Multiple Relaxation Time Lattice Boltzmann Research Articles

The computation of strain rate tensor in multiple-relaxation-time lattice Boltzmann model

The multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is an important class of LB model with lots of advantages over the traditional single-relaxation-time (SRT) LB model. Generally, the computation of strain rate tensor is crucial for the MRT-LB simulations of some complex flows. At present, only two formulae are available to compute the strain rate tensor in the MRT LB model. One is to compute the strain rate tensor using the non-equilibrium parts of macroscopic moments (Yu formula). The other is to compute the strain rate tensor using the non-equilibrium parts of density distribution functions (Chai formula). The mathematical expressions of these two formulae are so different that we do not know which formula to choose for computing the strain rate tensor in the MRT LB model. To overcome this problem, this paper presents a theoretical study of the relationship between Chai and Yu formulae. The results show that the Yu formula can be deduced from the Chai formula, although they have their own advantages and disadvantages. In particular, the Yu formula is computationally more efficient, while the Chai formula is applicable to more lattice patterns of the MRT LB models. Furthermore, the derivation of the Yu formula in a particular lattice pattern from the Chai formula is more convenient than that proposed by Yu et al.

Multiple Relaxation Time Lattice Boltzmann Simulation of 2D Natural Convection in a Square Cavity at High Rayleigh Numbers

Multiple Relaxation Time Lattice Boltzmann Simulation of 2D Natural Convection in a Square Cavity at High Rayleigh Numbers

Lattice Boltzmann model capable of mesoscopic vorticity computation.

It is well known that standard lattice Boltzmann (LB) models allow the strain-rate components to be computed mesoscopically (i.e., through the local particle distributions) and as such possess a second-order accuracy in strain rate. This is one of the appealing features of the lattice Boltzmann method (LBM) which is of only second-order accuracy in hydrodynamic velocity itself. However, no known LB model can provide the same quality for vorticity and pressure gradients. In this paper, we design a multiple-relaxation time LB model on a three-dimensional 27-discrete-velocity (D3Q27) lattice. A detailed Chapman-Enskog analysis is presented to illustrate all the necessary constraints in reproducing the isothermal Navier-Stokes equations. The remaining degrees of freedom are carefully analyzed to derive a model that accommodates mesoscopic computation of all the velocity and pressure gradients from the nonequilibrium moments. This way of vorticity calculation naturally ensures a second-order accuracy, which is also proven through an asymptotic analysis. We thus show, with enough degrees of freedom and appropriate modifications, the mesoscopic vorticity computation can be achieved in LBM. The resulting model is then validated in simulations of a three-dimensional decaying Taylor-Green flow, a lid-driven cavity flow, and a uniform flow passing a fixed sphere. Furthermore, it is shown that the mesoscopic vorticity computation can be realized even with single relaxation parameter.

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for axisymmetric thermal flows

Axisymmetric thermal flows in cylindrical systems are widely encountered in engineering practices. Typically, axisymmetric thermal flows belong in three-dimensional (3D) problems. However, taking advantage of the axisymmetric condition, the 3D axisymmetric flows can be reduced to quasi two-dimensional (2D) problems in the meridian plane, which significantly reduces the computational requirements and avoids treating the curved boundary. In recent years, various 2D lattice Boltzmann (LB) models, including single relaxation time LB (SRT-LB, or LBGK) and multiple relaxation time LB (MRT-LB) models, for axisymmetric thermal flows have been proposed. In the LB community, it is well accepted that the MRT-LB is superior to the LBGK in terms of numerical stability. The existing MRT-LB model for axisymmetric thermal flows are developed based on orthogonal basis vectors obtained from the combination of the lattice velocity components, i.e., the transform matrix in the existing MRT-LB is an orthogonal one. Unlike the existing MRT-LB model, in this paper, a non-orthogonal multiple-relaxation-time lattice Boltzmann (MRT-LB) method of simulating axisymmetric thermal flows is proposed. In the proposed MRT-LB method, the velocity field is solved by a D2Q9 discrete velocity set while the temperature by a D2Q5 discrete velocity set. The main advantage of the present MRT-LB model is that the transform matrix of the model is a non-orthogonal one, which is comprised of some proper non-orthogonal basis vectors obtained from the combination of the lattice velocity components. The non-orthogonal transform matrix of the present MRT-LB model contains more zero elements than the classical orthogonal transform matrix, and thus the present MRT-LB model is expected to be more efficient than the existing orthogonal-based MRT-LB model. The equilibrium velocity and temperature moments of the present MRT-LB model are expressed by mapping the equilibrium distribution functions onto their moment spaces through using the non-orthogonal transformation matrix. Also the vectors in the forcing term are modified according to the matrix mapping. Through the Chapman-Enskog analysis, it is demonstrated that the macroscopic governing equations in the cylindrical coordinate can be recovered from the present MRT-LB model. Then several numerical tests, including thermal Womersley flow, Rayleigh-Bnard convection in a vertical cylinder and natural convection in a vertical annulus, are conducted to validate the present model. It is found that the present numerical results are in good agreement with the analytical solutions and/or other numerical results reported in the literature. Numerical stability is also tested, and the results suggest that the present MRT model shows better numerical stability than its LBGK counterpart. Moreover, the numerical results also indicate that the present MRT-LB model is more computationally efficient than the existing MRT-LB model for axisymmetric thermal flow. These findings indicate that the present MRT-LB model can serve as a powerful method of computing the axisymmetric thermal flows.

A multiple-relaxation-time lattice Boltzmann model for the flow and heat transfer in a hydrodynamically and thermally anisotropic porous medium

A multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is proposed for simulating flow and heat transfer in the hydrodynamically and thermally anisotropic porous medium at the representative-elementary-volume (REV) scale. By selecting the appropriate equilibrium distributions, relaxation matrix and discrete force/heat source terms, the present MRT LB model can recover the correct Darcy–Brinkman–Forchheimer and energy equations with anisotropic permeability and thermal conductivity through the Chapman–Enskog procedure. Several natural convection problems in anisotropic porous medium are simulated to validate the present LB model. The corresponding numerical results are in good agreement with data in the available literature. Especially, natural convection in a cavity with two anisotropic porous layers is investigated. The numerical results indicates that, the use of anisotropic porous layer with some optimal parameters can produce higher rate of heat transfer compared with the isotropic porous layer.

Lattice Boltzmann Method for Simulating Disturbed Hemodynamic Characteristics of Blood Flow in Stenosed Human Carotid Bifurcation

The local hemodynamic factor plays a vital role in the formation and progression of atherosclerosis. In this study, we simulated pulsatile flow patterns in the three-dimensional stenosed and normal carotid artery bifurcations throughout a cardiac cycle using the multiple-relaxation-time lattice Boltzmann (MRT-LB) method. Additionally, we investigated the time-varied flow rate and its division ratios between the parent and daughter branches, the multidirectionality of the stress field, and the averaged local energy dissipation rate. The results can be used in computational modeling of carotid artery hemodynamics and further investigation of the relationship between hemodynamics and cardiovascular diseases.

Extended application of lattice Boltzmann method to rarefied gas flow in micro-channels

Abstract Simulation of rarefied gas flow in micro-channels is of great interest owing to its diverse applications in many engineering fields. In this study, a multiple-relaxation-time lattice Boltzmann (MRT-LB) model with a general second-order slip boundary condition is presented to investigate the behaviour of gas flow with a wide range of Knudsen number in micro-channels. With the aid of a Bosanquet-type effective viscosity, the effective relaxation time is correlated with local Knudsen number (Kn) to account for the varying degree of rarefaction effect. Unlike previous studies, the derived accommodation coefficient r for the combined bounce-back/diffusive reflection (CBBDR) boundary condition is dependent on the local Kn, which allows more flexibility to simulate the slip velocity along the channel walls. When compared with results of other methods, such as linearised Boltzmann equation, experimental data, direct simulation Monte Carlo (DSMC) and Information Preservation DSMC (IP-DSMC), it is found that the LB model is capable of capturing the flow behaviour, including the velocity profile, flow rate, pressure distribution and Knudsen minimum of rarefied gas with Kn up to 10. The effect of Knudsen layer (KL) on the velocity of gas flow with a wide range of Kn is also discussed. It is found that KL effect is negligible in the continuum flow and y -independent in the free molecular flow, while in the intermediate range, especially in transition flow, KL effect is significant and particular efforts should be made to capture this effect.

Total enthalpy-based lattice Boltzmann method with adaptive mesh refinement for solid-liquid phase change

A total enthalpy-based lattice Boltzmann (LB) method with adaptive mesh refinement (AMR) is developed in this paper to efficiently simulate solid-liquid phase change problem where variables vary significantly near the phase interface and thus finer grid is required. For the total enthalpy-based LB method, the velocity field is solved by an incompressible LB model with multiple-relaxation-time (MRT) collision scheme, and the temperature field is solved by a total enthalpy-based MRT LB model with the phase interface effects considered and the deviation term eliminated. With a kinetic assumption that the density distribution function for solid phase is at equilibrium state, a volumetric LB scheme is proposed to accurately realize the nonslip velocity condition on the diffusive phase interface and in the solid phase. As compared with the previous schemes, this scheme can avoid nonphysical flow in the solid phase. As for the AMR approach, it is developed based on multiblock grids. An indicator function is introduced to control the adaptive generation of multiblock grids, which can guarantee the existence of overlap area between adjacent blocks for information exchange. Since MRT collision schemes are used, the information exchange is directly carried out in the moment space. Numerical tests are firstly performed to validate the strict satisfaction of the nonslip velocity condition, and then melting problems in a square cavity with different Prandtl numbers and Rayleigh numbers are simulated, which demonstrate that the present method can handle solid-liquid phase change problem with high efficiency and accuracy.

GPGPU-based rising bubble simulations using a MRT lattice Boltzmann method coupled with level set interface capturing

A multiphase Lattice Boltzmann (LB) scheme coupled with a level set interface capturing model is used for the simulation of multiphase flows, and in particular, rising bubbles under moderate and high density and viscosity ratios. We make use of consistent time integration and force discretization schemes in particular for pressure forces along with using multiple relaxation time (MRT) form of the collision in the LB equation which enables us to preserve stability and accuracy for high density and critical Eo numbers. We first present the solution for the standard test of a static bubble in order to show the accuracy of the solution with respect to the Laplace law for pressure and also the spurious velocity level. We present quantitative benchmark computations and error analysis for the 2D rising bubble test cases being further validated against high precision finite element solutions in Hysing et al. (2009). Furthermore, by applying efficient multi-core and many core general purpose GPU (GPGPU) implementations outlines, we demonstrate that the desired parallel scaling characteristics of general LBM solutions are well preserved for the proposed coupled computations. The presented implementations are shown to outperform the available GPU-based phase-field LBM solvers in terms of computational time, turning the scheme into a desirable choice for massive multiphase simulations in three dimensions.

14-velocity and 18-velocity multiple-relaxation-time lattice Boltzmann models for three-dimensional incompressible flows

In this paper, 14-velocity and 18-velocity multiple-relaxation-time (MRT) lattice Boltzmann (LB) models are proposed for three-dimensional incompressible flows. These two models are constructed based on the incompressible LBGK model proposed by He et al. (2004) and the MRT LB model proposed by d’Humières et al. (2002), which have advantages in the computational efficiency and stability, respectively. Through the Chapman–Enskog analysis, the models can recover to three-dimensional incompressible Navier–Stokes equations in the low Mach number limit. To verify the present models, the steady Poiseuille flow, unsteady pulsatile flow and lid-driven cavity flow in three dimensions are simulated. The simulation results agree well with the analytical solutions or the existing numerical results. Moreover, it is found that the present models show higher accuracy than d’Humières et al. model and better stability than He et al. model.

Lattice Boltzmann Simulation of Mixed Convection Heat Transfer in a Driven Cavity with Non-uniform Heating of the Bottom Wall

The goal of this article is to study numerically the mixed convection in a differentially heated lid-driven cavity with non-uniform heating of the bottom wall. The velocity field is solved by a hybrid scheme with multiple relaxation time Lattice Boltzmann (MRT-LBM) model, while the temperature field is obtained by resolution of the energy balance equation using the finite difference method (FDM). First, the model is checked and validated using data from the literature. Validation of the present results with those available in the literature shows a good agreement. A good efficiency in time simulation is confirmed. Thereafter, the model has been applied to mixed convection in a driven cavity with non-uniform heating wall at the fixed Grashof number Gr = 106. It is found that, the heat transfer is weakened as the Richardson number is augmented. For Gr = 106, we note the appearance of secondary vortices at different positions of the cavity corners.

Simulations of Bingham plastic flows with the multiple-relaxation-time lattice Boltzmann model

Fresh cement mortar is a type of workable paste, which can be well approximated as a Bingham plastic and whose flow behavior is of major concern in engineering. In this paper, Papanastasiou’s model for Bingham fluids is solved by using the multiplerelaxation-time lattice Boltzmann model (MRT-LB). Analysis of the stress growth exponent m in Bingham fluid flow simulations shows that Papanastasiou’s model provides a good approximation of realistic Bingham plastics for values of m > 108. For lower values of m, Papanastasiou’s model is valid for fluids between Bingham and Newtonian fluids. The MRT-LB model is validated by two benchmark problems: 2D steady Poiseuille flows and lid-driven cavity flows. Comparing the numerical results of the velocity distributions with corresponding analytical solutions shows that the MRT-LB model is appropriate for studying Bingham fluids while also providing better numerical stability. We further apply the MRT-LB model to simulate flow through a sudden expansion channel and the flow surrounding a round particle. Besides the rich flow structures obtained in this work, the dynamics fluid force on the round particle is calculated. Results show that both the Reynolds number Re and the Bingham number Bn affect the drag coefficients C D , and a drag coefficient with Re and Bn being taken into account is proposed. The relationship of Bn and the ratio of unyielded zone thickness to particle diameter is also analyzed. Finally, the Bingham fluid flowing around a set of randomly dispersed particles is simulated to obtain the apparent viscosity and velocity fields. These results help simulation of fresh concrete flowing in porous media.

Two-dimensional MRT LB model for compressible and incompressible flows

In the paper we extend the Multiple-Relaxation-Time (MRT) Lattice Boltzmann (LB) model proposed in [Europhys. Lett., 2010, 90: 54003] so that it is suitable also for incompressible flows. To decrease the artificial oscillations, the convection term is discretized by the flux limiter scheme with splitting technique. A new model is validated by some well-known benchmark tests, including Riemann problem and Couette flow, and satisfying agreements are obtained between the simulation results and analytical ones. In order to show the merit of LB model over traditional methods, the non-equilibrium characteristics of system are solved. The simulation results are consistent with the physical analysis.

Investigation of Double Diffusive Natural Convection in Presence of Gray Gas Radiation Within a Square Cavity Using Multiple Relaxation Time Lattice Boltzmann Method

The present work proposes a coupling between the hybrid lattice Boltzmann, the finite difference method, and the discrete ordinates method to simulate combined double diffusive convection and volumetric radiation in a square cavity filled with an emitting and absorbing gray gas. The vertical walls are kept at different temperatures and pollutant concentrations, while the horizontal walls are insulated and impermeable. A parametric study is carried out to evaluate the influence of control parameters such as the Lewis number (Le), Planck number (Pl), and reference temperature ratio (Θ0) on the flow field as well as on heat and mass transfer. The results are presented in terms of isotherms, streamlines, isoconcentrations, average Nusselt, and Sherwood numbers. They show that the volumetric radiation accelerates the flow and significantly alters the structure of the dynamic, concentration, and temperature fields, especially when the thermal forces are dominant. On the other hand, when the mass forces are dominant, the flow is slowed down. The influence of radiation is greater on the thermal field than on the dynamic and concentration fields.

Study of double-diffusive natural convection and radiation in an inclined cavity using lattice Boltzmann method

This study deals with the presentation of a numerical investigation of coupled double diffusive convection and volumetric radiation in a tilted and differentially heated square enclosure filled with a gray fluid participating in absorption, emission and non scattering. The numerical procedure is based on a hybrid scheme with multiple relaxation time lattice Boltzmann (MRT-LB) and finite difference method (FDM). The fluid velocity is determined by D2Q9 MRT model and the energy equation is discretized by FDM to compute the temperature field, while the radiative part in the energy equation is calculated by the discrete ordinates method (DOM) with S8 quadrature. The enclosure walls are assumed to be opaque, diffuse and gray. The effects of various parameters such as the Rayleigh number (Ra), the buoyancy number (N), the optical thickness (τ0) and the angle of inclination (φ) on the flow structure and the heat and mass transfer are depicted. The results are reported in terms of streamlines, isotherms, iso-concentrations and averaged Nusselt (Nu) and Sherwood (Sh) numbers. The results show that the volumetric radiation modifies the temperature distribution and structure flow. Generally, the isotherms and iso-concentrations are inclined in the cavity, the flow is more stabilized in presence of cooperating flow case but the multicellular flow is favored in the opposite flows case.

Combined double-diffusive convection and radiation in a square enclosure filled with semitransparent fluid

A 2D numerical assessment of coupled double diffusive convection and volumetric radiation in a differentially heated square enclosure filled with a gray fluid participating in absorption, emission and nonscattering is carried out. Temperatures and concentrations are imposed at the vertical walls, whereas the horizontal walls are insulated and impermeable. All cavity walls are assumed to be opaque, diffuse and gray. The governing equations are solved by a hybrid scheme with multiple relaxation time lattice Boltzmann (MRT-LBM) and finite difference method (FDM). The velocity field is computed by D2Q9 MRT model while the temperature field is determined by resolution of the energy equation using FDM. The radiative source term in the energy equation is calculated by the discrete ordinates method (DOM) with S8 quadrature. A parametric study illustrating the influence of the buoyancy number (N) and the wall emissivity (εi) on the flow field and the heat and mass transfer is performed. Results are presented in terms of isotherms, streamlines, iso-concentrations, Nusselt and Sherwood numbers. Generally the isotherms and iso-concentrations are inclined in the core cavity, the flow is more stabilized in presence of cooperating flow case but opposing flows are found to be a catalyser of the multi-cellular structures.

An Improved MRT Lattice Boltzmann Model for Calculating Anisotropic Permeability of Compressed and Uncompressed Carbon Cloth Gas Diffusion Layers Based on X-Ray Computed Micro-Tomography

The gas diffusion layers (GDLs) in polymer proton exchange membrane fuel cells are under compression in operation. Understanding and then being able to quantify the reduced ability of GDLs to conduct gases due to the compression is hence important in fuel cell design. In this paper, we investigated the change of anisotropic permeability of GDLs under different compressions using the improved multiple-relaxation time (MRT) lattice Boltzmann model and X-ray computed micro-tomography. The binary 3D X-ray images of GDLs under different compressions were obtained using the technologies we developed previously, and the permeability of the GDLs in both through-plane and in-plane directions was calculated by simulating gas flow at micron scale through the 3D images. The results indicated that, in comparison with the single-relaxation time (SRT) lattice Boltzmann model commonly used in the literature, the MRT model is robust and flexible in choosing model parameters. The SRT model can give accurate results only when using a specific relaxation parameter whose value varies with porosity. The simulated results using the MRT model reveal that compression could lead to a significant decrease in permeability in both through-plane and in-plane directions, and that the relationship between the decreased permeability and porosity can be well described by both Kozeny-Carman relation and the equation derived by Tomadakis and Sotirchos (1993, “Ordinary and Transition Rdgime Diffusion in Random Fiber Structure,” AIChE J., 39, pp. 397–412) for porosity in the range from 50% to 85%. Since GDLs compression takes place mainly in the through-plane direction, the results presented in this work could provide an easy way to estimate permeability reduction in both through-plane and in-plane directions when the compressive pressure is known.

Numerical study on particle dispersion and deposition in a scaled ventilated chamber using a lattice Boltzmann method

This study numerically investigates particle dispersion and deposition in a scaled ventilated chamber. Three-dimensional airflow simulations at two different Reynolds numbers such as 150 and 300 were performed using a lattice Boltzmann (LB) method, while a Lagrangian particle tracking method was employed to compute particle dynamics in the airflow. Instead of the commonly used lattice Bhatnagar–Gross–Krook (LBGK) model, a massively parallel code using multiple-relaxation-time LB (MRT-LB) method due to its better performance on numerical stability was developed for three-dimensional particle dynamics problems. Good agreement was observed between present airflow simulation and FLUENT results. It was also found that the three-dimensional and two-dimensional airflow patterns in ventilated chamber were very distinct from each other, indicating that two-dimensional computation is not appropriate for such kind of problem, even at low Reynolds numbers. For the particle dispersion and deposition study, six particle size groups ranging from 0.051 to 10μm were used. The particle results were verified by comparing them with FLUENT discrete phase model (DPM) prediction. Then the characteristics of particle dispersion and deposition were analyzed.

Numerics of the lattice Boltzmann method: Effects of collision models on the lattice Boltzmann simulations

We conduct a comparative study to evaluate several lattice Boltzmann (LB) models for solving the near incompressible Navier-Stokes equations, including the lattice Boltzmann equation with the multiple-relaxation-time (MRT), the two-relaxation-time (TRT), the single-relaxation-time (SRT) collision models, and the entropic lattice Boltzmann equation (ELBE). The lid-driven square cavity flow in two dimensions is used as a benchmark test. Our results demonstrate that the ELBE does not improve the numerical stability of the SRT or the lattice Bhatnagar-Gross-Krook (LBGK) model. Our results also show that the MRT and TRT LB models are superior to the ELBE and LBGK models in terms of accuracy, stability, and computational efficiency and that the ELBE scheme is the most inferior among the LB models tested in this study, thus is unfit for carrying out numerical simulations in practice. Our study suggests that, to optimize the accuracy, stability, and efficiency in the MRT model, it requires at least three independently adjustable relaxation rates: one for the shear viscosity ν (or the Reynolds number Re), one for the bulk viscosity ζ, and one to satisfy the criterion imposed by the Dirichlet boundary conditions which are realized by the bounce-back-type boundary conditions.

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

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