Multiple-relaxation-time Lattice Boltzmann Model Research Articles

A phase-field-based multiple-relaxation-time lattice Boltzmann method for incompressible multiphase flows with density and viscosity contrasts

In this work, we present a unified framework of phase-field-based multiple-relaxation-time lattice Boltzmann (MRT-LB) method for incompressible multiphase flows with density and viscosity contrasts where a block-lower-triangular relaxation matrix is introduced. The present framework includes the classic MRT-LB model and central-moments-based LB model (CLBM, a particular version of the MRT-LB model). The governing equations of incompressible multiphase flows can be reproduced precisely through the direct Taylor expansion method at the second-order of expansion parameters. In this work, the pressure p is re-estimated, and we prove that the present pressure expression is suitable for classic MRT model, CLBM and various lattice structures with an accuracy of O(Δt2+ΔtMa2). In addition, the model construction and analysis are all carried out in the velocity space. As a result, the present model has a similar structure to the traditional Bhatnagar-Gross-Krook (BGK) model. Therefore the process of their analysis and implementation is also similar. We then test the present model by five classic physical problems. The results show that the present model is found to be accurate and has better numerical stability over the BGK model, and has the ability to simulate multiphase flow problems with large density ratio (1000).

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.

A coupled LES-LBM-IMB-DEM modeling for evaluating pressure drop of a heterogeneous alternating-layer packed bed

• A LES-LBM-IMB-DEM (LLID) model was established. • The LLID model was carefully calibrated and validated. • Interface characteristics and resistance loss mechanisms of HAL beds were revealed. • A new correlation was proposed for evaluating the pressure drop of HAL beds. Packed beds composed of a heterogeneous alternating-layer (HAL) granular exist in many practical applications, such as blast furnaces of the ironmaking industry. Insights into HAL beds are significant to process performance but are still challenging and not fully understood. A coupled LES-LBM-IMB-DEM (LLID) model is established to directly obtain the interphase closure of fluid–solid systems and evaluate the pressure drop of HAL beds. The LLID model particularly involves pores turbulent flow of particle-resolved scale using the multiple-relaxation-time lattice Boltzmann model (MRT-LBM) with large eddy simulation (LES), tracking the interaction of particle–particle/walls using discrete element method (DEM), and coupling fluid-granular system using immersed moving boundary (IMB). The model was carefully calibrated concerning lattice resolution, sub-grid numbers, collocations of collision term and weight function, relaxation time, Smagorinsky constant. Also, it was verified via flowing past a particle, regular packed bed, and HAL bed, and found reasonable agreements achieved. Furthermore, the model was applied to investigate HAL beds, especially the interface characteristics, velocity and pores distribution, mechanism of interface resistance loss, and revealed the relationship of specific pressure drop between the interface region and bulk layer section. Finally, a new correlation of pressure drops for HAL beds was proposed via linking the interface pressure drop to the bulk layer.

Improved curved-boundary scheme for lattice Boltzmann simulation of microscale gas flow with second-order slip condition

An improved curved-boundary scheme with second-order velocity slip condition for multiple-relaxation-time-lattice Boltzmann (MRT-LB) simulation of microgas flow is proposed. The proposed interpolation bounce-back (IBB)-explicit counter-extrapolation (ECE) scheme adopts the IBB method to describe the curved boundary, while the ECE method is employed to predict the slip velocity on gas-solid interface. To incorporate the effect of second-order velocity slip term and the influence of boundary curvature, a slip velocity model is also derived, from which the gas slip velocity is captured by the ECE discretization method. The influence of fictitious slip velocity can be eliminated by adopting the present ECE method, and the influence of actual offset between the lattice node and the physical boundary can be well considered by the IBB method. The proposed IBB-ECE boundary scheme is then implemented with the MRT-LB model and tested by simulations of force-driven gas flow in horizontal (inclined) microchannel, gas flow around a micro-cylinder, and Couette flow between two micro-cylinders. Numerical results show that the proposed IBB-ECE scheme improves the computational accuracy of gas slip flow (0.001<Kn≤0.1) when compared with other boundary schemes reported in the literature, and provides a precise and easy implementing scheme for curved boundary with second-order slip condition.

Modified phase-field-based lattice Boltzmann model for incompressible multiphase flows.

Based on the phase-field theory, a multiple-relaxation-time (MRT) lattice Boltzmann model is proposed for the immiscible multiphase fluids. In this model, the local Allen-Chan equation is chosen as the target equation to capture the phase interface. Unlike previous MRT schemes, an off-diagonal relaxation matrix is adopted in the present model so that the target phase-field equation can be recovered exactly without any artificial terms. To check the necessity of removing those artificial terms, comparative studies were carried out among different MRT schemes with or without correction. Results show that the artificial terms can be neglected at low March number but will cause unphysical diffusion or interface undulation instability for the relatively large March number cases. The present modified model shows superiority in reducing numerical errors by adjusting the free parameters. As the interface transport coupled to the fluid flow, a pressure-evolution lattice Boltzmann equation is adopted for hydrodynamic properties. Several benchmark cases for multiphase flow were conducted to test the validity of the present model, including the static drop test, Rayleigh-Taylor instability, and single rising bubble test. For the rising bubble simulation at high density ratios, bubble dynamics obtained by the present modified MRT lattice Boltzmann model agree well with those obtained by the FEM-based level set and FEM-based phase-field models.

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.

A multiple-relaxation-time lattice Boltzmann model for radiative transfer equation

A novel multiple-relaxation-time (MRT) lattice Boltzmann model is proposed for the radiative transfer equation (RTE). In this paper, the discussion and implementation are restricted to the grey (frequency-independent) radiative transfer equation. We establish this model by regarding the RTE as a particular convection–diffusion equation without the diffusion term. The equilibrium distribution function for the RTE is constructed, and the RTE is rigorously derived from the lattice Boltzmann equation via the Maxwell iteration under the diffusive scaling. Several numerical cases are conducted, including one-dimensional, two-dimensional and three-dimensional transient and steady cases. Results demonstrate that this MRT lattice Boltzmann model is accurate and stable for solving transient and steady radiative transfer problems in participating media. Besides, the second-order accuracy and convergence rate are observed for this model. This model may be further extended to various radiative transfer problems in participating media.

Study of pore-scale coke combustion in porous media using lattice Boltzmann method

In-situ combustion (ISC) for heavy oil recovery is a high-risk process and thus calls for a deep understanding of the coke combustion front. However, simulations for describing pore-scale coke combustion are challenging because of complicated combustion dynamics. This study proposes a multiple-relaxation-time (MRT) lattice Boltzmann (LB) method to simulate solid coke combustion in porous media at the pore scale. The MRT LB model can enhance numerical stability compared with the single-relaxation-time counterpart. This new LB model offers two major advances over existing LB models. First, thermal expansion effects are considered via temperature-dependent density, satisfying the low Mach number flow condition. Second, conjugate heat transfer and species conservation at the coke-fluid interface are modelled by a source term and a bounce-back scheme, respectively, without resorting to iterative methods. Simulations of pore-scale coke combustion are conducted to validate the superior performance of the developed LB model. The results show that this model can reproduce coke combustion dynamics and capture variations in isotherms from the fluid phase to the solid phase at the pore scale. In contrast, without considering effects of thermal expansion or conjugate heat transfer as in previous models, simulations would underestimate the burning temperature and even fail to predict instability caused by a high driving force. Moreover, using the proposed LB model, a parametric study is performed to assess the impact of the inlet air and porous structure on coke combustion. The findings suggest that inlet air temperature and driving force should be controlled within certain ranges, otherwise their influences may become negligible or even negative. Large porosity accelerates coke combustion and leads to a desired combustion front. Heterogeneous coke distribution affects pore-scale combustion details but not the overall combustion intensity, while a random matrix structure may influence combustion dynamics both locally and globally. As a whole, this study develops a reliable LB model to investigate pore-scale coke combustion, which contributes to advancing the knowledge base for heavy oil recovery using the ISC process.

Discrete unified gas kinetic scheme for nonlinear convection-diffusion equations.

In this paper, we develop a discrete unified gas kinetic scheme (DUGKS) for a general nonlinear convection-diffusion equation (NCDE) and show that the NCDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the present DUGKS through some classic convection-diffusion equations, and we find that the numerical results are in good agreement with analytical solutions and that the DUGKS model has a second-order convergence rate. Finally, as a finite-volume method, the DUGKS can also adopt the nonuniform mesh. Besides, we perform some comparisons among the DUGKS, the finite-volume lattice Boltzmann model (FV-LBM), the single-relaxation-time lattice Boltzmann model (SLBM), and the multiple-relaxation-time lattice Boltzmann model (MRT-LBM). The results show that the present DUGKS is more accurate than the FV-LBM, more stable than the SLBM, and almost has the same accuracy as the MRT-LBM. Moreover, the use of nonuniform mesh may make the DUGKS model more flexible.

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.

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 hydrodynamically-consistent MRT lattice Boltzmann model on a 2D rectangular grid

A multiple-relaxation time (MRT) lattice Boltzmann (LB) model on a D2Q9 rectangular grid is designed theoretically and validated numerically in the present work. By introducing stress components into the equilibrium moments, this MRT-LB model restores the isotropy of diffusive momentum transport at the macroscopic level (or in the continuum limit), leading to moment equations that are fully consistent with the Navier–Stokes equations. The model is derived by an inverse design process which is described in detail. Except one moment associated with the energy square, all other eight equilibrium moments can be theoretically and uniquely determined. The model is then carefully validated using both the two-dimensional decaying Taylor–Green vortex flow and lid-driven cavity flow, with different grid aspect ratios. The corresponding results from an earlier model (Bouzidi et al. (2001) [28]) are also presented for comparison. The results of Bouzidi et al.'s model show problems associated with anisotropy of viscosity coefficients, while the present model exhibits full isotropy and is accurate and stable.

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 simulation of the open channel flow connecting two cascaded hydropower stations

This paper examines the feasibility and the efficiency of a multiple-relaxation-time lattice Boltzmann model (MRT-LBM) for simulating open channel flows in engineering practice. A MRT-LBM scheme for 2-D shallow water flows taking into account of the bed slope and the friction is proposed. The scheme's reliability is verified by benchmark problems and the simulation capability is improved by implementing the scheme on a graphic processing unit (GPU). We use the method to analyze the flow characteristics in the connecting open channel of two cascaded hydropower stations. The flow fields and parameters such as the water depth, the flow rate, and the side-weir discharge, under different operating conditions, are analyzed. The factors affecting the accuracy and the efficiency are discussed. The results are found to be reasonable and may be used as a guidance in the project design. It is shown that the GPU-implemented MRT-LBM on a fine mesh can efficiently simulate two-dimensional shallow water flows in engineering practice.

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.

Three-dimensional simulations of Bingham plastic flows with the multiple-relaxation-time lattice Boltzmann model

ABSTRACTThis paper presents a three-dimensional (3D) parallel multiple-relaxation-time lattice Boltzmann model (MRT-LBM) for Bingham plastics which overcomes numerical instabilities in the simulation of non-Newtonian fluids for the Bhatnagar–Gross–Krook (BGK) model. The MRT-LBM and several related mathematical models are briefly described. Papanastasiou’s modified model is incorporated for better numerical stability. The impact of the relaxation parameters of the model is studied in detail. The MRT-LBM is then validated through a benchmark problem: a 3D steady Poiseuille flow. The results from the numerical simulations are consistent with those derived analytically which indicates that the MRT-LBM effectively simulates Bingham fluids but with better stability. A parallel MRT-LBM framework is introduced, and the parallel efficiency is tested through a simple case. The MRT-LBM is shown to be appropriate for parallel implementation and to have high efficiency. Finally, a Bingham fluid flowing past a square-based prism with a fixed sphere is simulated. It is found the drag coefficient is a function of both Reynolds number (Re) and Bingham number (Bn). These results reveal the flow behavior of Bingham plastics.

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.

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.