Gas-Kinetic Methods for Viscous Fluid Flows

by

Hongwei Liu

A Thesis Submitted to

The Hong Kong University of Science and Technology

in Partial Fulfillment of the Requirements for

the Degree of Doctor of Philosophy

in Mathematics

August 2007, Hong Kong

HKUST Library Reproduction is prohibited without the author’s prior written consent

Acknowledgments

First of all, I would like to express my sincere gratitude to my advisor, Prof. Kun Xu,

for his valuable guidance and advice during the past four years, and for his generous

support and constant encouragement. And also, as an excellent scholar, Prof. Xu

shows great zeal for knowledge and respectable perseverance in research, which sets a

very good example for me and influences me in many aspects.

Moreover, I wish to thank Prof. Mohamed S. Ghidaoui, Prof. Xiao-Ping Wang,

Prof. Mo Mu, and Prof. Yue-Hong Qian for serving as a member of my thesis exami-

nation committee and carefully reviewing this dissertation.

I am deeply indebted to Prof. Hua-Zhong Tang, Prof. Taku Ohwada, Prof. Li-Shi

Luo, Prof. Jing Fan, Prof. Wai-How Hui, and Prof. Jaw-Yen Yang for their kindly

help and warm encouragement during the period of my postgraduate study.

My special thanks go to Prof. Zhaoli Guo and Dr. Xin He. I wish to thank Zhaoli

for sharing his knowledge with me and many insightful discussions with me. I also

appreciate many helpful suggestions for my research from Xin.

My sincere appreciation goes to my other friends, research group members and

classmates: Dr. Changqiu Jin, Dr. Guoxi Ni, Dr. Manuel Torrilhon, Prof. Guiping

Zhao, Dr. Jianzheng Jiang, Yibing Chen, Yong Xiao, Qiaolin He, Xiaohong Zhu,

Junqing Yuan, Jue Wang, Lei Yang, Mingchao Cai, Chunyin Qiu. Thanks for making

my stay at the Department of Mathematics of HKUST a memorable and enjoyable

one, and for many help in both technical and mundane matters.

Finally, I would like to express my heartfelt gratitude to my family, especially to

my wife Ling Chen, for their care, support, understanding, and love. Their constant

cheers and encouragement have always helped me in many ways.

iv

Contents

Title Page i

Authorization Page ii

Signature Page iii

Acknowledgments iv

Table of Contents v

List of Figures ix

List of Tables xvi

Abstract xvii

1 Introduction 1

1.1 Mathematical models and classifications of fluid flows . . . . . . . . . . 1

1.2 Numerical methods for the simulation of fluid flows . . . . . . . . . . . 3

v

1.2.1 Numerical methods for Euler and Navier-Stokes equations . . . 3

1.2.2 Numerical approaches for continuum equations beyond Navier-

Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 Numerical solutions of the Boltzmann equation . . . . . . . . . 7

1.2.4 Molecular dynamics (MD) method . . . . . . . . . . . . . . . . 7

1.2.5 Direct simulation Monte Carlo (DSMC) method . . . . . . . . . 8

1.3 Objectives of the present study . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Gas-kinetic BGK method for inviscid and viscous flows 12

2.1 Boltzmann Equation and Bhatnagar-Gross-Kook (BGK) Model of the

Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Kinetic flux vector splitting (KFVS) method . . . . . . . . . . . . . . . 19

2.2.1 1st-order KFVS scheme . . . . . . . . . . . . . . . . . . . . . . 22

2.2.2 2nd-order KFVS scheme . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Gas-kinetic BGK method with the directional splitting fluxes . . . . . . 29

2.4 Multidimensional gas-kinetic BGK scheme for viscous flows . . . . . . . 38

2.5 Multidimensional BGK scheme for nearly incompressible viscous flows . 44

3 A Runge-Kutta discontinuous Galerkin method for viscous flow equa-

tions 48

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

vi

3.2 Runge-Kutta discontinuous Galerkin method . . . . . . . . . . . . . . . 51

3.2.1 DG Spatial Discretization in 1D Case . . . . . . . . . . . . . . . 51

3.2.2 Gas-Kinetic Flux Evaluation at a Cell Interface . . . . . . . . . 53

3.2.3 Extension to Multidimensional Cases . . . . . . . . . . . . . . . 58

3.2.4 Limiting Procedure and Boundary Conditions . . . . . . . . . . 60

3.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Multiscale gas-kinetic simulation for continuum and near continuum

flows 82

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2 Gas-kinetic equation and multiscale numerical formulation . . . . . . . 85

4.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Multiple temperature kinetic model for microscale flows 97

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2 Multiple translational temperature kinetic model . . . . . . . . . . . . 100

5.2.1 Standard BGK model and Navier-Stokes equations . . . . . . . 100

5.2.2 Multi-temperature gas-kinetic model and its corresponding Navier-

Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.2.3 Comparison between multi-T kinetic model and ES-BGK model 108

vii

5.3 Finite volume BGK scheme for the multi-T kinetic model . . . . . . . . 110

5.4 Effective viscosity approach and the generalized second-order slip bound-

ary condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6 Summary and conclusions 135

Bibliography 138

viii

List of Figures

2.1 Exact Euler solutions (solid line) and collisionless Boltzmann solutions

(2.17) (+ symbol) at time t = 0.15 units for the same initial condition. 21

2.2 Exact Euler solutions (solid line) and numerical solutions (+ symbol)

using the 1st-order KFVS scheme at time t = 0.15 units. . . . . . . . . 24

2.3 Initial gas distribution function for the 2nd−order KFVS scheme. . . . 26

2.4 Exact Euler solutions (solid line) and numerical solutions (+ symbol)

using the 2nd-order KFVS scheme at time t = 0.15 units. . . . . . . . . 28

2.5 The spatial distribution of the initial state f0 and the equilibrium state

g at t = 0 for the 2nd-order BGK scheme. . . . . . . . . . . . . . . . . 32

2.6 Exact Euler solutions (solid line) and numerical solutions (circle symbol)

using the 2nd-order gas-kinetic BGK scheme at time t = 0.15 units. . . 37

3.1 Temperature ratio (T − T0)/(T1 − T0) in the Couette flow with γ =

5/3, Ec = 50. The solid line is the analytical solution given by Eq.(3.40),

the plus symbol is the numerical solution by P 1 method and the circle

symbol is by P 2 one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

ix

3.2 Temperature ratio (T − T0)/(T1 − T0) in the Couette flow with γ =

7/5, P r = 0.72. The solid line is the analytical solution given by

Eq.(3.40), the plus symbol is the numerical solution by P 1 method and

the circle symbol is by P 2 one. . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 Navier-Stokes shock structure calculation, P 1 case. . . . . . . . . . . . 72

3.4 Navier-Stokes shock structure calculation, P 2 case. . . . . . . . . . . . 73

3.5 Shock tube test for the Navier-Stokes equations with kinematic viscosity

coefficient ν = 0.0005/ρ√λ. . . . . . . . . . . . . . . . . . . . . . . . . 74

3.6 The zoom-in views of the density distributions around the shock wave

in shock tube test with ν = 0.0005/ρ√λ. The upper one is the same

as that in Fig. 3.5, where only the values at cell centers are displayed.

The lower one presents the values at 15 equally spaced positions inside

each cell, the so-called subcell solution. . . . . . . . . . . . . . . . . . . 75

3.7 Shock tube test for the Navier-Stokes equations with kinematic viscosity

coefficient ν = 0.00005/ρ√λ. . . . . . . . . . . . . . . . . . . . . . . . . 76

3.8 Mesh distribution for laminar boundary layer problem. . . . . . . . . . 77

3.9 Laminar boundary layer problem. 100 equally spaced contours of the

fluid velocity U/U−∞ from 0 to 1.0061 from P 2 calculation. . . . . . . . 77

3.10 Laminar boundary layer problem. U velocity distributions along three

vertical lines by P 1 case. . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.11 Laminar boundary layer problem. U velocity distributions along three

vertical lines by P 2 case. . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.12 Laminar boundary layer problem. Skin friction coefficient distribution

along the flat plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

x

3.13 Laminar boundary layer problem. Logarithmic plot of the skin friction

coefficient distribution along the flat plate. . . . . . . . . . . . . . . . . 79

3.14 Shock boundary layer interaction. 30 equally spaced contours of pres-

sure p/p−∞ from 0.997 to 1.411 by P 1 (upper) and P 2 (lower) cases. . . 80

3.15 Shock boundary layer interaction. Skin friction (upper) and pressure

(lower) distributions at the plate surface. . . . . . . . . . . . . . . . . . 81

4.1 Temperature, density (left) and heat flux (right) distributions in a M =

8 argon shock structure with µ ∼ T 0.68. DSMC solution [83] vs. present

multiscale model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2 Velocity distribution in the Poiseuille flow for HS molecules µ ∼ T 0.5,

at three different Knudsen numebrs Kn = 0.1128 (upper, left), Kn =

0.2257 (upper, right) and Kn = 0.4514 (lower). Circles: solution of the

Boltzmann equation [84], solid line: present multiscale model. Ua is the

mean velocity across the channel. . . . . . . . . . . . . . . . . . . . . . 94

4.3 Density ratio versus position for helium gas in a channel with tem-

peratures TH = 294K, TC = 79K, and accommodation coefficients

αH = αC = 0.58, at three different Knudsen numbers Kn0 = 0.075

(upper, left), Kn0 = 0.118 (upper, right) and Kn0 = 0.399 (lower).

Circles: experimental data [85], dashed-line: Navier-Stokes solution,

solid line: present multiscale model. . . . . . . . . . . . . . . . . . . . 95

4.4 Unsteady Rayleigh problem at time t = 10τ . Solid lines: DSMC solu-

tions [98], circles: present model. . . . . . . . . . . . . . . . . . . . . . 96

4.5 Unsteady Rayleigh problem at time t = 100τ . Solid lines: DSMC

solutions [98], circles: present model. . . . . . . . . . . . . . . . . . . . 96

xi

5.1 Couette flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.2 Velocity U/Uupper (left) and temperature T/Tupper (right) distributions

in high speed Couette flow case for a gas with Pr = 2/3, µ ∼ T , and

Kn = 0.001, where the up-plate is moving with a speed ofMa = 3.0 and

the lower boundary is adiabatic. The circles are analytic Navier-Stokes

solutions provided in [94], and the solid lines are simulation results

from the current multi-temperature model. Multiple temperatures are

plotted in the above figure (right). . . . . . . . . . . . . . . . . . . . . . 123

5.3 Velocity u/Uw (left) and temperature T/Tw (right) for a gas with Pr =

0.68, µ ∼√T , and Kn = 0.01, where the up-plate is moving with a

speed of Uw = 300m/s. Both boundaries are isothermal with a temper-

ature Tw = 273K. The circles are DSMC solutions, and the solid lines

are simulation results from the current multi-temperature model. . . . . 123

5.4 Velocity u/Uw (left) and temperature T/Tw (right) for a gas with Kn =

0.1. The circles are DSMC solutions, and the solid lines are simulation

results from the current multi-temperature model. In terms of the tem-

perature distributions, the up one is Ty, the middle one is Tz, and the

low one is Tx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.5 Velocity u/Uw (left) and temperature T/Tw (right) for a gas with Kn =

0.5. The circles are DSMC solutions, and the solid lines are results

from the current multi-temperature model. In terms of the temperature

distributions, the up one is Ty, the middle one is Tz, and the low one is

Tx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.6 Force-driven Poiseuille flow (left) and pressure-driven Poiseuille flow

(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

xii

5.7 External force driven Poiseuille flow at Kn = 0.1 [114]. Density (left)

and velocity (right) distribution along the channel cross section, where

the circles are DSMC solution. . . . . . . . . . . . . . . . . . . . . . . . 125

5.8 External force driven Poiseuille flow at Kn = 0.1 [114]. Pressure (left)

and multiple temperatures (right) distributions, where the circles are

DSMC solutions. Both the curved pressure and the temperature min-

imum have been recovered from the multi-T model. Solid line on the

figure (right) is the averaged temperature, i.e., T = (Tx + Ty + Tz)/3. . 126

5.9 External force driven Poiseuille flow at Kn = 0.1 [114]. Pressure

(left) and temperature (right) distributions from BGK-NS method [107],

where the circles are DSMC solutions. The BGK-NS method basically

cannot capture the non-equilibrium effect in the Kn = 0.1 Poiseuille flow.126

5.10 Nondimensional density (left) and velocity (right) profiles in the cross-

stream direction at x = 0, solid line is multi-T model solution and circle

is DSMC data [115]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.11 Nondimensional temperature (left) and pressure (right) profiles in the

cross-stream direction at x = 0, solid line is multi-T model solution

and circle is DSMC data [115]. The averaged temperature T on the left

figure is defined by T = (Tx + Ty + Tz)/3. . . . . . . . . . . . . . . . . . 127

5.12 Nondimensional pressure profile in the cross-stream direction at x = 0,

solid line is BGK-NS solution and circle is DSMC data [115]. . . . . . . 128

5.13 Nondimensional density (left) and velocity (right) profiles in the stream-

wise direction at y = 0, solid line is multi-T model solution and circle

is DSMC data [115]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

xiii

5.14 Nondimensional temperature (left) and pressure (right) profiles in the

stream-wise direction at y = 0, solid line is multi-T model solution and

circle is DSMC data [115]. The averaged temperature T on the left

figure is defined by T = (Tx + Ty + Tz)/3. . . . . . . . . . . . . . . . . . 129

5.15 Nondimensional velocity profiles for force-driven Poiseuille flow at Knud-

sen numbers of 0.113, 0.226, 0.451, 0.677, 0.903, and 1.13. Circle is the

solution of the linearized Boltzmann equation by Ohwada et al. [84],

and solid line is from the multi-T kinetic model. . . . . . . . . . . . . . 130

5.16 Nondimensional flow rate Q as a function of the Knudsen number for

force-driven Poiseuille flow. Circle is the solution of the linearized Boltz-

mann equation by Ohwada et al. [84], and solid line is from the multi-T

kinetic model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.17 Cavity flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.18 Contours of the averaged temperature T , i.e. T = (Tx + Ty + Tz)/3, by

DSMC (left) and the current multi-T model (right). In both figures, 9

equally spaced contours from T = 350K to T = 530K are plotted. . . . 132

5.19 Contours of the temperature Tx by DSMC (left) and the current multi-

T model (right). In both figures, 9 equally spaced contours from Tx =

350K to Tx = 530K are plotted. . . . . . . . . . . . . . . . . . . . . . . 132

5.20 Contours of the temperature Ty by DSMC (left) and the current multi-

T model (right). In both figures, 9 equally spaced contours from Ty =

350K to Ty = 530K are plotted. . . . . . . . . . . . . . . . . . . . . . . 133

5.21 Contours of the temperature Tz by DSMC (left) and the current multi-

T model (right). In both figures, 9 equally spaced contours from Tz =

350K to Tz = 530K are plotted. . . . . . . . . . . . . . . . . . . . . . . 133

xiv

5.22 Velocity distributions. The left is the velocity U/Uw distributions along

the vertical line at system center x = L/2; the right is the velocity V/Uw

distributions along the horizontal line at system center y = L/2. . . . . 134

xv

List of Tables

3.1 The error and convergence order for P 1 case. . . . . . . . . . . . . . . . 69

3.2 The error and convergence order for P 2 case. . . . . . . . . . . . . . . . 69

3.3 The error and convergence order for P 3 case. . . . . . . . . . . . . . . . 70

xvi

Gas-kinetic Methods for Viscous Fluid Flows

Hongwei Liu

Department of Mathematics

The Hong Kong University of Science and Technology

Abstract

The gas-kinetic Bhatnagar-Gross-Krook (BGK) finite volume method has been suc-

cessfully proposed and developed for both inviscid and viscous flow simulations in

the past decade. In this thesis, the gas-kinetic method is extended in terms of three

aspects: the gas-kinetic Runge-Kutta discontinuous Galerkin (RKDG) finite element

method for viscous flow equations was successfully constructed and numerically tested;

a multiscale gas-kinetic approach was proposed and applied for continuum and near

continuum flow simulations; and a multiple translational temperature kinetic model

was presented for microscale flow problems.

In recent years, high-order numerical methods have been extensively investigated

and widely used in computational fluid dynamics (CFD), in order to effectively resolve

complex flow features using meshes which are reasonable for today’s computers. Here

we refer to high-order methods by those with order of accuracy at least three. Among

the most successful high-order formulations is the discontinuous Galerkin (DG) finite

element method. In the first part of this thesis, a RKDG method based on the gas-

kinetic formulation for viscous flow computation is proposed, which not only couples

the convective and dissipative terms together, but also includes both discontinuous and

continuous representation in the flux evaluation at a cell interface through a simple

hybrid gas distribution function. Due to the intrinsic connection between the gas-

kinetic BGK model and the Navier-Stokes (NS) equations, the NS flux is automatically

obtained by the present method. Numerical examples for both one-dimensional (1D)

and two-dimensional (2D) compressible viscous flows are presented to demonstrate the

xvii

accuracy and shock capturing capability of the current RKDG method.

In the near continuum regime, both solid mathematical models and efficient numer-

ical algorithms are highly desired. In the second part of this thesis, we investigate the

performance of the gas-kinetic approach with the generalized particle collision time,

which depends on both first- and second-order derivatives of the local flow variables,

for continuum and near continuum flow problems. Based on this model, the nonequi-

librium shock structure, Poiseuille flow, nonlinear heat conduction problems, and the

unsteady Rayleigh problem will be calculated. The numerical results are in good

agreement with those by both the direct simulation Monte Carlo (DSMC) method

and experience.

Microscale gas flow is a new rapidly growing research field being driven by mi-

crosystems technology. It has been shown that the fluid mechanics of microscale gas

flows are not the same as those experienced in the macroscopic world. In the third

part of the thesis, a gas-kinetic model with multiple translational temperature for

monatomic gas is proposed for microscale flow simulations. In the continuum regime,

the standard NS solutions are precisely recovered. For the microscale gas flows, the

simulation results are compared with both DSMC data and solutions of linearized

Boltzmann equation. Numerical experiments demonstrate that the multiple temper-

ature kinetic model has the advantage over the standard Navier-Stokes equations in

capturing the non-equilibrium physical phenomena for the microscale gas flow simula-

tion. It is clearly shown that many thermal nonequilibrium phenomena in microscale

flows can be well captured by modifying some assumptions in the standard NS equa-

tions.

xviii

Chapter 1

Introduction

1.1 Mathematical models and classifications of fluid

flows

Usually fluid is defined as a substance, which cannot withstand any tendency by ap-

plied forces to deform it in a way which leaves the volume unchanged [1]. Most liquids

and gases present such a feature, and are regarded as fluids. Different mathematical

models have been developed for describing fluid motion at different length scales, i.e.

macroscopic and microscopic levels. At the macroscopic level, the most commonly-

used mathematical description of viscous fluid flows is the Navier-Stokes (NS) equa-

tions equipped with the equation of state (EOS). The NS equations consist of the mass,

momentum and energy conservation equations, which are constructed based on three

important hypotheses: (i) continuum medium, (ii) linear constitutive relationship, and

(iii) Fourier’s law for heat conduction. In the limit of vanishing viscosity as well as

heat conduction, the NS equations reduce to the Euler equations which describe the

inviscid fluid flows. At the microscopic level, if a relatively large framework is chosen,

we can use the velocity distribution functions and derive their governing equations to

describe fluid behavior, such as the celebrated Boltzmann equation and Bhatnagar-

1

Gross-Krook (BGK) equation; or in a relatively small framework, we can assume the

fluid molecules subject to the Newtonian law of the classical mechanics and construct

the dynamic equation for each fluid molecule of the system; of course we can go further

to take into account the quantum effect if necessary. There are some intrinsic con-

nections between the macroscopic and microscopic descriptions of fluids, for example,

the Euler and Navier-Stokes equations can be derived from the Boltzmann or BGK

equation by the zeroth- and first-order Chapman-Enskog expansion, respectively.

Physically, fluids are described as a myriad of discrete molecules (monatomic, di-

atomic, or polyatomic). Physics has shown that both single molecules and inter-

molecular interactions are very complicated due to the inner structure of molecules.

However, in view of the large number of molecules in most cases, it is almost impos-

sible, and fortunately unnecessary, to address all the details of molecules and their

interactions for most engineering problems. Therefore, different mathematical models

and the corresponding numerical methods are suggested for different practical appli-

cations. Generally, many characteristic lengths are associated with the fluid flows: (i)

Λ, de Broglie thermal wavelength, (ii) d, diameter of molecules, (iii) δ, mean molecular

spacing, (iv) l, mean free path, and (v) L, characteristic length of the fluid system.

The fluid flows can be grossly classified according to the ratios of these characteristic

lengths, which will be discussed in the following paragraph.

The ratio of the mean free path to the characteristic length of the system, which is

called the Knudsen number (Kn = l/L), defines the degree of rarefaction of the flows.

According to the Knudsen number, fluid flows are generally classified into four regimes

[2]: the continuum flow regime (Kn ≤ 0.001); the slip flow regime (0.001 < Kn ≤ 0.1);

the transition flow regime (0.1 < Kn ≤ 10) and the free molecule flow regime (Kn >

10). The Knudsen number is usually regarded as an important parameter identifying

the validity of continuum approaches, such as the Navier-Stokes equations. It has been

shown [89] that the statistical fluctuations due to the motion of fluid molecules are

insignificant when the ratio of the characteristic length of a flow to the mean molecular

2

spacing L/δ is lager than 100, where the fluid behaviors can be characterized by some

macroscopic properties, such as the fluid density, velocity and temperature, which are

defined as the meaningful statistical average of the corresponding properties of fluid

molecules. It is also understood [3] that only when Λ/d ≈ 1 and Λ/δ ≈ 1, the quantum

effect should be taken into account. Fortunately, for most engineering applications,

the magnitude of Λ of fluids is much smaller than the corresponding parameters of d

and δ, so, the quantum effect can be neglected. A gas can be treated as a dilute gas

when the ratio of the mean molecular spacing to the diameter of molecules is large,

e.g. δ/d > 7. This means that it is very highly probable that only one other molecule

is involved with a molecule during collisions, therefore, for a dilute gas, molecular

interactions are commonly treated as binary collisions, where the Boltzmann equation

can be applied.

1.2 Numerical methods for the simulation of fluid

flows

Many numerical approaches have been designed and developed for simulating fluid

flows, and it is almost impossible to mention all of them here, therefore, in this section,

only several typical numerical methods for the simulation of fluid flows at different

length scales will be briefly reviewed.

1.2.1 Numerical methods for Euler and Navier-Stokes equa-

tions

The Euler and Navier-Stokes equations are the most commonly-used macroscopic gov-

erning equations for inviscid and viscous fluid flows, respectively. Various sophisticated

numerical techniques have been developed for solving these governing equations in the

3

literature, such as the finite volume method, the finite difference method, the finite

element method and many others. Detailed discussion on these numerical methods for

conventional macroscopic conservation equations can be found in [4]-[8].

In the last three decades, numerical algorithms for hyperbolic conservation laws,

e.g. the Euler equations, have been extensively investigated. One of the most suc-

cessful algorithms for the hyperbolic conservation laws is the Godunov method [9],

which laid a solid foundation for the development of modern upwind schemes includ-

ing MUSCL (Monotone Upstream Scheme for Conservation Laws) [10], TVD (Total

Variation Diminishing) [12, 13, 14], PPM (Piecewise Parabolic Method) [15], ENO

(Essentially Non-Oscillatory) [16, 17] and WENO (Weighted ENO) [18, 19] schemes.

The Godunov scheme was first extended to second-order by van Leer [10] with the in-

troduction of a limiter to remove spurious numerical oscillations near steep gradients.

Other techniques to eliminate spurious numerical oscillations were also developed, for

example the FCT (Flux-Corrected Transport) methods by Boris et al [11]. Meanwhile,

the “exact Riemann solver” used in the Godunov method was sometimes replaced by

“approximate Riemann solvers” [6] for better efficiency.

In recent years, high-order numerical methods capable of handling unstructured

grids for hyperbolic conservation laws are highly sought after to effectively resolve

complex flow structures with complex geometries, for example in computational aeroa-

coustics (CAA), direct numerical simulation (DNS) and large-eddy simulation (LES)

of turbulence, computational electromagnetics (CEM) and so on. The current state

of the art for high-order methods on unstructured grids includes the ENO [20] and

WENO [21] schemes, the discontinuous Galerkin (DG) methods [61, 56, 65, 68], the

spectral volume (SV) [22, 23, 24] and spectral difference (SD) [25, 26] methods, to

name just a few. A general review of these high-order methods can be found in [27].

Although great progress has been made for the numerical methods solving inviscid

flow equations, it is a different story for viscous flows, especially for the convection-

4

dominated problems, for example the high-speed, high Reynolds number flows. For the

Godunov-type methods, since the Riemann solutions for the Navier-Stokes equations

are not available, they cannot be used for viscous flows with a simple extension. One

technique that has been widely used is to treat the convection and dissipation effects

separately, for example, for a finite volume scheme, the interface flux from the inviscid

part of the Navier-Stokes equations is obtained by one of the numerical methods

developed for the Euler equations, and the numerical flux corresponding to the viscous

part of the governing equations is calculated by central-differencing approximation

with suitable accuracy requirement. In fact this is not a consistent way and sometimes

makes ambiguity from both physical and mathematical views although it works well

for many applications. Another technique that has been employed in the DG and SV

methods for NS equations is to rewrite the original governing equations into an enlarged

first-order system, then the DG or SV scheme is applied to the first-order system with

carefully chosen numerical fluxes from different terms of the system [56, 65, 24] to

achieve the expected order of accuracy. Accordingly, the computational efficiency is

reduced for this treatment, and also the way to determine the numerical fluxes of the

enlarged system seems to be an experimental one, except for the inviscid part.

The gas-kinetic BGK finite volume method has been successfully proposed and de-

veloped for both inviscid and viscous flow simulations in the past decade [77, 107,

80, 28, 29]. In this method, based on the gas-kinetic framework, the convection and

dissipation terms are coupled together and the interface fluxes corresponding to both

inviscid and viscous parts are evaluated with the same physical mechanism, the nu-

merical dissipation is controlled adaptively through the determination of the local

particle collision time. The gas-kinetic BGK scheme provides a consistent way from

both mathematical and physical views to accurately solve the Navier-Stokes equations.

In chapter 2, the formulation of the gas-kinetic BGK method will be presented and

the comparison with other methods will be given in detail.

In [81], for the first time, a gas-kinetic BGK method in the DG framework, i.e. the

5

DG-BGK method, was proposed, where the viscous flow equations are integrated in

time directly. For the DG-BGK method, there is no need to rewrite the Navier-Stokes

equations into a first-order system as done in [56, 65]. Numerical experiments have

shown that the DG-BGK method can give accurate viscous solutions in both high and

low Reynolds number flow simulations.

1.2.2 Numerical approaches for continuum equations beyond

Navier-Stokes equations

In hypersonic flows about space vehicles in low orbits or flows in micro-devices, such

as the micro-electro-mechanical systems (MEMS), the local Knudsen number lies in

the continuum-transition regime. The Navier-Stokes equations are not adequate to

model these flows since they are based on small deviation from local thermodynamic

equilibrium. To model these flows, many continuum models beyond NS equations

are proposed, such as the Burnett and super-Burnett equations [30, 31, 110] obtained

from the high-order Chapman-Enskog expansion of the Boltzmann or BGK equation,

Grad’s 13 moment equations [33], regularized 13 (R13) moment equations [34] and

many others.

Usually, for the Burnett equations, the finite volume or finite difference method can

be used to solve the macroscopic governing equations in a way similar to that for the

Navier-Stokes equations. For example, in the finite volume scheme, we can use the

Godunov-type methods for the inviscid flux calculation, and the central-differencing

approximation can be employed to evaluate the numerical flux corresponding to the

stress tensor and heat conduction terms [31, 32]. In [111], a gas-kinetic finite volume

BGK-Burnett solver was presented. One of the main advantages for this method is

that we do not need to evaluate the interface flux from the very complicated governing

equations at the macroscopic level, in fact the numerical flux is calculated by tak-

ing moments to the distribution function constructed at the microscopic level up to

6

O(Kn2). Also the kinetic boundary condition for the gas distribution function can be

easily implemented to obtain slip boundary automatically. The similar comments can

be applied to the gas-kinetic BGK supper-Burnett scheme in [110]. Another kinetic

method for the Burnett equations can be found in [96].

1.2.3 Numerical solutions of the Boltzmann equation

Many attempts to directly solve the Boltzmann equation have been carried out, how-

ever, it is still a challenge to achieve such a goal for general gas flows based on the

current computational capabilities. One major problem in doing this is that a large

number of elements or nodes are required to store the velocity distribution function in

the phase space, and another difficulty is the calculation of the collision term involves

a large number of operations. Therefore, direct solutions of the Boltzmann equation

have been limited to simple flow geometries for monatomic gases.

The first successful method for directly solving the Boltzmann equation was in-

troduced in a series of papers by Nordsieck, Hicks and Yen [35]. They dealt with

one-dimensional steady flow problems, and employed a conventional finite difference

technique to discretize the Boltzmann equation with a Monte Carlo sampling technique

for the collision term. Recently, Ohwada et al [84, 36] proposed another approach to

solve the Boltzmann equation, which was also a finite difference method with the col-

lision integral computed efficiently and accurately by their numerical kernel method.

However, application of this method so far is limited to 1D flow problems.

1.2.4 Molecular dynamics (MD) method

The molecular dynamics (MD) method simulates fluid flows by tracking the motions

of all molecules within a region of simulated physical space [37]. There are three main

aspects of a typical molecular dynamics simulation. First of all, the initial state of each

7

molecule is set by probabilistic approach, generally, molecules are randomly distributed

according to certain distribution functions for their location, velocity and internal

state. Next, some potential energy functions are needed to determine the force on a

fluid molecule from other fluid or wall molecules. Collisions occur whenever the spacing

between any pair of molecules decreases to the assumed cutoff limit of their force field.

Third, we let all simulated molecules obey the second law of Newtonian mechanics

and solve these dynamic equations by some numerical methods to redistribute the

molecules with new positions and velocities.

Macroscopic flow properties are obtained by averaging procedure from microscopic

states of molecules over a chosen space volume. Although the MD method is widely

recognized as a reliable tool to study complex fluid problems, it should be also noted

that the MD method is highly inefficient for most practical applications in comparison

with other numerical methods. As a result, molecular dynamics is usually limited to

flows where the continuum and statistical approaches are inadequate.

1.2.5 Direct simulation Monte Carlo (DSMC) method

The direct simulation Monte Carlo (DSMC) method is a widely used particle method

as an reliable numerical technique to simulate rarefied, nonequilibrium gas flows. This

method was innovated and mainly developed by Bird [38, 89] based on kinetic theories.

The fundamental idea of the DSMC method is to track a large number of statistical

representative particles in the computational domain, each of which represents the

extremely enormous number of real gas molecules. The motion and interactions of

each particle are used to modify particle positions, velocities, chemical species and so

on. The flow information is then sampled from the microscopic information about the

simulated particles.

The primary approximation of the DSMC method is to uncouple the particle stream-

ing and inter-particle collisions over small time intervals. Particle streaming is modeled

8

deterministically, whereas the collisions are treated statistically. The spacial cell size

used should be less than the mean free path but much larger than the mean molecu-

lar spacing, and the time step should be smaller than the mean collision time of gas

molecules. The limitations of DSMC method are the same as those of classical kinetic

theory: the assumption of molecular chaos and the restriction to dilute gases.

Although it is very difficult to fully prove the validity of the DSMC method by

strict mathematical reasoning, it has been demonstrated by some facts: many DSMC

calculations agree well with corresponding experimental data [38, 89]; excellent agree-

ment has been shown in the comparison of the DSMC with molecular dynamics (MD)

method for shock computations [39] and slip length calculations [40]; it has been shown

numerically by many researchers that the DSMC solutions approach the Navier-Stokes

solutions in the limit of very low Knudsen numbers.

The DSMC method is one of the most successful numerical approaches for rarefied

gas flows, however, when it is applied to simulate microscale gas flows, such as flows in

MEMS, the statistical scatter associated with this method prevents its application for

many practical flow problems, because of its huge numerical expense. Although some

attempts have been made to modify this method in order to simulate microscale gas

flows with affordable computational cost, for example the information preservation

(IP) method [90, 98, 99], the success of these modified methods is still limited at

present.

1.3 Objectives of the present study

It has been shown that the gas-kinetic BGK finite volume method is a very successful

method for both inviscid and viscous flow simulations [77, 80, 107], and the DG-BGK

finite element scheme [81] gives accurate numerical solutions for both low and high

Reynolds number flows. As a continuation of these studies, the first objective of

9

this thesis is to extend the gas-kinetic approach into the Runge-Kutta discontinuous

Galerkin (RKDG) framework and develop a gas-kinetic RKDG finite element method

for viscous flow equations.

The second objective of this thesis is to develop a multiscale gas-kinetic approach

for continuum and near continuum flow simulations based on an early investigation of

the argon shock structure calculation [108], where the particle collision time has been

generalized to depend on not only the local macroscopic flow variables but also their

derivatives. In other words, a generalized constitutive relationship as well as the heat

flux is proposed, which is applicable in both continuum and near continuum regimes.

The third objective of the thesis is to construct a multiple translational tempera-

ture kinetic model based on some previous work [109, 112, 113] and apply it to the

microscale flow problems. For the numerical simulations of microscale gas flows, some

available methods are too computationally expensive, such as the DSMC method, and

some are not able to capture physics accurately, for example the slip Navier-Stokes

solver, therefore, an efficient and reliable approach for modeling these flows is still

highly desired. The inaccuracy of the Navier-Stokes equations with slip boundary

conditions may be partially due to their single equilibrium temperature assumption,

so, we intend to throw away this assumption and construct a kinetic model that allows

multiple temperature for microscale flow simulations.

1.4 Organization of the thesis

The rest of the thesis is organized as follows:

Chapter 2 reviews the basic kinetic theory and some gas-kinetic numerical ap-

proaches for the inviscid and viscous flow equations, such as the kinetic flux vector

splitting (KFVS) scheme and the gas-kinetic BGK method. The comparison between

these two kinetic methods will be also given.

10

In Chapter 3, a gas-kinetic RKDG finite element method for viscous flow equa-

tions will be presented, which not only couples the convective and dissipative effects

together, but also includes both discontinuous and continuous representation in the

evaluation of interface flux. Six numerical examples for both one-dimensional (1D)

and two-dimensional (2D) flow problems will be given to demonstrate the validity and

accuracy of the proposed method.

In Chapter 4, a gas-kinetic multiscale approach will be introduced and applied for

continuum and near continuum flow simulations. The emphasis is on the determination

of the generalized local particle collision time based on a closed solution of the kinetic

equation. Numerical examples including shock structure, Poiseuille flow, nonlinear

heat conduction problem, and unsteady Rayleigh problem will be presented.

In Chapter 5, a multiple translational temperature kinetic model will be proposed,

the corresponding macroscopic governing equations up to O(Kn) will be given, and

the comparison of the current model with other kinetic models will be presented. It

will be clearly shown by the numerical examples the current multiple temperature

kinetic model has some advantages over the standard Navier-Stokes equations for

capturing many nonequilibrium physical phenomena in simulating microscale flows,

and the standard NS solutions can be precisely recovered in the continuum regime.

Chapter 6 summarizes our work and draws the main conclusions. Suggestions for

future research are also addressed.

11

Chapter 2

Gas-kinetic BGK method for

inviscid and viscous flows

Two ways among various mathematical models describing flow motion will be ad-

dressed in this chapter. The first one is based on macroscopic quantities, such as

mass, momentum and energy densities, as well as the physical laws governing these

quantities, such as the Euler, Navier-Stokes, Burnett, or high-order approximate equa-

tions supplied by the equation of state. The gas kinetic theory is another type of de-

scription coming from microscopic considerations. The fundamental quantity in this

description is the particle distribution function f(xi, ui, t), which gives the number

density of molecules in the six-dimensional phase space (xi, ui) = (x, y, z, u, v, w). The

evolution equation for the gas-distribution function f can be constructed, for exam-

ple the well-known Boltzmann equation. Physically, the gas kinetic equation provides

more information about the gas flow and has larger applicability than the macroscopic

counterpart.

The development of numerical schemes based on the gas-kinetic theory for com-

pressible flow simulations began in the 1960’s. The Chu’s method [41], based on

gas-kinetic Bhatnagar-Gross-Kook (BGK) model, with discretized particle velocity

12

space, is one of the earliest kinetic methods used for shock tube calculations. Another

kinetic scheme used in early 70’s is the beam scheme [42], which approximates the

gas distribution function as three delta functions in one-dimensional (1D) case and

the transport is based on the collisionless Boltzmann equation. Numerically, the beam

scheme is almost identical to the flux vector splitting (FVS) scheme of Steger-Warming

[43]. Unfortunately, the beam scheme is only known in the astrophysical society in

the 70’s. In the past three decades, many researchers have contributed to gas-kinetic

schemes. A partial list includes Reitz [44], pullin [45], Deshpande [46], Elizarova and

Chetverushkin [47], and many others.

The Boltzmann equation with vanishing collision term is called collisionless Boltz-

mann equation. The flux evaluation of most kinetic schemes introduced above is based

on the collisionless Boltzmann equation, the so-called Kinetic Flux Vector Splitting

(KFVS) scheme. In the past decade, the gas-kinetic methods based on BGK Model

[77, 107, 80] have also been developed to model the gas evolution process more pre-

cisely. The schemes of this class are named BGK-type schemes in order to distinguish

them from other Boltzmann-type schemes based on the collisionless Boltzmann equa-

tion. The BGK-type schemes take into account particle collisions in the whole gas

evolution process or flux evaluation process at a cell interface within a time step, from

which a time-dependent gas distribution function and the resulting numerical fluxes

at the cell interface can be obtained. Due to the intrinsic connection between the

BGK model and viscous governing equations, the BGK method gives Navier-Stokes

solutions directly in smooth regions. In the discontinuous regions, the scheme provides

a delicate dissipative mechanism to get a stable and crisp shock transition, and the

scheme becomes a shock capturing method. Since the gas evolution process in the

BGK scheme is a relaxation process from a non-equilibrium state to an equilibrium

one, it is expected that the entropy condition is satisfied in the BGK method. In

this chapter, the gas-kinetic BGK schemes developed in [77, 107, 80] for Navier-Stokes

equations will be introduced and compared with other methods.

13

2.1 Boltzmann Equation and Bhatnagar-Gross-Kook

(BGK) Model of the Boltzmann Equation

In the gas kinetic theory, the particle number density n, is approximated by a gas

distribution function f(xi, ui, t), where (xi, t) is the location of any point in space

and time, ui = (u, v, w) is particle velocity with three components in the x, y, and

z directions. The relation between the macroscopic fluid density ρ and f can be

expressed by the integral

ρ =

∫ ∫ ∫fdudvdw, (2.1)

in the particle velocity space. For molecules with internal degree of freedom, such

as rotation and vibration, the distribution function f can take these internal motion

into account as well through additional variables ξi, where ξi = (ξ1, ξ2, · · · , ξN) are

the components of the internal particle velocity in N dimensions. For monotonic

gas, the internal degree of freedom N is equal to 0. For diatomic gases, under the

normal pressure and temperature, N is equal to 2 which accounts for two independent

rotational degrees of freedom. Then, the ratio of the specific heat capacities Cv and

Cp for the gas in equilibrium state is commonly denoted by γ,

γ =Cp

Cv

=(N + 3) + 2

N + 3.

In order to understand the internal variable ξi inside the gas distribution function,

let’s first write down the Maxwell-Boltzmann distribution g for the equilibrium state,

g = ρ(λ

π)

N+3

2 e−λ((u−U)2+(v−V )2+(w−W )2+ξ21+···+ξ2

N), (2.2)

where ρ is the density, Ui = (U, V,W ), are the macroscopic velocity in the x−, y− and

z−directions, λ = m2kT

, where m is the molecular mass, k is the Boltzmann constant,

and T is the temperature. In the above equation, the parameters λ, Ui and ρ which

determine g uniquely are functions of space and time. Taking moments of equilibrium

state g, the mass, momentum and energy densities at any point in space and time can

14

be obtained. For example, the macroscopic and microscopic descriptions are related

by

ρ

ρUi

ρǫ

=

∫g

1

ui

12(u2

i + ξ2)

dΞ,

where dΞ = dudvdwdξ, dξ = dξ1dξ2 . . . dξN . If we re-define the internal variable ξi as

a vector in K dimensions, in three-dimensional case we have

K = N =5 − 3γ

γ − 1.

For one-dimensional gas flow, the random particle motion in y and z directions can be

included in the internal variable ξ of the molecules. As a result, the internal degree of

freedom becomes K = N +2. The distribution function g in the one-dimensional case

goes to

g = ρ(λ

π)

N+3

2 e−λ((u−U)2+v2+w2+ξ21+···+ξ2

N)

= ρ(λ

π)

K+1

2 e−λ((u−U)2+ξ2). (2.3)

In two-dimensional flow calculations, the random particle motion in the z direction is

included into the internal variable ξ, the total number of degrees of freedomK = N+1,

and the equilibrium distribution function is

g = ρ(λ

π)

N+3

2 e−λ((u−U)2+(v−V )2+w2+ξ21+···+ξ2

N )

= ρ(λ

π)

K+2

2 e−λ((u−U)2+(v−V )2+ξ2). (2.4)

Usually, we do not know the explicit form of the gas distribution function f in the

highly non-equilibrium flow regions, such as that inside a strong shock wave. What we

know is the dynamical evolution of the distribution function f , the so-called Boltzmann

Equation,

ft + uifxi+ aifui

= Q(f, f). (2.5)

Here f is the real gas distribution function, ai is the external forcing term acting on

the particle in i-th direction, and Q(f, f) is the collision operator. From the phys-

15

ical constraints of the conservation of mass, momentum and energy during particle

collisions, the following compatibility condition has to be satisfied,

∫ψQ(f, f)dΞ = 0, (2.6)

where dΞ = dudvdwdξ1dξ2 . . . dξK and ψ = (1, u, v, w, 12(u2 + v2 +w2 + ξ2))T . The gas

kinetic theory suggests that the Navier-Stokes equations are valid if the length scale

L of the flow is much larger than the mean free path l of the molecules, i.e.,

Kn =l

L≪ 1,

where Kn is defined as the Knudsen number. In most compressible flow problems, we

face the calculation of shock and boundary layers. For the shock waves and boundary

layers, the characteristic length scales are different. For example, in a boundary layer,

the significant length scale is the thickness of the boundary layer,

L ∼ 1

Re1/2,

where Re is the Reynolds number. On the other hand, for a shock wave the thickness

of the shock structure is the characteristic length scale,

L ∼ 1

Re.

Remarks:

1. In the Boltzmann equation (2.5), the advection term on the left hand side always

drives f away from local equilibrium distribution; the collision term on the right

hand side Q(f, f) pushes f back to equilibrium. Although, Q(f, f) does not

change the local mass, momentum and energy, it does re-distribute particles in

the phase space (ui, ξi), subsequently change the transport coefficients of the

particle system, e.g., viscosity and heat-conductivity. The real flow evolution is

governed by the competition and balance between the convection and collision

terms. The viscosity and heat conduction coefficients can be properly determined

through the determination of particle collision time.

16

2. For any shock capturing method, the numerical shock region usually spans over a

few mesh points. So, the mean free path of the ‘numerical fluid’ in these regions,

which is proportional to the shock thickness, is on the order of the cell size, i.e.

l ∼ x. As a result, the numerics amplifies the thickness of shock layer. The

BGK method, presented in this thesis, could consistently capture the amplified

numerical shock region from the controllable particle collision time τ , which also

ranges from the physical one τ ∼ µ to the numerical one τ ∼ t, where µ is

the viscosity coefficient and t is the time step. The robustness and accuracy

of the BGK method is mainly due to its ability to capture both equilibrium and

nonequilibrium gas flow.

One of the main functions of the particle collision term is to drive the gas distribu-

tion function f back to the equilibrium state g which has one to one correspondence

with the local mass, momentum and energy densities, (ρ, ρUi) and (ρǫ). Due to the

particle collision, during a time interval dt a fraction of dt/τ of molecules in a given

small volume undergoes collision, where τ is the average time interval between succes-

sive particle collisions, the so-called particle collision time. The collision term in the

BGK model alters the velocity-distribution function from f to g. With the assump-

tion of the rate of changes df/dt of f due to collisions is −(f − g)/τ , the Boltzmann

equation without external forcing term becomes [87],

∂f

∂t+ ui

∂f

∂xi

= −f − g

τ. (2.7)

This is so-called BGK model. At the same time, due to the mass, momentum and

energy conservation in particle collisions, the collision term (g − f)/τ satisfies the

compatibility condition, ∫ψg − f

τdΞ = 0. (2.8)

Eq.(2.7) is a nonlinear equation since the distribution function g depends on ρ, ρUi,

ρǫ, which is the integrals of the function f . The above BGK model is basically a

relaxation model.

17

If τ is a local constant, Eq.(2.7) may be written in integral form [48],

f(xi, t, ui, ξ) =1

τ

∫ t

t0

g(xi − ui(t− t′), t′, ui, ξ)e−(t−t′)/τdt′ +

e−(t−t0)/τf0(xi − ui(t− t0), ui, ξ), (2.9)

where f0 is the gas distribution function f at t0, and g is the equilibrium state in

(xi, t). As a special case, we consider an initial gas distribution function f(xi, t0)

which does not depend on the spatial coordinates. The corresponding macroscopic

mass ρ, momentum ρUi, and energy ρǫ are constant in space. Then, the corresponding

equilibrium state g will be a constant in both space and time. From Eq.(2.9), we have

f(t) = (1 − e−(t−t0)/τ )g + e−(t−t0)/τf0. (2.10)

The distribution function f tends to the equilibrium state g exponentially with a

characteristic relaxation time τ . For example, the smaller the τ is, the faster the

equilibrium will be reached. From this example, we can observe that the collision

time is related to the system dissipation and the gas-kinetic description provides more

information than the macroscopic one. Although, all macroscopic quantities are homo-

geneous and time independent, the particle distribution actually is a function of time.

Consequently, the dissipative property of the gas system is also a function of time.

The evolution from f to g is a process of increasing of entropy, which corresponds to

an irreversible process.

The particle collision time is related to both viscosity and heat conduction coeffi-

cients, and these two parameters are related through the Prandtl number,

Pr =ηCp

κ,

where η and κ are the coefficients of viscosity and thermal conductivity respectively.

The Prandtl number is almost a constant for air, and the value is about 0.72 at the

common temperature. From the BGK model, we can derive that the Prandtl number

becomes fixed with the value Pr = 1. Numerically, we can easily fix this number

through the modification of heat flux across a cell interface. Since in the continuum

18

regime the behavior of the fluid depends very little on the nature of individual particles,

the most important properties are: conservation, symmetry and dissipation. The BGK

model satisfies all these requirements [49]. In the following, we are going to introduce

a few shock capturing kinetic schemes.

2.2 Kinetic flux vector splitting (KFVS) method

It is well-known that the Euler equations can be derived from the Boltzmann equation

with a local equilibrium distribution function. For an equilibrium state, f is equal

to the maxwellian distribution function g, and the collision term Q(f, f) goes to zero

automatically, i.e., Q(g, g) ≡ 0. So, in the 1D case, once f = g holds, the Boltzmann

equation becomes

ft + ufx = 0, (2.11)

which is called collisionless Boltzmann equation. With the initial condition of the

gas distribution function f0(x) at time t = 0, the exact solution of the collisionless

Boltzmann equation is

f(x, t) = f0(x− ut). (2.12)

For example, for the same initial condition as the Riemann problem, two constant

equilibrium states at x ≤ 0 and x > 0 can be constructed around x = 0,

f0(x) = gl(1 −H(x)) + grH(x) =

gl , x ≤ 0,

gr , x > 0,(2.13)

where H(x) is the Heaviside function. The two equilibrium states gl and gr have one to

one correspondence with the macroscopic flow variables (ρl, ρlUl, ρlǫl), (ρr, ρrUr, ρrǫr)

on both sides and have the form

gl = ρl(λl

π)

K+1

2 e−λl((u−Ul)2+ξ2), gr = ρr(

λr

π)

K+1

2 e−λr((u−Ur)2+ξ2),

respectively. The exact solution of the collisionless Boltzmann equation is

f(x, t) = f0(x− ut) = gl(1 −H(x− ut)) + grH(x− ut). (2.14)

19

Since t > 0, the above equation can be reformulated as

f(x, t) = gl(1 −H(x

t− u)) + grH(

x

t− u) = f(

x

t) =

gl , u ≥ x

t,

gr, u < xt,

(2.15)

which is similar to the Riemann solution of the Euler equations. The collisionless

Boltzmann equation also provides a similarity solution for the case with two constant

initial states. Although this similarity solution is commonly referred to as an approx-

imate Riemann solution, it provides a different gas evolution picture, which can be

demonstrated by the following example. Firstly, we calculate the exact solutions of

Eq. (2.11) under the following initial condition

(ρ, ρU, ρǫ) =

(1.0, −2.0, 3.0), x ≤ 0,

(1.0, 2.0, 3.0), x > 0,(2.16)

from which two Maxwellians of gl and gr in Eq. (2.13)can be constructed. Based on

the solution (2.15), the mass, momentum and energy densities at any point in space

x and a fixed time t can be obtained by taking the moments of f(x, t),

ρ(x, t)

ρU(x, t)

ρǫ(x, t)

=

∫ ∫ ∞

−∞f(x, t)ψdudξ =

∫ ∫ ∞

xt

glψdudξ +

∫ ∫ xt

−∞grψdudξ

=

∫ ∫ ∞

xt

ρl(λl

π)

K+1

2 e−λl((u−Ul)2+ξ2)ψdudξ +

∫ ∫ xt

−∞ρr(

λr

π)

K+1

2 e−λr((u−Ur)2+ξ2)ψdudξ

=

∫ ∫ ∞

0

ρl(λl

π)

K+1

2 e−λl((u−Ul)2+ξ2)ψdudξ +

∫ ∫ 0

−∞ρr(

λr

π)

K+1

2 e−λr((u−Ur)2+ξ2)ψdudξ,

where u = u − xt, Ul = Ul − x

t, Ur = Ur − x

t, and ψ =

(1, u+ x

t, 1

2

((u+ x

t)2 + ξ2

))T.

20

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Den

sity

collisionless Boltzmann solution

exact Riemann solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

X

Vel

ocity

collisionless Boltzmann solution

exact Riemann solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

X

Pre

ssur

e

collisionless Boltzmann solution

exact Riemann solution

Figure 2.1: Exact Euler solutions (solid line) and collisionless Boltzmann solutions

(2.17) (+ symbol) at time t = 0.15 units for the same initial condition.

Then, the macroscopic variables in space and time can be obtained,

ρ(x, t)

ρU(x, t)

ρǫ(x, t)

= ρl

12erfc(−

√λlUl)

(12Ul + 1

2xt)erfc(−

√λlUl) + 1

2e−λlU

2l√

πλl

((Ul+

xt)2

4+ (K+1)

8λl)erfc(−

√λlUl) + ( Ul

4+ 1

2xt) e−λlU

2l√

πλl

+ρr

12erfc(

√λrUr)

(12Ur + 1

2xt)erfc(

√λrUl) − 1

2e−λrU2

r√πλr

((Ur+ x

t)2

4+ (K+1)

8λr)erfc(

√λrUr) − ( Ur

4+ 1

2xt) e−λrU2

r√πλr

, (2.17)

where the complementary error function is defined by

erfc(x) =2√π

∫ ∞

x

e−t2dt.

The density, velocity and pressure distributions in the above equation at time

t = 0.15 units are plotted in Fig. 2.1. The solid lines refer to the exact solution

from the similarity solution of the Euler equations and the + symbol are the solu-

tions from the collisionless Boltzmann equation (2.15). The above solution consists

21

of two strong rarefaction waves and a trivial stationary contact discontinuity. The

two rarefaction waves are quite smeared. This figure shows the difference between the

exact Euler solution and the exact collisionless Boltzmann solution under the same

initial condition. In other words, even with the initial equilibrium assumption, the

dynamics of the collisionless Boltzmann equation is different from that of the Euler

equations. The Eq.(2.11) is the correct description of the equilibrium flow motion if

the flow remains in a local equilibrium state. Nevertheless, even with an initial local

equilibrium state, the collisionless Boltzmann equation transports the gas distribution

function away from its equilibrium initial condition. Physically, the mechanism for

bringing the distribution function back to a maxwellian is the collision suffered by the

gas molecules, the so-called collision term in the Boltzmann equation. Therefore, the

collisionless Boltzmann equation without this mechanism, cannot keep the equilibrium

state, and theoretically cannot be used as a Euler solver. The collisionless mechanism

exists in many flux vector splitting schemes.

Based on the collisionless Boltzmann solution (2.11), the scheme based on the so-

lution (2.15) for the flux evaluation at a cell interface is so-called Kinetic Flux Vector

Splitting (KFVS) scheme. Although the KFVS scheme ignores particle collisions in

the gas evolution stage, it still gives a reasonable numerical solution, which is different

from the exact solution of the collisionless Boltzmann equation. It is the numerics

which adds pseudo-particle collisions in the scheme through the projection and re-

construction stages of a numerical method. In the following section, the first- and

second-order KFVS schemes are presented.

2.2.1 1st-order KFVS scheme

In 1D case, each cell occupies a small space x ∈ [xj−1/2, xj+1/2], where j+ 1/2 denotes

the cell interface between cells j and j + 1, and the cell center is located at xj . Wj =

(ρj , ρjUj, ρjǫj) is the initial conservative flow variables, i.e., the mass, momentum and

22

energy densities inside each cell j.

An equilibrium state gj has the following form

gj = ρj(λj

π)

K+1

2 e−λj((u−Uj)2+ξ2),

which can be uniquely determined from W . So, around a cell interface xj+1/2, we

have gl = gj and gr = gj+1 on the left and right hand sides. The solution f of the

collisionless Boltzmann equation at xj+1/2 becomes

f(xj+1/2, t) =

gj , u ≥ 0,

gj+1, u < 0.(2.18)

From the above distribution function, the numerical fluxes for the mass, momentum

and energy across the cell interface can be constructed, which are

Fj+1/2 =

Fρ

FρU

Fρǫ

j+1/2

=

∫uψf(xj+1/2, t)dudξ

=

∫

u>0

∫uψgjdudξ +

∫

u<0

∫uψgj+1dudξ (2.19)

where ψ = (1, u, 12(u2 + ξ2))T . Further, the fluxes can be expressed as

Fρ

FρU

Fρǫ

j+1/2

= ρj

Uj

2erfc(−

√λjUj) + 1

2e−λjU2

j√πλj

(U2

j

2+ 1

4λj)erfc(−

√λjUj) +

Uj

2e−λjU2

j√πλj

(U3

j

4+ K+3

8λjUj)erfc(−

√λjUj) + (

U2j

4+ K+2

8λj) e

−λjU2j√

πλj

+ρj+1

Uj+1

2erfc(

√λj+1Uj+1) − 1

2e−λj+1U2

j+1√πλj+1

(U2

j+1

2+ 1

4λj+1)erfc(

√λj+1Uj+1) − Uj+1

2e−λj+1U2

j+1√πλj+1

(U3

j+1

4+ K+3

8λj+1Uj+1)erfc(

√λj+1Uj+1) − (

U2j+1

4+ K+2

8λj+1) e

−λj+1U2j+1√

πλj+1

.

(2.20)

Using the above numerical fluxes, a finite volume method updates the cell averaged

flow variables W nj = (ρj, ρjUj , ρjǫj)

n inside each cell as

W n+1j = W n

j +tx(Fj−1/2 − Fj+1/2), (2.21)

23

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

XD

ensi

ty

1st−order KFVS scheme

exact Riemann solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

X

Vel

ocity

1st−order KFVS scheme

exact Riemann solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

X

Pre

ssur

e

1st−order KFVS scheme

exact Riemann solution

Figure 2.2: Exact Euler solutions (solid line) and numerical solutions (+ symbol) using

the 1st-order KFVS scheme at time t = 0.15 units.

where n is the time step number, t is time step, and x the cell size.

Let’s apply the 1st-order KFVS scheme to the same test case above Eq.(2.16). In

Fig. 2.2 the density, velocity and pressure distributions are plotted, where the solid

lines are the exact Riemann solutions and the + symbol are the numerical solutions

with 100 grid points using the 1st-order KFVS scheme. Obviously, we can observe the

difference between the numerical solutions and the exact solutions from the collisionless

Boltzmann equation by comparing Fig. 2.1 with Fig. 2.2. The difference is due to the

preparation of Maxwellian distribution functions at the beginning of each time step,

which is equivalent to adding pseudo-particle collisions into the collisionless Boltzmann

method. The diffusivity in the KFVS scheme is not only from the truncation error

of the numerical discretization, but also from the intrinsic nature of the governing

equation. The numerical dissipation is proportional to the time step, or the pseudo-

particle collision time.

24

2.2.2 2nd-order KFVS scheme

In order to develop a high-order scheme, we need first to reconstruct a high-order

initial condition. For example, the van Leer limiter [75],

Lj = (sign(s+) + sign(s−))|s+||s−|

|s+| + |s−|(2.22)

can be used for the reconstruction of initial conservative variables inside the jth cell,

where s+ and s− represent the possible slopes of conservative variables in the jth cell:

s+ =Wj+1 −Wj

xj+1 − xj, s− =

Wj −Wj−1

xj − xj−1

with Wl = Wj and Wr = Wj+1,∂Wl

∂x= Lj and ∂Wr

∂x= Lj+1 at the left and the right

hand sides of a cell interface xj+1/2. A second-order accurate initial condition can be

constructed,

W lj+1/2

W rj+1/2

=

Wl + ∂Wl

∂x(xj+1/2 − xj)

Wr + ∂Wr

∂x(xj+1/2 − xj+1)

, (2.23)

from which the equivalent initial gas distribution function f0 around a cell interface

xj+1/2 is obtained (see the schematic figure Fig.2.3)

f0(x) =

gl(1 + al(x− xj+1/2)) , x− xj+1/2 ≤ 0,

gr(1 + ar(x− xj+1/2)) , x− xj+1/2 ≥ 0.(2.24)

Based on the above initial condition, the solution f of the collisionless Boltzmann

equation (2.11) at xj+1/2 and time t becomes

f(xj+1/2, t) = f0(xj+1/2 − ut) =

gl(1 + al(−ut)) , u ≥ 0,

gr(1 + ar(−ut)) , u ≤ 0,(2.25)

where the terms al and ar are based on the Taylor expansion of the Maxwellian dis-

tribution function and have the form

al = al1 + al

2u+ al3

1

2(u2 + ξ2) , ar = ar

1 + ar2u+ ar

3

1

2(u2 + ξ2).

25

Figure 2.3: Initial gas distribution function for the 2nd−order KFVS scheme.

By using the relation between the microscopic gas distribution function and the macro-

scopic variables at xj+1/2, we get∫ψgldudξ = W l

j+1/2,

∫ψgrdudξ = W r

j+1/2,

∫ψglaldudξ =

W lj+1/2 −Wl

xj+1/2 − xj,

∫ψgrardudξ =

Wr −W rj+1/2

xj+1 − xj+1/2. (2.26)

Moreover, Eq.(2.26) on both sides of a cell interface can be simplified as

M

a1

a2

a3

=

1

ρ

∂ρ∂x

∂(ρU)∂x

∂(ρǫ)∂x

, (2.27)

where the symmetric matrix M has the form

M =

1 U 12(U2 + K+1

2λ)

U U2 + 12λ

12(U3 + (K+3)U

2λ)

12(U2 + K+1

2λ) 1

2(U3 + (K+3)U

2λ) 1

4(U4 + (K+3)U2

λ+ (K2+4K+3)

4λ2 )

.

(2.28)

These derivatives of macroscopic variables on the right side of Eq.(2.26) can be calcu-

lated by (2.22). The solutions of Eq.(2.27) are

a3 = 4λ2

K+1(B − 2UA),

a2 = 2λ(A− a3U2λ

),

a1 = 1ρ

∂ρ∂x

− a2U − a3(U2

2+ K+1

4λ),

26

where

A =1

ρ(∂(ρU)

∂x− U

∂ρ

∂x) , B =

1

ρ(2∂(ρǫ)

∂x− (U2 +

K + 1

2λ)∂ρ

∂x).

On the other hand, the values of (a1, a2, a3) can be also obtained directly by the

Taylor expansion of the Maxwellian distribution function in terms of the macroscopic

flow variables. From Eq.(2.25), the time−dependent numerical fluxes of the mass,

momentum and energy can be written as,

Fj+1/2 =

Fρ

Fρu

Fρǫ

j+1/2

=

∫

u>0

∫u

1

u

12(u2 + ξ2)

gl(1 − alut)dudξ +

∫

u<0

∫u

1

u

12(u2 + ξ2)

gr(1 − arut)dudξ. (2.29)

Then, the flow variables inside the jth cell can be updated through

W n+1j = W n

j +1

x

∫ t

0

(Fj−1/2 −Fj+1/2)dt. (2.30)

For the same test case shown above, the numerical results from the current 2nd-order

KFVS scheme are shown in Fig. 2.4. Comparing Fig. 2.4 with Fig. 2.2, the 2nd-order

KFVS scheme enhances the accuracy of the two strong rarefaction waves. But, the

KFVS scheme usually gives a poorer result than those obtained from the Godunov or

flux difference splitting (FDS) schemes. Even for the Euler solutions, the 2nd-order

FDS scheme usually gives less dissipative results than those from the 2nd-order FVS

scheme. On the other hand, the FDS schemes may be less robust than the FVS

methods. For example, the Roe’s scheme can get negative density or pressure easily

for the current test case.

Again, the gas evolution model in the KFVS scheme is based on the collisionless

Boltzmann equation. Therefore, in the gas evolution stage of the KFVS scheme, the

particles can transport freely. For example, gas in high temperature region is freely

moving into low temperature region without suffering any particle collisions. As a

27

result, the free penetration of particles strongly and easily smears any temperature

gradients and removes the possible formation of contact discontinuity wave. This can

be clearly seen in the Sod test case in [77] ( on pages 26-39 ). Similarly, the ‘shock’

from collisionless Boltzmann equation will also be smeared due to the free transport

of particles across the ”shock” front even though the shock has self-steepening mech-

anism. Numerically, the particles are not absolutely moving freely as described in

the collisionless Boltzmann equation. The numerical particles do suffer some kind of

collisions to reduce the dissipation or penetration to smear any gradient. In order

to reduce the numerical dissipation, the real particle collision has to be added in the

particle transport process, where the physical collision time τ is introduced into the

solution f around each cell interface. In the following, the gas-kinetic BGK scheme

will be introduced.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Den

sity

2nd−order KFVS scheme

exact Riemann solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

X

Vel

ocity

2nd−order KFVS scheme

exact Riemann solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

X

Pre

ssur

e

2nd−order KFVS scheme

exact Riemann sultion

Figure 2.4: Exact Euler solutions (solid line) and numerical solutions (+ symbol) using

the 2nd-order KFVS scheme at time t = 0.15 units.

28

2.3 Gas-kinetic BGK method with the directional

splitting fluxes

The 2D BGK equation can be written as [87]

ft + ufx + vfy =g − f

τ, (2.31)

where f is the gas distribution function and g is the equilibrium state approached by

f . Both f and g are functions of space (x, y), time t, particle velocities (u, v), and

internal variable ξ. The particle collision time τ is related to the viscosity and heat

conduction coefficients. The equilibrium state is a Maxwellian distribution,

g = ρ(λ

π)

K+2

2 e−λ((u−U)2+(v−V )2+ξ2),

where ρ is the density, U and V are the macroscopic velocities in the x and y directions,

and λ is related to the gas temperature m/2kT . For a 2D flow, the particle motion

in the z− direction is included into the internal variable ξ, and the total number of

degrees of freedom K in ξ is equal to (5 − 3γ)/(γ − 1) + 1. In the equilibrium state,

ξ2 is equal to ξ2 = ξ21 + ξ2

2 + ... + ξ2K . The relation between mass ρ, momentum

(m = ρU, n = ρV ), and energy densities ρǫ with the distribution function f is

w =

ρ

m

n

ρǫ

=

∫ψfdΞ, α = 1, 2, 3, 4, (2.32)

where ψ is the vector of moments

ψ = (ψ1, ψ2, ψ3, ψ4)T = (1, u, v,

1

2(u2 + v2 + ξ2))T ,

and dΞ = dudvdξ is the volume element in the phase space with dξ = dξ1dξ2...dξK .

Since mass, momentum and energy are conserved during particle collisions, f and g

satisfy the conservation constraint∫

(g − f)ψdΞ = 0, (2.33)

29

at any point in space and time.

We can use a directional splitting method to solve Eq. (2.31). For example, the

BGK model in the x−direction can be written as

ft + ufx =g − f

τ. (2.34)

For a local equilibrium state with f = g, the Euler equations can be obtained by

taking the moments of the above equation. This yields

∫ψ(gt + ugx)dΞ = 0, α = 1, 2, 3, 4, (2.35)

and the corresponding Euler equations in the x−direction are

ρ

ρU

ρV

ρǫ

t

+

ρU

ρU2 + p

ρUV

(ρǫ+ p)U

x

= 0, (2.36)

where ρǫ = 12ρ(U2 + V 2 + K+2

2λ) and p = ρ/2λ.

On the other hand, to the first order of τ , the Chapmann-Enskog expansion gives

f = g − τ(gt + ugx). Taking moments of ψ again to the BGK equation with the new

f , we get ∫ψ(gt + ugx)dΞ = τ

∫ψ(gtt + 2ugxt + u2gxx)dΞ, (2.37)

from which the Navier-Stokes equations with a dynamic viscous coefficient µ = τp can

be obtained,

ρ

ρU

ρV

ρǫ

t

+

ρU

ρU2 + p

ρUV

(ρǫ+ p)U

x

=

0

S1x

S2x

S3x

x

, (2.38)

30

where

S1x = τp

[2∂U

∂x− 2

K + 2

∂U

∂x

],

S2x = τp∂V

∂x,

S3x = τp

[2∂U

∂x+ V

∂V

∂x− 2

K + 2U∂U

∂x+K + 4

4

∂

∂x

(1

λ

)].

The derivation from the BGK model in 3D case to the NS equations for a monatomic

gas can be found in [77].

The general solution f of the BGK model Eq. (2.34) at a cell interface xj+1/2 and

time t is

f(xj+1/2, t, u, v, ξ) =1

τ

∫ t

0

g(x′, t′, u, v, ξ)e−(t−t′)/τdt′ + e−t/τf0(xj+1/2 − ut), (2.39)

where x′ = xj+1/2 − u(t− t′) is the trajectory of a particle motion and f0 is the initial

gas distribution function f at the beginning of each time step (t = 0). Two unknowns g

and f0 must be specified in Eq.(2.39) in order to obtain the solution f . The directional

splitting BGK-NS method is described in [107], where only the physical variables in

the normal direction of a cell interface are considered. The initial gas distribution

function f0 is assumed to have the form,

f0 =

gl[1 + al(x− xj+1/2) − τ(alu+ Al)] , x− xj+1/2 ≤ 0,

gr[1 + ar(x− xj+1/2) − τ(aru+ Ar)] , x− xj+1/2 ≥ 0.(2.40)

The terms proportional to τ represent the non-equilibrium parts in the Chapman-

Enskog expansion of the BGK model. The non-equilibrium parts have no direct con-

tribution to the conservative flow variables, i.e.,

∫(alu+ Al)ψgldΞ = 0,

∫(aru+ Ar)ψgrdΞ = 0.

(2.41)

After having f0, the equilibrium state g around (x = xj+1/2, t = 0) is constructed as

g = g0

[1 + (1 − H[x− xj+1/2])a

l(x− xj+1/2) + H[x− xj+1/2]ar(x− xj+1/2) + At

],

(2.42)

31

where H[x− xj+1/2] is the Heaviside function defined by

Figure 2.5: The spatial distribution of the initial state f0 and the equilibrium state g

at t = 0 for the 2nd-order BGK scheme.

H [x− xj+1/2] =

0 , x− xj+1/2 < 0,

1 , x− xj+1/2 ≥ 0.

Here g0 is a local Maxwellian distribution function located at x = xj+1/2. Even though

g is continuous at x = xj+1/2, it has different slopes at x < xj+1/2 and x > xj+1/2; see

Fig.2.5. In both f0 and g, al, Al, ar, Ar, al, ar , and A are related to the derivatives of a

Maxwellian in space and time. The dependence of al, ar, ..., A on the particle velocities

can be obtained from a Taylor expansion of a Maxwellian and has the following form,

al = al1 + al

2 + al3v + al

4

1

2(u2 + v2 + ξ2) = al

αψα,

Al = Al1 + Al

2 + Al3v + Al

4

1

2(u2 + v2 + ξ2) = Al

αψα,

A = A1 + A2 + A3v + A41

2(u2 + v2 + ξ2) = Aαψα,

where α = 1, 2, 3, 4, and all coefficients al1, a

l2, . . . , A4 are local constants.

In the reconstruction step described in the section 2.2.2, the distributions ρj(x), mj(x),

nj(x), and ρj ǫj(x) are obtained inside each cell xj−1/2 ≤ x ≤ xj+1/2. At the cell inter-

32

face xj+1/2, the left and right macroscopic states are

wj(xj+1/2) =

ρj(xj+1/2)

mj(xj+1/2)

nj(xj+1/2)

ρj ǫj(xj+1/2)

, wj+1(xj+1/2) =

ρj+1(xj+1/2)

mj+1(xj+1/2)

nj+1(xj+1/2)

ρj+1ǫj+1(xj+1/2)

(2.43)

By using the relation between the gas distribution function f and the macroscopic

variables Eq.(2.32), at xj+1/2, we get

∫glψdΞ = wj(xj+1/2),

∫glalψdΞ =

wj(xj+1/2) − wj(xj)

x− , (2.44)

∫grψdΞ = wj+1(xj+1/2),

∫grarψdΞ =

wj+1(xj+1) − wj+1(xj+1/2)

x+, (2.45)

where x− = xj+1/2 − xj and x+ = xj+1 − xj+1/2, and the Maxwellians are,

gl = ρl

(λl

π

)K+2

2

e−λl((u−U l)2+(v−V l)2+ξ2),

gr = ρr

(λr

π

)K+2

2

e−λr((u−Ur)2+(v−V r)2+ξ2).

All the parameters in gl and gr can be uniquely determined from Eq.(2.44) and (2.45),

and

ρl

U l

V l

λl

=

ρj(xj+1/2)

mj(xj+1/2)/ρj(xj+1/2)

nj(xj+1/2)/ρj(xj+1/2)

λl

,

ρr

U r

V r

λr

=

ρj+1(xj+1/2)

mj+1(xj+1/2)/ρj+1(xj+1/2)

nj+1(xj+1/2)/ρj+1(xj+1/2)

λr

,

where

λl =(K + 2)ρj(xj+1/2)

4(ρj ǫj(xj+1/2) − 1

2(m2

j (xj+1/2) + n2j (xj+1/2))/ρj(xj+1/2)

) ,

and

λr =(K + 2)ρj+1(xj+1/2)

4(ρj+1ǫj+1(xj+1/2) − 1

2(m2

j+1(xj+1/2) + n2j+1(xj+1/2))/ρj+1(xj+1/2)

) .

33

The slope al in Eq.(2.44) can be computed from,

wj(xj+1/2) − wj(xj)

ρlx+= M l

αβ

al1

al2

al3

al4

= M lαβa

lβ , (2.46)

where

M lαβ =

1

ρl

∫glψαψβdΞ.

The above matrix and the direct evaluation of the solution from the above equation

are similar to the one in section 2.2.2. For gr, the matrix M rαβ = 1

ρr

∫grψαψβdΞ has

the same structure as M lαβ , and the slope ar in Eq.(2.45) can be obtained similarly.

Further Al and Ar can be determined from Eq.(2.41), which are

M lαβA

lβ = − 1

ρl

∫glaluψαdΞ,

M rαβA

rβ = − 1

ρr

∫graruψαdΞ.

(2.47)

Since M lαβ , M r

αβ , and the right-hand sides of the above equations are known, the

parameters Al and Ar can be obtained subsequently using the previous method in

section 2.2.2.

In the equilibrium state g, g0 at the cell interface is defined by

g0 = ρ0

(λ0

π

)K+2

2

e−λ0((u−U0)2+(v−V0)2+ξ2).

The corresponding values of ρ0, U0, V0, and λ0 in g0 can be determined as follows.

Taking the limit t → 0 in Eq.(2.39) and substituting its solution into Eq.(2.33), the

conservation constraint at (x = xj+1/2, t = 0) gives

∫g0ψdΞ = w0 =

∫

u>0

∫glψdΞ +

∫

u>0

∫grψdΞ, (2.48)

34

where w0 = (ρ0, m0, n0, ρ0ǫ0)T is the macroscopic conservative variables located at the

cell interface at time t = 0. After having w0, λ0 in g0 can be found from

λ0 = (K + 2)ρ0

/(4

(ρ0ǫ0 −

1

2(m2

0 + n20)/ρ0

))

Then, al and ar of g in Eq.(2.39) can be calculated from

w0 − wj(xj)

ρ0x−= M0

αβ

al1

al2

al3

al4

= M0αβa

lβ , (2.49)

and

wj(xj+1) − w0

ρ0x+= M0

αβ

ar1

ar2

ar3

ar4

= M0αβ a

rβ , (2.50)

where the matrix 1ρ0M0

αβ =∫g0ψαψβdΞ is known, see Fig.2.5. Hence, (al

1, al2, a

l3, a

l4)

T

and (ar1, a

r2, a

r3, a

r4)

T can be found following the same procedure as al and ar.

Up to now, we have determined all parameters in the initial gas distribution function

f0 and the equilibrium state g at the beginning of each time step t = 0, the gas

distribution function f at a cell interface can be expressed as

f(xj+1/2, t, u, v, ξ) = (1 − e−t/τ )g0

+(τ(−1 + e−t/τ ) + te−t/τ )(alH[u] + ar(1 − H[u]))ug0

+τ(t/τ − 1 + e−t/τ )Ag0 + e−t/τ ((1 − (t+ τ)ual)H[u]gl

+(1 − u(t+ τ)ar)(1 − H[u])gr)

+e−t/τ (−τAlH[u]gl − τAr(1 − H[u])gr). (2.51)

The only unknown left in the above expression is A. Since both f and g contain A,

the integration of the conservation constraint Eq.(2.33) at xj+1/2 over the whole time

35

step t gives ∫ t

0

∫(g − f)ψdΞ = 0,

which goes to

M0αβAβ ≡ 1

ρ0(∂ρ/∂t, ∂m/∂t, ∂n/∂t, ∂E/∂t)T

=1

ρ0

∫[γ1g0 + γ2u(a

lH [u] + ar(1 −H [u]))g0 + γ3(H [u]gl + (1 −H [u])gr)

+γ4u(alH [u]gl + ar(1 −H [u])gr) + γ5((alu+ Al)H [u]gl

+(aru+ Ar)(1 −H [u])gr)]ψαdΞ, (2.52)

where

γ0 = t− τ(1 − e−t/τ ),

γ1 = −(1 − e−t/τ )/γ0,

γ2 = (−t+ 2τ(1 − e−t/τ ) −te−t/τ )/γ0,

γ3 = (1 − e−t/τ )/γ0,

γ4 = (te−t/τ − τ(1 − e−t/τ ))/γ0,

γ5 = −τ(1 − e−t/τ )/γ0,

Finally, the time-dependent numerical fluxes in the normal-direction across the cell

interface can be computed by

Fj+1/2 =

Fρ

Fm

Fn

Fρǫ

j+1/2

=

∫u

1

u

v

12(u2 + v2 + ξ2)

f(xj+1/2, t, u, v, ξ)dΞ, (2.53)

where f(xj+1/2, t, u, v, ξ) is given by Eq.(2.37). By integrating the above equation to

the whole time step, we can get the total mass, momentum and energy transport

across the cell interface xj+1/2. A similar procedure can be carried out for calculating

the numerical fluxes in y−direction. After that, we can use an explicit finite volume

scheme to update the averaged flow variables inside each cell. Usually, the above gas-

36

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

XD

ensi

ty

2nd−order BGK scheme

exact Riemann solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

X

Vel

ocity

2nd−order BGK scheme

exact Riemann solution

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

X

Pre

ssur

e

2nd−order BGK scheme

exact Riemann solution

Figure 2.6: Exact Euler solutions (solid line) and numerical solutions (circle symbol)

using the 2nd-order gas-kinetic BGK scheme at time t = 0.15 units.

kinetic BGK scheme for the Navier-Stokes equations is called the directional splitting

BGK-NS method, which is also referred in the following chapters.

The Sjogreen test case has been calculated by the exact solution from collisionless

Boltzmann equation (2.17), and the 1st and 2nd KFVS schemes respectively. The

results from the 2nd-order BGK scheme are shown in Fig.2.6, which are closer to

the exact Riemann solution by comparing it with Fig.2.1, Fig.2.2 and Fig.2.4. Note

that the BGK scheme solves the Navier-Stokes equations instead of the Euler ones.

As realized, the gas-kinetic BGK scheme has advantages for the capturing of strong

rarefaction waves.

For the simulation of viscous and heat conducting flow, a subject of debate on the

BGK scheme is that the scheme has a fixed Prandtl number with unity. This is a well-

known result for the BGK model. In order to change the above Prandtl number to any

realistic value, different methods to resolve this problem have been proposed. A simple

37

method is presented in [107]. In the BGK method, we have obtained explicitly the

time-dependent gas distribution function f at the cell interface Eq. (2.51). Therefore,

the time-dependent heat flux can be evaluated precisely,

q =1

2

∫(u− U)((u− U)2 + (v − V )2 + ξ2)fdΞ, (2.54)

where the average velocities U and V are defined by

U =

∫ufdΞ

/∫fdΞ , V =

∫vfdΞ

/∫fdΞ .

Then, the easiest way to fix the Prandtl number for the BGK scheme is to modify the

energy flux by subtracting the above heat flux (2.54) and adding another one with a

correct Prandtl number,

Fnewρǫ = Fρǫ +

(1

Pr− 1

)q, (2.55)

where Fρǫ is the energy flux in Eq.(2.53).

2.4 Multidimensional gas-kinetic BGK scheme for

viscous flows

In recent years, the gas-kinetic scheme based on the Bhatnagar-Gross-Krook (BGK)

model for the Navier-Stokes equations has been developed by implementing a mul-

tidimensional particle propagation mechanism in the flux evaluation [80], where the

gradients of flow variables in both normal and tangential directions of a cell interface

are explicitly included. The multidimensional gas-kinetic BGK scheme is to pave the

way to extend the current approach directly to the flow computation with unstructured

mesh, where the flow gradients in both parallel and perpendicular directions around a

cell interface can be explicitly taken into account in a viscous flow simulation. In [50],

the multidimensional gas-kinetic BGK scheme has been applied on unstructured mesh

for 3D computation of viscous flows. An important issue in the design and development

of aerospace vehicles is the accurate calculation of various types of flow phenomena on

38

aerodynamic performance and aero-thermal loads. Evaluation of aerodynamic heating

during reentry flight is one of the key issues, where complicated flow phenomena, i.e.,

shock boundary layer interaction, flow separation, and viscous/inviscid interaction,

will be encountered. To the current stage, the computation of the interaction between

a shock wave and a separated region in the hypersonic flow is still a very challeng-

ing problem in computational fluid dynamics. In terms of heat transfer, significant

differences between the computational results and experiments indicate that further

investigation and design of more accurate viscous flow solvers are needed to understand

these phenomena. The severe heating rates produced by the viscous/inviscid interac-

tions and by the shock/shock interactions can cause catastrophic failure for the vehicles

in the hypersonic flight. Over the past 15 years there have been concerted efforts both

in Europe and America to validate the Navier-Stokes and DSMC-based methods in

the description of complex hypersonic flows. Extensive codes validation have been

conducted. A number of experiments specifically designed with simple model configu-

ration in the laminar hyper velocity flow have been constructed and continuously being

used to examine complex flow phenomena, such as the viscous/inviscid interactions.

Based on the gas-kinetic theory, the Navier-Stokes equations can be derived from

the Boltzmann equation using the Chapman-Enskog expansion. In the gas-kinetic

representation, all flow variables become the moments of a single particle distribution

function. Since a gas distribution function is used to describe both equilibrium and

non-equilibrium states, the inviscid and viscous fluxes are obtained simultaneously

in the gas-kinetic scheme. However, in the traditional upwinding schemes for the

Navier-Stokes solutions, an operator splitting method is commonly adopted, where

the Riemann solver or equivalent flux evaluation based on two constant states is used

for the inviscid part and the central differences for the viscous and heat conduction

parts. Theoretically, the use of different flow distributions for the inviscid and viscous

parts is artificial, which may introduce numerical error in the hypersonic viscous flow

computation with strong coupling of the inviscid and viscous flow interactions. For

39

example, the dissipative characteristic of upwind schemes in the regions with sharp

gradients in the boundary layer may trigger unsteady mechanism to prevent from

obtaining steady state solution. Even though high-order discretization can be intro-

duced for the inviscid and viscous parts separately, an operator splitting error due

to the different initial condition (or equivalently kinematic dissipation) can be hardly

eliminated, especially in the cases with severe coupling between transport and dissi-

pative heating. On the contrary, for the gas-kinetic BGK scheme, both inviscid and

viscous parts are recovered in a single gas distribution function f , which is based on

the same initial condition. As pointed out earlier [107], the gas-kinetic description

is capable of giving a more complete description of the non-equilibrium flow. Even

though the non-equilibrium parts in a gas-distribution function have no direct contri-

bution to the macroscopic mass, momentum and energy, but they do contribute to the

higher moments, such as the fluxes. In [80], the multidimensional gas-kinetic BGK

scheme and its application to the hypersonic viscous flow are presented.

For a multidimensional scheme, the general solution f of the BGK model at a cell

interface xi+1/2, yj and time t is

f(xi+1/2, yj, t, u, v, ξ) =1

τ

∫ t

0

g(x′, y′, t′, u, v, ξ)e−(t−t′)/τdt+e−t/τf0(xi+1/2−ut, yj −vt),

(2.56)

where x′ = xi+1/2−u(t−t′), y′ = yj−v(t−t′) are the trajectory of a particle motion and

f0 is the initial gas distribution function f at the beginning of each time step (t = 0).

Two unknowns g and f0 must be specified in Eq.(2.56) in order to obtain the solution

f . Based on the Chapman-Enskog expansion of the BGK model Eq.(2.31), the gas

distribution function up to the Navier-Stokes order at the point (xi+1/2 = 0, yj = 0)

and time (t = 0) has the form [96],

fNS = g − τ(gt + ugx + vgy),

where φ1 = −τ(gt +ugx +vgy) has to satisfy the compatibility condition∫ψφ1dΞ = 0.

40

To the 2nd-order accuracy, the gas distribution function can be approximated as

fNS = g + gx(x− xi+1/2) + gy(y − yj) − τ(gt + ugx + vgy).

Therefore, in the multidimensional gas-kinetic scheme with the initial discontinuous

macroscopic variables at the left and right hand sides of a cell interface the initial gas

distribution function f0 has the form,

f0 =

gl(1 + al(x− xi+1/2) + bly − τ(al + bl + Al)) , x− xi+1/2 < 0,

gr(1 + ar(x− xi+1/2) + bry − τ(ar + br + Ar)), x− xi+1/2 ≥ 0,(2.57)

where glal, grar, glAl and grAr are related to the spatial and temporal derivatives

of the equilibrium states. Different from directional splitting method, glbl and grbr

correspond to the gradients in the tangential direction along the cell interface. The

terms proportional to τ represent the non-equilibrium parts in the Chapman-Enskog

expansion, and have no direct contribution to the conservative flow variables, i.e.,

∫(alu+ blv + Al)ψgldΞ = 0,

∫(aru+ brv + Ar)ψgrdΞ = 0,

(2.58)

which are the exact equations to determine Al and Ar. In the above f0, gl, gr, al ,and

ar have the same definition as the corresponding parameters in the previous direction

splitting BGK-NS method.

After having f0, the equilibrium state g around (x = xi+1/2, y = yj , t = 0) is

constructed as

g = g0(1+(1−H [x−xi+1/2])al(x−xi+1/2)+H [x−xi+1/2]a

r(x−xi+1/2)+by+At), (2.59)

where b is the additional term related to the flow variation in the tangential direction,

and g0 is a local Maxwellian distribution function located at (x = xi+1/2, y = yj).

In both f0 and g, al, Al, ar, Ar, al, ar, b, and A are related to the derivatives of a

Maxwellian in space and time. Similar to the previous BGK-NS method, al and

41

ar can be uniquely determined. With the definition of Maxwellian distributions

gl = ρl

(λl

π

)K+2

2

e−λl((u−U l)2+(v−V l)2+ξ2),

and

gr = ρr

(λr

π

)K+2

2

e−λr((u−Ur)2+(v−V r)2+ξ2),

in the tangential direction bl and br can be obtained from

∫glblψdΞ = ~t · ∇wl;

∫grbrψdΞ = ~t · ∇wr,

where ~t is the unit vector in the tangential direction along the cell interface, and ∇wl

and ∇wr are gradients of macroscopic variables w = (ρ, ρu, ρv, ρǫ)T on the left and

right hand sides of a cell interface. After determining the terms al, bl, ar, and br, Al

and Ar in f0 can be found from Eq.(2.58), which are

M lαβA

lβ = − 1

ρl

∫(alu+ blv)ψαg

ldΞ,

M rαβA

rβ = − 1

ρr

∫(aru+ brv)ψαg

rdΞ,(2.60)

where M lαβ =

∫glψαψβdΞ/ρ

l, and M rαβ =

∫grψαψβdΞ/ρ

r.

For the equilibrium state g in Eq.(2.59), g0 is still defined by Eq.(2.48), and al and

ar are also evaluated by Eq.(2.49) and (2.50) respectively. The term b is calculated

from

∫g0bψdΞ =

∫

u>0

glblψdΞ +

∫

u<0

grbrψdΞ. (2.61)

Now, we have determined all parameters in the initial gas distribution function f0

and equilibrium state g at the beginning of each time step t = 0. After substituting

Eqs.(2.57) and (2.59) into Eq.(2.56), the gas distribution f at a cell interface can be

42

expressed as,

f(xi+1/2, yj, t, u, v, ξ) = (1 − e−t/τ )g0

+(τ(−1 + e−t/τ ) + te−t/τ )(alH[u] + ar(1 − H[u]) + bv)ug0

+τ(t/τ − 1 + e−t/τ )Ag0 + e−t/τ ((1 − (t+ τ)(ual + vbl))H[u]gl

+(1 − (t+ τ)(uar + vbr)(1 − H[u])gr)

+e−t/τ (−τAlH[u]gl − τAr(1 − H[u])gr). (2.62)

The only unknown left in the above expression is A, which is contained in both

f (Eq.(2.62)) and g (Eq.(2.59)). The integration of the conservation constraint at

xi+1/2, yj over the whole time step t gives∫ t

0

∫(g − f)ψdtdΞ = 0,

from which A can be obtained. The multidimensional BGK scheme is similar to the

direction splitting gas-kinetic scheme except additional terms bl, br, and b related to

the flow variations in the tangential direction.

Finally, the time-dependent numerical fluxes in the normal-direction across the cell

interface can be computed by

Fi+1/2,j =

∫u

1

u

v

12(u2 + v2 + ξ2)

f(xi+1/2, yj, t, u, v, ξ)dΞ, (2.63)

where

Fi+1/2,j =

Fρ

Fm

Fn

Fρǫ

i+1/2,j .

The flow variables Wi,j = (ρ, (ρu), (ρv), (ρǫ))Ti,j inside cell (i, j) can be updated by

W n+1i,j = W n

i,j +

∫ t

0

(1

x(Fi−1/2,j − Fi+1/2,j) +1

y (Fi,j−1/2 −Fi,j+1/2)

)dt. (2.64)

43

In summary, the above multidimensional gas-kinetic BGK scheme includes both

normal and tangential slopes across a cell interface in the flux calculation. Theoreti-

cally, the above method can enhance the accuracy of the BGK-NS scheme although it

still remains second-order accuracy. Numerically, it has been shown, such as in [51],

that the multidimensional scheme can capture some physical phenomena which are

absent from the directional splitting method, such as the thermal creeping flow. The

advantages of the multidimensional gas-kinetic BGK scheme are the following:

1.The scheme is able to capture the characteristics flow more accurately with curved

boundary or nonuniform temperature, such as the inverted velocity distribution in

rarefied cylindrical Couette flow and the weak flow field induced by the temperature

gradient of a body;

2. It can also give better results in high-speed micro-channel flow and power-law

fluid flow between concentric rotating cylinders, when compared with direct simulation

Monte Carlo and analytic solutions.

2.5 Multidimensional BGK scheme for nearly in-

compressible viscous flows

As stated in the above two sections, the gas-kinetic BGK scheme is focusing on the high

speed compressible flows described by the Navier-Stokes equations. For some bench-

mark cases with the low Mach number, the above multidimensional gas-kinetic BGK

scheme can be much simplified to simulate the approximate incompressible isother-

mal viscous flows [52]. Since there is no discontinuity in the incompressible flow,

the distribution function f0 in Eq.(2.57) can be simplified as gl = gr, al = ar, and

bl = br. Consequently, Eq.(2.61) gives g0 = gl = gr. Similarly, al = ar = al = ar

and b = bl = br can be derived. The incompressible flow can be simulated using

BGK scheme at low Mach number. This is equivalent to the artificial compressibility

44

method. In this situation, the flow variables w = (ρ, ρu, ρv), where the flow energy ρǫ

can be ignored, is not necessary to be divided into left and right states with a discon-

tinuity at each cell interface. Here, with the definition ∂g/∂x = ag, ∂g/∂y = bg, and

∂g/∂t = Ag, we have a = a1 + a2u+ a3v, b = b1 + b2u+ b3v and A = A1 +A2u+A3v.

For the isothermal N-S equations, λ = m/2kT is a constant, where k, T , and m are

the Boltzmann constant, temperature, and molecular mass. In the 2D case, in order to

accurately capture the incompressible flow behavior the gas is limited to move only in

a plane without any random motion in the third direction. Then, the internal degree

of freedom K is assumed to be equal to 0, while γ = 2 is obtained with the relation

of γ = (K + 4)/(K + 2). Therefore, the local Maxwellian equilibrium distribution

function at the point t = 0, x0 = (x0, y0) with the particle velocity u = (u, v) is

defined by

g(x0,u, t = 0) = ρ(λ

π)e−λ((u−U)2+(v−V )2). (2.65)

With a given Mach number Ma, λ can be determined by

λ =M2

a

U2∞,

where U∞ = (U∞, V∞) is the upstream velocity, and U2∞ = U2

∞ + V 2∞.

The initial condition f0(x,u) at the beginning of each time-step t = 0 around x is

assumed to be a distribution function truncated to the Navier-Stokes order,

f0(x,u) = g(x0,u, 0) + f (1)(x0,u, 0) + (x − x0) · ∇g(x0,u, 0).

For the BGK model, the non-equilibrium state f (1) is f (1) = −τDg with D = ∂t+u·∇.

The specific form becomes

f (1) = −τ(∂g/∂t + u · ∇g) = −τ(au + bv + A)g.

Therefore, f0 at time t = 0 can be written as

f0(x,u) = g (1 − τ(au+ bv + A) + a(x− x0) + b(y − y0)) . (2.66)

45

The equilibrium distribution in space and time around (x0, t = 0) can be obtained

from the expansion of an equilibrium state

g(x,u, t) = g(x,u, 0) + (∂g/∂x)(x − x0) + (∂g/∂x)(y − y0) + (∂g/∂t)t

= g(1 + a(x− x0) + b(y − y0) + At).

Substituting both g and f0 into the general integral solution (2.56) in the above section,

the distribution function f at a cell interface (x = x0) becomes

f(x0,u, t) = g (1 − τ(au+ bv + A) + At) . (2.67)

In f(x0,u, t), the coefficients a1, a2, a3 and b1, b2, b3 can be found from the relation

∫ ∫

1

u

v

agdudv =

∂ρ/∂x

∂(ρu)/∂x

∂(ρv)/∂x

,

∫ ∫

1

u

v

bgdudv =

∂ρ/∂y

∂(ρu)/∂y

∂(ρv)/∂y

.

An algorithm similar to the one in previous section 2.2.2 can be used. The interpolation

of macroscopic variables ρ, ρu, ρv and their derivatives in space around x0 are presented

in Appendix A of [52]. After determining a and b, the term A is obtained from the

compatibility condition

∫ ∫

1

u

v

f (1)dudv =

∫ ∫

1

u

v

(−τ(au + bv + A)g)dudv = 0,

which gives

∫ ∫

1

u

v

Agdudv = −

∫ ∫

1

u

v

(au+ bv)gdudv.

Let’s define (xi, yj) as the center point of the space element Ωi,j with four cell

interfaces (xi−1/2, yj), (xi+1/2, yj), (xi, yj−1/2), and (xi, yj+1/2). Therefore, the fluxes

46

across the cell interface (xi+1/2, yj) in the x-direction can be evaluated by

Fi+1/2,j =

Fρ

Fρu

Fρv

i+1/2,j

=

∫ ∫u

1

u

v

f(xi+1/2,j ,u, t)dudv.

Similarly, Fi,j+1/2, the fluxes at the cell interface (xi, yj+1/2) in the y direction, can be

constructed. Therefore, the flow variables are updated by

ρ

ρu

ρv

n+1

Ωi,j

=

ρ

ρu

ρv

n

Ωi,j

+

∫ t

0

((Fi−1/2,j − Fi+1/2,j)

x +(Fi,j−1/2 − Fi,j+1/2)

y

)dt,

which presents an accurate Navier-Stokes flow solver in the incompressible limit.

In [52], the above gas-kinetic method and the lattice Boltzmann method (LBM) [53,

54] are compared numerically for the 2D steady cavity flow problem, which has been

extensively studied in the literature. It was found that for low Reynolds number case,

both of the methods can give equally accurate solutions, however, for high Reynolds

number case, the gas-kinetic scheme provides more accurate results than LBM. More

recently, the detailed comparison of the gas-kinetic method and the multi-relaxation-

time (MRT) lattice Boltzmann equation [55] for 2D flow around a square cylinder in

a channel has been made [124]. It has been shown that for both steady and unsteady

cases, the gas-kinetic scheme can give better predictions and is more robust than the

MRT method in high Reynolds number regime, and both of them provide similarly

accurate results in low Reynolds number regime although the MRT exhibits slightly

better computational efficiency for very low Reynolds number flows.

47

Chapter 3

A Runge-Kutta discontinuous

Galerkin method for viscous flow

equations

This chapter presents a Runge-Kutta discontinuous Galerkin (RKDG) method for

viscous flow computation. The construction of the RKDG method is based on a gas-

kinetic formulation, which not only couples the convective and dissipative terms to-

gether, but also includes both discontinuous and continuous representation in the flux

evaluation at a cell interface through a simple hybrid gas distribution function. Due

to the intrinsic connection between the gas-kinetic BGK model and the Navier-Stokes

equations, the Navier-Stokes flux is automatically obtained by the present method.

Numerical examples for both one-dimensional (1D) and two-dimensional (2D) com-

pressible viscous flows are presented to demonstrate the accuracy and shock capturing

capability of the current RKDG method.

48

3.1 Introduction

In the past decades, both the finite volume (FV) and the discontinuous Galerkin

(DG) finite element methods have been successfully developed for the compressible

flow simulations. Most FV schemes use piecewise constant representation for the flow

variables and employ the reconstruction techniques to obtain high accuracy. A higher-

order scheme usually has a larger stencil than a lower-order scheme, which makes

it difficult to be applied on unstructured mesh or complicated geometry. For the

DG method high-order accuracy is obtained by means of high-order approximation

within each element, where more information is stored for each element during the

computation. The compactness of the DG method allows it to deal with unstructured

mesh or complicated geometry easily. Now the DG method has served as a high-order

method for a broad class of engineering problems [60].

For viscous flow problems, many successful DG methods have been proposed in the

literature, such as those by Bassi and Rebay [56], Cockburn and Shu [65], Baumann and

Oden [68], and many others. In [59], a large class of discontinuous Galerkin methods

for second-order elliptic problems have been analyzed in a unified framework. More

recently, van Leer and Nomura [76] proposed a recovery-based DG method for diffusion

equation using the recovery principle. This method has deep physical insight in the

construction of a DG method for convection-diffusion problem.

The RKDG method for non-linear convection-dominated problems was first pro-

posed and studied by Cockburn, Shu, and their collaborators in a series of papers

[61]-[65], see [66] for a review of the method. The excellent results obtained by the

high-order accurate RKDG method demonstrated itself as a powerful tool in the com-

putational fluid dynamics. Recently, the gas-kinetic RKDG method proposed by Tang

and Warnecke [69] has been shown to be very accurate and efficient for inviscid flow

simulations.

49

In this chapter, a RKDG method for the viscous flow problems based on a gas-

kinetic formulation will be presented. Instead of treating the convection and dissipa-

tion effects separately, we use the gas-kinetic distribution function with both inviscid

and viscous terms in the construction of the numerical flux at the cell interface. Due

to the intrinsic connection between the gas-kinetic BGK model and the Navier-Stokes

equations, the Navier-Stokes flux is automatically obtained by the RKDG method.

The numerical dissipation introduced from the discontinuity at the cell interface is fa-

vored by the inviscid flow calculation, especially for the capturing of numerical shock

fronts. For viscous flow problems, it should be avoided because the artificial viscosity

from the discontinuity deteriorates the physical one [107]. A simplified gas-kinetic re-

laxation model, which plays the role of recovering the continuity of the flow variables

from the initial discontinuous representation, will be used in the present method. In

[81], the DG-BGK method has been developed which gives accurate solutions in both

high and low Reynolds number flow simulations. The spirit of the present RKDG

method in the construction of the viscous numerical flux is similar to that of the DG-

BGK method but with some significant modification and simplification. The RKDG

method uses Runge-Kutta or TVD-RK [73] method for the temporal discretization

and the DG-BGK method integrates the viscous flow equations in time directly. In

the DG-BGK method, only one-dimensional scheme was presented. However, in this

work the multidimensional RKDG method will be constructed as well. More impor-

tantly, the RKDG method for the viscous flows is more accurate than the DG-BGK

method, which is demonstrated by the numerical tests.

The outline of this chapter is as follows. In section 3.2 we describe the RKDG

method for the viscous flow equations. The one-dimensional formulation is given

in detail in the first two subsections, then the extension to multidimensional cases

is described. The limiting procedure and boundary conditions are also presented in

section 3.2. The performance of the method is illustrated in section 3.3 by six numerical

examples including both 1D and 2D problems. section 3.4 is the conclusion.

50

3.2 Runge-Kutta discontinuous Galerkin method

For the compressible flow simulations, a finite volume gas-kinetic BGK scheme has

been developed and applied to many physical and engineering problems [107]. Similar

to many other finite volume methods, the gas-kinetic scheme is mainly about the

flux evaluation at the cell interface. The distinguishable feature of the gas-kinetic

BGK scheme is that a Navier-Stokes flux is given directly from the MUSCL-type

reconstructed initial data [75], where both slopes at each cell interface participate the

gas evaluation. In the RKDG method [66], a high-order approximate solution inside

a cell is updated automatically and limited carefully to enforce the stability and to

suppress the numerical oscillations. In this section, we will present the RKDG method

for the Navier-Stokes equations by incorporating the gas-kinetic formulation.

3.2.1 DG Spatial Discretization in 1D Case

For a 1D flow, the BGK model in the x-direction is [87]:

ft + ufx =g − f

τ, (3.1)

where u is the particle velocity, f is the gas distribution function, and g is the equi-

librium state approached by f . The particle collision time τ is related to the viscosity

and heat conduction coefficients. The equilibrium state is a Maxwellian distribution,

g = ρ(λ

π)

K+1

2 e−λ[(u−U)2+ξ2], (3.2)

where ρ is the density, U is the macroscopic velocity, and λ is related to the gas

temperature T by λ = m/2kT , where m is the molecular mass, and k is the Boltzmann

constant. The total number of degree of freedom K in ξ is equal to (5−3γ)/(γ−1)+2.

In the above equilibrium state g, ξ2 is equal to ξ2 = ξ21 + ξ2

2 + · · · + ξ2K . The relation

between the macroscopic variables and the microscopic distribution functions is

W = (ρ, ρU, ρE)T =

∫ψfdΞ =

∫ψgdΞ, (3.3)

51

where ψ is the vector of moments

ψ = (1, u,1

2(u2 + ξ2))T , (3.4)

and dΞ = dudξ is the volume element in the phase space with dξ = dξ1dξ2 · · · dξK .

The compatibility condition can be obtained from Eq.(3.3), i.e.,

∫ψg − f

τdΞ = 0, (3.5)

where τ is assumed to be independent of individual particle velocity. Based on the

above BGK model, the corresponding Navier-Stokes equations can be derived. The

advantage of using the BGK equation instead of the Navier-Stokes equations is that

it is a first-order differential equation with a relaxation term.

By taking the moments ofψ to Eq.(3.1), due to the compatibility condition Eq.(3.5),

we have

∂

∂t

∫ψfdΞ +

∂

∂x

∫uψfdΞ = 0, (3.6)

or

Wt + Gx = 0, (3.7)

where G =∫uψfdΞ is the flux for the corresponding conservative variables W =

∫ψfdΞ. To the first order of τ , the Chapman-Enskog expansion shows that Eq.(3.7)

corresponds to the one-dimensional Navier-Stokes equations. Note that f in Eq.(3.6)

will include both equilibrium and non-equilibrium parts in a Chapman-Enskog expan-

sion. Therefore, the flux G will contain both inviscid and viscous terms accordingly.

For a given partition in 1D space, we denote the cells by Ii = [xi− 1

2

, xi+ 1

2

] and choose

the local Legendre polynomials φmi (x) as the basis functions, then the approximate

solution Wh can be expressed as,

Wh(x, t) =k∑

m=0

Wmi (t)φm

i (x), for x ∈ Ii. (3.8)

The initial value of Wmi (0) can be obtained by

Wmi (0) =

2m+ 1

∆xi

∫

Ii

W0(x)φmi (x)dx, (3.9)

52

for m = 0, · · · , k, where ∆xi = xi+1/2 − xi−1/2 and W0(x) is the initial condition. In

order to determine the time-dependent approximate solution, as given in [62], we can

enforce Eq.(3.7) cell by cell by means of a Galerkin method. More specifically, for each

cell Ii, we get

d

dtWm

i (t) +2m+ 1

∆xi(Gi+ 1

2

− (−1)mGi− 1

2

) =2m+ 1

∆xi

∫

Ii

G(x, t)d

dx(φm

i (x))dx, (3.10)

for m = 0, · · · , k, where Gi+ 1

2

is the flux at xi+ 1

2

, i.e., Gi+ 1

2

=∫uψf(xi+ 1

2

, t, u, ξ)dΞ,

and G(x, t) is the flux inside each cell. Eq.(3.10) is the system of ODEs for the degrees

of freedom Wmi (t) which can be solved by the Runge-Kutta (RK) or the TVD-RK time

stepping methods [73]. Given the continuous flow states inside the cells, we could use

G =∫uψfdΞ to calculate the integral on the right hand side of Eq.(3.10) by the

Gaussian rule consistent with the accuracy requirement for Navier-Stokes solutions.

In order to save the computational cost, we have also used the flux of the macroscopic

Navier-Stokes equations corresponding to Eq.(3.7) directly. Both of them work equally

well in our numerical tests. In the following, we are going to present the flux evaluation

at the cell interface xi+ 1

2

based on the gas-kinetic formulation for the viscous flow

equations.

3.2.2 Gas-Kinetic Flux Evaluation at a Cell Interface

In the finite volume BGK scheme [107], the flux at the cell interface is evaluated based

on the integral solution f of the BGK model (3.1),

f(xi+1/2, t, u, ξ) =1

τ

∫ t

0

g(x′

, t′

, u, ξ)e−(t−t′)/τdt

′

+ e−t/τf0(xi+1/2 − ut), (3.11)

where x′= xi+1/2 − u(t− t

′) is the trajectory of a particle motion and f0 is the initial

gas distribution function at the beginning of each time step (t = 0). The integration

part on the right hand side of Eq.(3.11) is the gain term due to the particle collision.

In order to figure out the gas distribution function at the cell interface xi+1/2, two

unknowns g and f0 in the above equation must be specified, see [107] for some details.

53

The time-dependent viscous flux given by the BGK scheme is accurate up to the

order of O(τ∆t2) in smooth regions [70]. In order to construct a simple formula of the

numerical flux for the RKDG method, we consider the hybridization of the loss and

gain terms in the gas distribution function in the present work. Similar to other hybrid

schemes, the BGK scheme presented in Eq.(3.11) can be further simplified. As shown

in [77], for the Navier-Stokes solutions the distribution function at the cell interface

can be constructed as

f = [1 − L(.)]f0 + L(.)fc, (3.12)

where f0 is the initial distribution function in Eq.(3.11). This is also the so-called

kinetic flux-vector splitting Navier-Stokes (KFVS-NS) distribution function proposed

by Chou and Baganoff [58]. In Eq.(3.12) fc is the distribution function due to the

collision effect, and L(.) is the relaxation parameter to determine the speed that a sys-

tem evolves into an equilibrium state and should be a function of local flow variables,

such as the flow jump around the cell interface. Inherently, the free transport mecha-

nism in f0 uses the time step ∆t as the particle collision time, the resulting numerical

viscosity coefficient is µf0≃ p∆t [79], where p is the pressure. The collision term fc

has the effect of recovering the continuous flow distribution from a discontinuous ini-

tial approximate solution, hence it will be helpful to reduce the numerical dissipation

introduced in a discontinuity. The principle of the hybridization is as follows. The

relaxation parameter L(.) should be determined in such a way that the contribution of

f0 becomes dominant in the non-equilibrium flow regions to provide enough numerical

dissipation while the physical scale cannot be resolved by the cell size. The term fc

contributes more in smooth regions to recover the physical dissipative effect.

There are many ways to construct the relaxation parameter L(.), see examples in

[77, 78]. In this work, one choice of L(.) is given. As we know, for shock wave, the

distribution function will stay in a non-equilibrium state along with the pressure jumps

across the shock, hence the parameter L(.) can be designed as a function of the local

54

pressure jumps around the cell interface, which is

L(.) = exp[−C| pl

i+1/2 − pri+1/2 |

pli+1/2 + pr

i+1/2

], (3.13)

where pl,ri+1/2 are the left and right values of pressure p at the cell interface xi+1/2.

Theoretically, C should depend on the physical solution, the numerical jumps at the

cell interface, and the cell size. Currently, it is still hard to give a general formulation

for it. Therefore, here C is a problem-dependent positive constant which ranges from

103 to 105 in our computation. In the regions with large pressure gradients, for example

in the numerical shock layer, when the shock is under-resolved, a small value of L(.)

should be used to add more numerical dissipation through the amplification of f0. In

this case, the numerical shock layer constructed will be much wider than the physical

one. On the other hand, if the shock structure is well-resolved, the distribution function

fc should be the dominant part. A similar switch function is also employed in the high-

resolution gas-kinetic schemes by Ohwada and Fukata [71].

In the FV BGK method [107], the initial macroscopic flow variables around a cell

interface are reconstructed by the MUSCL-type interpolation. But, for the RKDG

method, they are updated inside each cell. In the following, the location of the cell

interface xi+1/2 = 0 will be used for simplicity. With the initial macroscopic flow states

on both sides of a cell interface, to the Navier-Stokes order the initial gas distribution

function f0 is constructed as

f0 =

gl[1 + alx− τ(alu+ Al)], x ≤ 0,

gr[1 + arx− τ(aru+ Ar)], x ≥ 0,(3.14)

where gl and gr are the equilibrium states at the left and right hand sides of the cell

interface. The Maxwellian distribution functions gl and gr have unique correspondence

with the macroscopic variables there, i.e.,

gl = gl(Wli+1/2) and gr = gr(Wr

i+1/2). (3.15)

The additional terms −τgl(alu+Al) and −τgr(aru+Ar) are the non-equilibrium parts

obtained from the Chapman-Enskog expansion of the BGK model, which account

55

for the dissipative effects. Here al and ar in Eq.(3.14) are coming from the spatial

derivatives of the Maxwellian distribution functions, i.e.,

al = al1 + al

2u+1

2al

3(u2 + ξ2) and ar = ar

1 + ar2u+

1

2ar

3(u2 + ξ2), (3.16)

which can be uniquely evaluated from the spatial derivatives of the conservative vari-

ables at the left and right hand sides of the cell interface. Since there is no contribution

in the mass, momentum and energy from the non-equilibrium parts, the parameters

in Al = Al1 + Al

2u + 12Al

3(u2 + ξ2) and Ar = Ar

1 + Ar2u + 1

2Ar

3(u2 + ξ2) are uniquely

determined by the compatibility condition∫ψgl(alu+ Al)dΞ = 0 and

∫ψgr(aru+ Ar)dΞ = 0. (3.17)

Even though the non-equilibrium parts have no contribution to the conservative flow

variables (moments of ψ), they do have contribution to the flux (moments of uψ).

Note that for the initial distribution function f0, both the state and the derivative

have discontinuities at the cell interface.

The distribution function fc due to the collision effect can be constructed as

fc = g0[1 − τ(ualH[u] + uar(1 − H[u]) + A)], (3.18)

where H[u] is the Heaviside function, and g0 is a local Maxwellian distribution function

located at x = 0. The dependence of al, ar and A on the particle velocities are also

coming from a Taylor expansion of a Maxwellian distribution, which have the forms

al = al1+a

l2u+ 1

2al

3(u2+ξ2), ar = ar

1+ar2u+ 1

2ar

3(u2+ξ2), and A = A1+A2u+ 1

2A3(u

2+ξ2).

The determination of the parameters in Eq.(3.18) is as follows. The equilibrium state

g0 at the cell interface is constructed from the conservation constraint,

W0 =

∫ψg0dΞ =

∫

u>0

ψgldΞ +

∫

u<0

ψgrdΞ. (3.19)

For the distribution function fc, the terms related to the spatial and temporal gradients

have to be constructed as well. In the smooth flow cases, we can use the following

relation to calculate a continuous derivative a = al = ar of g0:∫ψg0adΞ =

∫

u>0

ψglaldΞ +

∫

u<0

ψgrardΞ. (3.20)

56

This relation has been tested in the first two numerical examples. Without using

limiter, the high-order accuracy of Eq.(3.20) has been clearly demonstrated. However,

for the cases where the limiting procedure is necessary for eliminating the numerical

oscillations the following approximation is used,

2(W0 − Wi)/∆xi =

∫ψg0a

ldΞ and 2(Wi+1 −W0)/∆xi+1 =

∫ψg0a

rdΞ, (3.21)

where two derivatives al and ar of g0 are evaluated, and Wi and Wi+1 are cell average

values. Numerically, for the solution with discontinuities, Eq.(3.21) performs better

than Eq.(3.20). Although Eq.(3.21) seems to be a second-order spacial approximation,

it still works well in the third-order P 2 scheme in our computation for the strongly

discontinuous tests. One of the reason for this is that W0 is constructed through the

higher-order approximation of gl and gr. The term A in Eq.(3.18) can be uniquely

determined from the compatibility condition at the cell interface, i.e.,

∫ψg0(ua+ A)dΞ = 0 (3.22)

for Eq.(3.20), and

∫ψg0(ua

lH[u] + uar(1 − H[u]) + A)dΞ = 0 (3.23)

for Eq.(3.21). Note that the distribution function fc plays the role of recovering the

continuity from the initial discontinuous representation. This procedure is critically

important to capture the viscous solution.

For the Navier-Stokes solutions, the viscosity and heat conduction coefficients are

related to the particle collision time τ . With the given dynamical viscosity coefficient

µ, the collision time can be calculated by τ = µ/p, where p is the pressure. Up to

this point, we have determined all parameters in the distribution functions f0 and

fc. After substituting Eqs.(3.13), (3.14) and (3.18) into Eq.(3.12), we can get the gas

distribution function f at the cell interface, then the numerical flux can be obtained by

taking the moments uψ to it. In order to get the heat conduction correct, the energy

flux in Eq.(3.12) can be modified according to the realistic Prandtl number [107].

57

3.2.3 Extension to Multidimensional Cases

The present RKDG method can be easily extended to multidimensional cases. Here we

describe the construction of the P 2 scheme for two-dimensional case with rectangular

elements. The 2D BGK model can be written as

ft + ufx + vfy =g − f

τ, (3.24)

where the equilibrium distribution g is

g = ρ(λ

π)

K+2

2 e−λ[(u−U)2+(v−V )2+ξ2]. (3.25)

By taking the moments of ψ = (1, u, v, 12(u2 + v2 + ξ2))T to Eq.(3.24), we can get the

governing equation

∂

∂t

∫ψfdΞ +

∂

∂x

∫uψfdΞ +

∂

∂y

∫vψfdΞ = 0, (3.26)

or

Wt + Gx + Hy = 0, (3.27)

which correspond to the 2D Navier-Stokes equations when f is approximated to the

first order of τ .

For a rectangular cell Ii,j = [xi−1/2, xi+1/2] × [yj−1/2, yj+1/2] in the computational

domain, we take the same local basis functions φmi,j as in [64],

φ0i,j(x) = 1, φ1

i,j(x) =2(x− xi)

∆xi

, φ2i,j(x) =

2(y − yj)

∆yj

,

φ3i,j(x) = φ1

i,j(x)φ2i,j(x), φ4

i,j(x) = (φ1i,j(x))2 − 1

3, φ5

i,j(x) = (φ2i,j(x))2 − 1

3,(3.28)

where x ≡ (x, y) ∈ Ii,j, ∆xi = xi+1/2 − xi−1/2 and ∆yj = yj+1/2 − yj−1/2. Then the

approximate solution Wh inside the element Ii,j can be expressed as

Wh(x, t) =5∑

m=0

Wmi,j(t)φ

mi,j(x), for x ∈ Ii,j. (3.29)

In order to determine the solution, we need to solve the following weak formulation of

Eq.(3.27):

58

ddtWm

i,j(t) + 1amS

(∫ yj+ 1

2y

j− 12

(G(xi+ 1

2

, y, t)φmi,j(xi+ 1

2

, y) − G(xi− 1

2

, y, t)φmi,j(xi− 1

2

, y))dy

+∫ x

i+ 12

xi− 1

2

(H(x, yj+ 1

2

, t)φmi,j(x, yj+ 1

2

) − H(x, yj− 1

2

, t)φmi,j(x, yj− 1

2

))dx

)

− 1amS

∫Ii,j

(G(x, y, t) ∂

∂xφm

i,j(x, y) + H(x, y, t) ∂∂yφm

i,j(x, y))dxdy = 0,

(3.30)

for m = 0, 1, · · · , 5, where am are the normalization constants as shown in [64] and

S = ∆xi∆yj. We use the 3-point Gaussian rule for the edge integral and a tensor

product of that with nine quadrature points for the interior integral in Eq.(3.30).

Again the flux G(x, y, t) and H(x, y, t) inside the element will be calculated from

the corresponding Navier-Stokes equations for efficiency consideration. The numerical

flux G(xi+ 1

2

, y, t) and H(x, yj+ 1

2

, t) at the cell interfaces will be calculated from the

gas-kinetic formulation which is presented in the following.

There are two approaches that can be used to construct the numerical flux: one

is the directional splitting method [107] and the other is the fully multidimensional

method [80]. In this work we employ the less costly splitting method to construct the

flux at the cell interfaces for efficiency and simplicity consideration. For the directional

splitting method, only the partial derivatives of the conservative variables in the normal

direction of the cell interface will be used in the flux evaluation, which is similar to

1D case. As an illustration, we give the detailed formulae for calculating the flux

in the x-direction G(xi+ 1

2

, y∗, t) at a Gaussian quadrature point (xi+ 1

2

, y∗), and the

similar calculations can be done in the y-direction. We will still use Eq.(3.1) with

the equilibrium state given by Eq.(3.25) and the simple hybrid distribution function

given by Eq.(3.12) to construct the flux G(xi+ 1

2

, y∗, t). The hybrid parameter L(.) in

Eq.(3.12) is determined by Eq.(3.13) again. With the assumption of xi+ 1

2

= 0, the

initial gas distribution function f0 in Eq.(3.12) is expressed as

f0 =

gl[1 + alx− τ(alu+ Al)], x ≤ 0, y = y∗,

gr[1 + arx− τ(aru+ Ar)], x ≥ 0, y = y∗,(3.31)

where gl and gr are the equilibrium states at the left and right hand sides of the

59

quadrature point, al and ar are expressed as

al = al1+al

2u+al3v+

1

2al

4(u2+v2+ξ2), ar = ar

1+ar2u+ar

3v+1

2ar

4(u2+v2+ξ2), (3.32)

which can be uniquely determined from the partial derivatives of the conservative vari-

ables with respect to x there. The terms Al = Al1 +Al

2u+Al3v+ 1

2Al

4(u2 +v2 + ξ2) and

Ar = Ar1 + Ar

2u + Ar3v + 1

2Ar

4(u2 + v2 + ξ2) can be calculated from the compatibility

condition Eq.(3.17). The collisional distribution function fc in Eq.(3.12) is written

as Eq.(3.18), where g0 is constructed by Eq.(3.19). Similar to 1D case, for the flow

problems with smooth solutions, we use the relation of Eq.(3.20) to calculate the con-

tinuous derivative a = al = ar of g0. For the viscous flows with strong discontinuities,

the limiting procedure becomes necessary, and the following approximation

2(W0 − Wi,j)/∆xi =

∫ψg0a

ldΞ and 2(Wi+1,j − W0)/∆xi+1 =

∫ψg0a

rdΞ

(3.33)

are used to obtain al and ar in Eq.(3.18), where Wi,j and Wi+1,j are cell average

values. Then we can employ the compatibility condition Eq.(3.22) or Eq.(3.23) to get

the term A in Eq.(3.18). Finally, the numerical flux G(xi+ 1

2

, y∗, t) can be obtained by

taking the moments of uψ to the distribution function f given by Eq.(3.12).

3.2.4 Limiting Procedure and Boundary Conditions

For the compressible flow simulations by the RKDG method, the direct update of the

numerical solution will generate numerical oscillations across strong shock waves. In

order to eliminate these oscillations, the non-linear limiter, usually used in the FV

method, has to be used in the RKDG method as well. In this work, in 1D case, we use

the Hermite WENO limiter [72] proposed by Qiu and Shu recently. In 2D case, for the

rectangular elements a similar limiting procedure to that in [64] is employed. Consider

a scalar equation wt+Gx+Hy = 0 in 2D space, for P 1 scheme the approximate solution

60

wh inside the element Ii,j is known as

wh(x, t) =

2∑

m=0

wmi,j(t)φ

mi,j(x), for x ∈ Ii,j . (3.34)

We use the following generalized slope limiter

w1i,j = M(w1

i,j, w0i+1,j − w0

i,j, w0i,j − w0

i−1,j), (3.35)

where the function M is the TVB corrected minmod function [74] defined by

M(a1, a2, a3) =

a1, if |a1| ≤ C∆x2

i ,

M(a1, a2, a3), otherwise,(3.36)

with the minmod function M defined by

M(a1, a2, a3) =

s min(|a1|, |a2|, |a3|), if s = sign(a1) = sign(a2) = sign(a3),

0, otherwise.

(3.37)

Similarly, w2i,j is limited by

w2i,j = M(w2

i,j, w0i,j+1 − w0

i,j, w0i,j − w0

i,j−1), (3.38)

with a change of ∆xi to ∆yj in Eq.(3.36). For P 2 scheme, we do not do any limiting if

w1i,j = w1

i,j and w2i,j = w2

i,j, otherwise we chop off the higher-order parts (w3i,j, w

4i,j, w

5i,j)

of the approximate solution. For both 1D and 2D cases, the componentwise limiting

operator is used after each Runge-Kutta or TVD-RK [73] inner stage.

Now we describe the treatment of boundary conditions. For the adiabatic wall, the

no-slip boundary condition for the velocity field is imposed by reversing the veloci-

ties in the ghost cell from the sate in the interior region, and the mass and energy

densities are put to be symmetric around the wall. For the isothermal wall, where

the boundary temperature is fixed, we use the condition of no net mass flux trans-

port across the boundary [107] to construct the flow states in the ghost cell, where

the spacial derivatives are calculated from the data around the wall by approximation

consistent with the accuracy requirement. For example, a parabolic reconstruction for

61

the flow variables around the wall has to be used for the P 2 scheme, whereas the lin-

ear approximation is implemented in the P 1 case. At the inflow/outflow boundaries,

the flow states at the external boundary are computed from the available data and

Riemann invariants, and the spacial derivatives there are also obtained by appropriate

approximation with the given boundary condition.

3.3 Numerical experiments

The present RKDG method will be tested in both 1D and 2D problems. We use the

two-stage TVD-RK time stepping method [73] for P 1 case and the three-stage one for

both P 2 and P 3 cases, the CFL number is 0.2 for P 1 case and 0.15 for P 2 case.

(1). Accuracy test

The first test is to solve the Navier-Stokes equations with the following initial data,

ρ(x, 0) = 1 + 0.2 sin(πx), U(x, 0) = 1, p(x, 0) = 1. (3.39)

The dynamical viscosity coefficient is taken by a value µ = 0.0005. The Prandtl

number is Pr = 2/3 and the specific heat ratio is γ = 5/3. The computational

domain is x ∈ [0, 2] and the periodic boundary condition is used. We compute the

viscous solution up to time t = 2 with a small time step to guarantee that the spatial

discretization error dominates. No limiter is used in this case. Since there is no exact

solution for this problem, we evaluate the numerical error between the solutions by two

successively refined meshes and use the error to estimate the numerical convergence

rates. The results are shown in Tables 3.1-3.3. From these results we can clearly notice

that a (k + 1)th-order convergence rate can be obtained for P k(k = 1, 2, 3) schemes

for smooth solutions.

(2). Couette flow

The Couette flow with a temperature gradient provides a good test for the RKDG

62

method to describe the viscous heat-conducting flow. With the bottom wall fixed, the

top boundary is moving at a speed U in the horizontal direction. The temperatures

at the bottom and top are fixed with values T0 and T1. Under the assumption of

constant viscosity and heat conduction coefficients and in the incompressible limit, a

steady state analytical temperature distribution can be obtained,

T − T0

T1 − T0=

y

H+PrEc

2

y

H

(1 − y

H

), (3.40)

where H is the height of the channel, Pr is the Prandtl number, Ec is the Eckert

number Ec = U2/Cp(T1 − T0), and Cp is the specific heat at constant pressure.

We set up the simulation as a 1D problem in the y-direction. There are 5 cells

used in this direction from 0 to 5 with H = 5 and ∆y = 1. The moving velocity at

the top boundary in the x-direction is U = 0.1. The dynamical viscosity coefficient

is taken as µ = 0.1. The initial density and Mach number of the gas inside the

channel are 1.0 and 0.1, respectively. In this case the fluid in the channel is almost

incompressible. The isothermal no-slip boundary conditions are implemented at both

ends. We have tested the RKDG method with a wide range of parameters. Here some

of them are presented: (i) specific heat ratio γ = 5/3, 7/5, (ii) different Prandtl number

Pr = 0.72, 1.0, (iii) different Eckert number Ec = 10, 50. The results without limiter

are shown in Figs. 3.1 and 3.2. From these figures, we see that the numerical results

recover the analytical solutions very well with the variations of all these parameters,

and the Prandtl number fix does modify the heat conduction term correctly. It is also

clearly shown that the higher-order P 2 scheme gives more accurate solutions than the

lower-order P 1 scheme with the same mesh size. If we further refine the mesh, the

difference between the numerical solution from P 1 and P 2 cases is indistinguishable

and both accurately recover the analytical solution.

(3). Navier-Stokes shock structure

The third test is the Navier-Stokes shock structure calculation. Although it is well

known that in the high Mach number case the Navier-Stokes solutions do not give the

63

physically realistic shock wave profile, it is still a useful case in establishing and testing

a valid solver for the Navier-Stokes equations. Even though the shock structure is well

resolved in this case, due to the highly non-equilibrium state inside the shock layer,

its accurate calculation bears large requirement on the accuracy and robustness of the

numerical method. The profile of a normal shock structure, and the correct capturing

of the viscous stress and heat conduction inside the shock layer represent a good test

for the viscous flow solver.

The shock structure calculated is for a monotonic gas with γ = 5/3 and a dynam-

ical viscosity coefficient µ ∼ T 0.8, where T is the temperature. The upstream Mach

number M = 1.5 and the Prandtl number Pr = 2/3 are used in this test. The dynam-

ical viscosity coefficient at the upstream keeps a constant value µ−∞ = 0.0005. The

reference solution is obtained by directly integrating the steady state Navier-Stokes

equations, and the Matlab programs are provided in Appendix C of [107]. Because

the normal stress and the heat flux seem to show the greatest numerical sensitivity,

these are selected to display. Therefore, the profiles of the temperature T and the

fluid velocity U across the shock layer, as well as the normal stress and the heat flux

defined by

τnn =4

3µUx

2p, qx = −5

4

µ

Pr

Tx

pc, (3.41)

versus fluid velocity U/U−∞, are calculated. In the above equation p is the pressure

and c is the speed of sound.

The mesh size used is ∆x = 1/800 for both P 1 and P 2 cases. The results calculated

by the second-order P 1 case are presented in Fig. 3.3 and those by the third-order P 2

case are shown in Fig. 3.4. From these results, we can see that the shock structure is

calculated accurately with a reasonable number of grid points inside the shock layer.

Moreover, the third-order scheme gives more accurate results than the second-order

scheme, especially in the normal stress and heat flux solutions.

(4). Shock tube problem

64

In the fourth example, in order to further test the RKDG method in capturing the

Navier-Stokes solutions in the unsteady case, we calculate the well-known Sod’s test

directly by solving the Navier-Stokes equations with γ = 1.4 and Pr = 2/3. The cell

size used here is ∆x = 1/200. Fig. 3.5 gives the results with a kinematic viscosity coef-

ficient ν = 0.0005/ρ√λ, where λ is related to the temperature in the local equilibrium

distribution function g0. The solid lines there are the reference solutions calculated by

the FV BGK scheme [107] with a much refined mesh size ∆x = 1/1200. In this case,

due to the large viscosity coefficient both the shock structure and the contact wave are

well resolved by the cell size used and both of them are captured accurately. Again the

higher-order scheme gives more accurate results than the lower-order scheme, which

can be clearly seen in both the velocity distributions in Fig. 3.5 and the zoom-in views

of the density distributions around the shock wave in Fig. 3.6. The subcell solution

presented in Fig. 3.6 also demonstrates this fact clearly. The results with a much

smaller viscosity coefficient ν = 0.00005/ρ√λ are also presented in Fig. 3.7. Here the

shock structure can not be resolved by the large cell size used, and the RKDG method

becomes a shock capturing scheme. The shock transition is purely constructed from

the numerical dissipation, which is much wider than the physical one determined from

the above physical viscosity.

(5). Laminar boundary layer

The next numerical example is the laminar boundary layer over a flat plate with the

length L. The Mach number is M = 0.2 and the Reynolds number based on the

upstream flow states and the length L is Re = 105. A rectangular mesh with 120× 30

cells is used and the mesh distribution is shown in Fig. 3.8. The mesh size ranges from

∆x/L = 1.0×10−3 at the leading edge to ∆x/L = 4.9×10−2 at the end of the plate in

the x-direction, and from ∆y/L = 6.6×10−4 near the wall to ∆y/L = 0.11 at the upper

boundary in the y-direction. The U velocity contours at the steady state computed by

the P 2 scheme are shown in Fig. 3.9. We have also compared the numerical results with

the theoretical ones given by the well-known Blasius solution in case of incompressible

65

flow. The U velocity distributions along three different vertical lines are shown in Figs.

3.10 and 3.11. From these figures, we can see that the numerical solutions by both

P 1 and P 2 schemes recover the theoretical solution accurately, even with as few as 4

grid points in the boundary layer. The computed skin friction coefficient along the flat

plate is shown in Fig. 3.12, where a very good agreement with the Blasius solution is

obtained. The logarithmic plot of the skin friction coefficient in Fig. 3.13 shows that

the higher-order P 2 scheme performs better than the lower-order P 1 scheme in both

the leading edge and the outflow region. Similar observation is obtained in [56].

(6). Shock boundary layer interaction

The final test deals with the interaction of an oblique shock with a laminar boundary

layer, which has been computed by Bassi and Rebay with an implicit high-order dis-

continuous Galerkin method [57]. The shock makes a 32.6o angle with the wall, which

is located at y = 0 and x ≥ 0, and hits the boundary layer on the wall at Xs = 10. The

Mach number of the shock wave is equal to 2 and the Reynolds number based on the

upstream flow condition and the characteristic length Xs is equal to 2.96 × 105. The

dynamical viscosity µ is computed according to the Sutherland’s law for the gas with

γ = 1.4 and Pr = 0.72. The computation was carried out on a rectangular domain

[−1.05 ≤ x ≤ 16.09]× [0 ≤ y ≤ 10.16]. A nonuniform mesh with 106×73 cells, similar

to the laminar boundary layer problem, is constructed. The mesh size varies from

∆x/Xs = 1.0 × 10−2 around x = 0 to ∆x/Xs = 2.6 × 10−2 at the end of the plate in

the x-direction, and from ∆y/Xs = 3.2 × 10−4 around y = 0 to ∆y/Xs = 7.6 × 10−2

at the upper boundary in the y-direction. The pressure contours computed by the P 1

and P 2 schemes are presented in Fig. 3.14. In this case the shock structure can not

be well resolved by the current mesh size and the RKDG method turns out to be a

shock capturing one in terms of the shock. As expected, the P 2 scheme gives a shaper

numerical shock transition than that from P 1 scheme due to the less numerical dissi-

pation introduced by weaker discontinuities at the cell interfaces. On the other hand,

the boundary layer can be resolved under the present mesh. The skin friction and

66

pressure distributions at the plate surface are shown in Fig. 3.15, where a fair agree-

ment with the experimental data [67] is obtained for both P 1 and P 2 schemes, and

the P 2 scheme performs slightly better than the P 1 scheme. Our numerical results are

comparable with those in [71]. The main discrepancy between the experimental and

numerical results is probably due to the assumption that the flow is laminar whereas

the real physical one could be turbulent.

3.4 Conclusions

In this chapter, a RKDG method for the viscous flow computation has been presented.

The construction of the RKDG method is based on a gas-kinetic formulation, which

combines the convective and dissipative terms together in a single gas distribution

function. Due to the intrinsic connection between the gas-kinetic BGK model and

the Navier-Stokes equations, the Navier-Stokes flux is automatically obtained by the

method. The current RKDG method has good shock capturing capacity, where the

numerical dissipation introduced from the numerical flux at the cell interface is con-

trolled adaptively by a hybrid parameter in the current approach. The RKDG method

works very well for all test cases presented. The higher-order P 2 scheme does give a

more accurate solution than that from the lower-order P 1 scheme, especially in the

well-resolved cases. In terms of the computational cost, the present RKDG method

is more expensive, especially in the multidimensional cases, than the finite volume

gas-kinetic BGK method for the Navier-Stokes equations. For the RKDG methods,

the limiting procedure plays an important role for the quality of numerical solutions

and its formulation is sophisticated in the cases with strong discontinuities. Generally

speaking, both the numerical flux and the limiting procedure are needed to bring nu-

merical dissipation into the viscous solutions. This kind of dissipation is unavoidable

because with limited cell size we can not fully resolve the physical solutions, such as the

shock structure, the leading edge of the boundary layer, or the small scale turbulent

67

flow, even though we are claiming to solve the viscous governing equations.

68

Table 3.1: The error and convergence order for P 1 case.

N L∞-error Order L1-error Order L2-error Order

10 3.05E-2 – 1.76E-2 – 1.99E-2 –

20 5.68E-3 2.42 3.36E-3 2.39 3.79E-3 2.39

40 1.03E-3 2.46 6.31E-4 2.41 7.07E-4 2.42

80 2.08E-4 2.31 1.28E-4 2.30 1.44E-4 2.30

Table 3.2: The error and convergence order for P 2 case.

N L∞-error Order L1-error Order L2-error Order

10 2.48E-3 – 1.51E-3 – 1.62E-3 –

20 2.76E-4 3.16 1.66E-4 3.18 1.86E-4 3.12

40 2.50E-5 3.47 1.54E-5 3.43 1.73E-5 3.42

80 2.54E-6 3.30 1.47E-6 3.39 1.64E-6 3.40

69

Table 3.3: The error and convergence order for P 3 case.

N L∞-error Order L1-error Order L2-error Order

10 9.05E-5 – 5.37E-5 – 5.67E-5 –

20 8.89E-6 3.35 4.23E-6 3.67 4.90E-6 3.53

40 4.90E-7 4.18 2.81E-7 3.91 3.17E-7 3.95

80 3.26E-8 3.91 1.42E-8 4.30 1.70E-8 4.22

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

y/H

(T−

T0)/

(T1−

T0)

Pr=1.0

Pr=0.72

Figure 3.1: Temperature ratio (T−T0)/(T1−T0) in the Couette flow with γ = 5/3, Ec =

50. The solid line is the analytical solution given by Eq.(3.40), the plus symbol is the

numerical solution by P 1 method and the circle symbol is by P 2 one.

70

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

y/H

(T−

T0)/

(T1−

T0)

Ec=50

Ec=10

Figure 3.2: Temperature ratio (T − T0)/(T1 − T0) in the Couette flow with γ =

7/5, P r = 0.72. The solid line is the analytical solution given by Eq.(3.40), the plus

symbol is the numerical solution by P 1 method and the circle symbol is by P 2 one.

71

−0.015 −0.01 −0.005 0 0.005 0.01

0.55

0.6

0.65

0.7

0.75

0.8

X

T

temperature

−0.015 −0.01 −0.005 0 0.005 0.01

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

X

U

velocity

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

U/U−∞

τ nn

, q x

normalstress

heat flux

Figure 3.3: Navier-Stokes shock structure calculation, P 1 case.

72

−0.015 −0.01 −0.005 0 0.005 0.01

0.55

0.6

0.65

0.7

0.75

0.8

X

T

temperature

−0.015 −0.01 −0.005 0 0.005 0.01

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

X

U

velocity

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

U/U−∞

τ nn ,

qx

normalstress

heat flux

Figure 3.4: Navier-Stokes shock structure calculation, P 2 case.

73

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

X

ρ

reference solution

RKDG−P1

RKDG−P2

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

X

U

reference solution

RKDG−P1

RKDG−P2

Figure 3.5: Shock tube test for the Navier-Stokes equations with kinematic viscosity

coefficient ν = 0.0005/ρ√λ.

74

0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

X

ρ

reference solution

RKDG−P1

RKDG−P2

0.32 0.325 0.33 0.335 0.34 0.345 0.35 0.355 0.36 0.365 0.37 0.3750.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

X

ρ

reference solution

RKDG−P1

RKDG−P2

Figure 3.6: The zoom-in views of the density distributions around the shock wave in

shock tube test with ν = 0.0005/ρ√λ. The upper one is the same as that in Fig. 3.5,

where only the values at cell centers are displayed. The lower one presents the values

at 15 equally spaced positions inside each cell, the so-called subcell solution.

75

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

X

ρ

reference solution

RKDG−P1

RKDG−P2

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8

1

X

U

reference solution

RKDG−P1

RKDG−P2

Figure 3.7: Shock tube test for the Navier-Stokes equations with kinematic viscosity

coefficient ν = 0.00005/ρ√λ.

76

−20 0 20 40 60 80

10

20

30

40

50

X

Y

Figure 3.8: Mesh distribution for laminar boundary layer problem.

X

Y

−20 0 20 40 60 80

10

20

30

40

50

Figure 3.9: Laminar boundary layer problem. 100 equally spaced contours of the fluid

velocity U/U−∞ from 0 to 1.0061 from P 2 calculation.

77

0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1

η

U/U

−∞

Blasius

x=3.28

x=26.67

x=63.59

Figure 3.10: Laminar boundary layer problem. U velocity distributions along three

vertical lines by P 1 case.

0 1 2 3 4 5 6 7 8

0

0.2

0.4

0.6

0.8

1

η

U/U

−∞

Blasius

x=3.28

x=26.67

x=63.59

Figure 3.11: Laminar boundary layer problem. U velocity distributions along three

vertical lines by P 2 case.

78

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.005

0.01

0.015

0.02

0.025

0.03

X/L

Cf

Blasius

RKDG−P1

RKDG−P2

Figure 3.12: Laminar boundary layer problem. Skin friction coefficient distribution

along the flat plate.

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−3

−2.5

−2

−1.5

−1

−0.5

Log10

(X/L)

Log 10

(Cf)

Blasius

RKDG−P1

RKDG−P2

Figure 3.13: Laminar boundary layer problem. Logarithmic plot of the skin friction

coefficient distribution along the flat plate.

79

X/Xs

Y/X

s

0 0.5 1 1.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

X/Xs

Y/X

s

0 0.5 1 1.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 3.14: Shock boundary layer interaction. 30 equally spaced contours of pressure

p/p−∞ from 0.997 to 1.411 by P 1 (upper) and P 2 (lower) cases.

80

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

−5

0

5

10

15

20

x 10−4

X/Xs

Cf

Experiment

RKDG−P1

RKDG−P2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

X/Xs

p/p −

∞

Experiment

RKDG−P1

RKDG−P2

Figure 3.15: Shock boundary layer interaction. Skin friction (upper) and pressure

(lower) distributions at the plate surface.

81

Chapter 4

Multiscale gas-kinetic simulation

for continuum and near continuum

flows

It is well known that for increasingly rarefied flow fields, the predictions from contin-

uum formulation, such as the Navier-Stokes equations, lose accuracy. The inclusion of

higher-order terms, such as Burnett or high-order moment equations, could improve

the predictive capabilities of such continuum formulations, but there has been only

limited success. Here, we present a multiscale model. On the macroscopic level, the

flow variables are updated based on the mass, momentum, and energy conservations

through the fluxes. On the other hand, the fluxes are constructed on the micro-

scopic level based on the gas-kinetic equation, which is valid in both continuum and

near continuum flow regime. Based on this model, non-equilibrium shock structure,

Poiseuille flow, nonlinear heat conduction problems, and unsteady Rayleigh problem

will be studied. In the near continuum flow regime, the current gas-kinetic simula-

tion is more efficient than the microscopic methods, such as the direction Boltzmann

solver and Direct Simulation Monte Carlo (DSMC) method. In the continuum flow

82

limit, the current formulation will go back to the gas-kinetic Navier-Stokes flow solver

automatically.

4.1 Introduction

The dynamical behavior of flows far from hydrodynamic equilibrium is an important

subject of non-equilibrium thermodynamics, with many applications in science and

engineering. The classification of various flow regimes is based on the dimensionless

parameter, i.e., the Knudsen number, which is a measure of the degree of rarefaction

of the medium. The Knudsen number Kn is defined as the ratio of the mean free

path to a characteristic length scale of the system. In the continuum flow regime

where Kn < 0.001, the Navier-Stokes equations with linear relations between stress

and strain and the Fourier’s law for heat conduction are adequate to model the fluid

behavior. For flows in the continuum-transition regime (0.01 < Kn < 1), the Navier-

Stokes equations are known to be inadequate. This regime is important for many

practical engineering problems, such as the simulation of microscale flows [92] and

hypersonic flow around space vehicles in low earth orbit [93]. Hence, there is a strong

desire and requirement for accurate models which give reliable solutions with lower

computational costs.

Currently, the DSMC method is the most successful technique in the numerical

prediction of low density flows [89]. However, in the continuum-transition regime,

especially for the low speed flow, the DSMC suffers from statistical noise in the bulk

velocity because of the random molecular motion. When the bulk velocity is much

slower than the thermal velocity, many independent samples are needed to eliminate

the statistical scattering, as for the micro-electro-mechanical system (MEMS) simula-

tion. In fact, for the air at room temperature, the standard deviation in the molecular

speed is about 300m/s, which would require approximately 9 million independent sam-

ples in DSMC to reduce the scatter in the bulk velocity to 0.1m/s. For MEMS gas

83

flows that operate in the mm/s range, the number of required samples can grow into

trillions. So, the DSMC method is impractical in these cases. Also, the requirement

of cell size being less than the particle mean free path in the DSMC method prevents

it from being widely applied in the continuum flow regime, especially for flows with

high Reynolds numbers.

Alternatively, many macroscopic continuum models have been intensively devel-

oped in the literature. These include the Navier-Stokes and the Burnett equations

from the Chapman-Enskog expansion, Grad’s 13 moment equations, the regularized

13 equations, and many others [92]. In order to assess these continuum models, a few

tests have been carried out [116]. It concluded that none of the models is commonly

acceptable for rarefied flow simulations. Overall, the small length scales and slow bulk

gas velocity combine to make continuum solution inaccurate, and particle solution

time consuming. Besides DSMC and continuum models, many alternative approaches

have also been proposed in recent years, such as the information preservation (IP)

method [90, 98] and the Lattice Boltzmann Method (LBM) [53, 54, 106]. However,

IP and LBM are mostly used for the isothermal flows. Recently, by considering only

the deviation from equilibrium, Baker and Hadjiconstantinou developed a variance

reduction method for Monte Carlo solution of the Boltzmann equation [82], where

significant computational savings can be obtained.

The goal of this study is to present a multiscale simulation, which uses both the

macroscopic (mass, momentum, and energy densities) and microscopic (gas distribu-

tion function and particle collision) descriptions. On the microscopic level, a flux is

constructed based on the close solution of gas-kinetic equation for the update of the

flow variables on the macroscopic level.

84

4.2 Gas-kinetic equation and multiscale numerical

formulation

The Boltzmann equation expresses the behavior of a many-particle kinetic system in

terms of the evolution equation for a single particle gas distribution function. The

simplification of the Boltzmann equation given by the BGK model is formulated as

[87],

∂f

∂t+ u · ∂f

∂x=g − f

τ, (4.1)

where f is the number density of molecules at position x and particle velocity u =

(u, v, w) at time t. The left hand side of the above equation represents the free stream-

ing of molecules in space, and the right side denotes the collision term. If the distri-

bution function f is known, macroscopic variables, such as mass, momentum, energy

and stress, can be obtained by integration over the moments of molecular velocity. In

the BGK model, the collision operator involves simple relaxation to a state of local

equilibrium, the distribution function given by g with a characteristic time scale τ .

For a monatomic gas, the equilibrium state is given by a Maxwellian,

g = ρ

(λ

π

) 3

2

e−λ(u−U)2 ,

where ρ is the density, U is the macroscopic fluid velocity, and λ is equal to m/2kT .

Here, m is the molecular mass, k is the Boltzmann constant, and T is the tempera-

ture. The relation between mass ρ, momentum ρU, and energy densities ρE with the

distribution function f becomes

ρ

ρU

ρE

=

∫ψfdu, (4.2)

where ψ is the vector of moments

ψ = (1,u,1

2u2)T ,

85

and the volume element in the phase space with du = dudvdw and u2 = u2 + v2 +w2.

Since mass, momentum and energy are conserved during particle collisions, f and g

satisfy the conservation constraint,∫ψ(g − f)du = 0, (4.3)

at any point in space and time. The BGK model was originally proposed to describe

the essential physics of molecular interactions with τ chosen as the molecular collision

time. Although the BGK model appears to describe only weak departures from local

equilibria, it has long been recognized that such an approximation works well beyond

its theoretical limits as long as the relaxation time is known for the physical process.

Based on the above BGK model, the Navier-Stokes equations can be derived with the

Chapman-Enskog expansion truncated to the 1st-order,

f = g + Knf1

= g − τ(∂g/∂t + u · ∂g/∂x). (4.4)

For the Burnett and super-Burnett solutions, the above expansion can be naturally

extended [96], such as f = g + Knf1 + Kn2f2 + Kn3f3 + ....

Even though, great progress has been made in the past two decades on the con-

struction and analysis of the extended hydrodynamics system, the Burnett, the Super-

Burnett equations, or regularization of the moment equations, the most successful

method to accurately capture rarefied gas effect is still the DSMC method. The

DSMC method is basically equivalent to solving the Boltzmann equation by simulat-

ing the free transport and collision steps in a particle transport process. The DSMC

solution has the same solution as the full Boltzmann equation. However, based on

the Chapman-Enskog expansion, only a limited number of truncated solutions are re-

tained. Different from the Chapman-Enskog expansion, our present model is based on

the closed solution of the kinetic equation. From the BGK model, we assume that it

has the following closed solution

f = g − τ∗(∂g/∂t+ u · ∂g/∂x), (4.5)

86

where τ∗ is the parameter to be determined. The difference between the above solution

and the 1st-order Chapman-Enskog expansion (4.4) is that a generalized collision time

τ∗ is introduced. Substituting the above equation into the BGK model (4.1), we can

obtain the relation between the generalized particle collision time τ∗ and the collision

time τ , which is well-defined in the continuum flow regime,

τ∗ =τ(1 −Dτ∗)

1 + τ(D2g/Dg), (4.6)

where D = ∂/∂t + u · ∂/∂x. To the leading order, a simplified local collision time,

τ∗ =τ

1 + τ(D2g/Dg), (4.7)

can be obtained. In the continuum flow limit, the modification term, τD2g/Dg ∼ Kn,

is expected to be small and τ∗ reverts back to τ , which is determined by τ = µ/p.

The dynamic viscosity coefficient µ can be obtained experimentally or theoretically,

such as the Sutherland’s law. In order to remove the dependence of the collision time

τ∗ on the individual molecular velocity, D2g/Dg can be evaluated by taking moment

φ, such as∫φD2gdudξ/

∫φDgdudξ. Since, the stress and heat conduction terms are

resulting from the different moments of the gas distribution function, the value of

τ∗ in the viscosity term τ∗p and heat conduction coefficient τ∗pCp/Pr are obtained

separately from different moments: φ1 = (u − U)2 for the viscosity coefficient and

φ2 = (u−U)[(u−U)2] for the heat conduction coefficient. A single moment was used

in an early investigation for the argon shock structure calculation [108]. Since both

D2g and Dg involve higher order spatial and temporal derivatives of an equilibrium

gas distribution function, a nonlinear limiter is imposed on the determination of τ ∗,

τ∗ =τ

1 + max[−αKn,min(τ(D2g/Dg), αKn)], (4.8)

where Kn is the Knudsen number and α takes the value 2. The reason for introducing

the above limiter is the following. Firstly, since there are several terms involved in

D2g/Dg, the mathematical evaluation of the ratio will be sensitive to the numerical

error and large fluctuations will be generated, especially in the flow region close to

87

the equilibrium, where both 1st- and 2nd-order derivatives tend to vanish. Also, the

difference between τ∗ and τ depends on the rarefaction of the flow. So, in order to

avoid numerical singularity, a Knudsen number dependent limiter is used. With the

above formulation, in the continuum flow limit, i.e., Kn → 0, the modification to τ

vanishes and the traditional definition τ is recovered.

The multiscale gas-kinetic scheme for the near continuum flow simulation is con-

structed based on the transfer between the macroscopic and microscopic flow descrip-

tions. On the macroscopic level, the time evolution of the macroscopic variables can

be achieved through a finite volume formulation. For each control volume, such as

x ∈ [xi−1/2, xi+1/2] in 1D case, the update of conservative flow variables is

Wn+1i = Wn

i +1

∆xi

∫ tn+1

tn(Fi−1/2(t) − Fi+1/2(t))dt, (4.9)

where Wni are the cell averaged mass, momentum, and total energy, and Fi+1/2 are the

corresponding fluxes at a cell interface, which is provided at microscopic level using

the solution of the kinetic model,

F =

∫uψfdu.

The time step ∆t is defined as ∆t = tn+1 − tn. As an explicit numerical scheme, the

time step of the gas-kinetic scheme for high Reynolds number flows is determined by

the Courant-Friedrichs-Lewy (CFL) condition, i.e., ∆t = ∆tCFL ≡ η∆x/(|U |max + cs),

where 0 < η ≤ 1 is the CFL number, ∆x is the minimum cell size, |U |max is the

maximum absolute value of the velocity, and cs =√γRT is the sound speed. For

low Reynolds number flows, the viscous term also influences the stability, and another

criterion, ∆t ≤ ∆x2/(2ν) with the kinematic viscosity ν = µ/ρ, should be imposed on

the time step. Therefore, a unified stability condition can be expressed as

∆t ≤ η∆x

(|U |max + cs)(1 + 2/Re∆x), (4.10)

where Re∆x = |U |max∆x/ν is the grid Reynolds number.

88

The microscopic model is mainly used to evaluate the gas distribution function at a

cell interface. Eq.(4.4) only presents a gas distribution function at an instant of time,

such as the beginning of each time step tn = 0. In order to obtain a high order accurate

solution in the whole time step from tn to tn+1, we need to evaluate a time-dependent

gas distribution function, which can be constructed in the following [107],

f = g − τ∗(∂g/∂t+ u∂g/∂x) + t∂g

∂t. (4.11)

The relation between τ∗ and τ is given in Eq.(4.7), where τ = µ/p and µ is given by the

Sutherland’s law. The time-dependent part t∂g/∂t coming from the Taylor expansion

with respect to t is used to account for the time evolution of the gas distribution

function. In the 1D case, for a monatomic gas the equilibrium state g is

g = ρ(λ

π)3/2 exp

(−λ((u− U)2 + ξ2

)), (4.12)

where ξ represents the particle random motion in y and z−directions, i.e., ξ2 = v2+w2.

The expansion ∂g/∂x can be expressed as

∂g

∂x=(a1 + a2u+ a3(u

2 + ξ2))g = ag.

Then, all the coefficients can be explicitly determined through the relation between the

microscopic and macroscopic variables, i.e., W =∫ψgdu and ∂W/∂x =

∫ψagdu,

where W = (ρ, ρU, ρE)T are the macroscopic mass, momentum, and energy densities.

The temporal variation of ∂g/∂t can be expanded similarly as a spatial expansion

and the corresponding coefficients can be obtained from the compatibility condition

Eq.(4.3), i.e., ∫ψ(∂g/∂t+ u∂g/∂x)du = 0.

Therefore, on the microscopic level, a time dependent gas distribution function can

be uniquely constructed, which can be subsequently used in the evaluation of the

corresponding fluxes.

In order to simulate the flow with any realistic Prandtl number, a modification of

the heat flux in the energy transport, such as that used in [107], is also implemented

in the present study.

89

4.3 Numerical experiments

To test the correctness of the above multiscale model, we have studied a few cases for

the near continuum flow, which are the argon shock structure, Poiseuille flow for hard

sphere (HS) molecules, nonlinear heat conduction problems for helium gas, and the

unsteady Rayleigh problem.

(1). Argon shock structure

For the non-equilibrium flow, one of the simplest and most fundamental gas dynamic

phenomena that can be used for the model validation is the internal structure of a

normal shock wave. There are mainly two reasons for this. Firstly, the shock wave

represents a flow condition that is far from thermodynamic equilibrium, and secondly

shock wave phenomenon is unique in that it allows one to separate the continuum

differential equations of fluid motion from the boundary conditions. The boundary

conditions for a shock wave are simply determined by the Rankine-Hugoniot relations.

For the Mach number 8 argon shock structure, Bird’s [83] DSMC method using an

inverse 11th power repulsive potential, µ ∼ T 0.68, gave a good agreement with the

experimental profile of argon gas. Fig. 4.1 shows the solution of temperature, density,

and heat flux from the current multiscale model compared with Bird’s DSMC solu-

tions. For this Mach 8 case, the shock thickness and the separation distance between

the density and temperature profiles by the current model compare well with DSMC

solution.

(2). Force-driven Poiseuille flow

The Poiseuille flow is also a fundamental flow problem of rarefied gas dynamics, which

is closely related to the flow in micro-channel devices. The accurate analysis of this

problem has been carried out by many authors [92]. One of the benchmark result is

the solution of the Boltzmann equation, which has been obtained by Ohwada et al.

[84] for hard sphere (HS) molecules. For HS molecules, we have µ ∼√T to determine

90

the dynamical viscosity coefficient. The boundary condition for the Poiseuille flow

here is the Maxwell’s diffusive boundary condition with accommodation coefficient 1.

The velocity distributions for both Boltzmann equation and the current multiscale

model are shown in Fig. 4.2 at three different Knudsen numbers. In the continuum

flow regime, i.e., Kn = 0.001, the current model automatically gives accurate Navier-

Stokes solutions.

(3). Nonlinear heat transfer problems

The next case is about the nonlinear heat transfer in the near continuum flow regime.

The experimental apparatus used in [85] consisted of two parallel flat plates made of

aluminum, one is cooled by liquid nitrogen and the other heated by electric heaters.

The temperatures of the hot and cold plates were 294K and 79K, respectively. Helium

gas was used as the test gas in the experiments. The Knudsen number is defined as

the ratio of mean free path at the center plane to the distance between the plates.

For the boundary condition, the thermal accommodation coefficients measured in the

free molecule regime in [86] were used with the assumption that their values remained

constant over the entire range (α = 0.58). The density distributions obtained from

the current model, and the continuum Navier-Stokes solution are compared with the

experimental data in Fig. 4.3.

(4). Rayleigh problem

The last test is the Rayleigh problem. The Rayleigh flow is an unsteady flow in which a

plate below a gas at rest suddenly acquires a constant parallel velocity and a constant

temperature. In the current calculation [98], the argon gas is at rest at t = 0 with a

temperature of 273K. When t > 0, the plate obtains a constant velocity 10m/s and a

constant temperature 373K. Figs. 4.4 and 4.5 show the simulated solution for density,

temperature, and velocities at two times of t = 10τ and 100τ , where the agreement

between the present model and the DSMC results is good at both times.

91

4.4 Conclusions

The multiscale method presented in this chapter is a nature extension of the gas-

kinetic method for the Navier-Stokes solutions to the near continuum flow regime.

The distinguishable point of the current kinetic model is that a closed solution of the

BGK equation is used in the evaluation of a generalized particle collision time, which

subsequently adjusts the values of the dissipative coefficients, such as the viscosity

and heat conduction coefficients, in the near continuum flow regime. This model has

been successfully applied to a few non-equilibrium flow problems, such as the shock

structure, Poiseuille flow, nonlinear heat conduction problems and unsteady Rayleigh

problem. The merit of a multiscale method is due to the coupling between macroscopic

and microscopic flow descriptions. The conservative flow variables are updated on the

macroscopic level through the fluxes, which are evaluated on the microscopic level,

where the flow physics can be much easily implemented. The current model can

be also naturally equipped with multiple translational, rotational, and vibrational

temperatures.

92

Figure 4.1: Temperature, density (left) and heat flux (right) distributions in a M = 8

argon shock structure with µ ∼ T 0.68. DSMC solution [83] vs. present multiscale

model.

93

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.2

0.4

0.6

0.8

1

1.2

1.4

y/H

U/U

a

Present ResultsLinear Boltzmann

Kn=0.1128

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.2

0.4

0.6

0.8

1

1.2

1.4

y/H

U/U

a

Present ResultsLinear Boltzmann

Kn=0.2257

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

y/H

U/U

a

Present ResultsLinear Boltzmann

Kn=0.4514

Figure 4.2: Velocity distribution in the Poiseuille flow for HS molecules µ ∼ T 0.5, at

three different Knudsen numebrs Kn = 0.1128 (upper, left), Kn = 0.2257 (upper,

right) and Kn = 0.4514 (lower). Circles: solution of the Boltzmann equation [84],

solid line: present multiscale model. Ua is the mean velocity across the channel.

94

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.7

0.8

0.9

1

1.1

1.2

1.3

1.4

x/L

ρ/ρ 0

Present ResultsNavier−StokesExperimental Data

Kn0=0.075

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.7

0.8

0.9

1

1.1

1.2

1.3

1.4

x/L

ρ/ρ 0

Present ResultsNavier−StokesExperimental Data

Kn0=0.118

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.7

0.8

0.9

1

1.1

1.2

1.3

1.4

x/L

ρ/ρ 0

Present ResultsNavier−StokesExperimental Data

Kn0=0.399

Figure 4.3: Density ratio versus position for helium gas in a channel with temperatures

TH = 294K, TC = 79K, and accommodation coefficients αH = αC = 0.58, at three

different Knudsen numbers Kn0 = 0.075 (upper, left), Kn0 = 0.118 (upper, right)

and Kn0 = 0.399 (lower). Circles: experimental data [85], dashed-line: Navier-Stokes

solution, solid line: present multiscale model.

95

0 0.1 0.2 0.3 0.4 0.5 0.60.8

0.9

1

1.1

1.2

1.3

y (m)

T/T

0 or

ρ/ρ 0

ρ/ρ0

T/T0

0 0.1 0.2 0.3 0.4 0.5 0.6−2

0

2

4

6

8

10

12

14

y (m)

Vx o

r V

y (m

/s)

Vy

Vx

Figure 4.4: Unsteady Rayleigh problem at time t = 10τ . Solid lines: DSMC solutions

[98], circles: present model.

0 0.1 0.2 0.3 0.4 0.5 0.60.8

0.9

1

1.1

1.2

1.3

y (m)

T/T

0 or

ρ/ρ 0

ρ/ρ0

T/T0

0 0.1 0.2 0.3 0.4 0.5 0.6−2

0

2

4

6

8

10

12

14

y (m)

Vx o

r V

y (m

/s)

Vy

Vx

Figure 4.5: Unsteady Rayleigh problem at time t = 100τ . Solid lines: DSMC solutions

[98], circles: present model.

96

Chapter 5

Multiple temperature kinetic

model for microscale flows

In this chapter, we propose a kinetic model with multiple translational temperature

for the microscale flow simulations. It is well recognized that the Navier-Stokes (NS)

equations lose accuracy for microscale flows, where the local Knudsen number could be

large. These inaccuracies may be partially due to the single temperature assumption

in the Navier-Stokes equations. Here, based on an extended gas-kinetic Bhatnagar-

Gross-Krook (BGK) equation with multiple temperature equilibrium state, a multi-

translational temperature model is proposed and its numerical scheme is also con-

structed. In the current approach, the energy exchange between x, y, and z directions

is modeled through the particle collision, and individual energy equation in different

directions is obtained. The newly constructed kinetic model is an enlarged system in

comparison with Holway’s Ellipsoid Statistical BGK model (ES-BGK). The detailed

difference will be presented. In the newly derived macroscopic governing equations

from the current model, all the viscous terms in the Navier-Stokes equations are re-

placed by the temperature relaxation terms. The constitutive relation in the standard

Navier-Stokes equations is recovered only in the limiting case when the flow is close

97

to the equilibrium with small temperature differences in different directions. In order

to validate the current model, we apply it to some microscale flow problems. In the

continuum flow regime, the Navier-Stokes solutions are precisely recovered from the

current model. As the Knudsen number increases, the simulation results are compared

with the solutions from both direct simulation Monte Carlo (DSMC) method and the

linearized Boltzmann equation. The numerical experiments validate the capacity of

the current method in capturing the thermal non-equilibrium effect for microscale

flows. It is clearly shown that many non-equilibrium physical phenomena in the near

continuum flow regime can be well captured by modifying some assumptions in the

standard Navier-Stokes equations.

5.1 Introduction

The transport phenomena, i.e., mass, heat, and momentum transfer, in the different

flow regime is of a great scientific and practical interest. The classification of various

flow regimes is based on the Knudsen number Kn, which is defined as the ratio of

the mean free path to a characteristic length scale of the system. Two cases could

make the local Knudsen number large: one is the low density of the medium, such as

the hypersonic flow around space vehicles in low earth orbit [93], another is the small

characteristic length of the flow system, for example the flow problems inside the micro-

electro-mechanical system (MEMS). It has been recognized that the fluid mechanics of

microscale fluid flows are not the same as those experienced in the macroscopic world.

Currently, our understanding of microscale flows lags far behind the rapid progress

in the microsystems. Hence, there is a strong desire and requirement for accurate

models which give reliable solutions with lower computational costs for microscale

flow problems.

The DSMC method makes great success in the numerical simulation of low density

flows [89]. However, for the microscale flows, the DSMC suffers from statistical noise

98

in the bulk velocity because of the random molecular motion. When the bulk velocity

is much slower than the thermal velocity, a large number of independent samples

are needed to eliminate the statistical scattering, which makes it impractical in these

cases. Although some attempts have been made to modify this method in order to

simulate microscale gas flows with affordable computational cost, for example the

information preservation (IP) method [90, 98], the success of these modified methods

is still limited at present. Alternatively, many macroscopic continuum models have

been intensively developed in the literature. These include the Navier-Stokes and the

Burnett equations from the Chapman-Enskog expansion [30, 31], Grad’s 13 moment

equations [33], the regularized 13 equations [34], and many others. In order to assess

these continuum models, a few test cases have been used [116], it seems that none

of the models is commonly acceptable as the Knudsen number increases. Recently,

the lattice Boltzmann method (LBM) has been extended to simulate microscale flows

[105, 106], however, it seems much more work is still needed to be done for improving

the accuracy of the prediction by the method. Overall, the small length scales and

slow bulk gas velocity combine to make continuum solutions inaccurate, and particle

solution time consuming. More references about the methods for the micro-flows can

be found in [92].

Based on a closed solution of the BGK model, a generalization of particle colli-

sion time or constitutive relationship has been obtained for the non-equilibrium flows

[108, 113]. The derived extended Navier-Stokes equations from the BGK model with

a generalized constitutive relationship, i.e. extended viscosity and heat conduction

coefficients to rarefied regime, have been successfully used in the calculation of argon

and nitrogen shock structures for a wide range of Mach numbers, i.e., 1.2 ≤ Ma ≤ 11.

The current study is to further modify the kinetic BGK model, construct new macro-

scopic governing equations with multiple translational temperature, and apply it to

capture thermal non-equilibrium phenomena in microscale flow problems. The ma-

jor point we will deliver is that all the viscous terms in the standard Navier-Stokes

99

equations will be replaced by the temperature relaxation terms in the newly derived

macroscopic governing equations from the multiple temperature kinetic model. In the

continuum flow regime, the temperature relaxation effect goes back automatically to

the Navier-Stokes assumptions.

In this chapter, section 5.2 provides details on the construction of the kinetic model

and its corresponding macroscopic governing equations. The gas-kinetic scheme to

solve the newly derived macroscopic governing equations will be presented in section

5.3. Section 5.4 describes the effective viscosity approach and the generalized second-

order slip boundary condition. Section 5.5 concerns the application of the current

method to microscale flow simulations. The numerical solutions from the current

model are compared with the exact Navier-Stokes solutions in the continuum flow

regime and the results from both direct simulation Monte Carlo (DSMC) method and

the linearized Boltzmann equation in the transition flow regime. The final section is

the conclusion.

5.2 Multiple translational temperature kinetic model

In this section, we first review the BGK equation and then construct the multi-

temperature (multi-T) kinetic model for monatomic gas, and derive its macroscopic

extended Navier-Stokes equations for microscale flow simulation.

5.2.1 Standard BGK model and Navier-Stokes equations

The Boltzmann equation expresses the behavior of a many-particle kinetic system in

terms of the evolution equation for a single particle gas distribution function. The

simplification of the Boltzmann equation given by the BGK model is formulated as

100

[87],

∂f

∂t+ u · ∂f

∂x=g − f

τ, (5.1)

where f is the number density of molecules at position x and particle velocity u =

(u, v, w) at time t. The left hand side of the above equation represents the free stream-

ing of molecules in space, and the right side denotes the collision term. If the distri-

bution function f is known, macroscopic variables, such as mass, momentum, energy

and stress, can be obtained by integration over the moments of molecular velocity. In

the BGK model, the collision operator involves simple relaxation to a state of local

equilibrium given by g with a characteristic time scale τ . Traditionally, the equilibrium

state is given by a Maxwellian,

g = ρ

(λ

π

)K+3

2

e−λ((u−U)2+ξ2),

where ρ is the density, U the macroscopic fluid velocity, and λ = m/2kT . Here, m

is the molecular mass, k is the Boltzmann constant, and T is the temperature. For

an equilibrium flow, the internal variable ξ accounts for the rotational and vibrational

modes, such as ξ2 = ξ21 + ξ2

2 + ...+ ξ2K , and the total number of degrees of freedom K

is related to the specific heat ratio γ. The relation between mass ρ, momentum ρU,

and energy densities ρE with the distribution function f becomes

ρ

ρU

ρE

=

∫ψfdudξ, (5.2)

where ψ is the vector of moments

ψ = (1,u,1

2(u2 + ξ2))T ,

and the volume element in the phase space is dudξ = dudvdwdξ1dξ2...dξK . Since

mass, momentum and energy are conserved during particle collisions, f and g satisfy

the conservation constraint, ∫ψ(g − f)du = 0, (5.3)

101

at any point in space and time.

The BGK model was originally proposed to describe the essential physics of molec-

ular interactions with τ chosen as the molecular collision time. Although the BGK

model appears to describe only weak departures from local equilibria, it has long been

recognized that such an approximation works well beyond its theoretical limits as

long as the relaxation time is known for the physical process. Based on the above

BGK model, the Navier-Stokes equations can be derived with the Chapman-Enskog

expansion truncated to the 1st-order,

f = g + Knf1

= g − τ(∂g/∂t + u · ∂g/∂x). (5.4)

For the Burnett and super-Burnett solutions, the above expansion can be naturally

extended [96], such as f = g + Knf1 + Kn2f2 + Kn3f3 + .... In the following of this

chapter, we only consider the monatomic gas with K = 0.

Based on the Chapman-Enskog expansion and the BGK model, in the continuum

flow limit the Navier-Stokes equations with the stress and Fourier heat conduction

terms can be derived. In this chapter, in order to make the expression clear, the

macroscopic governing equations in 1D space is presented, even though the multi-

temperature kinetic model is itself for multidimensional flows. The Navier-Stokes

equations derived from the BGK model for a monatomic gas in the 1D case can be

written as,

ρ

ρU

ρE

t

+

ρU

ρU2 + p

(ρE + p)U

x

=

0

43µUx

52µRTx + 4

3µUUx

x

, (5.5)

where p = ρRT is the pressure and µ = τp is the dynamical viscosity coefficient. With

the relation λ = m/2kT = 1/2RT and Cp = 5k/2m = 5R/2 for a monatomic gas,

the heat conduction coefficient in the above equations becomes κ = 5kµ/2m, and the

Prandtl number becomes fixed with the value Pr = µCp/κ = 1. This is a well known

102

result for the BGK model.

5.2.2 Multi-temperature gas-kinetic model and its correspond-

ing Navier-Stokes equations

Traditionally, the BGK model is considered suitable only for isothermal rarefied gas

flow. It does not provide reliable results for non-isothermal flows because it gives

incorrect Prandtl number. The disagreement between an exact solution based on

the Boltzmann equation and that obtained from the BGK model reaches 30% near

the hydrodynamic flow regime. In order to get the correct Prandtl number, many

modification of the BGK model have been proposed. One is the Ellipsoid-Statistical

BGK (ES-BGK) model of Holway [91], and the other is the S model of Shakhov [97].

In the ES-BGK model, the ”temperature” becomes a tensor and it is related to Prandtl

number. In the S-model, a heat flux term is added in the equilibrium state. In our early

BGK scheme for the continuum flow computation [107], the correct Prandtl number

is achieved through the modification of heat flux across a cell interface in a finite

volume scheme. In the following, we are going to propose a multi-temperature model.

The purpose of constructing this model is not for the Prandtl number correction, but

for the capturing of physical multi-translational temperature phenomena in the near

continuum flow regime, for example the microscale flows, where the accuracy of the

NS equations is not adequate.

This chapter mainly concerns the 2D flow simulation. In the following, a multi-T

model in 2D will be proposed. The generalized BGK model has the same form as the

original one,

∂f

∂t+ u

∂f

∂x+ v

∂f

∂y=g − f

τ, (5.6)

but the equilibrium state has multiple temperature,

g = ρ

(λx

π

)1/2(λy

π

)1/2(λz

π

)1/2

exp[−λx(u− U)2 − λy(v − V )2 − λzw

2]. (5.7)

103

Here λx = m/(2kTx), λy = m/(2kTy), and λz = m/(2kTz) are related to the transla-

tional temperature Tx, Ty, and Tz in x−, y− and z−directions. In order to determine

all unknowns in the corresponding macroscopic variables, such as ρ, U, V, Tx, Ty, and

Tz, we propose the following moments for the collision term in the BGK model,

∫φ

(∂f

∂t+ u

∂f

∂x+ v

∂f

∂y

)dudvdw =

∫φg − f

τdudvdw =

0

0

0

0

(ρEeqx − ρEx)/τ

(ρEeqy − ρEy)/τ

,(5.8)

where

φ =

(1, u, v,

1

2(u2 + v2 + w2),

1

2u2,

1

2v2

)T

.

The 1st four moments on the right hand side of Eq.(5.8) are the conservative moments

of the mass, momentum, and total energy. The last two moments are the newly

constructed models which simulate the energy exchange among different directions.

The equilibrium energies ρEeqx and ρEeq

y in Eq.(5.8) have the forms

ρEeqx =

1

2ρU2 +

ρ

4λeq,

and

ρEeqy =

1

2ρV 2 +

ρ

4λeq,

which are constructed based on the assumption that the system will approach to an

equilibrium state with equal temperature. The common equilibrium temperature in

all directions λeq is determined by equally distributing thermal energy in all degrees

of freedom,

ρ3

4λeq= ρE − 1

2ρ(U2 + V 2),

where ρE is the total energy, i.e.,

ρE =

∫1

2(u2 + v2 + w2)fdudvdw =

∫1

2(u2 + v2 + w2)gdudvdw.

104

The tendency for the gas distribution function to approach to a common Maxwellian

determined by (ρ, U, V, λeq) means that the H-theorem for the system (5.6) and (5.8)

is satisfied. Note that the last two moments on the right hand side of Eq.(5.8) cannot

be derived directly from the BGK equation (5.6) itself. It is a model we construct.

The basic consideration is that there needs particle collision to exchange energy in

different directions. The direct moments (12u2, 1

2v2) to the BGK equation (5.6) with

the multiple temperature equilibrium state g in Eq.(5.7) will give

[(1

2ρU2 +

ρ

4λx) − ρEx

]/τ

and [(1

2ρV 2 +

ρ

4λy) − ρEy

]/τ

for the two terms on the right hand side of Eq.(5.8), which are not adequate to close

the system due to the three unknowns (λx, λy, λz) instead of one. In other words, when

we introduce the multiple temperature equilibrium state in Eq.(5.6), we introduce two

more unknowns, such as λy and λz. In order to close the system to have a unique

solution, we have to introduce two additional equations or constraints, which are the

last two moments in (5.8), where λeq can be explicitly determined through the total

thermal energy in the system. So, our current multi-T BGK model is an extension

of the original BGK model and the nonconservative moments are modeled instead of

directly derived from the BGK collision term. Only through the new collision moments

(5.8), the corresponding equations are closed and the macroscopic mass, momentum,

total energy, and individual energy in each direction can be updated. In section 5.3,

we are going to present the numerical method based on Eq.(5.6) and (5.8) for the time

evolution of macroscopic physical quantities. The idea in our current model is that the

thermal equilibrium between x−, y−, and z−directions will be achieved through the

particle collisions, and there is a time delay to achieve the temperature equilibrium.

In the Navier-Stokes equations, it is assumed that the same equilibrium temperature

is obtained instantaneously. As shown in the following, the Navier-Stokes assumption

between the stress and the velocity gradient is only valid in the continuum flow limit.

105

The real viscosity terms in the NS equations are replaced by the temperature relaxation

term in the current model. The multi-T kinetic model is a more general equation than

the NS equations and it goes back to NS assumption when the temperature differences

among Tx, Ty, and Tz are small. In order to make the presentation clear, in the

following we are going to consider the 1D flow problems and derive the corresponding

macroscopic governing equations.

Based on the multi-T kinetic model, in the 1D case the equilibrium state can be

written as

g = ρ

(λ

‖

π

)1/2(λ

⊥

π

)

exp[−λ‖

(u− U)2 − λ⊥

ξ2],

where ξ represents the particle random motion in y and z−directions, i.e., ξ2 = v2+w2,

and the λ‖

= 1/2RT‖

and λ⊥

= 1/2RT⊥

represent the x-direction temperature T‖

and the temperature T⊥

in other directions. For the equilibrium flow with T‖

= T⊥,

the governing equations, i.e., the Euler equations, can be obtained from Eq.(5.6),

ρ

ρU

12ρ(U2 +RT

‖+ 2RT

⊥)

t

+

ρU

ρU2 + ρRT‖

12ρU(U2 + 3RT

‖+ 2RT

⊥)

x

=

0

0

0

.(5.9)

Even though T‖

= T⊥

in the above equilibrium flow, in order to distinguish the

different contribution to the energy and corresponding fluxes from the random motion

in different directions, the distinguishable temperatures T‖

and T⊥

are still used. By

using the 1st-order Chapman-Enskog expansion, the following dissipative governing

106

equations for the kinetic model (5.6) can be obtained,

ρ

ρU

12ρ(U2 +RT

‖+ 2RT

⊥)

ρRT⊥

t

+

ρU

ρU2 + ρRT‖

12ρU(U2 + 3RT

‖+ 2RT

⊥)

ρURT⊥

x

=

0

−23ρR(T

‖ − T⊥)

32τρR2T

‖(T

‖)x + τρR2T

‖(T

⊥)x − 2

3ρRU(T

‖ − T⊥)

τρR2T‖(T

⊥)x + 1

3ρRU(T

‖ − T⊥)

x

(5.10)

+

0

0

0

ρ3τR(T

‖ − T⊥)

.

The above equations are different from the standard Navier-Stokes equations (5.5) de-

rived from the BGK model in the 1D case. Instead of the viscous flux (4/3)µUx in (5.5),

the current model replaces it by the temperature relaxation term, i.e., −(2/3)ρR(T‖

−

T⊥

). Near to the thermal equilibrium limit, i.e., T‖

≃ T⊥

≃ T eq, the temperature

difference becomes

T‖ − T

⊥

= −2τT eqUx.

Therefore, the relaxation term in the momentum equation goes to

−2

3ρR(T

‖ − T⊥

) =4

3τρRT eqUx,

which exactly recovers the standard Navier-Stokes viscous term. Also, the total energy

dissipative flux becomes

4

3τρRT eqUUx +

5

2τρR2T eqT eq

x ,

where the pressure is p = ρRT eq. So, near the thermal equilibrium limit, the stan-

dard Navier-Stokes equations are obtained. Under this limit, for the thermal energy

107

equation in the last equation of (5.10), it will not become an independent equation

anymore, which can be derived from the total energy and momentum equations under

the condition T‖

= T⊥

= T eq. Hence, in terms of our multi-T kinetic model, we

have an enlarged system in comparison with the Navier-Stokes equations. But, this

system shrinks to the NS equations in the continuum limit. Also, as shown in the

numerical experiments, in the continuum flow regime, i.e., Kn ≤ 0.01, the solution

from the enlarged system will be the same as that from the Navier-Stokes equations.

The non-equilibrium thermal effect only takes places when Kn becomes large, such

as for the microscale flows. In order to validate the kinetic model, instead of solving

Eq.(5.10), we are going to solve numerically the kinetic model (5.6) directly through

a multiscale gas-kinetic scheme, which is presented in the section 5.3.

From the above relaxation model, we can realize that the viscous term approxima-

tion in the Navier-Stokes equations is not an intrinsic property of a gas, but rather, an

approximation designed to simulate the effect of thermal relaxation when the govern-

ing equations are cast in terms of a single temperature. This approximation is based

on the assumption that the time scale of the macroscopic gas motion is much larger

than the relaxation time for the thermal energy equilibrium. This local thermody-

namic equilibrium (LTE) assumption is a good approximation only for low Knudsen

number flows. When the characteristic time for temperature relaxation is comparable

to the characteristic flow time scale in the near continuum flow regime, for example

the microscale flows, the relaxation effect has to be considered.

5.2.3 Comparison between multi-T kinetic model and ES-

BGK model

Based on the 1st order Chapman-Enskog expansion, the original BGK model gives

the Navier-Stokes equations with unit Prandtl number. In order to obtain a proper

Prandtl number for a realistic flow, Holway suggested the ES-BGK model [91], where

108

the Maxwellian distribution in the BGK model is replaced by an anisotropic Gaussian

so that the collision term reads

Q = (gh − f)/τ,

where gh denotes the anisotropic Gaussian,

gh =ρ√

2πλij

exp[−1

2λ−1

ij CiCj],

and the matrix λij is given by

λij = RTδij + (1 − 1

Pr)pij

ρ.

Here λ−1ij denotes the inverse matrix. Same as the BGK model, the ES-BGK assumes

that the collision frequency is independent of the microscopic velocity. An entropy

condition for ES-BGK has been recently proved by Andries et al. [88]. The above

ES-BGK model is different from the kinetic model we proposed in the last subsection.

For example, for the 1D flow, the equilibrium state from the above ES-BGK model

has the form,

gh =ρ

(2π)3/2

1

λ1/211 λ22

exp[−(u − U)2

2λ11− (v2 + w2)

2λ22],

where

λ11 = (1 − Pr − 1

Pr)RT +

1

ρ

Pr − 1

Pr

∫(u− U)2fdudvdw,

and

λ22 = (1 +1

2

Pr − 1

Pr)RT − 1

2ρ

Pr − 1

Pr

∫(u− U)2fdudvdw,

are determined from the moments of f with a local temperature T . As shown above,

the Prandtl number involves in the determination of λ11 and λ22. Physically, λ11 and

λ22 in the ES-BGK model are only components of a matrix which have no direct

meaning to the temperature. The real temperature in ES-BGK is T , and λij is mainly

used for the recovery of correct Pr number by modifying the viscosity coefficient

to µ = τρRTPr. Also, the evaluation of λij is through the moments of f , which

is different from our model, where independent governing equations for the thermal

109

energy (or temperature) in different directions are proposed. In our multi-T model, the

modification of Pr number is not our concern. For a flow with Pr = 1, the ES-BGK

model will have a single ”temperature”, i.e., λ11 = λ22 = λ33. However, in our current

multi-T model, the multiple temperatures still exist at Pr = 1. In the continuum

flow limit, the multi-T model has intrinsically unit Prandtl number. As shown in the

next section, the Prandtl number fix will be done in the energy flux evaluation in the

numerical method, where the heat flux transport in a finite volume method will be

modified. As shown in the past literature, this Prandtl number fix is very accurate

and the solutions with different Prandtl number can be accurately captured.

5.3 Finite volume BGK scheme for the multi-T ki-

netic model

The kinetic model constructed in the previous section is solved based on the gas-

kinetic BGK scheme [107]. It is a conservative multiscale finite volume method, where

the update of the macroscopic flow variables is through the numerical fluxes at cell

interfaces which are evaluated based on the time-dependent gas distribution function.

Taking moments φ to Eq.(5.6) in a control volume (x, y) ∈ Ii,j = [xi−1/2, xi+1/2] ×

[yj−1/2, yj+1/2] and time interval t ∈ [tn, tn+1], the update of the macroscopic flow

variables, i.e., W = (ρ, ρU, ρV, ρE, ρEx, ρEy)T inside each numerical cell Ii,j from time

step tn to tn+1, becomes

Wn+1i,j = Wn

i,j +1

∆xi

∫ tn+1

tn

(Ei−1/2,j(t) − Ei+1/2,j(t)

)dt

+1

∆yj

∫ tn+1

tn

(Fi,j−1/2(t) − Fi,j+1/2(t)

)dt+ Sn

i,j∆t, (5.11)

where Ei+1/2,j and Fi,j+1/2 are the corresponding fluxes at the cell interfaces (xi+1/2, yj)

and (xi, yj+1/2), respectively, which are evaluated based on the gas distribution func-

110

tions fi+1/2,j and fi,j+1/2 there,

Ei+1/2,j =

∫uφfi+1/2,jdudvdw

and

Fi,j+1/2 =

∫vφfi,j+1/2dudvdw.

The source term is due to the moments of the collision term in Eq.(5.8) which has the

form

S =(0, 0, 0, 0, (ρEeq

x − ρEx)/τ, (ρEeqy − ρEy)/τ

)T.

For the current multi-T model, the evaluation of the gas distribution function f at

a cell interface is similar to the BGK-NS method in [107], where the only difference

between them is that three temperatures Tx, Ty, and Tz have to be accounted for. The

following is about the calculation of the gas distribution function fi+1/2,j based on the

multi-T model (5.6), and fi,j+1/2 can be obtained in a similar way.

Since we are going to develop a directional splitting method to solve Eq.(5.6), the

kinetic model in x−direction can be written as,

ft + ufx = (g − f)/τ,

where g is the multiple temperature equilibrium state (5.7). Based on the multi-T

kinetic model, up to the 1st-order expansion, we have the following gas distribution,

f(xi+1/2, yj, t, u, v, w) = g − τ(∂g/∂t+ u∂g/∂x) + t∂g

∂t, (5.12)

where −τ(gt + ugx) is the Chapman-Enskog expansion and tgt is the time evolution

part [96]. The relation between τ and µ is τ = µ/p, where µ is the dynamical viscosity

coefficient and p is the pressure. The connect between macroscopic variables and the

equilibrium state is

W = (ρ, ρU, ρV, ρE, ρEx, ρEy)T =

∫φgdudvdw.

So, from the reconstructed initial data W(xi+1/2, yj, tn) at the beginning of each time

step, the equilibrium state g in Eq.(5.12) can be uniquely determined at a cell interface.

111

Then, from the spatial derivative ∂W/∂x there, we can evaluate ∂g/∂x in Eq.(5.12) as

the following. Based on the Taylor expansion, the expansion ∂g/∂x can be expressed

as,

∂g

∂x= g

[1

ρ

∂ρ

∂x+

1

2λx

∂λx

∂x− ∂

∂x(λxU

2 + λyV2) +

1

2λy

∂λy

∂x+

1

2λz

∂λz

∂x

+2∂(λxU)

∂xu+ 2

∂(λyV )

∂xv − ∂λx

∂xu2 − ∂λy

∂xv2 −∂λz

∂xw2

]

= g[a1 + a2u+ a3v + a4u

2 + a5v2 + a6w

2]

= ga, (5.13)

where

a = a1 + a2u+ a3v + a4u2 + a5v

2 + a6w2.

All coefficients in a can be determined in the following. Since

λx = ρ/(4ρEx − 2ρU2),

we have

∂λx

∂x= −4λ2

x

ρ[∂(ρEx)

∂x− 1

2

∂(ρU2)

∂x] +

λx

ρ

∂ρ

∂x= −a4,

∂λy

∂x= −

4λ2y

ρ[∂(ρEy)

∂x− 1

2

∂(ρV 2)

∂x] +

λy

ρ

∂ρ

∂x= −a5,

∂λz

∂x= −4λ2

z

ρ[∂(ρE − ρEx − ρEy)

∂x] +

λz

ρ

∂ρ

∂x= −a6.

Let’s define

A =1

ρ[∂(ρU)

∂x− U

∂ρ

∂x],

and

B =1

ρ[∂(ρV )

∂x− V

∂ρ

∂x],

then,

a3 = −2a5V + 2λyB,

a2 = −2a4U + 2λxA,

and

a1 =1

ρ

∂ρ

∂x− a2U − a3V − a4(U

2 +1

2λx

) − a5(V2 +

1

2λy

) − a61

2λz

.

112

After the determination of ∂g/∂x = ga in Eq.(5.12), the term ∂g/∂t = gA with

A =[A1 + A2u+ A3v + A4u

2 + A5v2 + A6w

2],

can be obtained by requiring the non-equilibrium part in the Chapman-Enskog expan-

sion vanishing to the moments φ,

∫φ(∂g/∂t+ u∂g/∂x)dudvdw =

∫φ(A+ au)gdudvdw = 0,

where the six unknowns in A can be uniquely determined from the above six equations.

The procedure to get A is similar to obtaining a. Therefore, the gas distribution

function at the cell interface (5.12) is totally determined, which can be used to evaluate

the fluxes.

After the determination of f at a cell interface, we can explicitly evaluate the heat

flux there as well. In order to simulate the flow with any realistic Prandtl number,

a modification of the heat flux in the energy transport, such as that used in [107], is

also implemented in the present study. Therefore, the current model can simulate flow

with any Prandtl number.

5.4 Effective viscosity approach and the general-

ized second-order slip boundary condition

It is well known that the physical significance of the viscosity is a reflection of the

momentum exchange between the fluid molecules. From kinetic theory, the viscosity

µ is related to the mean free path l of gas molecules via µ = φcρl, where c is the mean

molecular speed, ρ is the gas density and φ is taken to be a constant with a value of

0.499 [117]. In the near-wall region, however, the presence of a solid boundary will

have a significant impact on the average distance a gas molecule can travel between

successive collisions with either another gas molecule or the solid wall. To enable

the Navier-Stokes equations to capture the velocity profile in the Knudsen layer, the

113

effective viscosity has been investigated in [118, 119]. Here we briefly review the method

recently proposed by Guo et al. [120]. Let us consider a gas bounded by two parallel

walls located at z = 0 and z = H , respectively. The effective viscosity µe is defined

by [120]

µe(z) = µ0

(1 +

1

2

((α− 1)e−α + (β − 1)e−β − α2Ei(α) − β2Ei(β)

)), (5.14)

where α = z/l0 and β = (H − z)/l0, and Ei(x) is the exponential integral function

defined by

Ei(x) =

∫ ∞

1

t−1e−xtdt. (5.15)

The generalized second-order slip boundary condition can be written as

u|s − u|w =

(l(∂u

∂z) − 1

2l∂

∂z(l∂u

∂z)

)

w

, (5.16)

here l = l0µ/µ0. The validity of the above effective viscosity method as well as the

generalized second-order slip boundary condition has been shown in [120].

5.5 Numerical experiments

(1). Shear driven Couette flows

Shear driven Couette flows are encountered in micromotors, comb mechanisms, and

microbearings. In the simplest case, the Couette flow can be used as a prototype flow

to model such flows driven by a moving plate. Since the Couette flow is shear driven,

the pressure does not change in the stream-wise direction. Hence, the compressibility

effects become important for large temperature fluctuations or at high speeds.

The Couette flow concerns a flow problem between two infinite parallel plates,

separately by a distance L. The schematic structure is shown in Fig. 5.1. In our

computation, the most cases we study are the hard sphere (HS) molecule and the

working gas is argon. The specific heat ratio is γ = 5/3 with molecular mass m =

114

6.63×10−26kg. The viscosity coefficient for HS is µ = 2.117×10−5√

(T/273)N ·s/m2.

The mean free path is defined as

l0 =16

5(

1

2πRT)1/2 µ

ρ0,

where R is the gas constant, T and ρ are temperature and density, respectively. In

most calculations, both surfaces maintain 273K and Maxwell diffusive kinetic reflection

boundary condition [111] is used. The density ρ0 has a value corresponding to the

pressure of 1atm (or 101325Pa) at T = 273K. The Knudsen number is defined as

Kn = l0/L, which increases as the length L decreases. In all computations, we use 50

cells in the one-dimensional computational domain.

In order to validate the multi-temperature model in the continuum flow limit, we

first apply it to the case, where the exact Navier-Stokes solutions are available. Under

the conditions of µ ∼ T ω with ω = 1 and of adiabatic lower wall condition, there is an

analytic solution in the compressible case [94],

τwx

µ∞Uw=

U

Uw+ Pr

γ − 1

2M2

∞

[U

Uw− 1

3(U

Uw)3

],

where Uw is the horizontal velocity of the upper wall and M∞ is the corresponding

Mach number, the shear stress τw is given by

τwL

µ∞Uw= 1 + Pr

γ − 1

3M2

∞.

In order to test the multi-T BGK scheme, we set up the upper wall with a speed of

M∞ = 3 and lower adiabatic wall with velocity zero, and Prandtl number Pr = 2/3.

The viscosity coefficient is set to be µ = 2.117 × 10−5(T/273)N · s/m2. The velocity

and temperature profiles in the channel are shown in Fig. 5.2, where the circles are

the exact NS solutions and the solid lines are from the current multi-T scheme. In

the current case, the Kn number has a value Kn = 0.001, which well belongs to the

NS flow regime. It is hard to distinguish the three temperatures in the x−, y−, and

z−directions in Fig. 5.2.

115

In the following, we simulate the Couette flow cases for the hard sphere (HS)

molecules with fixed upper wall velocity 300m/s. The use of this wall velocity is

from the consideration of two folds. One is the easy solution from DSMC simulation

and the other is the temperature deviation due to large shear. The Prandtl number

used is Pr = 0.68, which is consistent with the Prandtl number in the DSMC method

for the HS model. The Kundsen numbers simulated are Kn = 0.01, 0.1, and 0.5.

Fig. 5.3 shows the velocity and temperature profiles across the channel at Kn = 0.01,

where the solid lines are the current multi-T model results and circles are the DSMC

solutions. Note three temperatures are plotted for both DSMC and multi-T solutions,

even though they are indistinguishable. At this Knudsen number, the separation be-

tween the temperature is too small to be seen. As the Knudsen number increases to

0.1, the three temperatures can be clearly observed in Fig. 5.4, where the magnitudes

of the temperature are distributed from the highest Ty, to Tz, and to the lowest Tx. At

Kn = 0.1, both velocity and temperature from multi-T model have a fair agreement

with the DSMC results. As the Kn increases to 0.5, the deviation between different

temperature becomes more obvious. Fig. 5.5 shows the velocity and temperature

distributions. In this case, the slip velocity due to the kinetic diffusive boundary con-

dition becomes large, and both velocity and temperature distributions come to be

more flat in comparison with small Knudsen number results. The temperature in the

y−direction (same direction as the flow velocity) is higher than those in other two

directions. Even though there are deviations close to the boundary in the temperature

distributions, the overall match between the multi-T model and DSMC results are

fair. At this Knudsen number, the velocity profile is not a straight line. The slight

curvature near the wall may be due to the Knusden layer in the DSMC solution. In

terms of computational efficiency, the multi-T model takes one or two minutes in a

PC in all cases to get a steady state solution. Even though we concentrate on the

HS molecules in the above simulation, the multi-T model itself can be applied to any

molecular model with a generalized viscosity coefficient, such as the Sutherland’s law.

116

(2). Force-driven and pressure-driven Poiseuille flows at Kn = 0.1

It is generally recognized that in the slip flow regime with Knudsen number Kn≤ 0.1,

the Navier-Stokes equations with the slip boundary condition is capable to accu-

rately simulate the microchannel flow. However, for the simple both force-driven and

pressure-driven Poiseuille flows in the slip flow regime with relative small gradients

and Knudsen number, the Navier-Stokes equations give qualitatively incorrect predic-

tions [114, 115]. In the force-driven case, they fail to reproduce the central minimum

in the temperature profile and non constant pressure profile, which are both predicted

by the kinetic theory and observed in the DSMC simulation [100, 101, 102, 103, 104].

In the pressure-driven case, the pressure profiles in the cross-stream direction from

the Navier-Stokes solutions and DSMC data have the opposite curvature [115, 111].

Furthermore, it is not possible to correct this failure by modifying the equation of

state, transport coefficients or boundary conditions, and the discrepancy is caused by

the governing equations themselves. In order to understand these phenomena, many

analysis have been done. For example, the non-constant pressure is well explained

based on the Burnett equations [103], and the temperature minimum at the center is

explained only through the kinetic theory [100, 101, 102, 104], or the super-Burnett so-

lution [110]. As an excellent test for capturing non-equilibrium phenomena, the current

multi-temperature model will be used to study the force-driven and pressure-driven

Poiseuille flows at Kn = 0.1 as well.

The set up of the Poiseuille flows is given in [114], see Fig. 5.6 for the schematic

structures. The simulation fluid is a hard sphere gas with particle mass m = 1 and

diameter d = 1. At the reference density of ρ0 = 1.21 × 10−3, the mean free path

is l0 = m/(√

2πρ0d2) = 186. The distance between the thermal walls is Ly = 10l0

and their temperature is T0 = 1. The reference fluid speed is U0 =√

2kT0/m = 1,

so Boltzmann constant is taken as k = 1/2. The reference sound speed is c0 =√γkT0/m = 0.91 with γ = 5/3 for a monatomic gas. The reference pressure is

p0 = ρ0kT0/m = 6.05 × 10−4. The external force or the pressure gradient is chosen

117

so that the flow will be sub-sonic and laminar. Specifically, ρ0f = 8.31 × 10−8 for the

force-driven case, dp/dx ≈ 1.08×10−7 with pin = 32p0, pout = 1

2p0 and Lx = 3Ly = 30l0

for the pressure-driven case. In these cases the Knudsen number is Kn = l0/Ly = 0.1

and the Reynolds number is of order one. In all calculations, the cell size takes the size

of one fifth of the mean free path under the initial flow condition. Maxwell diffusive

kinetic boundary condition [111] with the accommodation coefficient σ = 1 is used.

Figs. 5.7 and 5.8 present the results for the force-driven case from the current

multi-T model. Besides the excellent match of density and velocity between the DSMC

and the multi-T results, the curved pressure and temperature distributions are well

captured as well. The temperature minimum in both Tx and the averaged temperature

T can be clearly observed in Fig. 5.8. This is surprising because the analysis in

[103] confirms that the temperature minimum does not appear even in the Burnett

solution. But, it can be recovered in the super-Burnett order [110]. However, based

on our current model, even with the inclusion of 1st-order derivatives in space and

time, see Eq.(5.12), the temperature minimun has been recovered. So, the use of

temperature relaxation in multi-T model to replace the stress and strain relation in

the Navier-Stokes equations has significant impact in capturing the non-equilibrium

physical phenomena in the near continuum flow regime. In order to further distinguish

the current multi-T model and the gas-kinetic BGK-NS method [107], the same test

case has been calculated by the BGK-NS method as well. As shown in Fig. 5.9, for

both pressure and temperature, the BGK-NS method with slip boundary condition

has no the capacity to capture the non-equilibrium effect. This is consistent with the

analysis in [114, 115].

Although both force-driven and pressure-driven Poiseuille flows appear to be equiv-

alent, they are in fact very different in many aspects as shown by both analysis and

numerical results in [115]. The results for the pressure-driven case from the multi-

T kinetic model are shown in Figs. 5.10-5.11 and 5.13-5.14. The comparison with

force-driven case is best made by considering the profiles of the flow variables in the

118

cross-stream direction measured at the center of the channel x = 0, see Figs. 5.10-

5.11. In force-driven case T/T0 > 1 (Fig. 5.8, right) due to viscous heating, but in

pressure-driven case T/T0 < 1 (Fig. 5.11, left) since the viscous heating is surpassed by

the expansion cooling. Although the Navier-Stokes equations give opposite-curvature

pressure profile compared with DSMC data, see Fig. 5.12, the agreement between the

results from the multi-T kinetic model and DSMC is quite good (Fig. 5.11, right),

which again clearly shows the multi-T model has the advantage over the Navier-Stokes

equations in capturing the non-equilibrium physical phenomena in the near continuum

flow regime. The distributions of flow variables along the stream-wise direction also

agree well with DSMC results as shown in Figs. 5.13-5.14.

(3). Force-driven Poiseuille flow at various Knudsen numbers

The force-driven Poiseuille flow at different Knudsen numbers is calculated using the

multi-T kinetic model. The accurate analysis of the problem has been carried out by

many authors [92]. One of the benchmark results is the solution of the Boltzmann equa-

tion, which has been obtained by Ohwada et al. [84] for hard-sphere (HS) molecules.

In our computation, the working gas is argon with molecular mass m = 6.63 × 10−26

kg. The dynamical viscosity coefficient for HS gas is µ = 2.117× 10−5√T/273Ns/m2.

The mean free path is defined as

l0 =16

5

(1

2πRT

)1/2µ

ρ0, (5.17)

where R is the gas constant, and T and ρ0 are temperature and density, respectively.

The density ρ0 has a value corresponding the pressure of 1 atm and T = 273K. The

accommodation coefficients are taken as σ = 1 for both walls, and the effective viscosity

method with the generalized slip boundary condition proposed by Guo et al. [120],

i.e. Eq.(5.14) and Eq.(5.16), are employed in our numerical simulation.

The velocity distributions for both the linearized Boltzmann equation by Ohwada

et al. [84] and the multi-T kinetic model at different Knudsen numbers are shown in

Fig. 5.15. For this problem, the nondimensional velocity is defined as U/Ua, where

119

Ua is the mean velocity across the channel. As expected, the discrepancy between

our predictions and the solution of the linearized Boltzmann equation increases with

the Knudsen number. However, our numerical results are comparable with those by

the wall-function approach in [119], which has a significant improvement over the

conventional Navier-Stokes slip-flow solution. The normalized mass flowrate is shown

in Fig. 5.16, from which we can see that the well-known Knudsen’s minimum is well

captured by the kinetic model.

(4). 2D cavity flow at Kn = 0.1

The 2D cavity flow will be simulated here. Fig. 5.17 shows the schematic graph for

this case: the upper wall keeps the temperature Tw = 273K and moves to the right

along positive x direction with the velocity Uw, and other three fixed walls keep the

temperature 2Tw = 2× 273K. The working gas is argon with the dynamical viscosity

coefficient

µ = 2.117 × 10−5

(T

Tw

)ω

, (5.18)

where ω = 0.81. Initially, the gas is static with temperature Tw and pressure 1atm (or

101325Pa), and the density can be obtained from the equation of state for perfect gas

p = ρRT , where R is the gas constant defined by R = k/m with Boltzmann constant

k and molecular mass m. Our numerical results are compared with DSMC solutions

from the variable hard sphere (VHS) model. For this model, the mean free path of

the gas can be calculated from

λ =4(7 − 2ω)(5 − 2ω)

30√π

× µ

ρc, (5.19)

where c is the most probable speed c =√

2RT . From the initial condition, we can get

the corresponding mean free path λ0 = 4.82× 10−8m, then the side length of cavity is

determined by L = λ0/Kn with a given Knudsen number Kn. Here we let Kn = 0.1.

The velocity Uw of the upper moving wall is chosen so that the corresponding Mach

number Ma = Uw/√γRTw takes the value of 0.3, where γ = 5/3.

The numerical results from both the current multi-T kinetic model and DSMC are

120

shown in Figs. 5.18-22. From the temperature contours in Figs. 5.18-21, we can see

the agreement between the results by multi-T model and DSMC solutions is satis-

factory. In this case, the different temperature (Tx, Ty, Tz) distributions are clearly

distinguished, especially for the temperature Tx and Ty. The velocity distributions are

shown in Fig. 5.22. Here even the Knudsen number is not large (Kn = 0.1), some

discrepancy between the results from multi-T model and DSMC is still considerable,

especially for the velocity V/Uw distributions along the horizontal line at system cen-

ter y = L/2 (the right, Fig. 5.22), which indicates that the flow structure is quite

complicated for this case due to the large shear and temperature gradient inside such

a microscale system.

5.6 Conclusions

In this chapter, a gas-kinetic model with multi-translational temperature is proposed

and its corresponding Navier-Stokes equations up to the first order of Kn are derived.

The difference between the current kinetic model and the ES-BGK mode of Holway is

explicitly pointed out. In the corresponding macroscopic governing equations of the

current multi-T kinetic model, the assumption between stress and strain in the stan-

dard Navier-Stokes equations is replaced by the temperature relaxation term. Based

on the numerical examples, it becomes evident that besides modeling slip boundary

condition, for the near continuum flow some basic assumptions in the Navier-Stokes

equations need to be modified. The current model presents such a step to go beyond

the Navier-Stokes formulation.

Numerical experiments demonstrate the validity of the current kinetic method in

microscale flow simulations. In the continuum flow regime, the Navier-Stokes solutions

are precisely recovered from the current model. It is clearly shown that the multi-T

model has the advantage over the standard Navier-Stokes equations in capturing the

non-equilibrium physical phenomena as the Knudsen number increases. The current

121

Figure 5.1: Couette flow.

multiple temperature kinetic model and its numerical method provide an alternative

effective tool for the study of microscale flows, where the DSMC method can be very

expensive.

122

x/L

U/U

uppe

r

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

BGK-3T

Exact

x/L

T/T

uppe

r

0 0.25 0.5 0.75 11

1.5

2

2.5

3

BGK-3T

Exact

Figure 5.2: Velocity U/Uupper (left) and temperature T/Tupper (right) distributions in

high speed Couette flow case for a gas with Pr = 2/3, µ ∼ T , and Kn = 0.001, where

the up-plate is moving with a speed of Ma = 3.0 and the lower boundary is adiabatic.

The circles are analytic Navier-Stokes solutions provided in [94], and the solid lines are

simulation results from the current multi-temperature model. Multiple temperatures

are plotted in the above figure (right).

x/L

u/U

w

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

BGK-3T

DSMC

x/L

T/T

w

0 0.25 0.5 0.75 11

1.025

1.05

1.075

1.1

BGK-3T

DSMC

Figure 5.3: Velocity u/Uw (left) and temperature T/Tw (right) for a gas with Pr =

0.68, µ ∼√T , and Kn = 0.01, where the up-plate is moving with a speed of Uw =

300m/s. Both boundaries are isothermal with a temperature Tw = 273K. The circles

are DSMC solutions, and the solid lines are simulation results from the current multi-

temperature model.

123

x/L

u/U

w

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

BGK-3T

DSMC

x/L

T/T

w

0 0.25 0.5 0.75 11

1.025

1.05

1.075

1.1

BGK-3T

DSMC

Figure 5.4: Velocity u/Uw (left) and temperature T/Tw (right) for a gas withKn = 0.1.

The circles are DSMC solutions, and the solid lines are simulation results from the

current multi-temperature model. In terms of the temperature distributions, the up

one is Ty, the middle one is Tz, and the low one is Tx.

x/L

u/U

w

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

BGK-3T

DSMC

x/L

T/T

w

0 0.25 0.5 0.75 11

1.05

1.1

1.15

1.2

BGK-3T

DSMC

Figure 5.5: Velocity u/Uw (left) and temperature T/Tw (right) for a gas withKn = 0.5.

The circles are DSMC solutions, and the solid lines are results from the current multi-

temperature model. In terms of the temperature distributions, the up one is Ty, the

middle one is Tz, and the low one is Tx.

124

Figure 5.6: Force-driven Poiseuille flow (left) and pressure-driven Poiseuille flow

(right).

y

ρ

-1000 -500 0 500 10001.20E-03

1.21E-03

1.22E-03

1.23E-03

1.24E-03

BGK-3TDSMC

y

U

-1000 -500 0 500 10000.1

0.2

0.3

0.4

0.5

BGK-3TDSMC

Figure 5.7: External force driven Poiseuille flow at Kn = 0.1 [114]. Density (left)

and velocity (right) distribution along the channel cross section, where the circles are

DSMC solution.

125

y

P

-1000 -500 0 500 1000

6.26E-04

6.28E-04

6.30E-04

6.32E-04

6.34E-04

6.36E-04

BGK-3TDSMC

y

T,T

x,T

y,T

z

-1000 -500 0 500 1000

1.03

1.04

1.05

T, BGK-3TTx, BGK-3TTy, BGK-3TTz, BGK-3TT, DSMC

Figure 5.8: External force driven Poiseuille flow at Kn = 0.1 [114]. Pressure (left) and

multiple temperatures (right) distributions, where the circles are DSMC solutions.

Both the curved pressure and the temperature minimum have been recovered from

the multi-T model. Solid line on the figure (right) is the averaged temperature, i.e.,

T = (Tx + Ty + Tz)/3.

y

P

-1000 -500 0 500 1000

6.26E-04

6.28E-04

6.30E-04

6.32E-04

6.34E-04

6.36E-04

BGK-NSDSMC

y

T

-1000 -500 0 500 1000

1.03

1.04

1.05

BGK-NSDSMC

Figure 5.9: External force driven Poiseuille flow at Kn = 0.1 [114]. Pressure (left) and

temperature (right) distributions from BGK-NS method [107], where the circles are

DSMC solutions. The BGK-NS method basically cannot capture the non-equilibrium

effect in the Kn = 0.1 Poiseuille flow.

126

−1000 −800 −600 −400 −200 0 200 400 600 800 10001.3

1.35

1.4

1.45density

y

ρ

−1000 −800 −600 −400 −200 0 200 400 600 800 10000.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5velocity

y

U

Figure 5.10: Nondimensional density (left) and velocity (right) profiles in the cross-

stream direction at x = 0, solid line is multi-T model solution and circle is DSMC

data [115].

−1000 −800 −600 −400 −200 0 200 400 600 800 10000.94

0.95

0.96

0.97

0.98

0.99

1temperature

y

T, T

x, T

y, T

z

T, BGK−3T

Tx, BGK−3T

Ty, BGK−3T

Tz, BGK−3T

T, DSMC

−1000 −800 −600 −400 −200 0 200 400 600 800 10001.09

1.095

1.1

1.105

1.11

1.115

1.12pressure

y

P

Figure 5.11: Nondimensional temperature (left) and pressure (right) profiles in the

cross-stream direction at x = 0, solid line is multi-T model solution and circle is

DSMC data [115]. The averaged temperature T on the left figure is defined by T =

(Tx + Ty + Tz)/3.

127

−1000 −800 −600 −400 −200 0 200 400 600 800 10001.09

1.095

1.1

1.105

1.11

1.115

1.12pressure

y

P

Figure 5.12: Nondimensional pressure profile in the cross-stream direction at x = 0,

solid line is BGK-NS solution and circle is DSMC data [115].

−3000 −2000 −1000 0 1000 2000 3000

0.8

1

1.2

1.4

1.6

1.8

2density

x

ρ

−3000 −2000 −1000 0 1000 2000 30000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8velocity

x

U

Figure 5.13: Nondimensional density (left) and velocity (right) profiles in the stream-

wise direction at y = 0, solid line is multi-T model solution and circle is DSMC data

[115].

128

−3000 −2000 −1000 0 1000 2000 30000.75

0.8

0.85

0.9

0.95

1

1.05

1.1temperature

x

T, T

x, T

y, T

z

T, BGK−3T

Tx, BGK−3T

Ty, BGK−3T

Tz, BGK−3T

T, DSMC

−3000 −2000 −1000 0 1000 2000 30000.4

0.6

0.8

1

1.2

1.4

1.6

1.8pressure

x

P

Figure 5.14: Nondimensional temperature (left) and pressure (right) profiles in the

stream-wise direction at y = 0, solid line is multi-T model solution and circle is

DSMC data [115]. The averaged temperature T on the left figure is defined by T =

(Tx + Ty + Tz)/3.

129

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

y/H

U/U

a

Kn=0.113

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

y/H

U/U

a

Kn=0.226

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

y/H

U/U

a

Kn=0.451

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

y/H

U/U

a

Kn=0.677

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

y/H

U/U

a

Kn=0.903

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

y/H

U/U

a

Kn=1.13

Figure 5.15: Nondimensional velocity profiles for force-driven Poiseuille flow at Knud-

sen numbers of 0.113, 0.226, 0.451, 0.677, 0.903, and 1.13. Circle is the solution of

the linearized Boltzmann equation by Ohwada et al. [84], and solid line is from the

multi-T kinetic model.

130

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Kn

Q

Ohwada et al.

Multi−T model

Figure 5.16: Nondimensional flow rate Q as a function of the Knudsen number for

force-driven Poiseuille flow. Circle is the solution of the linearized Boltzmann equation

by Ohwada et al. [84], and solid line is from the multi-T kinetic model.

y

x

TwUw

2Tw

2Tw 2Tw

Figure 5.17: Cavity flow.

131

x/L

y/L

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x/L

y/L

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 5.18: Contours of the averaged temperature T , i.e. T = (Tx + Ty + Tz)/3, by

DSMC (left) and the current multi-T model (right). In both figures, 9 equally spaced

contours from T = 350K to T = 530K are plotted.

x/L

y/L

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x/L

y/L

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 5.19: Contours of the temperature Tx by DSMC (left) and the current multi-

T model (right). In both figures, 9 equally spaced contours from Tx = 350K to

Tx = 530K are plotted.

132

x/L

y/L

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x/L

y/L

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 5.20: Contours of the temperature Ty by DSMC (left) and the current multi-

T model (right). In both figures, 9 equally spaced contours from Ty = 350K to

Ty = 530K are plotted.

x/L

y/L

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x/L

y/L

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Figure 5.21: Contours of the temperature Tz by DSMC (left) and the current multi-

T model (right). In both figures, 9 equally spaced contours from Tz = 350K to

Tz = 530K are plotted.

133

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

y/L

U/U

w

BGK−3T

DSMC

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

x/L

V/U

w

BGK−3T

DSMC

Figure 5.22: Velocity distributions. The left is the velocity U/Uw distributions along

the vertical line at system center x = L/2; the right is the velocity V/Uw distributions

along the horizontal line at system center y = L/2.

134

Chapter 6

Summary and conclusions

This thesis reports on some progresses in investigating the gas-kinetic approaches

for numerical simulations of viscous fluid flows. In the continuum regime where the

Navier-Stokes equations are accurate enough to capture the fluid physics, high-order

numerical schemes for the NS equations are strongly desired to effectively resolve

complex flow structures. We carried out some study in this direction and proposed

a high-order gas-kinetic RKDG method for the Navier-Stokes equations. In the near

continuum regime where the NS equations are known to be inadequate, a multiscale

gas-kinetic approach based on the generalized particle collision time is introduced and

numerically tested. The microscale fluid flow is a rapidly growing research field being

driven by microsystems technology, a multiple translational temperature kinetic model

has been presented for numerically simulating these microscale gas flows. The major

conclusions of these studies are given as follows:

(1) The gas-kinetic RKDG finite element method newly developed for Navier-Stokes

equations combines the convective and dissipative effects together in a single gas dis-

tribution function, which includes both discontinuous and continuous representation

in the flux evaluation at a cell interface. The method is tested by many numerical

examples including both 1D and 2D flow problems. It has been shown that the higher-

135

order P 2 scheme does give a more accurate solution than that from the lower-order P 1

scheme, especially in the well-resolved cases. Numerical experiments also show that

the gas-kinetic RKDG method has a good shock capturing capacity.

(2) In the multiscale gas-kinetic approach proposed for continuum and near con-

tinuum flows, based on the closed solution to the BGK equation, the local particle

collision time is generalized to depend on both the local macroscopic flow variables and

their derivatives. In the continuum regime, the generalized particle collision time goes

back to the definition from classical kinetic theory automatically. Numerical examples

demonstrate the validity and efficiency of the current method in both continuum and

near continuum regimes.

(3) For the multiple translational temperature kinetic model, the corresponding

macroscopic governing equations up to O(Kn) are presented. In the newly derived

macroscopic governing equations, the constitutive relation in the standard Navier-

Stokes equations is replaced by the temperature relaxation terms. Numerical experi-

ments show that in the continuum flow regime, the standard Navier-Stokes solutions

are precisely recovered from the current model. For simulating the microscale gas

flows, the current multiple temperature kinetic model has some obvious advantages

over the standard Navier-Stokes equations for capturing many nonequilibrium physical

phenomena.

Generally, the gas-kinetic BGK method is a very useful and powerful tool to study

the fluid flows. In the continuum regime, the gas-kinetic BGK scheme provides a

consistent way from both mathematical and physical views to accurately solve the

Navier-Stokes equations. In the flow regime where new mathematical models and

the corresponding numerical methods are needed, some modifications to the original

method can be made in order to capture these nonequilibrium physical phenomena.

Specifically, several aspects related to the work presented in this thesis can be further

studied in the future:

136

(1) For the gas-kinetic RKDG finite element method, first, new ways to construct

the relaxation parameter in the hybrid distribution function may be required to make

the formulation problem-independent. Second, the extension of the current gas-kinetic

RKDG method to three-dimensional flow problems is straight forward. Third, the

current method can be extended on unstructured mesh without evident difficulties.

(2) The limiting procedure of the multiscale approach based on the generalization

of the particle collision time can be further investigated in order to get the optimal

one. Various moments taken to the derivatives of the distribution function in de-

termining the generalized particle collision time can be tested and compared. The

multidimensional instead of splitting version of this method may be developed.

(3) The multiple translational temperature kinetic model can be tested by more

numerical examples in order to get more understanding about this new method, and

it may be extended for the diatomic gases with both translational and rotational

temperatures. The boundary conditions for the current method may be further in-

vestigated since they are so important to the microscale flow problems. The idea of

the generalization of the particle collision time may be implemented into the multiple

temperature model in some ways. Since the current model presents the first step to go

beyond the Navier-Stokes equations by throwing away the single equilibrium temper-

ature assumption, more kinetic models inspired by the current one may be developed

in the future.

137

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