Galilean invariant viscosity term for an athermal integer lattice Boltzmann automaton in three dimensions M. Geier, A. Greiner, and J. G. Korvink Imtek, Albert-Ludwigs University of Freiburg Georges-K¨ohler-Allee103, 79110 Freiburg, Germany, geier@imtek.de ABSTRACT The athermal lattice Boltzmann automata (LBA) are promising replacements for Navier Stokes solvers in computational fluid dynamics (CFD). They are inher- ently parallel, scale exactly linear with the number of computational elements and can be applied to arbitrary geometries. The integer LBA presented here adds un- conditional stability, roundoff-error freeness, and exact fulfillment of conservation laws to the list of benefits. The original LBA had artifacts not acceptable for indus- trial application. With ongoing research most of them could be removed. One of the artifacts was that the vis- cosity of the simulated fluid depended on the flow speed in an anisotropic manner. Previous work concentrated on removing this artifact by adding degrees of freedom to the unit cell. Here we present a method reducing the errors in the viscosity term without the need for new degrees of freedom. No computational overhead is in- troduced by this method. Keywords: lattice Boltzmann, cellular automaton, cen- tral moments, Galilean invariance, viscosity 1 INTRODUCTION The athermal LBA provide an alternative to solving the Navier Stokes equations for fluid dynamic problems [1]. The computational domain is divided into unit cells of equal shape and most often equal size. Each cell is either an ordinary fluid site or a boundary cell defining walls or inflows. Instead of obtaining a set of equations like in other numerical approaches, the LBA uses a cel- lular automaton as computational engine. Every cell is connected to a finite number of neighbors via links. The underling physics of a fluid is modeled as particles mov- ing along these links in a synchronous streaming step and a subsequent mass and momentum conserving scat- tering step at the centers of the cells. The original LBA uses fuzzy particles to remove noise from the solution. With the applied floating point arithmetic it introduces numerical roundoff-errors that accumulate very fast due to the explicitness of the algorithm. They make ordinary LBA incapable of long time simulation. The cellular au- tomaton description of the algorithm, however, makes it straightforward to get rid of the need for floating point arithmetic since no matrix has to be inverted and no equation needs to be solved. Instead, we allow a link to be occupied by a large number of particles modeled by an integer value. By further restricting the arithmetic to certain bounds we also exclude over- and underflow. The integer or digital LBA is hence unconditionally stable and fulfills conservation laws exactly (not only within machine precision). It comes with the additional benefit of linear complexity in space and time. Hence, it is ob- vious that the digital LBA will play an important role in the future when we encounter larger problems that have to be simulated for longer times. Only uncondition- ally stable and roundoff-error free methods, preventing us from the need of ever higher precision numbers for longer simulations, can survive in the long run. In order to prepare this method for its promising future ongoing research is yet to handle some teething problems. 2 ARTIFACTS We simplify the collective behavior of all particles in a fluid to two basic operations: free streaming and rear- rangement of particles according to entropy maximiza- tion. Particles cannot miss colliding with each other like in molecular dynamics since motion is restricted to links and scattering to sites. The computational efficiency with respect to memory consumption and speed is or- ders of magnitude better than for solving the Boltzmann Transport Equation directly. However, the restriction to a lattice introduces some unnatural artifacts, most no- table the lack of Galilean invariance [2]. This means that some fluid properties depend on whether the frame of reverence is moving relative to the fixed lattice or not. In a preceding method, called the lattice gas automaton [3][4][5], this problem was so severe that Galilean invari- ance was associated with a Galilean breaking factor g in front of the convective term of the Navier Stokes equa- tion derived from the lattice gas automaton. It took years of research to find a way to force g = 1. The re- sult of this research was the LBA [1] that is hence said to be Galilean invariant. This is achieved by forcing all second order central moments (κ xx , κ yy , κ zz , κ xy , κ xz , κ xy ) of the single site particle distribution function to isotropically approach a constant value. We define NSTI-Nanotech 2004, www.nsti.org, ISBN 0-9728422-7-6 Vol. 1, 2004 255