DirectSimulationMonteCarloApplication ......DeclarationofAuthorship I, Naveen Kumar KULAKARNI, declare that this thesis titled, ’Direct Simulation Monte CarloApplicationforAnalysisofIso-thermalGasFlowinMicro

INSTITUTE OF MECHANICSBULGARIAN ACADEMY OF SCIENCES

Doctoral Thesis

Direct Simulation Monte Carlo Applicationfor Analysis of Isothermal Gas Flow in

Microchannel Configurations

Author:Naveen Kumar KULAKARNI

Supervisor:Prof. DSc. Stefan K. STEFANOV

A thesis submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy

in the Scientific field

01.01.13 “Mathematical modelling and application of Mathematics in Mechanics”

Sofia, December 2014

ii

Publicly defended before a jury composed of:

Internal members:

Prof. DSc. Stefan K. STEFANOVAssoc. Prof. Dr. Slavtcho SLAVTCHEV

Institute of Mechanics - Bulgarian Academy of Sciences, Bulgaria

External members:

Prof. D.Sci. Zapryan ZapryanovSofia University, Bulgaria

Prof. D.Sci. Ivan DimovInstitute of Information and Communication Technologies - Bulgarian Academy of

Sciences, Bulgaria

Prof. Dr. Aneta KaraivanovaInstitute of Information and Communication Technologies - Bulgarian Academy of

Sciences, Bulgaria

Declaration of Authorship

I, Naveen Kumar KULAKARNI, declare that this thesis titled, ’Direct Simulation MonteCarlo Application for Analysis of Iso-thermal Gas Flow in Micro-channel Configurations’ andthe work presented in it are my own. I confirm that:

This work was done wholly or mainly while in candidature for a research degree atInstitute of Mechanics-BAS

Where any part of this thesis has previously been submitted for a degree or any otherqualification at this Institute or any other institution, this has been clearly stated.

Where I have consulted the published work of others, this is always clearly attributed.

Where I have quoted from the work of others, the source is always given. With theexception of such quotations, this thesis is entirely my own work.

I have acknowledged all main sources of help.

Where the thesis is based on work done by myself jointly with others, I have madeclear exactly what was done by others and what I have contributed myself.

Signed:

Date:

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Preface

This research work is focused on the application of the direct simulation Monte Carlo method(DSMC) and its efficient implementation for the solution of the dynamics of the gas flow inthe microchannel configurations considering the isothermal pressure driven gas flows. Theaims of the dissertation may be defined as follows:

Implementation of pressure boundary conditions for the direct simulation Monte Carlomethod for a study of time dependent and time independent problems of the gas flowthrough various geometrical configurations of microchannels.

Implementation of efficient collision algorithms with improved stochastic properties.

Investigation of the effects of the geometrical shape of the microchannels on the gasflow through them.

Investigation of and mass transfer through 900 bend microchannel using experiments.

Development of a 2D and 3D DSMC codes using standard MPI and OpenMP forinvestigation of gas flows in microchannel configurations.

The dissertation is structured in seven chapters which may be outlined as follows:

The Chapter 1 is devoted to the general concepts, theoretical background and stateof the art in the field of gas microflows and the numerical simulation. The numericalmethods which are employed for this investigation are also introduced.

In Chapter 2, various techniques for the application of a pressure boundary for thestudy of the pressure driven flows were discussed.

In Chapter 3, the time dependent transient driven gas flow through some microchannelconfigurations were presented; a shear driven Couette flow, and a pressure driven gasflow past a backward facing step were shown.

v

in Chapter 4, discussed are a parallel implementation of the DSMC code created forthe simulation of gas flows through microchannel configurations using the standardMPI and OpenMP.

in Chapter 5, the mass flow rate through microchannels was studied using the DSMCmethod considering straight and 90 degree bend microchannels.

in Chapter 6, the effects of the finite distance between the transverse walls of themicrochannel configuration was investigated, considering a straight, a 90 degree bend,and a T-junction of microchannels.

the Chapter 7 presents the experimental measurements of the mass flow rate througha 90 degree bend microchannel, carried out at the IUSTI/UP laboratory.

The dissertation is concluded with the scientific contributions and possible future ex-tensions of the present work.

This dissertation consists of 7 chapters written on 133 pages. The results are presented in 60figures (of which 16 illustrate the implementation details), 14 tables and a 105 journal papers,theses, books, and other sources were cited. A part of this investigation was accomplishedfor three years in the framework of the GASMEMS project: an initial training networkfinanced by the FP7 of the European Commission with thirteen participants and six associatedpartners from all over the European Community. Its aim is to provide fundamental andapplied research in the fields of dynamics of gas microflows and of microstructure design andmanufacturing. The main part of the research was carried out at the Institute of Mechanicsof the Bulgarian Academy of Sciences under the supervision of Prof. D.Sc. Stefan Stefanov,but this project offered the opportunity to make two secondment periods: the first one atthe AOES-Netherlands under the supervision of Dr. G. N. Markelov and the second one atIUSTI/UP, Marseille, France under the supervision of Prof. I. Graur. The main results ofthis dissertation were reported in several workshops, such as the GASMEMS Workshops in2010 and 2011, and in several international conferences on ICMAR-2010, AMITANS-2013,and the 1st European Conference on Gas Micro Flows, June 2012 in Skiathos (Greece),and several times at Institute of Mechanics-BAS, in the years 2011, 2012, 2013, and 2014.Moreover, the main results of the dissertation have been accepted for publication, submittedand are under review and are in the process of submission to international scientific journalswith impact factor and published in conference proceedings of international conferences.

INSTITUTE OF MECHANICSBULGARIAN ACADEMY OF SCIENCES

Abstract

01.01.13 “Mathematical modelling and application ofMathematics in Mechanics”

Doctor of Philosophy

Direct Simulation Monte Carlo Applicationfor Analysis of Iso-thermal Gas Flow in

Micro-channel Configurations

by Naveen Kumar KULAKARNI

The research work presents a study of the application of the Direct Simulation Monte Carlo(DSMC) to the isothermal and near-isothermal gas flow in micro-channel configurations.The numerical calculations were performed to study steady state and unsteady state gasflows through several (2D and 3D) microchannel junctions and configurations resulting fromcombination of several microchannels. The subject of this study includes a vast variety ofpossible flow configurations and some of them are illustrated here. Thus, the combinations ofmicrochannels resulting in a vertical bend, a long channel, and a T-junction, etc., have beenstudied, and the dynamic response of some suddenly triggered pressure driven unsteady-stategas flows were analyzed. A parallel implementation of the DSMC code(s) was performed in2D and in 3D using standard MPI and OpenMP. Pressure boundary treatments were dis-cussed and effects of the rarefaction, pressure drop, and geometrical shape on the gas flowthrough microchannel configurations were investigated. The implicit boundary conditionswere successfully implemented in conjunction with the DSMC to match the flow characteris-tics with those of a long channel geometry while simulating a relatively short microchannels.Effects of cross sectional shape, and transverse walls on the gas flow through microchannelconfigurations were studied, for some selected values of rarefaction parameter in the slipregime and in the transition regime. Some experiments were carried out to measure themass flow rate through a 90 degree bend microchannel.

Acknowledgements

I express my gratitude to my advisor Prof. D.Sc. S. Stefanov for his valuable guidance,encouragement, support and patience during my work at Institute of Mechanics-BAS. I amalso grateful to Dr. G. Markelov, and to Prof. I. Graur and Prof. P. Perrier for their Valuableadvice during my stays at AOES-Netherlands and at IUSTI-UP, respectively.

I am grateful to GASMEMS coordinators Prof. S. Colin, Prof. L. Baldas, and C. Prado,group and task coordinators Prof. D. Valougeorgis and Prof.Y. Zhang, and to Prof. J.Brandner, and Prof. A. Frijns for their suggestions during the course of the project.

I appreciate the support of the IMech-BAS director Prof. D.Sc. V. Kavardjikov, the HoD,the Scientific Council, Prof. D.Sc. D. Danchev, and Prof. E. Manoach. I also acknowledgethe support from the fellow members of the IMech-BAS Asst. Prof. V. Rusinov, Asc. Prof.N. Pesheva, Asc. Prof. M. Datcheva, Prof. D.Sc. N. K. Vitanov, G. Valchev, and Sr. Asst.Prof. K. Shterev.

I thank Prof. D.Sci. Z. Zapryanov, Prof. D.Sci. I. Dimov, Prof. Dr. A. Karaivanova, Prof.D.Sci. S. Stefanov, and Asc. Prof. Sl. Slavtchev for their Valuable advice on this thesis ascommittee members.

My special thanks to Ms. Elena Stefanov for the warmth and hospitality shown in the earlydays of my stay in Sofia, and to Alice for friendship during my stay in Marseille.

I also appreciate the friendship and questions from my fellow GASMEMS members M. Var-gas, Yongli Li, J. Kim, A. Vittoriosi, Y. Yang, A. Dinler, O. Buchina, M. Hadj-Nacer, O.Rovenskaya, M. Rojas, E. Arlemark, A. Mikos-Wojsznis, N. Dongari, L. Szálmas, H. Chalabi,V. Leontidis, G. Dumazer, and F. Samouda.

I acknowledge the financial support from the Institute of Mechanics-BAS that provided methe opportunity to pursue my doctoral study. The research leading to these results hasreceived funding from the European Community’s Seventh Framework Programme (ITN -FP7/2007-2013) under Grant Agreement No. 215504.

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Contents

Declaration of Authorship iii

Preface v

Abstract vii

Acknowledgements viii

Contents ix

List of Algorithms xiii

List of Figures xiii

List of Tables xix

Abbreviations xxi

Physical Constants xxiii

1 Introduction 11.1 Direct Simulation Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Hard Sphere model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Collision approaches: Greedy on time Vs Greedy on memory . . . . . . . . . 7

1.3.1 Greedy on simulation time . . . . . . . . . . . . . . . . . . . . . . . 81.3.1.1 No Time Counter scheme . . . . . . . . . . . . . . . . . . 81.3.1.2 Majorant collision scheme . . . . . . . . . . . . . . . . . . 9

1.3.2 Greedy on memory: Bernoulli Trial scheme . . . . . . . . . . . . . . . 101.3.3 Greedy on time: Simplified Bernoulli Trials scheme . . . . . . . . . . 10

1.4 Repeated collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 Strang splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Moments of the distribution function . . . . . . . . . . . . . . . . . . . . . 131.7 Time average and Ensemble average . . . . . . . . . . . . . . . . . . . . . . 131.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Pressure Boundaries 17

ix

Contents x

2.1 Techniques for the application of a specified pressure . . . . . . . . . . . . . 172.1.1 Using Maxwellian distribution and injecting particles at a constant rate 172.1.2 Implicit boundary conditions . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Unsteady state DSMC simulations 253.1 A DSMC code for unsteady state simulations . . . . . . . . . . . . . . . . . 263.2 Unsteady state DSMC simulation of plane Couette flow . . . . . . . . . . . . 273.3 Unsteady state DSMC simulations of the gas flow past a backward facing step 28

3.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Parallel implementation of the DSMC code for gas flow simulation in mi-crochannel configurations 394.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Parallelization using the OpenMP . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 Parallel implementation of a 3D DSMC code for gas flow simulationin microchannel using OpenMP . . . . . . . . . . . . . . . . . . . . . 41

4.3 Parallel implementation of the DSMC code for gas flow simulation in mi-crochannel configurations using the MPI . . . . . . . . . . . . . . . . . . . . 47

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5 Mass flow rate through straight and 90 degree bend micro channels in slipand transitional flow regimes 515.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Problem formulation and computational consideration . . . . . . . . . . . . . 535.3 Details of the DSMC technique . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.3.1 Inlet-outlet boundary conditions . . . . . . . . . . . . . . . . . . . . 565.3.2 Definition of the flow parameters . . . . . . . . . . . . . . . . . . . . 56

5.4 Simulation of the flow through a straight microchannel . . . . . . . . . . . . 575.4.1 Flow rate through a straight microchannel . . . . . . . . . . . . . . 575.4.2 Influence of the relative dimensions of the cross section on the flow

rate through a straight microchannel . . . . . . . . . . . . . . . . . . 595.5 Simulation of the flow through a 90 degrees bend microchannel . . . . . . . 61

5.5.1 Reduced flow rate through a 90 degrees bend microchannel . . . . . 615.5.2 Effect of the cross section on the flow rate through a 90 degrees bend

microchannel and effective length of a 90 degrees bend microchannel 645.5.3 Comparison of the flow rate through a straight microchannel to flow

rate through a 90 degrees bend microchannel . . . . . . . . . . . . . 685.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6 Effects of finite distance between transversal dimensions in microchannel con-figurations: DSMC analysis 716.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2 Computational consideration . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2.1 Details of the DSMC technique . . . . . . . . . . . . . . . . . . . . . 736.2.2 Definition of the flow parameters . . . . . . . . . . . . . . . . . . . . 74

Contents xi

6.3 Results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.3.1 Effects of finite distance between opposite tansversals in straight and

90 degree bend microchannels . . . . . . . . . . . . . . . . . . . . . 756.3.2 Effects of finite distance between opposite tansversal walls in T-junction 81

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7 Experimental measurements 87

A Appendix 93A.1 Box-Muller algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Concluding remarks 951 Scientific contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 952 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

List of publications 97I International Conference Publications . . . . . . . . . . . . . . . . . . . . . . 97II International Journal Publications . . . . . . . . . . . . . . . . . . . . . . . . 97

Bibliography 99

List of Algorithms

1 An iteration over the core steps of the DSMC . . . . . . . . . . . . . . . . . . 4

2 A DSMC algorithm for time-averaged simulation . . . . . . . . . . . . . . . . 6

3 Knuth’s algorithm for a Poisson distributed random variable [1] . . . . . . . . 9

4 A general flowchart of the Unsteady state simulations using DSMC . . . . . . 25

5 An iteration over the core steps of the DSMC for unsteady state simulation . . 26

xiii

List of Figures

1.1 Time averaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Ensemble averaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 The established pressure distribution in a straight channel (top at Kn = 0.1and bottom at Kn = 0.5 for pressure ratio Pin/Pout = 3 . . . . . . . . . . . 19

2.2 Time evolution of the number fluxes through a straight channel(flow rates)at inlet and outlet calculated by DSMC for a pressure ratio Pin/Pout = 3 . . 19

2.3 The established pressure distribution in a vertical bend . . . . . . . . . . . . 192.4 Time evolution of the number fluxes at inlet and outlet of a vertical bend

calculated by DSMC for a pressure ratio Pin/Pout = 3 . . . . . . . . . . . . 202.5 The established pressure distribution in a T-junction . . . . . . . . . . . . . . 202.6 Time evolution of the number fluxes at inlet and outlet of a T-junction

calculated by DSMC for a pressure ratio Pin/Pout = 3 . . . . . . . . . . . . 202.7 The velocity components in a T-junction . . . . . . . . . . . . . . . . . . . . 212.8 The mass flow rate through straight micro channels of various lengths, cal-

culated using two approaches for the sound speed, Viz: using a =√γRT ,

and using a = dpdρ

(1 + dρ

ρ

). The diffuse reflection with complete energy

accommodation was considered on the solid walls [2]. . . . . . . . . . . . . 24

3.1 The construction of a 2D-DSMC software . . . . . . . . . . . . . . . . . . . 273.2 A schematic of the Couette flow . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Velocity profiles in impulsively started shear flow (Kn = 1, U = 0.05Vmp) at

various times(500∆t, 1000∆t, 1500∆t, and 2000∆t) . . . . . . . . . . . . . 283.4 The backward facing-step. . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Flow rate of mass flux for a fixed inlet Kn . . . . . . . . . . . . . . . . . . . 313.6 Growth rate of mass flux rate for a fixed inlet rarefaction . . . . . . . . . . . 323.7 Flow rate of mass flux and the growth rate of mass flux rate for a fixed outlet

rarefaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.8 The reduced flow rate past a backward facing step for Knout = 0.05. . . . . 343.9 Scalar pressure iso-lines for fixed inlet rarefaction Knin = 0.2 . . . . . . . . . 343.10 Scalar pressure iso-lines for fixed outlet rarefaction Knout = 0.125 . . . . . . 343.11 Shear stress (cell volume based) iso-lines for fixed inlet rarefaction Knin = 1.0 353.12 Shear stress (cell volume based) iso-lines for fixed outlet rarefaction Knin =

0.125 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 The shared memory(fork-join) model . . . . . . . . . . . . . . . . . . . . . . 404.2 The structure of the DSMC code implemented . . . . . . . . . . . . . . . . . 414.3 The data flow diagram of the DSMC code implemented . . . . . . . . . . . . 42

xv

List of Figures xvi

4.4 Mach Mach number along the 90 degree bend microchannel with a squarecross section, simulated using the OpenMP parallel version of the 3D DSMCcode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5 A DSMC algorithm (for time-averaged simulation) parallelized using theOpenMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.6 The flowchart of the DSMC code(for unsteady state simulations) parallelizedusing the standard MPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.7 validation of the MPI implementation considering a straight microchannel . . 49

5.1 The general view of 3D-microchannel geometry. . . . . . . . . . . . . . . . . 545.2 A typical measurement of the flow rate in the steady-state. . . . . . . . . . . 555.3 The straight microchannel. . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.4 The effect of the relative dimensions of the cross section on the flow rate

via a relative flow rate (G/GW=H) through a straight microchannel comparedto Sharipov [3] at δ = 1 (in the left), and at δ = 10 (in the right). Thecharacteristic dimension was D = H. . . . . . . . . . . . . . . . . . . . . . . 60

5.5 Relative flow rates (G/Gδa=0.01) in a straight microchannel (in the left), and acomparison of the straight microchannel with a square section with that ofa rectangular cross section with (W/H →∞) (2D simulation) (in the right).The characteristic dimension was D = H. . . . . . . . . . . . . . . . . . . . 60

5.6 The mass flow rate through a 900 bend microchannel with a square cross-section vs pressure drop between inlet and outlet cross-sections for the rar-efaction parameter δ = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.7 Mass flow rate through a 900 bend microchannel with a square cross section,for different pressure drops, simulated using the DSMC, and from the Croceet al [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.8 The 900 bend microchannel. . . . . . . . . . . . . . . . . . . . . . . . . . . 635.9 The geometry of a 900 bend microchannel, considered for studying the effect

of relative dimensions of its cross section on the flow rate . . . . . . . . . . . 655.10 Relative flow rates (G/Gδa=0.01) through a 900 bend microchannels with a

square cross section, a rectangular cross section with H = 9W , and a crosssection with W/H → ∞ (2D) (in the left), and a zoom around the minima(in the right). The characteristic dimension was D = minW,H. . . . . . . 65

5.11 A comparison of the 900 bend microchannel with a square section with thatof a rectangular cross section with H = 9W , and another with W/H → ∞(2D simulation), via mass flow rates in the free molecular, the transitionaland the slip regimes. The characteristic dimension was D = minW,H. . . 66

5.12 The effect of the cross section on the reduced flow rate (G ) through a 900

bend micro channel for a constant hydraulic diameter, for various pressuredrops, with CL = 21H. Shown are G/GW=H; for δa = 1 (up), and δa = 10(down). The characteristic dimension was Dh . . . . . . . . . . . . . . . . . 67

5.13 A comparison of the straight and 90 degree bend microchannels consideringthe mass flow rate, comparison was performd considering the argon gas aswell as nitrogen gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.1 The general view of 3D-microchannel geometry. . . . . . . . . . . . . . . . 736.2 The straight microchannel. . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.3 Pressure deviation in a straight microchannel from the 2D analytical expres-

sion (6.4) by Arkillic et al [5], for a Knout = 0.072, L = 9H and Pin/Pout = 3. 76

List of Figures xvii

6.4 Pressure profiles along the streamwise direction in a straight microchannel,for a Knout = 0.072, L = 9H, from the DSMC and the 2D analyticalexpression (6.4) by Arkillic et al [5]. . . . . . . . . . . . . . . . . . . . . . . 76

6.5 The temperature profile along a straight microchannel, for a Knout = 0.072and Pin/Pout = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.6 Temperature profiles at the exit of a straight microchannel (length L = 9H),for a Knout = 0.072, L = 9H and Pin/Pout = 3. . . . . . . . . . . . . . . . 77

6.7 The stream wise component of the velocity (u) at the exit, at half the width,along the height; for a Knout = 0.072 and Pin/Pout = 3. . . . . . . . . . . . 78

6.8 The stream wise component of the velocity (u) at the exit, at half the height,along the width; for a Knout = 0.072, L = 9H and Pin/Pout = 3. The caseW = 10H was simulated considering a plane of symmetry at half of the width. 78

6.9 Scaling of the stream wise component of the velocity (u) with the wideningof the straight microchannel, for a Knout = 0.072, L = 9H and Pin/Pout = 3. 79

6.10 The 900 bend microchannel. . . . . . . . . . . . . . . . . . . . . . . . . . . 796.11 The effect of the width to height ratio in a straight and a 900 bend mi-

crochannels with a constant Dh, for a pressure drop of 2.0 (relative to outletpressure). The characteristic dimension was the hydrauilic diameter (Dh). . . 81

6.12 A T-junction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.13 Difference in shares (% of the mass flow rate through the straight arm - %

of the mass flow rate through the vertical arm) of the outbound arms in thetotal outward mass flow rate in the T-junction. . . . . . . . . . . . . . . . . 84

6.14 Share of the vertical arm (% of the mass flow rate) in the outbound totalflow rate in T-junction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.1 Schematic diagram of the experimental setup (present at UP/IUSTI) usedfor measuring the gas flows through a 90 degree bend micro-channel . . . . 87

7.2 A geometry of the 90 degree bend micochannel. . . . . . . . . . . . . . . . . 887.3 The reduced flow rate through the 90 degree bend micochannel . . . . . . . 897.4 A comparison of the non- dim. flow rate (S) through the 90 degree bend

micochannel with that of second order expression for the straight channel(Eq. 7.7) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

List of Tables

3.1 The gap values corresponding to Kninitial . . . . . . . . . . . . . . . . . . . 30

5.1 The grid dependence of the reduced mass flow rate through a straight microchannel with a square cross section for various lengths, δa = 1. Grid A:15×15×15 cells with 20 PPC, grid B: 11×11×11 cells with 50.8 PPC, gridC: 7× 7× 7 cells with 196.8 PPC. The characteristic dimension was D = H. 57

5.2 The reduced mass flow rate (G) through a straight micro channel with asquare cross-section and various lengths; Pin/Pout = 2, δa = 10. The grid Aof 15× 15× 15 cells with 20 PPC is used. The characteristic dimension wasD = H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3 The mass flow rate through straight micro channels of various cross sections,for a pressure difference: (Pin−Pout)/Pout = 1 , and with a length L = 21H.The characteristic dimension was D = H. . . . . . . . . . . . . . . . . . . . 59

5.4 Effect of the cross-section shape on the reduced mass flow rate G/GW=Hthrough a straight micro channel of a constant hydraulic diameter, for a pres-sure ratio Pin/Pout = 3 and channel length equal to 21H. The characteristicdimension was D = H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.5 Effect of pressure drop and channel length on the reduced flow rate G througha 900 bend micro channel of the squared cross section. The characteristicdimension was D = H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.6 The reduced flow rate, through a 900 bend micro channel with a characteristicdimension D = W (the width of the microchannel), for various heights, fora pressure drop ( (Pin − Pout)/Pout) = 1 at δa = 11.52. The characteristicdimension was D = W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.7 The reduced flow rate, through a 900 bend micro channel with a characteristicdimension D = H (the height of the microchannel), for various widths, fora pressure drop: ( (Pin − Pout)/Pout) = 1, at δa = 11.52. The characteristicdimension was D = H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.8 The reduced mass flow rate (G) through a 900 bend microchannel, andthrough a straight microchannel, for a pressure drop: ( (Pin − Pout)/Pout) = 1.The characteristic dimension was D = H. . . . . . . . . . . . . . . . . . . . 66

6.1 The effect of the two finite dimensions of the cross section of a microchan-nel on the mass flow rate through straight micro channel, for various crosssections, for a pressure difference: (Pin−Pout)/Pout = 1 , and with a lengthL = 21H. M is the mass flow rate per unit cross sectional area. . . . . . . . 80

6.2 The influence of the transversal walls on the mass flow rate through 900 bendmicrochannel, for various cross sections, for a pressure difference: (Pin −Pout)/Pout = 1 , and with a length L = 15H. M is the mass flow rate perunit cross sectional area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xix

List of Tables xx

6.3 The distribution of the flow rate between the outbound arms of a T-junction,for various widths, for CL = 15H. Where, Mtotal = Mstraight + Mvertical.M is the mass flow rate per unit cross sectional area. The cross sectionalarea of both the outbound arms was the same. . . . . . . . . . . . . . . . . 82

6.4 The influence of the transversal walls on the outbound flow rate through aT-junction, at δa = 1. Where, Mtotal = Mstraight+Mvertical. M is the massflow rate per unit cross sectional area. The cross sectional area of both theoutbound arms was the same. . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.5 The influence of the transversal walls on the outbound flow rate through aT-junction, at δa = 10. Where, Mtotal = Mstraight + Mvertical. M is themass flow rate per unit cross sectional area. The cross sectional area of boththe outbound arms was the same. . . . . . . . . . . . . . . . . . . . . . . . 83

6.6 The distribution of the flow rate between the outbound arms of a T-junction,for various values of rarefaction parameter (δa), for a pressure difference:(Pin − Pout)/Pout = 2 and CL = 15H with H = 19.83 × 10−6 m. Where,mtotal = mstraight+mvertical. The cross sectional area of both the outboundarms was the same. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Abbreviations

BFS Backward Facing Step

BT Bernoulli Trial scheme for computing binary collisions

CL Center line Length of the microchannel configuration

CVT Constant Volume Technique

CDG Capacitance Diaphragm vacuum Gauge

DSMC Direct Simulation of Monte Carlo

HS Hard Sphere

I/O Inlet / Outlet boundary

Kn Knudsen number

MEMS Micro Electro Mechanical Systems

MPI Message Passing Iinterface, a communication system designed to enable-

parallel programming for distributed memory architectures

NTC No Time Counter scheme for computing binary collisions

OpenMP Open Multi Processing application program interface that supports multi-platform-

shared memory multiprocessing programming

PBC Periodic Boundary Condition

PPC Particles Per Cell

SBT Simplified Bernoulli Trial scheme for computing binary collisions

xxi

Physical Constants

Boltzmann constant kB = 1.3806488× 10−23 JK−1

Gas constant R = 8.3144621 JK−1mol−1

Avogadro constant NA = 6.02214129× 1023 mol−1

xxiii

To myFather and Sister. . .

xxv

Chapter 1

Introduction

Gas flows are an inherent part of various technological processes and applications that areimplemented in micro-electro-mechanical systems (MEMS). Typically, MEMS devices havea characteristic length between 1 × 10−6 m and 1 × 10−3 m. Due to a very small size ofthe micro-systems rarefaction effects have to be taken into account, which often requiresan application of both continuum and kinetic numerical approaches. The Knudsen number(Kn) is a measure of rarefaction and is defined as the ratio of the mean free path to acharacteristic length scale of the micro device. The flow regimes are conventionally classifiedas:

• Kn → 0, inviscid limit where the transport term vanish. The Euler equations areapplicable.

• Kn < 0.01, the Hydrodynamic regime, where transport collisions are dominated. Thegas may be treated as continuum medium and the Navier-Stokes equations with no-slipboundary conditions are applicable.

• 0.01 6 Kn 6 0.1, the Slip flow, non-equilibrium phenomena appear in the bound-ary regions. The Navier-Stokes equations with the slip boundary conditions may beapplied.

• 0.1 6 Kn 6 10, the Transition regime, a kinetic description of the gas is necessary.Inter-molecular collisions are reduced and gas-surface interaction play important role.

• Kn > 10, the Free molecular flow, ballastic motion of molecules.

The computational analysis of gaseous flows in MEMS devices operating in flow regimesat large Knudsen numbers (Kn > 0.1) cannot be based on classical continuum models offluid motion because they are not valid for non-equilibrium flow conditions, although several

1

Chapter 1. Introduction 2

attempts have been made to extend their validity [6–8] into the early transitional regime. Insuch flows, the mean free path of the gas molecules is comparable to the characteristic sizeof the micro device and the kinetic effects of rarefaction and non-equilibrium must be takeninto consideration. The rarefied flows can be analyzed in detail using the direct simulationMonte Carlo (DSMC) method [2, 9, 10] and kinetic models. The DSMC technique usesmodel particles that move and collide in physical space to perform a direct simulation of themolecular gas dynamics.

Understanding of the gas flow behavior is very important for improvement of the MEMS andin some other industrial applications. Some popular implementations of the DSMC includethe studying the MEMS applications and aerospace and vacuum applications that fall in thezones (of rarefaction) where the Navier-Stokes approach breaks down.

The miniaturization has emphasized on the of micro- and nanometer-sized devices as theyoffer increased reliability, low cost and high efficiency. Similarly, Vacuum flows are encoun-tered in many applications, such as a simple pressure sensor, the vacuum systems of fusionreactors, etc. High altitude aerodynamics should to be investigated thoroughly for a properoperation of spacecrafts and satellites [11]. The DSMC method is often utilized to studythese conditions. Thus, the accuracy and efficiency of simulations is of high importancefor the design, manufacturing and optimization of these devices. MEMS have been used assensors for mass flow, velocity and sound, pressure, temperature; as actuators for linear andangular motions, and as simple components for complex systems such as micro-heat-enginesand micro-heat-pumps; also, as micro-resonators and comb drive sensors. The torques andforces induced by the rarefied gas ambient can significantly affect performance and sensi-tivity characteristics. The flows through porous media are also found to be rarefied. Theread-head sliders in hard disk drives operate under conditions where the Knudsen number inthe micro gap is relatively large. Some of the applications of the rarefied gases include: hy-personic flows around space vehicles and satellites during re-entry in rarefied conditions, theconstruction of micro-scale propulsion devices, for a detailed analysis of vacuum pumps andgas separators, multi-layer insulation blankets, Vacuum deposition systems that are used forthe fabrication of thin-film materials, MEMS and nanocomposites. The rarefied or vacuumconditions are maintained in a particle accelerators, and fusion reactors. [12–31].

The microscopic models consider the gas as a set of discrete molecules providing theirposition and velocity at all times. The Boltzmann equation (Eq. 1.1) is the mathematicalmodel at microscopic level [10]. The current work is limited to the case where no externalfields are present; extension of this work to the presence of external fields is straightforward.In the absence of external fields, the Boltzmann equation can be written in the following


form:

∂f

∂t+ ξ · ∂f

∂X=[df

dt

]coll

=∫ ∫

(f ′∗ f ′ − f∗ f)B(g, θ)d2Ω(θ)d3ξ∗. (1.1)

Therefore, the target is to obtain the distribution function, which defines the fraction ofparticles in the phase space, that have approximately the velocity ξ and position X at timet.

1.1 Direct Simulation Monte Carlo

The Monte Carlo methods utilize the random processes to obtain an approximate solutionto problems of computational Physics. The Monte Carlo approximation thus obtained wouldbe treated as a solution within some specified limits with a certain probability [32]. Thecentral limit theorem gives the information about the accuracy of the estimate after a finitenumber of draws, and the law of large numbers makes sure that the estimate converges tothe correct value with the increase of the number of draws.

The direct simulation Monte Carlo (DSMC) [2, 9, 10] method proposed by Bird, has beenextensively applied in aerospace engineering, in vacuum technology and to study the gasmicro flows. The basic assumptions of the DSMC method are:

• the assumption of molecular chaos – the probability distribution function of a twoparticle configuration is product of the individual one particle probability distributionfunctions.

• the gas is dilute, i.e., under the sufficiently low densities, the mean molecular spacingin such gas would be much larger than the effective molecular diameter- which allowsthe collisions between particles to be considered as binary collisions.

A dilute gas [2] is such gas where only a very small proportion of space is occupied bymolecules (δ >> d, where δ = n(−1/3) is the mean molecular spacing and d is the effectivediameter of the molecule, n is the number density of the molecules), and each moleculewill move outside the range of influence of other molecules. Under these conditions, it isconvenient to assume that only binary collisions happen within the gas.

The DSMC method utilizes the statistical approach for computing collisions between par-ticles. Wagner [33] provided a proof of convergence of the DSMC to the solution of theBoltzmann equation in the limit of infinite number of model particles in a cell.


The standard DSMC [2] makes use of the following time splitting scheme, to determine theevolution of the distribution function f(tk) from a time tk to f(tk+1) (at time tk+1) anapproximate solution of the physical process, as given in the following two steps:

Collisions between model particles, where the molecular velocities may be changed whilethe molecular position is kept the same

f [(tk + τ),X(tk),Ξ(tk + τ)] = Sτ,∆xQ

f [(tk + τ),X(tk),Ξ(tk)]

. (1.2)

Movement: A convective motion of the particles, without changing their velocities

f [(tk + τ),X(tk + τ),Ξ(tk + τ)] = SτD

f [(tk),X(tk),Ξ(tk + τ)]

. (1.3)

Where Sτ,∆xQ and SτD are the operators for simulating the action of the collision and theconvective movement of the model particles, τ is the time step and ∆x is the cell size.Therefore, considering a global operator which evaluates the kinetic equation solution at tk+1

from the previous state defined as Sτ,∆xQ+D = Sτ,∆xQ SτD, and as was shown by Bobylev andOhwada [34] that such a splitting scheme delivers an approximate solution of the Boltzmannequation with an accuracy of first order with respect to cell size and time step.

The flow chart in the algorithms 2 and 1 shows an application of collision and movementoperators in the DSMC simulation. The process of implementation may be described as

Algorithm 1: An iteration over the core steps of the DSMC

follows:


• The time interval [0,T], over which the solution is sought, is subdivided into sub-intervals with step τ

• For each element of the junction configuration, solid boundaries and open boundariesof elements of the construction are defined according to the chosen geometry. Thespace domain of each element is subdivided into cells with sides ∆x and ∆y (∆x, ∆yand ∆z in case of 3D simulation).

• The gas molecules are simulated in each of the element with N model particles havingpositions (Xi(t)) and velocities ξi.

• Within each time step there are Nm particles in the mth cell; this number is varied bycomputing its evolution in the following two stages:

Stage 1. The binary collisions in each cell are calculated without moving the particlesby using the hard sphere collision model.

Stage 2. All particles in the computational domain are moved at a distance propor-tional to their new velocities without colliding. If the distance, at which eachparticle is shifted, is larger than the distance to the solid boundary, then an in-teraction with the boundary occurs. A list of particles that crossed the openboundaries between each two neighboring elements is created and saved in abuffer. These particles are removed from the particle list of the current element.Each element receives a number of particles listed in the corresponding buffer.According to pressure maintained at the external end, a number of particles hav-ing semi-Maxwellian velocity distribution enters the computational domain fromthe corresponding external end.

• Stages 1 and 2 are repeated until end of computation time (T)

• The important moments of the distribution functions are calculated by ensemble aver-aging over a number of process trajectories and over a long interval of time from onetrajectory for the case of time-averaging.

1.2 Hard Sphere model

The inverse power law model, which is a point centric repulsive model expressed as

F ∝ 1rη. (1.4)

Where, r is distance between the interacting particles, with a repulsive force F acting betweenthem. The hard sphere (HS) model [2] may be classified as a special case of the inverse


Algorithm 2: A DSMC algorithm for time-averaged simulation

power law model with η =∞, the interaction becomes effective at r = 12(d1 + d2), denoted

as r = d12, with the d1 and d2 being the respective diameters of the inteacting particles.For an apse line chosen through the centers of the spheres, the impact factors were shownto be

b = d12sinθA, (1.5)∣∣∣∣∣ dbdχ∣∣∣∣∣ = d12

2 sin

(χ

2

). (1.6)

Where, χ = 2θA is the angle between the incident and reflected directions of model particlewith respect to a stationary collision partner, and b is the (closest approach) distance of theincident particle from the apse line. The collision cross section is given by σ = d2

124 , therefore,

in the center of mass frame of reference, the scattering from the HS molecules is isotropic(i.e., independent of the χ). The total collision cross section is

σT =∫ 4π

0σdΩ = πd2

12. (1.7)


1.3 Collision approaches: Greedy on time Vs Greedy on mem-ory

There were a number of binary collision schemes proposed and implemented for the DirectSimulation Monte Carlo (DSMC) of a rarefied gas flow; a few of them are listed below:

• Time Counter scheme (G. Bird [2])

• No Time Counter scheme (G. Bird [2, 9])

• Majorant Frequency scheme (M. Ivanov et. al. [35, 36])

• Null Collision scheme (K. Koura [37])

• Modified Nanbu scheme (K.Nanbu and Babovsky [38])

• Ballot Box scheme(V. Yanitskiy [39])

• Bernoulli trials scheme (V. Yanitskiy and S. Stefanov [40])

• Simplified Bernoulli trials scheme (S. Stefanov [40])

From the implementation point of view, some of these popular collision schemes for theDSMC method for the gas flow simulation in microchannels may be classified into thefollowing two approaches

Greedy on the simulation time

Greedy on the memory

The first approach includes the collision schemes like No Time Counter (NTC) scheme,Majorant Collision scheme, and Simplified Bernouli Trial scheme(SBT), etc. Whereas, theBernouli Trial scheme(BT) falls under the second category.

For a given accuracy, based on the implementation restrictions and requirements, if one hasto solve an Engineering or Scientific problem that requires an amount of memory that can beallotted, then the obvious choice would be the collision schemes that fall under the ‘Greedyon the simulation time category’. However, if the adequate amount of the memory is notavailable for using a large number of particles for the simulation, then one has to choose fromthe ‘Greedy on the memory category’. The first category uses a considerably large numberof particles for the the gas flow simulation in microchannels using the DSMC method, e.g.,about 20 particles per cell are used while using the schemes like NTC. However, when the


collison schemes from the second category were chosen, a very few particles per cell wouldbe utilized, e.g., as few as about 2-3 particles per cell might be used while using the schemeslike BT. The collision schemes from the second category take a relatively larger amount oftime (with a lower number of particles) to achieve the same level of accuracy in the solution.Therefore, the trade off is between the simulation time and the memory allocation.

1.3.1 Greedy on simulation time

The implemented versions of the NTC collision scheme (in the current work) uses the fol-lowing equation for a collision frequency: ν = 0.5N(N −1)[gσT (g)]max. This is identical tothe corresponding equations in the Null-collision DSMC scheme with excluding self-collisions[41] and in the majorant collision frequency DSMC scheme [35, 36]. A comparison of theNTC and majorant collision schemes was performed by Maltsev [42], analyzing the errorsconsidering the Fourier problem.

1.3.1.1 No Time Counter scheme

The details of the NTC scheme and its recent development are given in the references [2]and [9]. A regular NTC method employs a collision frequency

ν = 0.5NNFN (σT gij)max∆t/Vc. (1.8)

The potential collision pairs are selected from a collision cell containing N model particlesat each time step ∆t, and the collisions occur between each pair with probability

σT gij(σT gij)max

or〈σT gij〉〈σT gij〉max

. (1.9)

Here, FN is the number of real molecules represented by each model particle, Vc is thevolume of the collision cell, σT is the total collision cross-section, and gij is the relativespeed in the collision. The (σT gij)max is the value that has occurred in the collision cell.

The version of the NTC scheme proposed in the Ref. [9], uses the following collision fre-quency:

ν = 0.5N(N − 1)FN (σT gij)max∆t/Vc. (1.10)

The time complexity of the NTC scheme is O(N (l)), where the superscript l refers to the

cell.


1.3.1.2 Majorant collision scheme

Ivanov et al [35] have shown that the Majorant Frequency Schemes are mathematicallystrictly derived from the Master Kinetic Equation (MKE). The majorant frequency algorithmfor the master kinetic equation was presented as follows:

Step 1 Selection of the initial state (t0,C0) of N-particle trajectory from the probabilitydensity f0

NC0; the number of state is k = 0.

Step 2 Sampling the random variable τ from νmexpνmt, and tk+1 = tk + τ .

Step 3 If tk+1 > T , then the trajectory is terminated (go to Step 1). If tk+1 < T , then thepair i, j is randomly selected from N(N − 1)/2 possible pairs and if

Rf <g′i, jσt(g

′i, j)

[g′i, jσt(g′i, j)]max

, (1.11)

where Rf is uniformly distributed over (0, 1), then the collision occurs (go to Step 4).Otherwise collision is observed (go to Step 2).

Step 4 Post-collisional particle velocities are calculated. Afterwards, the transition to Step2. Now the state of N-partcle system is . . . (tk+1,Ck+1)

In the current work, the following algorithm by Knuth [1] was used to follow the Poissondistribution performing the collisions.

Input: (δτ × νm)Output: int(n− 1)L← exp(−δτ × νm);a← 1;n← 1;while a > L do

Un← rand(seed);a← a× Un;n← n+ 1;

endAlgorithm 3: Knuth’s algorithm for a Poisson distributed random variable [1]

Where, νm is majorant collision frequency, and δτ is the uniformly chosen from (0 , ∆t), andUn is uniformly selected from (0 , 1).

The time complexity of the majorant collision frequency scheme is O(N (l)), where the

superscript l refers to the cell.


1.3.2 Greedy on memory: Bernoulli Trial scheme

A two variants of the BT scheme were proposed by V. Yanitskiy and S. Stefanov [40] Viz:Bernoulli Trials scheme and Simplified Bernoulli Trials scheme.

The first variant of the BT scheme consists of the following steps.

Step 1 For each cell l (l = 1, ..., L) with volume V (l)c : Each particle in the cell l with

velocities (ξi , ξj), i < j = 1, . . . , N (l) is checked with the probability

Pij = ∆tσijcr,ijV

(l)c

. (1.12)

Step 2 The velocities of the particles would be updated to a post collision value, when thecollision is accepted; otherwise, remain the same.

This version of the BT scheme requires a number of operations O(N (l)2) in cell l, because

the number of collision pairs that are checked for collision every time step is N (l)(N (l)−1)/2.This BT method allows to work with a smaller number of particles in the grid cells, as fewas 2-3 per cell. However, in the current work, a particle number (per cell) ranging from 5-20were utilized.

An efficient implementation of the BT scheme would be realized, using the Strang splittingmethod, which allows the potential collisions between particles of neighboring cells. Thisnumber of collisions becomes more important as the number of particles per cell is decreased.

1.3.3 Greedy on time: Simplified Bernoulli Trials scheme

The Simplified Bernoulli Trials (SBT) scheme realizes a number of computations proportionalto the number of particles per cell N (l), as the number of collision pairs which are checkedfor collision every time step is (N (l) − 1)/2. Its description may be expressed as:

Step 1 the sequence of particle pairs i = 1, . . . , (N (l) − 1) is chosen from N (l) particles incell l as follows:

1. The first particle i is the particle with index i in the particle list for cell l.

2. The second particle j ∈ [(i + 1), . . . , N (l)] is chosen with probability 1/k fromk = (N (l) − i) taking place in the list after particle i.


Step 2 then the particle pair (i , j) is checked for collision with probability

Pij = k∆tσijcr,ijV

(l)c

(1.13)

if the collision is accepted then velocities (ξi, ξj) are changed to their post-collisionvalues (ξ′i, ξ

′j), otherwise they remain unchanged.

A peculiar property of the BT and SBT is that, they avoid the repeated collisions between thesame pair of particle, hence a reduction in the local collision frequency. The time complexityof the SBT scheme is O

(N (l)) .

The authors of the Ref. [43] have proposed a combination of the SBT collision schemewith the transient adaptive subcell (TAS) technique implemented in the direct simulationMonte Carlo (DSMC) for calculation of non-equilibrium gas flows with reduced computationalresources.

1.4 Repeated collisions

It has been proved that the effect of a number of repeated consecutive collisions of the samepair is equivalent to the realization of only one collision, which effects in the reduction ofthe collision rate.

Let us assume, (ξ(0)i , ξ

(0)j ) are the initial velocities of the molecules i and j, and after n

repeated collisions their velocities will be given by:

ξ′(n)i = 1

2[(ξ(n−1)i + ξ(n−1)

j ) + |ξ(n−1)i − ξ(n−1)

j |ω(n)] (1.14)

ξ′(n)j = 1

2[(ξ(n−1)i − ξ(n−1)

j ) + |ξ(n−1)i − ξ(n−1)

j |ω(n)] (1.15)

Where, the ω denotes a isotropic unit vector. It is to note that, the sum of their velocitiesis conserved ξ(k)

i + ξ(k)j = ξ

(m)i + ξ

(m)j , ∀ k, m ∈ [0 , n]; as well as the magnitude of their

relative velocity |ξ(k)i − ξ

(k)j | = constant, ∀ k ∈ [0 , n], therefore, it can be deduced that the

effect is the same as only one collision:

ξ′(n)i = 1

2[(ξ(0)i + ξ(0)

j ) + |ξ(0)i − ξ

(0)j | ω

(n)] (1.16)

ξ′(n)j = 1

2[(ξ(0)i − ξ

(0)j ) + |ξ(0)

i − ξ(0)j | ω

(n)] (1.17)


1.5 Strang splitting

In order to allow the collisions between particles of neighboring cells, a two-step collisionalgorithm applied twice on a dual grid, the following splitting scheme was realized: Collision1

- Move - Collision2 - Sampling. Where, the Collision1 was performed for half of the timestep on the main grid, while the Collision2 was performed for another half of the time stepon a shifted grid. This shifted grid is the original grid which was moved half of the cell sizein all the three directions for a 3D simulation (in two directions for a 2D simulation). Thefollowing are benefits from the applied splitting strategy:

• It improves the accuracy of the (movement-collision) splitting scheme, which becomesof second order with respect to the time step O

([∆t]2

).

• The stochastic properties of the collision algorithm are improved, due to the collisionsbetween particles in neighboring cells.

If we denote the numerical algorithms approximating the action of the collision and convectiveterms by Sτ,hQ and SτD, respectively; and if we denote by Sτ,hQ+D the operator evaluating thesolution of the Boltzmann eq. at tk + 1 from the state at tk, then the splitting method isexpressed with the approximation

Sτ,hQ+D ≈ SτDS

τ,hQ (1.18)

Using the result obtained by Bobylev and Ohwada [34], one can show that the splittingmethod approximates the Boltzmann equation with accuracy O

(τ + h

). The accuracy with

respect to time step can be improved by using Strang splitting symmetric scheme:

Sτ,hQ+D = Sτ/2Q S

τD[Sτ/2Q (f0)]+ O

(τ3). (1.19)

During the particle simulation the following two stages are performed over each time step(tk, tk+1), k = 1, . . . ,K:

Stage 1. Operator Sτ,hQ (the standard NTC collision procedure). Three steps are includedin the “No Time Counter" collision procedure performed in each cell l, ∀ l = 1, . . . ,M :

• computing the number of particle pairs Nc to be checked for a collision;

• acceptance-rejection of each pair (i, j), 1 6 i < j 6 N (l), chosen at randomfrom the particle subset N (l);

• if the collision is accepted then the particle velocities are changed to their post-collision values. During stage 1, the particle positions are not changed.


Stage 2. Operator SτD (Free molecular motion), for each particle X ′i = Xi + ξiτ . Theboundary conditions are also simulated within Stage 2.

1.6 Moments of the distribution function

Some of the important moments of the distribution function in terms of velocities are givenbelow [10, p. 13] :

ρ(x, t) =∫mf d3c, (1.20)

u(x, t) = 1ρ(x, t)

∫mc d3c, (1.21)

T (x, t) = 13kBn(x, t)

∫m(c− u(x, t))2f d3c, (1.22)

T ij(x, t) =∫m(ci − ui(x, t))(cj − uj(x, t))f d3c. (1.23)

The pressure tensor is given by: p = nm c′ c′ and the components of the pressure are bypij = nmc′ c′. The scalar pressure is calculated as: p = 1

3ρ(u2 + v2 + w2).

Whereas, the flux vector for any quantity Q is nQc · e (or) nQc, therefore, the flux vectorcorresponding to mass transfer is: nmc.

1.7 Time average and Ensemble average

Figure 1.1: Time averaging.

In practice, a measurement of any macroscopic property is performed in discrete intervals oftime, and is carried out over some finite time, thus the quantity obtained is a time-averagedvalue of the macroscopic property of interest. Therefore, the time average is from a singletrajectory over a long time interval, whereas the ensemble average is performed over manyidentical trajectories at a certain instance of time. An ‘ensemble’ is a collection of largenumber of systems which are macroscopically equivalent to the system under consideration.The microscopic description of these replica systems can not be specified and that will differ


Figure 1.2: Ensemble averaging.

a lot between them, as there are large numbers of microscopic states corresponding to anygiven macroscopic state. Therefore, all macroscopic properties of a given system can bedescribed in terms of the microscopic state of the system, at various instances of time,by performing the average over the ensemble of replica systems. However, in equilibriumsystems, time and ensemble averages of physical quantities are equivalent [44].

The current work, presents an application of the ensemble-averaging for the cases wherethe dynamics and evolution of a system in time were to be studied, and time-averaging forthe cases where only the macroscopic properties of the molecules in a given volume ele-ment in steady state were of interest, where the final steady state solution of the problemis independent of time. Therefore, time-averaging is performed by accumulating the impor-tant moments over one trajectory and averaging over many time steps after allowing thesimulation to reach the steady state.

Thus, in stationary problems time averaging could be employed for obtaining results witha low statistical errors. When a stationary flow is reached, various molecular properties(number of molecules, velocity, squared velocity, etc.) are sampled and averaged in each cellof the computational domain., these average properties are used to obtain estimates of flowmacro-parameters (moments of the distribution function in terms of velocities and internalmodes of molecules).

In contrast to methods of continuum gas dynamics, a statistical error in the results is intrinsicto the DSMC, which is inversely proportional to the square root of the sampling volume.It is commonly believed that the statistical error of the DSMC method is determined onlyby the overall volume of the sampling, (N · L), and the statistical error is proportional to


√N · L. For finite N , however, there is always a statistical dependence of particles; and,

hence, the statistical error is proportional to√L.

1.8 Summary

The direct simulation Monte Carlo method is introduced, the popular collision approacheswere classified into ‘Greedy on time’ and ‘Greedy on memory’ categories, based on theirimplementation; the flow sampling and averaging techniques were discussed.

Chapter 2

Pressure Boundaries

2.1 Techniques for the application of a specified pressure

The following techniques were implemented in conjunction with the DSMC method for ap-plying a specified pressure at the open boundaries. Where, the number density correspondingto a given pressure was calculated from the state equation

p = ρRT (2.1)

2.1.1 Using Maxwellian distribution and injecting particles at a constantrate

At the inlet and outlet boundaries, which are present at the ends of the micro-channelgeometry, the particles are injected according to a ‘semi-Maxwellian velocity distributionwith zero bulk velocity’ for particle flux at a constant rate as [2, 45]:

N

A= nVmp

exp(−s2) +√πs[1 + erf(s)]

2√π

(2.2)

Where N is the rate of particles injected, s is the speed ratio, calculated as s = V cos(θ)Vmp

, n isthe number density, Vmp =

√2kBT/m is the most probable velocity, kB is the Boltzmann

constant, T is the temperature, m is the molecular mass of the gas, and A is the area ofan element. A particle would be removed from the list of particles whenever it is supposedto leave the computational domain via an open boundary. For the zero bulk velocity, thisrelation given in the Eq. 2.2 becomes

N

A= nVmp

2√π

(2.3)

17

Chapter 2. Pressure Boundaries 18

The particles may be injected directly or via reservoirs into the computational domain.For a direct injection, the particles are injected according to the exponential distribution:< N > exp(− < N > τ), where < N > is the mean number of particles to be injectedwithin a time step. This is the simplest approach for applying a given pressure at the openboundaries; however, this approach gives acceptable values of pressures at the input/output(I/O) boundaries when applied for relatively longer channels, and for much shorter channelsthis approach may not provide the desired value of pressure at I/O boundaries (the channelends).

Injecting the particles via a particle reservoir

The second approach is to inject the particles via reservoirs [46]. The particle reservoirsare maintained at the ends of the micro-channel geometry, with the size of the reservoirsappreciably larger than the characteristic length (e.g., the height or hydraulic diameter ofthe micro-channel) [46]. The model particles are generated at the outer open walls ofthe reservoirs from a Maxwellian velocity distribution according to the applied pressure.Obviously, this is not accurate when the borders (the particle injection points) are close tothe inlet and outlets of the channel. A set of computations has been performed to find asize L, of the reservoirs that allow one to mitigate an effect of such boundary conditionson the numerical solution. This approach is understandably a realistic one. However, itrequires additional memory allocation and costlier in terms of simulation time and effort,while it gives a different value from the desired pressure to be applied the I/O boundaries(the channel ends).

Application to microchannels

The following graphics show an application of the above discussed pressure boundary treat-ment (in the Subsection 2.1.1) considering straight microchannel, vertical bend and a T-junction, at rarefaction Kn = 0.1 and Kn = 0.5. The results were obtained from Unsteady-state simulations in 2D using the DSMC. The diffuse reflection with complete energy ac-commodation was considered on the solid walls [2].

The pressure immediately next to the inlet is less than the applied pressure, this might beattributed to the “open end" effects as the microchannel is not connected to some reservoirand sharply ends at the outlets. The pressure varies in non- linear fashion along the flowdirection in the channel, and pressure at the outlet is lowered because outlet is in the normaldirection to the flow from the inlet. An interesting observation on the flow rates can bedone from Fig. 2.6. It is clearly seen that the flow is not symmetric and flow rate at the


Figure 2.1: The established pressure distribution in a straight channel (top at Kn = 0.1and bottom at Kn = 0.5 for pressure ratio Pin/Pout = 3

(a) Kn = 0.1 (b) Kn = 0.5

Figure 2.2: Time evolution of the number fluxes through a straight channel(flow rates)at inlet and outlet calculated by DSMC for a pressure ratio Pin/Pout = 3

(a) Kn = 0.1 (b) Kn = 0.5

Figure 2.3: The established pressure distribution in a vertical bend


(a) Kn = 0.1 (b) Kn = 0.5

Figure 2.4: Time evolution of the number fluxes at inlet and outlet of a vertical bendcalculated by DSMC for a pressure ratio Pin/Pout = 3

(a) Kn = 0.1 (b) Kn = 0.5

Figure 2.5: The established pressure distribution in a T-junction

(a) Kn = 0.1 (b) Kn = 0.5

Figure 2.6: Time evolution of the number fluxes at inlet and outlet of a T-junctioncalculated by DSMC for a pressure ratio Pin/Pout = 3


(a) Vx, at Kn = 0.1 (b) Vx, at Kn = 0.5

(c) Vy, at Kn = 0.1 (d) Vy, at Kn = 0.5

Figure 2.7: The velocity components in a T-junction

horizontal element outlet is larger than the flow rate trough the vertical one. The flow rateat the lower Kn = 0.1 is larger than the one at the higher Kn = 0.5.


2.1.2 Implicit boundary conditions

This approach is based on the injection of particles as per the macroscopic flow propertiesinside the micro-channel. Implicit boundary conditions [45, 47–50] may be applied to main-tain a given pressure at the inlet and outlet sections, which allows to a large extent to avoidthe effects arising at the micro-channel ends. We denote by subscript ‘in’ all parametersrelated to the upstream conditions at the micro-channel extremity with the largest pressure(the entrance of element 1 with pressure pin in our case). The inlet temperature Tin ismaintained to be the same during the simulation. The outlet parameters for elements 2,3, and 4 are denoted by subscript ‘out’. The temperature at the outlet is not imposed,rather calculated from the other flow properties, such as pressure and density. The velocityV = u, v, w at both inlet and outlet sections is unknown. In this approach, the treat-ment of the upstream boundary is different from the downstream boundary. It is based onthe first-order extrapolation to determine the inlet mean velocity from the flow inside thecomputational domain. The implicit boundary conditions come from the dynamics of idealgas. As these conditions realize a negative feedback to keep a given pressure drop, they canbe used effectively for subsonic flows. The role of the characteristic equations is to ensurefaster convergence of the iteration process. The first Monte Carlo (MC) sweep is as givenunder step 1 with zero bulk velocity, and at the beginning of each of the subsequent MCsweeps the rate of the incoming particles is calculated using an updated speed of the flowand the flow properties at the open boundaries corrected using the step 2.

Step 1) The step 1 is same as the procedure described in the Subsection 2.1.1, with theparticles injected directly into the computational domain.

Step 2) Upstream boundary: At the upstream boundary, the pressure pin and tempera-ture Tin are the given parameters to the flow and the number density nin couldbe obtained from the equation of state as following: By using the 1D char-acteristic equation du/a = −dρ/ρ, where a is the sound speed, the followingimplicit boundary conditions [47, 49, 50] are obtained for the outlet cells: Theinlet boundary conditions for density and velocity components are computed asfollows:

ρin,j = pinRTin

,

uin,j = uin+1,j + pin−pin+1,j

ρin+1,jain+1,j,

vin,j = vin+1,j ,

win,j = win+1,j .

(2.4)

Here, the subscript (in, j) represents the cells located on the inlet boundarysurface. Where the subscript j denotes average cell values and the subscript ‘in’


signifies the inlet boundary, R is the gas constant. The inlet arm of the channelwas assumed to be in horizontal direction.

Downstream boundary: At the downstream boundary, the approach is similar to theone developed by Wang and Li [47], Fang et al [48], based on the given flowparameter, pressure pe at the exit. The other mean properties of the flow are tobe determined from the preceding calculations, using the following equations forthe outlet cells; e.g., for the outlet arm present in the horizontal direction, theimplicit boundary conditions were:

ρout,j = ρout−1,j + pout−pout−1,j

a2out−1,j

,

Tout,j = pout

ρout−1,jR,

uout,j = uout−1,j + pout−1,j−pout

ρout−1,jaout−1,j,

vout,j = vout−1,j ,

wout,j = wout−1,j .

(2.5)

aj is the local sound speed, and ρ is the density the subscript ‘out’ signifiesthe outlet boundary. These equations are similar to the one developed by Wangand Li [47], Fang et al [48] in 2D. In the previous relations u is the stream wisecomponent of velocity; local sound speed is equal to aj =

√γRTj , where γ is the

specific heat ratio, R is the specific gas constant. The variables with subscript(out, j) represent the macroscopic quantities averaged over a given computationaltime in the cell j, which belongs to the outlet boundary surface. However, forthe outlet arm in the vertical direction, the following boundary conditions wereemployed:

ρout,j = ρout−1,j + Pout−pout−1,j

a2out−1,j

,

Tout,j = Poutρout−1,jR

,

uout,j = uout−1,j ,

vout,j = vout−1,j + pout−1,j−Pout

ρout−1,jaout−1,j,

wout,j = wout−1,j .

(2.6)

The sound speed can be calculated as a =√γRT , where γ is the ratio of the specific heats,

R is the gas constant, and T is the temperature. Alternatively, we may calculate the soundspeed from the momentum balance equation as:

a = dp

dρ

(1 + dρ

ρ

). (2.7)

The Fig. 2.8 shows that for the gas flow simulation through a straight microchannel, boththese approaches for calculating the sound speed may give equivalent results, which is testedfor various lengths microchannel.


Further, the application of the pressure boundary conditions is shown in the Chapters chap-ter 3, chapter 4, chapter 5, and chapter 6.

2 4 6 8 10 12 14 161

2

3

4

5

6

7

Micro channel legth (in units of height)

Mas

s flo

w r

ate

(Kg/

m2 /s

)

Straight, sound speed from momentum balance Eq.

Straight, sound speed from (γRT)12

Figure 2.8: The mass flow rate through straight micro channels of various lengths,calculated using two approaches for the sound speed, Viz: using a =

√γRT , and using a =

dpdρ

(1 + dρ

ρ

). The diffuse reflection with complete energy accommodation was considered

on the solid walls [2].

The implicit boundary conditions (2.5), (2.4) allow to avoid the reservoir simulations and toconsider the gas flows through relatively short microchannels by relating the results to thoseof the limit case of long channel.

2.2 Summary

Some techniques for an application of a pressure boundary at the inlet/outlet section forstudying of the pressure driven flows were discussed. The implicit boundary conditions wereintroduced.

Chapter 3

Unsteady state DSMC simulations

Algorithm 4: A general flowchart of the Unsteady state simulations using DSMC

For an unsteady state simulation, the properties of interest are not only the macro-parametersbut also the evolution of the distribution function at certain points of space. The algorithm4 shows the flow chart for unsteady simulation of the gas flow using the DSMC method.Where nj is the total number of trajectories simulated. T is the simulation time of eachof the trajectory. The DSMC core consists of the indexing the particles according to theirposition in the computational domain, computing collisions, injecting/removing the particlesat the open boundaries, and movement of the particles over a time step dt. A sequence these

25

Chapter 3. Unsteady state DSMC simulations 26

Algorithm 5: An iteration over the core steps of the DSMC for unsteady state simulation

steps is shown in the Fig. 5. Ensemble averaging of the sampled properties is performed forthe unsteady state simulations.

3.1 A DSMC code for unsteady state simulations

The Fig. 3.1 shows the structure of the software developed in 2D which is as follows: anelement was created with the possibility to attach micro-channels around it in four directionsdepending upon the required geometry of the micro-channel combination. Each of theelements had its own local coordinate (x , y) system and dimensions with a given length. Themodel particles enter from one end of the resulting geometry, called the inlet and may leavethe structure from the other external outlets, based on the flow direction (resulting from thedriving pressures, maintained at the external ends of the construction). A DSMC code wasalso developed in 3D on the similar lines that simulates the unsteady state gas flow throughmicrochannel configurations in 3D. The construction of a junction configuration consistsof several elements. The unit element of micro-channel is constructed as square/rectanglewith upper and lower walls diffusely reflecting at wall temperature and has two openings onthe other two sides of the square/ rectangle with serving as inlet/outlet with the possibilityto keep them open or closed depending upon the required geometric construction. Theseelements were attached to a central element, which binds all the micro-channels togetherforming a cross-junction, and facilitates the wings to be attached or detached from it. Inother words, the central element may have each of its four sides open or closed with a solidwall diffusely reflecting depending on the considered configuration. Each element of the


Figure 3.1: The construction of a 2D-DSMC software

junction had its own system of local coordinate axes (xi vs yi, ∀ i = 0, . . . , 4) and the flowwas simulated with respect to these axes (locally). The overall flow was transformed intothe global (frame of reference) system of axes. Each element of the micro-channel spannedin the first quadrant of the local coordinate axes, starting at the origin of the local system.

3.2 Unsteady state DSMC simulation of plane Couette flow

This section presents an example of the unsteady state DSMC simulation. The Fig. 3.2shows the schematic of the Couette flow, simulated is an impulsively started shear drivenflow with the walls of a infinitely long plane- microchannel moving with a speed ±U , whereU = 0.05Vmp, Vmp =

√2RT , R is the gas constant and T is the temperature. The velocity

profile is shown at various times in the Fig. 3.3. The diffuse reflection with complete energyaccommodation was considered on the solid walls [2].

Figure 3.2: A schematic of the Couette flow


0 0.2 0.4 0.6 0.8 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

y / H

u / V

mp

Kn = 1

Figure 3.3: Velocity profiles in impulsively started shear flow (Kn = 1, U = 0.05Vmp)at various times(500∆t, 1000∆t, 1500∆t, and 2000∆t)

3.3 Unsteady state DSMC simulations of the gas flow past abackward facing step

In this section, the unsteady state DSMC simulation of the gas flow behavior past a backwardfacing step in micro-channels is discussed in two dimensions. Junctions are an important partof micro-channel assembling, and a joining of micro-channels with different cross sectionalarea is very common. The backward facing step [51–53] is an assembly of micro-channels,in which a micro-channel with smaller cross sectional area joins in the downstream directionwith another micro-channel with larger cross sectional area. Quite often, this problem isconsidered as a benchmark problem to test software(s) developed for studying fluid (gas)flows in channels at the micro and nano level. We attempt to study the gas flow in a micro-channel with a backward-facing step in two dimensions, assuming the uniformity of flow inthe third dimension.

According to the Ref. [54] and the references there in, the transition mechanisms in planechannels [7, 46, 55–57] and pipes and cylinders [58, 59] have received much attention overthe past three decades. These basic flows are understood considerably better than non-parallel flows arising in more complex geometries [60, 61] Therefore, the basic flows in microchannels are understood considerably better than non-parallel flows arising in more complexgeometries. In this context, the flow over a backward-facing step has emerged as a prototypeof a non-trivial yet simple geometry in which to examine the onset of turbulence. However,for the cases considered in the present chapter, the Reynolds number is much lower thanthe minimum value(s) stated in the above cited paper in order to observe the turbulentmotion inside this geometry. Our focus is on studying the gas flows past backward-facingstep flow with initial value of Kninitial in the range 0.2 ÷ 1.0 in the narrow region of thechannel, and with initial values of Kninitial in the range 0.05÷0.25 in the wider region (past


the backward-facing step). This Section is organized in the following fashion: Firstly, theproblem is outlined, followed by a brief description of the present code. Secondly, the DSMCsimulation results, obtained on a uniform rectangular grid for the gas flow past backwardfacing step are presented.

3.3.1 Problem formulation

The pressure driven (nearly) isothermal rarefied gas flow was considered through a combina-tion of micro-channels that form a backward-facing step, see Fig. 3.4. Uniform- rectangulargrids were used for computing collisions (a finer grid), and accumulating important mo-ments of flow (a coarser grid). The coarser grid cell side was five times as that of finergrid cell. Initially, the computational domain was filled with the gas at the uniform pressurep0 equal to atmospheric pressure, 1.e5 Pa. The unsteady simulation of rarefied gas underthe influence of the suddenly triggered external pressure at the inlet was carried out in twodimensions. The pressure maintained at the inlet was pin, equal to three times pout, and apressure pout equal to atmospheric pressure (1.e5 Pa) was maintained at the other externalend, the outlet. The applied temperature at the inlet Tin, at the outlet, Tout and initialtemperature in the micro-channel was equal to 273 K. The diffuse reflection with completeenergy accommodation was considered on the solid walls [2].

The Knudsen number (Kn), which is the ratio of the mean free path to the characteristiclength, was used to determine the rarefaction of the gas; Kn = λ/H, λ is the gas mean freepath and H is the characteristic length. The mean free path is defined as λ = 1/(

√2πd2n),

where d is the diameter of the molecule, and n is the number density corresponding to initialequilibrium. The gap in the vertical direction (the height) is taken as the characteristiclength. The considered micro-channel may be classified into two parts according to the gapin vertical direction (the height), pursuant to this, we have two different values of initialKn in the micro-channel corresponding to each height. The value of Kninitial (further on,the subscript initial is omitted for the sake of simplicity, and throughout this paper all thevalues of Kn correspond to initial time) may be varied by changing the gap (the height),while keeping other parameters as the same. The time step dt was less than the meancollision time and the total length of micro-channel was taken to be the same for all thecases equal to 0.94995µm. In this chapter all macroscopic variables are given in the non-dimensional form by using the scales: mean free path of gas molecules for length, thermalvelocity (

√2RT ) for velocity, p0 for pressure and T0 for temperature. The time values are

given in non-dimensional form scaled by the ratio of mean free path and molecular thermalvelocity .


Table 3.1: The gap values corresponding to Kninitial

Gap µ m 0.06333 0.10555 0.158325 0.25332 0.31665Kninitial 1 0.6 0.4 0.25 0.2Gap µ m 0.05664 0.6333 0.75996 1.2666 -Kninitial 0.125 0.1 0.0833 0.05 -

A hard sphere monatomic gas and NTC scheme were applied to model inter-molecularcollisions. The implemented version of the NTC scheme uses the following equation for acollision frequency: ν = 0.5N(N − 1)(gσt)/V ; where N is the number of model particles inthe cell, σt is the total collision cross section, g is the relative velocity of a pair of particlesand V is cell volume. This is identical to the correspondent equations in the Null-collisionDSMC scheme with excluding self-collisions [41] and in the majorant collision frequencyDSMC scheme [36]. Flat velocity profiles with a semi Maxwellian distribution [45, 62] ofmolecular velocities are used at the ends, inlet and outlet, for simulation of the incomingmolecules.

Unsteady state simulations [63] were performed, and sampling was done over an ensembleof trajectories.

3.3.2 Results

The DSMC simulations were performed according to the following: first the parameters arechosen such that initial Kn in the narrow part of channel was equal to 1.0, by varying thegap h2 of the micro-channel the initial value of Kn in the wider part of channel was chosenaccording to ratios of initial values of Kn, Knnarrowpart : Knwiderpart :: 4 : 1, 8 : 1, 12 : 1,and 20 : 1 (Table 3.1). Then for all the values of initial Knwiderpart, Knnarrowpart wasvaried from 0.2 through 1.0, by varying the gap h1.

Figure 3.4: The backward facing-step.

Figure 3.5(a-d) and Fig. 3.5(e-h) show the time evolution (the time is scaled with thesimulation time interval) of the normalized flow rates mAK (m is flow rate, A is the cross-sectional area andK is normalization constant), and their respective growth rates (derivativesof the flow rates) for fixed values of the inlet Kn. For inlet Kn = 0.2 the growth rate of


flux rate at the outlet has a maximum around time 12% of the total time interval ttotal, andthis maximum is shifted towards slightly larger values of time for inlet Kn 0.4 through 1.0.In all cases the height of the maximum increases with the Knudsen number. The relaxationin the growth rate of flow rates was quicker for lower values of the inlet Kn.

0 0.2 0.4 0.6 0.8 1−5

0

5

10

15

20x 10

−8

t

mas

s flu

x

Kn0.125 Kn0.0833Kn0.05

(a) Flow rate of flux at inlet and outlet forinlet Kn = 0.2

0 0.2 0.4 0.6 0.8 1−2

0

2

4

6

8

10

12x 10

−8

t

mas

s flu

x

outlet Kn0.25outlet Kn0.0833outlet Kn0.05

(b) Flow rate of flux at inlet and outlet forinlet Kn = 0.4

0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5

6x 10

−8

t

mas

s flu

x

outlet Kn0.25outlet Kn0.125outlet Kn0.0833outlet Kn0.05

(c) Flow rate of flux at inlet and outlet forinlet Kn = 0.6

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2

2.5

3x 10

−8

t

mas

s flu

x

Kn0.25Kn0.125Kn0.0833Kn0.05

(d) Flow rate of flux at inlet and outlet forinlet Kn = 1.0

Figure 3.5: Flow rate of mass flux for a fixed inlet Kn

Figures 3.7(a-h) show the time evolution of the normalized flow (mAK) rates and theirrespective growth rates for fixed values of outlet Kn. In all cases the height of the maximumgrowth rate was increasing with the 1/Kn. In all the cases the steady-state value of flowrate was scaling with 1/Kn, as shown in Fig. 3.7(e - h). The reduced flow rate past theback ward facing step in microchannel is shown in the Fig. 3.8.

Figures 3.9(a-c) show the final pressure fields: when the Kn in narrow part of channel wasequal to 0.2 and the Kn in the wider part was 0.125, there was a region of minimum inscalar pressure at the step and for cases with lower Kn in the wider part from 0.125 to 0.05,this region was close to the step corner and relatively smaller for lower values of Knwiderpart,the pressure drop across the micro-channel was higher for lower Knwiderpart.

Figure 3.10(a-c) and 3.9(a) show the final pressure fields when the Kn in wider part ofchannel was kept equal to 0.125, when the Kn in the narrow part was 1.0, the pressure inthe wider part was almost uniform, and has a minimum value, and for the lower Kn in the


0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

0.2

t

grow

th r

ate

of fl

ux

Kn 0.05Kn 0.0833Kn 0.125

(a) Growth rate of flux rate at out-let for inletKn 0.2

0 0.2 0.4 0.6 0.8 1−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

t

grow

th r

ate

of fl

ux

Kn 0.05Kn 0.25Kn 0.0833

(b) Growth rate of flux rate at out-let for inletKn 0.4

0 0.2 0.4 0.6 0.8 1−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

t

grow

th r

ate

of fl

ux

Kn 0.05Kn 0.25Kn 0.0833Kn 0.125

(c) Growth rate of flux rate at out-let for inletKn 0.6

0 0.2 0.4 0.6 0.8 1−2

0

2

4

6

8

10x 10

−3

t

grow

th r

ate

of fl

ux

Kn 0.05Kn 0.25Kn 0.0833Kn 0.125

(d) Growth rate of flux rate at out-let for inletKn 1.0

Figure 3.6: Growth rate of mass flux rate for a fixed inlet rarefaction

narrow part of micro-channel the region with minimum value of pressure starts reducing insize, we also see the formation of secondary regions of minimum along the step for Kn 0.4,and finally a secondary region of minimum was seen spanning up to the foot of the step forKn 0.2. The pressure drop decreased for the lower Kn in the narrow part.

The following observations may be made from the shear stress studies for fixed Kn in widerpart at 0.05 and various values of the Kn in the narrow part of the micro-channel at timettotal, When the Kn in narrow part of channel was equal to 1.0 the shear stress at the top ofthe channel was more or less constant (with an exception at the inlet and at the step corner)until the step is reached, and then it dropped to a small value, slightly above zero as theout-let is reached. There is an indication of symmetry (with an exception of a small jumpin around the step corner) in shear stress about a small value above zero, along the channelabove the step. When the Kn in the narrow part was decreased to 0.6, the shear stressat the top of micro-channel in the narrow part starts becoming non-linear, the shear stressafter the step corner falls slower than in the case of Kn 1.0, it is non-linear in nature andbecomes small but slightly above zero as the out-let is reached. There is an increase in thevalue of shear stress at the corner of step compared to the previous case. When the Kn inthe narrow part was decreased further to 0.4, the shear stress changes in the behavior were


0 0.2 0.4 0.6 0.8 1−2

0

2

4

6

8

10

12x 10

−8

t

mas

s flu

x

Kn 1 Kn0.4 Kn 0.6

(a) Flux rate at inlet and outlet for Knout = 0.25

0 0.2 0.4 0.6 0.8 1−2

0

2

4

6

8

10

12

14

16

18x 10

−8

t

mas

s flu

x

Kn 1 Kn 0.2 Kn 0.4 Kn 0.6

(b) Flux rate at inlet and outlet for Knout =0.125

0 0.2 0.4 0.6 0.8 1−2

0

2

4

6

8

10

12

14

16

18x 10

−8

t

mas

s flu

x

Kn 1 Kn 0.2 Kn 0.4 Kn 0.6

(c) Flux rate at inlet and outlet for Knout =0.0833

0 0.2 0.4 0.6 0.8 1−2

0

2

4

6

8

10

12

14

16

18x 10

−8

t

mas

s flu

x

Kn 1 Kn 0.2 Kn 0.4 Kn 0.6

(d) Flux rate at inlet and outlet for Knout =0.05

(e) Growth rate of flux rate at out-let forKnout = 0.25

(f) Growth rate of flux rate at out-let for Knout =0.125

(g) Growth rate of flux rate at out-let for Knout =0.0833

(h) Growth rate of flux rate at out-let forKnout = 0.05

Figure 3.7: Flow rate of mass flux and the growth rate of mass flux rate for a fixed outletrarefaction


Figure 3.8: The reduced flow rate past a backward facing step for Knout = 0.05.

(a) Scalar pressure iso-lines forKnin = 0.2 and Knout = 0.125

(b) Scalar pressure iso-lines forKnin = 0.2 and Knout =0.0833

(c) Scalar pressure iso-linesfor Knin = 0.2 andKnout = 0.05

Figure 3.9: Scalar pressure iso-lines for fixed inlet rarefaction Knin = 0.2

(a) Scalar pressure iso-lines forKnin = 1.0 and Knout =0.125

(b) Scalar pressure iso-lines forKnin = 0.6 and Knout =0.125

(c) Scalar pressure iso-lines forKnin = 0.4 and Knout = 0.125

Figure 3.10: Scalar pressure iso-lines for fixed outlet rarefaction Knout = 0.125

similar as in case of Kn = 0.6 in narrow part, but with increased in intensity. When the Knin the narrow part was decreased further to 0.2 the shear stress at the top of micro-channelin the wider part had lost its non-linearity, its value at the out-let more than in all previouscases. When the Kn in the wider part was kept at 0.05, in all the cases for Kn in the narrowpart from 1.0 through 0.4, the shear stress at the bottom of wider part of micro-channelwas close to zero, but a negative value, but was dropping to be -0.02 when Kn 0.2, in thenarrow part. Now, fixing the Kn in the narrow part as 0.2 and varying the Kn in the widerpart from 0.05 through 0.125, we observe that the non-linearity in shear stress was regainedat the top and the difference in the value of shear stress at the beginning of wider part and


at the outlet was increased.

Figures 3.11(a-d), show iso-lines of shear stress at time tfinal for Kn 1.0 in the narrow part ofmicro-channel for various values of the Kn in the wider part 0.25 through 0.05, we observethe following: there was a region with minimum value of shear stress, and initially at Kn0.25 this region of minimum was larger and bent towards the step, and as the Kn in thewider part is decreased to 0.05, this region was diminishing in size and also was turning itselfaway from the step. Figure 3.12(a-d), show iso-lines of shear stress at time tfinal, when theKn in the wider part of micro-channel was fixed at 0.125 and the Kn in the narrow partof micro-channel was varied from Kn 0.2 to 1.0, the following was observed: There was aregion of a minimum present at the step corner in the wider part of micro-channel.

(a) Shear stress iso-lines for Knin = 1.0 andKnout = 0.25

(b) Shear stress iso-lines for Knin = 1.0 andKnout = 0.125

(c) Shear stress iso-lines for Knin = 1.0 andKnout = 0.0833

(d) Shear stress iso-lines for Knin = 1.0 andKnout = 0.05

Figure 3.11: Shear stress (cell volume based) iso-lines for fixed inlet rarefaction Knin =1.0


(a) Shear stress iso-lines for Knin = 0.2 andKnout = 0.125

(b) Shear stress iso-lines for Knin = 0.4 and Knout =0.125

(c) Shear stress iso-lines for Knin = 0.6 andKnout = 0.125

(d) Shear stress iso-lines for Knin = 1.0 andKnout = 0.125

Figure 3.12: Shear stress (cell volume based) iso-lines for fixed outlet rarefaction Knin =0.125


3.4 Summary

Unsteady-state gas flow simulations were performed using the DSMC method considering thebackward-facing step flow as benchmark problem. At the first the rarefactionKninitial in thenarrow part of channel was considered to be 1.0, and the rarefaction Kninitial in the widerpart of channel was chosen according to ratios of Kninitial, Knnarrowpart : Knwiderpart ::4 : 1, 8 : 1, 12 : 1, and 20 : 1. Then for all the values of rarefaction Knwiderpart, intialrarefaction Knnarrowpart was varied from 0.2 through 1.0. The behavior of flow rates, scalarpressure, and shear stress is depicted. A qualitative analysis of flow rates and their growthrates was done for fixed values of inlet rarefaction (Knin) and for fixed values of outletrarefaction (Knout). Scalar pressure behavior was qualitatively shown for fixed values ofinlet Kn for fixed values of outlet Kn. Shear stress behaviour qualitatively shown for fixedvalues of inlet Kn for fixed values of outlet Kn.

Unsteady-state gas flow simulations were performed using the DSMC method consideringthe straight micro channel, vertical bend and a T-junction, and the results are shown in theChapter 2.

Chapter 4

Parallel implementation of theDSMC code for gas flow simulationin microchannel configurations

4.1 Introduction

An implementation of the DSMC algorithm, that utilizes a large memory and simulationtime is worth done in parallel, in order to meet its requirements of memory and reducingthe simulation time. A two of the popular methods for parallelization are discussed in thischapter, viz.: parallelizing a DSMC code using the message passing interface (MPI), andparallelizing a DSMC code using the OpenMP. Both the MPI and the OpenMP are portableacross a variety of platforms from laptops to some supercomputers [64–67].

The Ref.s [68–73] concern some of the strategies for a parallel DSMC implementation usingstructured and unstructured grids.

Using the MPI, a code can be distributed among the MPI processes, with each of the MPIprocesses running a chunk of the problem, which is dynamically decomposed among them.The MPI processes usually have their own (distributed) memory. Where as, the OpenMPdirectives specify the way parallel sections or the loop iterations have to be split among thethreads, with a team of threads sharing a single memory space, however, each of the memberthread may also have some of their own private variables.

For a better performance, OpenMP parallelism would be generally achieved at the highestlevel possible in the parallel code. Similarly, MPI would be used with a minimal communica-tion(s), usually by decomposing the task such that most of the work is done independently,

39

Chapter 4. Parallel Implementation of the DSMC code for gas flow simulation inMicrochannel configurations 40

and possibly by MPI processes over a subset of data. However, occasionally it’s very effectiveto use MPI and OpenMP to parallelize processing at the same level.

When subtasks must communicate many times per second in an application, its called fine-grained parallelism, otherwise if its subtasks communicate more frequently, it exhibits coarse-grained parallelism, and if they rarely or never communicate its called embarrassingly parallel.Embarrassingly parallel applications are considered the easiest to parallelize.

In the current implementation, the MPI-paralleized version of the DSMC code does theensemble averaging of the flow properties, and the OpenMP-parallelized version of the DSMCcode does the time averaging of the flow properties.

4.2 Parallelization using the OpenMP

A shared memory architecture consists of a team of processors that can access the globalmemory by means of a bus or some interconnection. The processes read and write asyn-chronously, and the accessing time of any piece of data is the same, as all of the communi-cation goes through the bus.

Each of the processors can access the address space (memory) with the same ease as itsco-processors.

OpenMP is the proposed industry standard Application Program Interface (API) for sharedmemory programming, and is based on a combination of compiler directives, library routinesand environment variables that can be used to describe a shared memory parallelism.

Figure 4.1: The shared memory(fork-join) model

The OpenMP utilizes a fork-join model, resulting in spawning a team of threads at thebeginning of a parallel region and joins them at the end of the parallel region. An OpenMPprogram begins on the master thread as a single process. The master thread executes theprogram sequentially until A parallel region construct is encountered. The master threadthen employs (FORK) a team of parallel threads. The part of the program that is enclosedby the parallel region construct is then executed in parallel by this team of threads, whichincludes the master thread. Once all threads of this the team complete the execution of the


statements in the parallel region construct, they synchronize and terminate (JOIN), leavingonly the master thread.

However, a drawback of a shared memory architectures is that, they do not scale very well.The main problem occurs when a number of processors attempt to access the global addressspace at the same time, leading to a condition called racing.

Data environment constructs:

FIRSTPRIVATE ( list ): The private copies of the variables listed under FIRSTPRIVATEare initialised from the original variables existing before the construct. The scope of thevariables listed under the FIRSTPRIVATE clause is local to a thread in a team;

SHARED ( list ): This clause causes all the variables that appear in the list to be sharedamong all the threads in the team. Therefore, each thread within the team have access tothe same storage area for SHARED data.

DEFAULT ( PRIVATE | SHARED | NONE ): The DEFAULT clause allows the user todetermine the attributes for all the variables in a parallel region. Variables in THREADPRI-VATE common blocks are not affected by this clause. E.g., DEFAULT (SHARED): declaresall the variables in the parallel region as shared among the threads of the team.

4.2.1 Parallel implementation of a 3D DSMC code for gas flow simulationin microchannel using OpenMP

Figure 4.2: The structure of the DSMC code implemented

The software structure is shown in the Fig. 4.2 and the data flow between the blocks (ele-ment) of the microchannel construction is shown in the Fig. 4.3. A domain decomposition


Figure 4.3: The data flow diagram of the DSMC code implemented

technique was employed which allows running each of the domain on a separate processor.The following listing gives a part of the directives used in the parallellization of the DSMCcode using the OpenMP:

Step 1 Taking the structure of the code in to account, the given geometry (e.g. a T-junction) would be parallelized such that each element of the geometry would run ona separate processor (thread) with the help of work-sharing constructs within a parallelregion .

Step 2 An additional speedup would be achieved by the load balancing, by further de-composing a problem: e.g., the tasks under each of the element would be furtherdistributed among the available processors by means of a dynamical allocation of achunk of the sub-problem among the processes. The description of the dynamicalallocation of the particle movement and collisions schemes is given below:

• The following directives describe a parallel execution of the movement of the listof the particles in an element by dynamically allocating them in units of ‘CHUNK’among the threads.!$OMP PARALLEL SHARED(CHUNK) PRIVATE(n)!$OMP DO SCHEDULE(DYNAMIC,CHUNK)

do n=NumberofPartcles,1,(-1)moving nth particle

end do!$OMP END DO NOWAIT!$OMP END PARALLEL

• the outermost loop of the collision routine would be dynamically decomposedamong the threads, by allocating ‘CHUNK’ iterations to each of them


!$OMP PARALLEL SHARED(CHUNK) PRIVATE(k)!$OMP DO SCHEDULE(DYNAMIC,CHUNK)

do k=1,NumberofCellsInZdirectionperforming collisions in the cells: grid(k,j,i) ∀j, i

end do!$OMP END DO NOWAIT!$OMP END PARALLEL

Parallel region construct

The following construct describes the parallel region for the kinetic steps in the DSMCalgorithm.

!$OMP PARALLEL DEFAULT(SHARED), FIRSTPRIVATE(dt)A sections work-sharing construct...

!$OMP END PARALLEL

Work Sharing constructs

SECTIONS directive identifies a non-iterative work-sharing construct that specifies a setof constructs to be divided among threads in a team. As each section is executed once bya member-thread in the team, the tasks such as reading inputs, and setting the read theinput data, compute constants, setting up the initial state, converting all variables to non-dimensional ones, initialization of the arrays, that accumulate macroscopic quantities, etccorresponding to each of the element of the geometry were work-shared among the threadsusing the SECTIONS construct(s). The following describe work-sharing of the collision andmovement processes, where the suffix i∀0to4 correspond to each element of the microchannelconfiguration):

!$OMP SECTIONS!$OMP SECTION

call collision0

call onestep0

!$OMP SECTIONcall collision1

call onestep1



call onestep2


call onestep3


call onestep4

!$OMP END SECTIONSWhere the routines collision0, etc include the routines for indexing the particles indices inacordance with their positions, and those that perform the collisions among the potentialcollision partners. The routines onestep0 etc, include the functions and routines that performmovement of all the particles in the list during the time step dt, and also call the routinesfor the injection of particles at a specified pressure boundary.

The following gives the calls for the routines for inter transfer of particles among the ele-ments of the channel geometry, and sampling - that accumulate sums for calculation of themacroscopic properties.

!$OMP SECTIONS!$OMP SECTION

call receive-particles0call sampling0

!$OMP SECTIONcall receive-particles1call sampling1




!$OMP END SECTIONS


A similar work-sharing was be performed for indexing the particles according to their position.The results obtained from the OpenMP parallel version of the DSMC code are presented inthe Chapter 6, and are submitted to Int. J. of Heat and mass Transfer.

The Fig. 4.4 shows the Mach number along the 90 degree bend microchannel with a squarecross section, simulated using the OpenMP parallel version of the 3D DSMC code, forpin/pout = 3 and Knout = 0.072. The diffuse reflection with complete energy accommoda-tion was considered on the solid walls [2]. A similar parallel implementation was performed

0 0.2 0.4 0.6 0.8 1

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

x / CL

mac

h nu

mbe

r

Figure 4.4: Mach Mach number along the 90 degree bend microchannel with a squarecross section, simulated using the OpenMP parallel version of the 3D DSMC code

for the 2D DSMC code, using the OpenMP.

The flow chart in the Fig. 4.5 shows A DSMC algorithm (for time-averaged simulation)parallelized using the OpenMP.


Figure 4.5: A DSMC algorithm (for time-averaged simulation) parallelized using theOpenMP


(a) The flowchart of the DSMCcode

(b) An iteration overthe core steps of theDSMC

Figure 4.6: The flowchart of the DSMC code(for unsteady state simulations) parallelizedusing the standard MPI

4.3 Parallel implementation of the DSMC code for gas flowsimulation in microchannel configurations using the MPI

The 3D-DSMC code for gas flow simulation in microchannel and a 2D-DSMC code for gasflow simulation in microchannel were parallelized using the MPI, each does the ensembleaveraging of the flow properties. An embarrassingly parallel approach was followed. TheDSMC code utilizes a number of independent trajectories, that form an ensemble. Hence,it is quite efficient to distribute each of these trajectories among the MPI-processes. Thus ahigh speedup close to the number of processors may be achieved. However, a small percentless due to the MPI-communications that distribute the task among the MPI-processes, andreduce the sampled properties from the MPI-processes on to the root-process, and writingoutput to the files.The flow chart in the Fig. 4.6 shows A DSMC algorithm (for unsteady state simulation)parallelized using the MPI standard.

• The following statement invokes the MPI-library:include ‘mpif.h’integer status(MPI_STATUS_SIZE)


call MPI_COMM_RANK(MPI_COMM_WORLD,my_id,ierr)

• The following communications initialize and define the MPI-parallel environment

call MPI_INIT(ierr)

call MPI_COMM_RANK(MPI_COMM_WORLD,my_id,ierr)

call MPI_COMM_SIZE(MPI_COMM_WORLD,num_proc,ierr)

• The following communications apply a barrier for all the MPI processes to be synchro-nized.CALL MPI_BARRIER(MPI_COMM_WORLD,ierr)

• A typical reduction of the sampled flow properties is shown below:

CALL MPI_REDUCE(dnsty,sdnsty,nele,MPI_DOUBLE_PRECISION,MPI_SUM,

nroot,MPI_COMM_WORLD,status,ierr)

CALL MPI_REDUCE(flux_in_1,sflux_in_1,nrun,MPI_DOUBLE_PRECISION,

MPI_SUM,nroot,MPI_COMM_WORLD,status,ierr)

• The following communications distribute the arrays with input parameters, and logicalvariables of an element ’i’ from the root to other processes.

call MPI_BCAST(ip_int_i,12,MPI_INTEGER,0,MPI_COMM_WORLD,ierr)

call MPI_BCAST(ip_real_i,24,MPI_DOUBLE_PRECISION,0, MPI_COMM_WORLD,ierr)

call MPI_BCAST(ip_bool_i,8,MPI_LOGICAL,0,MPI_COMM_WORLD,ierr)

call MPI_BCAST(cha,1, MPI_CHARACTER, 0, MPI_COMM_WORLD, ierr)

call MPI_BCAST(c,kount,MPI_DOUBLE_PRECISION,0,MPI_COMM_WORLD,ierr)

The Fig. 4.7 presents a validation of the the MPI implementation with the scalar versionof the software, the number flux at the entrance and exit of the microchannel, and thescalar pressure along the center line of the straight microchannel were shown to be in veryagreement, tested for the outlet rarefaction Kn = 0.5, in 2D. The diffuse reflection withcomplete energy accommodation was considered on the solid walls [2].


0 5 10 15 200.25

0.3

0.35

0.4

0.45

0.5

t

num

ber−

flux

mpi−#proc:1mpi−#proc:2mpi−#proc:3Serial

(a) non-dim. number flux at the entrance forKnout = 0.5

0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t

num

ber−

flux


(b) non-dim. number flux at the entrance forKnout = 0.5

0 0.2 0.4 0.6 0.8 11.4

1.6

1.8

2

2.2

2.4

2.6

2.8

x / L

non

dim

. sca

lar

pres

sure


(c) Scalar pressure along the channel for Knout =0.5

Figure 4.7: validation of the MPI implementation considering a straight microchannel

4.4 Summary

A parallel implementation of the (2D and 3D) DSMC code(s) created for the simulation of gasflows through microchannel configurations using the standard MPI, and using the OpenMPwere discussed. The MPI was opted for an ensemble averaging of the flow properties, forstudying the unsteady-state gas flow in microchannel configurations; whereas the OpenMPwas utilized for a time-averaging of the flow properties, for studying the steady-state gasflow in microchannel configurations.

Chapter 5

Mass flow rate through straight and90 degree bend micro channels inslip and transitional flow regimes

5.1 Introduction

In many micro and nano scale devices such as the micro-electro-mechanical systems (MEMS)a gas is rarefied, that means that the molecular mean free path is comparable or larger thanthe characteristic dimension of a micro device. The applications of these micro scale de-vices may be found in a wide range of engineering products (e.g., fluidic micro-actuators,resonators, vacuum generators, mass flow and temperature micro-sensors, pressure gauges,and micro heat-exchangers, etc). The straight microchannels and also more complex con-figurations like the bend channel, converging/diverging channel etc. are the basic elementsin such a micro-system. To a large extent, the flow regime in the microsystems falls beyondthe continuum fluid dynamic description, therefore, the models based on the kinetic theorymust be applied to simulate of the gas flows [2, 10, 18]

Recently some attempts were made to study the gas flow through the channel of variousconfigurations like the bend channel, T-shape channel and zigzag channel [74–76]. Thestraight channels with various cross section shape, such as hexagonal, trapezoidal, triangularetc., were also studied [46, 56, 77]. In addition, the analytical expressions for the massflow rate through straight channel with circular, rectangular cross-sections may be found in[6–8, 78–84].

The flow through the bend channel was studied by Raju and Roy [85]. The authors simulatedthe flow in a microchannel with two 90 degree bends using a finite element method. Agarwal

51

Chapter 5. Mass flow rate through straight and 90 degree bend micro channels in slip andtransitional flow regimes 52

et al [74] have used the lattice Boltzmann method to study the isothermal gas flow througha microchannel with a single 90 degrees bend. The computations were performed for thetwo-dimensional geometry considering some typical Reynolds and Knudsen numbers.

The authors of Ref. [76] have performed a numerical investigation of a gas flow throughmicrochannels with a sharp, 90 degrees bend using the viscous, two-dimensional compressibleNavier-Stokes (NS) equations with the classical first-order slip boundary condition. The flowstructure near the corner was analyzed and the effects of rarefaction and compressibility onthe flow were investigated. The presence of a recirculation was detected on both the innerand outer walls of the corner for larger Reynolds number. The authors have also shownthat the mass flux through a bend microchannel can be slightly larger than that through astraight microchannel of the similar length and driven by the same pressure difference.

The authors of Ref. [4] have performed a numerical investigation of pressure-driven gas flowin three-dimensional bent microchannels using the NS equations subjected to the with firstorder slip boundary condition. The effects of the channel cross-section aspect ratio (widthto height ratio), of the exit Knudsen number, of compressibility, and of the bend curvaturewere investigated. The authors have shown that the mass flow through a bend microchannelcan be slightly larger than that through an equivalent straight microchannel.

The DSMC method was used in [86] to simulate the gas flow through microchannel elementssuch as 900 bend and T-junctions. The authors have also shown that the mass flow ratethrough a 900 bend microchannel can be slightly higher compared to that through a straightmicrochannel, for a certain inlet Knudsen numbers.

This chapter presents the work the systematic comparative study of the flow propertiesthrough the straight and 900 degree bend channels is carried out in the slip and transitionalflow regimes using the DSMC method. This method is computationally more expensivecompared to other methods which can be implemented for the simulations in the slip andhydrodynamic flow regimes. In addition, when the flow trough the channels (short or long)is simulated by using the DSMC the large tanks are usually used to avoid the entranceeffects [46, 87, 88]. Usually the size of the tanks, which is chosen depending on value of theKnudsen number, is taken equal to several times (from 5 to 10) of the characteristic lengthsof the problem. Therefore the simulation requires the high memory and computational time.

The present work concerns the application of the DSMC method with the implicit inlet/outletboundary conditions. It was shown that, it is sufficient to simulate a moderately long channelwith these implicit boundary conditions to get the same flow characteristics (e.g. the massflow rate) as in the long channel. The implicit boundary conditions allow to reduce thecomputational time and they do not introduce the end effects.


The first goal of this chapter is to examine the application of the implicit boundary conditionswith the DSMC to simulate the gas flows through relatively long microchannels. Differentchannel lengths were tested and the minimal channel length was found which allows toreproduce the properties of the very long micro channel. A study of the effect of the crosssection by keeping the hydraulic diameter to be a constant was also performed. This sortof the study would allow us to relate a rectangular channel with a variable width to heightratio (W/H) to a pipe of a similar hydraulic diameter, or a pipe of similar throughput.This approach facilitates one to compare the channel geometry with that of the pipe. Thiswill also enable to interchange the channels with pipes or vice-versa, in the micro systemswithout altering the throughput. Likewise, using this sort of data one can choose amongthe rectangular microchannels straight and/or with 90 degrees bend in the middle, and thepipes while designing a micro device.

An assessment of the influence of the transverse walls on the stationary isothermal gas flowthrough a rectangular microchannel, via the reduced mass flow rate, considering straight and900 bend micro channels,was performed.

The outline of this numerical results sections is as follows: in the section 5.4, the the flowthrough a straight microchannel is investigated considering the (reduced) flow rate, andstudy the influence of the relative dimensions of the cross section on the flow rate througha straight micro channel, and compare the current DSMC simulated results to an analyticalstudy performed by Meolans et al [89] in for the slip regime, and with the Sharipov [3]. In thesection 5.5, the dependence of the reduced flow rate (G) through a 900 bend microchannelon the center line length, and on the pressure difference between its inlet and outlet werestudied. The reduced flow rate (G) through a 900 bend microchannel, and through astraight microchannel, in the free molecular, slip and transitional regimes of the gas flowwas studied. A study of cross sectional effects on the reduced flow rate (G) through a 900

bend microchannel, with a constant hydraulic diameter was performed.

5.2 Problem formulation and computational consideration

The in-house DSMC software is developed following the classical Monte Carlo approach [2]for the geometrical configuration shown in Fig. 5.1. This configuration consists on fourmicro channel elements (1, 2, 3, and 4) with corresponding length Li, attached to a centralelement (0), whose sides can be open or closed. This configuration allows to simulate variouschannel configurations: straight and bend channels, the T-junction, the cross-junction andetc. The present chapter presents only the results for the straight and bend micro channels.


A pressure driven isothermal gas flows through the three-dimensional straight and 900 bendmicrochannels are studied. The both types of channels have the rectangular cross-sectionsand various height H to width W ratios are considered together with different channelslengths L.

Figure 5.1: The general view of 3D-microchannel geometry.

To simulate the straight channel configuration the elements 1, 0 and 3 were used, see Fig.5.1. The elements 1, 0 and 4 were needed for 900 bend channel configuration, Fig. 5.1. Thewalls of all elements were maintained at constant temperature Tw and they were consideredas diffuse reflective with complete energy accommodation [2]. The inlet Pin and outlet Poutpressures were kept constant. For the relatively small pressure differences between the inletand outlet cross-sections, (Pin − Pout) /Pout ≤ 2, considered here, the flow velocity throughthe channel is small and the flow is assumed to be isothermal. The inlet temperature Tin isfixed equal to the wall temperature Tw. The outlet temperature Tout is calculated by usingthe characteristic technique, explained in details in Section 2.1.2.

One of the goals of the present chapter is to find the minimum channel length which allowsto consider the channel as long enough, so that the gas flow rate acquires self-similarity forthe channel lengths longer that this minimum length. The criterion for the choice of thisminimum length is the asymptotical behavior of the reduced flow rate G, which is definedin analogy with the reduced flow rate in Ref. [3]

G = L

DHW (Pin − Pout)

√2kBTm

M. (5.1)

In the previous expression T is the temperature, m is the molecular mass of the gas, kBis the Boltzmann constant, M is the mass flow rate through a cross section of a channel.The D represents the characteristic dimension, and W , H are the width and height of the


microchannel respectively. In case of the 900 bend micro channel the centerline length CL,see Fig. 6.10, is considered as the equivalent length of the channel (L = LCL = L1+L2+H).

The following options were considered to define the characteristic length(D) of a microchan-nel: the hydraulic diameter(Dh), and the minW,H; therefore, the simulations were per-formed with the two choices for the case of a straight channel (Dh andH, withH 6W ), andthree choices (Dh,W , and H) for the case of a 900 bend micro channel (due to asymmetry),for the characteristic length.

0 200 400 600 800 1000−2

0

2

4

6

8

10x 10

−10

simulation−time intervals

mas

s flo

w r

ate

(Kg/

sec.

)

steady−state considered for the time averaging

at the inletat the outlet

Figure 5.2: A typical measurement of the flow rate in the steady-state.

5.3 Details of the DSMC technique

The DSMC method in its classical version [2] is implemented. To model the binary collisionstwo collision schemes are examined: the traditional "No Time Counter" [2] and the "Simpli-fied Bernoulli Trials" [40, 90]. The various tests have shown that for considered isothermalflows the both schemes give equivalent, with respect to accuracy, results for the wide rangeof the computational parameters. Thus, the results presented in this chapter are calculatedby the slightly more efficient NTC scheme. The mesh and the computational time step arechosen such that the particle does not cross in average more than 1 cell per kinetic step.Typically, the simulations are carried out with a number of particles per cell, at the beginningof the simulation, ranging from 20 to 40, which is further raised as the simulation progressed.


The time step is chosen to be smaller than mean collision time. A uniform grid is used forcomputing the binary collisions. The mass flow rate in each simulation is time-averaged afterreaching its steady state value, over a very long interval of (simulation) time to reduce thescatter associated with the calculation.

A typical measurement was shown in the Fig. 5.2, where each of the simulation time intervalspans 72 kinetic steps.

5.3.1 Inlet-outlet boundary conditions

Implicit boundary conditions (Chapter 2, Subsection 2.1.2) [45, 47–50] were applied tomaintain a given pressure at the inlet and outlet sections, which allows to a large extent toavoid the effects arising at the micro-channel ends. The notation is as follows: subscript ‘in’is for all the parameters related to the upstream conditions at the micro-channel extremitywith the largest pressure (the entrance of element 1 with pressure pin in our case). Theinlet temperature Tin is maintained to be the same during the simulation. The outletparameters are denoted by subscript ‘out’. The temperature at the outlet is not imposed,rather calculated from the other flow properties, such as pressure and density. The velocityV = u, v, w at both inlet and outlet sections is unknown.

The implicit boundary conditions Eq. (2.5), Eq. (2.4) (see Chapter 2) allow to avoid thereservoir simulations and to consider the gas flows through relatively short microchannels byrelating the results to those of the limit case of long channel.

5.3.2 Definition of the flow parameters

In this chapter, the rarefaction parameter δ is used and was defined as

δ = PD

µ√

2RT(5.2)

to characterize the level of the gas rarefaction. Here D is the characteristic dimension ofthe problem. When using the Hard Sphere (HS) model

the Knudsen number is related to the rarefaction parameter δ as

Kn =√π

21δ. (5.3)

In the current study the average rarefaction parameter is used and is defined as

δa = 12 (δin + δout) , (5.4)


Table 5.1: The grid dependence of the reduced mass flow rate through a straight microchannel with a square cross section for various lengths, δa = 1. Grid A: 15× 15× 15 cellswith 20 PPC, grid B: 11× 11× 11 cells with 50.8 PPC, grid C: 7× 7× 7 cells with 196.8

PPC. The characteristic dimension was D = H.

grid GL = 3H L = 5H L = 7H L = 9H L = 15H

A 0.8002 0.7894 0.7852 0.7811 0.7782B 0.7577 0.7618 0.7670 0.7656 0.7652C 0.7607 0.7652 0.7606 0.7580 0.7546

where δin and δout are calculated according to eq. (6.1ng the respective macroscopic valuesat the inlet and outlet cross-sections.

5.4 Simulation of the flow through a straight microchannel

5.4.1 Flow rate through a straight microchannel

The characteristic dimension of a channel D is used in the definition of the rarefactionparameter, Eq. (6.1). The height of the microchannel H is taken as the characteristicdimension of the problem in the simulations of the flow through the straight channel, soD = H. The micro channel height is fixed equal to 19.83µm, this value correspondsto a microchannel used in the experimental set-up in the IUSTI laboratory, Aix MarseilleUniversity, France. A comparison with the experimental measurements is planned in thenear future. The simulations of the flow through a straight microchannel are carried out fordifferent length of the channel varying from 3H to 21H. The simulations are performedfor the rarefaction parameter δa = 1 and 10 and by applying the inlet-outlet pressure ratioPin/Pout equal to 3. The simulations are conducted at a temperature of 300 K.

Figure 5.3: The straight microchannel.

The simulations carried out with initial number of particles equal to 67500 per H×H×H .Three different grids are used with the same initial number of particles. The grid A consistsin 15× 15× 15 cells per a volume H ×H ×H corresponding to 20 model particles per cells(PPC) in mean. The grid B has the 11 × 11 × 11 cells per H ×H ×H corresponding to50.7 PPC in mean. The rougher grid C has 7× 7× 7 cells per H ×H ×H correspondingto 196.8 PPC in mean.


The reduced mass flow rate (G) is shown in Table 5.1 for δa = 1 and various channellength L. For each fixed channel length the reduced mass flow rate increases with the gridrefinement. The difference between the mass flow rates for the fixed length obtained withthe grids A and C is around 3% except the smalls length L = 3H.

For the finest grid A the reduced mass flow rate decrease in 2.7% when the channel lengthincreases from 3H to 15H.

Table 5.2: The reduced mass flow rate (G) through a straight micro channel with a squarecross-section and various lengths; Pin/Pout = 2, δa = 10. The grid A of 15× 15× 15 cells

with 20 PPC is used. The characteristic dimension was D = H.

grid GL = 3H L = 5H L = 7H L = 9H L = 15H L = 21H

A 1.3008 1.3001 1.2971 1.2960 1.2908 1.2941

It is possible to compare the reduced mass flow rate with the results of Ref. [3], where theisothermal gas flow through a long channel of various rectangular cross-section is simulatedby using the linearized BGK kinetic equation. The authors of Ref. [3] carried out thesimulation of the gas flow through a rectangular long channel of various cross-section aspectratios driven by small pressure gradient. Therefore in his simulation the outlet rarefactionparameter is only slightly different from the inlet one. It is not the case for the presentsimulations. Therefore, in order to compare both results the expression proposed in Ref. [3]is used

G (δa) = Q

(δin + δout

2

), (5.5)

which allows to taken into account the larger pressure gradient and consequently the deltavariations. The values of the reduced mass flow rate Q for various values of δ may be foundin Ref. [3]. The mass flow rate through the channel of length equal to 15H differs from thecorresponding value obtained in Ref. [3] in 1.5%, therefore one can consider the channellength as long enough to reproduce the properties of a long channel.

Table 5.2 shows the results for the rarefaction parameter δa = 10 and various lengths ofmicrochannel. The simulations are performed using the finest grid A. For this larger valueof rarefaction parameter δ the decreasing in the mass flow rate is smaller than in the case ofδa = 1 and it is smaller than 1% when the channel length varying from 3H to 21H. Whencomparing the value of the reduced mass flow rate for the channel with L = 21H to thecorresponding results of Ref. [3] anew the difference in 1.5% is found.


Table 5.3: The mass flow rate through straight micro channels of various cross sections,for a pressure difference: (Pin − Pout)/Pout = 1 , and with a length L = 21H. The

characteristic dimension was D = H.

W/HM/M∞

δa = 1.125 δa = 11.25 [89] slip regime1 0.5141 0.4741 0.4792 0.6938 0.7065 0.7123 0.7821 0.8023 -4 0.8329 0.8510 -5 0.8644 0.8708 0.8846 0.8854 0.8991 -8 0.9118 0.9212 -16 0.9485 0.9288 -∞ 1.0000 1.0000 1.000

5.4.2 Influence of the relative dimensions of the cross section on the flowrate through a straight microchannel

The effect of the cross-section aspect ratio on the mass flow rate is studied for two valuesof the average rarefaction parameter δa = 1.125 and 11.25. The present results on thereduced mass flow rate M/M∞ together with the corresponding analytical results, obtainedin [89] for the long channel in slip flow regime, are shown in Table 5.3. Here M is the massflow rate through a straight microchannel per unit cross sectional area, and M∞ refers tothe corresponding value for the microchannel having a cross section with the infinite widthW →∞ (two dimensional case: the flow between two parallel plates). From this comparisonit is clear that the simulation in the straight channel channel of the length L/H = 21reproduce very well the mass flow rate obtained analytically in [89]. The maximal differencebetween the obtained here results for δa = 11.25 and the results of [89] is less than 1.5%.For δa = 11.25 this difference increases because the results in Ref. were obtained only forthe slip flow regime.

Figure 5.4 shows a comparison of the mass flow rate G for various cross-section aspect ratioW/H, reduced by the mass flow rate obtained for the square cross section GW=H . Thisreduced mass flow rate G/GW=H calculated in the present work is plotted together withthe results obtained in [3]. As in previous Section, expression 5.5 was used to taken intoaccount the pressure variation along the channel. However, as it was mentioned in [3] thisapproximative expression (5.5) works well only for the cross-section close to the square one.As it is clear from the comparison, Fig. 5.4, the both results are very close up to the cross-section aspect ratio equal to 5, after that the approximative relation (5.5) overestimate themass flow rate.


0 5 10 15 20

1

1.2

1.4

1.6

1.8

2

2.2

W/H

G/G

W=

H

0 5 10 15 20

1

1.2

1.4

1.6

1.8

2

2.2

W/H

G/G

W=

H

Sharipov δ=1

DSMC δa=1 Sharipov δ=10

DSMC δa=10

Figure 5.4: The effect of the relative dimensions of the cross section on the flow ratevia a relative flow rate (G/GW=H) through a straight microchannel compared to Sharipov[3] at δ = 1 (in the left), and at δ = 10 (in the right). The characteristic dimension was

D = H.

10−2

100

102

0.8

1

1.2

1.4

1.6

1.8

2

2.2

δa

G:G

δ=0.

01

10−2

100

102

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

δa

2DW=H

G2D

:GW=H

Figure 5.5: Relative flow rates (G/Gδa=0.01) in a straight microchannel (in the left), anda comparison of the straight microchannel with a square section with that of a rectangularcross section with (W/H →∞) (2D simulation) (in the right). The characteristic dimension

was D = H.

Figure 5.5(left) shows the mass flow rate through the channel with the square cross-section(W = H) and with the very large cross-section (W →∞, 2D case) for various rarefactionparameters. The both mass flow rates are normalized by the corresponding values of themass flow rate for the rarefaction parameter equal to 0.01 (G/Gδa=0.01). It is to note thatthe Knudsen minimum is deeper for the square cross-section channel than that for the verylarge cross-section channel. On Fig. 5.5 (right) the ratio of the mass flow rates between theinfinite cross-section (2D case) G2D and the square cross-sections GW=H is presented. Itcan be seen that, the ratio of the reduced flow rates G2D/GW=H varies significantly withthe rarefaction. The maximal difference between the mass flow rates is in the free molecular


regime (δa = 0.01) and the minimal one is found in the transitional regime for δa ∼ 1.

Table 5.4: Effect of the cross-section shape on the reduced mass flow rate G/GW=Hthrough a straight micro channel of a constant hydraulic diameter, for a pressure ratioPin/Pout = 3 and channel length equal to 21H. The characteristic dimension was D = H.

W/H 1 2 3 4 5 6 8 16

Gδa=1/G(W=H,δa=1) 1.0 1.3656 1.5415 1.6409 1.7019 1.7427 1.7944 1.8724

Gδa=10/G(W=H,δa=10) 1.0 1.4448 1.6211 1.7126 1.7711 1.8103 1.8659 1.9631

In what follows is an investigation of the effect of the cross section shape on the reduced massflow rate G through a straight micro channel with a constant hydraulic diameter, defined asDh = 2WH/(W +H). Table 5.4 shows the effect of the cross section shape on the reducedmass flow for the pressure ratio Pin/Pout = 3 and for a channel length equal to 21H. Thecharacteristic dimension was D = H. The simulations are performed for the width to heightratio varying from 1 to 16 and for two values of the rarefaction parameter δa = 1 and 10.It can be seen that the reduced mass flow rate through increases more rapidly, with W/Hincreasing, in the slip regime compared to the transitional one.

5.5 Simulation of the flow through a 90 degrees bend mi-crochannel

5.5.1 Reduced flow rate through a 90 degrees bend microchannel

In this Section the flow through the 900 bend channel with the square cross-section is studied.Two rarefaction parameters are considered δa = 1 and 10. For these rarefaction parametersthe simulations are carried out for the pressure drop (Pin−Pout)/Pout equal to 0.2; 0.5 and1. The diffuse reflection with complete energy accommodation was considered on the solidwalls [2]. Influence of the channel length is analyzed. As it was mentioned for the first timein Section 5.2 when the flow through the 900 bend channel is analyzed the centerline lengthis used as the channel length in eq. (5.1). Table 5.5 summarizes the dependence of thereduced flow rate G through a 900 bend micro channel on the channel length and on thepressure drop. As it is clear from the provided data, for a fixed pressure ratio the mass flowrate G changes very slightly (less than 1%) when the channel centerline length increases.Therefore, it may concluded that a centerline length of 9H is long enough for obtaining amass flow rate similar to a long micro channel when applying the implicit boundary conditionsand for considered range of the rarefaction parameter. However, for the further study wasperformed with a microchannel length equal to 21H.


Table 5.5: Effect of pressure drop and channel length on the reduced flow rate G througha 900 bend micro channel of the squared cross section. The characteristic dimension was

D = H.

CL/H (Pin − Pout)/Pout δa = 1 δa = 10

90.2 0.721 1.16930.5 0.7420 1.23551.0 0.7537 1.2629

150.2 0.719 1.17950.5 0.7444 1.23511.0 0.7544 1.2619

210.2 0.724 1.17280.5 0.7456 1.23501.0 0.7543 1.2561

0 0.5 1 1.5 20

1

2

3

4

5

6x 10

−11

pressure drop

mas

s flo

w r

ate

(Kg/

s)

δa=1, pressure drop: (p

in−p

out)/p

out

δin

=1, pressure drop: (pin

−pout

)/pin

δa=1, pressure drop: (p

in−p

out)/p

mean

Figure 5.6: The mass flow rate through a 900 bend microchannel with a square cross-section vs pressure drop between inlet and outlet cross-sections for the rarefaction parameter

δ = 1

On Fig. 5.6 the dimensional mass flow rate M through the 900 bend channel with the squarecross-section is plotted as a function of the pressure drop (Pin − Pout)/Pout varying from0.2 to 2. The rarefaction parameter δa is equal to 1. The non-linear increasing of the massflow rate is observed analogous to that found in Refs. [4, 47, 76].

The comparative study of the mass flow rate behavior as a function of the pressure dropis carried out under the conditions used in Ref. [4]. To do that the outlet rarefactionparameter was fixed, therefore the outlet pressure is also fixed. The pressure drop varieswith the inlet pressure varying. The present DSMC simulations are performed with anoutlet Knudsen number Knout equal to 0.071, the corresponding rarefaction parameter maybe calculated from eq. (5.3). The channel height is equal, as in [4] to 1µm. For thiscomparison the channel centerline length is taken to be equal 9H and the channel cross-section has the square shape. The working gas for the DSMC simulations was the argon;


0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.42

4

6

8

10

12

14

x 10−11

pressure drop

mas

s flo

w r

ate

( K

g/s)

DSMC, Bend, HS, Kne 0.71DSMC,Straight, HS, Kne 0.71N−S slip, Bend, Croce et alN−S Slip, staright, Croce et al

Figure 5.7: Mass flow rate through a 900 bend microchannel with a square cross section,for different pressure drops, simulated using the DSMC, and from the Croce et al [4]

Figure 5.8: The 900 bend microchannel.

whereas the authors of Ref. [4] have considered the nitrogen flow with the same outletKnudsen numberKnout = 0.071. Figure 5.7 shows the mass flow rate through a 900 bendmicrochannel and through a straight microchannel obtained in the present study togetherwith the results of Ref. [4]. It can be seen that, the mass flow rates through a 900 bendmicrochannel and through a straight microchannel are identical to each other. The similarresults are obtained in Ref. [4] where the Navier-Stokes equations with the first order slipboundary conditions were used. The mass flow rate predicted by the DSMC is marginallyhigher than one predicted by the Navier-Stokes slip solution obtained in [4].

It is to note that, for a study of effect of the pressure drop at a given rarefaction, whendefinition of the rarefaction and the pressure drop were consistent (i.e., when both of them


were defined wrt the same metric of pressure, viz.ly Pout for δout and Knout; or Pin for δin;or Pmean for δa), the the behavior of the mass flow rate was linear, otherwise nonlinear. Astudy was conducted under isothermal conditions, considering the outlet pressure, the inletpressure, and the mean pressure (to define the rarefaction and to measure the pressure drop)as in the Fig. 5.6, and the Fig. 5.7.

5.5.2 Effect of the cross section on the flow rate through a 90 degrees bendmicrochannel and effective length of a 90 degrees bend microchannel

Table 5.6: The reduced flow rate, through a 900 bend micro channel with a characteristicdimension D = W (the width of the microchannel), for various heights, for a pressure drop

( (Pin − Pout)/Pout) = 1 at δa = 11.52. The characteristic dimension was D = W .

H/D 1 2 3 4 5 6 8

G at δa = 11.52 1.3074 1.9515 2.2587 2.4419 2.5706 2.6703 2.8228

Table 5.7: The reduced flow rate, through a 900 bend micro channel with a characteristicdimension D = H (the height of the microchannel), for various widths, for a pressure drop:

( (Pin − Pout)/Pout) = 1, at δa = 11.52. The characteristic dimension was D = H.

W/D 1 2 3 4 5 6 8

G at δa = 11.52 1.3074 1.9040 2.1523 2.2823 2.3579 2.4095 2.4740

The reduced flow rate through a 900 bent micro channel, for various heights and widths,for a non-dimensional pressure drop (Pin − Pout)/Pout = 1, at δa = 11.52 is shown in theTable 5.6, and the Table 5.7, respectively. From Table 5.6, and the Table 5.7 it maybe inferred that at a rarefaction δa = 11.52, a reduced flow rate increases at a larger ratewith increase of W , compared to a similar increase in H. Here, it must be noted that, forthe computational considerations, the arm lengths were kept the same (at Larm = 9D, withthe D as the characteristic length, where Larm is both the L1 and L2 in the Fig. 6.10)for all the cases considered; as a result of which, the centerline length (CL) of the channeldoes not change when W > H, and was constant at CL = 19D. However, it varies fromCL = 19D when H = W , to CL = 26D when H = 8W when H > W .


(a) A 900 bend microchannel, considered forstudying the effect of the height (H ≥ Ws)on the flowrate

(b) A 900 bend microchannel, consid-ered for studying the effect of the width(W ≥ Hs) on the flowrate

Figure 5.9: The geometry of a 900 bend microchannel, considered for studying the effectof relative dimensions of its cross section on the flow rate

10−2

100

102

0.8

1

1.2

1.4

1.6

1.8

2

2.2

δa

G:G

δ=0.

01

2D,(W/H)−>inf.H=WH=9W

10−2

100

0.85

0.9

0.95

1

1.05

1.1

a zoom around minima

δa

G:G

δ=0.

01

Figure 5.10: Relative flow rates (G/Gδa=0.01) through a 900 bend microchannels witha square cross section, a rectangular cross section with H = 9W , and a cross sectionwith W/H → ∞ (2D) (in the left), and a zoom around the minima (in the right). The

characteristic dimension was D = minW,H.


10−2

10−1

100

101

102

1.9

2

2.1

2.2

2.3

2.4

2.5

δa

G2D

:GH=W

GH=9W

:GH=W

Figure 5.11: A comparison of the 900 bend microchannel with a square section withthat of a rectangular cross section with H = 9W , and another with W/H → ∞ (2Dsimulation), via mass flow rates in the free molecular, the transitional and the slip regimes.

The characteristic dimension was D = minW,H.

Table 5.8: The reduced mass flow rate (G) through a 900 bend microchannel, and througha straight microchannel, for a pressure drop: ( (Pin − Pout)/Pout) = 1. The characteristic

dimension was D = H.

δa 900 bendwith W=H

900 bendwithW →∞

straightwith W=H

straightwithW →∞

0.01 0.7534 1.6082 0.8083 2.06680.02 0.7520 1.5901 0.8046 2.01840.04 0.7490 1.5774 0.8017 1.94770.08 0.7470 1.5482 0.7931 1.84580.16 0.7425 1.5026 0.7830 1.72370.32 0.7391 1.4577 0.7753 1.60950.64 0.7432 1.4316 0.7724 1.53561.28 0.7654 1.4621 0.7946 1.54212.56 0.8380 1.6146 0.8616 1.68165.12 0.9907 1.9780 1.0185 2.048110.24 1.2694 2.6524 1.3039 2.731520.48 1.6319 3.4943 1.6829 3.6545


The Fig. 5.10 shows the relative flow rates ( G/Gδa=0.01 ) through a 900 bent microchannelswith a square cross section, a rectangular cross section with H = 9W , and a 2D crosssection. The minimum of the mass flow rate through a 900 bent microchannel has a smallshift towards a relatively lower δa, when the cross section has changed to square from theone with W/H →∞, a 2D simulation. The minimum of the mass flow rate is more prominentin the case of the cross section with W/H →∞ compared to a square cross section as seenin the Fig. 5.10 and in the Table 5.8. There was no shift of the minimum observed incase of a straight microchannel. The simulations were performed using a uniform grid with15×15×15 cells per every volume of H×H×H for computing the collisions. The behaviorof the flow rates in the considered extreme cross sections was studied as shown in the Fig.5.10, Fig. 5.11, and Fig. 5.5. The effect of the cross section on the reduced flow rate forthe considered cross sections (viz.ly square and rectangular ) was shown in the Fig. 5.11 inthe slip and the transitional regimes for various values of δa, and it is seen that, the ratio ofthe flow rates has a minimum around the δa = 1.28, and is increasing towards a lower anda larger value of δa in the range δa ∈ [0.01, 20.48]. Which implies that the cross sectionaleffect on the mass flow rate varies with the rarefaction parameter (δa), in the slip and thetransitional regimes. Hence it is found that, the ratio of the flow rates cannot be predictedprecisely, calculating/simulating at a single representative value of δa for the whole regimeof the gas flow, be it the slip regime or the transitional regime of the gas flow. However, ithas a tendency to become a constant for the free molecular regime and saturating in the sipregime. The Fig. 5.10 and Fig. 5.11 also contain a study of the mass flow rate through a900 bent microchannel with the height to width ratio of 9 : 1, and a comparison with a 900

bend microchannel with square cross section. It is seen that the behavior is similar to thecomparison of 2D simulation with the square cross section, however, magnitudes differ.

0 2 4 6 8 10 12 14 16 180.5

1

1.5

2

relative width

G/G

W=

H

δa=1

0 2 4 6 8 10 12 14 16 180.5

1

1.5

2

relative width

G/G

W=

H

δa=10

(dp/pout

)=0.2

(dp/pout

)=1.0

(dp/pout

)=2.0

(dp/pout

)=0.2

(dp/pout

)=1.0

(dp/pout

)=2.0

Figure 5.12: The effect of the cross section on the reduced flow rate (G ) through a900 bend micro channel for a constant hydraulic diameter, for various pressure drops, withCL = 21H. Shown are G/GW=H ; for δa = 1 (up), and δa = 10 (down). The characteristic

dimension was Dh


In what follows, we investigate, the effect of the cross section on the reduced flow rate (G) through a 900 bend micro channel for a constant hydraulic diameter, for various pressuredrops, studied with a center line length CL = 21H. It is to note that for this studythe rarefaction parameter is calculated basing on the hydraulic diameter as the characteristicdimension of the problem, D = Dh. The Table Fig. 5.12 presents a study of the dependenceof the reduced flow rate through a 900 bend micro channel on the cross section, for a constanthydraulic diameter. It is observed that the dependence is non-linear in nature, with a kneebetween W = 2H and W = 8H, for both the rarefactions considered δa = 1 and δa = 10.This was illustrated in the Fig. 5.12.

5.5.3 Comparison of the flow rate through a straight microchannel to flowrate through a 90 degrees bend microchannel

0.05 0.1 0.15 0.20.97

0.98

0.99

1

1.01

1.02

1.03

1.04

Knout

Mbend/M

straight

Argon,(Pin

/Pout

)=2.8817, W=H

Nitrogen, (Pin

/Pout

)=3.06, W=H

Figure 5.13: A comparison of the straight and 90 degree bend microchannels consideringthe mass flow rate, comparison was performd considering the argon gas as well as nitrogen

gas.

Our results suggest that the flow rate though a 900 bend microchannel in the transitionaland slip regimes has been lower than the flow rate though a straight microchannel, however,under certain cicumstances, the opposite might be obtained. E.g., in the Fig. 5.13 it canbe seen that the mass flow rate through a 900 bend microchannel can be slightly higherthan the flow rate through a straight microchannel. This phenomenon was earlier shown byAgarwal et al [74] using the lattice Boltzmann method (in 2D); by Rovenskaya et al [76]


(in 2D) and Croce et al [4] (in 2D and 3D) using the N-S solver; and by White et al [86]using the DSMC method (in 2D), in a similar range of Kn.

The DSMC simulations (shown in the Fig. 5.13) were performed considering argon andnitrogen. For the case of argon a non dimensional pressure drop of (Pin − Pout)/Pout = 1.8817applied at a temperature 300 K to a channel of length CL = 9H, with a square crosssection. A time step of 3.2157 × 10−11 sec., and a mesh of 22 × 22 × 22 cells per everyvolume of H ×H ×H were used; whereas, for the simulations considering the nitrogen gas,a non dimensional pressure drop of (Pin − Pout)/Pout = 2.06 applied at a temperature 300 Kto a channel of length CL = 9H, with a square cross section. A time step of 2.6928×10−11

sec., and a mesh of 44×44×44 cells per every volume of H×H×H were used. The diffusereflection with complete energy accommodation was considered on the solid walls [2].

5.6 Conclusions

A study of the pressure driven isothermal gas flows through different three-dimensional mi-crochannel configurations such as straight micro-channels, 900 bends, all with a rectangularcross-section was performed. Implicit inlet/outlet boundary conditions have been applied atthe channel ends in order to avoid the ends effects and study the transition from short tolong channel configuration when the results of the linear kinetic theory can be used. It wasfound that a channel length of 9 to 15 times of the channel height gave reasonable resultsthat deviated from the theoretical values with 2 to 3%s. The presented results are illustrativeand quantitative in nature and are limited to around the transitional regime of the gas flow.The results are subjected to a statistical fluctuation. An analysis was performed in detail forthe influence of the geometry of short channel configurations on the gas flow characteristics,over a wider range of the rarefaction.

The results suggest that application of the implicit boundary conditions with the DSMC tosolve the flow through a microchannel that mimics a flow though a long microchannel, for alow pressure difference between its inlet/outlet, is appropriate over an almost entire regimeof rarefaction, when the channel has a square cross section, however, when the cross sectionof the channel is rectangular, the main flow parameter, the reduced mass flow rate starts todeviate from the desired value in the free molecular regime of the gas flow, say, for. The meshresolution affects the simulated reduced flow rate, and the stream wise velocity componentof the simulated gas micro-flow. The reduced flow rate increases with an increase of thepressure difference between the inlet and the outlet.

The study was conducted for a pressure difference, in the range of 20% to 200% relative tothe outlet pressure (for a given outlet rarefaction), in the range of 20% to 90% relative to


the inlet pressure (for a given inlet rarefaction), and in the range of 20% to 190% relative tothe mean pressure (for a given mean rarefaction). A linear increment was was observed inthe mass flow rate, when the rarefaction and pressure drop were defined wrt the same metricof pressure; otherwise the increment was nonlinear, however, which tends to saturate.

The reduced flow rate was presented for the gas micro-flow though a 900 bent microchan-nel for a three cross sections in the free molecular regime, in the slip regime, and in thetransitional regime. A small shift in the minimum value of the reduced mass flow rate wasobserved when the cross section was changed from square to a plane (2D). The ratio of thereduced flow rates (GW 6=H/GW=H) varies with the rarefaction and, we cannot have a singlerepresentative value for this ratio for the entire regime of the gas flow, be it the slip or thetransitional regime. However, it has a tendency to become a constant for the free molecularregime and saturating in the sip regime.

The effect of the relative dimensions of the cross section was studied for a straight and a 900

bent microchannel. In case of the 900 bent micro-channels the height affects the reducedflow rate relatively more than the width. The reduced flow rate tends to saturate relativelyquickly with the increase of the width compared to the increase of the height. A study wasalso conducted for the effect of the relative dimensions of the cross section by keeping thehydraulic diameter constant, for three values of pressure differences.

A comparison of the mass flow rate through a straight microchannel with the mass flow ratethrough a 900 bend microchannel was presented. The investigations were performed withthree possible characteristic dimensions (viz.ly height, width, and hydraulic diameter) forthe case of a 900 bend microchannel, and with two possible characteristic dimensions (viz.lyheight, and hydraulic diameter) for the case of a straight microchannel.

Chapter 6

Effects of finite distance betweentransversal dimensions inmicrochannel configurations: DSMCanalysis

6.1 Introduction

The operating conditions of micro and nano systems often lie in and around the transitionalregime and in the slip regime of the gas flow [6], and the kinetic methods and modelshave been quite useful in predicting the flow properties in this regime of flow [2, 10, 18].Straight microchannels and relatively more complex microchannel configurations are someof the basic elements in the micro-electro-mechanical systems (MEMS). Hence, much of theattention has been paid in recent times to a study of the micro geometries such as: a pipe,a long channel, a short channel, a channel bend at various angles (e.g., bend at 90 degrees),zigzag channels, and channels with various cross sectional aspect ratios and shapes suchas hexagonal, trapezoidal, triangular, etc[46, 56, 74–77]. The isothermal/non-isothermalpressure driven flows, thermal driven flows, and shear driven flows were investigated. Theflow fields, stresses, heat and mass fluxes were extensively studied. Analytical and numericalinvestigations were performed to study the gas flow though microchannels in the past fewyears [6–8, 78–84, 91–95]. The studies were demonstrative, illustrative, and comparativein nature and a few databases were created for the rarefied gas flows through the micro-channels.

71

Chapter 6. Effects of finite distance between transversal dimensions in microchannelconfigurations: DSMC analysis 72

The authors of the paper [3] studied the mass flow rate through a long rectangular channel,using the calculations based on the model kinetic equation in the wide range of the height-to-width ratio, the paper [96] is related to the study of the gas flow through a zigzag channelusing the linearized Kinetic equation, investigating the flow over a wide range of rarefactionand for several cross sectional aspect ratios.

The papers [4, 74, 76, 85] were devoted to studying 900 degree bend microchannels. In thepaper [4] using the Navier-Stokes (N-S) equations with the slip boundary conditions, theeffects of the channel aspect ratio (width to height) were studied in slip regime, and haveshown that, for larger cross section aspect ratios flow distributions in a 3D microchannels,get closer to two dimensional ones.

Aubert and Colin [97] have studied the influence of the height to width ratio on the normalizedmass flow rate and pressure profile in a rectangular duct, at the outlet Knudsen numberKnout = 0.1 for height to width ratios 1 and 0.1. Meolans et al [89] have proposed ananalytical approach for the slip regime to take into account of the influence of the transversewalls on the stationary isothermal gas flow through a long rectangular microchannel, andhave studied the effect of the two transversal finite dimensions on the mass flow rate. Thepapers [98–100] too concern the study the flow through a rectangular channel.

The work [101] concerns a comparative study of the mass flow rate through a straight anda 900 bend microchannels studying the cross sectional effects in detail, using the DSMCmethod and implicit boundary conditions. The prime advantages of applying the implicitboundary conditions are that it avoids the utility of the particle reservoirs for applying thepressure at the I/O, besides the obtained flow properties are akin to those from a long channelgeometry, hence reducing the computational time, and they also avoid the end effects toa large extent. Moreover, as the present study concerns the effects of a finite distance(between the transeversal dimensions of a microchannel geometry, say width) in a normaldirection to the flow, we could relate our results to a long channel geometry.

The present chapter is devoted to a DSMC analysis of the influence of the transversal wallson the flow considering the mass flow rate as the property of interest. An in-house DSMCcode was developed implemented in conjunction with the implicit boundary conditions for thispurpose. A straight microchannel, a 900 bend microchannel, and a T-junction are consideredfor studying the effects of the finite distance between a pair of opposite transversal walls, onthe mass flow rate. A rectangular cross section was considered for this purpose. It should benoted that, in case of the 900 bend microchannel and the T-junction, the geometry has twoopposite pairs of transversal walls in the vertical arm; while the distance between one pair ofthem (the height) defines the outlet-rarefaction, the effects of the distance between the otherpair of transversal walls is the topic of the current work, however, the study was conductedfor various values of rarefaction. The chapter is organized as the following: the Section 6.2


describes the software structure, the details of the DSMC technique, definition of the flowparameters, and the implicit boundary conditions. The Section 6.3 presents a qualitativeand quantitative analysis of the influence of the finite separation between a pair of oppositedimensions on the pressure, temperature, and velocity profiles in a straight microchannel,and on the mass flow rate through a straight, 900 bend microchannels, and T-junction, inthe late slip regime and in the transitional regime.

6.2 Computational consideration

The structure of the in-house DSMC software developed follows the classical Monte Carloapproach [2]. This geometrical configuration allows to simulate various channel configura-tions: straight and bend channels, the T-junction, the cross-junction and etc. The currentchapter presents only the results for the straight, the 900 bend, and the T-junction of microchannels.

Figure 6.1: The general view of 3D-microchannel geometry.

In case of the 900 bend micro channel and T-junction the centerline length CL (see Fig.6.10 and Fig. 6.12), is considered as the equivalent length of the channel (L = LCL =L1 + L2 + H). The length of the inbound and outbound arms was considered to be thesame in a given channel geometry.

6.2.1 Details of the DSMC technique

The DSMC method in its classical version [2] is implemented. To model the binary collisionsthree collision schemes are examined: the traditional “No Time Counter" (NTC) [2, 9], the“Majorant collision scheme" [36], and the “Simplified Bernoulli Trials" (SBT)[40, 90]. Incase of a non iso-thermal flow simulation, for an implementation with restrictions memory,


the SBT scheme gives a relatively more accurate results with a relatively low number particlesper cell compared to the NTC; however, in the present work we mainly study the iso-thermalflow. The results presented in this chapter are calculated by the NTC scheme.

The mesh and the computational time step are chosen such that the particle does not crossin average more than 1 cell per kinetic step. The time step is chosen to be smaller thanmean collision time. A uniform grid is used for computing the binary collisions. Typically,the simulations are carried out with a number of particles per cell, at the beginning of thesimulation, ranging from 20 to 40, which is further raised as the simulation progressed.The flow properties (e.g., mass flow rate) in each simulation is time-averaged after reachingits steady state value, over a very long interval of (simulation) time to reduce the scatterassociated with the calculation. The diffuse reflection with complete energy accommodationwas considered on the solid walls [2].

6.2.2 Definition of the flow parameters

In this paper, we use the rarefaction parameter δ, which is related to the Knudsen numberas

δ =√π

21Kn

. (6.1)

to characterize the level of the gas rarefaction. When using the Hard Sphere (HS) modelthe Knudsen number (Kn) is

Kn = kBT√2πd2P

1H. (6.2)

Where the H is the characteristic length (e.g., the height or the hydraulic diameter) of themicrochannel, d is the diameter of the particle, kB the Boltzmann constant, P is the pressure,and T is the temperature. The rarefaction parameter is used in this chapter (alternatively,the Knudsen number), to identify the level of the gas rarefaction. In the following we usethe average rarefaction parameter which is defined as

δa = 12 (δin + δout) , (6.3)

where δin and δout are calculated according to Eq. (6.1) by using the respective macroscopicvalues at the inlet and outlet cross-sections.

Inlet-outlet boundary conditions

Implicit boundary conditions [45, 47, 48] are applied to maintain a given pressure at the inletand outlet sections. The following notation was followed: the subscript ‘in’ all parametersrelated to the upstream conditions at the micro-channel extremity with the largest pressure


(the entrance of element 1 with pressure pin in our case). The inlet temperature Tin ismaintained to be the same during the simulation. The outlet parameters for elements aredenoted by subscript ‘out’. The temperature at the outlet is not imposed, rather calculatedfrom the other flow properties, such as pressure and density.

6.3 Results and analysis

A pressure driven near-isothermal gas flows through the three-dimensional straight, 900 bendmicrochannels, and T-junction are studied. A rectangular cross-section with various widthW to height H ratios are considered. The walls of all elements are maintained at constanttemperature Tw and they are considered as diffuse reflective. The inlet Pin and outletPout pressures are kept constant. A pressure difference between the inlet and outlet cross-sections, (Pin − Pout) /Pout = 1 and 2 are considered here. The inlet temperature Tin isfixed equal to the wall temperature Tw. The outlet temperature Tout is calculated by usingthe characteristic technique, explained in details in Subsection 6.2.2. The working gas forthe DSMC simulations was the argon. The simulations under the Subsection 6.3.1 and theSubsection 6.3.2 were conducted at a temperature of 300 K. The grid consists of 15×15×15cells per a volume H ×H ×H corresponding to 20 model particles per cells in mean wasutilized.

6.3.1 Effects of finite distance between opposite tansversals in straight and90 degree bend microchannels

Figure 6.2: The straight microchannel.

Effects of a finite width in straight microchannels

The following expression (6.4) for the pressure distribution along the microchannel by Arkillicet al [5], was used as a reference in the 2D case, and the relative deviation was shown in theFig. 6.3 to analyze the effects of the separation between the transversal walls, consideringfour cross sections of a straight microchannel calculated using the DSMC, viz.: square(W = H), with W = 2H, W = 5H, and W = 10H. A maximum of about 4.2% ofthe deviation was observed in case of a square cross section, while it was around 1.5% for


0 0.2 0.4 0.6 0.8 1−1

0

1

2

3

4

5

6

pres

sure

dev

iatio

n fr

om A

rkill

ic e

t al e

xpr.

(%)

x / L

W=H, DSMCW=2H, DSMCW=5H, DSMCW=10H, DSMC

Figure 6.3: Pressure deviation in a straight microchannel from the 2D analytical expres-sion (6.4) by Arkillic et al [5], for a Knout = 0.072, L = 9H and Pin/Pout = 3.

0 0.2 0.4 0.6 0.8 14000

6000

8000

10000

12000

14000

pres

sure

(P

a)

x/L

W=H, DSMCW=H, SIMPLE−TSW=2H, DSMCW=5H, DSMC2D, Arkillic et al

Figure 6.4: Pressure profiles along the streamwise direction in a straight microchannel,for a Knout = 0.072, L = 9H, from the DSMC and the 2D analytical expression (6.4) by

Arkillic et al [5].

the cross section with W = 10H, and was around 2.5%, 1.7% for the cross sections withW = 2H and W = 5H, respectively. The corresponding pressure profiles were comparedagainst the 2D profile from the expression (6.4) in the Fig. 6.4.

P (x) = Pout[−S +√S2 + (1 + 2S)x+ (P 2

i + 2SPi)(1− x)], (6.4)

Where, x = x/L, S = 6σKnout with the σ = 1, Knout = 0.072 was the outlet Knudsennumber, and Pi = 3 was the ratio of the inlet pressure and the outlet pressure.

The Fig. 6.4 shows that, the pressure profiles are approaching the Arkillic et al’s 2D profileas the channel widens, however, a difference between the pressure profiles of the DSMCand the SIMPLE-TS may be attributed to a different boundary conditions applied. The


0 0.2 0.4 0.6 0.8 1235

240

245

250

255

260

265

270

275

Tem

pera

ure

(K)

x / L

W=H, DSMCW=2H, DSMCW=5H, DSMCW=10H, DSMC

Figure 6.5: The temperature profile along a straight microchannel, for a Knout = 0.072and Pin/Pout = 3.

0 0.2 0.4 0.6 0.8 1235

240

245

250

255

260

265

270

275

Tem

pera

ure

(K)

y / H

W=H, DSMCW=2H, DSMCW=5H, DSMC

Figure 6.6: Temperature profiles at the exit of a straight microchannel (length L = 9H),for a Knout = 0.072, L = 9H and Pin/Pout = 3.

SIMPLE-TS calculations use velocity-slip and temperature jump boundary conditions [81].It should be noted that, the Arkillic et al 6.4 use the velocity-slip boundary condition. TheSIMPLE-TS (see [81]), is a pressure based, iterative finite volume method, utilized for thecalculation of unsteady, compressible, viscous and heat-conductive gas flows. The streamwise temperature profiles are shown in the Fig. 6.5, a cooling of about 35K was observedat the exit for a case of W = 5H as shown in the Fig. 6.5 and Fig. 6.6, the temperatureprofiles for the casesW = 5H andW = 10H were identical along the stream wise direction.The temperature profiles show a cooling along the channel, and there was a relatively morecooling observed for a wider microchannel. The Fig. 6.6 shows the temperature profiles atthe exit, there was a temperature difference (jump) of about 3 − 5K near the walls in theDSMC results. The Fig. 6.7 shows that the DSMC simulated velocity profile reaches a 2Dprofile by the width W = 5H, and a similar observation can be made from the Fig. 6.8.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

20

40

60

80

100

120

140

160

180

200

220

u (m

/ s)

y/H

W=H, DSMCW=H, SIMPLE−TSW=2H, DSMCW=5H, DSMC2D, DSMCW=10H, DSMC

Figure 6.7: The stream wise component of the velocity (u) at the exit, at half the width,along the height; for a Knout = 0.072 and Pin/Pout = 3.

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

160

180

200

220

u (m

/ s)

z / W

W=H, DSMCW=H, SIMPLE−TSW=2H, DSMCW=5H, DSMC2D, DSMCW=10H, DSMC

Figure 6.8: The stream wise component of the velocity (u) at the exit, at half the height,along the width; for a Knout = 0.072, L = 9H and Pin/Pout = 3. The case W = 10H

was simulated considering a plane of symmetry at half of the width.

The Fig. 6.8 shows a flat velocity profile, for W = 5H observed in a normal plane along thewidth from about 25% to 75% of the width, where as, for the case W = 10H the velocityprofile in observed the normal plane along the width was flat from about 12.5% to 87.5% ofthe width (by virtue of symmetry). Which means, the velocity profile was flat about 50% ofthe width in the middle of the channel for the case W = 5H and about 75% of the width inthe middle of the channel for the caseW = 10H. Both the cases (W = 5H andW = 10H)were matching very well with the 2D profile, along the width in the above stated ranges (ofwidth), however, the 2D profile was flat for almost in the whole range of width.

The scaling of the stream wise component of the velocity at the exit, was presented in theFig. 6.7 and Fig. 6.8, and along the channel was shown in the Fig. 6.9. In the process, acomparison of the DSMC produced results with those obtained from the SIMPLE-TS was


0 0.2 0.4 0.6 0.8 150

100

150

200

250

stre

amw

ise

com

pone

nt o

f vel

ocity

(m

/s)

x / L

W=H, DSMCW=H, SIMPLE−TSW=2H, DSMCW=5H, DSMCW=10H, DSMC

Figure 6.9: Scaling of the stream wise component of the velocity (u) with the wideningof the straight microchannel, for a Knout = 0.072, L = 9H and Pin/Pout = 3.

performed in the Fig. 6.7, Fig. 6.8, and Fig. 6.9, a very good agreement of the velocityprofiles from the DSMC and SIMPLE-TS was shown, for a square cross section. The DSMCresults were obtained using an OpenMP parallel version of the software. The simulations wereperformed at a temperature 273K, using the argon gas. The grid consisting of 26× 26× 26cells per a volume H ×H ×H corresponding to 20 model particles per cells in mean wasutilized.

Influence of tansversal walls on flow rate through straight and 90 degree bend mi-crochannels

Figure 6.10: The 900 bend microchannel.

The Table 6.1 shows the influence of the transversal walls on the near-isothermal pressuredriven mass flow rate through a straight channel, at two values of the rarefaction parameterδa = 1 and δa = 10, also listed are the results from [89] for the slip regime, a good agreementwas shown (for δa = 10), and a small difference was due to the fact that [89] corresponds


Table 6.1: The effect of the two finite dimensions of the cross section of a microchannelon the mass flow rate through straight micro channel, for various cross sections, for apressure difference: (Pin − Pout)/Pout = 1 , and with a length L = 21H. M is the mass

flow rate per unit cross sectional area.

W/HM/M∞

δa = 1. δa = 10. [89] slip regime1 0.5145 0.4738 0.4792 0.6939 0.7062 0.7123 0.7821 0.8021 -4 0.8329 0.8506 -5 0.8645 0.8704 0.8846 0.8856 0.8985 -8 0.9124 0.9217 -16 0.9516 0.9281 -∞ 1.0000 1.0000 1.000

to purely slip regime, and the δa = 10 falls very close to the transitional regime. However,it is worth mentioning that the influence of the transversal walls on the gas flow, varies withthe rarefaction in the transitional regime [101], thus the influence is different for both thevalues of rarefaction considered. The Table 6.2 shows the influence of the transversal wallson the isothermal pressure driven mass flow rate through a 900 bend micochannel, at twovalues of the rarefaction parameter δa = 1 and δa = 10, and as evident from the results, thetransversal wall influence is marginally lower in the case of a 900 bend micochannel than itscounterpart in a straight microchannel (the transversal wall influence at δa = 1 and δa = 10was studied using the DSMC in [101] considering the straight microchannel).

Table 6.2: The influence of the transversal walls on the mass flow rate through 900 bendmicrochannel, for various cross sections, for a pressure difference: (Pin − Pout)/Pout = 1 ,

and with a length L = 15H. M is the mass flow rate per unit cross sectional area.

W/HM/M∞

δa = 1. δa = 10.1 0.5295 0.48172 0.7118 0.71413 0.8004 0.80864 0.8504 0.85685 0.8811 0.88296 0.8998 0.90138 0.9276 0.927916 0.9661 0.9583∞ 1.0000 1.0000

The Figure 6.11 shows a study of the effects of the separation between a pair of transversalwalls in a straight and a 900 bend micochannels, for a constant hydraulic diameter (Dh) (seeEq. 6.5). It is seen that the flow rate (per unit cross sectional area) increases upto W = 8H


0 5 10 150.8

1

1.2

1.4

1.6

1.8

2

relative width

M/M

W=H

(Pin

−Pout

)/pout

=2

δa=1, straight

δa=10, straight

δa=1, bend

δa=10, bend

Figure 6.11: The effect of the width to height ratio in a straight and a 900 bend mi-crochannels with a constant Dh, for a pressure drop of 2.0 (relative to outlet pressure).

The characteristic dimension was the hydrauilic diameter (Dh).

and saturates thereafter. The effects of the tranversal wall separation on flow rate (per unitarea) were quite similar at the δa = 10 for both the configurations, however, were differentat δa = 1, with the flow rate (per unit area) through a straight micochannel growing largerthan the corresponding flow rate (per unit area) through a 900 bend microchannel. Thus arelatively lower influence was observed at a lower rarefaction.

Dh = 2WH

W +H(6.5)

Where W and H are the width and the height of the microchannel, respectively.

6.3.2 Effects of finite distance between opposite tansversal walls in T-junction

Figure 6.12: A T-junction.

In T-junction, (see Table 6.3) a majority of the outbound mass flows through the straightarm, for all the rectangular cross sections considered, with a small growth in the share as


the channel widens. A 1.67% of the growth in the share was observed at the δa = 1, whileas it was 0.73% at the δa = 10. Therefore, a similar decrease in the shares was observed forthe vertical arm, as the channel widens. The share (read as: the % of the mass flow rate inthe mass total flow rate) of the vertical arm in the outbound flow rate was relatively higherat δa = 10 compared to at δa = 1.

Table 6.3: The distribution of the flow rate between the outbound arms of a T-junction,for various widths, for CL = 15H. Where, Mtotal = Mstraight + Mvertical. M is the massflow rate per unit cross sectional area. The cross sectional area of both the outbound arms

was the same.

δa = 1 δa = 10

Relativewidth:W/H

Mout,total

×10−2

(Kg/m2/s)

% of theflow ratethroughstraightarm

% of theflow ratethroughverticalarm

Mout,total

(Kg/m2/s)



1 5.15 52.27 47.73 0.8583 51.98 48.022 6.96 52.39 47.61 1.2763 52.13 47.873 7.84 52.55 47.45 1.4484 52.16 47.844 8.35 52.75 47.25 1.5365 52.23 47.775 8.67 52.89 47.11 1.5889 52.26 47.746 8.87 52.87 47.13 1.6127 51.98 48.028 9.14 53.37 46.63 1.6636 52.55 47.4516 9.54 53.57 46.43 1.7247 52.74 47.26∞ (2D) 9.95 53.94 46.06 1.7997 52.71 47.29

The Tables 6.4 and 6.5 show that, in T-junction, at δa = 1 & 10, the effects of thetransversal walls on the flow rate were higher on the flow through the outbound straight armcompared to on the flow through the outbound vertical arm. The difference in the effects(difference in the influences with respect to the influence in straight arm) was as high as6.9% at the δa = 1, and 2.96% at the δa = 10, when the cross section was a square, however,diminish to zero as the cross section widens towards 2D. The effects of the transversal wallswas higher at δa = 10 compared to at δa = 1.

As seen in the Tables 6.4, 6.5, and 6.2 a small difference in the respective influences ofthe transversal walls on the flow rate can be seen at the δa = 1, between the 900 bendmicrochannel and the vertical arm of the T-junction. However, it becomes negligibly smallat δa = 10.

The Table 6.6 shows that, in T-junction, the mass flow rate has been higher throughthe outbound straight arm than the flow rate through the outbound vertical arm, at anyrarefaction, for all δa considered in the range δa ∈ [0.16, 10.24]. The respective differences inthe shares of the flow rates of the both the outbound arms were higher at a lower rarefaction,which decreases from a maximum value at δ1 = 0.16 to a minimum value at δ1 = 5.12. The


Table 6.4: The influence of the transversal walls on the outbound flow rate through aT-junction, at δa = 1. Where, Mtotal = Mstraight + Mvertical. M is the mass flow rateper unit cross sectional area. The cross sectional area of both the outbound arms was the

same.

Relativewidth: W/H

Mtotal

Mtotal, ∞

Mstraight

Mstraight, ∞

Mvertical

Mvertical, ∞Relativeinfluence:Mvert./Mvert.,∞

Mstr./Mstr.,∞

1 0.5171 0.5012 0.5358 1.06902 0.6990 0.6789 0.7225 1.06423 0.7877 0.7675 0.8114 1.05724 0.8395 0.8210 0.8612 1.04905 0.8709 0.8539 0.8908 1.04326 0.8908 0.8732 0.9113 1.04368 0.9183 0.9087 0.9295 1.022916 0.9581 0.9515 0.9658 1.0150∞ (2D) 1.0000 1.0000 1.0000 1.0000

Table 6.5: The influence of the transversal walls on the outbound flow rate through aT-junction, at δa = 10. Where, Mtotal = Mstraight + Mvertical. M is the mass flow rateper unit cross sectional area. The cross sectional area of both the outbound arms was the

same.

Relativewidth: W/H

Mtotal

Mtotal, ∞

Mstraight

Mstraight, ∞

Mvertical

Mvertical, ∞Relativeinfluence:Mvert./Mvert.,∞

Mstr./Mstr.,∞

1 0.4769 0.4703 0.4842 1.02962 0.7092 0.7014 0.7179 1.02353 0.8048 0.7965 0.8141 1.02214 0.8538 0.8460 0.8623 1.01935 0.8829 0.8754 0.8913 1.01826 0.8961 0.8837 0.9099 1.02968 0.9244 0.9217 0.9274 1.006216 0.9583 0.9590 0.9576 0.9985∞ (2D) 1.0000 1.0000 1.0000 1.0000

differences between the shares was higher in the 2D cases compared to their counterpartswith the square cross section, the differences between the shares in the outbound flow rateare shown in the Fig. 6.13. The share of the vertical arm in the total mass flow rateincreases upto a δa = 3.84 and saturates thereafter.


10−1

100

101

2

4

6

8

10

12

14

16

18

δa

diffe

renc

e in

% o

f flo

w in

out

boun

d ar

ms

W=H2D

Figure 6.13: Difference in shares (% of the mass flow rate through the straight arm - %of the mass flow rate through the vertical arm) of the outbound arms in the total outward

mass flow rate in the T-junction.

Table 6.6: The distribution of the flow rate between the outbound arms of a T-junction, for various values of rarefaction parameter (δa), for a pressure difference:(Pin − Pout)/Pout = 2 and CL = 15H with H = 19.83 × 10−6 m. Where, mtotal =mstraight + mvertical. The cross sectional area of both the outbound arms was the same.

Square cross section(W = H) 2D (W/H →∞)

Rarefactionparameterδa

mout,total

(Kg/s)



mout,total

(Kg/s)



0.16 4.81× 10−12 52.74 47.26 5.11× 10−7 58.26 41.740.32 9.56× 10−12 52.30 47.70 9.76× 10−7 56.27 43.730.64 1.92× 10−11 51.88 48.12 1.90× 10−6 54.44 45.561.28 3.96× 10−11 51.52 48.48 3.84× 10−6 53.18 46.821.92 6.18× 10−11 51.32 48.68 6.03× 10−6 52.69 47.312.56 8.63× 10−11 51.24 48.76 8.46× 10−6 52.42 47.583.84 1.41× 10−10 51.15 48.85 1.41× 10−5 52.21 47.795.12 2.04× 10−10 51.14 48.86 2.07× 10−5 52.12 47.887.84 3.61× 10−10 51.18 48.82 3.77× 10−5 52.15 47.8510.24 5.23× 10−10 51.20 48.80 5.55× 10−5 52.29 47.71


2 4 6 8 10 12

42

44

46

48

50

52

% o

f th

e fl

ow th

roug

h ve

rtic

al a

rm

δa

W=H2D

Figure 6.14: Share of the vertical arm (% of the mass flow rate) in the outbound totalflow rate in T-junction.


6.4 Conclusions

A qualitative and quantitative analysis of the effects of finite distance between the transversaldimensions on the gas flow was performed using the DSMC method, considering a pressuredriven near-isothermal gas flow through different three-dimensional microchannel configura-tions, with a rectangular cross-section. The implicit boundary conditions were utilized, thathelped to avoid large end effects; and also due to the fact that our dimension of interest (say,width) was normal to the flow direction, we could relate our results to a long microchannelconfiguration(s). The effects of finite width on the gas flow through a straight microchannelwere analyzed considering the flow velocity, temperature, and pressure. The influence of thetransversal wall on the flow rate was studied considering a straight, a 900 bend microchannel,and a T-junction. The effects of finite distance between the transversal transversal walls wasstudied for a constant hydraulic diameter, in a straight and a 900 bend microchannels, andshowed that the mass flow rate grows with the increase of the width, upto a width W = 8Hand saturates thereafter. The influence of the distance between the transversal walls wasrelatively higher at δa = 10 compared to δa = 1.

In case of T-junction with a square cross-section, the difference between the respective influ-ences (of the finite distance between the transversal dimensions on the flow rate) observedin the straight and vertical arms has been as high as 6.9% at δa = 1 and 2.96% at δa = 10,relative to the corresponding influence in the vertical arm; however, it diminishes to zero fora cross-sectional aspect ratio W/H →∞.

The distribution of the mass flow between the outbound straight and vertical arms of aT-junction was presented, corresponding to the considered flow conditions; for various cross-sections and for various values of rarefaction parameter (δa) in the transitional regime. It isobserved that, for the considered flow conditions, the difference in mass flowing through thestraight and vertical arms was higher in the 2D cases compared to their counterparts witha square cross-section; likewise, as the aspect ratio (W/H) is increased, the percentage ofthe mass flowing through the vertical arm of the T-junction out of the total outbound massflow was significantly reduced; this feature becomes more evident as the gas becomes morerarefied, below δa = 5.12 in the range δa ∈ [0.16, 10.24]. However, the share of the outboundvertical arm in the total flow mass rate increased upto a δa = 3.84, and experienced a slightreduction thereafter, for both the square and 2D cross-sections.

Chapter 7

Experimental measurements

Experimentalsetup

Figure 7.1: Schematic diagram of the experimental setup (present at UP/IUSTI) usedfor measuring the gas flows through a 90 degree bend micro-channel

The gas flows through a 90 degrees bend micro-channel were measured using the setuppresent at Universite de provence/ Ecole Polytechnique universitaire de Marseille (UP-/IUSTI), France. The schematic of the setup is as shown in the Fig. 7.1. A constantvolume technique (CVT) [82, 102–105] based procedure had been used to measure the massflow rate through a bend micro-channel. The pressure variation in the outlet tank is setat 1% of the mean of the tank pressure over the experiment. Two capacitance diaphragmvacuum gauge (CDG)s, one for the inlet and the other for the outlet were used simulta-neously to measure the respective pressures. Several combinations of CDGs were used tomeasure the pressure according to the range of each gauge and the pressure to be measured.The capacitance diaphragm vacuum gauges that were used in the measurement processcould measure a pressure of 100 milli Torr, 1 Torr, 10 Torr, 100 Torr, and 1000 Torr, re-spectively. Combinations of these gauges were used in measuring the gas flow through themicro-channel.

87

Chapter 7. Experimental measurements 88

The following relation shown in the Eq. 7.2 for the mass flow Qm were used in calculatingthe gas flow, which was derived from the state equation:

PoutV = mRT (7.1)

Qm = V

RT

δPoutτ

(7.2)

Where V is the tank volume, R is the specific gas constant, and T is the temperature. TheδPout/τ is the rate of change of pressure (slope of the pressure vs time) in the outlet/inletsection. The reduced flow rate G was calculated using the Eq. 7.3 using the experimentallymeasured mass flow rate M through a cross section of the channel.

G = L

DHW (Pin − Pout)

√2kBTm

M. (7.3)

In the previous expression T is the temperature, m is the molecular mass of the gas, kBis the Boltzmann constant, M is the mass flow rate through a cross section of a channel.The D represents the characteristic dimension, and W , H are the width and height of themicrochannel respectively. In case of the 900 bend micro channel the centerline length CL,is considered as the equivalent length of the channel (L = LCL = L1 + L2 + H, See Fig.7.2).

Figure 7.2: A geometry of the 90 degree bend micochannel.


The dimensions of the microchannel were as following: it has two arms with the length ofone arm equal to 14 mm and the length of the other arm equal to 7 mm. The height of eachof the arms was 1 mm and the width of the each of the arms was 19.83 microns. Nitrogen,Argon, and Helium gases were used as working gas and the measurements were done fora range of pressure difference between inlet and outlet sections. Typically, for the pressureat the outlet section was varied from 10 milli Torr to 650 Torr. The pressure at the inletsection was varied from 15 milli Torr to 1300 Torr, for a various ratios of initial pressure inthe inlet and outlet sections. The measurements are shown in the Fig. 7.3, Fig. 7.4. Thetank volumes of the inlet and the outlet section were precisely measured with respect to areference tank of known volume of 55.3 cm3 .The volume of the inlet section was 224.757cm3 (including the sensor), and that of the outlet section was 241.54 cm3 (including thesensor).

The measurements were done at room temperature and in the stationary regime of flow, fora difference of pressure in the outlet section of about 1% from the beginning of an intervalof measurement.

10−1

100

101

102

103

0

5

10

15

20

25

30

35

40

45

50

δ

G

ArHeN2

Figure 7.3: The reduced flow rate through the 90 degree bend micochannel

The phenomenon of mass minimum over a range of rarefaction was shown by the measure-ments for the considered bend micro-channel. The Fig. 7.3 shows the measured dimension-less flow rates G in the range of rarefaction δ ∈ [0.1, 225] through 900 bend micro-channelfor various pressure drops between the inlet and the outlet considering the Helium as workinggas for a number of ratios of the inlet to the outlet pressures. Where δ = 0.5

√π/Knmean.


The Fig. 7.4 shows the non-dimensional mass flow rate (S), obtained for various differencesof pressure between inlet and outlet, for the case of gas flow through 90 degree bend micro-channel, considering Argon, Helium, and Nitrogen. The S approaching the value 1 as theKnmean tends to zero. The non-dimensional mass flow rate S is defined as

S = MH2W∆PPm

12µRTL(7.4)

The dynamic viscosity of the gas was calculated as

µ

µ0=(T

T0

)ω(7.5)

S = 1 + 6A1Knm (7.6)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.41

2

3

4

5

6

7

Knmean

S

Ar, 90 deg. bend experimentHe 90 deg. bend experimentN2 90 deg. bend experiment

straight, analytical 2nd order

Figure 7.4: A comparison of the non- dim. flow rate (S) through the 90 degree bendmicochannel with that of second order expression for the straight channel (Eq. 7.7)

The non-dimensional mass flow rate S was observed to be non-linear in nature Here Knmeanwas calculated from the mean pressure at the I/O boundaries.

S = 1 +AtheorKnm +BtheorKn2m (7.7)


The tests for the leak were carried out and the leaks were found to be O(1.e− 15Kg/sec)to O(1.e − 14Kg/sec), which are quite low compared to the measured gas flow O(1.e −13Kg/sec) to O(1.e− 9Kg/sec).

Appendix A

Appendix

A.1 Box-Muller algorithm

To get a standard normal from a standard uniform by inverting the cumulative distributionfunction of the Gaussian distribution function is not possible, therefore, to overcome thisdifficulty, the following technique is used obtained by change of variables, which allows usto get the normal distribution with zero mean and unit variance:y1 =

√−2 ln(z1)cos(2πz2)

y2 =√

−2 ln(z1)sin(2πz2)To make the Box-Muller transform to produce a Gaussian distribution with non unit varianceis by using a invariant relation asx = yσ + µ

we havex1 =

√−2 ln(z1)cos(2πz2)σ + µ

x2 =√

−2 ln(z1)sin(2πz2)σ + µ

93

Concluding remarks

1 Scientific contributions

• The Unsteady state DSMC simulations studied have given light to the gas flow past abackward facing step in a microchannel configuration and the behavior of the growthrate of mass flux can be related to the response of a micro-system.

• The implicit boundary conditions were implemented for the gas flows through mi-crochannel configurations, and it was found that for a microchannel length say, for acenter line length of the channel equal to nine times the channel height (CL = 9H)and more the flow properties were matching very well with those of a long channel.The implicit boundary conditions avoid the use of the particle reservoirs for maintainingthe pressure at the I/O boundaries, hence reduce the simulation time considerably.

• Different microchannel geometries, viz: straight, 900 bend, T-junction, channels withsudden expansion in cross sectional area, etc, were investigated in the slip and transi-tional regimes of rarefaction using the DSMC in 2D and 3D, and the rarefaction effectson the gas flow through micro channel configurations were presented.

• The influence of the cross sectional shape and finite distance between the lateraland transverse walls was investigated in the slip and transitional regimes, for themicrochannel configurations such as the straight, 90 degree bend, and T-junction.Therefore, it is possible to predict the flow properties (such as mass flow rate, orvelocity) in the rectangular microchannels of finite cross sectional aspect ratio (widthto height ratio) with those of a infinite aspect ratio. However, the effects of the crosssection of the microchannel were found to vary with the gas rarefaction.

• The measurements of the mass flow through a 900 bend were carried out (using anexperimental bench present at the IUSTI/UP).

• The DSMC code(s) were created for simulation of gas flows through microchannelconfigurations, and were parallelized using the MPI standard and OpenMP.

95

Concluding remarks 96

2 Future work

Some of the possible future tasks are:

• Developing Variance Reduction Schemes for the gas micro flow simulations

• Solving Heat Transfer problems: effect of thermal creep using the DSMC

• GPU computing using the DSMC

• Investigation of gaseous mixtures using the DSMC

• Unsteady calculations on more complex geometries

• Investigation on collision schemes that use small number of particles in cells and im-proving their accuracy

List of publications

I International Conference Publications

Kulakarni,N. K.∗, Stefanov, S., and Markelov,G. N., “DSMC Simulation of RarefiedGas Flow through Microchannel Junctions” in 2nd GASMEMS workshop , edited byA. Frijns, (2010)

Markelov, G. N., Kulakarni, N.K.∗, and Stefanov, S.K. “DSMC Simulation of UnsteadyRarefied Gas Flows through Microchannel Combinations” in 15th Int. Conf. on theMethods of Aerophysical Research (ICMAR 2010) , 1-8, Novosibirsk, Russia, November1-6, 2010

Kulakarni, N. K.∗, Stefanov, S. K., and Markelov, G. N., “DSMC simulation of gasflow past a backward facing step in micro-channels”, in Proceedings on CDROM of the3rd International GASMEMS Workshop (GASMEMS11), Bertinoro, June 9-11, 2011.GASMEMS11-17:1-9.

Stefanov, S. K.∗, Kulakarni,N. K., and Shterev,K. S., "Modeling of Gas Flows throughMicrochannel Configurations" ,AIP Conference Proceedings 1561, 59 (2013); doi:10.1063/1.4827214

II International Journal Publications

Kulakarni, N. K.∗, Shterev, K., and Stefanov, S. K., “ Effects of finite separation be-tween a pair of opposite transversal dimensions in microchannel configurations: DSMCanalysis in transitional regime", accepted for publishing in the journal ‘Int. Journal ofHeat and Mass Transfer’.

Kulakarni, N. K.∗, and Stefanov, S. K., “DSMC simulation of gas flow through a shortand a bend micro-channel”, in Proceedings of the 1st European conference on gasmicro flows, Skiathos, Greece, June 6-8, 2012. Journal of Physics: Conference Series362 (2012) 012014, doi:10.1088/1742-6596/362/1/012014

97

List of publications 98

Kulakarni, N. K.∗, Perrier, P., Graur, I.,and Stefanov, S. K., “ Mass Flowrate throughMicrochannel Configurations in the Slip and the Transitional Regimes", work in progress.

* Corresponding author

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Documents

DirectSimulationMonteCarloApplication ......DeclarationofAuthorship I, Naveen Kumar KULAKARNI, declare that this thesis titled, ’Direct Simulation Monte CarloApplicationforAnalysisofIso-thermalGasFlowinMicro