Multiscale and Multiphysics Flow Simulations of Using the Boltzmann Equation: Applications to Porous Media and MEMS

171 Pages·2020·6.311 MB·English

by  Jun Li

Jun Li
Multiscale and
Multiphysics
Flow Simulations
of Using the
Boltzmann Equation
Applications to Porous Media and MEMS
Multiscale and Multiphysics Flow Simulations
of Using the Boltzmann Equation
Jun Li
Multiscale and Multiphysics
Flow Simulations of Using
the Boltzmann Equation
Applications to Porous Media and MEMS
123
Jun Li
Centerfor Integrative Petroleum Research
Collegeof Petroleum Engineering andGeosciences
KingFahd University of Petroleum andMinerals
Dhahran, SaudiArabia
ISBN978-3-030-26465-9 ISBN978-3-030-26466-6 (eBook)
https://doi.org/10.1007/978-3-030-26466-6
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Preface
This book presents a concise introduction to the kinetic theory, based on which
different simulation methods that were independently developed for solving prob-
lemsofdifferent fieldscanbenaturallyrelatedtoeachother.Then,theadvantages
and disadvantages of different methods will be discussed with reference to each
other.Itmainlycovers four advanced simulationmethods based ontheBoltzmann
equation (i.e., direct simulation Monte Carlo method, direct simulation BGK
method, discrete velocity method, and lattice Boltzmann method) and their appli-
cations with detailed results.
I tried to present an introduction to different theories offluid dynamics starting
fromthetraditionalmacroscopicdescription(i.e.,Navier-Stokes-Fourierequations),
which ends with discussion on its limitation in solving gas flows of high Knudsen
number (Kn), including low-speed gas flows in vacuum systems, micro/nano-
electro-mechanical systems and unconventional rock, as well as supersonic gas
flows. This limitation is usually due to the invalidity of the ordinary constitutive
equations and boundary conditions at high Kn. Subsequently, the microscopic sta-
tisticaldescriptionbasedontheBoltzmannequationatthemolecularscaleaswellas
therelevant simulation methods with applications isintroduced.
In discussing the existing simulation methods, it was necessary to omit some
topics,whicharedetailedelsewhereasindicatedinthereferences,tomakethebook
concise. This omission does not compromise the aim of the book, which is to
provide a basic understanding of the potentials of different simulation methods
based on the Boltzmann equation in solving different problems at multiscale. The
choiceofreferencecoveragecertainlyreflectsapersonalpreferenceandIapologize
foranyomissionofotherimportantandrelevantwork.Thisbookcanservereaders
whoareinterestedinthemathematical derivationsofdifferent theoriesinChaps.1
and 2, or the relevant simulation methods and applications in the subsequent
chapters.
Dhahran, Saudi Arabia Jun Li
May 2019
vii
Contents
1 Fluid Mechanics Based on Continuum Assumption . . . . . . . . . . . . . 1
1.1 Continuum Assumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Macroscopic Descriptions of Flow Phenomenon. . . . . . . . . . . . . 3
1.3 Derivations of Macroscopic Governing Equations. . . . . . . . . . . . 5
1.3.1 Force and Heat Interactions Between Fluid Parcels . . . . 5
1.3.2 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.5 Entropy Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.6 Convection-Diffusion Equation . . . . . . . . . . . . . . . . . . . 13
1.4 Transformation to Arbitrary Non-inertial Reference Frame . . . . . 14
1.5 Component Form in Arbitrary Orthogonal Curvilinear
Coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Limitations of Continuum Assumption. . . . . . . . . . . . . . . . . . . . 21
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Distribution Function in the Phase Space . . . . . . . . . . . . . . . . . . 25
2.2 Intermolecular Collision Dynamics . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Molecular Models for Transport Coefficients . . . . . . . . . . . . . . . 33
2.4 Derivation of the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . 38
2.5 Calculations of Macroscopic Properties . . . . . . . . . . . . . . . . . . . 41
2.6 Moment Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.7 H-Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.8 Properties at Equilibrium State . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.9 Simplification to Model Equation . . . . . . . . . . . . . . . . . . . . . . . 53
2.10 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.10.1 Specular Reflection Model . . . . . . . . . . . . . . . . . . . . . . 58
ix
x Contents
2.10.2 Diffuse Reflection Model . . . . . . . . . . . . . . . . . . . . . . . 59
2.10.3 CLL Reflection Model . . . . . . . . . . . . . . . . . . . . . . . . . 61
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3 Simulation Methods for Rarefied Gas Flows. . . . . . . . . . . . . . . . . . . 67
3.1 Direct Simulation Monte Carlo Method . . . . . . . . . . . . . . . . . . . 67
3.2 Direct Simulation BGK Method . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.1 Initialization Process. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2.2 Molecular Motion and Intermolecular Collision . . . . . . . 74
3.2.3 External Body Force . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.2.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.2.5 Summary of the DSBGK Algorithm . . . . . . . . . . . . . . . 83
3.2.6 Extension to Gas Mixtures . . . . . . . . . . . . . . . . . . . . . . 84
3.3 Discrete Velocity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.4.1 Couette Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.4.2 Lid-Driven Cavity Flows . . . . . . . . . . . . . . . . . . . . . . . 92
3.4.3 Thermal Transpiration Through Micro-channel. . . . . . . . 100
3.4.4 Apparent Permeability of Shale Gas . . . . . . . . . . . . . . . 104
3.4.5 Scaling Law of Gas Permeability in the Slip Regime . . . 113
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4 Multiscale LBM Simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.1 Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2 Chapman–Enskog Expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3 Shan–Chen Model for Multiphase Multicomponent Systems . . . . 126
4.3.1 Two-Phase Flows of Single Component . . . . . . . . . . . . 128
4.3.2 Flows of Two Immiscible Components . . . . . . . . . . . . . 131
4.4 Upscaled Lattice Boltzmann Method . . . . . . . . . . . . . . . . . . . . . 132
4.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.5.1 Large Eddy Simulation of Turbulence. . . . . . . . . . . . . . 134
4.5.2 Intrinsic Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.5.3 Buoyant Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.5.4 Two-Phase Flow Inside Real Digital Rock. . . . . . . . . . . 148
4.5.5 Upscaled Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.6 Introduction to Fortran Code. . . . . . . . . . . . . . . . . . . . . . . . . . . 160
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Acronyms
BGK Bhatnagar-Gross-Krook
Ca Capillary number, the ratio of viscous force to interfacial tension
CFD Computational fluid dynamics
CFL Courant-Friedrichs-Lewy
D2Q9 Two-dimensional nine-velocity-grids model
DSBGK Direct simulation BGK
DSMC Direct simulation Monte Carlo
DUGKS Discrete unified gas-kinetic scheme
DVM Discrete velocity method
EOS Equation of state
FEM Finite element method
GDVM Godunov-type discrete velocity method
GHS Generalized hard sphere
GSS Generalized soft sphere
HPC High-performance-computing
HS Hard sphere
IFT Interfacial tension coefficient
Kn Knudsen number, the ratio of molecular mean free path to flow
characteristic length
LBM Lattice Boltzmann method
LES Large eddy simulation
LVDSMC Low-variance deviational simulation Monte Carlo
Ma Mach number, the ratio offlow speed to sound speed
MD Molecular dynamics
MEMS Micro-electro-mechanical systems
NEMS Nano-electro-mechanical systems
N–S–C–H Navier–Stokes and Cahn–Hilliard
N–S–F Navier–Stokes–Fourier
N–S–K Navier–Stokes–Korteweg
NTC No-time-counter
xi
xii Acronyms
PDE Partial differential equation
Pr Prandtl number, the ratio of momentum diffusivity rate to thermal
diffusivity rate
Re Reynolds number, the ratio of inertial force to viscous force
Rf Random fraction uniformly distributed between 0 and 1
RHS Right hand side
STP Standard temperature and pressure, i.e., 273.15 K and 1 atm
UGKS Unified gas-kinetic scheme
VHS Variable hard sphere
VSS Variable soft sphere