Lattice Boltzmann Method Scheme Research Articles

Derivation and analysis of lattice Boltzmann form of the mild slope equation

The present study investigates the performance of the Lattice Boltzmann Method (LBM) as an alternative numerical method for simulating wave propagation in coastal zones. While formulations like the Mild-Slope Equation (MSE) remain efficient for coastal engineering applications due to their simplicity, LBM offers an explicit numerical solver for partial differential equations (PDEs). LBM utilizes local operations, making it well-suited for Graphics Processing Unit (GPU) acceleration and consequently efficient simulations. This study examines LBM's performance in several benchmark cases, including wave propagation behind a single breakwater, harbor resonance, refraction of long waves over a parabolic shoal, and wave propagation over an elliptic shoal. The Chapman-Enskog multi-scale expansion was used within a popular 2D LBM scheme to derive the relaxation time and equilibrium distribution function. The results showed a good agreement between LBM predictions and benchmark data. Furthermore, LBM demonstrated efficiency in reducing the computation time, achieving a notable 30–50% decrease in CPU time and an 85% reduction when utilizing a GPU.

Comprehensive performance analysis and structural improvement of latent heat thermal energy storage (LHTES) unit using a novel parallel enthalpy-based lattice Boltzmann model

Latent heat thermal energy storage (LHTES) utilizing phase change material (PCM) is one of the critical enablers in developing sustainable and low-carbon energy systems. To fill the knowledge gap, this paper presents an enthalpy-based solid–liquid model through the lattice Boltzmann method (LBM) with a multi-relaxation-time (MRT) approach, aiming to simulate convective phase change in LHTES units with and without porous media in Cartesian or axisymmetric domains. To improve accuracy and efficiency, the model integrates a differential scanning calorimetry (DSC) correlated equation for enthalpy modeling, couples with a 1D heat-transfer-fluid (HTF) model for boundary treatment of HTF side, and employs a parallel LBM scheme for efficient parametric studies. The validation demonstrates the model’s success in predicting PCM phase change, with errors below 10%. A comprehensive numerical analysis is then conducted to quantitatively evaluate the effect of convection on PCM melting. Novel metrics, such as acceleration rates (ac) of PCM melting and threshold Rayleigh numbers (Radc) at various aspect ratios, are introduced. Furthermore, PCM melting in the porous cylindrical unit is explored. Findings reveal up to 86% acceleration in melting compared to pure PCM at 80% energy storage, and the porous media with porosity above 0.9 is recommended for thermal enhancement. Moreover, this paper analyzes the negative effect of uneven temperature distributions caused by convection on LHTES unit efficiency. A modified LHTES unit geometry is proposed to offset this negative effect, and the study demonstrates successful mitigation of uneven temperature distributions, achieving up to 57 % acceleration in PCM melting.

Stable LBM schemes for acoustic scaling simulations under high Reynolds to Mach ratio: Introduction to the DM-TS operator

Hydroacoustic analyses using direct methods are challenging in the field of computational physics. To date, hybrid methods are an alternative approach adopted for performing hydroacoustic analysis, although it demonstrates clear limitations in analyzing nonlinear turbulent noise. The lattice Boltzmann method (LBM) provides a solution to address the limitations of the hybrid method and is frequently applied in aeroacoustics. However, existing methods, such as the Bhatnagar–Gross–Krook (BGK) operator, suffer from instabilities that manifest in an oscillatory fashion in underwater conditions. Hydroacoustic situations are characterized by high Reynolds number (Re) to Mach number (Ma) ratios, and these conditions correspond to cases where stability problems occur. In this study, a direct-method-based two-step (DM-TS) LBM collision operator is introduced to solve the stability problem. The stability, accuracy, and consistency of the scheme are investigated and compared with the BGK-LBM. The DM-TS operator is developed to provide the same accuracy order as the BGK scheme while addressing the stability problem. Vortex shedding from a two-dimensional cylinder is simulated using the DM-TS operator to prove that the proposed method could be used to solve hydroacoustic problems. The results of the analysis indicate that dynamic and acoustic pressures can be obtained simultaneously using the DM–TS operator.

Revisiting the second-order convergence of the lattice Boltzmann method with reaction-type source terms

This study analyses an approach to consistently recover the second-order convergence of the lattice Boltzmann method (LBM), which is frequently degraded by an improper discretisation of the required source terms. The current work focuses on advection-diffusion models, in which the source terms are dependent on the intensity of transported fields. Such terms can be observed in reaction-type equations used in heat and mass transfer problems or multiphase flows. The investigated scheme is applicable to a wide range of formulations within the LBM framework. All considered source terms are interpreted as contributions to the zeroth-moment of the distribution function. These account for sources in a scalar field, such as density, concentration, temperature or a phase field. Further application of this work can be found in the method of manufactured solutions or in the immersed boundary method. This paper is dedicated to three aspects regarding proper inclusion of the source term in LBM schemes. Firstly, it identifies the differences observed between the ways in which source terms are included in the LBM schemes present in the literature. The algebraic manipulations are explicitly presented in this paper to clarify the observed differences, and to identify their origin. Secondly, it analyses in full detail, the implicit relation between the value of the transported macroscopic field, and the sum of the LBM densities. This relation is valid for any source term discretization scheme. It is a crucial ingredient for preserving the second-order convergence in the case of complex source terms. Moreover, three equivalent forms of the second-order accurate collision operator are presented. Finally, closed form solutions of this implicit relation are shown for a variety of common models, including general linear and second order terms; population growth models, such as the Logistic or Gompertz model and the Allen-Cahn equation. The second-order convergence of the proposed LBM schemes is verified on both linear and non-linear source terms. The pitfalls of the commonly used acoustic and diffusive scalings are identified and discussed. Furthermore, for a simplified case, the competing errors are shown visually with isolines of error in the space of spatial and temporal resolutions.

DEM-LBM coupling for partially saturated granular assemblies

In this paper, we propose a phase-field-based Lattice Boltzmann Method (LBM) model coupled with the Discrete Element Method (DEM) for simulating unsaturated granular media over a large range of degrees of saturation. The formation of capillary bridges between grains is computed via an LBM scheme that solves Navier–Stokes and Allen–Cahn equations for multi-phase flow in complex geometries. As for the motions of the grains, these are computed within DEM where the contact behavior between spherical particles is based on a simple elastic–plastic contact law. Using an efficient DEM-LBM coupling framework, capillarity effects are explored at the pore scale with the collapse of a sandcastle as a classic illustration. This is done by considering an assembly of several thousands of spherical particles connected by capillary bridges for different degrees of saturation. It is shown that the mean capillary stress increases with the degree of saturation up to a certain threshold beyond which capillary stress drops until complete saturation. In addition, the Soil Water Characteristic Curve is qualitatively recovered and compared to a theoretical model. As such, the proposed DEM-LBM coupling scheme becomes a viable numerical tool that can explicitly model and explore pore-scale phenomena in unsaturated soils.

Lattice Boltzmann method with effective correction of phonon properties for nano/microscale heat transfer

The significant self-heating effect severely restricts the performance and reliability of nano-electronic devices. Accordingly, it is very important to understand the process and mechanism of nano/microscale heat transfer for thermal management and thermal design of devices. In this work, we propose a new Lattice Boltzmann Method (LBM) scheme with effective correction of phonon mean free path (MFP) and relaxation time to study phonon heat transfer in silicon thin films and silicon medium with defects, where the correction factor is dependent on the lattice structure of LBM. The transformation analysis of phonon transfer mechanism at different scales shows that the size effect of cross-plane thermal conductivity is more remarkable than that of in-plane thermal conductivity. And the thermal conductivity of silicon medium with defects decreases exponentially as defect density increases. The proposed new LBM scheme can generate more accurate results than the traditional ones in the heat conduction simulations of different nano/microscale structures.

A New Lattice Boltzmann Scheme for Photonic Bandgap and Defect Mode Simulation in One-Dimensional Plasma Photonic Crystals

A novel lattice Boltzmann method (LBM) with a pseudo-equilibrium potential is proposed for electromagnetic wave propagation in one-dimensional (1D) plasma photonic crystals. The final form of the LBM incorporates the dispersive effect of plasma media with a pseudo-equilibrium potential in the equilibrium distribution functions. The consistency between the proposed lattice Boltzmann scheme and Maxwell’s equations was rigorously proven based on the Chapman–Enskog expansion technique. Based on the proposed LBM scheme, we investigated the effects of the thickness and relative dielectric constant of a defect layer on the EM wave propagation and defect modes of 1D plasma photonic crystals. We have illustrated that several defect modes can be tuned to appear within the photonic bandgaps. Both the frequency and number of the defect modes could be tuned by changing the relative dielectric constant and thickness of the defect modes. These strategies would assist in the design of narrowband filters.

Lattice Boltzmann approach for the fluid-structure interaction of a neo-Hookean medium.

In this paper, a lattice Boltzmann method (LBM) for the fluid-structure interaction of an incompressible neo-Hookean medium is proposed. The objective is to use the lattice Boltzmann method to model the fluid and the solid domains at the same time, and thus avoid coupling two solvers, one for the fluid and one for the structure. The specific case of a neo-Hookean incompressible medium allows us to use a Eulerian formulation for the structure problem, which resembles the Navier-Stokes equation. Then, a macroscopic multiphase equationcan be used to model the fluid and structure problems together. Next, the LBM approach is deduced from this macroscopic multiphase formulation of the fluid-structure interaction problem. It consists in extending the LBM to the solid domain by adding a tensor term in the equilibrium function of the collision operator. The effect of the added tensor term is to cancel in the solid domain the viscous fluid constraints and add neo-Hookean constraints. Thus, only the third moment of the LBM is modified, the first two being conserved, and only the constraints in the macroscopic Navier-Stokes equationare changed. The LBM scheme obtained can then model the interaction of a fluid and a structure composed of an incompressible neo-Hookean medium with a single solver for the fluid and the solid. Two additional equationsare used, one to track the fluid and solid domains with the Cahn-Hilliard equation, and the other to compute the solid displacement field by a finite-difference scheme. The proposed method is applied successfully on three cases from the literature.

Restoring the conservativity of characteristic-based segregated models: Application to the hybrid lattice Boltzmann method

A general methodology is introduced to build conservative numerical models for fluid simulations based on segregated schemes, where mass, momentum, and energy equations are solved by different methods. It is especially designed here for developing new numerical discretizations of the total energy equation and adapted to a thermal coupling with the lattice Boltzmann method (LBM). The proposed methodology is based on a linear equivalence with standard discretizations of the entropy equation, which, as a characteristic variable of the Euler system, allows efficiently decoupling the energy equation with the LBM. To this extent, any LBM scheme is equivalently written under a finite-volume formulation involving fluxes, which are further included in the total energy equation as numerical corrections. The viscous heat production is implicitly considered thanks to the knowledge of the LBM momentum flux. Three models are subsequently derived: a first-order upwind, a Lax–Wendroff, and a third-order Godunov-type schemes. They are assessed on standard academic test cases: a Couette flow, entropy spot and vortex convections, a Sod shock tube, several two-dimensional Riemann problems, and a shock–vortex interaction. Three key features are then exhibited: (1) the models are conservative by construction, recovering correct jump relations across shock waves; (2) the stability and accuracy of entropy modes can be explicitly controlled; and (3) the low dissipation of the LBM for isentropic phenomena is preserved.

Безынтерполяционный LBM на неравномерных сетках

Lattice Boltzmann method (LBM) is a numerical scheme for solving fluid dynamics problems. One of the important and actively developing areas of LBM is the correct construction of the scheme on non-uniform spatial grids. With non-uniform grids the total number of calculations can be significantly reduced. However, at the moment, the construction of an LBM scheme near the boundary of grids with different spatial steps inevitably requires data interpolation, which can reduce the LBM approximation order and lead to violation of conservation laws. In this work, for the first time, we have developed and tested a method for constructing an athermal node-based LBM on non-uniform grids without interpolation, with the same time step for grids of different scales. The method based on a two-stage transformation of populations corresponding to different on-grid stencils.

LBM simulation of heat transfer processes inside GIS capsule filled with different insulation gas mixtures

Lattice Boltzmann method (LBM) scheme is adopted to simulate the heat transfer processes inside isolated phase and three-phase enclosure gas insulated switchgear (GIS) capsules. D2Q9 fluid flow and thermal Lattice Boltzmann equations (LBEs) are solved through streaming and collision processes. Buoyancy force driven by natural convection is considered by the Boussinesq model. Power losses and thermal boundary conditions are obtained from closed formulas. Numerical results by LBM simulations are compared with those from physical experiments and conventional CFD simulations. Results show that LBM scheme could provide a simple and efficient alternative tool in the flow-thermal simulation of GIS equipment with desirable accuracy. Thermal performances of GIS capsule filled with different insulation gas mixtures (SF6, N2, CF4, CO2, CF3I) are then analyzed. Results show that upper conductors of three-phase GIS has larger temperature rise than lower conductor due to the natural convection effect. Conductor temperature rise inside gas mixture increases with the increase of second gas (N2, CO2, CF4) fraction. The thermal performance of GIS capsule could be reduced with the presentation of insulation gas mixtures.

Lattice Boltzmann Simulation of Plane Strain Problems

AbstractThe Lattice Boltzmann Method (LBM), e.g. in [3] and [4], can be interpreted as an alternative method for the numerical solution of partial differential equations. The LBM is usually applied to solve fluid flows. However, the interpretation of the LBM as a general numerical tool, allows to extend the LBM to solid mechanics as well. In this spirit, the LBM has been studied in recent years. First publications [5], [6] present a LBM scheme for the numerical solution of the dynamic behavior of a linear elastic solid under simplified deformation assumptions. For so‐called anti‐plane shear deformation, the only non‐zero displacement component is governed by the two‐dimensional wave equation. In this work, the existing LBM for the two‐dimensional wave equation is extended to more general plane strain problems. The algorithm reduces the plane strain problem to the solution of two separate wave equations for the volume dilatation and the non‐zero component of the rotation vector, respectively.

Lattice Boltzmann method for simulation of diffusion magnetic resonance imaging physics in multiphase tissue models.

We report an implementation of the lattice Boltzmann method (LBM) to integrate the Bloch-Torrey equation, which describes the evolution of the transverse magnetization vector and the fate of the signal of diffusion magnetic resonance imaging (dMRI). Motivated by the need to interpret dMRI experiments in biological tissues, and to offset the small time-step limitation of classical LBM, a hybrid LBM scheme is introduced and implemented to solve the Bloch-Torrey equation. A membrane boundary condition is presented which is able to accurately represent the effects of thin curvilinear membranes typically found in biological tissues. As implemented, the hybrid LBM scheme accommodates piece-wise uniform transport, dMRI parameters, periodic and mirroring outer boundary conditions, and finite membrane permeabilities on non-boundary-conforming inner boundaries. By comparing with analytical solutions of limiting cases, we demonstrate that the hybrid LBM scheme is more accurate than the classical LBM scheme. The proposed explicit LBM scheme maintains second-order spatial accuracy, stability, and first-order temporal accuracy for a wide range of parameters. The parallel implementation of the hybrid LBM code in a multi-CPU computer system, as well as on GPUs, is straightforward and efficient. Along with offering certain advantages over finite element or Monte Carlo schemes, the proposed hybrid LBM constitutes a flexible scheme that can by easily adapted to model more complex interfacial conditions and physics in heterogeneous multiphase tissue models and to accommodate sophisticated dMRI sequences.

Discrete effects on the forcing term for the lattice Boltzmann modeling of steady hydrodynamics

This work concerns with the clean inclusion of the forcing term in the lattice Boltzmann method (LBM) for the modeling of non-uniform body forces in steady hydrodynamics. The study is conducted for the two-relaxation-time (TRT) scheme. Here, we consider a simple, but yet sufficiently generic, flow configuration driven by a spatially varying body force based on which we derive the analytical solution of the forced LBM-TRT and compare two force strategies set by first- and second-order expansions. This procedure exactly establishes the macroscopic system satisfied by each force formulation at discrete level. The obtained theoretical results are further verified in two distinct benchmark channel flow problems. Overall, this study shows that the spatial discrete effects posed by the LBM modeling of the force term may come in through two sources. The first one is a defect inherent to LBM, arising from the non-local spatial discretization of the forcing term, and given by the discrete Laplacian of the body force. While it corrupts the discrete momentum balance with a non-linear viscosity dependent term in the single-relaxation-time schemes, this inconsistency is avoided with the TRT scheme, through its free-tunable relaxation degree of freedom Λ. The other error source is unique to the use of second-order force expansions, where its non-zero second-order velocity moment interferes with the discrete momentum balance through a spurious first-order derivative term, leading to several forms of inconsistency in the LBM steady solution. For simulations under the convective scaling it makes the LBM scheme a numerical representation of a differential system distinct from the physical one, whereas under the diffusive scaling it leads to viscosity-dependent numerical errors, which corrupt the otherwise consistent structure of TRT steady-state solutions. In contrast, the defects reported herein are absent with the first-order force expansion scheme operated within the scope of the LBM-TRT model for steady hydrodynamics.

Lattice Boltzmann simulation of wind-driven rainfall on a slope surface

In overland flow modeling, the influences of the slope degree, rain intensity and wind condition can be significant. In this study, the effects of the slope gradient and wind-driven rain were integrated in a nonlinear shallow water model based on the lattice Boltzmann method (LBM). Since slope gradient and wind condition can not only affect the flow rate, but also its momentum, we enhanced the mathematical model and derived two LBM schemes that include their hydrodynamic influence on the surface flow. Then, this model was verified by several typical cases. The results indicated that the momentum component of rainfall should be considered during the overland flow simulation influenced by wind, otherwise it would result in an intrinsic error of approximately [Formula: see text] in simulated results.

Aeroacoustic simulation of broadband sound generated from low-Mach-number flows using a lattice Boltzmann method

The present paper demonstrates the capability of a numerical method based on the lattice Boltzmann method (LBM) with wall-resolved grid to predict the broadband sound generated from the turbulent boundary layer at low Mach numbers. The present method is based on the lattice BGK equation with the D2Q9 and D3Q15 models, and a multi-scale approach using hierarchically refined grids is proposed to efficiently and simultaneously capture the multi-scale phenomena of turbulent eddies near walls and far-field sound waves. Numerical instabilities caused by the lack of grid resolution are suppressed with a fourth-order implicit filtering scheme. This numerical method is discussed in two benchmark problems and an application to the prediction of the broadband sound generated from the turbulent boundary layer. First, the computational accuracy and speed of the LBM scheme are assessed with a pulse-propagating problem. The results indicate that the LBM can achieve accuracy comparable to the fourth-order central scheme with the four-stage Runge-Kutta method for the compressible Navier-Stokes (N-S) equations and compute 12.3 times faster. These findings suggest that the LBM is an efficient computational method for aeroacoustic simulations. Second, the proposed method is validated by simulating the Aeolian tone generated by the flow past a circular cylinder at Reynolds number of 150 and Mach number of 0.2. The present simulation is compared with a compressible N-S simulation using a high-order finite difference scheme in terms of the wave profile and the propagation speed of the tonal sound. This validation result suggests that the present method is available for direct aeroacoustic simulations of low-Mach-number flows. Finally, the capability of the present method to predict the broadband sound is demonstrated by conducting a wall-resolved simulation for the turbulent flow generated by a short separation bubble over an isolated airfoil at Reynolds number of 2.0×105 and Mach number of 0.058. This simulation shows a good agreement with measurements of the surface pressure distributions, the wake velocity profiles, and the far-field sound spectrum. In contrast to hybrid approaches based on the incompressible N-S equations, the present method can accurately predict the broadband sound in the high-frequency range by simulating the acoustic scattering on the airfoil.

Design and thermal insulation performance analysis of endothermic opacifiers doped silica aerogels

In order to reduce the thermal radiation and further deter the heat transfer in silica aerogel thermal insulation materials, the endothermic opacifier is designed with the shell of SiC and the core of phase change material (PCM) in this paper. Then the transient heat transfer characteristics of the endothermic opacifiers doped silica aerogel (EOSA) are numerically investigated and its short-term thermal insulation performance is estimated with a modified lattice Boltzmann method (LBM) model. The modified LBM scheme is established with a new source term to remedy the effect of the variation in the volumetric heat capacity of the specimen, and this scheme is fully validated based on the analytical results and available experimental data. The results demonstrate that adding the endothermic opacifier to silica aerogels results in a significant heat transfer delay and an improvement in short-term thermal insulation performance by suppressing radiation heat transfer and solid-liquid phase change heat absorption. With the same doping amount, the endothermic opacifiers doped silica aerogel exhibits more excellent short-term thermal insulation performance and a lower density than that of SiC opacifiers doped silica aerogels (SOSA). It indicates that the EOSA possesses great potential to be a candidate as the lightweight and efficient thermal insulation material at high temperature. Besides, exhaustively parametric studies based on the EOSA, including doping amount, latent heat of PCM, phase change temperature and endothermic opacifier particle size are conducted, and several design rules are established. This analysis could serve as a framework to develop and design silica aerogel composites with low-density and excellent short-term thermal insulation performance.

Modelling the Sintering Neck Growth Process of Metal Fibers under the Surface Diffusion Mechanism Using the Lattice Boltzmann Method

In this paper, the sintering neck growth process of metal fibers under the surface diffusion mechanism is simulated by using the Lattice Boltzmann method (LBM). The surface diffusion model is developed considering the geometrical characteristic of metal fibers. Then, the LBM scheme is constructed for solving the developed surface diffusion model. The sintering neck growth process of two metal fibers with different fiber angles is simulated by LBM. The simulated morphologies of sintering metal fibers well agree with ones obtained by experiments. Moreover, the numerical simulation results show that the sintering neck radius of two metal fibers is increased with the increase of fiber angle, which implies that the initial geometrical characteristic plays an important role in the sintering neck formation and growth of metal fibers.

A comparative study between D2Q9 and D2Q5 lattice Boltzmann scheme for mass transport phenomena in porous media

Characterization of different transport phenomena through porous media represents a key factor to improve mechanical or electrical properties of materials. Porous materials have several applications in different fields such as geological sciences, energy sciences or biological applications. Considering the difficulty to carry out experimental studies in porous media, these transport phenomena are more feasible to describe with computational tools, i.e., to compute the involved parameters. In this scenario, the Lattice Boltzmann Method (LBM) appears as a powerful tool to solve different transport phenomena at micro- and meso-scale. The fluid flow behavior, analyzed with LBM, is commonly solved using the D2Q9 scheme. This scheme has shown a reliable solution for fluid flow problems. On the other hand, the mass transport phenomena are recommended to be solved using the D2Q5 scheme. However, there is not yet a comparative, detailed and complete study of the impact of using such schemes in porous media. The purpose of this study is to analyze the impact of using the D2Q5 and D2Q9 LBM scheme in the computation of mass concentration within porous media considering a constant particle size. Parameters such as porosity and tortuosity are also considered in this study.

Coupled lattice Boltzmann simulation method for bidomain type models in cardiac electrophysiology with multiple time-delays

In this work, we propose a mathematical model of the cardiac electrophysiology which take into account time delays in signal transmission, in order to capture the whole activities of macro- to micro-scale transport processes, and use this model to analyze the propagation of electrophysiological waves in the heart by using a developed coupling Lattice Boltzmann Method (LBM). The propagation of electrical activity in the heart is mathematically modeled by a modified bidomain system. As transmembrane potential evolves, the domain has anisotropical properties which are transposed into intracellular and extracellular conductivity. The new bidomain system is a multi-scale, stiff and strongly nonlinear coupled reaction-diffusion model in the shape of a set of ordinary differential equations coupled with a set of partial differential equations with multiple time delays. Due to delays, dynamic and geometry complexity, numerical simulation and implementation of this type of coupled systems are very ambitious mathematical and computational problems but are crucial in several biomedical applications. We introduce a modified LBM scheme, reliable, efficient, stable and easy to implement in the context of such bidomain systems with multiple time delays. Numerical tests to confirm effectiveness and accuracy of our approach are provided and, the influence and impact of delays to restore normal heart rhythm are analyzed.