Dominique d'Humières

Lattice BGK Models for Navier-Stokes Equation

We propose the lattice BGK models, as an alternative to lattice gases or the lattice Boltzmann equation, to obtain an efficient numerical scheme for the simulation of fluid dynamics. With a properly chosen equilibrium distribution, the Navier-Stokes equation is obtained from the kinetic BGK equation at the second-order of approximation. Compared to lattice gases, the present model is noise-free, has Galileian invariance and a velocity-independent pressure. It involves a relaxation parameter that influences the stability of the new scheme. Numerical simulations are shown to confirm the speed of sound and the shear viscosity.

Multiple-relaxation-time lattice Boltzmann models in three dimensions

This article provides a concise exposition of the multiple–relaxation–time lattice Boltzmann equation, with examples of 15–velocity and 19–velocity models in three dimensions. Simulation of a diagonally lid–driven cavity flow in three dimensions at Re = 500 and 2000 is performed. The results clearly demonstrate the superior numerical stability of the multiple–relaxation–time lattice Boltzmann equation over the popular lattice Bhatnagar–Gross–Krook equation.

Lattice Gas Models for 3D Hydrodynamics

The 3D Navier-Stokes equations are obtained from two different lattice gas models. The first one has its sites on a cubic lattice and has particle speeds zero, one and √2. The second one is a 3D projection of a lattice gas implementation of the 4D Navier-Stokes equations, residing on a face-centred hypercubic lattice.

Simulating Fully Three-Dimensional External Flow by Lattice Gas Methods

We have built a three-dimensional 24-bit lattice gas algorithm with improved collision rules. Collisions are defined by a look-up table with 224 entries, fine-tuned to maximize the Reynolds number. External flow past a circular plate at Reynolds number around 190 has been simulated. The flow is found to evolve from axi-symmetric to fully 3D. Such simulations take a few minutes of CRAY-2 per circulation time (based on plate diameter and upstream velocity).

A Lattice-Boltzmann Model for Visco-Elasticity

The classical lattice-Boltzmann scheme is extended in an attempt to represent visco-elastic fluids in two dimensions. At each lattice site, two new quantities are added. A suitable coupling of these quantities with the viscous stress tensor leads to a nonzero shear modulus and visco-elastic effects. A Chapman–Enskog expansion gives us the equilibrium populations and conditions for isotropy of the model. A finite wave vector analysis is needed to study the relaxation of sound waves and to determine the dependence of the transport coefficients upon the frequency.

On the small-scale dynamical behavior of lattice BGK and lattice Boltzmann schemes

In this paper, we report some comparative simulations between lattice BGK and lattice Bolzmann schemes for two-dimensional fluid flows. A quantitative assessment of the validity of the lattice BGK and lattice Bolzmann schemes is presented for the two-dimensional weakly compressibleKolmogorov flow. We use this flow to study the difference of the two schemes at small scales. A lowReynolds (Re ∼ 300) number simulation shows the almost identical energy spectra for both schemes except for the small-scale dynamics of lattice Bolzmann which is more noisy. Because of the intrinsic difficulties of nonlinear stability analysis, we use numerical simulations to investigate which scheme is more stable. It turns out the lattice BGK is more stable. It turns out the lattice BGK is more robust than lattice Bolzmann by increasing theReynolds numbers. Detailed comparison with other methods (e.g., spectral method) remains to be done in the near future.

Three-dimensional lattice gas with minimal interactions

Abstract An interaction has been added to the classical lattice gas model that exchanges momentum between sites. The hydrodynamic limit of the model can be obtained from a Chapman-Enskog expansion and all the coefficients can be expressed explicitly. The interaction contributions to viscous terms are independent of the initial interaction-free model and grow like r2. When the interaction range r is large enough, the pressure acquires a negative slope for certain densities. The model then has a phase transition, observed in simulations. Some examples of a three-dimensional phase separation are shown. Whatever the dimension of space, the model can be represented by a cellular automaton with only nearest neighbor communications, using messenger particles or photons. In 2 dimensions of space, this allows fitting the model into a 16 bit table adequate for a special purpose cellular automaton machine like the RAP1.

Editorial: Dedication to Pierre Lallemand on the occasion of his retirement

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Lattice-gas hydrodynamics on the IBM 3090 vector facility

After a brief review of the means for characterizing lattice gases using cellular automate rules, the authors discuss the implementation of the rules for simulating hydrodynamic phenomena which can be described by the Navier-Stokes equations. Special emphasis is placed on data-mapping strategies and implementation through the use of the high speed and large memory resources offered by vector multiprocessors such as the IBM 3090 Vector Facility. The authors present performance data which pertain to square and hexagonal lattice gases, and discuss the limits of the approach used and its potential extendability to other areas.

Lattice gases and parallel processors

This paper introduces the notion of lattice gas as a new medium to perform fluid dymanics simulations. After giving the definition of lattice gases, the results of statistical analyses of their macroscopic behaviour are reviewed. It is shown that Navier-Stokes equations can be simulated. Some information is given concerning the algorithms and their possible adaptation to parallel hard-ware, including cellular automata. Finally the construction of a specialized machine for lattice gas simulations will be presented.