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2D & 3D Unstructured Simulations and Coupling Techniques for

Micro-geometries and Rarefied Gas Flow

by

Hsinchih Frank Liu

B.S., Culture University, Taipei, Taiwan 1987-1991 M.S.,State University of New York, Buffalo, NY, USA 1993-1994

M.ENG., Cornell University, Ithaca, NY, USA 1994-1995 Sc.M., Brown University, Providence, RI, USA 1998

Thesis

Submitted in partial fulfillment o f the requirements for the Degree of Doctor of Philosophy

in the Division of Engineering at Brown University

PROVIDENCE, RHODE ISLAND

May 2000


UMI Number 9987799

Copyright 2000 by Liu, HsinChih Frank

All rights reserved.

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Copyright by Hsinchih Frank Liu, 2000. All Rights Reserved.


This dissertation by Hsinchih Frank Liu is accepted in its present form by the Division of Engineering as satisfying the

dissertation requirement for the degree of Doctor of Philosophy

Date 0*51 i j / 2 . Q C Q ________ ^Prof. George Em Kamiadakis, Director

Recommended to the Graduate Council

Date 5 - U - ^ - O O O ____________ gaT^oSVigC"'V |Prof. Martin Maxey, Reader

m U aDate_____________________________ _____________Prof. Chau-Hsing Sii, Reader

Approved by the Graduate Council

D ate C/W t e n .Peder J. EstrupDean of the Graduate School and Research

i i J .


Abstract of “2D & 3D Unstructured Simulations and Coupling Techniques for Micro

geometries and Rarefied Gas FLow”, by Hsinchih Frank Liu, Ph.D., Brown University,

May 2000

The direct simulation of Monte Carlo methods for 2D and 3D using triangular and tetrahe

dral meshes are developed in this research work for microflow and rarefied gas flow. A set

of optimized algorithms are considered and applied to improve the computational time and

accuracy when unstructured meshes are used. A couple of the validations for the internal

flow and external flow are also accomplished in the 2D and 3D unstructured domains,

including Couette flow, pressure-driven pipe flow, and external sphere case. A series of

applications with complex geometry are also demonstrated by utilizing different boundary

conditions such as riblets with moving wall, pressure-driven channel flow with curvatures

(rough channel) and 3D conduit flow (sudden expansion flow). The research work on the

spinning rotor gauge (SRG) project is also presented. This project covers the entire Kn

range including the continuum and rarefied gas flow simulated under a complicated geom

etry environment such as a sphere in a pipe. In SRG project, we demonstrate a drag com

putation and consider the effect o f Kn, Re, and Ma. The blockage effect is also studied in

the SRG project. With the combination o f the developed unstructured DSMC and the

spectral element algorithm used in the SRG project, one is capable of handling the gas

flow within any Kn range in an arbitrary geometry. The fourth piece in this research work

includes the development of various coupling techniques between the continuum NS

solver and DSMC. The coupling techniques can be used in the continuum break-down

which occurs inside o f a domain, or during the phase change, etc. A successful coupling

case handled by the spectral element method with slip model and unstructured DSMC is

demonstrated. Their coupling and iteration process are also studied in this research work.

Specifically, the coupling techniques developed in this work include the flux properties

control, the particle capture algorithm, and the artificial moving wall algorithm.


Acknowledgements

I would like to thank my advisor, George Em Kamiadakis, for his continuos encourage

ments, supports and advices for this challenging research work, and especially the strength

and momentum I got from him for accomplishing the projects. I also thank him for intro

ducing me into this promising and interesting research topic, microflows. I would like to

thank my thesis readers, Martin Maxey and Chau-Hsing Su for their valuable suggestions

about this thesis work. I also thank them for their inspiring teaching and discussions to

stimulate my research ideas. I would like to thank Nikolaos Gatsonis for his tremendous

supports and advices while I stayed in his lab. I also thank him for many precious discus

sions in F3 code. I would like to thank Ali Beskok for sharing his expertise in microflow

with me and providing mu-flow code in projects. I would like to thank Ron Henderson

and David Newman for providing PRISM and the help from Constantinos Evangelinos for

the drag computation. I would like to thank Kenny Breuer for taking the time to give me

his feedback on this thesis draft. I would like to thank Robert Hurt, William Wolovich,

Virginia Novak, and Joseph Calo for their academic advices during the course o f my

studying at Brown. I also thank Virginia Novak for giving me her helps and encourage

ments through this study. I would like to thank Margaret Gidley for her kindness and tak

ing care of me in past 5 years. I would like to thank my sister and brother, Janet and

James, for their endless loves and supports.


To my mother, father, and grandmothers

vi


Contents

1. Introduction 1

1.1 Introduction 1

1.2 Outline 1

1.3 Objectives 2

2. Classification and Approaches of the Gas Flow 4

2.1 Role of the Knudsen Number 4

2.2 Is DSMC Trustworthy? 5

2.3 Comparison Between DSMC and Molecular Dynamics 7

2.4 Structure of the DSMC Method 8

2.5 Bibliography 9

3. 2D Unstructured Direct Simulation of Monte Carlo Method 10

3.1 Unstructured DSMC Procedure 10

3.2 Group Establishment 13

3.3 Particles Initiation 14

3.4 Index Establishment 14

3.5 Flux Properties on the Boundary 14

3.6 Collision Sampling 16

3.7 Periodic Condition 17

3.8 From Microscopic to Macroscopic Properties 18

vii


3.9 Simulation and Application 19

3.9.1 Box Case 19

3.9.2 Pressure-driven Channel Flow Case 20

3.9.3 Couette Flow Case 20

3.9.4 Riblets Case 21

3.10 Bibliography 27

4. Flow Simulations and Models in the Spinning Rotor Gauge 28

4.1 Introduction 28

4.2 Simulation Result Using F3 29

4.2.0 Brief Introduction for F3 29

4.2.1 Validation of F3 30

4.2.2 Computational Domain for Sphere-in-a-pipe 33

4.2.3 Simulation Results for a Sphere in a Pipe 36

4.3 Simulation Result from PRISM 43

4.3.1 Introduction 43

4.3.2 A Brief o f the Galerkin Formulation and the Spectral Element

Basis Functions 43

4.3.3 Validation of Prism 47

4.3.4 Simulated Cd for the Sphere in a Pipe Using Stokes Equation 50

4.3.5 Simulated Cd for the Sphere in a Pipe Using NS Equation 52

4.4 Simulation Result from (l Flow 56

4.4.1 Formulation for 2-D Compressible Navier-Stokes Equation 56

4.4.2 Simulation with Slip Model 57

4.5 Conclusions for SRG 60

4.6 Bibliography 61

viii


5. Coupling Techniques 63

5.1 Introduction 63

5.2 Coupling Issues 64

5.3 Control o f the Flux Properties 65

5.4 NS and DSMC Coupling 69

5.5 Particle Capture Algorithm 76

5.6 Artificial Moving Wall Algorithm 85

5.7 Bibliography 89

6. 3D Unstructured Direct Simulation of Monte Carlo Method 6 90

6.1 Introduction 90

6.2 Unstructured DSMC Procedure 91

6.3 Particles Initiation 91

6.4 Flux Properties on the Boundary 92

6.5 Validation and Application 93

6.5.1 Box Case 93

6.5.2 Pipe Flow Case 93

6.5.3 Sphere in the External Flow 94

6.5.4 Applications 95

6.6 Bibliography 114

7. Summary 115

ix


List of Figures

2.1 Categories of gas flow and the approaches 6

3.1 Flow chart of the unstructured DSMC procedure 12

3 .2 Contour profiles of the box case. Using 2D unstructured DSMC. 22

3.3 Contour profiles of the channel flow. Using the 2D unstructured DSMC. 23

3.4 Contour profiles of 2D Couette flow with wall velocity 300 m/s and

periodic condition. 24

3.5 Physical properties of the cross section in the 2D Couette flow. 25

3.6 Contour profile of the riblets with a moving wall. This simulation is using

2D unstructured DSMC with moving wall condition and periodic condition. 26

4.1 Cd profile of the sphere case by using F3 with different speed ratios 31

4.2 Drag of the sphere case by using F3 with different speed ratios when

Kn=3.5 32

4.3 Velocity contour of a sphere in a pipe case with a extension region by

using F3 34

4.4 Density contour of a sphere in a pipe case with a extension region by

using F3 35

4.5 Cd profile with different H/D by using F3 when Kn=l and Kn=3.5 39

4.6 Cd ratio with different H/D by using F3 when Kn=l and Kn=3.5 42

4.7 Mesh of the simulation domain in Prism 46

x


4.8 Velocity-u contour of the Stokes flow past a sphere 48

4.9 Velocity-v contour of the Stokes flow past a sphere 48

4.10 Pressure contour of the Stokes flow past a sphere 49

4.11 Cd profile of the Stokes flow past a sphere in a pipe with different H/D

and different Re. 51

4.12 Contour of the velocity-u in a sphere in a pipe case 52

4.13 Cd profile with the different H/D in a sphere in a pipe case 54

4.14 Cd ratio profile with the different H/D in a sphere in a pipe case.

(Navier-Stokes flow. Re = 0.25) 55

4.15 A profile of the velocity vector near the sphere, by using p. flow with

slip model. 58

4.16 Cd profile with different Kn by using p flow 59

5.1 Extracted profile from the inlet of the channel flow case by using the

parabolic distribution of the flux property 67

5.2 An arbitrary shape of the flux distribution applied in the inlet 68

5.3 A combination of approaches in one domain 69

5.4 NS solver and DSMC coupling with an overlapping area 70

5.5 Contour profile of the velocity-U in a coupled domain by using the 2D

unstructured DSMC and p flow. 73

5.6 Extracted velocity profile from Fig 5.5 in the coupled domain 74

5.7 A comparison in the interface from the coupled domain. The velocity-u is

extracted from Fig 5.5 75

5.8 Infinite possible combinations exist in the information exchange process 79

xi


5.9 An illustration o f the combination of PC A and BCCP 80

5.10 Contours o f the channel flow with curvatures (rough channel) 81

5.11 Contour profiles of the coupled domain by different approaches: BCCP

only and BCCP+PCA 82

5.12 Comparison from the extracted data (center line) between BCCP and

BCCP+PCA 83

5.13 Comparison from the extracted data (top line) between BCCP and

BCCP+PCA 84

5.14 Comparison of the physical properties between coupled domain and the

entire domain. Single interface is applied in this case. 87

5.15 Comparison of the physical properties between coupled domain and the

entire domain with two interfaces. 88

6.1 Contour of the physical properties by using 3D unstructured DSMC in

the box case 97

6.2 Velocity-u contour in the pipe flow and the normalized velocity-u

comparison with analytical solution 98

6.3 Velocity-v and velocity-w contours in the 3D pipe flow 99

6.4 Temperature and density contours in the 3D pipe flow 100

6.5 Velocity scaling in the 3d pipe flow 101

6.6 Normalized mass flow rate comparison with the models in the 3D pipe

flow 102

6.7 Velocity-u, Kn and temperature contours in the 3D sphere external flow 103

6.8 Cd of the sphere case in the varied Ma range 104

xii


6.9

6.10

6.11

6.12

6.13

6.14

6.15

6.16

6.17

6.18

Contours o f the physical properties by using 3D unstructured DSMC in

the Couette flow case with specular condition and periodic condition 105

Contours o f the velocity-U and the meshes in the computational domain

by using 3D unstructured DSMC in the square-pipe with a bump case 106

Contours of the velocity-V and the velocity-W by using 3D unstructured

DSMC in the square-pipe-with-a-bump case 107

Contours of the density and the temperature by using 3D unstructured

DSMC in the square-pipe-with-a-bump case 108

Contours o f the sphere-pipe case 109

Mesh structure of the 3D conduit flow. The symmetric condition is applied

in the central cross section of the conduit 110

Velocity-u contour of the 3d conduit flow 111

Eddy center and separation point in the 3D conduit flow 111

Comparison between bubble length and bubble center. The lines shows

the difference between continuum flow and rarefied gas flow. 112

Extracted profile from 3D conduit flow. 5 sets of data are presented

from different locations. 113

xiii


Chapter 1

Introduction

1.1 Introduction

The Micro-geometry and rarefied gas flow related problem is the one of the most

promising and important topics in the next decades. Micro (or nano) technology has

become the main science for the future of many electrical and mechanical systems. At the

same time, the difficulty to accomplish the experiments in the micro scale makes the

computational simulation a powerful and economical approach. This research work

emphasizes on developing the 2D & 3D unstructured algorithm and coupling techniques

for the simulation of the microflow and rarefied gas flow.

1.2 Outline

The two- and three-dimensional unstructured Direct Simulation of Monte Carlo method

(DSMC) are developed in this thesis. The unstructured grids can deal with any arbitrary

geometric problem and therefore enables Monte Carlo approach to be applied to the

engineering practices. In chapter 2, the general concepts about DSMC will be discussed.

The unstructured algorithms will be demonstrated in chapter 3.


The simulation results in the Spinning Rotor Gauge (SRG) project will be shown in

chapter 4. The SRG Project covers the entire range of the Knudsen number (Kn) with

geometric complexity. Three methods are used to conquer the difficulties: incompressible

axisymmetric spectral element method, compressible axisymmetric spectral element

method with slip models, and 3D DSMC.

In the continuum break-down region and multiple phase flow, developing the coupling

techniques becomes a necessary approach. On the other hand, since the DSMC method

consumes more computational time, applying coupling techniques between DSMC and

NS solver becomes an economic way. A couple of algorithms are developed in this thesis,

including Artificial Moving Wall (AMW), Particle Capture Algorithm (PCA), Boundary

Condition Control Process (BCCP), and interface coupling by controlling number flux.

All of them are valid for both structured and unstructured grids. Moreover, AMW, PCA,

and BCCP all guarantee the convergence in the simulation results.

1.3 Objectives

. Construct and develop 2D unstructured DSMC procedures and algorithms

. Verify and demonstrate the 2D Unstructured results

. Develop special boundary conditions such as periodic condition in 2D simulation

. Modify 2D unstructured procedures and algorithms for connecting to 3D

unstructured DSMC

. Construct and develop 3D unstructured DSMC procedures and algorithms

2


. Verify and demonstrate the 3D unstructured results

. Develop special boundary conditions such as periodic condition in 3D simulation

. Verify and Identify “F3” with different Ma numbers.

. Identify “F3” by extra free stream buffer for low Re number.

. Verify and Identify “Prism” with different Re numbers.

. Verify and Identify “Mu flow” with different Re numbers.

. Construct project process in SRG based on the Knudsen number range and

discuss conclusions

. Construct the Artificial Moving Wall (AMW) algorithm.

. Match and verify AMW in ID structured and 2D unstructured couette flow.

. Construct interface coupling by controlling number flux

. Match and apply parabolic number flux distribution in the inlet channel flow

. Construct the Boundary Condition Control Process (BCCP)

. Match “Mu Flow” and 2D unstructured DSMC in the channel flow

. Demonstrate the “updating simultaneously NS solver and DSMC procedure”

. Construct the Particle Capture Algorithm (PCA).

. Demonstrate and match an arbitrary geometry case, channel with curvatures,

by full DSMC, BCCP DSMC and PCA+BCCP DSMC

3


Chapter 2

Classification and Approaches of the Gas Flow

2.1 Role of the Knudsen Number

Both global and local knudsen number are important indicators in the rarefied flow

simulation (Fig. 2.1). When Kn is very small, the traditional assumption about the

continuum is sustained. If Kn is located between 0.001 and 0.1, the slip effect on the solid

body needs to be considered in the continuum region. When Kn is bigger than 0.1, the gas

flow is considered to be within the break-down region. More details will be discussed in

the chapter 4.

The global knudsen number is defined as Kn = X /L , where X is the mean free path and

L defines the characteristic length. Any length in the simulation domain can be chosen as

L, and the free stream condition can be used to determine the mean free path. For the local

knudsen number, L is chosen from the scale length of the macroscopic gradients, L = q /

(dq/dx). Both global and local Kn numbers will not involve in the computational process.


The local knudsen number is computed from the local physical properties. After

simulation, the local Kn number becomes an important indicator to remove the

discrepancy between the different choices of the characteristic lengths.

2.2 Is DSMC Trustworthy?

DSMC has been a powerful tool to simulate the rarefied gas flow and micro-geometries

which is in the bigger Knudsen number [2.1 ] range. DSMC also has been approved by

many scientists as a very trustworthy numerical method for verifying the simulation

results. The latest published papers and algorithms devoted to the bigger Kn range are

using the DSMC for the verification and comparison [2.2].

Most of the numerical methods such as finite difference and finite element methods are

computing the physical properties on the grid edges in the continuum region. The

boundary conditions and the specific data can be directly assigned to the edges. These

numerical methods with “direct assignment” characteristic have the advantages of setting

the boundary conditions easily, but will require lengthy verifications from many

theoretical and experimental data.

By contrast, the nature of DSMC requires the probabilistic assignments. The physical

data is coming from integrating all o f probabilistic particles in each cell not from the grid

edges. Any specific or deterministic assignment will bring down the accuracy. Unless

procedures and algorithms are correct, DSMC will not produce a correct result by using

any deterministic or direct assignment. Therefore, DSMC is a trustworthy method, but

5


setting boundary conditions becomes a difficult task.

12/97 by Frank Liu

(1) The Knudsen number limits on the mathematical models

DISCRETE PARTICLE OR MOLECULAR

MODEL

BOLTZMANN EQUATIONCOLLIS ION LES SBOLTZMANNEQUATION

CONTINUUMMODEL

EULEREQS.

NAVIER-STOKESEQUATIONS

CONSERVATION EQUATIONS DO NOT FORM A

CLOSED SET

LOCALKNUDSENNUMBER

-a/V-0.01 0.1 10

INVISCIDL iM rr

COFREE-MOLECULE

LIMIT

(2) The classification and approach

CLASSIFICATION

LOCALKNUDSENNUMBER

APPROACH

Figure 2.1: Categories of gas flow and the approaches

Continuum Slip Transition Free Molecularregion region region region

0.001 0.1 10

N-S EQS. N-S EQS. DIRECT SIMULATION(No-Slip) (Slip) MONTE CARLO

6


2.3 Comparison Between DSMC and Molecular Dynamics

Direct Simulation Monte Carlo (DSMC'):

(1) The intermolecular collisions are dealt with on a probabilistic basis rather than a

deterministic basis; it also requires an assumption of molecular chaos.

(2) DSMC’s initial condition permits exact specification, like uniform equilibrium flow.

(3) DSMC’s boundary condition are specified by the behavior of the individual molecules

which could be based on probability density function, that causes the difficulty in

controlling boundary condition.

(4) The steady state and the local thermodynamic equilibrium can be achieved in the long

period of computational time.

(5) The cell is used to sample the macroscopic flow properties. The cell size is also related

to the computational time step.

(6) The computational time is directly proportional to the number of simulated molecules.

The collisional pairs will be randomly chosen in the collision stage.

Molecular-Dynamics CMP):

(1) Simultaneously following the trajectories o f a large number of the molecules within a

region of simulated physical space

(2) Probabilistic procedures are required for the initial stage, but all the subsequent

procedures are deterministic.

(3) The trajectory for a particular molecule requires one to consider all other molecules as

the collisional partners within the assumed cut-off radius; the amount of the

computational work is proportional to the square o f the number of the simulated

particles.

7


(4) The collisions between molecules would be dominated by potential energy in the short

range and Coulomb interactions in the long range.

(5) This method have been proved to be a valuable method for the dense gas and liquid but

inappropriate for the dilute gas.

(6) MD can do one-to-one correspondence between real and simulated molecules, this

means that not talcing samples is an available way.

2.4 Structure of the DSMC Method

The required model for the molecular behavior in DSMC primarily comes from the

classical kinetic theory o f gases. The popular model called VHS [2.1 ] will be used in the

thesis. The main procedure in structured DSMC consists of the three parts: particle

collisions, interaction with the solid body, and new particle assignments. The extra

procedure such as cell-group and index procedure will be added onto the unstructured

simulation. Basically, there is no fixed procedure for DSMC.

8


2.5 Bibliography

[2.1 ] Bird G. A., Molecular Gas Dynamics and the Direct Simulation o f Gas

Flows. Clarendon Press, Oxford, 1994.

[2.2] Daniel W. Mackowski, Dimitrios H. Papadopoulos, and Daniel E. Rosner,

Comparison of Burnett and DSMC predictions of pressure distributions and

normal stress in one-dimensional, strongly nonisothermal gases. Physics o f

Fluids, Vol. 11, No. 8, August 1999.

9


Chapter 3

2D Unstructured Direct Simulation of Monte Carlo Method

3.1 Unstructured DSMC Procedure

The flow chart of the procedure is shown at Figure 3.1. Building a group of neighboring

cells for each cell is the first step in this procedure. The radius of the group needs to be

bigger than the particle travelling distance. For each time step, a new index between

particle and cells will be searched only inside of the group range. This will reduce the

computational time. On the other hand, the chances that the particle travels to outside of

the group without being considered as a collisional partner is eliminated. The approach of

building group is a reasonable assumption to simulate the real physical space.

In each time step, a particle moves with a random distance to a new position in the move

stage. If it encounters the wall, the reflectional stage will be executed. The collisional

stage will be carried out in each cell. Particles that goes through the move and reflectional

stages and ends up at outside of the domain will be removed and accumulated for new

assignments. The new particle stage will assign new molecules on the boundary to

10


maintain that the total number of particles is constant. A certain amount of particles need

to be assigned to each cell to assure statistical accuracy. After going through the previous

stages, the sampling stage will take part in to interpret and average the particle properties.

In the beginning of simulation process, the time averaging stage can also used to eliminate

the noise o f the particle motion and smooth the data. After a long period time of running

or in the status of the steady state, it is not necessary to use the time average stage.

11


DSMC procedure for triangular cell

No

No t = t (stop) ?

j Repeat run [ L until steady state j

YesHsinChih Frank Liu

t = t (sample) ?

^ ^ T Y e sSample flow properties

Set sample molecular initial state

Perform intermolecular collisions

Introduce new sample moleaular in the inflow / outflow boundaries

Build a group-cells for each cell Bulid a layer-cells for each boundary cell

Move sample moleculars along their trajectories in each time step, Computing the Multi-reflections with solid boundaries as they occur

Figure 3.1: Flow chart of the unstructured DSMC procedure


3.2 Group Establishment

The first layer of the group will be built by the neighboring cells. The neighboring cells of

the neighboring cells will keep forming the next consecutive layers.

X n ( i ) is defined as the coordinates of the N lh cell's edge. X n( i ) 6 9v, / = 1, 2, 3 .

is defined as the nth cell’s k ‘h layer. Q represents the computational domain.

Definition 3.1 G ^ = {m: 3 j = 1,2,3 , such that X ^ { j ) = X n ( i ) }> n e Q

Definition 3.2 G ^ = G^ . w ^ ^

m e Gn

For the cells that are adjacent to a boundary such as a wall or inlet/outlet condition, the

specific group of the boundary cell will also be built. The particle inside of the boundary

£cell's group will be checked for the possibility of reflection. B is defined as the nth

cell’s group formed by the other boundary cells by k layers. describes the cells

k Qlocated at the boundary. A n is defined as the group formed by the B . Where n denotes

the cell, k and q denote the layers of the group.

k kDefinition 3.3 Bn = {m: m e G n n m e }, n e

Definition 3.4 A ^ = LJ (7 ^ , n e Q , n m om e B „

13


3.3 Particles Initiation

Initially the particle positions are assigned randomly through each cell in the entire

domain. (XI, x2, x3) and (y I, y2, y3) denote the coordinate of a cell edge. X and y

indicate the particle position.

x = X * xl + |i * x 2 + ( l - X - p ) *x3 (3.1)

y = X * y 1 + p. * y2 + (1 - X - p) *y3 (3.2)

X > 0, p > 0 . (X + p) < I (3.3)

where X and p are determined by a random function. If (X + p.) > 1 , X and p need to be

re-sampled. The alternative way is to mirror the X and p. along the (X + p.) = 1 line.

3.4 Index Establishment

For each time step, the index routine is needed for each particle and cell. An easy and fast

algorithm will reduce the computational time such as (3.4).

A , = det ( -r / — x ) (> ,-->•)

( -r i + i - -r ) ( y I + i - y ), i = 1-3 (3.4)

x ( + ! = .r j , v, + , = y , , when (i+ 1) > 3. When A , , A 2 and A3 have the same sign, the

index is built.

3.5 Flux Properties on the Boundary

Flux property assignment plays an important role in the boundary condition control

process (BCCP). In here, the flux properties will be assigned to each cell and can be

14


adjusted by the free stream condition and a probability density function, F(i), i e .

F(i) is consistent with the boundary condition. F(i) can represent a continuous or a

discrete function. Fx(i) and Fy(i) represent the x and y components of F(i). To sample

continuous F(i), the general method such as acceptance-rejection method can be used.

F(i) will be sampled twice in the simulation process: (1) the amount of particle which is

used to inject into each cell, (2) the velocity that each particle carries besides free stream

velocity and thermal velocity. The inward flux of some quantity Q is

J J J Q u f dudvdw . where n is the number density. For the equilibrium gas.

3 3 * * I/ = (P /7 t " )exp(~P c~) , where c is the thermal velocity. P = (m / ( 2 k T )) “ , where

m is the molecular mass.

The inward number flux in each cell N(i) is obtained by setting

Q = i (3.5)

The inward normal momentum flux in each cell P(i) is obtained by setting

Q = m(u + cocos(0) + Fx{i)) (3.6)

The inward translational energy flux q(i) is obtained by setting

Q = ( 1 / 2 ) m c~ . (3.7)

The flux properties are calculated as the following:

15


J (m' + cocos(0) + Fx(i))e.xp(-$~u'~)du’ (3.8)-(c„ cos0 + Fx(i))

3 oo oo

P(i) = nm J exp(-(3~w'2)dw' f e x p (-p 2v '“)Jv '7t

J (m' + cocos(0) + F .r(/))“e x p (-P - M'~)^//' (3.9)-(C„COS0 + Fx(i))

q(i) =

OO OO oo

n- l~ f f f {(// ' + cocos(0) + F x( i ) )2 + (v ' + cosin(0) + F y( i ) )2 + w '2}~>K -«o -oo - ( c„ cos 9 + Fx( i ))

*

(//' + cQcos(0) + F x (/) )e x p { -P “(w'“ + v'~ + w'~)}du 'dv 'dw ' (3.10)

3.6 Collision Sampling

The NTC method and VHS model are used in this work [2.1][3.1][3.2]. The selected pairs

for collision in each cell at each time step are computed by:

( ^ N N F ( a TC r) A t ) / V c (3.11)l 2 n max J

where /V is the average number of molecules in the cell, F n is the real molecules

represented by a single sample particle, Vc is the cell volume. (<yr^ r) ,„ ar W*N

recorded and updated in each time step.

The first particle will be picked randomly in the cell. The second particle will be picked

16


(<*TCr )randomly inside of the cell’s group range. If ----——----- is bigger than the chosen random

(<JTCr) v ‘ r ' m a x

number, the collision is accepted. The post-collision velocity will be computed and the

rotational energy will be adjusted. Otherwise, the first and the second particle will be re

picked.

In the unstructured grid domain, the cell group has circular-like size with radius Rc. Rc is

bigger than first particle travelling distance. Therefore the range o f the cell group range

covers the collisional possibility between the first particle with other articles. This

situation is similar to the real molecular behavior. As for structured grid. The formed

group has a square-like size which is not as realistic as real particle motion.

3.7 Periodic Condition

Developing an appropriate periodic condition is very important in the Couette flow

simulation. A similar concept can also be used in the coupling techniques. Intuitively, in

the molecular simulation, the particle moving out from the upstream of the domain will be

shifted to the downstream of the domain; the particle moving out from the downstream of

the domain will be shifted to the upstream of the domain.

In order to get accurate results, three critical characteristics need to be addressed. First,

the properties of shifted particles by applying periodic condition must be assigned as

deterministic. This is very different from the regular routines for the new particle

assignment on the boundary, which are assigned with probabilistic properties. Secondly, a

17


new cell group which is close to the domain edge will be built for the multiple-reflection

between the particles and the solid boundaries. £1E describes the wall cells located at the

edge of the domain.

k IrDefinition 3.5 E = KJ B

n o mm e L ie

Thirdly, after the shifting of the particles, the new particle position is located at outside of

the original group. The indices need to be refreshed.

3.8 From Microscopic to Macroscopic Properties

There are a couple of macroscopic properties of interest which are related to transport of

mass, momentum, and energy caused by the particle motion. S denotes the total chemical

species. P denotes the particular species [3.3].

S

(1) Density: p = £ Onpn p) = n m (3.12)p = I

where n: Number density, m: molecular mass

(2) Mass average velocity: c = [ - ) Y ( m nn c n) = ^ (3.13)vpJ *-* p >’ •’ mp= 1

S ___ ______

(3 ) Pressure: P = - £ ^Onpnp)(c2p - c2a) = | n { m c 2 - m c 20) (3.14)p = I

1 i ->(4) Translational kinetic temperature: T [r = - ( m c ‘ - m c ‘ ) (3.15)

where k: Boltzmann constant, ca : mean or stream mass velocity

18


(5) Stress tensor: x tr = —n(m uv - m u0v0) (3.16)

(6) Mean free path: X = -----------J l nK cl"

(3-17)

(7) Knudsen number: Kn = ~ = Const xL Rc (3.18)

where y = — : Specific heat ratio

3.9 Simulation and Application

3.9.1 Box Case

The box is demonstrated in Figure 3.2. The particle is initially distributed randomly in the

box with isolated walls. The temperature of the walls are 273 K. The most probable

speed for each particle is around 340 m/s. No extra forces are applied to the particles. This

case can be used to verify the code since it comes with exact solutions before computation.

Most of the subroutines will be tested including Move stage. Reflection stage. Index stage,

and Collisional stage, etc. The only one subroutine can not be tested is the New particle

assignment stage. The New particle assignment stage can be tested in the pressure-driven

channel flow or in the no pressure-driven channel with specular walls.

The final results of the box case show:

. The average velocity component is zero.

. The flow domain temperature is the same as wall temperature.

19


The average density is uniformly distributed.

3.9.2 Pressure-driven Channel Flow Case

The contours of the physical properties of the pressure-driven channel flow is shown in

Figure 3.3. The Knudsen number in the cross section from the center of the channel is

around 0.1. The velocity is increased along with channel due to the pressure variance.

The density is decreased along with the channel. The slip velocity is proportional to the

Kn and the mean free path. Therefore, the slip velocity at each surface is also increased,

since the lower density and higher mean free path. The total cell number is 4504. For each

cell, two layers o f the group cells are formed. The initial free stream speed is 250 m/s. The

time step is IE-6. Nitrogen is used as particles.

3.9.3 Couette Flow Case

The final results o f Couette case are demonstrated in Figure 3.4 and Figure 3.5. Figure 3.4

shows the contour profile of its physical properties. Figure 3.5 is the extracted data from

Figure 3.4 along the y axis. This case is running with the periodic condition. It is using

the same setup and parameter as [2.1], from page 262 to 264, which use 1-D structured

code, for comparison. The velocity of the moving wall is 300 m/s. The total cell number

is 16384. The total sample particle is 327680. Two layers are used in each cell group.

The time step is 3.5E-6.

The slip of the velocity and temperature at the surfaces are very close to the product of the

Kn with the velocity gradient and the temperature gradient, respectively. The velocity slip

20


is around 3.5 m/s at each surface, the temperature jump is over 0.6 K at each surface.

In the Figure 3.4 and Figure 3.5, the velocity U is linearly distributed. Both temperature

and density are parabolicaily distributed. They matches the results with [1] very well.

The approach of the periodic condition is verified in this case.

3.9.4 Riblets Case

The riblets case, demonstrated in Figure 3.6, is used to demonstrated the unstructured

algorithm and periodic condition applied on an arbitrary geometry. The sharp angle in the

riblet edge can also show the ability of algorithm to handle the multiple reflection between

particle and wall which has the arbitrary angle.

The velocity of the upper wall is 300 m/s. The riblet-size wall is fixed without moving.

The inlet and outlet are using the periodic boundary condition. The total number of

sample particle is 19180. The time step is 5E-7.

21


Box Case (DSMC) 392 Cells

Hsin-Chih Frank Liu 1997

Density

4 9S46E-074 9S2C2E-074 945436 074 9336SE-074 93227E-074 92S69E-074 9191E-074 912S2E-C74 90S 94 E-074 *90366-074 89277E-074 «*619€-074 *7961E-C7> 0 64 *73©3E-074 06644E-O7

0.0

0.6

1 61095 t 387 1 163C5

j 0939092 [ 0 715139 0.491166

| 0-267233 I 0.0432805

I -0.160672 I -0 404625 I -0 626578 | -0*52531

•t 07646 1-1 30044

52439

Velocity - U

!Hi

0 8

> 0 6

0 4

0.2

0-02

O T274 161273 926 273 e92 273 457 273 222 272 997 272 752 272 517 272 2*2 272 04* 271 *13 271 57* 271 3*3 271 10*

[ 270 *73

Temperature0 8

. 0 6

0 936965 0796356 0 65574*0 515139 0 374531 0 233922 0 093314

-0 0472944 -0 1*7903 •0 328511 -0 46912 •0 609728 -0 750337 •0 *90945 •1 03155

Velocity - V

Figure 3.2: Contour profiles o f the box case

22


0.9

0 8

0.7

0.6

0.5

0.4p-

0.3

0.2

0.1

0-0.1

*0.2

2D Unstructured DSMC Channel Flow (Pressure-D riven) Triangular Cells: 4504

HsinChih Frank Liu

Velocity - U

8 236 .263 '

220 .553 | 204 .337 I

— ? 189 122 173 406 157 691 j

141.975 126-26 110-544 i

94 8287 : 79 1132 I 63-3976 ! 47 6821 :

Temperature302.013298.082294 .151

286.289262.358278.427274.496270.565266.634262.703258.772254.841

Density 1.39933E-061.34105E-06128278E-061224SE-061.16622E-061.10794E-06I.04967E-06

9 3 3 1 13E-078.74836E-07

I8.16559E-07

7.00004E-07

Figure 3.3: Contour profiles of channel flow

23


DSMC2D Couette F]ow Periodic Boundary Condition Wall Velocity 300 m/s

Hstn-CfUh Frank Liu 1998

Triangular oeUc: 16384

M B « 7 I

V ododty-V

Temperature Velocity - V

Figure 3.4: Contour profiles of 2D Couette flow

24


DSMC (Unstructured)2D Couette Flow Periodic Boundary Condition Wall Velocity 300 m/s

Velocity - U

U

0 2

0 1 ► 0

-0 t

-02

-03

-04

-05

Temperature

X .

OT

Density

V .

9 16-Ofl 9 26-Ofl 93E-04 9 4E-06 9 SE-06 9 6E-060

Hsin-Chih Frank Liu 1998

Figure 3.5: Physical properties of the cross section in the Couette flow

25


DSMC Moving Wall with Periodic condition Riblets ( 959 Cells )

Moving Wall ( U = 300 )

Periodic 'v- Periodic

HsinChih Frank Liu 1999Fixed Wall ( U = 0 )

Velocity - U I277 £39 28147 249 301 235 t3 t 220 962 206.793 192 624 179 4S5 164 296 ISO 117 135 947 121 778 107 609 934399 7927C6 65.1017 SO 9325 36.7634 22.5942 942S07

22.5065

16290161669

Velocity - V 14CS3711 94059 627297 714065 600663 467661 J7447

-0 736728

7 07633» 191S411 3047

15 5311

300 919299 361297 903296 445Temperature 294 967293 529292.071290 612299 154297 696766 236284 76763 322

260.406779 946277 49276 032274 574273.116

Density I0000131001 0 00011*9570 ooo:i*as«0 00011775 0 00011*447 0 000115443 0 00011* oooeits0 00011223? 0 000111 0 000110025 0 000106021 0000107017 0000106714 000010641 0000104507 0 000103403 0 0001032*0 0 00010*11 0 000100002

Figure 3.6: Contour profile of the riblets with a moving wall

26


3.10 Bibliography

[3.1 ] Bird G. A. Monte Carlo simulation in an engineering context. Progr. Astro.

Aero. 74, 239-255, 1981.

[3.2] Bird G. A. Perception of numerical methods in rarefied gas dynamics. Progr.

Astro, and Aero. 118, 211-226, 1989.

[3.3] Vincenti, W.G. and Kruger, C.H. Introduction to physical gas dynamics,

Wiley, New York, 1965.

27


Chapter 4

Flow Simulations and Models in the Spinning Rotor Gauge

4.1 Introduction

The spinning rotor gauge (SRG) device has been used broadly in industry for measuring

the properties of the rarefied gas flow. This rotor of the SRG is magnetically suspended

without mechanical contact on the axis of a nonmagnetic pipe. The magnets supporting

the suspension of the rotor is properly adjusted to keep the rotor constantly staying on the

axis of the pipe. After the rotor is energized to reach a given rotational speed, the drop of

rotational speed caused by the gas friction is measured as a function o f time. The relevant

experimental studies have been addressed by Fremerey who has first proposed the specific

configuration used in the NIST PC-controlled experiment and shows the SRG is reliable

from the experimental and theoretical point o f view [4.1][4.2][4.3].

Computational results for the gaseous drag force of the spinning rotor gauge are presented

here in full Knudsen number (Kn) ranges from continuum flow to rarefied gas flow. The

study is divided into three categories based on the Knudsen number: (1) the noslip region

28


(Kn < 0.001) is implemented by incompressible axisymmetric code called Prism which

uses spectral element algorithm. (2) the slip region (Kn = 0.001-0.1) is implemented by

compressible axisymmetric code called |I Flow which uses spectral element algorithm

and slip models, (3) the transitional and free molecular region (Kn = 0.1-10) is

implemented by the three-dimensional direct-simulation-monte-carlo code called F3

which is capable of handling arbitrary geometry [4.4][2.1].

Here, we simplify the main part of SRG into a stationary sphere in a pipe. The drag

coefficient of the sphere is calculated by varying the ratio of the diameter o f the pipe (H) to

the diameter of the sphere (D) at the low Reynold number in each Kn category. The main

objective of this work is to investigate the blockage effect in the pipe and to provide a set

of reference data of drag coefficients in a broad Kn range for further SRG applications.

In the following, we first describe high Kn simulations with the code F3. We then

examine the opposite limit of the incompressible flow with the code Prism. Finally, we

present results using the code Flow for the slip/transitional flow region.

4.2 Simulation Result Using F3

4.2.0 Brief Introduction for F3

F3 is the DSMC program developed by Bird. The F3 builds the surface element near the

solid body to handle the arbitrary geometry, besides the regular cells and subcells in the

flow domain [4.5],

29


4.2.1 The Validation of F3

The drag coefficient on sphere is used for verifying the F3 (see the Table 4.1 and Figure

4.1); the drag is shown in Figure 2. F3 results is compared to the following data in five

different speed ratio [4.6][4.7][4.8]:

(1) Bird’s analytical model:

I4 2

2 5 + 1 , „2, 45 + 4 5 -Cd = ; — ~ exp ( 5 ) + ---------- -------1 /2 .3 K 5 25

f t x 2(1 - e )k / 2 ( T w \ 2+ " L— 5 5 ---------( H (4.D

where £ : Fraction of specular reflection, £ = 0 for diffuse reflection

S = Speed ratio, 7* = Wall Temperature, T x = Free Stream Temperature

(2) Baily’s experimental data for subsonic flow in the transitional region,

(3) Kinslow’s experimental data for supersonic flow in the transitional region,

(4) Henderson’s numerical model in the transitional region.

Table 4.1

Case Re Ma S Cd (F3)Cd

(Henders-on)

Cd(Baily)

Cd(Kinslo

w)

1 0.125 0.36 0.3 14.626 12.4 13.8 Na

2 0.2075 0.6 0.5 9.07 8.12 7.9 Na

3 0.3458 1 0.836 5.7445 5.725 5.1 Na

4 1.037 3 2.508 2.892 1.87 Na 1.99

5 1.723 5 4.18 2.504 1.68 Na 1.8

30


Drag Coefficient on Sphere, Kn=3.5

14

F3BirdHendersonBailyKinslow

■Oo

.□__

Speed Ratio (s)

Figure 4.1: Cd of the sphere case by using F3


Drag

Drag on Sphere, Kn=3.50.0007

0.0006

0.0005

0.0004

0.0003

0.0002

0.0001

0 1 2 3 4Speed Ratio (s)

Figure 4.2: Drag of the sphere case by using F3


4.2.2 Computational Domain for Sphere-in-a-pipe

In the transitional region, the mean free path of molecular is much bigger than the cell

size. Therefore the disturbance in the pipe will propagate to the outside region of the pipe,

especially for the subsonic flow with large Knudsen number. Thus, If the length of the

DSMC domain is chosen to be the exact length of the pipe, the error will occur on the inlet

of the pipe. A remedy to avoid this is to generate the larger domain which is much longer

than the length of the pipe, and apply the free stream condition far from the inlet of the

pipe. The bigger domain will not only generate accurate results but also make the inlet

Reynold number proportional to the free stream velocity and density.

Owing to the occurrence of the big gradient o f the velocity and density in the inlet of the

pipe, an extra length is added on the original length of the pipe. The extra length is chosen

to be the three times of the sphere radius. Then one can have the properties in the

reference point which is close to the expected conditions. The distance between the

reference point and the sphere center is the original length of the pipe we want to study.

Figure 4.3 and Figure 4.4 show the local velocity and density for each location in the

sphere-pipe case.

33


( 3 0 ) » S « p 1 H T C ^ r o l p ta t to r F l

1100

> 6 0 0

Free Stream, U = 176

U = 115 U = 49 Inlet, U = 54

Reference inlet point, U = 49.35, Re = 0.125

Sphere in a pipe

Reference outlet, U = 21 Outlet, U = 15.5

Velocity - U170.22 158.133 146.046 133.958121.871 109.784 97.6971 85.61 73.5228 61.4357 49.3485 37.2614 25.174313.0871 2 1

500 1000

Figure 4.3: Velocity contour of a sphere in a pipe case

34


( 3 0 ) 2 J S « p 1 f 7 , E t r t g o t h y r j

1200

1100

1000

> 6 0 0

Free Stream, q = 4.70E-06

q = 4. IE-06 q = 3.20E-06

Inlet, q = 2.86E-06

Reference inlet point, q = 3.38E-06, Re = 0.125

Density - q

Sphere in a pipe

Reference outlet, q = 8.3E-06 Outlet, q = 7.5E-06

8.36464E-067.807E-067.24935E-066.69171E-066.13407E-065.57643E-065.01878E-064.46114E-063.9035E-063.34586E-062.78821 E-062.23057E-061.67293E-061.11529E-065.57643E-07

500 1000

Figure 4.4: Density contour of a sphere in a pipe case

35


4.2.3 Simulation Results for a Sphere in a Pipe

There are four cases simulated in this study. Case (1) represents Kn = 3.5 and H/D = 4.

Case (2) represents Kn = 3.5 and H/D = 10. Case(3) represents Kn = 1 and H/D = 2.

Case(4) represents Kn = 1 and H/D = 4. Table 4.2 illustrates the free stream conditions

utilized in the F3 code for sphere-pipe cases. Table 4.3 and Figure 4.5 show the calculated

drag and Cd for sphere-pipe case. The data o f the reference point in Table 4.3 will be used

as the free stream condition in the sphere-only case. Table 4.4 illustrates the free stream

condition utilized in the sphere-only case. Table 4.5 shows the calculated drag, Cd (in

sphere-only case) and Cd-ratio. Cd-ratio is defined as the Cd of the sphere in the sphere-

pipe case divided by the Cd of the sphere in the sphere-only case; the Cd calculated from

both cases are based on the reference point. Figure 4.6 shows the Cd-ratio for all four

cases. Table 6 shows that the drag is increased by the free stream velocity, with fixed Kn

and H/D.

Table 4.2Case

Number (1) (2) (3) (4)

Kn 3.5 3.5 1 1

H /D 4 10 2 4

Free Stream Velocity in X

0 0 0 0

Free Stream Velocity in Y

0 0 0 0


Table 4.2

CaseNumber (1) (2) (3) (4)

Free Stream Velocity in Z

317 150 176 72

Number of cells in X- division

23 38 20 23

Number of cells in Y- division

23 38 20 23

Number of cells in Z- division

124 84 200 124

Total number of the surface element

8000 11000 11000 8000

Number density of the stream condition

2.94E+19 2.94E+19 1.029E+20 1.029E+20

Number of the real molecular represented by one sample molecular

2 .108E+10 6.7246E+10 1.736E+10 7.379E+10

Total Number of the Molecules per cell

20 20 20 20

Reflectiontype

Diffuse Diffuse Diffuse Diffuse

Minimum x coordinate

-0.028 -0.068 -0.015 -0.028

37


Table 4.2

CaseNumber (1) (2) (3) (4)

Maximum x coordinate

0.028 0.068 0.015 +0.028

Minimum y coordinate

-0.028 -0.068 -0.015 -0.028

Maximum y coordinate

0.028 0.068 0.015 +0.028

Maximum z coordinate

-0.08 -0.08 -0.08 -0.08

Minimum Z coordinate

0.22 0.22 0.22 0.22

Time step 3.0E-6 3.0E-6 3.0E-6 3.0E-6

SurfaceTemperature

273 273 273 273

Table 4.3

CaseNumber (1) (2) (3) (4)

Kn 3.5 3.5 I 1

H/D 4 10 2 4

reference point velocity in Z

140 119 49.35 38

reference point density

1.21E-6 I.39E-6 3.38E-6 4.58E-6

ReferenceRe

0.125 0.125 0.125 0.125

Drag 2.096E-5 2.102E-5 2.6866E-5 1.9969E-5

Cd 13.01 14.696 49.31 44.454

38


60

55

5045

40

35

30

25

20

15

10

5

00 2 4 6 8 10 126HID

Figure 4.5: Cd with different H/D by using F3

Table 4.4

CaseNumber (1) (2) (3) (4)

Number Density of the free stream condition

2.515E+19 2.94E+19 7.027E+19 9.52E+19

Free Stream Velocity in Z

140 120 49.35 38


Table 4.4

CaseNumber (1) (2) (3) (4)

Number of Cells in X division

46 40 46 46

Number of Cells in Y division

46 40 46 46

Number of Cells in Z division

38 40 38 38

Total number of the molecular per cell

20 20 20 20

Minimum X coordinate

-0.069 -0.06 -0.069 -0.069

Maximum X coordinate

0.069 -0.06 +0.069 0.069

Minimum Y coordinate

-0.069 -0.06 -0.069 -0.069

Maximum Y coordinate

0.069 -0.06 +0.069 0.069

Minimum Z coordinate

-0.057 -0.06 -0.057 -0.057

Maximum Z coordinate

0.057 -0.06 0.057 0.057

Number of the real molecular represented by one sample molecular

3.395E+I0 7.938E+10 9.4863E+11 1.285E+11

40


Table 4.4

CaseNumber (1) (2) (3) (4)

Time step 3.0E-6 5.0E-6 3.0E-6 3.0E-6

Surfacetemperature

273 273 273 273

Reflectiontype

Diffuse Diffuse Diffuse Diffuse

Table 4.5

CaseNumber (1) (2) (3) (4)

Drag 2.087E-5 2.057E-5 1.9063E-5 1.938E-5

Cd 12.7 14.626 33.5258 42.422

Cd Ratio 1.024 1.00485 1.602 1.056

Table 4.6

Kn H/D Free Stream Velocity Drag

3.5 4 255 1.588E-5

3.5 4 312 2.042E-5

3.5 4 317 2.096E-5

1.0 4 78 2.19E-5

1.0 4 100 2.736E-5

1.0 2 100 1.787E-5

1.0 2 140 2.34E-5

1.0 2 176 2.6866E-5

41


.75

OCD

DC 1.25*oo

0.75

0.510

H / D

Figure 4.6: Cd ratio with different H/D by using F3


4.3 Simulation Result from PRISM

4.3.1 Introduction

Prism is a incompressible spectral element code which can handle any 2D, axisymmetric

and 3D arbitrary geometry. The Prism code simulates the axisymmetric sphere-pipe case

as shown in Figure 7 [4.9][4.10].

4.3.2 The Galerkin Formulation and the Spectral Element Basis

Functions

The theory of the spectral element basis function is originally derived from the Galerkin

finite element method. A Galerkin finite element method consists o f 3 steps: (i) deriving

the equivalent weak form from the strong form of a well-posed boundary value problem,

(ii) approximating the weak form with a finite collection of functions, and (iii) solving the

matrix to determine the basis coefficients. The key to the success o f the Galerkin method

is the selection of the basis functions. A good choice of basis functions can reduce the

complexity of the matrix to be computable and without losing the accuracy [4.9].

One can consider a problem: u " + f = 0 on Q , where Q. is the unit interval 0 < x < 1

and f: [0 , 1 ] —> , is a given smooth function. The boundary conditions are:

m(0) = g , u ' ( l ) = h . Then one will have the residual R(u) = jw (u " + f ) dxQ

= 0 , where w is a test function which satisfies w(0) = 0. Define

43


We have J w 'u ' d x = J w f d x + w ( 1) h . Notice that Cl Cl

a ( w , u ) = (w , / ) + w ( 1) h is still an infinite-dimensional problem. Now, reducing

h han infinite-dimensional problem to an n-dimensional problem. Let S C S and V Cl V

h h h hbe any finite dimensional subspace. Find U E S such that for every w E V . One

will have a ( w , u = (w ^ , / ) + w ^ ( 1) h . Choosing a set of n basis functions

h h<f) j ,<(>•■>, • • •, to represent each number of S and V . One will have

w h = C j(j)1 + C2({)2 + . . . + c w(j)w,(j) (0 ) = 0 , ^ ( 0 ) = 1 and

h , x-'' , x h hu = g (p /J+j + y ^ d p t y p . Substitute u for u and w for w , the weak form

p= 1

n

becomes > c G = 0 , and jL* p pp = 1

n

G p = a ( § p ’ § q ) - ( (b p ’ f ) - (b p ( l ) h - a ( § p ’ ibn+0 8 - Thisrequiresq= 1

G p = 0 for any choice of the . Finally, one need to solve the matrix problem: Ad = F,

d = A F. Where d: d p , Apq: a(<j) , 0 ) , and

44


F . = U l ) » -

The spectral element method is accomplished by using Guess-Lobito-Rhinelander

interpolants. First, partitioning the domain into subintervals (or elements) where is given

by Q,k = [Cl , b ] . On each k'h element we define a set o f the N+l nodes,

/ k kk b - a , (

X i = CL + ---- (Z, + 1) , where Z, are the zeros of the Legendre polynomial of

order N. With this choice, the local basis function by using Lagrangian interpolant

( i _through these nodes can be written as H i ( t . ) — ------------------------ —----------------- .By

N ( N + l ) L n ( S f.) ( $ - £ , )

taking (J)^(.v) = , the global basis function <J) becomes an N f/‘order polynomial

on Q * but vanishes outside o f this element.

45


PRISM

Sphere in a Pipe

A' \ \ -i

:; i d Hi ii\ / V ! .

Axisymmetric Mesh Domain ( Sphere in a Pipe )8 -

6 -

0X

Figure 4.7: Mesh of the simulation domain in Prism

46


4.3.3 Validation of Prism

The stokes flow passing through a sphere is used for verification. The simulated results

are shown in Figure 4.8, Figure 4.9 and Figure 4.10. The fluid forces on the sphere are a

function of the pressure and velocity gradient [4.10]:

Fo = jt f d u- p n x + 2 v -=r-nx + d s , and C d — Fd

| p rL d U 2. The

calculated drag is 26.66 which is identical to the analytical solution, 67CJJ.R U . Table 4.7

shows the error norm of velocity components between Prism and analytical results.

Table 4.7: Errors in the sphere case of the Stokes flow

Error Type Magnitude

Error in U (L-infinity norm) 6.242434E-06

Error in U (L2 norm) 7.50862E-09

Error in U (HI norm) 9.237644E-08

Error in V (L-infinity norm) 6.951837E-06

Error in V (L2 norm) 8.929860E-09

Error in V (H 1 norm) 1.216868E-07

47


Stokes flow past a sphere, Velocitv-U Profile

o*rtoi20612945 <OTvtrr

Figure 4.8: Velocity-u contour o f the Stokes flow past a sphere

Stokes flow past a sphere, 2 5 L Velocity-V Profile

20

V

B 0.126245 C10621 00001747 0 0721)97

1—4 006*1046 (— 4 0 0360600 U 0 01600*9

m -0 01603*9 5 3 -0.5360699 Hj| -0 0641046 ■jjj -O 0721397 H -0 09017*7 ■ -O 10621 H O 1262*5

X

Figure 4.9: Velocity-v contour of the Stokes flow past a sphere

48


Stokes flow past a sphere, Pressure Profile

asms0*0*425 0671442 0530251 C 40*67* 027409 01*1507

0X

Figure 4.10: Pressure contour o f the Stokes flow past a sphere

49


4.3.4 Simulated Cd for the Sphere in a Pipe Using Stokes

Equation

The Cd for Low Re with varied H/D in Stokes equation is calculated. Figure 4.11 shows

the Cd in two Re regions. Re = 0.125 and Re = 0.25 respectively; the data is also shown in

Table 4.8. From the plot, one can conclude that the smaller H/D or Re is, the larger the Cd

becomes.

Table 4.8

H/D Cd (Re=0.25) Cd (Re=0.125)

2 950.38792 1900.7758

2.5 615.17752 1230.355062

3 477.31404 954.628096

3.5 407.0825983 814.1651393

4 362.5782295 725.1563699

4.5 334.1696657 668.3384465

5 312.6822151 625.3638444

10 230.3938 460.7747796

50


1800Stokes flow past a sphere in a pipe

1600

1400

Stokes, Re=0.125 Stokes, Re=0.251200

■O 1 0 0 0

800

600

400

200

10H /D

Figure 4.11: Cd of the Stokes flow past a sphere in a pipe with different H/D and Re


4.3.5 Simulated Cd for the Sphere in a Pipe Using NS Equation

Figure 4.12 shows a velocity U-component profile in the sphere-pipe case with Navier-

Stokes equation. Table 4.9 and Figure 4.13 show the Cd for the different H/D in Re=0.125

and Re=0.25 cases. The Cd-Ratio is shown in Figure 4.14. When H/D=60, Cd-ratio

equals to one. At Lower Re cases, the Cd in the Stokes equation and Navier-Stokes

equation follow the same trend (Figure 4.11 and Figure 4.13).

Navier-Stokes flow past a sphere in a pipe Velocity-U profile

U1.558111.734241.610361.486491.362611.238741.114870.9909920.8671180.7432440.619370.4954960.3716220.2477480.123874

Figure 4.12: Contour of the velocity-u in a sphere in a pipe case

52


Table 4.9

H/D Cd (Re = 0.25) Cd (Re = 0.125)

2 950.40 1900.7829

2.5 615.21 1230.37076

3 477.37063 954.6559821

3.5 407.1729 814.2109

4 362.64 725.224

4.5 333.987 668.429969

5 312.292 625.4722

10 228.61960 459.4546

20 159.9671 325.6166578

30 130.235 264.23933

40 117.93 237.00538

50 112.9 225.6

60 110.3171 220.7523815

53


1800 Navier-Stokes flow past a sphere in a pipe

1600

1400

1200

■0 1000

800

600

400

200

20 40 50 6030H /D

Figure 4.13: Cd profile with the different H/D in a sphere in a pipe case with different

Re


Cd

Rati

o

8Navier-Stokes flow , Re = 0.25

7

6

5

4

3

2

1

00 10 20 5030 40 60

H / D

Figure 4.14: Cd ratio profile with the different H/D in a sphere in a pipe case


4.4 Simulation Result from |J,Flow

4.4.1 Formulation for 2-D Compressible Navier-Stokes

Equation

P p u p v

3_ p u 3 + ~—

2pM + p + 3

pwv

d t p v d x pw v 3 y ( p v 2 + p )£ _(£ + /?) • u _(£ + />) • v

_L a_ R e d x

2 (_ du d v

o

3 ^ 1 d v

f 3 a 3v

. 3m dvu + m ^ + t ,

V +icy d T P r d x

_1_ _3_R e d y

0

*■(£*£2 f n 3v duI ' V T y - T x .

_ , _3v 3 iA (d u dvd y d x

KY 9 7 M + P r ' 3 ^

1 2 2 .where Energy, E = p[ T + - ( m + v ) |;Pressure, P — (y — l ) p 7 ;

56


^ UoReference Temperature, T0 =

4.4.2 Simulation with Slip Model

The \lF low [4.11][4.12][4.13 J[4.14] is a compressible spectral element code with slip

model which can handle 2D and axisymmetric flow in the continuum and slip regions.

2 - 0The slip model is described as: f / - U = -----K S W ( J

f 7 > “a u

2> Js_, where Us:

Slip velocity, Uw: wall velocity, C5v Accommodation coefficient, <j^ = I for the diffuse

reflection. The Figure 4.15 shows the velocity profile in vector field. The slip velocity is

generated on the surface of the pipe and the sphere. Figure 4.16 plots the Cd versus

different Kn in the H/D = 2, Re = 0.25 case.

57


Velocity Vector with Slip Model

0.5

1 0 1X

Figure 4.15: A profile of the velocity vector near the sphere

58


540

520

500

480

460

440

O 420

400

380

360

340

320

300

O o NS (No-Slip), Re=0.25♦ NS (Slip), Re=0.25

♦

J 1 I 1 1 1 ' *0 0.05 0.1

Kn

Figure 4.16: Cd with different Kn

59


4.5 Conclusions for SRG

The main conclusions o f this study are:

(1) The drag coefficient Cd is inversely proportional to the Reynolds number in the

Stokes flow region.

(2) The drag coefficient Cd is inversely proportional to the pipe to sphere diameter

ratio H/D in the continuum and slip flow region.

(3) The drag coefficient Cd is inversely proportional to the Knudsen number as verified

by slip flow and transitional flow simulation.

(4) The drag coefficient Cd in the transitional flow region is inversely proportional to

the pipe to sphere diameter ratio H/D. However, this dependence is much weaker

than the continuum flow case.

(5) For high Knudsen number flows such as Kn is bigger than one, the blockage effects

is negligible for H/D bigger than 4.

(6) For external rarefied flow past a sphere, the drag coefficient Cd is inversely

proportional to the speed ratio (and hence the Mach number).

60


4.6 Bibliography

[4.1 ] Fremerey J.K. The spinning rotor gauge. J. Vacuum Science Technology A,

3(3): 1715-1720, 1985.

[4.2] Fremerey J.K. Spinning rotor vacuum gauges Vacuum, 32 (10/11):685-690,1982.

[4.3] Reich G. Spinning rotor viscosity gauge: A transfer standard for thelaboratory

or an accurate gauge for vacuum process control. J. Vacuum Science Technology A,

3(3):1715-1720, 1985.

[4.4] Kennard E.H. Kinetic Theory o f Gases. McGraw-Hill Book Co. Inc., New York,

1938.

[4.5] Gatsonis N.A., Maynard E.P., and Erlandson R.E. Monte Carlo modeling and

analysis o f pressure sensor measurements during suborbital flight. J.

Spacecraft and Rockets, 34, No. 1, January 1997.

[4.6] Baily A.B. Sphere drag coefficient for subsonic speeds in continuum and

free-molecular flows. J. Fluid Mechanics, 65, part 2:401-410,1974.

[4.7] Kinslow M. and Potter J.L. Drag of sphere in rarefied hypersonic flow. AIAA

Journal, l,N o . 11, Nov. 1963.

[4.8] Henderson C.B. Drag coefficients of sphere in continuum and rarefied flows.

AIAA Journal. 14, No.6 June 1976.

[4.9] R. D. Henderson. Unstructured Spectral Element Methods: parallel Algorithms

and Simulations. Ph.D., Princeton University, 1994.

[4.10] David Newman. A Computational Study o f Fluid/Structure Interactions:

Flow-Induced Vibrations o f a Flex Cable. Ph.D., Princeton University, 1996.

[4.11 ] Besekok A. and Kamiadakis G. E. Simulation of heat and momentum transfer

61


in complex micro geometries. AIAA J. Thermophysics & Heat Transfer,

8 (4):647-655, 1994.

[4.12] Beskok A. Simulations and Models fo r Gas Flows in Micro geometries. Ph.D,

Princeton University, 1996.

[4.13] Beskok A. and Kamiadakis G. E. A model for internal micro gas flows in the

entire Knudsen region. Submitted to Microscale Thermophysical Engineering,

1998.

[4.14] Wai Sun Don Theory and Application o f Spectral Methods fo r the Unsteady

Compressible Wake Flow Past a Two-Dimensional Circular Cylinder. Ph.D,

Brown University, 1989.

62


Chapter 5

Coupling Techniques

5.1 Introduction

The main application of the coupling techniques is comprising the regions o f the

continuum and the rarefaction. For example, the coupling techniques are useful during

continuum flow breakdown and phase change [5.1][5.2]. When the condition of

Kn <= 0.1 exists in the computational domain, instead of using DSMC only, applying the

NS solver with slip condition and the DSMC by coupling techniques can also reduce the

computational time.

Before the coupling techniques are applied, results from both NS solver and DSMC must

be verified. There are a couple o f basic cases that can be run on the DSMC for

verification. For example, using the box case and the free stream case allows one to check

results against their known solutions. Furthermore, using channel and pipe cases allows

one to check results against the theoretical solutions. Then the verified DSMC program

can be used to check the NS solvers with slip condition in different Kn ranges.


The main difficulties in the coupling issues will be discussed and a couple of effective and

powerful numerical methods such as AMW, PCA and flux property assignment will be

developed and demonstrated in this research work.

5.2 Coupling Issues

The first difficulty existing in the coupling of the DSMC and the NS solver is controlling

the boundary condition. If the boundary data can be controlled as expected, then the

coupling process and the full domain matching procedures become possible. The second

difficulty is combining the two distinct domains into one. The combined domain requires

the gradient of all physical properties such as velocity components, density, and

temperature to change smoothly. The success on interface matching cannot guarantee a

combined domain with smooth gradient of the physical properties. Thus, seeking a

matching interface and controlling boundary conditions are the first steps in the coupling

issue. Then, matching the two distinct domains into one single domain is the second step.

Usually the boundary value in most of the NS solvers can be assigned directly to each

grid. The extra necessity in the NS solver for coupling is to develop the arbitrary

boundary conditions to replace the regular inlet or outlet boundary conditions. On the

Contrast, setting the appropriate particle motions and controlling boundary conditions in

DSMC is a very difficult task. There are at least four difficulties: (1) in DSMC domain,

only very small amount of the particles are connected to the boundary, (2) inside of the

boundary cells, only small part of the particles encounter with the boundary and the

information is updated from the boundary condition, (3) the updated particles will collide

64


with the rest majority particles to exchange the information, and (4) the final physical data

is coming from the center of the cell by integrating total particles in each cell, not directly

from a single particle. All of the above make the direct assignment on the center of a cell

(or a grid edge) in DSMC impossible and controlling boundary condition becomes very

difficult. The difficulties in controlling boundary conditions can be conquered by

adjusting the flux properties such as number flux, or assigning the appropriate information

to each particle, etc.

In the particle simulation, one single particle carries all o f the information which means all

of the physical data coupling together. Moreover, after averaging and integrating all o f the

particles, it is very hard to satisfy and match all o f the physical information like density,

velocity, and temperature. Therefore, successfully matching the interface can not assure

that all of physical data will match up in the combined domain.

By combining and applying the methods developed in this research work, such as AMW,

PCA, and boundary condition control process, one can successfully overcome these

obstacles and eventually couple all of the physical data including velocity components,

density, temperature, and other properties.

5.3 Control of the Flux Properties

Assigning the flux properties appropriately to the boundary can make the final physical

data meet the expectation. Intuitively, assigning the physical data directly on the particle

seems to be effective. However, after the updated particles have been diluted by the

65


majority part of the particles in the domain, the effect o f the assignment on specific

particles is very small. Flux control plays more important role than applying specific form

directly on the particles.

It is an important step to get a parabolic velocity profile in the inlet o f the pressure-driven

channel flow. In a regular DSMC case, usually the uniform flux properties and

probabilistic particle motion are assigned in the inlet. Therefore, all of the physical data

are uniformly distributed. In this thesis, by using equations (3.5) to (3.7), Fx(i) has the

parabolic distribution as the incompressible flow with no slip condition.

Fx(i)=Fmax*4* ^ ) 2j ,[5.3], H is the width of the channel. The central part of the

channel has the biggest Fx(i), which means the biggest number flux will happen in the

center of the channel, and the smallest number flux is near the wall of the channel. Figure

5.1 shows the simulation results of the extracted velocity U profile from the inlet and the

theoretical solution. In this case, after using the parabolic distribution of the flux in the

inlet, new particles in the inlet can be applied with either a probabilistic distribution, a

deterministic distribution or a distrbution with probabilistic and deterministic combined.

One has to notice that the number flux effect still will dominate the flow domain and

control the physical properties to meet the expectation.

In order to demonstrate the effect of controlling flux properties, an arbitrary shape of the

number flux shown in Figure 5.2 is applied in the inlet. The input has highest value and is

sharply reduced to the lowest value in the center. This input is the opposite performance

66


of the regular internal flow which has the highest value in the center and the lowest near

the wall. In Figure 5.2, the simulation shows the correspondence to the initial input. The

flow on the wall has the slip velocity. The neighboring area to the wall initially has the

highest value, it is then sharply decreased to the lowest region.

I

Velocity-U extracted from the inlet

Flux control simulation Theoretical parabolic profile

0.4

0.2 f

-5.72%23E-06 0.499994Y

0.999994

H sin C h ih F ra n k L iu 1 9 9 7

Figure 5.1: Extracted profile from the inlet of the channel flow case by using a

parabolic distribution of the flux property


2D unstructured DSM CHsinChih Frank U u 1997

Inputted Number Flux in the inlet

Velocity - U profile h Extracted from the inlet ^70 -

60 f

50 -

40 -

30 -

20 -

- 0.2 - 0.1 0Y

0.1 0.2

Figure 5.2: An arbitrary shape of the flux distribution applied in the inlet

68


5.4 NS and DSMC Coupling

A typical domain shown in Figure 5.3 will explain the coupling of different approaches.

For a internal flow, when the density is decreased, Kn will be increased, and velocity will

be increased along the channel. The later section of the channel might local at the

different Kn range and needs different numerical methods. In Figure 5.3, the beginning

section uses the NS solver without slip condition. The next section uses NS solver with

the slip model. The DSMC method is used in the last region due to rarefaction.

NS NS ( Sl i p ) DSMC

Figure 5.3: A combination of approaches in one domain

Usually, the coupling process requires the information from both numerical approaches

which is updated simultaneously. Eventually the overlapping area or interface will

converge to one information. To achieve these, one needs two requirements discussed in

the previous sections. Firstly, both numerical methods must be correct or verified in all of

the physical properties. Secondly, both numerical methods are capable of controlling the

boundary condition or applying any arbitrary boundary condition.

6 9


In the case shown in Figure 5.4, a NS solver (p. flow) and unstructured DSMC are coupled

in the internal flow. The velocity U is verified in p flow. The unstructured DSMC are

capable of controlling the boundary condition. A overlapping area is used to increase the

chance of success in the coupling process. The Kn of the overlapping area is around 0.1.

In the overlapping area or the interface region, both sides are using similar amount of cell

and grid. Initially, the physical parameters such as dynamic viscosity, specific heat, and

wall temperature, etc. are set to be the same in both sides, and the boundary condition is

assumed known or arbitrary in the inlet or outlet o f the entire domain.

NS ( Spectral Element - Slip Model ) DSMC

: ; • • ; ;

: “ ~ — f - 7 - • -------;— 7 -

... . .. .... .• r r - f i

• ... ; : •...... -----.............. : : : s | p p H g g

: r: : : -----------------

Overlapping

Figure 5.4: NS solver and DSMC coupling with an overlapping area

70


The NS solver produces a set of non-dimensional data. The DSMC produces dimensional

data. Since one can always rescale the dimensions during the coupling process, therefore

it is not necessary to non-dimensionalized DSMC data. The coupling error can be reduced

by: (1) increasing the total particles and cells number in the DSMC, and (2) increasing the

polynomial orders and elements numbers in the NS.

The coupling process is listed below:

(1) after running NS and DSMC separately, extract the center line in both domains,

(2) connect the two extracted center line data and calculate the “scaling factor”,

(3) start doing the global iteration, and the scaling factor will be used to convert the NS

data; the outlet of the NS will match the DSMC data, and the inlet of the DSMC will

match the NS data,

(4) for each global matching step, DSMC will go through the boundary condition control

process; the uniform flux is employed in the inlet and a probabilistic velocity

distribution is applied to the particles. The outlet boundary condition o f NS will be

updated by equation (5.1).

t . .P ns — out le t dsrnc - c e n t e r— = ^P n s - o u t l e t n s - c e n t e r

where p lns _ outlet represents the t‘h step outlet density boundary condition, and

U n s - c e n t e r denotes the extracted center line of the velocity U in NS domain, and

U , denotes the extracted center line o f the velocity U in DSMC domain.dsnic — center


Figure 5.5 to Figure 5.7 demonstrate the coupling results. After 12 iterations, two domains

match very well and the velocity U distribution becomes very smooth. Figure 5.5 shows

the velocity U contour profile. Figure 5.6 shows the extracted the data from three different

positions along the wall. Figure 5.7 shows the extracted data from the outlet o f the NS and

the same position from DSMC. Figure 5.6 shows that the velocity increases as the distance

into the channel increases at three different cross sections. In Figure 5.7, the data of NS

and DSMC match very well with an equal parabolic shape including the slip velocity.

72


1.75

1.5

1.25

1

*0.75

0.5

0.25

0

I

U167.46159.122150.784142.446134.108125.769117.431109.093100.75592.416784.078575.740367.402259.06450.7258

2D Channel Flow Coupling

Overlapping

DSMC

-1.4 -1.2 -1 -0.8 -0.6 -0 .4 -0 .2 0 0 .2 0 .4 0.6 0 .8x

H sinC hih Frank Liu 1998

Figure 5.5: Contour profile of the velocity-u in the coupled domain

73


HsinChih Frank Liu 1998

2D Channel Flow Coupling

160

Center150

140 Wall

130

120

110

NS100

90

80 DSMC70

60

50

40

30

201 - 0.5 0 0.5

X

Figure 5.6: Extracted velocity profile from Fig 5.5 in the coupled domain


U /U

-cent

er2D Channel F L ow V eloc ity profile at the Interface

DSM C1 NS ( 5th order)

0.9

0.8

0.7

0.6

0.5

0.4

0 0.25 0.5 0 .75 1


Figure 5.7: A comparison in the interface from the coupled domain. The velocity-u is

extracted from Figure 5.5


5.5 Particle Capture Algorithm

The difficulty in the DSMC coupling is shown in Figure 5.8. In the coupling process, the

information interchange between DSMC and NS solver is between grid edges and the cell

center. However, there exists infinitely possible combinations of the particles to

correspond to the known physical data from the cell center. The challenge lies in finding

the correct combination to match all of the physical data in the interface and fit into the

entire domain. The Particle Capture Algorithm (PCA) is developed in this thesis to

enforce the effect o f the previous developed algorithms. The explanation of PCA can be

found in Figure 5.9. PCA is keeping the properties o f the particles which leave the outlet

and applying them in the inlet at each time step.

In cases without complicated geometry such as channel flow, using single interface

without the overlapping area is possible. However, the cases with complicated geometry

usually require the overlapping area. A bigger overlapping area will increase the chance

of success in coupling, but also cost more computational time. By adding the particle

capture algorithm, using the smaller overlapping area becomes possible.

The flux properties by adding the particle capture algorithm is defined below,

3 oo oo

N(i) = — J e x p w'~)dw' J exp(-|3~v'~)dv' *K

— o o — o o

00

1 (« ' + cocos(0) + F x(i) - F xoullel) e x p ( -$ 2u '2)du (5.2)(c„cos9 + Fx(i) - Fxuullrt)

76


J (« ' + cocos(0) + F x (i) - Fxoutlet )"exp(~P ~u'~)du (5.3)-(C„COS0 + F x( I ) - Fxou„rl)

out let-(c„cos0 + Fx(i) - Fxnu„rt)

( v ' + cosin(0) + F y(i) - Fyoutlef + w ' 2 } *

(u ’ + cocos(0) + Fx(i) - Fxolt[lel)exp{-$~ (u '~ + v'~ + w '~ )}du 'dv 'dw ' (5 .4 )

where Fxout[et denotes the properties distribution of the outlet in the x-components, and

Fyoutlet denotes the properties distribution of the outlet in the y-components.

An arbitrary and complicated geometry case, a channel with curvatures (rough channel), is

shown in Figure 5.10 and Figure 5.11 to demonstrate the effectiveness of the particle

capture algorithm. DSMC is used to get the full domain results and is used to verify the

PCA data. The full domain and the coupled domain have to use same constrains such as

fixed molecular model and fixed molecular properties, etc. One can notice that the total

number of constrains used in PCA will be more than the ordinary approach of validation

like using theoretical or empirical results. Thus, after the validation process, this

algorithm will be more trustworthy. Figure 5.10 is a full domain results using DSMC.

77


Figure 5.11 shows the coupled DSMC domains by two different algorithms: one is using

the boundary condition control process only, the other is adding the particle capture

algorithm. Figure 5.12 and Figure 5.13 show the extracted data from the cross sections in

the domain. From the plots, CPA shows the excellent ability to catch the velocity-V

profile in the top line. All of the physical properties match very well, but density in the

inlet is a little bit off (see Figure 5.12 and Figure 5.13). On the contrast, without using

CPA, we will need to use bigger overlapping domain in the coupling process since the data

close to the interface doesn’t match very well.

78


Hsinchih Frank Liu 1999

NS Grid DSMC Cell

Fig. 1 Exchange Information

DSMC Cell

Fig. 2 Infinite Possible Particle Combinations

Figure 5.8: Infinite possible combinations exist in the information exchange process


B oundary C ondition C ontro l Process

New Particles

CoupledDSMC

Particle C apture A lgorithm

Fig. C om bina tion o f B oundary C ondition C ontrol and Partic le C ap tu re A lgorithm

Hsinchih Frank Liu 1999

Figure 5.9: Combination of PCA and BCCP

80


DSM C 2D Channel Flow with Curvatures Triangular cells : 4426


V elocity - U

V elocity - V

D ensity

• 73M1C-C7 T M r m « r

« IMT447 « o t9 ? i t a r

X

T em perature

Ml MI I n « M i 1 » r » ? i ;140 MI I r%4041 |mw In o «it )m « o a j

Figure 5.10: Contours of the channel flow with curvatures (rough channel)

81


Coupled DSMC PCA + BCCP

Coupled DSMC BCCP only

Velocity - U

Velocity - V

Density

Velocity - U

lc tVT'»

Velocity - V

Density

Temperature

I

Temperature

i n tr .7 !

<«n t t a | I f l O I


Figure 5.11: Contour profiles of the coupled domain

82


E xtra ct C en ter l in e

Full DSMC Coupled DSMC ; bccpBCCP

350

Full DSMC300 Coupled DSMC

( PCA + B C C P )

Coupled DSMC ( BCCP)

250

150

100

Velocity - U50

0.5 1.5i

Full DSMC

Coupled DSMC ( PCA + BCCP )

Coupled DSMC ( B C C P )

Velocity - V

1X

1 8E-06

Density1 6E-06

1 4E-06

IE-06Full DSMC

Coupled DSMC ( PCA +■ BCCP )Coupled DSMC ( B C C P)

6E-07

4E-07

0.5 1 1.5

310

Temperature300

290

280

270

0260Full DSMC

250

Coupled DSMC ( PCA + BCCP >

Coupled DSMC ( B C C P) t

240

230

220

2100.5 1 1.5


Figure 5.12: Comparison from the extracted data

83


E xtra ct Top L ine

Full DSM C Coupled DSM C I £ £ ; BCCP

300 Full DSMC



250

200

150

100

50 Velocity - U

0.5 1.5t

Full DSMC


Coupled DSMC ( B C C P)

> 30

Velocity - V

1.8E-06Density

1 6E-06

1.2E-06

1E-06Full DSMC

8E-07Coupled DSMC ( PCA + BCCP )Coupled DSMC ( B C C P )

6E-07

4E-07

0.5 1.5

310

300 Temperature290

280

270

0 2 6 0Full DSMC

250



240

230

220

0.5 1 1.5

HsinChih Frank Uu 1999

Figure 5.13: Comparison from the extracted data

84


5.6 Artificial Moving Wall Algorithm

Artificial Moving Wall (AMW) algorithm is developed in this thesis to deal with the

coupling interface existing at the tangential direction of a flow velocity such as in the

couette flow. In the previous coupling approaches, the inward normal flux is bigger than

zero in the coupling interface. However, in the couette flow, the inward number flux in the

tangential direction of the flow velocity is very close to zero. If using the previous

algorithm, the rate o f the convergence will be very slow due to very few new particles can

be assigned in the interface. The effects o f the boundary condition control is limited.

The AMW algorithm is using an artificial wall in the interface. The artificial wall blocks

the information exchanging and directly provides the information to the coupled domain.

Because the factor that the samll number flux causes lower efficiency of the information

exchange doesn't exist, AMW will have a faster rate of convergence.

The advantages of AMW compared to the other published papers [5.4][5.5] are

(1) all of the physical properties in the coupled domain can be matched including velocity,

density and temperature, (2) AMW has faster rate of convergence, (3) the interface can be

chosen in any location, (4) one can have one or two interfaces in one domain, (5) AMW

can be applied to both structured and unstructured domain, and (6) the matching results

will not be affected by the number of particles.

Two cases are chosen to demonstrate the ability of the AMW:

(1) ID domain matching with big coupling region (see Figure 5.14).

85


(2) ID domain matching with two interfaces (see Figure 5.15).

Figure 5.14 and Figure 5.15 are using ID structured program [2.1] for the coupling

process. The full domain is using ID DSMC with solid walls as the boundary conditions.

In the coupled domain, one boundary is using solid wall; the other side is using AMW.

The case shown in Figure 5.14 has a very big coupling region. AMW demonstrates the

excellent matching ability without regarding the size o f the coupling region. Besides the

linear distribution of velocity is matching, the nonlinear profile in the density and

temperature are also matching well. The case shown in Figure 5.15 demonstrates that one

can choose two interfaces in the domain by using the AMW. All of the physical properties

match very well.

86


Couette Flow (ID)DSMC coupling (AMW)

Moving Wall

250i Velocity

200DSMC

3 150

100InterfaceDSMC (AMW)

Full domain DSMC (Exact Solution)

50Exact solution

0.25 0.5 0.75

Fixed Wall

28a2 6 7

2 86

2 8 5

2 84

2 83

2 82

281

2 80

2 7 9

2 78

2 7 7

2 7 6

2 7 5

Temperature

9.6E-06

9.55E-06

9.5E-06

9 45E-06

9.4E-06 j»

— 9.35E-06

9.3E-06

9-25E-06

9.2E-06

9.15E-06

Density

/

0 .5Y

0.5Y


Figure 5.14: Comparison of the physical properties between coupled domain and the

entire domain. Single interface is applied in this case.

87


Couette Flow (ID)DSMC coupling (AMW)

Moving Wall

Exact solution 2 5 0

Interface 200

DSMC

100InterfaceDSMC (AMW)

Full domain DSMC (Exact Solution)

50Exact solution

0.25 0.5 0 .75

Fixed Wall

2ea2 87

2 66

285

2B4

2 83

2 82

261

2 80

2 79

278

2 77

2 76

2 7 5 ■

Temperature

0 5Y

9.6E-06

9.55E-06

9.5E-06

945E -06

9.4E-06

— 9.35E-06

9.3E-06

9.2SE-06

92E -06

9.15E-06

Density

v

0.5Y


Figure 5.15: Comparison of the physical properties between coupled domain and the

entire domain with two interfaces.

88


5.7 Bibliography

[5.1 ] Hash, D. B. and Hassan, H. A. Two-Dimensional Coupling issues o f Hybrid

DSMC / Navier Stokes Solvers. AIAA 97-2507, 32nd Thermophysics

Conference. June 23-25,1997. Atlanta, GA.

[5.2] Wadsworth D. C. and Erwin. D. A. Two-Dimensional Hybrid Continuum /

Particle Approach for Rarefied Flows. AIAA 23rd Plasmadynamics & Lasers

Conference. July 6-8, 1992. Nashville, TN.

[5.3] Panton, R.L. Incompressible Flow. John Wiley & Sons, Inc. 1996

[5.4] Hash, D. B. and Hassan, H. A. Assessment o f Schemes for Coupling Monte

Carlo and Navier-Stokes Solution Methods. J. Thermophysics & Heat

Transfer, Vol. 10, No. 2, April-Jun2 1996,

[5.5] Hash, D. B. and Hassan, H. A Hybrid DSMC / Navier-Stokes Solver. AIAA

95-0410, 33rd Aerospace Sciences Meeting and Exhibit. Reno, NV. Jan. 9-

12,1995

89


Chapter 6

3D Unstructured Direct Simulation of Monte Carlo Method

6.1 Introduction

The 3D unstructured simulation is the extension of the 2D simulation. In the previous

simulation, most of the algorithms used in 2D are ready to be applied in 3D. In 2D

simulation, the z component of the physical properties are still calculated, and the length

of the z-coordinate is treated as zero; the volume is treated as area. In 3D simulation, the

real geometry needs to be considered, therefore the cell volume and the area of the surface

element on the cell will be calculated distinctively.

Different forms of cell can be used in 3D simulation, such as regular cubic cell, irregular

cubic cell, hexahedral cell, tetrahedral cell etc. Tetrahedral cells are used in this thesis.

Some 2D approaches, equations, and algorithms developed in this thesis can be easily

extended onto tetrahedral cell domain.

Usually the 3D computation takes much longer computational time than 2D. In many

90


published papers, parallel computations are popularly used in the 3D simulation.

However, in the regular industrial environment, the parallel computing machine is not as

prevail as single processor due to cost. Moreover, the parallel computing sometimes

allows slower algorithms to be used in the program. Since, One single processor will be

used for demonstrating the simulation result in this thesis, programming structure has

been optimized for speed.

6.2 Unstructured DSMC Procedure

The 3D procedure is the same as 2D but uses the tetrahedral cells instead. In 2D

simulation, the formed group will be a circle; with 3D, the formed group will be a sphere.

The process of the group establishment is similar to the one defined in section 3.2.

Definition 6.1 G ^ = {m: 3 i, j = 1, 2, 3, 4 . such that X m ( j ) = X n ( 0 K n € Q.

The definition from (3.2) to (3.4) will be applied to three dimensional simulation as well.

6.3 Particles Initiation

Similar method used in 2D is also used in 3D for initiating the particles. (XI, x2, x3, x4),

(y 1, y2, y3, y 4 ) , and (z l, z2, z3, z4) denote the cell edge coordinate.

x = X * x 1 + p * x 2 + k *x3 + ( l - ? i - p . - K ) *x4 (6.1)

y = A. * y l + p. * y2 + K *y3 + (1 - X - p - k ) *y4 (6.2)

z = k * z l + p * z2 + ic *z3 + (1 - A. - - k) *z4 (6.3)

? i > 0 , p . > 0 , t c > 0 , ( A . + p + i c ) <l (6.4)

91


where X , p., and k are determined by a random function, and then is substituted into (6.1),

(6.2). and (6.3) to define x, y, and z. If (A. + ji + tc) > 1, then X , p., and k need to be re

sampled.

6.4 Flux Properties on the Boundary

The flux properties calculation in 3D is similar to 2D, but one needs to consider the effects

from the components in the z axis of the free stream condition and boundary condition

control process in each cell. The probability density function F(i) has three different

components in each axis, Fx(i), Fy(i), and Fz(i).

K

(« ' + c'l + F x(i))e xp (—$~u'~)du' (6.5)-(c* +• F x ( /))

p(i) = ' ' 3^ J exp(-P~vv/“)i/vv/ J exp(—$~v'~)dv' *71

(6.6 )

- { c l + F x ( i ) )

_(c‘ + F x ( , ) )

(u ' + cxQ +• F x :(/))exp{-p2(M/2 + v '2 + w '2)}d u 'd v 'd w ' (6.7)

92


6.5 Validation and Application

6.5.1 Box Case

The box case is the first testing case to verify most o f the algorithm and the program. The

box case is demonstrated in Figure 6.1. Each wall of the box has the same temperature,

and the diffusive reflection is applied to the particles when the interaction between a

particle and a solid body occurs. The results match the expectation: the velocity

components are zero. The temperature and density are uniformly distributed through out

the entire box.

6.5.2 Pipe Flow Case

A case of pressure driven pipe flow is demonstrated in Figure 6.2. 6.3 and 6.4. The

extracted velocity data is compared with the analytical data [6.1 ] with Kn=0.1. 14479 cells

are applied to this case. 20 particles per cell are initially distributed in the simulation. The

diameter of the pipe is 1 meter and the length is 2.4 meters. In order to compare to the

analytical solution, the initial number density is adjusted to achieve the appropriate

Knudsen range after the computation.

The further validation is demonstrated in Figure 6.5 and Figure 6.6 by using the model

proposed by [4.12]. More cells and a longer pipe are used in the bigger Kn range case.

For instance, 111878 cells and a 5 meter long pipe are applied to the Kn=l case. Figure

6.5 shows the velocity scaling validation by using

93


U(y)/U =

where a is the pipe radius, and the Knudsen number is defined as Kn = - . b= -1 is chosena

in this case; the velocity scaling of b=-l is validated by structured DSMC results and

analytical solutions [4.12].

Figure 6.6 shows the normalized flowrate in a pipe as a function of Knudsen number. The

minimum of the flow rate occurs at around Kn =1 or Kn = 3 (see studies

[6 .1][6.2][6.3][4.4]). The asymptotic limit of flowrate in a pipe in a free molecular flow

region is given by [4.4], which is chosen to compute the normalized flow rate.

l^ _ 4 3A/Y27CV . . . .M fm ~ 1 a ~l \ r t ) (6'9)

The numerical model proposed by [4.12] and the formula derived by Knudsen [4.4] are

used for the comparison in Figure 6.6. Kn is computed at the average pressure

P = (Pt + P0)/2 .

M = - ^ L ( 1 + a K n )( 1 + — f o O M fm 64 Kn V 1 — bKn*

(6 . 10)

M Kn _ 3K [ H +2.507(1 / ( K n ) ) ' ( 6 U )M fm 64Kn Vi +3.095 ( l / ( K n ) )

a = 1 .3 5 8 -0 .22 (Kn)~°'5 and b = - I are chosen in (6.11).

6.5.3 Sphere in the External Flow

94


As the same process in the section 4.2.1, the validation of F3, the Cd computation in the

external sphere case is also used for validation in here. Figure 6.7 shows the physical

properties in the 3D contour, and Figure 6.8 shows the Cd comparison. In the simulation

process, the free stream condition are applied to the boundary. The initial conditions and

boundary conditions are also fixed except the free stream velocity which is varied in the

different Ma range: 0.36, 0.6, 1, 3 and 5. The magnitude of lift is much smaller than the

drag in this case. For example, in Ma=3 case, the drag is 10-2 and the lift is 10-5 .

Compared to F3 code, there are 16500 cells and 8000 surface elements used in section

4.2.1, and there are 92495 tetrahedral cells used in this computation.

6.5.4 Applications

Figure 6.9 is demonstrating the Couette flow case. A couple of the boundary conditions

are applied in this case: moving wall condition, fixed wall condition, specular reflection,

and periodic conditions. The moving wall velocity is the same as 2D, 300 m/s. The results

are also consistent with the 2D simulation. The velocity component U is linearly

distributed. The values of the velocity V and W are close to zero. The temperature and

the density profiles are parabolic distributed.

A pressure-driven square pipe with a bump case is shown in Figure 6.10, 6.11, and 6.12.

This case contains a non-symmetric arbitrary geometry. 13492 tetrahedral cells are used

in this case.


The sphere in a pipe case is shown in Figure 6.13. The geometry ratio is the same as

section 4.4 with H/D=2 and Kn= 1. The inlet and outlet conditions are similar to the

pressure-driven pipe cases. 41105 tetrahedral cells are used in this case.

The 3D conduit flow case is shown in Figure 6.14, 6.15 and 6.16. 89346 tetrahedral cells

are used. The symmetric conditions are applied in the central cross section to reduce the

usage of the tetrahedral cells in this case. The Reynold number is computed from mean

velocity and mean density. The diameter of the inlet is chosen as the characteristic length.

Figure 6.16 shows the non-dimensional reattachment length which is also called bubble

length, L/Do, and the center of the eddy which is also called bubble center, L2/Do, as a

function of the Re. Figure 6.17 shows the comparison between the 3D unstructured

DSMC simulation with an outlet condition Kn=0.1 and the continuum region results

including the 3D numerical simulation using the vorticity-velocity solver, IW A [6.4], 3D

numerical simulation using the tetrahedral velocity-pressure solver, NEKTAR [6.5] and the

experiment results [6.6]. For both bubble length and bubble center, the effect that the

rarefaction suppresses the separation is shown. Figure 6.18 shows the extracted data from

5 different locations. It includes velocity-u, temperature and density profiles.

96


DSMC 3D Box 55 Tetrahedral Cells

HsinChih Frank Liu

Tetrahedral Mesh

S 6592

| 3 33605

C 796696

-1 73425

I <42654

I -6 .79655

| -6 3277

Velocity - U

279 797

277 701

275 605

773 509

271 412

269316

Temperature

306636E-09

3 0243IE -09

2 992266-09

2.640216-06

2*96166-00

2. *56116-09

2614066-09

Density

3 45556

1 30475

•0 646056

2 99667

7 29646

-9 44929

Velocity - V

Z

w■ 7 60421

543005

| 3 25562

1 06159

1 09264

I -326667

I 5 4411

Velocity - W

Figure 6.1: Contour o f the physical properties by using 3D unstructured DSMC in the

box case

97


Poiseullle Flow in a Cylindrical Tube UDSM C (3 D )

Veloclty-U

0.5

0.25

-0.25

-0.5 •0.5

-0.25'

^ 0'o5F0 5 ^ 2.5

Hsin-Chih Frank Liu

Flow

u397.538351.47305.403259.335213.267157.199121.13175.063328.9954

0.9

0.8

0.7

XI 0.6 33 0.5

UDSMC (Kn=0.1)

0.4Analytical Solution Loyal lea 19900.3

0.2

- 0.75 -0.5 - 0.25 0y/R

Figure 6.2: Velocity-u contour in the pipe flow and the normalized velocity-u comparison

with analytical solution

98


r

Velocity-VF lo w

I 0.25

V36 .562428.528120 .493912.45964 .42535

-3.60891-11 .6432-19 .6774-27 .7117

Velocity-W

w3 3 .4272 2 4 .9417 16.4563 7 .9708

-0 .51466 -9 .00012 -17 .4856 -25.971 -34 .4565

Figure 6.3: Velocity-v and velocity-w contours in the 3D pipe flow

9 9


Ii

Temperature

10 .2 5 1

0-S 2 5

OT

311.396301.291291.187281.082270.977260.873250.768240.664230.559

XDensity

8.73891 E-07 7.89291 E-07 7.04691 E-07 6.20091 E-07 5.35491 E-07 4.50891 E-07 3.66291 E-07 2.81691 E-07 1.97091 E-07

Figure 6.4: Temperature and density contours in the 3D pipe flow

100


IJ / I

I-av

erag

e

Velocity Scaling at Wall and Center-line of Pipe Flows for Slip and Transitional Flows

□ UDSMC ( 3D )Loyalka & Hamoodi

Beskok & Kam iadakis1.75

CTr1.25

0.75

0.5

0.25

10' 10' 10Kn

Figure 6.5: Velocity scaling in the 3d pipe flow


\

5i-c

3

2.75

2.5UDSMC (3D) Knudsen’s Model Beskok & Kamladaklsxs

2.25

2

1.75

1.5

1.25

1

Kn

Figure 6.6: Normalized mass flow rate comparison with the models in the 3D

pipe flow

102


6

Velocity-U

Sphere in Rarefied Flow UDSMC (3D) 92495 Tetrahedral Cells Kn=3.5Ma=3

Flow

I—• a n a a s ;

H IM 3 871 I

| v

Temperature

Figure 6.7: Velocity-u, Kn and temperature contours in the 3D sphere external flow

case

103


16

14

The Drag Coefficient on Sphere, Kn=3.5

12

10

S 8

6

4

2

D>□

\

oj i ' ' 1 ■ 1 ■ ■ ■

□ UDSMC(3D)- Bird

> F3Baily

o Kinslow

o

Ma

Figure 6.8: Cd of the sphere case in the varied Ma range

104


DSMC Couette Flow Tetrahedral Cells : 4206

M oving Wall U = 3 0 0

Periodic

Fixed Wall


Periodic

Specular

244 622

172 996

101 37

29 7442

Velocity - U

2 74 .1 6 4

•0 .129062

3.8306

Temperature Velocity - V

2 91717E-07

Density Velocity - W

Figure 6.9: 3D unstructured Couette flow

105


3D unstructured DSMC Square Pipe with Bump 13492 tetrahedral cells

- 0.2 n

323.887290.792257.697224.602191.508158.413125.31892.223359.128526.03377.06109

Velocity - U

0.2 n i

Figure 6.10: 3D mesh in the computational domain and velocity-u contour

106


Velocity - V

! -3

51.672 43.6338 35.5957 27.5576 19.5194 11.4813 3.4432

I -4.59493 -12.6331

I -20.6712 -28.7093

-0.2 n

29.991522.9476

Velocity - W 15.90378.859781.81586

*5.22805•12.272•19.3159 !-26.3598-33.4037-40.4476

-0 .2 N

HsinChih Frank Liu

Figure 6 .11: Velocity-v and velocity-w contours in the 3D square pipe with bump

case

107


1 74004E-071.59009E-071.44013E-071.29017E-071.14021 E-079.90254E-088.40296E-086.90338E-085.4038E-083.90423E-082.40465E-08

Density

-0.2 n

Temperature300.667293.426286.186278.946271.705264.4652S722S249984242.744235.504228263

-0 .2 N

HsinChih Frank Liu

Figure 6.12: Density and temperature contours in the 3D square pipe with bump

case

108


UDSMC (3D) Sphere in a Pipe Kn=341105 CellsVelocity-U profile

Velocity - U

6.8158

Temperature 302.076

294 765

207 4S3

280 141

265 517

258 206

250 894

243 582

228.958i i ti r , i i AS?

0 5

Density 100992E-08 ,

945316-09 :

8 807016-09 1

8.160916-09 j

7514826-09 !

' 6 868736-09

622264E-09

5576546-09 1

4.930456-09 I4284366-09

3 638266-09 !

Hsin-Chih Frank Liu

Figure 6.13: Contours of the sphere-pipe case

109


Figure 6.14: Mesh structure of the 3D conduit flow. The symmetric condition is applied in

the central cross section o f the conduit.

110


3D Conduit89346 Tetrahedron cells

velodiy-U

Figure 6.15: Velocity-u contour of the 3d conduit flow

1.8

1.6

1.4

Eddy Center Separation PointIf

0.8 -

423.187396.409389.832342.854316.077289.299

235.744208.967182.189155.411128.634101.85675.078948.301421.5238

Velocity-U (Z=0)

Fig 6.16: Eddy center and separation point in the 3D conduit flow

111


oQfNJCQ

4 i -

3.5 -

-□ M acagno & Hung -D ye Exp

- ------ •>— - IV V A .L /D o> IV V A . L2/D o

_ Nektar. L/Do- < Neklar. L2/Do- ------- A----- - U D SM C(3D ), Kn=0. Kouilci). L/Do

- C' U D SM C(3D ). Kn=0.1 (outlet). L2/Do3 -

Figure 6.17: Comparison between bubble length and bubble center. The lines shows the

difference between continuum flow and rarefied gas flow.

112


3D Conduit Flow UDSM C (3D)

Flow

Wall

C enter Line

Velocity-U

D I K j / m )

3.SE-06 Density

3E-06

2.5E-06

l 5 c -06

IE-06

5E-07

3tQ

300

290

2802H 270

260

250

240

230

Temperature

!"• / - •I v .I / " \K ' V '\ , v .

Figure 6.18: Extracted profile from 3D conduit flow. 5 sets of data are presented from

different locations.

113


6.6 Bibliography

[6.1] Loyalka S.K. and Hamoodi S. A. Poiseuille flow of a rarefied gas in a

cylindrical tube: Solution of linearized Boltzmann equation. Physics o f

Fluids A, 2 (1 1):2061-2065, 1990.

[6.2] Cercignana C. Plane Poiseuille flow and Knudsen minimum effect.

Proceedings or the Third International Symposium on Rarefied Gas

Dynamics, Edited by J. A. Laurmann, 1:92-101, 1963

[6.3] Ohawada T., Sone Y., and Aoki K. Numerical analysis of the Poiseuille and

thermal transpiration flows between two parallel plates on the basis o f the

Boltzmann equation for hard sphere molecules. Physics o f Fluids A, 1, no

12:2042-2049, 1989

[6.4] James Trujillo. Effective high-order vorticity-velocity formulation. Ph.D

thesis, Princeton University, 1998.

[6.5] Sherwin, S. J. Triangular and tetrahedral spectral/h-p finite element

methods fo r fluid dynamics. Ph.D. thesis, Princeton University, 1995.

[6.6] Macagno, E. O. and hung, T. Computational and experimental study o f a

captive annular eddy. Journal o f Fluid Mechanics 28 (1967), 43.

114


Chapter 7

Summary

We conclude this thesis by summarizing the accomplishments of the present research:

. An optimized unstructured DSMC procedures and algorithms are necessary for

reducing the computational time and improving the accuracy, especially in the higher

dimensional computation.

. To obtain the correct DSMC results, there are a couple of factors that one needs to

pay attention to: the interaction between particles and a solid body, the diffusion

effect from rarefaction, the appropriate total particle numbers for statistical basis,

the appropriate procedure and molecular models, and an appropriate time step, etc.

. In the SRG project, developing an understanding from the effects of Re, Kn, Ma

and H/D is critical for the drag calculation.

. In the bigger Knudsen number range, the buffers in the inlet and outlet of the domain

are necessary to be added, in order to reduce the diffusion effect by the rarefaction

which occurs on the boundaries.

. Flux properties control is crucial in the DSMC boundary condition control process.

Flux properties can be used to get the anticipated profile of the physical properties

during the interface matching process.

115


Once the boundary conditions can be controlled, a successful coupling process will

be easier to be developed. Adding the overlapping area can also help to get the

matched results.

The artificial moving wall algorithm is an optimized and effective method for the

cases where the coupled domain or interface has small inward number flux.

The particle capture algorithm is capable of matching all of the physical properties

such as density, velocity components, and temperature in the complicated geometry

without using overlapping area.

In the 3D conduit case, rarefaction suppresses the bubble length and bubble center.

116


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INFORMATION TO USERS - Brown University · 2017-08-21 · The direct simulation of Monte Carlo methods for 2D and 3D using triangular and tetrahe dral meshes are developed in this