Discrete Models of Fluid Dynamics

The following sections are included:

• Organizing Committees

• ACKNOWLEDGEMENTS

• PREFACE

• CONTENTS

The aim of this paper is to describe a method allowing the determination of summational invariants in discrete kinetic theory. With the velocity vectors, we construct two linear spaces : one over ℚ (ℚ is the set of rational numbers) and the other over ℝ (ℝ is the set of real numbers). We prove that, if we take into account all the collisions, the possible presence of spurious invariants results from the fact that the dimension of the ℚ-vectorial space generated by a family of vectors is greater than the dimension of the ℝ-vectorial space generated by the same vectors. Moreover, we give a physical interpretation to all the invariants.

The trend to equilibrium can be completely different for the usual kinetic theory with continuous velocity and discrete velocity models. Here a theorem of trend to equilibrium for continuous velocities is proved and a few cases of absence of trend to a homogeneous equilibrium for kinetic models are discussed.

The discrete Boltzmann equation for a mixture of four gases undergoing to chemical reactions of bi-molecular type is derived. Thermodynamical equilibrium for such gases is investigated, deriving from the kinetic equations the mass-action law.

The following sections are included:

• Introduction

• The Euler and the Navier-Stokes equations

• Basic assumption

• Diffusion waves

• Shock waves (single mode)

• Rarefaction waves (single mode)

• Superposition of shock or rarefaction waves

• References

The following sections are included:

• Introduction

• Green-Kubo theory, hydrodynamics for Ginzburg-Landau models

• Kawasaki evolutions in the van der Waals limit

• REFERENCES

We prove that for some discrete models of the Boltzmann equation, the initial value problem possesses a global solution in time, even for partially negative initial data.

We show that for all usual 2D hexagonal models, the only conserved quantities being the total mass, the total momentum and the total energy, one can define a temperature as a function of the density and the internal energy, in such a way that for low mean velocities the heat flux exactly obeys a Fourrier law. This expression for the heat flux is derived from the Boltzmann equations using a Chapman-Enskog method. In order to illustate this property we propose numerical simulations of heat diffusion experiments.

We apply the general scheme given in [7] for discrete Boltzmann equations on a tangent fiber bundle to obtain the Maxwell-Boltzmann system satisfied by a plasma where the velocities take only a finite number of values. We show that the system is non strictly hyperbolic in the sense of Leray-Ohya [6]. We consider then the case of a plasma in a given electromagnetic field (E,H). We construct solutions with E = 0, H the magnetic field of a rotating body.

In the recent years, a new type of fully discrete Boltzmann equation, arising from the micro-dynamics of Lattice Gas, i.e. a special a class of cellular automata specifically designed to simulate hydrodynamical behaviour [1], has proved to perform quite competitively for the numerical simulation of a wide variety of complex hydrodynamical phenomena. This special type of Boltzmann equation, normally referred to as Lattice Boltzmann Equation (LBE), has in fact successfully been employed in the simulation of a wide span of hydrodynamical regimes, ranging from laminar flows in porous media to fully developed turbulent flows [2]. In this paper, we present some recent developments in the theory of LBE together with a quantitative discussion on the possibility to extend LBB to the domain of two-dimensional magneto-hydrodynatnics.

The objective of this article is to present recent progress on a paradigmatic boundary value problem for the steady Boltzmann equation. A comparison with older results for discrete velocity models is also done, and some open problems are suggested.

Existence and uniqueness of solutions of a boundary value problem (BVP) with general boundary data for the Carleman model of the Boltzmann equation are investigated. Necessary and sufficient conditions for existence of the solutions, and results of a numerical study of a trend to equilibrium of the underlining initial boundary value problem (IBVP) are discussed. Explicit examples of the nonunique positive solutions are given.

We derive a local existence and uniqueness result for the Broadwell model in a plane box with specular reflection, by using the classical approximation method of the fractional steps.

For the discrete Boltzmann models in R^q+1 q=1,2,3, with two speeds ,1 or 2 we consider the shock waves along an axis for both the square 8v_i, cubic 14v_i and hypercubic 24v_i models. They satisfy two classes of q-dependent non linear equations and the main difference comes from the projections of the speeds along the axis which are either 1 or 1,2. First we compare the two methods of shock waves solutions: either the Rankine-Hugoniot relations for traveling waves or the similarity solutions. The first method does not always lead to positive cross-sections. Second we study the local entropy and temperature overshoots across the shock.

This paper deals with the coupling of the nonlinear heat equation with the discrete Boltzmann equation in a class of initial–boundary value problems. The starting point is the formulation of the problem, then suitable coupling equations are used in order to derive a self-consistent set of equations for the whole system.

An extended FHP lattice gas model with two kinds of particles is introduced in order to simulate the dynamics of a frontal interface between a salt and fresh water body. First, the hydrodynamical similarity has been checked for the given lattice gas flow whereby the overall Richardson number Ri is verified to be the governing dimensionless parameter. From the time development of the flow pattern it is examined how the convergence of the flow is build up and how the stability of the front and its velocity depend on Ri for 0 ≤ Ri ≤ 1.

The velocity autocorrelation function (VACF) in a 1-D 5-bits model, the 2-D FHP-III model and the quasi 3-D FCHC model are investigated using mode coupling theory accounting for finite size corrections and for all possible pairs of hydrodynamic modes. This extended mode coupling theory describes the molecular dynamics data of Frenkel c.s. quantitatively from about three mean free times on. For times upto two mean free times the M.D. data agree with the predictions from the Boltzmann approximation. However, in the quasi 3-D model, the M.D.-results exhibit unexplained deviations in the early stages from the Boltzmann values and in the intermediate stage from those obtained with the extended mode coupling theory.

A plane four velocity model of the Uehling-Uhlenbeck equation is used to study the Couette flow. Assuming the Maxwellian law of reflection of molecules from the boundaries we solve the problem explicitly. Graphs representing the profiles of the mean velocity are given.

After introduction of the concept temperature, consistent with thermodynamics and irreversible thermodynamics, Green-Kubo formulas are derived for the heat conductivity and the viscosities and evaluated in Boltzmann approximation. The bulk viscosity vanishes identically. The conservation laws determine which gradients of thermodynamic variables should be considered as driving forces, and guarantee transport coefficients consistent with irreversible thermodynamics.

The following sections are included:

• INTRODUCTION

• KINETIC PROPAGATOR

• HYDRODYNAMIC MODES

• GEOMETRIC STAGGERED MODES

• DYNAMIC STAGGERED MODES

• ACKNOWLEDGEMENTS

• REFERENCES

Using the microcanonical ensemble formalism we obtain the equilibrium entropy density of a nine velocities lattice gas automaton on a square lattice. From the entropy all the thermodynamic equilibrium properties follow but we focus on the temperature. Our results are exact, so that any definition of temperature used when studying the hydrodynamic behavior of the model should reduce to the thermodynamic temperature.

We find conditions under which a discrete lattice introduced in [1] converges for a finite time to the plane Broadwell model in a square box.

We construct a class of probabilistic automata for the Ginzburg-Landau equation : , with x real and where f(x) is a polynomial whose maximum degree is set by the lattice symmetry. This class of Ginzburg-Landau equations is commonly used for the phenomenological description of dynamical phase transitions with a non-conserved order parameter, and in particular for reaction-diffusion systems. The lattice gas automaton method provides a microscopic approach to this class of systems, with the virtue that the automaton constitutes a minimal model (that is with highly simplified microdynamics) which preserve the full system dynamics…

In this paper, we report recent lattice gas simulations for single-phase and two-phase fluid flows for two dimensional problems using the Connection Machine-2. For the single-phase fluid problem, we use the standard 7-bit lattice gas model with the maximum collision rules. The velocity and vorticity field of the Kelvin-Helmholtz instability is studied. It is shown that the lattice gas method preserves the main properties of the flow patterns observed in other numerical simulations. Using colored particles and holes, the lattice gas method is extended to simulate immiscible fluids with adjustable surface tension, using a purely local collision scheme. The locality of this model allows us to implement a very fast and parallel algorithm on the Connection Machine-2. Because this new model correctly describes short-range particle-particle interactions between liquids and also particle-solid interactions between the fluid and the wall, cohesion and wettability can be simulated. Applications of the current model to several physical systems including spinodal decomposition, Rayleigh-Taylor flows and wettability in two-phase flows through porous media are discussed.

The following sections are included:

• LIST OF PARTICIPANTS

• Series on Advances In Mathematics for Applied Sciences