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## Über dieses Buch

This volume contains the proceedings of the symposium held on Friday 6 July 1990 at the University Pierre et Marie Curie (Paris VI), France, in honor of Professor Henri Cabannes on the occasion of his retirement. There were about one hundred participants from nine countries: Canada, France, Germany, Italy, Japan, Norway, Portugal, the Netherlands, and the USA. Many of his past students or his colleagues were among the participants. The twenty-six papers in this volume are written versions submitted by the authors and cover almost all the fields in which Professor Cabannes has actively worked for more than forty-five years. The papers are presented in four chapters: classical kinetic theory and fluid dynamics, discrete kinetic theory, applied fluid mechanics, and continuum mechanics. The editors would like to take this opportunity to thank the generous spon­ sors of the symposium: the University Pierre et Marie Curie, Commissariat a l'Energie Atomique (especially Academician R. Dautray and Dr. N. Camarcat) and Direction des Recherches et Etudes Techniques (especially Professor P. Lallemand). Many thanks are also due to all the participants for making the symposium a success. Finally, we thank Professor W. Beiglbock and his team at Springer-Verlag for producing this volume.

## Inhaltsverzeichnis

### Trend to Equilibrium in a Gas According to the Boltzmann Equation

Abstract
A proof that a gas in a container kept at constant and uniform temperature reaches a Maxwellian state is given. The cases of specularly reflecting walls and velocity reversing walls, previously considered by Desvillettes, are singular, in the sense that the Maxwellian is not uniquely determined by the boundary conditions.
C. Cercignani

### The Dirichlet Boundary Value Problem for B.G.K. Equation

Abstract
We prove the global existence of a solution to the B.G.K. model of Boltzmann Equation set in a bounded domain with Dirichlet boundary condition. Our proof combines the averaging lemmas and estimates on the second moment in v of the solution in order to pass to limit (strongly) in the temperature.
B. Perthame, A. Pham Ngoc Dinh

### Asymptotic Theory of a Steady Flow of a Rarefied Gas Past Bodies for Small Knudsen Numbers

Abstract
A survey is made of the asymptotic behavior for small Knudsen numbers of the time-independent solution of the boundary-value problem of the Boltzmann equation over a general domain. Included is the hydrodynamic system (hydrodynamic type equations and their slip boundary conditions) describing the asymptotic behavior, with several new results.
Y. Sone

### Computation of Transitional Rarefied Flow

Abstract
Rarefied flows around reentry vehicles can be evaluated by using a Monte-Carlo numerical simulation. Results for Hermes vehicle are presented and agree fairly well with wind-tunnel data.
As the gas density increases and the flow becomes more transi­tional, the Monte-Carlo simulation becomes much more expensive both in computational time and memory requirement. When rarefaction effects are effective only in the immediate vicinity of the vehicle, in the Knudsen layer, the flowfield can be numerically determined through the continuum approach by the Navier-Stokes equations with slip boundary conditions at the wall. The variational method deve­loped by F.Golse [1] and S.K.Loyalka [2] gives a practical way to obtain these boundary conditions in the case of gas with internal energy.
F. Coron, J. P. Pallegoix

### Gas Flows Around the Condensed Phase with Strong Evaporation or Condensation — Fluid Dynamic Equation and Its Boundary Condition on the Interface and Their Application —

Abstract
A gas in contact with its condensed phase is considered, and a steady gas flow around the condensed phase, on the surface of which strong evaporation or condensation is taking place, is investigated on the basis of the kinetic theory. The system of fluid dynamic equation and its boundary condition on the interface of the gas and its condensed phase that describes the gas flow in the continuum flow limit is derived systematically with the aid of the recent results of numerical analysis of the half-space problem of evaporation and condensation. The effect of condensation factor in the kinetic boundary condition on the fluid dynamic boundary condition is also discussed. As an application of the system, the gas flow between two parallel condensed phases with different temperatures is investigated.
K. Aoki, Y. Sone

### Fluid Dynamic Limits of Discrete Velocity Kinetic Equations

Abstract
The connection between discrete velocity kinetic theory and fluid dynamics is systematically described. Conditions that formally lead to generalized compressible Euler equations or to generalized incompressible Navier-Stokes equations are given. These conditions are related to an H-theorem. It is proven that a large class of polynomial collision operators in semidetailed balance satisfies this H-theorem. Finally, results are given concerning the global validity in time of the convergence for the case where the formal scaling of the kinetic equation leads to the linearized incompressible Navier-Stokes limit.
C. Bardos, F. Golse, D. Levermore

### On the Euler Equation in Discrete Kinetic Theory

Abstract
The aim of this note is to investigate some mathematical structure of the Euler equation derived from the discrete Boltzmann equation as the first order approximation of the Chapman-Enskog expansion. The analysis developed in this article would be a basis for the study of nonlinear waves in discrete kinetic theory (cf. [2,7,3] for shock waves, [10] for rarefaction waves, and [6] for diffusion waves).
S. Kawashima, N. Bellomo

### Existence globale et diffusion en théorie cinétique discrète

Résumé
L’objet de ce travail est de présenter, en renvoyant éventuellement à [3] [4] [5] pour les démonstrations, les résultats d’existence de solutions globales bornées du problème de Cauchy que nous avons obtenus pour les modèles généraux de la théorie cinétique discrète des gaz dans les deux cas suivants: données petites en dimension quelconque d’espace et données quelconques en dimension 1.
J.-M. Bony

### On the Cauchy Problem for the Semidiscrete Enskog Equation

Abstract
We prove that the semidiscrete Enskog equation, an analog of the semidiscrete Boltzmann equation introduced by H.Cabannes [1], has a global mild solution when the initial data are such that $[1 + {\left| {\vec x} \right|^\alpha } + Log\phi ]\phi \in {L_1}$.
G. Toscani, G. Borgioli, A. Pulvirenti

### On Uniform Boundedness of Solutions to Discrete Velocity Models in Several Dimensions

Abstract
The global existence question for discrete velocity models in more than one space dimension remains unsolved, except for initial values which are small in some sense ([2], [6], [7]). The crucial difficulty is that we do not seem to have the tools to obtain uniform L — bounds in time on the local solution in terms of the initial values (this, of course, would entail global existence of a mild solution). The purpose of this article is to compare the situation with the better understood one-dimensional case, spell out some crucial differences, and point out a possible way to progress.
R. Illner

### Temperature and Local Entropy Overshoots for the Second Fourteen-Velocity Cabannes Model

Abstract
For the Cabannes 14 velocities model with speeds $\sqrt 3$, 2, we construct different classes of similarity shock waves solutions and study temperature and local entropy overshoots. We observe temperature overshoots only when the shock front speed is higher than the speed of the slow particles. On the contrary the local entropy overshoot can always be present when the scaling parameter of the solutions is close to a critical value which limits the local entropy increase.
H. Cornille

### A One Dimensional Lattice Boltzmann Equation with Galilean Invariance

Abstract
A three-velocity one-dimensional Lattice Boltzmann Equation is presented. The stability of the equilibrium distribution is studied and the dynamical equations are given. It is shown that Galilean invariance can be recovered for the proper choice of the collision operator. This particular model is then used to compare the results of numerical simulations to the behavior of a shock tube.
Yue-Hong Qian, D. d’Humières, P. Lallemand

### The Euler Description for a Class of Discrete Models of Gases with Multiple Collisions

Abstract
Two difficulties arise in discrete kinetic theory with binary collisions only. The first one is the existence of some macroscopic variables other than mass, momentum and energy, in relation to parasite summational invariants. The second one concerns the anisotropic character generally related to the models. In order to eliminate these difficulties, multiple collisions are introduced, and some symmetry properties on the models are adopted. The Euler equations are then given for discrete models with different moduli.
P. Chauvat, F. Coulouvrat, R. Gatignol

### On the Semidiscrete Boltzmann Equation with Multiple Collisions

Abstract
This paper deals with the derivation, in the framework of the discrete kinetic theory, of an evolution equation for gas particles undergoing multiple collisions. In particular, we consider a gas of particles moving in all directions of the plane with only one velocity modulus and undergoing binary and triple collisions. An evolution equation is derived, the analysis of the Maxwellian state is studied in details, a formal H-theorem is derived and the formal derivation of the hydrodynamical equations is dealt with.
E. Longo, N. Bellomo

### A Discrete Velocity Model for Gases with Chemical Reactions of Dissociation and Recombination

Abstract
In the last years, discrete velocity models in kinetic theory of gases have had a great development. A rather large bibliography on this subject can be found in the review paper [1], This sucess is mainly due to the fact that discrete models can be used for the mathematical of rather complex physical phenomenologies such as multiple collisions between gas-particles or collisions between different gas-species (see, for instance, refs. [2, 3]).
R. Monaco, M. Pandolfi Bianchi

### Frozen and Equilibrium Speeds of Sound in Non-equilibrium Flows

Abstract
Efficient techniques have been developed, in the last decade, to achieve numerical solutions for the system of conservation laws that describe compressible inviscid flows. These techniques are founded on upwind formulations, such as the flux-vector or the flux-difference splitting. The high quality of the numerical results is related to the respect of the domains of dependence of propagating signals, a physical feature that is retained and preserved in the numerical procedure. The speed of sound plays a preminent role in describing the propagation of signals.
M. Pandolfi, R. Marsilio

### Nonlinear Propagation of Acoustic and Internal Waves in a Stratified Fluid

Abstract
The present article is a theoretical study of the combined propagation of acoustic and internal waves of finite amplitude in a (horizontally, say) stratified fluid. The linear theory of small amplitude waves predicts that the two waves (i.e., the fluctuations in acoustic pressure and in vertical particle velocity or displacement) are in general coupled. The waves are decoupled whenever the typical wavenumbers are large compared with the inverse of the scale height for the stratification. Furthermore, a horizontally propagating acoustic wave with zero vertical particle displacement (or velocity), when this is possible, is always decoupled from the internal waves, according to linear theory. In a weakly nonlinear theory, however, the waves may be coupled. A time independent radiation pressure is formed within a standing, horizontally directed acoustic wave, which produces a stationary vertical displacement of the particles. Experimental evidence of this effect has been reported elsewhere [1].
J. Naze-Tjøtta, S. Tjøtta

### A Higher Order Panel Method for Nonlinear Gravity Wave Simulation

Summary
We present an efficient higher order panel method for the numerical simulation of nonlinear gravity waves. The method is based on a Green’s formulation for the velocity potential that is introduced under the assumptions of an ideal fluid and an irrotational flow. This panel method gives accurate results for both linear and highly nonlinear waves. Test results are shown for a highly nonlinear Stokes wave and an overturning sinusoidal wave.
J. Broeze, E. F. G. van Daalen, P. J. Zandbergen

### Numerical Reliability of MHD Flow Calculations at High Hartmann Numbers

Abstract
In this paper, a new numerical investigation is performed for the two-dimensional MHD flow in a rectangular duct and an error analysis of the traditional calculation of solution is given.
Arbitrary values of the flow parameters: Hartmann number and wall conduction ratio can now be chosen, and the singular perturbation problem solved and analysed using the interval mathematics and verified inclusion methods (Ε-Methods). Furthermore, the error analysis of the traditional calculation applied to this MHD flow shows that there is a lack of reliability of the known numerical result of this physical problem, even for Hartmann numbers M ≤ 1000. These results indicate that the reliability of numerical data should, at least, be proved for any calculation with the control of rounding errors using an accurate floating-point arithmetic and inclusion methods of interval mathematics.
K. G. Roesner, W. U. Würfel

### Interaction Between an Oblique Shock and a Detached Shock Upstream of a Cylinder in Supersonic Flow

Abstract
This paper describes an experimental study concerning the interaction, in supersonic flow, between an oblique shock, generated by a wedge, and a detached shock upstream of a cylinder. The experiments were performed in the Berkeley supersonic wind tunnel at a Mach number of 2.4. The experiments differ from those carried out earlier in the application of new interferometric techniques to measure the density. These experimental results were compared with established numerical results. The data obtained were used in order to understand the phenomena which appear, for example, when an oblique shock is generated at the entry to a duct on the aerospace plane. It is known that these interactions can cause significant increases in pressure and heat transfer rate.
M. Holt, M. P. Loomis

### The Design of Super-Concorde and Space Vehicles Using Global Optimization Techniques

Abstract
The optimum-optimorum configuration of the space vehicle is the configuration for which the shapes of its surface and also of its planprojection are simultaneously determined in such a manner that its drag attains its minimum at a given cruising Mach number M. The problem of the determination of the optimum-optimorum configuration of a space vehicle of variable geometry which presents a minimum drag at two cruising Mach numbers M and M* are here also considered.
A. Nastase

### Vibrating Strings with Obstacles: The Analytic Study

Abstract
It is a very pleasant task for me to be here in Paris to pay honour to Professor Henri Cabannes on the occasion of his retirement. I remember with a great pleasure many opportunities of scientific contact with him in the last years, and above all the period I spent in this University three years ago. I thank him for his kindness, and I wish him to take again many satisfactions in continuing his activity.
C. Citrini

### Vibrations of Euler-Bernoulli Beams with Pointwise Obstacles

Abstract
In this paper, we discuss the numerical simulation of the vibrations of a beam in the presence of pointwise obstacles. We suppose that these vibrations are modeled by the Euler-Bernoulli equation for linear beams and that one extremity of the beam is clamped while the other may be rigidly attached to a rigid body. The numerical methodology is based on the following techniques: Hermite cubic finite elements for the space discretization, an energy preserving finite difference time discretization scheme, and a penalty treatment of the inequalities associated to the obstacles. The resulting methodology is robust, seems to be accurate and is easy to implement. The results of numerical experiments show the possibilities of the methods discussed here.
H. Carlsson, R. Glowinski

### Contribution to the Fracture Analysis of Composite Materials

Abstract
Within the generalized plane elasticity framework, the computation of singularities (stress concentration), together with a matched asymptotic expansion method, allow to build solutions to various problems involving a small perturbation: rigid inclusion, micro-void,… Taking a micro-crack as small perturbation brings to the analysis of brittle fracture and to the asymptotics of the energy release rate. A Griffith criterion divides the singularities into two classes, namely the strong and weak singularities. As an example, the case of a cracked ply embedded in a stratified composite is examined. With given assumptions on the toughness of the components and of the interface, the crack can propagate to become a delamination crack (i.e. lying at the interface between the components).
D. Leguillon, E. Sanchez-Palenia

### Homogenization Method Applied to Porous Media

Abstract
Homogenization method ([1], [2]) is used in the study of media with a microstructure on a scale which is very much smaller than the macroscopic scale of interest, the dimension of a specimen for instance. Under the assumption that the spacial distribution of the heterogeneities is, in some sense, periodic, it gives the passage from a microscopic description to a macroscopic description of a problem. Periodicity is an hypothesis which is convenient in order to obtain results in a precise mathematical form, and which may be used as a simplified model for more general situations. The fine periodic structure of the medium is associated with a small parameter ε. The homogenization process is an asymptotic study, as ε tends to zero, which gives firstly the local variations, in the period, of the field quantities and then leads to a rigourous deductive procedure for obtaining the macroscopic equations of the bulk behaviour.
T. Lévy

### Modal Analysis of Flexible Multibody Systems

Summary
We present a distributed-element method for vibration analysis of flexible multibody systems modelled by a chain of rigid and elastic bodies with tree structure. The method is based on the impedance matrix, which defines in frequency domain the linear transformation between the resultant forces and torques exerted on the boundaries of each body in the chain and the displacements of these boundaries. This impedance matrix is obtained by a spectral expansion in terms of a set of component modes.
M. Pascal

### Backmatter

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