Multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: Modeling, analysis, and elements

Acoustic cavitation dynamics of bubble clusters near solid wall: A multiphase lattice Boltzmann approach

Understanding the behavior of cavitation bubble clusters in an acoustic field is crucial for advancing the study of acoustic cavitation. This study uses the multi-relaxation time lattice Boltzmann method (MRT-LBM) to simulate the dynamics of cavitation bubble clusters near a wall, offering new insights into complex cavitation phenomena. The effectiveness of MRT-LBM was verified through thermodynamic consistency, mesh independence, and comparison with the K-M equation solution. The study focuses on the effects of bubble cluster position, acoustic frequency, amplitude, and bubble number on cavitation dynamics. The results found that the impact of bubble cluster proximity to solid boundaries, where smaller offsets result in stronger cavitation effects, significantly increasing wall pressure and jet velocity. The analysis also reveals that low frequencies promote complete bubble collapse, while high frequencies enhance jet velocity but weaken pressure waves. Additionally, higher amplitudes increase jet velocity but disperse energy, reducing wall pressure. Frequency spectrum analysis of wall pressure pw and velocity uw further uncovers significant differences in their spectra and how they influence cavitation intensity, finding that frequency and amplitude are key factors in balancing pressure and jet velocity. These findings underscore the importance of optimizing frequency and amplitude to enhance cavitation effects, which can improve applications relying on acoustic cavitation.

Investigating Cell-Induced Mixing Dynamics in Microfluidic Droplets Using the Lattice Boltzmann Method

This study investigates the impact of cell dynamics on mixing efficiency within a microfluidic droplet, emphasizing the relationship between cell motion, deformability, and resultant asymmetry in velocity and concentration fields. Simulations were conducted for droplets containing encapsulated cells at varying Peclet numbers (Pe = 100-800) and coupling constants (kt = 0.0025, 0.005, 0.0075). The mixing index was significantly enhanced by the presence of the encapsulated cell, particularly at high Peclet numbers, where cell-induced disturbances in the velocity field disrupted symmetrical flow patterns, improving mixing. An asymmetry index quantified deviations in velocity and concentration fields caused by cell motion. Results revealed a complex interplay between cell deformability and fluid-cell interactions. Lower coupling constants corresponded to weaker velocity field asymmetry but, paradoxically, higher mixing indices. This counterintuitive finding was examined by analyzing the asymmetry in the x-component of the velocity field, aligned with the primary concentration gradient. Disturbances in this direction enhanced convective transport across the diffusion interface, crucial for efficient mixing. The study also examined cell trajectory and membrane deformation during droplet generation. Cells with moderate deformability exhibited greater off-center movement, leading to increased rotational dynamics and chaotic flow patterns conducive to enhanced mixing. In contrast, cells with higher or lower deformability followed more constrained paths, resulting in less effective mixing. These findings suggest optimal mixing within microfluidic droplets occurs when cells exhibit moderate deformability, balancing fluid-cell coupling and the ability to induce flow field asymmetry. These insights could inform the design of microfluidic systems for applications requiring precise mixing control, such as biomedical diagnostics and chemical synthesis.

Subgrid-scale model for large eddy simulations of incompressible turbulent flows within the lattice Boltzmann framework

Large eddy simulations are a popular method for turbulent simulations because of their accuracy and efficiency. In this paper, a coupling algorithm is proposed that combines nonequilibrium moments (NM) and the volumetric strain-stretching (VSS) model within the framework of the lattice Boltzmann method (LBM). This algorithm establishes a relation between the NM and the eddy viscosity by using a special calculation form of the VSS model and Chapman-Enskog analysis. The coupling algorithm is validated in three typical flow cases: freely decaying homogeneous isotropic turbulence, homogeneous isotropic turbulence with body forces, and incompressible turbulent channel flow at Re_{τ}=180. The results show that the coupling algorithm is accurate and efficient when compared with the results of direct numerical simulations. Using calculation format of the eddy viscosity, a uniform calculation format is used for each grid point of the flow field during the modeling process. The modeling process uses only the local distribution function to obtain the local eddy viscosity coefficients without any additional processing on the boundary, while optimizing the memory access process to fit the inherent parallelism of the LBM. The efficiency of the calculation is improved by about 20% compared to the central difference method within the lattice Boltzmann framework for calculating the eddy viscosity.

Three-dimensional solidification modeling of various materials using the lattice Boltzmann method with an explicit enthalpy equation

Based on the mesoscopic scale, the lattice Boltzmann method (LBM) with an enthalpy-based model represented in the form of distribution functions is widely used in the liquid-solid phase transition process of energy storage materials due to its direct and relatively accurate characterization of the presence of latent heat of solidification. However, since the enthalpy distribution function itself contains the physical properties of the material, these properties are transferred along with the enthalpy distribution function during the streaming process. This leads to deviations between the enthalpy-based model when simulating the phase transition process of different materials mixed and the actual process. To address this issue, in this paper, we construct an enthalpy-based model for different types of materials. For multiple materials, various forms of enthalpy distribution functions are employed. This method still uses the form of enthalpy distribution functions for collisions and streaming processes among the same type of substance, while for heat transfer between different materials, it avoids the direct transfer of enthalpy distribution functions and instead applies a source term to the enthalpy distribution functions, characterizing the heat transfer between different materials through the energy change before and after mixing based on the temperature. To verify the accuracy of the method proposed in this paper, a detailed solidification model for two different materials is constructed using the example of water droplets solidifying in air, and the results are compared with experimental outcomes. The results of the simulation show that the model constructed in this paper is largely in line with the actual process.

Phase-field-based lattice Boltzmann method for two-phase flows with interfacial mass or heat transfer

In this work, we develop a phase-field-based lattice Boltzmann (LB) method for a two-scalar model of the two-phase flows with interfacial mass or heat transfer. Through the Chapman-Enskog analysis, we show that the present LB method can correctly recover the governing equations for phase field, flow field, and concentration or temperature field. In particular, to derive the two-scalar equations for the mass or heat transfer, we propose a new LB model with an auxiliary source distribution function to describe the extra flux terms, and the discretizations of some derivative terms can be avoided. The accuracy and efficiency of the present LB method are also tested through several benchmark problems, and the influence of mass or heat transfer on the fluid viscosity is further considered by introducing an exponential relation. The numerical results show that the present LB method is suitable for the two-phase flows with interfacial mass or heat transfer.

Macroscopic finite-difference scheme and modified equations of the general propagation multiple-relaxation-time lattice Boltzmann model

In this paper we first present the general propagation multiple-relaxation-time lattice Boltzmann (GPMRT-LB) model and obtain the corresponding macroscopic finite-difference (GPMFD) scheme on conservative moments. Then based on the Maxwell iteration method, we conduct the analysis on the truncation errors and modified equations (MEs) of the GPMRT-LB model and GPMFD scheme at both diffusive and acoustic scalings. For the nonlinear anisotropic convection-diffusion equation (NACDE) and Navier-Stokes equations (NSEs), we also derive the first- and second-order MEs of the GPMRT-LB model and GPMFD scheme. In particular, for the one-dimensional convection-diffusion equation (CDE) with the constant velocity and diffusion coefficient, we can develop a fourth-order GPMRT-LB (F-GPMRT-LB) model and the corresponding fourth-order GPMFD (F-GPMFD) scheme at the diffusive scaling. Finally, three benchmark problems, the Gauss hill problem, the CDE with nonlinear convection and diffusion terms, and the Taylor-Green vortex flow in two-dimensional space, are used to test the GPMRT-LB model and GPMFD scheme, and it is found that the numerical results not only are in good agreement with corresponding analytical solutions, but also have a second-order convergence rate in space. Additionally, a numerical study on one-dimensional CDE also demonstrates that the F-GPMRT-LB model and F-GPMFD scheme can achieve a fourth-order accuracy in space, which is consistent with our theoretical analysis.

Auto-ejection of liquid from a nozzle

Auto-ejection of liquid is an important process in engineering applications, and is also very complicated since it involves interface moving, deforming, and jet breaking up. In this work, a theoretical velocity of meniscus at nozzle exit is first derived, which can be used to analyze the critical condition for auto-ejection of liquid. Then a consistent and conservative axisymmetric lattice Boltzmann (LB) method is proposed to study the auto-ejection process of liquid jet from a nozzle. We test the LB model by conducting some simulations, and find that the numerical results agree well with the theoretical and experimental data. We further consider the effects of contraction ratio, length ratio, contact angle, and nozzle structure on the auto-ejection, and observe some distinct phenomena during the ejection process, including the deformation of meniscus, capillary necking, and droplet pinch off. Finally, the results reported in the present work may play an instructive role on the design of droplet ejectors and the understanding of jetting dynamics in microgravity environment.

Macroscopic finite-difference scheme based on the mesoscopic regularized lattice-Boltzmann method

In this paper, we develop a macroscopic finite-difference scheme from the mesoscopic regularized lattice Boltzmann (RLB) method to solve the Navier-Stokes equations (NSEs) and convection-diffusion equation (CDE). Unlike the commonly used RLB method based on the evolution of a set of distribution functions, this macroscopic finite-difference scheme is constructed based on the hydrodynamic variables of NSEs (density, momentum, and strain rate tensor) or macroscopic variables of CDE (concentration and flux), and thus shares low memory requirement and high computational efficiency. Based on an accuracy analysis, it is shown that, the same as the mesoscopic RLB method, the macroscopic finite-difference scheme also has a second-order accuracy in space. In addition, we would like to point out that compared with the RLB method and its equivalent macroscopic numerical scheme, the present macroscopic finite-difference scheme is much simpler and more efficient since it is only a two-level system with macroscopic variables. Finally, we perform some simulations of several benchmark problems, and find that the numerical results are not only in agreement with analytical solutions, but also consistent with the theoretical analysis.

Theoretical and numerical study on the well-balanced regularized lattice Boltzmann model for two-phase flow

In the multiphase flow simulations based on the lattice Boltzmann equation (LBE), the spurious velocity near the interface and the inconsistent density properties are frequently observed. In this paper, a well-balanced regularized lattice Boltzmann (WB-RLB) model with Hermite expansion up to third order is developed for two-phase flows. To this end, the equilibrium distribution function and the modified force term proposed by Guo [Phys. Fluids 33, 031709 (2021)1070-663110.1063/5.0041446] are directly introduced into the regularization of the transformed distribution functions when considering the LBE with trapezoidal integral. First, to give a detailed comparison of the well-balanced lattice Boltzmann equation (WB-LBE), WB-RLB, and second-order mixed difference scheme (SOMDS) proposed by Lee and Fischer [Phys. Rev. E 74, 046709 (2006)1539-375510.1103/PhysRevE.74.046709], the theoretical analyses on the force balance of LBE with two different gradient operators, isotropic central scheme (ICS) and SOMDS, as well as the numerical simulations of the stationary droplet are carried out. The force analysis shows that SOMDS can achieve a higher accuracy than ICS for the force balance, which has been validated in the simulations of stationary droplet cases. For the stationary droplet cases, all three models (WB-LBE, WB-RLB, and SOMDS) can capture the physical equilibrium state even at a large density ratio of 1000. Also, the numerical investigations of the WB-RLB model with third-order expansion (WB-RLB3) demonstrate that adjusting the relaxation parameters of the third-order moment can further improve the accuracy and stability of the WB-RLB model. Then, both the droplet coalescence and the phase separation cases are investigated with considering the effect of different interface thickness, which demonstrates that the performance of the WB-RLB for the two-phase dynamic problems is still quite well, and it exhibits better numerical stability when compared with the WB-LBE. In addition, the contact angle problem is investigated by the present WB-RLB model; the numerical results show that the predicted values of the contact angles agree well with the analytical solutions, but the well-balance property is not validated, especially near the three-phase junction. Overall, the present WB-RLB model exhibits excellent numerical accuracy and stability for both static and dynamic interface problems.

Rectangular multiple-relaxation-time lattice Boltzmann method for the Navier-Stokes and nonlinear convection-diffusion equations: General equilibrium and some important issues

In this paper, we develop a general rectangular multiple-relaxation-time lattice Boltzmann (RMRT-LB) method for the Navier-Stokes equations (NSEs) and nonlinear convection-diffusion equation (NCDE) by extending our recent unified framework of the multiple-relaxation-time lattice Boltzmann (MRT-LB) method [Chai and Shi, Phys. Rev. E 102, 023306 (2020)10.1103/PhysRevE.102.023306], where an equilibrium distribution function (EDF) [Lu et al., Philos. Trans. R. Soc. A 369, 2311 (2011)10.1098/rsta.2011.0022] on a rectangular lattice is utilized. The anisotropy of the lattice tensor on a rectangular lattice leads to anisotropy of the third-order moment of the EDF, which is inconsistent with the isotropy of the viscous stress tensor of the NSEs. To eliminate this inconsistency, we extend the relaxation matrix related to the dynamic and bulk viscosities. As a result, the macroscopic NSEs can be recovered from the RMRT-LB method through the direct Taylor expansion method. Whereas the rectangular lattice does not lead to the change of the zero-, first- and second-order moments of the EDF, the unified framework of the MRT-LB method can be directly applied to the NCDE. It should be noted that the RMRT-LB model for NSEs can be derived on the rDdQq (q discrete velocities in d-dimensional space, d≥1) lattice, including rD2Q9, rD3Q19, and rD3Q27 lattices, while there are no rectangular D3Q13 and D3Q15 lattices within this framework of the RMRT-LB method. Thanks to the block-lower triangular relaxation matrix introduced in the unified framework, the RMRT-LB versions (if existing) of the previous MRT-LB models can be obtained, including those based on raw (natural) moment, central moment, Hermite moment, and central Hermite moment. It is also found that when the parameter c_{s} is an adjustable parameter in the standard or rectangular lattice, the present RMRT-LB method becomes a kind of MRT-LB method for the NSEs and NCDE, and the commonly used MRT-LB models on the DdQq lattice are only its special cases. We also perform some numerical simulations, and the results show that the present RMRT-LB method can give accurate results and also have a good numerical stability.

Lattice Boltzmann finite-difference-based model for fully nonlinear electrohydrodynamic deformation of a liquid droplet

Electric field-induced flows involving multiple fluid components with a range of different electrical properties are described by the coupled Taylor-Melcher leaky-dielectric model. We present a lattice Boltzmann (LB)-finite difference (FD) method-based hybrid framework to solve the complete Taylor-Melcher leaky-dielectric model considering the nonlinear surface charge convection effects. Unlike the existing LB-based models, we treat the interfacial discontinuities using direction-specific continuous gradients, which prevents the miscalculation arising due to volumetric gradients without directional derivatives, simultaneously maintaining the electroneutrality of the bulk. While fluid transport is recovered through the LB method using a multiple relaxation time (MRT) scheme, the FD method with a central difference scheme is applied to discretize the charge transport equation at the interface, in addition to the electric field governing equations in the bulk and at the interface. We apply the developed numerical model to study the different regimes of droplet deformation due to an external electric field. Similar to the existing analytical and other numerical models, excluding the surface charge convection (SCC) term from the charge transport equation, the present methodology has shown excellent agreement with the existing literature. In addition, the effect of SCC in each of the regimes is analyzed. With the present numerical model, we observe a strong presence of SCC in the oblate deformation regime, contrary to the weak effect on prolate deformations. We further discuss the reason behind such differences in the magnitude of nonlinearity induced by the SCC in all the regimes of deformation.

Fourth-order multiple-relaxation-time lattice Boltzmann model and equivalent finite-difference scheme for one-dimensional convection-diffusion equations

In this paper, we first develop a fourth-order multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the one-dimensional convection-diffusion equation (CDE) with the constant velocity and diffusion coefficient, where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is used. We also perform the Chapman-Enskog analysis to recover the CDE from the MRT-LB model. Then an explicit four-level finite-difference (FLFD) scheme is derived from the developed MRT-LB model for the CDE. Through the Taylor expansion, the truncation error of the FLFD scheme is obtained, and at the diffusive scaling, the FLFD scheme can achieve the fourth-order accuracy in space. After that, we present a stability analysis and derive the same stability condition for the MRT-LB model and FLFD scheme. Finally, we perform some numerical experiments to test the MRT-LB model and FLFD scheme, and the numerical results show that they have a fourth-order convergence rate in space, which is consistent with our theoretical analysis.

Improved hybrid Allen-Cahn phase-field-based lattice Boltzmann method for incompressible two-phase flows

In this work we develop an improved phase-field based lattice Boltzmann (LB) method where a hybrid Allen-Cahn equation (ACE) with a flexible weight instead of a global weight is used to suppress the numerical dispersion and eliminate the coarsening phenomenon. Then two LB models are adopted to solve the hybrid ACE and the Navier-Stokes equations, respectively. Through the Chapman-Enskog analysis, the present LB model can correctly recover the hybrid ACE, and the macroscopic order parameter used to label different phases can be calculated explicitly. Finally, the present LB method is validated by five tests, including the diagonal translation of a circular interface, two stationary bubbles with different radii, a bubble rising under the gravity, the Rayleigh-Taylor instability in two-dimensional and three-dimensional cases, and the three-dimensional Plateau-Rayleigh instability. The numerical results show that the present LB method has a superior performance in reducing the numerical dispersion and the coarsening phenomenon.

Unified directional parabolic-accurate lattice Boltzmann boundary schemes for grid-rotated narrow gaps and curved walls in creeping and inertial fluid flows

The goal of this work is to advance the characteristics of existing lattice Boltzmann Dirichlet velocity boundary schemes in terms of the accuracy, locality, stability, and mass conservation for arbitrarily grid-inclined straight walls, curved surfaces, and narrow fluid gaps, for both creeping and inertial flow regimes. We reach this objective with two infinite-member boundary classes: (1) the single-node "Linear Plus" (LI^{+}) and (2) the two-node "Extended Multireflection" (EMR). The LI^{+} unifies all directional rules relying on the linear combinations of up to three pre- or postcollision populations, including their "ghost-node" interpolations and adjustable nonequilibrium approximations. On this basis, we propose three groups of LI^{+} nonequilibrium local corrections: (1) the LI_{1}^{+} is parametrized, meaning that its steady-state solution is physically consistent: the momentum accuracy is viscosity-independent in Stokes flow, and it is fixed by the Reynolds number (Re) in inertial flow; (2) the LI_{3}^{+} is parametrized, exact for arbitrary grid-rotated Poiseuille force-driven Stokes flow and thus most accurate in porous flow; and (3) the LI_{4}^{+} is parametrized, exact for pressure and inertial term gradients, and hence advantageous in very narrow porous gaps and at higher Reynolds range. The directional, two-relaxation-time collision operator plays a crucial role for all these features, but also for efficiency and robustness of the boundary schemes due to a proposed nonequilibrium linear stability criterion which reliably delineates their suitable coefficients and relaxation space. Our methodology allows one to improve any directional rule for Stokes or Navier-Stokes accuracy, but their parametrization is not guaranteed. In this context, the parametrized two-node EMR class enlarges the single-node schemes to match exactness in a grid-rotated linear Couette flow modeled with an equilibrium distribution designed for the Navier-Stokes equation (NSE). However, exactness of a grid-rotated Poiseuille NSE flow requires us to perform (1) the modification of the standard NSE term for exact bulk solvability and (2) the EMR extension towards the third neighbor node. A unique relaxation and equilibrium exact configuration for grid-rotated Poiseuille NSE flow allows us to classify the Galilean invariance characteristics of the boundary schemes without any bulk interference; in turn, its truncated solution suggests how, when increasing the Reynolds number, to avoid a deterioration of the mass-leakage rate and momentum accuracy due to a specific Reynolds scaling of the kinetic relaxation collision rate. The optimal schemes and strategies for creeping and inertial regimes are then singled out through a series of numerical tests, such as grid-rotated channels and rotated Couette flow with wall-normal injection, cylindrical porous array, and Couette flow between concentric cylinders, also comparing them against circular-shape fitted FEM solutions.

Multiple-distribution-function lattice Boltzmann method for convection-diffusion-system-based incompressible Navier-Stokes equations

In this paper, a multiple-distribution-function lattice Boltzmann method (MDF-LBM) with a multiple-relaxation-time model is proposed for incompressible Navier-Stokes equations which are considered as coupled convection-diffusion equations. Through direct Taylor expansion analysis, we show that the Navier-Stokes equations can be recovered correctly from the present MDF-LBM, and additionally, it is also found that the velocity and pressure can be directly computed through the zero and first-order moments of the distribution function. Then in the framework of the present MDF-LBM, we develop a locally computational scheme for the velocity gradient in which the first-order moment of the nonequilibrium distribution is used; this scheme is also extended to calculate the velocity divergence, strain rate tensor, shear stress, and vorticity. Finally, we also conduct some simulations to test the MDF-LBM and find that the numerical results not only agree with some available analytical and numerical solutions but also have a second-order convergence rate in space.

Thermal lattice Boltzmann model for liquid-vapor phase change

The lattice Boltzmann method is adopted to solve the liquid-vapor phase change problems in this article. By modifying the collision term for the temperature evolution equation, a thermal lattice Boltzmann model is constructed. As compared with previous studies, the most striking feature of the present approach is that it could avoid the calculations of both the Laplacian term of temperature [∇·(κ∇T)] and the gradient term of heat capacitance [∇(ρc_{v})]. In addition, since the present approach adopts a simple linear equilibrium distribution function, it is possible to use the D2Q5 lattice for the two-dimensional cases considered here. Thus, the present model is more efficient than previous models in which the lattice is usually limited to the D2Q9. The proposed model is first validated by the problems of droplet evaporation in open space and droplet evaporation on a heated surface, and the numerical results show good agreement with the analytical results and the finite difference method. Then it is used to model the nucleate boiling problem, and the relationship between detachment bubble diameter and gravitational acceleration obtained with the present approach fits well with previous works.

Consistent and conservative phase-field-based lattice Boltzmann method for incompressible two-phase flows

In this work, we consider a general consistent and conservative phase-field model for the incompressible two-phase flows. In this model, not only the Cahn-Hilliard or Allen-Cahn equation can be adopted, but also the mass and the momentum fluxes in the Navier-Stokes equations are reformulated such that the consistency of reduction, consistency of mass and momentum transport, and the consistency of mass conservation are satisfied. We further develop a lattice Boltzmann (LB) method, and show that through the direct Taylor expansion, the present LB method can correctly recover the consistent and conservative phase-field model. Additionally, if the divergence of the extra momentum flux is seen as a force term, the extra force in the present LB method would include another term which has not been considered in the previous LB methods. To quantitatively evaluate the incompressibility and the consistency of the mass conservation, two statistical variables are introduced in the study of the deformation of a square droplet, and the results show that the present LB method is more accurate. The layered Poiseuille flow and a droplet spreading on an ideal wall are further investigated, and the numerical results are in good agreement with the analytical solutions. Finally, the problems of the Rayleigh-Taylor instability, a single rising bubble, and the dam break with the high Reynolds numbers and/or large density ratios are studied, and it is found that the present consistent and conservative LB method is robust for such complex two-phase flows.

Multiple-relaxation-time finite-difference lattice Boltzmann model for the nonlinear convection-diffusion equation

In this paper, a multiple-relaxation-time finite-difference lattice Boltzmann method (MRT-FDLBM) is developed for the nonlinear convection-diffusion equation (NCDE). Through designing the equilibrium distribution function and the source term properly, the NCDE can be recovered exactly from MRT-FDLBM. We also conduct the von Neumann stability analysis on the present MRT-FDLBM and its special case, i.e., single-relaxation-time finite-difference lattice Boltzmann method (SRT-FDLBM). Then, a simplified version of MRT-FDLBM (SMRT-FDLBM) is also proposed, which can save about 15% computational cost. In addition, a series of real and complex-value NCDEs, including the isotropic convection-diffusion equation, Burgers-Fisher equation, sine-Gordon equation, heat-conduction equation, and Schrödinger equation, are used to test the performance of MRT-FDLBM. The results show that both MRT-FDLBM and SMRT-FDLBM have second-order convergence rates in space and time. Finally, the stability and accuracy of five different models are compared, including the MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method [H. Wang, B. Shi et al., Appl. Math. Comput. 309, 334 (2017)10.1016/j.amc.2017.04.015], and the lattice Boltzmann method (LBM). The stability tests show that the sequence of stability from high to low is as follows: MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, the previous finite-difference lattice Boltzmann method, and LBM. In most of the precision test results, it is found that the order from high to low of precision is MRT-FDLBM, SMRT-FDLBM, SRT-FDLBM, and the previous finite-difference lattice Boltzmann method.

Multiple-relaxation-time lattice Boltzmann model-based four-level finite-difference scheme for one-dimensional diffusion equations

In this paper, we first present a multiple-relaxation-time lattice Boltzmann (MRT-LB) model for one-dimensional diffusion equation where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is considered. Then through the theoretical analysis, we derive an explicit four-level finite-difference scheme from this MRT-LB model. The results show that the four-level finite-difference scheme is unconditionally stable, and through adjusting the weight coefficient ω_{0} and the relaxation parameters s_{1} and s_{2} corresponding to the first and second moments, it can also have a sixth-order accuracy in space. Finally, we also test the four-level finite-difference scheme through some numerical simulations and find that the numerical results are consistent with our theoretical analysis.

Lattice Boltzmann model for conjugate heat transfer across thin walls

A lattice Boltzmann (LB) model with an efficient and accurate interface treatment for conjugate heat transfer across a thin wall between two different media is developed. The proposed interface treatment avoids fine meshing and computation within the thin layer; instead, the energy balance within the thin layer and the conjugate conditions on each interface are utilized to construct explicit updating schemes for the microscopic distribution functions of the LB model at the interior lattice nodes of the two media next to the thin layer. The proposed interface schemes reduce to the standard interface scheme for conjugate conditions in the literature in the limit of zero thickness of the thin layer, and thus it can be considered a more general interface treatment. A simplified version of the interface treatment is also proposed when the heat flux variation along the tangential direction of the thin layer is negligible. Three representative numerical tests are conducted to verify the applicability and accuracy of the proposed interface schemes. The results demonstrate that the intrinsic second-order accuracy of the LB model is preserved with the proposed interface schemes for thin layers with constant tangential fluxes, while for general situations with varying tangential fluxes, first-order accuracy is obtained. This interface treatment within the LB framework is attractive in conjugate heat transfer modeling involving thin layers for its simplicity, accuracy, and significant reduction in computational resources.