10.2 The fhp lattice gas cellular automaton for simulating fluid

Introduction

Solving fluid flows is an important everyday task for engineers, physicists, and applied mathematicians. It can also be rather complex, especially when simulating flows with high Reynolds numbers. The classic approach to tackling this problem is to solve the Navier-Stokes equations directly using, for example, finite element or finite difference methods, but for domains with complicated geometries, it can be difficult to find and implement a suitable meshover which to apply these techniques. In cases like these, it may be more desirable to accurately simulate the behavior of the flow instead of solving for it outright. To this end, scientists and mathematicians have devised lattice gas and lattice Boltzmann methods for modeling fluid flow. In our VIGRE seminar this semester, we implemented a basic 2-D version of the FHP lattice gas cellular automaton (LGCA) with an eye towards extending this model to simulatestring motion in fluid flow.

Brief description of the fhp model

The LGCA approach to simulating fluid flow involves defining a lattice on which a large number of simulated particles move. Each particle has a mass and a velocity and therefore a momentum. The rules for particle interaction are chosen so that, in the macroscopic limit (i.e., when the momentum vectors are averaged over relatively large subdomains of the entire lattice), the resulting flow obeys the Navier-Stokes equations.

For the basic FHP model in 2-D, the lattice is chosen to have hexagonal symmetry. This means that a particle in the FHP LGCA can move from one node to another with six possible lattice velocities. (These velocities are equal in magnitude but vary with direction. So-called "multi-speed" FHP models allow for even more possibilities, but we do not discuss these here.) The reason for the hexagonal symmetry is that it has been shown that certain key tensorsfail to be isotropic on any lattice with a lesser degree of symmetry (e.g., a cubic lattice), which prevents the model from yielding the Navier-Stokes equations in the macroscopic limit ( [link] , pgs. 38, 51)

When particles meet at a node, it is possible for a collision to occur. The basic FHP model defines only two- and three-particle collisions, but it turns out that this is sufficient to yield the desired behavior. (More sophisticated models may define more complicated collision interactions.) If two particles travelling in opposite directions meet at a node, then the particle pair is randomly rotated either clockwise or counterclockwise by sixty degrees. If threeparticles meet at a node in a symmetric configuration, then they collide in such a way that this configuration is “inverted." Perhaps an illustration will help clarify: