We present an overview of the lattice Boltzmann method (LBM), a parallel and efficient algorithm for simulating single-phase and multiphase fluid flows and for incorporating additional physical complexities. The LBM is especially useful for modeling complicated boundary conditions and multiphase interfaces. Recent extensions of this method are described, including simulations of fluid turbulence, suspension flows, and reaction diffusion systems.

Lattice boltzmann method for fluid flows

Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

Pressure (density) and velocity boundary conditions inside a flow domain are studied for 2-D and 3-D lattice Boltzmann BGK models (LBGK) and a new method to specify these conditions are proposed. These conditions are constructed in consistency of the wall boundary condition based on an idea of bounceback of non-equilibrium distribution. When these conditions are used together with the improved incompressible LBGK model by Zou et al., the simulation results recover the analytical solution of the plane Poiseuille flow driven by pressure (density) difference with machine accuracy. Since the half-way wall bounceback boundary condition is very easy to implement and was shown theoretically to give second-order accuracy for the 2-D Poiseuille flow with forcing, it is used with pressure (density) inlet/outlet conditions proposed in this paper and in Chen et al. to study the 2-D Poiseuille flow and the 3-D square duct flow. The numerical results are approximately second-order accurate. The magnitude of the error of the half-way wall bounceback is comparable with that using some other published boundary conditions. Besides, the bounceback condition has a much better stability behavior than that of other boundary conditions.

https://arxiv.org/pdf/comp-gas/9611001.pdf

On pressure and velocity flow boundary conditions and bounceback for the lattice Boltzmann BGK model

The generalized hydrodynamics (the wave vector dependence of the transport coefficients) of a generalized lattice Boltzmann equation (LBE) is studied in detail. The generalized lattice Boltzmann equation is constructed in moment space rather than in discrete velocity space. The generalized hydrodynamics of the model is obtained by solving the dispersion equation of the linearized LBE either analytically by using perturbation technique or numerically. The proposed LBE model has a maximum number of adjustable parameters for the given set of discrete velocities. Generalized hydrodynamics characterizes dispersion, dissipation (hyper-viscosities), anisotropy, and lack of Galilean invariance of the model, and can be applied to select the values of the adjustable parameters which optimize the properties of the model. The proposed generalized hydrodynamic analysis also provides some insights into stability and proper initial conditions for LBE simulations. The stability properties of some 2D LBE models are analyzed and compared with each other in the parameter space of the mean streaming velocity and the viscous relaxation time. The procedure described in this work can be applied to analyze other LBE models. As examples, LBE models with various interpolation schemes are analyzed. Numerical results on shear flow with an initially discontinuous velocity profile (shock) with or without a constant streaming velocity are shown to demonstrate the dispersion effects in the LBE model; the results compare favorably with our theoretical analysis. We also show that whereas linear analysis of the LBE evolution operator is equivalent to Chapman-Enskog analysis in the long wave-length limit (wave vector k = 0), it can also provide results for large values of k. Such results are important for the stability and other hydrodynamic properties of the LBE method and cannot be obtained through Chapman-Enskog analysis.

/pdf/theory-of-the-lattice-boltzmann-method-dispersion-pv854qk9i2.pdf

Theory of the lattice boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability

Pressure ~density! and velocity boundary conditions are studied for 2-D and 3-D lattice Boltzmann BGK models ~LBGK! and a new method to specify these conditions is proposed. These conditions are constructed in consistency with the wall boundary condition, based on the idea of bounceback of the non-equilibrium distribution. When these conditions are used together with the incompressible LBGK model @J. Stat. Phys. 81 ,3 5 ~1995!# the simulation results recover the analytical solution of the plane Poiseuille flow driven by a pressure ~density! difference. The half-way wall bounceback boundary condition is also used with the pressure ~density! inlet/outlet conditions proposed in this paper and in Phys. Fluids 8, 2527 ~1996! to study 2-D Poiseuille flow and 3-D square duct flow. The numerical results are approximately second-order accurate. The magnitude of the error of the half-way wall bounceback boundary condition is comparable with that of other published boundary conditions and it has better stability behavior. © 1997 American Institute of Physics. @S1070-6631~97!03406-5#

http://phelafel.technion.ac.il/~drorden/project/ZouHe.pdf

On pressure and velocity boundary conditions for the lattice Boltzmann BGK model

We propose the lattice BGK models, as an alternative to lattice gases or the lattice Boltzmann equation, to obtain an efficient numerical scheme for the simulation of fluid dynamics. With a properly chosen equilibrium distribution, the Navier-Stokes equation is obtained from the kinetic BGK equation at the second-order of approximation. Compared to lattice gases, the present model is noise-free, has Galileian invariance and a velocity-independent pressure. It involves a relaxation parameter that influences the stability of the new scheme. Numerical simulations are shown to confirm the speed of sound and the shear viscosity.

https://iopscience.iop.org/article/10.1209/0295-5075/17/6/001/pdf

Lattice BGK Models for Navier-Stokes Equation

Evolutionary, pattern forming partial differential equations (PDEs) are often derived as limiting descriptions of microscopic, kinetic theory-based models of molecular processes (e.g., reaction and diffusion). The PDE dynamic behavior can be probed through direct simulation (time integration) or, more systematically, through stability/bifurcation calculations; time-stepper-based approaches, like the Recursive Projection Method [Shroff, G. M. & Keller, H. B. (1993) SIAM J. Numer. Anal. 30, 1099–1120] provide an attractive framework for the latter. We demonstrate an adaptation of this approach that allows for a direct, effective (“coarse”) bifurcation analysis of microscopic, kinetic-based models; this is illustrated through a comparative study of the FitzHugh-Nagumo PDE and of a corresponding Lattice–Boltzmann model.

https://www.pnas.org/content/pnas/97/18/9840.full.pdf

"Coarse" stability and bifurcation analysis using time-steppers: a reaction-diffusion example.

Recent Advances in Lattice Boltzmann Computing

In this letter, a nonlinear deviation from the Navier-Stokes equation is obtained from the recently proposed LBGK models, which are designed as an alternative to lattice gas or lattice Boltzmann equation. The classical Chapman-Enskog method is extended to derive the nonlinear-deviation term as well as its coefficient. Their analytical expression is derived for the first time, thanks to the simplicity of the LBGK models. A numerical simulation of a shock profile is presented. The influence of the correction on the kinetics of compressible flow is discussed. A complete analysis of the thermodynamics including the temperature will be presented elsewhere.

http://iopscience.iop.org/article/10.1209/0295-5075/21/3/001/pdf

Lattice BGK Models for the Navier-Stokes Equation: Nonlinear Deviation in Compressible Regimes

In this paper, we propose new lattice BGK models for one-, two- and three-dimensional thermohydrodynamics. A proper internal energy is introduced and the energy equation is obtained. The derivation of the thermohydrodynamic equations is systematic and numerical simulations are carried out to verify the theoretical values of the sound speed, the shear viscosity and the conductivity. One-dimensional shock-tube problem and two-dimensional Rayleigh-Benard convection are simulated. Good performance and satisfactory results are obtained. These models can be applied to many interesting cases, in particular, the transonic regimes where the compressibility can not be neglected.

Yuehong Qian

Papers

Lattice BGK Models for Navier-Stokes Equation

"Coarse" stability and bifurcation analysis using time-steppers: a reaction-diffusion example.

Recent Advances in Lattice Boltzmann Computing

Lattice BGK Models for the Navier-Stokes Equation: Nonlinear Deviation in Compressible Regimes

Simulating thermohydrodynamics with lattice BGK models