The Cumulant Method

117 Pages·2003·2.1 MB·English

The Cumulant Method
von der
Fakult¨at fu¨r Naturwissenschaften
genehmigte
Dissertation
zur Erlangung des akademischen Grades
doctor rerum naturalium
(Dr. rer. nat.)
vorgelegt von Dipl.-Phys. Steffen Seeger,
geboren am 20. April 1974 in Karl-Marx-Stadt,
eingereicht am 20. Mai 2003.
Gutachter: Prof. Dr. Karl Heinz Hoffmann
Prof. Dr. Michael Schreiber
Prof. Dr. Dr.h.c. Ingo Mu¨ller
Tag der Verteidigung: 9. September 2003
http://archiv.tu-chemnitz.de/pub/2003/0120
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Bibliographische Beschreibung
Seeger,Steffen
The Cumulant Method
Technische Universit¨at Chemnitz, Fakult¨at fu¨r Naturwissenschaften
Dissertation, 2003 (in englischer Sprache)
117 Seiten, 20 Abbildungen, 9 Abbildungen im Text, 76 Literaturzitate
Referat
IndieserArbeitwirdeineneueMethodezurReduktionderBoltzmann-Gleichung
auf ein System partieller Differentialgleichungen diskutiert. Nach einer kurzen
Einfu¨hrungindiekinetischeTheorieeinerMischunginerterGasewirdeinU¨berblick
in die aus der Literatur bekannten Momentenmethoden gegeben.
Der anschließend vorgestellten Kumulantenmethode liegt die Annahme zugrunde,
daß durch Stoßprozesse in einem Gas Korrelationen h¨oherer Ordnung schneller
abgebaut werden als solche niedrigerer Ordnung. Basierend auf dieser Annahme
werden die Bewegungsgleichungen fu¨r die Kumulanten und die Produktionsterme
der resultierenden Bilanzgleichungen fu¨r eine Mischung inerter Maxwell-Gase
berechnet.DieUntersuchungderRelaxationzumGleichgewichterlaubtdenBezug
zu bekannten Modellen der Kontinuumsmechanik und untermauert die Gu¨ltigkeit
der Annahme fu¨r die Begru¨ndung des o.g. Ansatzes in diesem Fall.
ImzweitenTeilderArbeitwerdendieErgebnissenumerischerUntersuchungenvor-
gestellt, wobei Simulationen mit verschiedenen Randbedingungen fu¨r Couette-
und Poiseulle-Str¨omungen durchgefu¨hrt wurden. Es werden verschiedene Eigen-
schaften von Modellen fu¨r verdu¨nnte Gase als auch des Navier-Stokes-Modells
beobachtet. Dabei ist jedoch eine sehr starke Abh¨angigkeit von den angewende-
ten Randbedingungen festzustellen. Abschließend werden Momentenmethoden als
eine besondere Form von Diskretisierungen der Boltzmann-Gleichung nach der
Methode der gewichteten Residuen diskutiert, was einen Ausblick auf zuku¨nftige
Arbeiten erlaubt.
Schlagworte
Statistische Physik, Kinetische Theorie, Boltzmanngleichung, Momentenmethode,
Kumulantenmethode, Gasdynamik, Fluiddynamik, Bilanzgleichungen, Hyperboli-
sche Differentialgleichungen
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Contents
List of Figures 7
1 Introduction 9
1.1 Ideal Gas in Equilibrium . . . . . . . . . . . . . . . . . . . . . 11
1.2 Standard Conditions and Dimensions . . . . . . . . . . . . . . 12
2 Kinetic Theory 15
2.1 The Boltzmann Equation . . . . . . . . . . . . . . . . . . . 17
2.2 The Collision Operator . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Entropy and Equilibrium . . . . . . . . . . . . . . . . . . . . . 25
2.4 The Moment Method(s) . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 General Properties . . . . . . . . . . . . . . . . . . . . 28
2.4.2 Maxwell’s Moment System . . . . . . . . . . . . . . 30
2.4.3 Grad’s Method of Moments . . . . . . . . . . . . . . . 31
2.4.4 Extended Irreversible Thermodynamics . . . . . . . . . 33
2.4.5 Eu’s Modified Moment Method . . . . . . . . . . . . . 35
2.4.6 Levermore’s Moment System . . . . . . . . . . . . . 36
2.4.7 Lattice-Boltzmann Methods . . . . . . . . . . . . . . 37
2.4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 The Cumulant Method 41
3.1 Interpretation of the Moments . . . . . . . . . . . . . . . . . . 43
3.2 The Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . 44
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6 CONTENTS
3.3 The Ansatz Function ϕCM . . . . . . . . . . . . . . . . . . . . 45
s
3.4 The Equations of Motion for C . . . . . . . . . . . . . . . . . 49
s
3.5 Production Terms for ϕCM . . . . . . . . . . . . . . . . . . . . 53
s
3.5.1 2D Maxwell Mixture . . . . . . . . . . . . . . . . . . 53
3.5.2 2D Maxwell Gas . . . . . . . . . . . . . . . . . . . . 56
3.5.3 2D Maxwell Mixture, Close to Equilibrium . . . . . 57
3.5.4 Classic Linearization . . . . . . . . . . . . . . . . . . . 60
3.5.5 2D BGK Mixture . . . . . . . . . . . . . . . . . . . . . 60
3.5.6 2D BGK Gas . . . . . . . . . . . . . . . . . . . . . . . 62
3.6 Approach To Equilibrium . . . . . . . . . . . . . . . . . . . . 62
3.7 Relation to the Navier-Stokes Model . . . . . . . . . . . . . 67
3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Numerical Simulation 71
4.1 A Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . 76
4.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.1 Adiabatic Slip . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.2 Adiabatic No-Slip . . . . . . . . . . . . . . . . . . . . . 79
4.2.3 Thermal No-Slip . . . . . . . . . . . . . . . . . . . . . 79
4.2.4 Navier-Stokes . . . . . . . . . . . . . . . . . . . . . 80
4.3 Simulation of a Gas in a Slab . . . . . . . . . . . . . . . . . . 80
4.3.1 Couette Flow . . . . . . . . . . . . . . . . . . . . . . 81
4.3.2 Poiseulle Flow . . . . . . . . . . . . . . . . . . . . . 88
4.4 A Weighted Residual Formulation . . . . . . . . . . . . . . . . 92
5 Conclusions 97
A Adiabatic Boundaries 101
Bibliography 105
Zusammenfassung 112
List of Figures
1.1 Ideal Gas contained in a domain Ω . . . . . . . . . . . . . . . 11
2.1 Notation and geometry for a collision of two particles . . . . . 21
2.2 Scattering angle vs. impact parameter . . . . . . . . . . . . . 25
2.3 Phase space density approximated by Grad-ansatz . . . . . . 32
3.1 Relation of the Mathematica notebooks . . . . . . . . . . . 42
3.2 Collision operator eigenvalues vs. approximation order. . . . . 66
4.1 Convection tensor eigenvalues vs. approximation order . . . . 72
4.2 Convection tensor eigenvalues vs. velocity . . . . . . . . . . . . 73
4.3 Convection tensor eigenvalues vs. specific energy . . . . . . . . 73
4.4 Convection tensor eigenvalues vs. normal stress . . . . . . . . 74
4.5 Convection tensor eigenvalues vs. energy flux . . . . . . . . . . 75
4.6 Couette flow: results for adiabatic no-slip boundary . . . . . 82
4.7 Couette flow: heat flux predicted by molecular dynamics . . 83
4.8 Couette flow: results for isodense algorithm . . . . . . . . . 84
4.9 Couette flow: results for isobar algorithm . . . . . . . . . . 85
4.10 Couette flow: convection tensor eigenvalues . . . . . . . . . 87
4.11 Poiseulle flow: temperature predicted by molecular dynamics 88
4.12 Poiseulle flow: convection tensor eigenvalues . . . . . . . . . 89
4.13 Poiseulle flow: results for isodense algorithm . . . . . . . . 90
4.14 Poiseulle flow: results for isobar algorithm . . . . . . . . . . 91
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8 LIST OF FIGURES
Chapter 1
Introduction
In the past decades computational gas dynamics has become an indispens-
able tool in science and engineering. Understanding the underlying effects
and mechanisms of momentum, heat and mass transfer is a requirement for
succeeding in use, control and optimization of these processes. Mathemat-
ical and computational techniques have made fast progress over the years,
so that numerical solutions for complex problems can be obtained today. It
is interesting that despite the breathtaking development of computing ma-
chinery, the major gains in complexity of solvable problems are achieved by
sophisticated methods of solution, and not so much due to faster computers.
Most methods start from partial differential equations that pose a particular
physical model of the gas and try to successively reduce the complexity of
the problem from a set of partial differential equations to a set of ordinary
differential equations to a linear system of algebraic equations. These models
assume that a continuum description can be applied, even though we know
that gases are composed of ‘particles’ interacting with each another. These
particles are mainly atoms and molecules, a ‘few’ ions but also significant
numbers of larger particles such as pollen, bacteria, dust and others arising
from a number of natural sources.
So it is not surprising that the models used as starting points (such as the
Euler or Navier-Stokes equations) are valid only for particular flow con-
ditions: the average distance between two subsequent particle collisions (the
mean free path) must be small compared to a typical length of the flow
(e.g. radius of an object in the flow). If this is not the case (e.g. the flow
of rarefied gases), the use of statistical descriptions as in the kinetic theory
is required. Leading to an integro-differential equation proposed by Boltz-
mann, kinetic theory became a practical tool for aerospace engineers first.
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10 CHAPTER 1. INTRODUCTION
Today there is a wide field of application: environmental problems, aerosol
and micro-pore reactors, physical and chemical vapor deposition and many
more. Last but not least the recent development of micro-machines with
typical sizes of a few µm to a few mm can make phenomena of rarefied gas
flow a basis of important systems of such scale. For these problems, the flow
conditions are often between the extremes of small mean free paths and col-
lisionless (so-called free-molecular) flow. This poses the question if – similar
to adaptive discretization in modern methods of solution for partial differ-
ential equations – one can adapt the underlying physical model of the flow.
This leads to many non-trivial questions [01] which are the topic of inten-
sive, current research: How to derive a consistent series of approximations
between the Euler equations on one hand and the Boltzmann equation
on the other? How to determine the validity of the various models? How to
know when we have to improve the physical model or when it is safe to avoid
computations by using a simpler one?
The present work discusses some of these questions by presenting a new
method to reduce the Boltzmann equation to a series of partial differential
equations, applying this method to the model of particles with Maxwell in-
teraction and a single relaxation time ansatz, where analytical and numerical
results can be obtained.
This chapter recalls some results of the statistical description of an ideal
gas in equilibrium and defines the conventions used for the presentation of
numeric results. Chapter 2 introduces a set of Boltzmann equations for a
non-reacting mixture of gases and briefly reviews so-called moment methods,
used to reduce the Boltzmann equation to a set of partial differential equa-
tions. Chapter 3 discusses the mathematical significance of moments and
presents the cumulant method. This new ansatz, made in velocity Fourier
space, was developed by the author and allows derivation of a set of moment
equations in their full non-linear form. To validate the basic assumption for
the ansatz – that high-order parameters characterize faster decaying correla-
tions – the method is applied to derive equations for Maxwell and BGK
gases and the resulting equations are discussed for states close to thermody-
namic equilibrium where we find the principal assumption confirmed. Then
their relation to the Navier-Stokes model of fluids is shown. Chapter 4
presents some numerical results regarding the domain of validity, as well as
results from numerical solution of some flow problems. It shows that a de-
scription of the interaction of the gas particles provided by the Boltzmann
equation is only one part. A description of the interaction of gas particles
with the solid and the liquid surfaces surrounding it appears to be equally
important. Chapter 5 closes with conclusions from and review of the results.