The Lattice Boltzmann Equation for Fluid Dynamics and Beyond , Sauro Succi Princeton U. Press, New York, 2001. $100.00 (288 pp.). ISBN 0-19-850398-9
The Navier–Stokes equations describe the flow of simple fluids. The equations can be derived easily by applying the laws of mass conservation and Newton’s second law to an elementary fluid volume (“volumelet,” in the charming terminology of Sauro Succi’s The Lattice Boltzmann Equation for Fluid Dynamics and Beyond) incorporating the assumption that stress and strain are proportional.
Despite their apparent simplicity, the Navier–Stokes equations describe rich physics, some of which is still not understood in detail. For example, at high Reynolds numbers (low viscosities, high velocities, and large length scales) flow becomes turbulent, as exemplified by waterfalls, blood flow, and atmospheric flows. In such flows, the nonlinear nature of the Navier–Stokes equations is making itself felt; many different length scales contribute to the fluid motion.
Solving the Navier–Stokes equations preoccupies many scientists and engineers. Geophysicists need to predict the flow of ocean currents, engineers designing cars aim for streamlining to reduce turbulence, and reactor physicists model flow and heat transfer under extreme conditions. For computational fluid dynamics, commercial codes that solve the Navier–Stokes equations are well tested and widely used.
The codes are excellent for simple flow in simple geometries, but life becomes more tricky when tortuous boundaries, turbulence, or multiphase flows become important. And, of course, the investigator’s goal is always to encompass ever larger systems on increasingly fine grids.
Hence the excitement when, in 1986, Uriel Frisch, Brosl Hasslacher, and Yves Pomeau proposed a lattice-gas cellular automaton able to solve two-dimensional flow. Particles hopping on a hexagonal lattice, interacting at the nodes via simple collision rules, were shown to obey the Navier–Stokes equations for sufficiently long length and time scales.
Here was a new approach to computational fluid dynamics that was free of round-off errors and ideally suited to parallel computers. But there were problems: statistical noise, a lack of Galilean invariance, and limitations to low Reynolds number.
In the Frisch-Hasslacher-Pomeau model, the particle collisions conserve mass and momentum and the hexagonal lattice has just enough symmetry to reproduce the rotational isotropy of 2D space. Essentially, the conservation laws are fed in, and the Navier–Stokes equations drop out. This idea has spurred the development of a wide range of novel Navier–Stokes solvers. Perhaps most successful of those is the lattice Boltzmann algorithm of Succi’s book. The book’s first few chapters give an account of the way in which lattice Boltzmann ideas developed from lattice-gas cellular automata. Succi was closely involved in those developments and one of the driving forces behind them. The evolution of the different approaches and the excitement of the research come across clearly in his book.
How does lattice Boltzmann compare to more conventional computational fluid dynamics approaches? The second part of the book discusses the technical details of irregular grids and boundary conditions and outlines applications of lattice Boltzmann algorithms to flow in porous media and to turbulence. Succi has experience with a wide range of Navier–Stokes solvers and presents a balanced view, although he is obviously fond of the lattice Boltzmann approach and pleased when it performs well. For standard applications, lattice Boltzmann is unlikely to displace more conventional methods, but for certain problems, such as multiphase flow in porous media, it is a strong contender for the best fluid-simulation approach currently available.
The lattice Boltzmann equation can also be thought of as a discrete, simplified version of the continuum Boltzmann equation of nonequilibrium statistical mechanics. This connection has the tantalizing consequence that lattice Boltzmann may contain more mesoscopic physics than the Navier–Stokes equations themselves. For example, the lattice Boltzmann algorithm is particularly adept at describing the flow of complex fluids in which thermodynamic and hydrodynamic information comes into play.
Applications to multiphase fluids, colloids, reaction–diffusion equations, and other problems at the forefront of research, such as a lattice Boltzmann description of quantum mechanics, are summarized in the final third of the book. Inevitably, the treatment is already somewhat dated and superficial, but it provides a flavor of the possible and the exciting.
This is a book for the enthusiast that reminds the expert how much fun he can have investigating the physics and applications of this rich field.
