We develop, after Dellar (P. J. Dellar, Phys. Rev. E. 65, 036309 (2002), J. Comput. Phys. 190, pp351 (2003)), a multiple-relaxation time (MRT), chromodynamic, multi-component lattice Boltzmann equation (MCLBE) scheme for simulation of isothermal, immiscible fluid flow with a density contrast. It is based on Lishchuk's method (J. U. Brackbill, D. B. Kothe and C. Zemach, J. Comp. Phys. 100, 335-354 (1992), S. V. Lishchuk, C. M. Care and I. Halliday, Phys. Rev. E. 67(3), 036701(2), (2003)) and the segregation of d'Ortona et al. (U. D'Ortona, D. Salin, M. Cieplak, R. B. Rybka and J. R. Banavar Phys. Rev. E. 51, 3718, (1995)). We focus on fundamental model verifiability but do relate some of our data to that from previous approaches, due to Ba et al. (Y. Ba, H. Liu, Q. Li, Q. Kang and J. Sun, Phys. Rev. E 94, 023310 (2016)) and earlier Liu et al. (H. Liu, A. J. Valocchi and Q. Kang, Phys. Rev. E 85, 046309 (2012)), who pioneered large density difference chromodynamic MCLBE and showed the practical benefits of a MRT collision model. Specifically, we test the extent to which chromodynamic MCLBE MRT schemes comply with the kinematic condition of mutual impenetrability and the continuous traction condition by developing analytical benchmarking flows. We conclude that our data, taken with those of Ba et al., verify the utility of MRT chromodynamic MCLBE.
Electroconvective flow between two infinitely long parallel electrodes is investigated via a multiphysics computational model. The model solves for spatiotemporal flow properties using two-relaxation-time Lattice Boltzmann Method for fluid and charge transport coupled to Fast Fourier Transport Poisson solver for the electric potential. The segregated model agrees with the previous analytical and numerical results providing a robust approach for modeling electrohydrodynamic flows.
A novel Lattice Boltzmann method is derived using the Principle of Minimum Cross Entropy (MinxEnt) via the minimization of Kullback-Leibler Divergence (KLD). By carrying out the actual single step Newton-Raphson minimization (MinxEnt-LBM) a more accurate and stable Lattice Boltzmann Method can be implemented. To demonstrate this, 1D shock tube and 2D lid-driven cavity flow simulations are carried out and compared to Single Relaxation Time LBM, Two Relaxation Time LBM, Multiple Relaxation Time LBM and Eherenfest Step LBM.
A closure for the effective relaxation time of the Boltzmann-BGK kinetic equation for fluid turbulence is presented, based on a double-averaging procedure over both kinetic and turbulent fluctuations. The resulting effective relaxation time appears to agree with values obtained via a renormalization group treatment of the Navier-Stokes equation only at low values of $k/T$, the ratio of turbulent kinetic energy to fluid temperature. For $k/T>0.1$ the kinetic treatment delivers a significantly longer effective relaxation time.
In this paper, three-dimensional (3D) multi-relaxation time (MRT) lattice-Boltzmann (LB) models for multiphase flow are presented. In contrast to the Bhatnagar-Gross-Krook (BGK) model, a widely employed kinetic model, in MRT models the rates of relaxation processes owing to collisions of particle populations may be independently adjusted. As a result, the MRT models offer a significant improvement in numerical stability of the LB method for simulating fluids with lower viscosities. We show through the Chapman-Enskog multiscale analysis that the continuum limit behavior of 3D MRT LB models corr…
In this paper, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is developed for simulating incompressible axisymmetric thermal flows in porous media at the representative elementary volume (REV) scale. In the model, a D2Q9 MRT-LB equation is proposed to solve the flow field in addition to the D2Q5 LB equation for the temperature field. The source terms of the model are simple and contain no velocity and temperature gradient terms. The generalized axisymmetric Navier-Stokes equations for axisymmetric flows in porous media are correctly recovered from the MRT-LB model through the C…
In this paper, a non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann (LB) method for simulating incompressible thermal flows is presented. In the method, the incompressible Navier-Stokes equations and temperature equation are solved separately by two different MRT-LB equations, which are developed based on non-orthogonal basis vectors obtained from the combinations of the lattice velocity components. The macroscopic governing equations of incompressible thermal flows can be recovered from the method through the Chapman-Enskog analysis in the incompressible limit. Numerical simulati…
Saturated vapor condensation on the homogeneous and heterogeneous sub-cooled wall under gravity was numerically simulated by the improved pseudo-potential phase-change multiple-relaxation-time lattice Boltzmann (LB) method. Firstly, the phase-change LB method coupling finite different method to solve the energy equation was verified by the D-2 law of the droplets evaporation, which improve the model could precisely realize the numerical simulation of phase change.
The development of multiple-relaxation-time (MRT) Lattice Boltzmann method (LBM) is a significant contribution in improving the numerical behavior, revealing the math and physics mechanism and extending the application of LBM. However, some of the MRT schemes proposed previously are not physically-consistent. In this work, we take D2Q9 as a example to show how to derive physically-consistent MRT-LBM schemes by eigenvalue decomposition of the collision operator. In addition, the scheme is validated by the equivalence to Navier-Stokes equations and numerical simulations.
In this article, a coupled Two-relaxation-time Lattice Boltzmann-Volume penalization (TRT-LBM-VP) method is presented to simulate flows past obstacles. Two relaxation times are used in the collision operator, of which one is related to the fluid viscosity and the other one is related to the numerical stability and accuracy. The volume penalization method is introduced into the TRT-LBM by an external forcing term. In the procedure of the TRT-LBM-VP, the processes of interpolating velocities on the boundaries points and distributing the force density to the Eulerian points are unneeded. Performi…
A multiple-relaxation-time discrete Boltzmann model (DBM) is proposed for multicomponent mixtures, where compressible, hydrodynamic, and thermodynamic nonequilibrium effects are taken into account. It allows the specific heat ratio and the Prandtl number to be adjustable, and is suitable for both low and high speed fluid flows. From the physical side, besides being consistent with the multicomponent Navier-Stokes equations, Fick's law and Stefan-Maxwell diffusion equation in the hydrodynamic limit, the DBM provides more kinetic information about the nonequilibrium effects. The physical capabil…
Double-diffusive convection in porous media is a common phenomenon in nature, and has received considerable attention in a wide variety of engineering applications. In this paper, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is developed for simulating double-diffusive convection in porous media at the representative elementary volume scale. The MRT-LB model is constructed in the framework of the triple-distribution-function approach: the velocity field, the temperature and concentration fields are solved separately by three different MRT-LB equations. The present model has tw…
We develop an improved lattice-Boltzmann numerical scheme to solve magnetohydrodynamic (MHD) equations in the regime of low magnetic Reynolds numbers, grounded on a manifestly Galilean covariant modeling of the Navier-Stokes equations. The simulation of the magnetic induction equation within the lattice-Boltzmann approach to MHD has been usually devised along the lines of the simplest phenomenological description, the single relaxation time (SRT) model. In order to deal with well-known stability difficulties of the SRT framework, we introduce, alternatively, a multi-relaxation-time technique f…
We show that, when a single relaxation time lattice Boltzmann algorithm is used to solve the hydrodynamic equations of a binary fluid for which the two components have different viscosities, strong spurious velocities in the steady state lead to incorrect results for the equilibrium contact angle. We identify the origins of these spurious currents, and demonstrate how the results can be greatly improved by using a lattice Boltzmann method based on a multiple-relaxation-time algorithm. By considering capillary filling we describe the dependence of the advancing contact angle on the interface ve…
A novel Lattice Boltzmann Method applicable to compressible fluid flows is developed. This method is based on replacing the governing equations by a relaxation system and the interpretation of the diagonal form of the relaxation system as a discrete velocity Boltzmann system. As a result of this interpretation, the local equilibrium distribution functions are simple algebraic functions of the conserved variables and the fluxes, without the low Mach number expansion present in the equilibrium distribution of the traditional Lattice Boltzmann Method (LBM). This new Lattice Boltzmann Relaxation S…
Callaway's dual relaxation times model, which takes into account the normal and resistive scatterings of phonon, is used to describe the heat conduction in materials like graphene. For steady-state problems, the Callaway model is usually solved by the conventional iterative scheme (CIS), which is efficient in the ballistic regime, but inefficient in the diffusive/hydrodynamic regime. In this paper, a general synthetic iterative scheme (GSIS) is proposed to expedite the convergence to steady-state solutions. First, macroscopic synthetic equations are designed to guide the evolution of equilibri…
In this paper, a two-dimensional eight-velocity (D2Q8) multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is proposed for incompressible porous flows at the representative elementary volume scale based on the Brinkman-Forchheimer-extended Darcy formulation. In the model, the porosity is included into the pressure-based equilibrium moments, and the linear and nonlinear drag forces of the porous media are incorporated into the model by adding a forcing term to the MRT-LB equation in the moment space. Through the Chapman-Enskog analysis, the generalized Navier-Stokes equations can be rec…
In this paper, a double multiple-relaxation-time lattice Boltzmann model is developed for simulating transient solid-liquid phase change problems in porous media at the representative elementary volume scale. The model uses two different multiple-relaxation-time lattice Boltzmann equations, one for the flow field and the other for the temperature field with nonlinear latent heat source term. The model is based on the generalized non-Darcy formulation, and the solid-liquid phase change interface is traced through the liquid fraction which is determined by the enthalpy method. The model is valid…
We present three dimensional realizations of the model introduced recently by (Karlin, Bösch, Chikatamarla, Phys. Rev. E 2014) and review the role of the entropic stabilizer. The presented models achieve outstanding numerical stability in presence of turbulent high Reynolds number flows. We report accurate results for low order moments for homogeneous isotropic decaying turbulence and second order grid convergence for most assessed statistical quantities. The explicit and efficient nature of the scheme renders it a very promising candidate for both engineering and scientific purposes in the vi…
The LBM is combined with the Volume Penalization (VP LBM) approach to simulate flows in the presence of obstacles. The single relaxation time LBM and the multiple relaxation time LBM are used. For cases where the fluid motion is enhanced by moving boundaries, the comparison of our results with analytical results is satisfactory. The flow around a cylinder at a relatively high Reynolds number is also calculated. For that case,the VP LBM method was parallelized. A comparison of our results with those found in the literature, demonstrates the reliability of the VP LBM to treat complex flows.
In this paper, an efficient three-dimensional lattice Boltzmann (LB) model with multiple-relaxation-time (MRT) collision operator is developed for the simulation of multiphase flows. This model is an extension of our previous two-dimensional model (H. Liang, B. C. Shi, Z. L. Guo, and Z. H. Chai, Phys. Rev. E. 89, 053320 (2014)) to the three dimensions using the D3Q7 (seven discrete velocities in three dimensions) lattice for the Chan-Hilliard equation (CHE) and the D3Q15 lattice for the Navier-Stokes equations (NSEs). Due to the smaller lattice-velocity numbers used, the computional efficiency…
In this paper, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is proposed for convection heat transfer in porous media under local thermal non-equilibrium (LTNE) condition. The model is constructed within the framework of the three-distribution-function approach: two temperature-based MRT-LB equations are proposed for the temperature fields of fluid and solid phases in addition to the MRT-LB equation of a density distribution function for the velocity field described by the generalized non-Darcy model. The thermal non-equilibrium effects are incorporated into the model by adding…
In this paper, we demonstrate a set of fundamental conditions required for the formulation of a thermohydrodynamic lattice Boltzmann model at an arbitrary Prandtl number. A specific collision operator form is then proposed that is in compliance with these conditions. It admits two independent relaxation times, one for viscosity and another for thermal conductivity. But more importantly, the resulting thermohydrodynamic equations based on such a collision operator form is theoretically shown to remove the well known non-Galilean invariant artifact at non-unity Prandtl numbers in previous therma…
A new multiple-relaxation-time lattice Boltzmann scheme for compressible flows with arbitrary specific heat ratio and Prandtl number is presented. In the new scheme, which is based on a two-dimensional 16-discrete-velocity model, the moment space and the corresponding transformation matrix are constructed according to the seven-moment relations associated with the local equilibrium distribution function. In the continuum limit, the model recovers the compressible Navier-Stokes equations with flexible specific-heat ratio and Prandtl number. Numerical experiments show that compressible flows wit…
We analyse a linear lattice Boltzmann (LB) formulation for simulation of linear acoustic wave propagation in heterogeneous media. We employ the single-relaxation-time Bhatnagar-Gross-Krook (BGK) as well as the general multi-relaxation-time (MRT) collision operators. By calculating the dispersion relation for various 2D lattices, we show that the D2Q5 lattice is the most suitable model for the linear acoustic problem. We also implement a grid-refinement algorithm for the LB scheme to simulate waves propagating in a heterogeneous medium with velocity contrasts. Our results show that the LB schem…
We propose a multiple relaxation time entropic realization of a two-phase flow lattice Boltzmann model we introduced in earlier works arXiv:2112.01975 S.A. Hosseini, B. Dorschner, and I. V. Karlin, arXiv preprint, arXiv:2112.01975 (2021). While the original model with a single relaxation time allows us to reach large density ratios, it is limited in terms of stability with respect to non-dimensional viscosity and Courant--Friedrichs--Lewy number. Here we show that the entropic multiple relaxation time model extends the stability limits of the model significantly, which allows us to reach large…
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. (Chin. Phys., 2004, 13: 40-46) and the MRT LB model proposed by d'Humières et al. (Philos. Trans. R. Soc., A, 2002, 360: 437-451), 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 …
The non-equilibrium gas dynamics is described by the Boltzmann equation, which can be solved numerically through the deterministic and stochastic methods. Due to the complicated collision term of the Boltzmann equation, many kinetic relaxation models have been proposed and used in the past seventy years for the study of rarefied flow. In order to develop a multiscale method for the rarefied and continuum flow simulation, by adopting the integral solution of the kinetic model equation a DVM-type unified gas-kinetic scheme (UGKS) has been constructed. The UGKS models the gas dynamics on the cell…
We present an a priori derivation of the force scheme for lattice Boltzmann method based on kinetic theoretical formulation. We show that the discrete lattice effect, previously eliminated a posteriori in BGK collision model, is due to first-order space-time discretization and can be eliminated generically for a wide range of collision models with second-order space-time discretization. Particularly, the force scheme for the recently developed spectral multiple-relaxation-time (SMRT) collision model is obtained and numerically verified.
The dynamics of the cavitation bubble collapse is a fundamental issue for the bubble collapse application and prevention. In present work, the modified forcing scheme for the pseudopotential multi-relaxation-time lattice Boltzmann model developed by Li Q. et al. is adopted to develop a cavitation bubble collapse model. In the respects of coexistence curves and Laplace law verification, the improved pseudopotential multi-relaxation-time lattice Boltzmann model is investigated. The independence between the kinematic viscosity and the thermodynamic consistency, surface tension is founded. By homo…
Multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is an important class of LB model with lots of advantages over traditional single-relaxation-time (SRT) LB model. In addition, the computation of strain rate tensor is crucial in MRT-LB simulations of some complex flows. Up to now, there are only two formulas to compute the strain rate tensor in the MRT LB model. One is to compute the strain rate tensor by using non-equilibrium parts of macroscopic moments (Yu formula). The other is to compute the strain rate tensor by using non-equilibrium parts of density distribution functions (Cha…
We propose a novel fluid-structure interaction (FSI) scheme using the entropic multi-relaxation time lattice Boltzmann (KBC) model for the fluid domain in combination with a nonlinear finite element solver for the structural part. We show validity of the proposed scheme for various challenging set-ups by comparison to literature data. Beyond validation, we extend the KBC model to multiphase flows and couple it with FEM solver. Robustness and viability of the entropic multi-relaxation time model for complex FSI applications is shown by simulations of droplet impact on elastic superhydrophobic s…
We apply lattice Boltzmann methods to study the relaxation of the velocity profile in binary fluids under shear during spinodal decomposition. In simple fluids, when a shear flow is applied on the boundaries of the system, the time required to obtain a triangular profile is inversely proportional to the viscosity and proportional to the square of the size of the system. We find that the same behaviour also occurs for binary mixtures, for any component ratio in the mixture and independently from the time when shear flow is switched on during phase separation.
The Navier-Stokes equations, are understood as the result of the low-order expansion in powers of dimensionless rate of strain $η_{ij}=τ_{0}S_{ij}$, where $τ_{0}$ is the microscopic relaxation time of a close-to- thermodynamic equilibrium fluid. In strongly sheared non-equilibrium fluids where $|η_{ij}|\geq 1$, the hydrodynamic description breaks down. According to Bogolubov's conjecture, strongly non-equlibrium systems are characterized by an hierarchy of relaxation times corresponding to various stages of the relaxation process. A "hydro-kinetic" equation with the relaxation time involving b…
A discrete Boltzmann model with state-dependent power-law relaxation time for nonequilibrium transport in compressible flows
Fujian Normal University（福建师范大学）；North China Institute of Aerospace Engineering（北华航天工业学院）；Laoshan Laboratory（崂山实验室）；Institute of Atmospheric Physics, Chinese Academy of Sciences（中国科学院大气物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）
A lattice Boltzmann (LB) theory, analytical characteristic integral (ACI) LB theory, is proposed in this paper. ACI LB theory takes Bhatnagar-Gross-Krook (BGK) Boltzmann equation as the exact kinetic equation behind Navier-Stokes continuum and momentum equations and constructs LB equation by rigorously integrating BGK-Boltzmann equation along characteristics. It's a general theory, supporting most existed LB equations including the standard lattice BGK (LBGK) equation inherited from lattice-gas automata, whose theoretical foundation had been questioned. ACI LB theory also indicates that the ch…
Starting from Boltzmann equation with relaxation time approximation for the collision term and using Chapman-Enskog like expansion for distribution function close to equilibrium, we derive hydrodynamic evolution equations for the dissipative quantities directly from their definition. Although the form of the equations is identical to those obtained in traditional Israel-Stewart approaches employing Grad's 14-moment approximation and second moment of Boltzmann equation, the coefficients obtained are different. In the case of one-dimensional scaling expansion, we demonstrate that our results are…
The first-order approximation to the solution of the Boltzmann--Curtiss transport equation is derived. The resulting distribution function treats the rotation or gyration of spherical particles as an independent classical variable, deviating from the quantum mechanical treatment of molecular rotation found in the Wang Chang-Uhlenbeck equation. The Boltzmann-Curtiss equation, therefore, does not treat different rotational motions as separate molecular species. The first-order distribution function yields momentum equations for the translational velocity and gyration that match the form of the g…
In the present paper a lattice Boltzmann scheme is presented which exhibits an increased stability and accuracy with respect to standard single- or multi-relaxation-time (MRT) approaches. The scheme is based on a single-relaxation-time model where a special regularization procedure is applied. This regularization is based on the fact that, for a-thermal flows, there exists a recursive way to express the velocity distribution function at any order (in the Hermite series sense) in terms of the density, velocity, and stress tensor. A linear stability analysis is conducted which shows enhanced dis…
To probe both the Hydrodynamic Non-Equilibrium (HNE) and Thermodynamic Non-Equilibrium (TNE) in the combustion process, a two-dimensional Multiple-Relaxation-Time (MRT) version of Lattice Boltzmann Kinetic Model(LBKM) for combustion phenomena is presented. The chemical energy released in the progress of combustion is dynamically coupled into the system by adding a chemical term to the LB kinetic equation. Beside describing the evolutions of the conserved quantities, the density, momentum and energy, which are what the Navier-Stokes model describes, the MRT-LBKM presents also a coarse-grained d…
In this mini-review we summarize the progress of Lattice Boltzmann(LB) modeling and simulating compressible flows in our group in recent years. Main contents include (i) Single-Relaxation-Time(SRT) LB model supplemented by additional viscosity, (ii) Multiple-Relaxation-Time(MRT) LB model, and (iii) LB study on hydrodynamic instabilities. The former two belong to improvements of physical modeling and the third belongs to simulation or application. The SRT-LB model supplemented by additional viscosity keeps the original framework of Lattice Bhatnagar-Gross-Krook (LBGK). So, it is easier and more…
A modified lattice Boltzmann model with a stochastic relaxation mechanism mimicking "virtual'' collisions between free-streaming particles and solid walls is introduced. This modified scheme permits to compute plane channel flows in satisfactory agreement with analytical results over a broad spectrum of Knudsen numbers, ranging from the hydrodynamic regime, all the way to quasi-free flow regimes up to Kn 30.
We present new analytic solutions to the relativistic Boltzmann equation within the relaxation time approximation. We first obtain spherically expanding solutions which are the kinetic counterparts of the exact solutions of the Israel-Stewart equation in the literature. This allows us to compare the solutions of the kinetic and hydrodynamic equations at an analytical level. We then derive a novel boost-invariant solution of the Boltzmann equation which has an unconventional dependence on the proper time. The existence of such a solution is also suggested in second order hydrodynamics and fluid…
