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

日本物理学会誌及びJournal of the Physical Society of Japanの月刊について

Monthly Notices is one of the three largest general primary astronomical research publications. It is an international journal, published by the Royal Astronomical Society. This article 1 describes its publication policy and practice.

Monthly Notices of the Royal Astronomical Society

A new and very general technique for simulating solid–fluid suspensions is described; its most important feature is that the computational cost scales linearly with the number of particles. The method combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-flow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic fluctuations in the fluid stress tensor, rather than from random forces or displacements applied directly to the particles. In this paper, the theoretical foundations of the technique are laid out, illustrated by simple analytical and numerical examples; in a companion paper (Part 2), extensive numerical tests of the method, for stationary flows, time-dependent flows, and finite-Reynolds-number flows, are reported.

Numerical simulations of particulate suspensions via a discretized boltzmann equation: part 1. theoretical foundation

We present evidence from observations and simulations to show that the fluctuations in the interplanetary magnetic and velocity fields are better described as dynamically evolving turbulence than as a superposition of noninteracting MHD waves. Spectral evolution, fluctuation anisotropics, and cross-helicity evolution are all better understood from the turbulence viewpoint. The picture that emerges from this work is that the interplanetary fluctuations are initially produced near the Sun as outward propagating Alfven waves, but are then mixed with inward propagating waves by a cascade from non-Alfvenic low wave numbers and dissipated at high wave numbers. In the outer heliosphere the cascade and dissipation rates decrease and become more nearly equal, producing a quasi-steady spectrum with a Kolmogoroff-like spectral index. The cascade from low wave numbers seems to be related to shear between and within interplanetary streams. The compressive fluctuations may be understood as largely consisting of density variations induced by the incompressive pressure fluctuations, with an additional component due to fast mode fluctuations.

Observation and simulation of mhd turbulence in the solar wind

The Outer Heliosphere: The Next Frontiers

Solar wind research and studies of charged particle propagation often assume that the interplanetary magnetic field represents a stationary random process. The extent to which ensemble averages of the solar wind magnetic fields follow the asymptotic behavior predicted by the ergodic theorem was investigated. Several time periods, including a span of nearly two years, are analyzed. Data intervals which span many solar rotations satisfy the conditions of weak stationarity if the effects of solar rotation are included in the asymptotic analysis. Shorter intervals which include a small integral number of interplanetary sectors also satisfy weak stationarity. The results are illustrated using magnetometer data from the ISEE-3, Voyager and IMP spacecraft.

Stationarity of magnetohydrodynamic fluctuations in the solar wind

We study one-point statistical properties of the induced turbulent electric field for a magnetohydrodynamic (MHD) plasma under the quasinormal approximation. Assuming exact Gaussianity for both the velocity field and the magnetic field, and different degrees of correlations between their Cartesian components, we derive the probability distribution function (PDF) for the Cartesian components of the electric field e i . We show that the PDF reduces in some canonical cases to an exponential function of the form exp(-‖e i ‖). To study deviations from these results in the more realistic case in which the velocity and magnetic fields are not exactly normal but quasinormal instead, we perform three-dimensional numerical simulations of the MHD equations at moderate Reynolds numbers. For turbulent relaxation from an initial condition, we find that the analytical results give a very good first-order approximation to the computed PDF.

One-point statistics of the induced electric field in quasinormal magnetofluid turbulence.

During the last four decades, a considerable number of investigations has been carried out into the evolution of turbulence in two dimensional Navier-Stokes flows. Much of the information has come from numerical solution of the (otherwise insoluble) dynamical equations and thus has necessarily required some kind of boundary conditions: spatially periodic, no-slip, stress-free, or free-slip. The theoretical framework that has proved to be of the most predictive value has been one employing an entropy functional (sometimes called the Boltzmann entropy) whose maximization has been correlated well in several cases with the late-time configurations into which the computed turbulence has relaxed. More recently, flow in the unbounded domain has been addressed by Gallay and Wayne who have shown a late-time relaxation to the classical Oseen vortex (also sometimes called the Lamb-Oseen vortex) for situations involving a finite net circulation or non-zero total integrated vorticity. Their proof involves powerful but...

William H. Matthaeus

Papers

Observation and simulation of mhd turbulence in the solar wind

The Outer Heliosphere: The Next Frontiers

Stationarity of magnetohydrodynamic fluctuations in the solar wind

One-point statistics of the induced electric field in quasinormal magnetofluid turbulence.

Oseen vortex as a maximum entropy state of a two dimensional fluid