triple, etc.) vanishes separately, because of the exclusion principle. So the solution is a Fermi-Dirac distribution. This proof also shows that as long as conservation laws of any kind are embodied in the collision term, each type of collision is separately zero under the Fermi-Dirac distribution. Accordingly, the Fermi-Dirac solution is universal across collision types. This implies that one cannot alter the character of the Fermi-Dirac distribution in the lattice gas by adding collision types that respect collision invariants. Since f is now assumed to be a Fermi-Dirac distribution, we take it as

(Here we have returned to our original conventions for Î.) The equilibrium value for ƒ at v = 0, namely No, is where p is the density. Expanding the Fermi-Dirac form for fo about this equilibrium value gives us

eq

(16)

the same form as the Chapman-Enskog expansion (Eq. 15). To fix a and ẞ we use number and momentum conservation as constraints, so that ƒ becomes

For general single-speed models with particle speed c and b velocity vectors in D dimensions, the result above generalizes to

These forms depend only on the structure of tensor products of îa in D dimensions. When we discuss the full Navier-Stokes equations, we will show how to absorb the g(p) Galilean-invariance-breaking term in Eq. 17 into a rescaling of variables.

Isotropy and The Momentum Tensor. We will go on to discuss viscosity and the lattice form of the Navier-Stokes equation, but first we comment briefly on how the structure of the momentum tensor depends on the geometry of the lattice. Those interested in all the details can find them discussed from several viewpoints in Frisch, d'Humières, Hasslacher, Lallemand, and Pomeau 1987.

By definition II;; = Σ(); (3);fs, where fa is determined by the ChapmanEnskog, or direct, expansion (Eq. 15). Isotropy implies invariance under rotations and reflections; tensors that are isotropic are proportional to a scalar. Define the tensors E(") = Σ(3), (3), For E(") with regular b-sided polygons, we can derive conditions on b for E(") to be isotropic. These conditions are

...

For b = 4, the case of the HPP (Hardy, de Pazzis, and Pomeau) square lattice, E (4) is not isotropic. For b = 6, the hexagonal neighborhood case, all tensors up to n = 5 are isotropic.

Using the Chapman-Enskog expansion for f and the notation above for tensors, II;; has the following tensor structure.

where we are following the discussion of Wolfram. The momentum stress tensor must be isotropic up to E(4) in order that the leading terms in the momentum equation (corresponding to the convective and viscous terms in the Navier-Stokes equation) be isotropic. For the square model, the original discrete-lattice model, we have nonisotropy manifested in two places through the momentum flux tensor.

The nonisotropy implies that we do not get a Navier-Stokes type equation for the square lattice. For the hexagonal model, ẞ = 6, isotropy is maintained through order ß E(4). By using general considerations on tensor structures for polygons and polyhedra in D-dimensional space, one can quickly arrive at probable models for Navier-Stokes dynamics in any dimension. The starting point is that isotropy, or the lack of it, in both convective and viscous terms (the Euler and the Navier-Stokes equations), is controlled completely by the geometry of the underlying lattice. This crucial point was missed by all earlier workers on lattice models who thought that the geometry of the underlying lattice was irrelevant.

Viscosity for Lattice Gas Models. In "The Continuum Argument" we saw that the general form of the compressible Navier-Stokes equation with bulk viscosity ( = 0 is

where

is the kniematic shear viscosity. To derive this form for the discrete model, one must solve for II; using both the Chapman-Enskog approximation for f3 and the momentum-conservation equation. We noted earlier that the momentum equation contained corrections as powers of the lattice spacing but chose to ignore these at first pass. However, if we use the full Taylor expansion developed in the lattice-size scaling, we find that the contribution to the viscous term of the momentum equation is – {μV2v. Note that the correction to the viscosity is a constant (see Eq. 19) that depends only on the lattice and dimension and is independent of the scattering-rule set. This extra noncovariant-derivative contribution to the viscosity must be subtracted from the bare viscosity calculated from the normal perturbation expansion to get the renormalized viscosity, which is the one actually measured in the lattice gas. In other words, the bare coupling constant of the lattice gas model gets "dressed" by this constant amount, owing to the discrete vacuum that the particle must pass through, to become the physical lattice-gas viscosity.

Viscosity is a coupling constant and can be found by any method that can isolate the B1 term in the Chapman-Enskog expansion. The simplest methods involve solving for the eigenvalues and right eigenvectors of the linearized collision operator, which is a tedious exercise in linear algebra. Using the results of such a calculation, we can write the Navier-Stokes form of the momentum equation in which the viscosity v(p) appears explicitly:

We end this theoretical analysis by showing under what conditions we recover these equations for lattice gases. One way is to freeze the density everywhere except in the pressure term of the momentum equation (Eq. 18). Then, in the low-velocity limit, we can write the lattice Navier-Stokes equations as

-1

To be more precise, we do an e expansion of the momentum equation, where € is the same order as the global lattice size Lg (see Frisch et al. for details), and rescale the variables as follows:

of variables keeps the Reynolds number fixed.) Now all the relevant terms in the momentum equation are of O(e3) and higher order terms are O(1) or smaller. So to leading order (where V means) we get

ƏTV + V · V1V = -V1P' + 'VV and V1V = 0.

.

Thus we recover the incompressible Navier-Stokes equations. To obtain this result, we have done a fixed-Reynolds-number, large-scale, low-Mach-number expansion and Galilean invariance has been restored, at least formally, by a time rescaling.

Simulations of Fluid Dynamics

with the Hexagonal Lattice Gas Automaton

In the last two years several groups in the United States and France have done simulations of fluid-dynamical phenomena using the hexagonal lattice-gas automaton. The purpose of these simulations was twofold: first, to check the internal consistency of the automaton, and second, to determine, by both qualitative and quantitative measures, whether the model behaves the same or nearly the same as the known analytic and numerical solutions of the Navier-Stokes equations.

The classes of experiments done can be grouped roughly as free flows, flow instabilities, flows past objects, and flows in channels or pipes. These simulations were run in a range of Reynolds numbers between 100 and 700 (and for relatively low mean flow velocities, so that the fluid is nearly incompressible). We first checked to see whether the automaton developed various classic instabilities when triggered by two types of mechanisms, external perturbations and internal noise. The two classic instabilities studied were the Kelvin-Helmholtz instability of two opposing shear flows and the Rayleigh-Taylor instability. We describe the Kelvin-Helmholtz instability in some detail.

In the Kelvin-Helmholtz instability one is looking for the development of a finalstate vortex structure of appropriate vortex polarity. From an initial state of two opposing flows undergoing shear, the detailed development of the instability depends on the initial perturbation of the flows. Left unperturbed, except by internal noise in the automaton, at first the two opposing flows develop velocity fields that signal the development of a boundary layer, then sets of vortices develop in these boundary layers, and finally vortex interactions occur that trigger a large-scale instability and the development of large-vortex final states. The same pattern appears in standard two-dimensional numerical simulations of the Navier-Stokes equations near the incompressible regime. No pathological non-Navier-Stokes behavior was observed. These results extend over the entire range of Reynolds numbers (100-700) run with the simple hexagonal model. It is notable that the Kelvin-Helmholtz instability is self-starting due to the automaton internal noise, and the instability proceeds rapidly.

The Rayleigh-Taylor instability was simulated by a French group in a slightly compressible fluid range, where it behaves like a Navier-Stokes fluid with no anomalies.

These global topological tests check whether automaton dynamics captures the correct overall structure of fluids. In general, whenever the automaton is run in the Navier-Stokes range, it produces the expected global topological behavior and correct