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

[1] A transport theory including cross helicity, magnetohydrodynamic (MHD) turbulence, and driving by shear and pickup ions, is applied to the radial evolution of the solar wind. The radial decrease of cross helicity observed in the solar wind can be accounted for when sufficient driving is included to overcome the inherent tendency for MHD turbulence to produce Alfvenic states.

Transport of cross helicity and the radial evolution of Alfvenicity in the solar wind

The perpendicular diffusion of charged test particles in a static representation of low-frequency, weakly turbulent magnetic fields superimposed on a steady background field is investigated with numerical simulations. The magnetic field variation is purely one-dimensional, consistent with an ensemble of field-aligned Alfv?n waves in the limit of zero Alfv?n speed. The diffusion coefficient ?? obtained from the numerical simulations is compared with Bieber & Matthaeus's recent theory of spatial diffusion, specialized to one-dimensional field geometry, for a number of particle energies/rigidities. It is found that the recent theory provides a better fit to the numerical data for ?? > 2, where ? is the particle gyrofrequency and ? is a rigidity-dependent decorrelation time, than does the well-known quasi-linear theory, or field line random walk limit. However, while the Bieber & Matthaeus theory fails to fit the simulation data for ?? < 1, the theory corresponding to the field line random walk limit exhibits only a 20% error here. Possible reasons for the discrepancy between the theory and simulations are discussed.

Numerical Investigation of Perpendicular Diffusion of Charged Test Particles in Weak Magnetostatic Slab Turbulence

An incompressible, dissipative numerical code of the spectral type is used to follow the nonlinear evolution of a magnetohydrodynamic sheet pinch in two spatial dimensions. The evolution involves considerable turbulent activity in the electric current field, with the excited spatial scales ranging from the size of the containing volume down to the dissipation lengths of the magnetic and velocity fields. Strong current filamentation near magnetic X-points is observed, as is 'jetting', or expulsion of magnetofluid from the vicinity of the X-point parallel to the current sheet.

Nonlinear evolution of the sheet pinch

[1] A nonlinear guiding center (NLGC) theory for diffusion of charged particles perpendicular to the mean magnetic field was recently proposed. Here, we draw attention to a number of attractive features of this theory: (1) The theory provides a natural mechanism to connect the perpendicular mean free path with the parallel mean free path. In fact, the parallel mean free path is the only particle property required to determine uniquely the perpendicular mean free path. (2) Under a broad range of conditions, the theory predicts that the perpendicular mean free path will be of order one percent or a few percent of the parallel mean free path, in agreement with numerical simulations of particle transport. (3) For conditions representative of the inner heliosphere, the theory predicts values of the perpendicular mean free path in agreement with observational determinations from Jovian electrons and from modeling Ulysses observations of Galactic protons.

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Nonlinear guiding center theory of perpendicular diffusion: General properties and comparison with observation

As Voyager 2 approached Jupiter's bow shock, large-amplitude fluctuations were seen in both the magnetic field and plasma fluid velocity. These fluctuations generally coincided with the occurrence of long-lived energetic particle events similar to the upstream waves often observed near the earth's bow shock. In this paper an analysis of the magnetic field and plasma observations using spectral methods is presented. The characteristic spectral features related to the upstream waves are generally seen near 1 mHz. The measured correlation lengths of these fluctuations suggest that they are coherent over only a few wavelengths. The analysis is consistent with the hypothesis that these fluctuations are driven by streaming ions, possibly protons. No evidence for the existence of whistler waves is found. It is argued that some of the observed spectral features suggest that dynamical turbulent processes are occurring in the uptream wave region, including a possible observation of an inverse cascade of magnetic helicity to large spatial scales.

William H. Matthaeus

Papers

Transport of cross helicity and the radial evolution of Alfvenicity in the solar wind

Numerical Investigation of Perpendicular Diffusion of Charged Test Particles in Weak Magnetostatic Slab Turbulence

Nonlinear evolution of the sheet pinch

Nonlinear guiding center theory of perpendicular diffusion: General properties and comparison with observation

Turbulence Analysis of the Jovian Upstream 'Wave' Phenomenon