Cellular-automaton simulations of simple boundary-layer problems

Abstract

A lattice-gas automaton is a variant of a cellular automaton. Its cellular universe is a regular triangular lattice, and particles reside on the lattice nodes. The time evolution of this discrete dynamical system of particles proceeds in two alternating phases: collision and propagation. Such a model, though very simple and deterministic, is capable of producing very complex behaviors. A boundary layer develops whenever a real, viscous fluid flows along a solid boundary. We simulate boundary-layer and related problems in the incompressible limit of fluid dynamics using a lattice-gas automaton. Our lattice-gas automaton simulations show that viscosity effects on Couette flows (flows between parallel plates), Stokes flows, and Blasius flows (flows across a plate) give results as predicted by the Navier-Stokes equations. By considering different geometries and by carefully varying the gas properties, we obtain in particular the time-dependent velocity profiles, which are in good agreement with theoretical predictions. These inferences may be viewed as further support for the internal consistency of the lattice-gas approach, and they also substantiate the belief that the lattice-gas automaton can be a useful, viable tool for simulating fluid dynamics.