Heat and Mass TransferSeries Editors: D. Mewes and E Mayinger

Ching Shen

Rarefied Gas DynamicsFundamentals, Simulations and Micro Flows

With 53 Figures

fyj Springer

Series EditorsProf. Dr.-Ing. Dieter Mewes Prof. em. Dr.-Ing. E.h. Franz MayingerUniversität Hannover Technische Universität MünchenInstitut für Verfahrenstechnik Lehrstuhl fur ThermodynamikCallinstr. 36 Boltzmannstr. 1530167 Hannover, Germany 85748 Garching, Germany

AuthorProf. Ching ShenChinese Academy of SciencesInstitute of MechanicsZhongguancun Rd. 15100080 BeijingPeople's Republic of China

Library of Congress Control Number: 2004116858

ISBN 3-540-23926-X Springer Berlin Heidelberg New YorkThis work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication ofthis publication or parts thereof is permitted only under the provisions of the German Copyright Lawof September 9, 1965, in its current version, and permission for use must always be obtained fromSpringer. Violations are liable to prosecution act under German Copyright Law.

Springer is a part of Springer Science + Business Media

springeronline.com

© Springer-Verlag Berlin Heidelberg 2005Printed in Germany

The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.

Typesetting: camera ready copy supplied by authorCover-Design: deblik, BerlinProduction: medionet AG, Berlin

Printed on acid-free paper 62/3141 Rw 5 4 3 2 10

PREFACE

Aerodynamics is a science engaged in the investigation of the motion of air and

other gases and their interaction with bodies, and is one of the most important

bases of the aeronautic and astronautic techniques. The continuous improvement

of the configurations of the airplanes and the space vehicles aid the constant

enhancement of their performances are closely related with the development of the

aerodynamics. In the design of new flying vehicles the aerodynamics will play

more and more important role.

The undertakings of aeronautics and astronautics in our country have gained

achievements of world interest, the aerodynamics community has made

outstanding contributions for the development of these undertakings and the

science of aerodynamics. To promote further the development of the

aerodynamics, meet the challenge in the new century, summary the experience,

cultivate the professional personnel and to serve better the cause of aeronautics

and astronautics and the national economy, the present Series of Modern

Aerodynamics is organized and published.

The Series of Modern Aerodynamics consists of about 20 monographs divided

into theoretical and experimental parts. The theoretical part includes: Theory of

transonic aerodynamics, Theory of inviscid hypersonic aerodynamics, Rarefied

gas dynamics, Computational fluid dynamics-fundamentals and applications of

finite difference methods, Spectral methods of computational fluid dynamics,

Finite element methods of fluid mechanics, Heat transfer of hypersonic gas and

ablation thermal protection, Dynamics of multiphase turbulent reacting fluid

flows, High temperature nonequilibrium air flows, Turbulence, Vortex stability,

Wing engineering and industrial aerodynamics, Aerodynamics for airplane design,

and others. The experimental part includes: Wind tunnel testing, Wind tunnel

balance, Interference and correction on wind tunnel testing, Impulsive wind

vi PREFACE

tunnels, Modern flow visualization, and others. The editors and writers of this

series uphold the following principles of writing. Firstly, it serves as a bridge from

the fundamental aerodynamics to the frontiers of the modern aerodynamics.

Secondly, it is a series of special monographs each devoted to a specialized topic.

Thirdly, the series pays attention not only to the already existing achievements but

also to the modern developments. Fourthly, each monograph introduces

systematically the knowledge and development of the special topic. Fifthly, the

series forms a complete whole, the monographs combined together cover various

fields of modern aerodynamics. To organize and promote the writing of the series

an editorial board with academician Zhuang Fenggan as its president was formed

in charge of working out the plane of writing, selecting writers, examining and

approving the manuscripts, recommending to the Judging Panel of the Publication

Foundation of National Defense Technical Books to apply for the financial

support. The Chinese Aerodynamics Research and Development Center supported

the work of the editorial board in the aspects of personnel and expenditure. Each

monograph of this series after applying and receiving the support from the

Publication Foundation of National Defense Technical Books was chosen and

edited by the National Defense Industry Press.

This English version of Rarefied Gas Dynamics published by Springer-Verlag,

Berlin/Heidelberg, is a translation from its Chinese edition revised and updated by

the author. In particular, the 7.7 section is rewritten as a new Chapter 8

'Microscale Slow Gas Flows, Information Preservation Method' to give more

comprehensive account of the subject and reflect some advances obtained after the

publication of the Chinese edition.

The flight, maneuver and braking of aerospace vehicles at high altitudes

demands from the gas dynamics the answer on questions of force and heat action

of the low density gas flow. When the density is lowered to a level that the mean

free path of the gas molecules is not a small magnitude in comparison with the

characteristic length of the flow, the ordinary methods of the continuum gas

dynamics are no longer suitable, the methods of discrete molecular gas dynamics,

or, of rarefied gas dynamics are required. Meanwhile the condition of high speed

flight leads to the necessity of consideration of the physical processes taking place

PREFACE vii

inside the molecules such as the excitation of the internal energies, the chemical

reactions and the excitation and transition of electronic levels. This leads to the

development and expansion of the gas dynamics towards micro scale scopes in

two aspects, i.e. the research of the gas flows with the discrete molecular effect

taken into account by involving the methods of rarefied gas dynamics, and the

consideration of the internal structure of the molecules. The application areas of

the rarefied gas dynamics besides aeronautics and astronautics include some

frontier realms of the advanced technical development such as: plasma material

processing in vacuum, micro-electronic etching, micro-electronic mechanic

systems and chemical industry. This makes the research on low speed rarefied gas

dynamics very important. In the world literature the research on gas dynamics on

the molecular level and on the internal physical processes inside molecules in the

gas flows is very active. Relatively the domestic works on these aspects is fewer.

From the viewpoint of the prospects of the subject and the demands of the

development of science and technology research on gas dynamics on the

micro-scale is a direction needs to be strengthened and further developed in our

country, and it is appropriate to advocate and promote in the scientific research

layout and education arrangement.

This book elucidates the methods of molecular gas dynamics or rarefied gas

dynamics which treat the problems of gas flows when the discrete molecular

effects of the gas prevail under the circumstances of low density, the emphases

being stressed on the basis of the methods, the direct simulation Monte Carlo

method applied to the simulation of non-equilibrium effects and the frontier

subjects related to low speed microscale rarefied gas flows. As the basis of the

discipline two chapters on molecular structure and basic kinetic theory are

introduced. The first chapter devotes a minimum space in brief and summarized

description of the molecular energy state structure and energy distribution as the

necessary basis for the investigation of the non-equilibrium in high enthalpy

rarefied gas flows. The second chapter discusses the basis of the kinetic theory

focusing on binary collisbns, Boltzmann equation and the equilibrium state of the

gas, including the phenomenological molecular models: the VHS model of G. A.

Bird and also the VSS model, the GHS model and the GSS model. The third

chapter discusses various realistic models of gas surface interactions, including the

viii PREFACE

reciprocity principle reflecting the detailed equilibrium, the CLL model based on

this principle and the application in direct simulation in the cases including

incomplete energy accommodation and internal energy exchange. The fourth

chapter deals with the free molecular flows. Chapter 5 discusses the continuum

equations and slip boundary conditions applied for slip flow regime, including

Burnett equation the usability of which has been proved to be penetrated to more

rarefied range. This chapter also includes the discussion of some simple problems

solved by the Navier-Stokes equation plus slip boundary conditions and the

problem of thermophoresis. Chapter 6 introduces with fair comprehensiveness and

generality various analytical and numerical methods developed in transition

regime. Chapter 7 introduces the direct simulation Monte Carlo (DSMC) method

with emphasis stressed on the specific issues encountered in dealing with

non-equilibrium rarefied gas dynamics, including the work of the author and his

colleague in treating the excitation and relaxation of the internal energy, the

chemical reactions and the general coding of simulation of the flow field around

complex configurations, i.e. the generalized acceptance-rejection method in

sampling from the distribution with singularities, the sterically dependent

chemical reaction model and the derivation of the Ahrrenius-Kooij form of

reaction coefficient and the new version of the position element algorithm of the

general code of the DSMC simulation etc. Chapter 8 is dedicated to the simulation

of low velocity micro scale rarefied gas flows which becomes an actual and urgent

task encountered by the rarefied gas dynamics in the 21st century with the rapid

development of the micro-electro-mechanical system (MEMS). Some methods of

solution of the rarefied flow problems is examined from the point of view of

utilization in simulating the flows in MEMS. The information preservation (IP)

method is introduced with a general description, some validation of the method

and a program demonstrating the method. The resolving of the boundary condition

regulation problem in MEMS by using the conservative scheme and the super

relaxation method is illustrated on the example of flow in long micro diannels.

The thin film air bearing problem with authentic length of the write/read head of

the Winchester hard disc drive is solved, and the use of the degenerated Reynolds

equation is suggested to solve the microchannel flow and to serve as a criterion of

PREFACE ix

the merit of strict kinetic theory for testing various methods intending to treat the

rarefied gas flows in MEMS.

In choosing the contents of the book the author proceeds out of the

consideration to elucidate various essential basics of the subject and is influenced

by his own interests, so the quotation of literature is far from complete, hoping the

forgiveness and understanding of all the scientists having made contributions to

the development of the subject.

This book provides a solid basis for engaging in the studying of the molecular

gas dynamics for the senior students and graduates in the aerospace and the

mechanical engineering departments of universities and colleges, giving them an

overall acquaintance of the modern development of rarefied gas dynamics in

various regimes and leading them to reach the frontier topics of the

non-equilibrium rarefied gas dynamics and the low speed microscale gas

dynamics. It will be also of benefit to the scientific and technical workers engaged

in the aerospace high altitude aerodynamic force and heating design and in the

research on gas flow in the MEMS when treating practical gas dynamics

problems.

The author would appreciate Prof. Hu Zhenhua and Prof. Fan Jing for many

helpful discussions concerning the contents of some sections of the book. The

author thanks Mr. Tian Dongbo, Mr. Jiang Jianzheng, Mr. Xie Chong, and Dr. Liu

Hongli who helped in making the drawings and layout of the book.

Zhongguancun, Beijing, China C. Shen

September, 2004

NOMENCLATURE

a sound speed, =*JykT/m ; radius of a sphere particle

ad dynamic factor

b distance of the closest approach, miss-distance

c the magnitude of the molecule velocity

c molecule velocity

c molecule thermal velocity

c() molecule average velocity, average velocity

cr relative velocity

c average thermal speed, = •J^kT/nm

cm most probable molecular thermal speed, =\J2kT/m

cr mean value of the relative speed

c's mean squire root thermal speed, = yßkT/m

cp specific heat at constant pressure

cv specific heat at constant volume

C flow conductance, =Q/Ap

Cm velocity slip coefficient

C thermal creep coefficient

Ct temperature jump coefficient

d molecular diameter

dHS molecular diameter of the hard sphere model

dms molecular diameter of the VHS model

dvss molecular diameter of the VSS model

DT thermal diffusion coefficient

xii NOMENCLATURE

e specific energy

E activation energy

/ velocity distribution function

fK another definition of velocity distribution function, = fl n

Fm single particle distribution function, = f/ N

F. incident flux

F specularly reflected flux

Fw diffusely reflected flux; impact stress originated from collision

F external force

g equilibrium state Maxwellian distribution

gj degeneracy

G dimensionless temperature gradient, = | V r | al To

h Planck constant, =6.6260755xKT 3 4 . / s ;

height of the channel; height of the write/read head over the drive

platter

ho height of the write/read head over the drive platter at the outlet

H Bol tzmann H function; dimensionless height =hlht

I moment of inertia

J rotational quantum number

k Boltzmann constant, = 1.380658xl CT23 JK~';

the ratio of the conductivity of gas to that of the particle, =Kgl Kp ,

k force constante

k, forward reaction rate constant

fr reverse reaction rate constant

K conductivity

Kn Knudsen number, =XIL

I unit normal vector

L characteristic length of the flow field; length of the write/read head in

the hard disc drive

NOMENCLATURE xiii

m molecular mass

mr reduced mass

Ma Mach number, —UI a

M molecular weight

mol gram molecular weight

n number density; vibrational quantum number

n0 Loschmidt number, = 2 .68666x l0 2 5 m 3

N total number of degrees of freedom of the internal degree variable £, ,

N Avogadro number, = 6 . 0 2 2 1 3 7 X 1 0 2 3 O T O / '

p pressure

psig pound per square inch, gauge, = 6.895kPa above aim

P pressure tensor

Pr Prandtl number, =cpfi/K

P dimensionless pressure, p/p

ql heat flux vector

q =q.

Q function of molecular velocity; partition function;

flow rate

Qc flow rate of the Couette flow

QF flow rate of the Poiseuille flow

R gas constant, =R IM = kl m

ranf random fraction between 0 and 1

Re Reynolds number, =pUL/n

R universal gas constant, =E.3\45\\Jmor'K '

S entropy; speed ratio, =Ulcm = UZ-JlRT

S(cr) ~CPT

T temperature

U velocity of oncoming flow

xiv NOMENCLATURE

us gas velocity near the surface, gas velocity at outer edge of Knudsen

layer

V potential energy of system

W dimensionless variable, =blr

X =x/L

Z relaxation collision number

Zv vibrational relaxation collision number

Z. rotational relaxation collision number

a thermal accommodation coefficient; the power exponent of the cosine

of the deflection angle in VSS scattering model

B beta function

ß reciprocal of the most probable thermal speed, =(2RT)~U2

Y ratio of specific heats

F gamma function

S spacing between molecules

<5;/ Kronecker tensor

£ molecular energy

Ej molecular dissociation energy

£ B molecular electronic energy

er molecular rotational energy

Et molecular translational energy

£v molecular vibrational energy

£ negative power exponent in the dependence of oT on £(

C, number of degrees of freedom; slip coefficient

f] power exponent in the inverse power law model

0 characteristic temperature

K constant in the force power law

A molecular mean free path

NOMENCLATURE xv

A mean free path in collision of the scattered molecules with the

oncoming molecules

A bearing number, =6/J,UL/p JiJ

jX viscosity, for hard sphere model ß ~ 1 /2(pcA)

v collision frequency

p density

a tangential momentum accommodation coefficient

a' normal m o m e n t u m accommodat ion coefficient

oD diffusion collision cross section

aM momen tum collision cross section

<7„ reaction collision cross sectionK

<JT total collision cross section

0^ viscosity collision cross section

T viscous stress tensor , = Ttj

T mean collision t ime, = A I c ; temperature

Tl 7 shear stress tensor

0 vibrational energy exchange probability

X deflection angle in collision

I// wave function

ft) power exponent of temperature in the viscosity law;

super relaxation factor

Q, number of ways of distributions of particles on different levels

SUPERSCRIPTS AND SUBSCRIPTS

e emitted, scattered; electronic

g gas

/ incident; internal

xvi NOMENCLATURE

mt

/

P

r

t

V

oo

*

a

M

internal

left

particle

rotational; right

translational

vibrational

oncoming flow value

values after collision

average of a

cut off value of a

ENGLISH ABBREVIATIONS

AFE vehicle aero-assisted flight experiment vehicle

BGK equation Bhatnagar-Gross-Krook equation

BKW equation Boltzmann-Krook-Welender equation

CALTECH California Institute of Technology

CFD computational fluid dynamics

CLL model Cercignani-Lampis-Lord model

DSMC method direct simulation Monte Carlo method

EDM electric discharge machining

FORTRAN language formula translation language

GHS model generalized hard sphere model

GSS model generalized soft sphere model

HS model hard sphere model

IP method information preservation method

LB method Larsen-Borgnakke method

LBM lattice Boltzmann method

NOMENCLATURE xvii

LIGA Lithographie Galvanoformung Abformung

(German), Lithographic electroforming

MD method molecular dynamics method

MEMS micro-electro-mechanical system

MIT Massachusetts Institute of Technology

NEMS nano-electro-mechanical system

NTC method no time counter method

RSF method randomly sampled frequency method

STP standard temperature and pressure

SSTO vehicle single stage to orbit vehicle

TC method time counter method

UCLA University of California, Los Angeles

VHS model variable hard sphere model

VSS model variable soft sphere model

CONTENTS

Preface v

Nomenclature xi

0 Introduction 1

0.1 The Conception of Rarefied Gas Dynamics 1

0.2 The Molecular Model of Gases 3

0.3 Mean Free Path of Molecules 4

0.4 Division of Flow Regimes 5

0.5 NonequilibriumPhenomena and Rarefied Gas Dynamics 9

0.6 Similarity Criteria 13

References 18

1 Molecular Structure and Energy States 21

1.1 Diatomic Molecules 21

1.2 Energy Distribution of Molecules 30

1.2.1 Boltzmann's Relation 32

1.2.2 Calculation of The Number £1 of Microscopic States 34

1.2.3 Boltzmann Distribution 37

1.3 Internal Energy Distribution Functions 43

References 50

2 Some Basic Concepts of Kinetic Theory 51

2.1 The Velocity Distribution Function 51

2.2 Macroscopic Properties 53

2.3 Binary Elastic Collisions of Molecules 61

2.4 Collision Cross-sections and Molecule Models 69

2.4.1 Hard Sphere Model 72

xx CONTENTS

2.4.2 The Inverse Power Law Model 74

2.4.3 Maxwell Model 76

2.4.4 Variable Hard Sphere (VHS) Model 77

2.4.5 Variable Soft Sphere (VSS) Model 80

2.4.6 Generalized Hard Sphere (GHS) Model 85

2.4.7 Generalized Soft Sphere (GSS) Model 87

2.5 The Eight Velocity Gas Model 88

2.6 Boltzmann Equation 92

2.7 Collis ion Integral and The Total Number of Collisions 98

2.8 Evaluation of Collision Integrals 100

2.9 The Maxwell Transport Equation - The Moment Equation 104

2.10 Maxwell Distribution 106

2.11 Equilibrium State of Gases 112

2.11.1 Some Peculiar Speeds of Gas 112

2.11.2 Molecular Collision Frequency and The Mean Free Path.... 115

2.11.3 The Mean Value of Collision Quantities 120

2.11.4 The Reference Diameter of The VSS Model and The VHS

Model 123

2.12 Gas Mixture 124

2.12.1 The Macroscopic Properties 124

2.12.2 The Boltzmann Equations 126

2.12.3 Number of Collisions, Collision Frequency and Mean Free

Path 126

2.12.4 Collision Frequency of a Molecule of species A with

Molecules of Species B in Gas Mixture of VSS (or VHS)

Molecules 127

References 128

Interaction of Molecules with Solid Surface 131

3.1 Introduction 131

3.2 Specular and Diffuse Reflection 132

3.3 The Reciprocity Principle 140

3.4 The CLL Gas Surface Interaction Model 142

References 158

CONTENTS xxi

Free Molecular Flow 159

4.1 The Number Flux and The Momentum Flux of Molecules in Gases

160

4.2 The Aerodynamic Forces Acted on Bodies 163

4.3 Heat Transfer to Surface Element 170

4.4 Free Molecular Effusion and Thermal Transpiration. 174

4.5 Couette Flow and Heat Transfer between Plane Plates 177

4.6 The general solutions, unsteady flow 182

References 190

Continuum Models 191

5.1 Introduction 191

5.2 Basic Equations 192

5.2.1 Equations of Mass, Momentum and Energy Conservation.. 192

5.2.2 Chapman-Enskog Expansion 193

5.2.3 Euler Equation 194

5.2.4 Navier-Stokes Equations 194

5.2.5 Burnett Equations 196

5.2.6 Grad's Thirteen Moment Equations 201

5.2.7 The Asymptotic Theory for Small Knudsen Numbers 203

5.3 Slip Boundary Conditions 204

5.3.1 The Simple Derivation 204

5.3.2 The Conservation of Momentum and Energy Fluxes in The

Knudsen Layer 206

5.3.3 The Derivation of The Slip Velocity Formula 207

5.3.4 The Derivation of The Temperature Jump Expression 209

5.3.5 The Extension to Cases of Multi-component Gases and

Non-equilibrium Flows 212

5.4 The Solution of Some Simple Problems 212

5.4.1 Couette Flow 213

5.4.2 The Poiseuille Flow 215

5.4.3 The Rayleigh Problem 218

5.5 Thermal Creep and Thermophoresis 220

5.6 Second Order Slip-jump Conditions 227

xxii CONTENTS

References 228

Transitional Regime 231

6.1 General Overview 231

6.2 Linearized Boltzmann Equation 233

6.3 The Moment Method 239

6.4 Model Equations 247

6.5 The Finite Difference Method 255

6.6 Discrete Ordinate Method 257

6.7 Integral Methods 263

6.8 Direct Simulation Methods 264

References 269

Direct Simulation Monte-Carlo (DSMC) Method 275

7.1 Introduction 275

7.2 Sampling of Collisions 278

7.3 Example of Solution of Problem by The DSMC Method 281

7.4 The Excitation and Relaxation of The Internal Energies 288

7.4.1 Introduction of Phenomenological Models 288

7.4.2 Implementation of Larsen-Borgnakke Model 289

7.4.3 Cases of Distributions with Singularities, Generalized

Acceptance-rejection Method 293

7.4.4 Larsen- Borgnakke Method for Discrete Energy Levels 295

7.4.5 Relaxation Collision Number and Vibrational Exchange

Probability 297

7.5 Simulation of Chemical Reactions 299

7.5.1 Chemical Reaction Rate Coefficient 299

7.5.2 Phenomenological Chemical Reaction Model of Bird 300

7.5.3 A Sterically Dependent Chemical Reaction Model 302

7.6 Computation of Comp Heated Flow Fields 310

References 313

Microscale Slow Gas Flows, Information Preservation Method 317

8.1 Introduction 317

CONTENTS xxiii

8.2 Methods for Solving The Rarefied Gas Flows in MEMS 321

8.3 Information Preservation (IP) Method 326

8.3.1 The Description of The Method 326

8.3.2 The Validation of The Method 329

8.3.3 Program Demonstrating The Method 332

8.4 Unidirectional Flows 333

8.5 The MicroChannel Flow Problem 338

8.6 Thin Film Air Bearing Problem 348

8.7 Use of Degenerated Reynolds Equation in Channel Flow 355

8.8 Some Actual Problems and Concluding Remarks 360

References 363

Appendix I Gas Properties 367

References 368

Appendix II Some Integrals 369

II. 1 The gamma Function and Error Function 369

11.2 Some Definite Integrals 370

11.3 The beta Function 373

References 374

Appendix III Sampling from a Prescribed Distribution 375

111.1 Inversion of Cumulative Distribution Function 375

111.2 Acceptance-rejection Method 378

111.3 Generalized Acceptance-rejection Method 378

References 381

Appendix IV Program of The Couette Flow 383

Subject Index 399