Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
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A lattice-gas automaton is a form of a cellular automaton that has been used quite successfully to simulate fluid flows in the limits of incompressible fluid dynamics. Its cellular universe is a regular lattice and particles reside on the lattice nodes. At each time step, this discrete dynamical system of particles is updated in two alternating phases: collision and propagation. The question is thus does repeated iterations of these phases possess the power to produce the solution of a complex problem? A few examples from lattice-gas automaton simulations are presented. The cellular universe used is a 6-bit triangular lattice model in two dimensions so that there are a total of sixty-four possible states at each lattice node. The deterministic updates are symmetric two-particle and three-particle collisions. A test is first presented in the regime of wave propagation by considering the propagation of particle density gradient. It is demonstrated that the density contour propagates outwards with a velocity of ~ 0.709, in agreement with the theoretical prediction of 1@/2 for a triangular lattice. Simulations of simple boundary layer and related problems in the incompressible limits of fluid dynamics are then presented. The simulations results show that viscosity effects on adjacent opposing streams, Couette flows, Stokes flows and Blasius flows give results as predicted by the Navier-Stokes equations. It is seen that as the geometry of the cellular universe or as the gas properties are varied, the time-dependent velocity profiles vary accordingly, in good agreement with theoretical predictions.