinetic scheme

This work presents an interface-resolved model for simulating the solid-liquid phase transition problems. In the method, the natural convection flow is handled by the recently proposed discrete unified gas kinetic scheme (DUGKS). Compared with the conventional lattice Boltzmann equation, DUGKS has more advantages on numerical stability and multiscale computation. On the other hand, the ghost-cell (GC) immersed boundary scheme is applied to capturing the solid-liquid interface of pure material. In contrast to the phase-field or enthalpy-based method, the interface here with zero thickness is resolved and explicitly tracked on the fixed Cartesian grid. It is worth noting that due to the adsorption or release of latent heat, the phase front should be evolved according to the Stefan condition. Since compatible with the GC method, the position of the melting front is directly identified by the intersection point, and the complicated treatment of marker inserting and/or deleting in the previous studies is removed. Three numerical examples, including the one-dimension conduction melting, convection melting in a square cavity and melting in a circular cavity with natural convection are conducted to test the present model. The result demonstrates the accuracy of the method for solid-liquid phase change problems.

PDE‐based event‐triggered containment control of multi‐agent systems with input delay under spatial boundary communication

AbstractThis article studies the containment control problem for multi‐agent systems with input delay under spatial boundary communication by employing an event‐based approach. Firstly, the collective dynamics of multi‐agent systems are described as parabolic partial differential equations (PDEs). Applying the integral transformation method developed in PDEs, the delayed parabolic PDEs are transformed into a series of new coupled PDE‐PDE systems. Then, by taking the spatial boundary communication scheme into account and using the local boundary information, two boundary containment control protocols together with a dynamic event‐triggered scheme (DETS) are designed, such that the states of all followers converge to a convex hull formed by multiple leader agents with and without input delay. The optimal protocols are given by minimizing the 2‐norm of the designed control gain matrix, and the exclusion of the Zeno behavior is analyzed. Finally, a numerical simulation example is provided to support the main results.

Improvement of the stability and accuracy of solid-wall immersed boundary schemes for the linearized Euler equations using boundary constraints

A Cartesian cut-cell method with explicit high-order finite-difference stencils for solid walls at internodal wall positions has been developed to solve the linearized Euler equations. For this, additional boundary constraints besides the physical impermeability condition have been incorporated into Hermite-based finite-difference stencils near the boundary. We demonstrate that our methodology and the ghost point formulations of Tam & Dong (1994) and Gloerfelt (2001) are equivalent when a wall intersects with a grid point. Other than Tam & Kurbatskii's (1997) extrapolation-based cut-cell treatment, our formulation can be regarded as a straightforward generalization of the above-mentioned ghost point formulations. A numerical reflection problem demonstrates improved accuracy in 1D for high-order schemes of sixth- and 12th-order. The scattering of a plane wave with 8 PPW for a 7-point-, and 5 PPW for an 11-point boundary scheme at a 2D-cylinder in fluid at rest demonstrates the extension of the proposed method to higher dimensions. For the 1D case, stability analysis has been carried out to show the significant impact of using boundary constraints to improve the numerical stability of a boundary scheme. In the multidimensional case, weak instabilities have to be suppressed by selective filtering. Our new formalism for boundary methods enables the assessment of the propagation characteristics of modified 1D-boundary stencils for first derivatives by Fourier analysis. It is shown that the incorporation of boundary constraints improves the modified wavenumber signature of the boundary stencils significantly.

Numerical simulation of mixed aerosols deposition behavior on cylindrical cross fibers

In the actual dust removal process, aerosol particles with distinct physical properties often coexist. Most studies on fiber filtration focused on the particles of a single species, resulting in a poor understanding of the deposition behavior of mixed aerosols. In this study, the dynamic deposition behavior of mixed aerosol particles containing silica (SiO2) and polystyrene (PS) on cylindrical cross fibers was numerically studied using a lattice Boltzmann model coupled with discrete element method (LB-DEM). The mutual influences between the dynamic particle transport/deposition and the airflow were considered using immersed moving boundary (IMB) scheme. The mixed particles exhibit different deposit morphology and filtration performance from single particulate species mainly in the dendritic filtration stage, where the mixed particles tend to deposit on the lateral sides rather than the front of fibers. The filtration efficiency and pressure drop of mixed aerosols are not a superposition of the individual components. At high particle Stroke number (St), the synergistic deposition of SiO2 particles and PS particles of low elastic modulus occurs, resulting in filtration efficiencies that are approximately-four times greater than any single particulate species. When St is below 1.0, the filtration efficiency is improved with increasing St, which is similar to that of single species. Moreover, at the same flow velocity, the pressure drop rises faster and the flow is easier to be clogged for mixed particle filtration. The pressure drop rising rates of the mixed filtration at a high St range from 0.013 to 0.020, which is more than three times the rates for PS (0.006) and SiO2 (0.003).

Impacts of solid wall boundary conditions in the lattice Boltzmann method on turbulent outdoor flow: A case study of a single 1:1:2 building model

Solid wall boundary condition is an important topic in outdoor wind environment simulation represented by single building based on lattice Boltzmann method (LBM), but there is a lack of quantitative comparative analysis of simulation results under these boundary conditions. This paper conducted LBM-based large-eddy simulation (LBM-LES) of turbulent flow around a 1:1:2 single building, applying both half-way bounce-back and interpolation-based bounce-back (the Bouzidi–Firdaouss–Lallemand scheme, BFL) to solid wall boundaries. The simulation accuracy of different solid wall boundary conditions was examed with consideration of grid resolutions and relative positions between grid points and solid boundaries. Data obtained via experiments and the finite volume method-based large-eddy simulation (FVM-LES) technique were used to examine the flow structure and distribution of the time-averaged velocity and turbulent kinetic energy (TKE). The computational costs were also discussed. Results demonstrated that when the all the boundaries fell on the grid and orthogonal to the grid, the half-way bounce-back scheme attained a high accuracy. On the contrary, the BFL scheme achieved a higher accuracy if the solid wall boundaries did not fall on the grid. Under both boundary schemes, the grid resolutions of b/32 and b/16 (b: building width) could satisfy the acceptable range, but the b/32 grid resolution facilitated a better performance near the wall. The BFL scheme was more sensitive to the grid resolution and the accuracy improved faster with increasing grid resolution than half-way. As for the calculation costs, the calculation time of BFL is more than twice that of half-way bounce-back. However, BFL is still recommended if the wall boundaries are curved and complex in the actual built environment.

Research on the gas migration trend and mechanism of the transition flow regime in coal based on MRT-LBM simulation

In order to reveal the process and mechanism of gas flow in a low-permeability coal seam, a new multiple-relaxation-time lattice Boltzmann method (MRT-LBM) model of gas migration in coal micro/nanopores based on Langmuir monolayer adsorption theory, the slip boundary scheme and Bosanquet effective viscosity was established. The software MATLAB was used to carry out the simulation study of uniform pore gas flow based on the MRT-LBM model, and the results were compared and verified with the porous anodic alumina membrane gas flow experimental results. On this basis, the gas flow in coal pores with different micro/nanopore sizes considering adsorption was simulated. The results show that the dimensionless permeability coefficient increases with decreasing pore size under the same pressure, which reflects the subsequent enhancement of the microboundary constraint effect and reveals that the pore system becomes the main controlling factor of coal seam permeability within the coal matrix in the middle and late stages of coal seam gas extraction, while the role of the microboundary constraint effect needs to be considered. The gas adsorption layer weakens the pore gas flow capacity, but for pores with a radius greater than 16 nm, the apparent change in permeability caused by the adsorption layer is less than 5%, and the adsorption effect can be ignored. N2, CH4, and CO2 enter the transition flow regime under different pressure conditions; with gas extraction, the gas pressure decreases, and the difference in the gas flow characteristics of the three gases increases.

Two pressure boundary conditions for multi-component multiphase flow simulations using the pseudo-potential lattice Boltzmann model

In this study, two boundary condition (BC) schemes are proposed for the pseudo-potential (PP) lattice Boltzmann method (LBM). The two BC schemes include a modified Convective Boundary Condition (mCBC) and a modified Non-equilibrium Bounceback (mNEBB), and their development is aimed at treating pressure-driven outflow problems for fluids with similar and different density ratios in multi-component multiphase (MCMP) environments. It is the first attempt to incorporate interaction force at boundaries for MCMP fluids. Key limitations of existing LBM boundary schemes for MCMP flow are alleviated. In particular, the constraint of zero-velocity specification at tangential direction is removed by applying the proposed mCBC scheme. In addition, it is demonstrated that multiphase outflow performance is improved with the application of either scheme, while alleviating the nonphysical distortions introduced by the available BC schemes. Three numerical case studies have been conducted to test the proposed scheme. Results have been validated using uncut-off extended domains. These cases are designed to test the induced distortion to the fluids exiting boundaries, examine the scheme’s capability to treat the pressure boundary condition, and explore whether the model can make physically meaningful predictions when these modified BCs are applied. The mCBC scheme is shown to be more capable to correctly treat such pressure specified boundary conditions.

Spectrally accurate solutions to inhomogeneous elliptic PDE in smooth geometries using function intension

We present a spectrally accurate embedded boundary method for solving linear, inhomogeneous, elliptic partial differential equations (PDE) in general smooth geometries in two dimensions, focusing in this manuscript on the Poisson, modified Helmholtz, and Stokes equations. Unlike several recently proposed methods which rely on function extension, we propose a method which instead utilizes function intension, or the smooth truncation of known function values. Similar to those methods based on extension, once the inhomogeneity is truncated we may solve the PDE using any of the many simple, fast, and robust solvers that have been developed for regular grids on simple domains. Function intension is inherently stable, as are all steps in the proposed solution method, and can be used on domains which do not readily admit extensions. We pay a price in exchange for improved stability and flexibility: in addition to solving the PDE on the regular domain, we must additionally (1) solve the PDE on a small auxiliary domain that is fitted to the boundary, and (2) ensure consistency of the solution across the interface between this auxiliary domain and the rest of the physical domain. We show how these tasks may be accomplished efficiently (in both the asymptotic and practical sense), and compare convergence to two recent high-order embedded boundary schemes. Finally, to demonstrate the wide applicability of the method, we solve a nonlinear predator-prey model, achieving rapid convergence in both space and time.

Slip velocity boundary conditions for the lattice Boltzmann modeling of microchannel flows

AbstractSlip flows in ducts are important in numerous engineering applications, most notably in microchannel flows. Compared to the standard no‐slip Dirichlet condition, the case of slip formulates as a Robin‐type condition for the fluid tangential velocity. Such an increase in mathematical complexity is accompanied by a more challenging numerical transcription. The present work concerns with this topic, addressing the modeling of the slip velocity boundary condition in the lattice Boltzmann method (LBM) applied to steady slow viscous flows inside ducts of nontrivial shapes. As novelty, we extend the newly revised local second‐order boundary (LSOB) Dirichlet fluid flow method [Philos. Trans. R. Soc. A 378, 20190404 (2020)] to implement the slip velocity condition within the two‐relaxation‐time (TRT) framework. The LSOB follows an in‐node philosophy where its operation principle seeks to explicitly reconstruct the unknown boundary populations in the form of a third‐order accurate Chapman–Enskog expansion, where the wall slip condition is built‐in as a normal Taylor‐type condition. The key point of this approach is that the required first‐ and second‐order momentum derivatives, rather than computed through nonlocal finite difference approximations, are locally determined through a simple local linear algebra procedure, whose formulation is particularly aided by the TRT symmetry argument. To express the obtained derivatives, two approaches are considered, called and , which operate with node and wall variables, respectively. These two formulations are developed to prescribe the physical slip condition over plane and curved walls, including the corners. Their consistency and accuracy characteristics are examined against alternative linkwise strategies to impose the wall slip velocity, such as the kinetic‐based diffusive bounce‐back scheme, the central linear interpolation slip scheme, and the multireflection slip scheme. The several slip schemes are tested over different 3D microchannel configurations, with walls not conforming with the LBM uniform mesh. Numerical tests confirm the advanced accuracy characteristics of the proposed LSOB slip boundary scheme, revealing the added challenge of the wall slip modeling, and that parabolic accuracy is a necessary requirement to reach second‐order accuracy within this problem class.

Shakedown analysis and assessment method of four-stress parameters Bree-type problems

For pressure equipment in the industrial field, the shakedown design methods presented in the ASME and other international design codes are mainly based on the Bree problem. The original Bree problem and its related modified Bree problems usually contain two types of stress parameters, whereas design by analysis based on the stress categorisation of the codes actually includes multiple types of stress parameters. In this paper, the four-stress parameters Bree-type shakedown problems (involving mechanical membrane and bending as well as thermal membrane and bending) have been studied under three typical thermo-mechanical loading conditions for the first time to reveal the mechanisms of the four-dimensional Bree diagram. A detailed discussion on the mechanical models for various uniaxial Bree-type problems is conducted to clarify the scope of application and mechanical assumptions for each problem. A common finite element model is developed for all discussed Bree-type problems. The Linear Matching Method (LMM) is utilised to construct the shakedown boundaries of three examples under three loading conditions, and the semi-analytical parametric equations of the shakedown boundaries are derived based on the numerical results and four-dimensional ratcheting boundary theory. Using the idea of a minimum shakedown boundary, a unified and conservative shakedown assessment scheme is proposed for the four-stress parameters Bree-type problems. The newly presented results can enhance the understanding of the structural behaviour of pressure equipment for many loading conditions and guide shakedown design and ratcheting assessment of practical industrial components.

Numerical simulations of polygonal particles settling within non-Newtonian fluids

The settling of circular and polygonal particles within non-Newtonian fluids is investigated by combining the lattice Boltzmann method (LBM) and the discrete element method (DEM). The immersed moving boundary (IMB) scheme with good numerical stability is adopted to couple LBM and DEM. To efficiently calculate the solid coverage ratio in IMB, a novel method is developed, which simply involves judging whether the square is fully occupied by the particle and subdividing the square crossed by the fluid–solid boundary. After validations, the drafting–kissing–tumbling dynamics of two particles settling in the Newtonian and power-law fluids are studied first. It shows that the shear-thickening fluid has a longer kissing duration than the Newtonian and shear-thinning fluids. The kissing duration of squared particles (0.29–0.41 s) is shorter than triangular particles (0.32–0.84 s) and much shorter than circular particles (0.61–0.98 s). Then, the settling of multiple and multi-shape particles in a closed cavity is analyzed. The disturbed area of kinematic viscosity induced by particle motion in the shear-thinning fluid is 21.0–22.5 cm2, significantly larger than in the shear-thickening fluid (10.1–10.8 cm2). Circular particles have a larger disturbed area than the polygonal particles due to the larger settling velocity. Moreover, compared with the Newtonian and shear-thinning fluids, the shear-thickening fluid has a smaller vertical length of particle cluster, meaning a positive influence on the agglomeration of particles.

A modified lattice Boltzmann model for microcylindrical Couette gas flows

We develop an axisymmetric lattice Boltzmann (LB) model to simulate microcylindrical Couette gas flows (MCGF) in the slip regime and transition regime, respectively. A kinetic boundary scheme in the cylindrical coordinate system is proposed to fulfill the second-order slip boundary condition at the cylindrical wall. To consider the effect of the Knudsen layers for transition flows, local effective Knudsen numbers are introduced into the kinetic boundary scheme and relaxation time. Numerical tests are executed to acquire the velocity distributions of the time-independent and time-dependent MCGF. Comparisons with the analytical solution and direct Monte Carlo data are also implemented. The simulation results demonstrate that the developed LB model can successfully acquire the velocity distribution of the MCGF with an intermediate Knudsen number in the transition regime.

Lattice Boltzmann study of dissolution in porous media: Comparison of VOP with VOF-curved boundary coupling

In dissolution processes, during geometry evolution, the two-phase interface is continuously changed. For tracking the position of the interface, the method of combining VOF with curved boundary schemes was presented in this research. Since the combination would be rather complex, in addition to the original approach, a simplified approximate approach was also introduced. Further, for simulating heterogeneous dissolution in lattice Boltzmann (LB) framework, the volume of pixel (VOP) method has been widely used by researchers. Since the area of the solid-fluid interface is not captured by this method, the study of its validity was also presented in this research. For this purpose, simulations of calcium carbonate dissolution by hydrochloric acid in simple and complex porous media were performed. In addition to the VOP method, the volume of fluid (VOF) method in which the solid-fluid interface is tracked, was also applied for all cases of simulations. The results revealed that from a dissolution perspective, the performance of the VOF-curved boundary combination is essentially similar to that of the VOF-bounce back combination. However, the algorithm of bounce back was found to be more than 10% computationally efficient. Also, comparison of the outcomes from VOP and VOF indicated that VOP intrinsically overestimates the surface area of reactions by about 25–35%, which fictitiously leads to 20–30% higher reaction rates and nearly 20–30% less dissolution times. Hence, it was deduced that the temporal outcomes of VOP may not be valid. However, from a non-temporal perspective, the dissolution patterns of both methods were found to be essentially similar.

Numerical study on the drag and flow characteristics of porous particles at intermediate Reynolds numbers

Flow entrained with permeable particles is frequently encountered in natural and engineering applications. However, the lack of knowledge about the drag on the porous spheres and the corresponding complex flow structures has been a hindrance for better understanding of these flow systems In this work, we perform a comprehensive numerical simulation of flow past a porous particle with a wide range of Reynolds number (5 ≤ Re ≤ 200) and porosity (0.1 ≤ ɛ ≤ 0.95). Furthermore, considering that few studies has reported the flows with moving porous spheres, the simulation of both a single and two porous spheres settling in an enclosure is conducted for further evaluating the effect of moving porous media on the fluid–solid flows. The lattice Boltzmann method (LBM) is applied to the fluid flow, and particularly, the immersed moving boundary (IMB) scheme describes the interaction between the fluid and porous solid body. The accuracy of that the hybrid IMB-LBM for modeling porous sphere flows, to our knowledge, is first explored and established by the present study. The results suggest that both Re and ɛ have a significant influence on the drag and recirculation flow of porous particles. At Re ≤ 75, the porous particle experiences a lower resistance than that of the impermeable counterpart. When Re exceeds 75, the drag ratio is higher than unity for ɛ ≤ 0.85. With increasing porosity, the recirculation wake penetrates the particle, then detaches from the particle, and eventually disappears in the particle rear. Based on the simulated results, a new drag correlation for porous spheres is developed. As for the moving porous spheres, they fall slower than the solid counterparts, and their drafting, kissing and tumbling (DKT) process can also be slowed down.

Estimation of parameters of gas storage operation in inhomogeneous aquifers

The purpose of the article is development and testing of a mathematical model of gas storage in a layered aquifer with a low permeabilityinterlayer for a case of plane-parallel and axial-symmetric filtration. Methodology. One of the most common models of anisotropy of rocks is a model of a layered seam, which is explained by geological conditions of sedimentation, which lead to stratification of layers with different collecting properties. In the practice of underground gas storage, consideration of such a model is of particular importance.This is due to possible significant difference in advancing a boundary of gas-water contact through the interlayers with different filtration characteristics caused by changes in a position of a gas zone. A comprehensive approach is applied, which includes collection, systematization and analysis of actual data on filtration and physical and mechanical properties of host rocks that affect the formation of natural and technogenic deposits, as well as analytical and numerical methods for solving equations of gas-water contact in different conditions. Results. Gas-hydrodynamic model of underground gas storage in an inhomogeneous aquifer is justified for calculation of its cyclic operation in a three-layer seam considering cross-flows through a low permeability interlayer.The results can be used in evaluation calculations at a design stage of gas storage facilities in aquifers. Scientific novelty. A mathematical model of gas storage in a layered aquifer with a low permeability interlayerfor a case of plane-parallel and axial-symmetric filtration is developed and tested. A new method of linearization of a system of differential equations for determining pressures in a collectingseamis obtained in the article and it is a generalization of previously used methods, with an introduction of "boundary schemes". Practical significance. Calculation results indicate a significant influence of characteristics of a layered porous medium on the advance of gas-water contact along individual layers. The results can be used in the evaluation calculations at a design stage of gas storage facilities in aquifers.

Comparative investigation of a lattice Boltzmann boundary treatment of multiphase mass transport with heterogeneous chemical reactions.

Multiphase reactive transport in porous media is an important component of many natural and engineering processes. In the present study, boundary schemes for the continuum species transport-lattice Boltzmann (CST-LB) mass transport model and the multicomponent pseudopotential model are proposed to simulate heterogeneous chemical reactions in a multiphase system. For the CST-LB model, a lattice-interface-tracking scheme for the heterogeneous chemical reaction boundary is provided. Meanwhile, a local-average virtual density boundary scheme for the multicomponent pseudopotential model is formulated based on the work of Li etal. [Li, Yu, and Luo, Phys. Rev. E 100, 053313 (2019)10.1103/PhysRevE.100.053313]. With these boundary treatments, a numerical implementation is put forward that couples the multiphase fluid flow, interfacial species transport, heterogeneous chemical reactions, and porous matrix structural evolution. A series of comparison benchmark cases are investigated to evaluate the numerical performance for different pseudopotential wetting boundary treatments, and an application case of multiphase dissolution in porous media is conducted to validate the present models' ability to solve complex problems. By applying the present LB models with reasonable boundary treatments, multiphase reactive transport in various natural or engineering scenarios can be simulated accurately.

Influence of the wettability on the liquid breakup in planar prefilming airblast atomization using a coupled lattice Boltzmann–large eddy simulation model

In this paper, an efficient coupled lattice Boltzmann–large eddy simulation model [X. An et al., “Coupled lattice Boltzmann-large eddy simulation model for three-dimensional multiphase flows at large density ratio and high Reynolds number,” Phys. Rev. E 104, 045305 (2021)] based on the Allen–Cahn phase-field theory is introduced for simulating the liquid breakup in planar prefilming airblast atomization. This is the first time that the lattice Boltzmann method is used in the three-dimensional numerical investigation of prefilming airblast atomization to the best of our knowledge. The present model utilizes two evolution equations: one is used to capture the fluid interface, and another is adopted to solve hydrodynamic properties. An advanced multiple-relaxation-time scheme is also applied for the collision operator to enhance the numerical stability. To investigate the influence of the wettability on the liquid breakup accurately, a simple and efficient wetting boundary scheme is delicately designed and strictly validated. Additionally, to evaluate the atomization quality intuitively, an atomization efficiency coefficient is proposed for characterizing the liquid breakup process. The numerical results reveal that the influence of the wettability lies in the liquid accumulation phenomenon at the edge of the prefilmer and the droplet movement in the vertical direction. The atomization quality adopting a non-wetting prefilmer is better than other cases, according to the atomization efficiency coefficients, the mean droplet equivalent diameters with their size distribution proportions, and the atomization angles in the vertical direction. In addition, it is also found that the droplet proportion above the prefilmer increases as the contact angle increases, and the proportions on both sides of the prefilmer account for 50% at the contact angle of 90°.

3D Lattice Boltzmann Method-Discrete-Element Method with Immersed Moving Boundary Scheme Numerical Modeling of Microparticles Migration Carried by a Fluid in Fracture

SummaryBased on the LBM-IMB-DEM (coupled lattice Boltzmann method-discrete-element method with immersed moving boundary scheme) model, the 3D microparticles’ migration carried by a fluid in fracture is numerically investigated both considering and not considering the van der Waals adhesive force. A boundary cells’ tracing method called the layer-by-layer tracing method both applicable to 2D and 3D situations for IMB with high efficiency and accuracy is proposed, and based on its searching results, the contacts in DEM could be detected straightforwardly with a finite amount of computation. In the layer-by-layer tracing method, a limited number of points, including some cell centers and nodes, near the particle boundary are determined whether they are covered by the particle, and then the other cells in one layer could be precisely classified. In the contact detecting process for DEM, based on the boundary cells tracing results in IMB, the cells that are covered by no less than two particles are the potential cells where there may be contacts. For wall boundaries with irregular shapes, the wall solid boundary (WSB) cells are converted to circumcircle (or circumscribed sphere in 3D situation), and then the solid particle-wall interactions could be replaced by the interactions between the solid particles and those circumcircles or circumscribed spheres. Two cases, including single-particle sedimentation in a viscous fluid, two-particle Drafting-Kissing-Tumbling (DKT) simulation for both nonadhesive and adhesive particles, are used to validate the LBM-IMB-DEM method. Besides, multiparticle sedimentation tests for different particle radiuses are conducted to present the advantages of the layer-by-layer tracing method. At last, nonadhesive and adhesive microparticles’ liberation, transport, and retention carried by a fluid in fracture with irregular shapes are simulated. The numerical results show that the adhesive force that plays a dominated role for microparticles has significant effects on the mechanics of solid particles migration. Under the influence of adhesive forces, the microparticles tend to form a stable agglomerate and migrate as a whole, which is different from the situation for nonadhesive particles where they are relatively independent and basically migrate by layer. Besides, even if the fluid velocity is much lower than the critical velocity evaluated through force or torque analysis on a single particle, when the total hydrodynamic forces exerted on the agglomerate overcome the total adhesive force between the wall and the lower layer of particles, the agglomerate can start to move and liberate from the wall.

Characteristic boundary conditions for magnetohydrodynamic equations

In the present study, a characteristic-based boundary condition scheme is developed for the compressible magnetohydrodynamic (MHD) equations in the general curvilinear coordinate system, which is an extension of the characteristic boundary scheme for the Navier–Stokes equations. The eigenstructure and the complete set of characteristic waves are derived for the ideal MHD equations in general curvilinear coordinates (ξ,η,ζ). The characteristic boundary conditions are derived and implemented in a high-order MHD solver where the sixth-order compact scheme is used for the spatial discretization. The fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme is also employed for the spatial discretization of problems with discontinuities. In our MHD solver, the fourth-order Runge–Kutta scheme is utilized for time integration. The characteristic boundary scheme is first verified for the non-magnetic (i.e., B=0) Sod shock tube problem. Then, various in-house test cases are designed to examine the derived MHD characteristic boundary scheme for three different types of boundaries: non-reflecting inlet and outlet, solid wall, and single characteristic wave injection. The numerical examples demonstrate the accuracy and robustness of the MHD characteristic boundary scheme.