AB-INITIO COLLISION MODELS FOR DSMC AND

AB-INITIO COLLISION MODELS FOR DSMC AND THEIR APPLICATIONS TO REACTING FLOWS

A Dissertation Submitted to the Faculty of Purdue University by Israel Borges Sebasti˜ao

In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

August 2017 Purdue University West Lafayette, Indiana



   

ProQuest Number: 10605581

  

All rights reserved

 

INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion.







   

ProQuest 10605581 Published by ProQuest LLC (2017 ). Copyright of the Dissertation is held by the Author.



All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. 



ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346

ii

THE PURDUE UNIVERSITY GRADUATE SCHOOL STATEMENT OF DISSERTATION APPROVAL

Dr. Alina A. Alexeenko, Chair School of Aeronautics and Astronautics Dr. Li Qiao School of Aeronautics and Astronautics Dr. Serguey O. Macheret School of Aeronautics and Astronautics Dr. Michael A. Gallis Engineering Sciences Center, Sandia National Laboratories

Approved by: Dr. Weinong Wayne Chen Head of the School Graduate Program

iii

This dissertation is dedicated to my parents and sisters – C´assia, Rog´erio, Mariane, and Mayara – for their unconditional love and support, and to my wife – Jessica – for sharing life’s journey and its marvels with me.

iv

ACKNOWLEDGMENTS There are many people who helped me throughout my academic and professional path and, therefore, somehow contributed to this dissertation. First, I need to thank my parents for teaching me that the real wisdom and serenity are things cannot be taken away from you. I am really grateful for all the lessons and opportunities I received from my teachers, professors, and coworkers at SENAI, ETEC, Villares, Orbital, SEW, and UMC. Special thanks should be given to my mentors and friends, Jeronimo Travelho and Wilson Santos from INPE. Jeff was the first to show me the beauty of science and walk me through academic life. Studying abroad would never be possible without Wilson’s help and motivation. I will always be indebted with my PhD advisor, Alina Alexeenko, for giving me the opportunity to join her group and learn more about rarefied gases, teaching me the importance of asking the right questions on research, and sharing her academic and professional experiences. I thank my committee members, Li Qiao, Serguey Macheret, and Michael Gallis for their insightful comments and suggestions on my work. I also want to express my gratitude to all Purdue AAE faculty and staff for maintaining the best atmosphere a student could ask for. I would like to thank my research group colleagues for their friendship and fruitful discussions: A. Pikus, A. Ibrayeva, A. Strongrich, A. Weaver, A. Ganguly, C. Pekardan, D. Parkos, G. Shivkumar, H. Luo, K. Fowee, N. Varma, M. Kulakhmetov, S. Jaiswal, S. Tholeti, T. Zhu, T. Cofer, V. Ayyaswamy, and W. O’Neill. I was extremely lucky meeting you all. Likewise, I owe special thanks to some Brazilian friends who continued helping me despite the distance: Fachini, Luthi, Max, Mineiro, and Rodrigo. Finally, I need to acknowledge CNPq-Brazil for funding my undergraduate PIBIC program, master’s studies at INPE, and the present PhD research under grant No. GDE/201444/2012-7.

v

TABLE OF CONTENTS Page LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 1 INTRODUCTION . . . . . . . 1.1 Nonequilibrium Flows . . 1.2 The DSMC Method . . . . 1.3 Goals and Structure of the

. . . .

. . . .

1 1 5 7

2 POST-REACTION ENERGY REDISTRIBUTION MODELING . . 2.1 Motivation and Background . . . . . . . . . . . . . . . . . . . 2.2 DSMC Collision Models . . . . . . . . . . . . . . . . . . . . . 2.2.1 Collision Sampling and Internal Energy Modes . . . . . 2.2.2 Chemical Reactions . . . . . . . . . . . . . . . . . . . . 2.2.3 Energy Redistribution in Nonreacting Collisions . . . . 2.3 Conventional LB Post-Reaction Energy Redistribution . . . . 2.4 A TCE-Based Approach for Post-Reaction Vibrational Energy tribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Exchange Reactions . . . . . . . . . . . . . . . . . . . . 2.4.2 Recombination Reactions . . . . . . . . . . . . . . . . . 2.5 A Modified LB Post-Reaction Energy Redistribution . . . . . 2.5.1 Constant ZV and ZR Thermochemical Relaxation . . . 2.5.2 Realistic ZV (T) and ZR (T) for H2 -O2 Combustion . . . 2.5.3 ZV (T) and ZR (T) Thermochemical Relaxation . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Redis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

11 12 15 15 16 20 24

. . . . . . . .

28 29 31 35 37 39 44 45

3 LOW-SPEED COMBUSTION FLOW MODELING . . . . . 3.1 Motivation and Background . . . . . . . . . . . . . . . 3.2 Deflagration Waves . . . . . . . . . . . . . . . . . . . . 3.3 Assessment of H2 -O2 Reaction Mechanisms . . . . . . . 3.4 DSMC Calculations of H2 -O2 Deflagration Waves . . . 3.4.1 Flow and Numerical Conditions . . . . . . . . . 3.4.2 DSMC Estimation of Deflagration Wave Speed . 3.4.3 Verification and Validation Analysis . . . . . . . 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

47 47 50 51 55 55 57 59 67

. . . . . . . . . . . . . . . . . . . . . Dissertation

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . . . . . . .

. . . .

. . . . . . . . .

. . . .

. . . . . . . . .

. . . .

. . . . . . . . .

. . . .

. . . .

. . . . . . . . .

. . . .

. . . . . . . . .

. . . . . . . . .

vi Page 4 IMPLEMENTATION OF COMPACT QCT-BASED COLLISION MODELS 4.1 Motivation and Background . . . . . . . . . . . . . . . . . . . . . . . . 4.2 State-Specific VT Relaxation and Reaction Models . . . . . . . . . . . 4.2.1 ME-QCT-VT Model . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 SSD and SSE Models . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 DSMC Implementation of State-Specific Models . . . . . . . . . 5 COMPARING AB-INITIO AND PHENOMENOLOGICAL MODELS 5.1 DSMC Numerical Models . . . . . . . . . . . . . . . . . . . . . 5.2 O2 -O Vibrational Relaxation Time . . . . . . . . . . . . . . . . 5.2.1 0-D VT Relaxation . . . . . . . . . . . . . . . . . . . . . 5.2.2 1-D Nonreacting Shockwaves . . . . . . . . . . . . . . . . 5.3 N2 -O Vibrational Relaxation Time . . . . . . . . . . . . . . . . 5.3.1 0-D VT Relaxation . . . . . . . . . . . . . . . . . . . . . 5.4 O2 -O Dissociation Rates . . . . . . . . . . . . . . . . . . . . . . 5.4.1 0-D Thermochemical Relaxation . . . . . . . . . . . . . . 5.4.2 1-D Reacting Shockwaves . . . . . . . . . . . . . . . . . 5.5 N2 -O Dissociation and Exchange Rates . . . . . . . . . . . . . . 5.5.1 Dissociation Rate Coefficients . . . . . . . . . . . . . . . 5.5.2 Exchange Rate Coefficients . . . . . . . . . . . . . . . . . 5.5.3 0-D Thermochemical Relaxation . . . . . . . . . . . . . . 5.5.4 1-D Reacting Shockwaves . . . . . . . . . . . . . . . . . 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . .

6 DSMC SIMULATION OF O2 SHOCKWAVES BASED ON HIGH-FIDELITY MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 O2 -O2 VT Relaxation and Dissociation Models . . . . . . . . . . . . . 6.1.1 VT Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Equilibrium Dissociation Rates . . . . . . . . . . . . . . . . . 6.1.3 TCE+MF DSMC Implementation . . . . . . . . . . . . . . . . 6.2 DSMC Simulation of O2 Shockwaves . . . . . . . . . . . . . . . . . . 6.2.1 Comparing M=9.3 Cases . . . . . . . . . . . . . . . . . . . . 6.2.2 Comparing M=13.4 Cases . . . . . . . . . . . . . . . . . . . . 6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 71 72 76 77 81 82 83 87 90 94 96 97 103 106 108 109 112 114 117 119 121 122 122 123 124 127 129 131 135

7 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

vii

LIST OF TABLES Table

Page

2.1

H2 -O2 combustion reaction mechanism [47]. . . . . . . . . . . . . . . . . . 19

2.2

VHS model kinetic parameters for Tref = 273 K. . . . . . . . . . . . . . . . 19

2.3

Estimate for collision-specific ZVC (T ) based on Equation 2.28 fit. . . . . . . 44

3.1

Investigated H2 -O2 reaction mechanisms. . . . . . . . . . . . . . . . . . . . 52

3.2

Comparison of different reaction mechanismsa in predicting the adiabatic flame temperature of a H2 -O2 stoichiometric mixture at 1 atm. . . . . . . . 53

3.3

Impact of DSMC numerical parameters on Sf lame and corresponding standard deviation σSf lame for under stoichiometric conditions. . . . . . . . . . 65

5.1

Collision-specific O2 -O system VHS model kinetic parameters for Tref = 273 K [136]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2

Air-system VHS model kinetic parameters for Tref = 273 K. . . . . . . . . 82

5.3

Arrhenius parameters for the reaction rate coefficients used in the present TCE calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.1

O2 freestream shockwave conditions. . . . . . . . . . . . . . . . . . . . . 127

6.2

Summary of the DSMC modeling frameworks employed in this work. . . 128

viii

LIST OF FIGURES Figure 1.1

Page

Flow macroscopic properties depend on the dynamic behavior of microscopic particles (left) that can store energy in their translational and internal modes (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Physical models and flow regimes in terms of the Knudsen number. . . . .

3

1.3

Base DSMC algorithm flowchart and its connection to Boltzmann equation. Right hand side figure is adapted from Ref. [18]. . . . . . . . . . . . .

6

2.1

Comparison of corrected and uncorrected TCE rates for the H2 +H2 O=H+H+H2 O reaction pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2

0-D DSMC adiabatic ro-vibrational relaxation via H2 O-H collisions (left) and the corresponding equilibrium vibrational populations (right) for H2 O at n = 1020 molec/m3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3

LB thermochemical relaxation via H2 +H2 O H+H+H2 O reactions with Zv = 1 (left) and Zv = 104 (right). . . . . . . . . . . . . . . . . . . . . . . 26

2.4

Vibrational relaxation number as a function of temperature for N2 -N2 and O2 -O2 collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5

Solutions of the chemical kinetic equations for the thermochemical relaxation of O2 -O-H-OH mixtures. These results are used to guide the selection of TREF values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6

DSMC distribution functions for the O2 and OH vibrational levels in equilibrium mixtures at 2400 K (left) and 3600 K (right). . . . . . . . . . . . . 30

2.7

Thermochemical relaxation of the O2 +H OH+O system using LB model (left) and proposed approach (right) for case 1. . . . . . . . . . . . . . . . 32

2.8

Pre- and post-reaction vibrational level distributions of O2 and OH species using LB model (left) and proposed approach (right) for case 1. . . . . . . 32

2.9

Thermochemical relaxation of the O2 +H OH+O system using LB model (left) and proposed approach (right) for case 2. . . . . . . . . . . . . . . . 33

2.10 Pre- and post-reaction vibrational level distributions of O2 and OH species using LB model (left) and proposed approach (right) for case 2. . . . . . . 33

ix Figure

Page

2.11 Thermochemical relaxation for H2 +M H+H+M reactions using proposed approach for H (left), H2 (right), and H2 O (bottom) third-bodies. . . . . . 34 2.12 Thermochemical relaxation of H2 -O2 combustion system using the conventional (left) and modified (right) LB implementaion for ZV = 1. . . . . 38 2.13 Thermochemical relaxation of H2 -O2 combustion using the conventional (left) and modified (right) LB implementaion for ZV = 104 . . . . . . . . . . 39 2.14 0-D isothermal VT relaxation via O2 -O collisions for constant ZV = 50 and ZR = 1: SPARTA vs DS1V implementations. . . . . . . . . . . . . . . 42 2.15 A survey of VT relaxation numbers relevant to H2 -O2 combustion. . . . . . 43 2.16 H2 -O2 thermochemical relaxation system using the modified LB implementaion for ZV (T ) and ZR (T ). . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1

0-D adiabatic Chemked simulations to compare different reaction mechanisms in predicting the self-ignition of a H2 -O2 stoichiometric mixture at 1 atm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.2

1-D PREMIX simulations of H2 -O2 laminar premixed flames based on different reaction mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.3

Comparing temperatures and flame propagation speeds (Sf lame ) predicted by PREMIX for different reaction mechanisms. . . . . . . . . . . . . . . . 55

3.4

Schematic representation of the flow domain, initial, and boundary conditions.56

3.5

Atomic-level 2-D representation of flame propagation toward a H2 -O2 premixed gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.6

Schematics of the proposed method to calculate deflagration wave speed under the DSMC framework. The reference point (left) is used to track the flame position evolution (right). . . . . . . . . . . . . . . . . . . . . . . 58

3.7

DSMC calculations to assess the impact of ZV and ZR values on the investigated 1-D flame structure. . . . . . . . . . . . . . . . . . . . . . . . . 59

3.8

DSMC parametric study of the impact of Ncells (top), Nparticles (middle), and LX (bottom) on flame structure. Shown profiles correspond to t = 30 µs.60

3.9

Impact of the DSMC domain size (LX ) on the propagation of pressure and gas velocity waves. Arrows indicate current direction of the wave propagation.61

3.10 DSMC predicted flame front evolution in terms of temperature (top) and pressure waves (bottom) for the standard length domain (LX = 2 mm) case. 62

x Figure

Page

3.11 DSMC predicted flame front evolution in terms of temperature (top) and pressure waves (bottom) for the extended length domain (LX = 20 mm) case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.12 DSMC parametric study of the impact of Ncells , Nparticles (left), and LX (right) on flame position evolution. . . . . . . . . . . . . . . . . . . . . . . 64 3.13 Impact of the DSMC domain size (LX ) on the flame and gas velocities. Horizontal line indicates the time-averaged deflagration wave speed Sf lame . 65 3.14 Verification analysis of impact of the different DSMC numerical parameters on Sf lame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.15 Comparison of present DSMC-based Sf lame calculations against thirdparty solutions and experimental data. . . . . . . . . . . . . . . . . . . . . 66 4.1

Typical framework for the development of QCT-based models. . . . . . . . 71

4.2

O2 +O state-to-state vibrational cross-sections obtained from QCT calculations [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3

DSMC collision procedure for the ME-QCT-VT and SSD models. . . . . . 79

5.1

0-D DSMC isothermal vibrational relaxations using the ME-QCT-VT model for O2 -O system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2

Vibrational relaxation time based on decay of the normalized energy difference (left) and corresponding O2 -O mean collision time τc (right). . . . . 85

5.3

Vibrational relaxation times for O2 -O collisions: results in terms of pτ (left) and the relaxation collision number ZVC (right). . . . . . . . . . . . . 86

5.4

0-D isothermal VT relaxation via O2 -O collisions; DSMC solutions with ME-QCT-VT and LB models, and master equation solution using the complete set of state-to-state rates [27]. . . . . . . . . . . . . . . . . . . . 88

5.5

0-D isothermal VT relaxation via O2 -O collisions for different TRT values; results are presented in a normalized form. . . . . . . . . . . . . . . . . . . 88

5.6

0-D isothermal VT relaxation via O2 -O collisions for a vibrationally cold mixture that resembles a Mach 8 normal shockwave. . . . . . . . . . . . . 89

5.7

0-D isothermal VT relaxation via O2 -O collisions for a vibrationally hot mixture that resembles a nozzle expansion. . . . . . . . . . . . . . . . . . . 89

5.8

Mach 8 shockwave stabilization using the stagnation streamline approach.

5.9

Temperature and vibrational population across a 1-D nonreacting shockwave with M1 =8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

91

xi Figure

Page

5.10 1-D nonreacting shockwave for a M1 =8 freestream. . . . . . . . . . . . . . 93 5.11 0-D DSMC isothermal vibrational relaxations using the ME-QCT-VT model for N2 -O system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.12 Vibrational relaxation time based on decay of the normalized energy difference (left) and corresponding N2 -O mean collision time τc (right). . . . . 95 5.13 Vibrational relaxation times for N2 -O collisions: results in terms of pτ (left) and relaxation collision numbers ZVC (right). . . . . . . . . . . . . . . 95 5.14 0-D isothermal VT relaxation via N2 -O collisions; DSMC solutions with ME-QCT-VT, calibrated LB, and third-party master equation solutions using the complete set of state-to-state [28]. . . . . . . . . . . . . . . . . . 96 5.15 0-D isothermal VT relaxation via N2 -O collisions; DSMC solutions with ME-QCT-VT and master equation solution using the complete set and ME-QCT-VT fit state-to-state rates [28]. . . . . . . . . . . . . . . . . . . . 97 5.16 Comparison of different O2 +O→3O equilibrium dissociation rate coefficients: QCT calculations versus experimental data. . . . . . . . . . . . . . 98 5.17 Comparison of different O2 +O→3O equilibrium dissociation rate coefficients: QCT calculations versus numerical models (left) and corresponding relative errors (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.18 Comparison of different O2 +O→3O nonequilibrium dissociation rate coefficients. Ro-translational temperatures TRT equal to 5,000 (left), 10,000 (right), and 20,000 K (bottom) are considered. Corresponding thermal equilibrium conditions (TRT = TV ) are denoted by the vertical lines. . . . 102 5.19 QCT-based calibration of Kuznetsov and Macheret-Fridman (MF) models for O2 +O→3O under TV > TT conditions. Ro-translational temperatures TRT equal to 5,000 (left), and 10,000 K (right) are considered. Corresponding thermal equilibrium conditions (TRT = TV ) are denoted by the vertical lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.20 0-D adiabatic thermochemical relaxation via O2 -O collisions for mixtures initiated with TV = TRT (top), TV < TRT (middle), and TV > TRT (bottom).104 5.21 Temperatures (left) and vibrational populations (right) across a M1 =8 normal shockwave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.22 Temperatures and mole fraction distributions across normal shockwaves with M1 =12 and M1 =16. . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.23 Comparison of different N2 +O→2N+O equilibrium dissociation rate coefficients: QCT calculations versus experimental data and numerical models. 110

xii Figure

Page

5.24 Comparison of different N2 +O→2N+O reaction nonequilibrium factors Z(TT , TV ) = k(TT , TV )/k0 (TT ). Ro-translational temperatures TRT equal to 5,000 (left), 10,000 (right), and 15,000 K (bottom) are considered. Corresponding thermal equilibrium conditions (TRT = TV ) are denoted by the vertical lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.25 Comparison of different N2 +O→NO+O equilibrium exchange rate coefficients: QCT calculations versus experimental data (left) and numerical models (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.26 Comparison of different N2 +O→NO+O reaction nonequilibrium factors Z(TT , TV ) = k(TT , TV )/k0 (TT ). Ro-translational temperatures TRT equal to 5,000 (left), 10,000 (right), and 15,000 K (bottom) are considered. Corresponding thermal equilibrium conditions (TRT = TV ) are denoted by the vertical lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.27 0-D thermochemical relaxation for a 99%N2 and 1%O initial mixture. Only the N2 +O→2N+O reaction pathway is considered. Two constant ZV values are considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.28 0-D thermochemical relaxation for a 75%N2 , 20%O, and 5%NO initial mixture. Only the N2 +O→NO+O reaction pathway is considered. Two constant ZV values are considered. Post-reaction NO states are based on LB model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.29 Temperature and mole fraction distributions across normal air shockwaves with M1 = 16 (left) and M1 = 24 (right). Only Table 5.3 reaction patways are considered. Vertical lines indicate mixture TT 6= TV regions. . . . . . 118 6.1

Vibrational relaxation collision numbers for O2 -O2 collisions. . . . . . . . 122

6.2

Comparison of different O2 +O2 → 2O+O2 equilibrium dissociation rate coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.3

Comparison of different O2 +O2 → 2O+O2 nonequilibrium dissociation rate coefficients. Ro-translational temperatures TRT equal to 5,000 (left), 10,000 (right), and 20,000 K (bottom) are considered. Corresponding thermal equilibrium conditions (TRT = TV ) are denoted by the vertical lines.126

6.4

Parametric study of the impact of δ, Npart , and Ncell on the steady state temperature distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.5

TV , XO ) disTemperature and atomic oxygen mole fraction ( TT , tributions across the M = 9.3 shockwave (left) and comparison of model 4 against third-party solutions for the corresponding TV distributions. . . . 130

xiii Figure

Page

6.6

Temperature distributions ( TT , TV ) across the M = 13.4 shockwave: comparison of models 1 and 3 (left) and models 2 and 4 (right) against experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.7

Comparison of models 3 and 4 against third-party solutions for the corresponding TV distributions (left) and distribution of the atomic oxygen mole fraction (right) across the M = 13.4 shockwave. . . . . . . . . . . . 133

6.8

Vibrational energy populations at different x−coordinates across the M = 13.4 shockwave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

xiv

ABBREVIATIONS AHO

anharmonic oscillator

CFD

computational fluid dynamics

DMBE

double many-body expansion

DSMC

direct simulation Monte Carlo

FHO

forced harmonic oscillator

KSS

Kuznetsov state-specific

LB

Larsen-Borgnakke

LT

Landau-Teller

ME-QCT

maximum entropy quasi-classical trajectory

MD

molecular dynamics

MF

Macheret-Fridman

MW

Millikan-White

NTC

no-time-counter

PES

potential energy surface

QC

quasi-classical

QCT

quasi-classical trajectory

QK

quantum-kinetic

RT

rotational-translational

SHO

simple harmonic oscillator

SSD

state-specific dissociation

SSE

state-specific exchange

STS

state-to-state

TCE

total collision energy

VHS/VSS

variable hard/soft sphere

VT

vibrational-translational

xv

ABSTRACT Borges Sebasti˜ao, Israel PhD, Purdue University, August 2017. Ab-initio Collision Models for DSMC and Their Applications to Reacting Flows. Major Professor: Alina Alexeenko. The direct simulation Monte Carlo (DSMC) method is an atomistic-level technique for modeling nonequilibrium flows appearing in the fields of gas dynamics and physical chemistry. Development of DSMC collision models, in its vast majority, has traditionally focused on high-enthalpy and high-Knudsen number reentry flows. As pointed out in this work, standard DSMC approaches for post-reaction energy redistribution may not satisfy detailed balance when recombination/exchange reactions play an important role in the flow energy balance. This issue can be even more critical in reacting mixtures involving polyatomic species such as combustion. As the first goal of this dissertation, we address this issue and propose new strategies for post-reaction energy redistribution that ensure reacting mixtures relax to complete thermochemical equilibrium. Second, we apply these strategies to model atmospheric low-speed combustion problems with DSMC. In particular, we model the 1-D laminar flame structure of H2 -O2 premixed systems. This is also intended to illustrate how evolution of high-performance computational platforms can be used to extend the conventional DSMC range of applicability. The third and last part of the dissertation focuses on the implementation of compact high-fidelity collision models, based on ab-initio data, to accurately describe strong nonequilibrium flows. These new reaction and energy exchange models are consistently compared with the standard DSMC phenomenological framework.

xvi

1

1. INTRODUCTION 1.1

Nonequilibrium Flows From a classical and simplistic point of view, all matter is composed of atoms

and molecules that move around in perpetual motion, attracting each other at larger distances but repelling as they get closer [1]. In a dilute gaseous medium, in particular, these particles spend most of the time in a free flight without strongly interacting with each other. Based on the kinetic theory of gases [2], macroscopic properties such as pressure p, number density n, and temperature T correspond to the observable effects of the fluid-flow microscopic behavior. For instance, the translational temperature is a measurement of the mean energy of the molecular thermal motion. These particles, as illustrated in Fig. 1.1, also have their own internal structure. Atoms are composed of nucleus surrounded by an electron cloud that can exchange energy and molecules, which consist of electronically bonded atoms, can further store energy in rotational and vibrational modes. Likewise, rotational, vibrational, and electronic temperatures can be defined to quantify the average energy in each of these internal modes. In this context, thermal equilibrium means that there is, statistically, complete energy equipartition between the translational and active internal modes. It also implies that all directions are equally likely for the thermal velocity components. Thermal nonequilibrium takes place when, for some reason, the average amount of energy in a specific mode differs from the others. The intermolecular interactions are a natural and spontaneous mechanism for thermal relaxation in a fluidic system. The successive collisions stimulate the energy exchange between the translational and internal modes. However, for each species, every internal mode requires a different amount of energy and characteristic time to be excited or de-excited. Therefore, under some conditions, the lack of collisions that contribute to thermal relaxation can

2

Figure 1.1.: Flow macroscopic properties depend on the dynamic behavior of microscopic particles (left) that can store energy in their translational and internal modes (right).

lead the gas to an apparent nonequilibrium state. Quantification of the nonequilibrium degree is important when choosing a specific framework to model the fluid-flow dynamics. The ratio of the molecular mean free path λ to the flow characteristic length L, defined as Knudsen number [3, 4], Kn =

λ , L

(1.1)

is typically used to characterize the rarefaction and, consequently, nonequilibrium degrees. Although L can be chosen as an overall flow dimension, a better description is obtained using the gradient length scale of a macroscopic quantity ϕ, L=

ϕ , |∇ϕ|

(1.2)

where ϕ can represent the local velocity, density, pressure, temperature, and so forth. Sometimes, instead of the Kn, it is more appropriate to use the ratio of some molec-

3 ular relaxation time τrelax and the flow characteristic time τf low , known as Deborah number, De =

τrelax . τf low

(1.3)

Note that τrelax can represent different molecular time scales such as the mean collision time or the rotational, vibrational, and electronic relaxation times. In any case, Kn and De numbers indicate whether the flow modeling should be based on kinetic/molecular or continuum approaches. In the kinetic approach, which mainly relies on Boltzmann equation formulation, the fluid is treated as an ensemble of particles whose velocities and internal states are governed by classical and quantum dynamics [2]. On the other hand, within the continuum formulation, the molecular structure of the fluid is neglected and the flow is assumed as a continuum medium whose variations in its macroscopic properties respect the conservation of mass, momentum, and energy. This approach further requires the use of constitutive relations to establish a closed set of equations. These relations describe the fluid response to gradients in the macroscopic properties and the complexity of their functional form depends on the nonequilibrium degree [5]. As illustrated in Fig. 1.2, application of the well known Navier-Stokes (NS) equations

Figure 1.2.: Physical models and flow regimes in terms of the Knudsen number.

4 with no-slip conditions is indicated only for Kn < 0.01 and even the Burnett equations, which involve nonlinear constitutive stress-strain relations, may no longer be valid for Kn > 0.2. Hypersonic flows, high-altitude spacecraft propulsion, microflows, and reactors for thin-film processing are examples of applications where the gas thermochemical nonequilibrium is significant. Despite the fact that these examples encompass a broad spectrum of time and spatial scales, they are all connected by the same nature of rarefied flows, i.e., high-Knudsen conditions. In such flows, as aforementioned, the continuum hypothesis and equilibrium assumption may not be appropriate and a kinetic description of the flow via Boltzmann equation,     ∂ ∂ ∂ ∂f , +u· +F· f (t, x, u) = ∂t ∂x ∂u ∂t coll

(1.4)

becomes necessary to account for microscopic effects. This integro-differential equation describes the evolution of the molecular velocity, u, distribution function f (t, x, u) for a single system of particles. It states that the rate of change of the number of particles in a phase space element, dx du, is affected by the following processes: (i) convection of particles across the physical space volume element dx due to the molecular motion; (ii) change of the molecular velocities due to external forces F per unity mass; and (iii) depletion and replenishment of particles with velocities within the u + du range due to collisions. This latter process is originally described by a
nonlinear integral, called collision integral and herein indicated by ∂f . ∂t coll Among others [6], the direct simulation Monte Carlo (DSMC) method has become the standard approach to solving the Boltzmann equation for nonequilibrium flows of scientific and engineering interest. Both the DSMC and the Boltzmann equation derivation rely on the assumption of molecular chaos and, in their standard form, are limited to dilute gases. While modeling of collisions processes is the major challenge in the direct solution of Boltzmann equation, DSMC is a discrete technique that deals with individual particles into a direct simulation framework. Further details and suggestions for improvement of the DSMC method are given throughout this dissertation.

5 1.2

The DSMC Method The direct simulation Monte Carlo (DSMC) method, first proposed and applied

by Bird [7,8], is a stochastic atomistic technique for numerical modeling of nonequilibrium dilute gas flows [3,4]. Examples of DSMC applications include problems arising in planetary sciences [9], spacecraft aerothermodynamics [10], microflows [11–15], and low-pressure manufacturing processes [16]. This method has developed as a synthesis of kinetic theory of gases, Monte Carlo methods, particle dynamics and physical chemistry. Its basic direct simulation algorithm shares several assumptions that are embedded in the formulation of the N-particle and Boltzmann equations [2]. For simple cases, in particular, it can also be shown that the DSMC algorithm can be constructed from the exact solution of these equations [17]. However, the relatively simple implementation of high-fidelity molecular interaction models into the DSMC framework has extended its capabilities beyond Boltzmann equation. In practice, the DSMC method employs thousands to billions of representative computational particles to reproduce the real gas behavior. It assumes that, over sufficiently small time intervals, the molecular motion and interactions can be decoupled into two subsequent processes: molecular ballistic movement and collision relaxation. As summarized in Fig. 1.3, DSMC main algorithm consists in tracking the trajectories and store the thermal states of all computational particles as they move and interact within a cell-discretized physical domain. Particles are moved deterministically according to their velocities and a discrete time step while the collisions are described by statistical models. For each cell of the spatial grid, the microscopic quantities of these particles are sampled and then averaged in order to calculate the corresponding macroscopic flow properties. The DSMC capability of predicting the fluid-flow dynamics based on the microscopic behavior of representative particles allows and simplifies the implementation of realistic collision models that are mathematically intractable for Boltzmann equation. In contrast to other techniques that solve Eq. 1.4, even using simplified collision mod-

6

Figure 1.3.: Base DSMC algorithm flowchart and its connection to Boltzmann equation. Right hand side figure is adapted from Ref. [18].

els, DSMC can reproduce temperature-dependent data for diffusion, viscosity, and thermal conductivity coefficients of real gases. Another interesting DSMC feature is that its computational cost is independent of the number N of chemical species in the mixture. This contrasts with conventional continuum solvers, in which one transport equation is solved for each one of the mixture species, resulting in a computational time that typically scales with N 2 [19]. This issue becomes even more critical in master equation solvers, in which one conservation equation has also to be solved for each one of the tracked internal molecular states. All these features make DSMC a powerful and robust tool for nonequilibrium flow modeling. In principle, the DSMC method could consistently account for all types of molecular interactions using only appropriate cross-sectional data [20–23]. However, such data are rarely available for all possible molecular processes occurring in real gas mixtures and even the few available cross-sections are typically associated with large degrees of uncertainty. To overcome this challenge, DSMC takes advantage of phenomenological models that can reproduce the macroscopic gas behavior using elementary kinetic data and are still computationally efficient. Examples of these models include the variable hard/soft sphere (VHS/VSS) models for elastic scattering [4,24],

7 the Larsen-Borgnakke (LB) model for internal energy exchange [25], and the total collision energy (TCE) model for chemical reaction rates [26]. The relevant details of these standard DSMC models are discussed in next chapters.

1.3

Goals and Structure of the Dissertation In this dissertation, our efforts are concentrated on improved DSMC collision

models for reacting flows. In particular, we have three main goals: (A) develop and implement post-reaction energy redistribution strategies that satisfy detailed balance in flows dominated by exchange/recombination reactions; (B) use these newly proposed strategies to evaluate how DSMC performs in modeling atmospheric low-speed combustion flows; and (C) develop and implement a DSMC framework to integrate the high-fidelity, ab-initio relaxation and reaction models recently developed by Kulakhmetov [27] and Luo [28] for atom-diatom systems. The new approaches are consistently compared against the corresponding phenomenological models that are traditionally employed in standard DSMC simulations. In order to achieve each of these goals, we established the following steps and rationale. • Post-Reaction Energy Redistribution Modeling A1) Choose a reduced H2 -O2 combustion reaction mechanism available in literature as the benchmark system in which recombination and exchange reactions play an important role. Correct all the Arrhenius coefficients to account for discrete vibrational levels and verify that TCE model reproduces the given equilibrium rates. A2) Simulate 0-D adiabatic thermochemical relaxation cases with DSMC and verify these calculations against corresponding solutions of the chemical kinetic equations. Identify why standard LB formulation does not lead the reacting mixture to full thermal equilibrium under large vibrationaltranslational (VT) relaxation number Zv conditions.

8 A3) Propose and implement TCE-based approaches, that can be extended to other chemistry models, to populate post-reaction energy states. Evaluate these new strategies against the corresponding available phenomenological methodologies. • Low-Speed Combustion Flow Modeling B1) Based on the continuum modeling framework, briefly compare how different H2 -O2 combustion mechanisms available in literature perform in predicting ignition times and laminar premixed flame structures. B2) Propose an empirical methodology to estimate the deflagration wave propagation speed within the DSMC framework. B3) Using the post-reaction energy redistribution strategies developed in the first part of this dissertation, simulate the 1-D laminar flame structure of H2 -O2 premixed systems with DSMC. Then, compare these results against solutions by well-established continuum solvers and experimental data available in the literature. • Implementation of Compact Ab-initio Collision Models C1) Implement the new ab-initio VT energy exchange models into DSMC. Simulate 0-D thermal relaxation cases to extract the vibrational relaxation times and use them to validate the new models against published master equation and experimental data. C2) Based on the vibrational relaxation times, calibrate the VT relaxation numbers Zv (T ) and consistently use them in the LB model. Simulate 0-D and 1-D hypothetical thermal relaxation cases and compare the calibratedLB results with those obtained by the ab-initio models. C4) Implement the new ab-initio dissociation and exchange models into DSMC. Calculate the corresponding equilibrium and nonequilibrium reaction rate coefficients and validate the new models against published data.

9 C5) Calibrate the TCE model to match the ab-initio equilibrium rate coefficients. Simulate 0-D and 1-D hypothetical thermochemical relaxation cases to compare the calibrated TCE-LB results against the new models. C6) Simulate reacting shockwave experimental conditions to asses performance of the new high-fidelity models against the calibrated TCE-LB results, published master equation, and experimental data. Based on these three goals and steps, the rest of the dissertation is organized as follows. Chapter 2 deals with the first goal, i.e., the DSMC modeling of post-reaction energy redistribution. The evaluation of these new models applied to low-speed combustion flows is addressed in Chapter 3. The third goal, however, is split into the three subsequent chapters. Chapter 4 discusses the new ab-initio collision models and their DSMC implementation in details. Verification and validation of these models with 0-D and 1-D simulations involving hypothetical mixtures are covered in Chapter 5. DSMC calculations of experimental shockwave conditions based on the new high-fidelity models are presented in Chapter 6. The corresponding motivation and background are opportunely provided in each of these chapters. Finally, Chapter 7 outlines the main findings and conclusions drawn in this dissertation. Suggestions for future work are also listed in order to clarify open question that could not be answered with present studies.

10

11

2. POST-REACTION ENERGY REDISTRIBUTION MODELING The direct simulation Monte Carlo (DSMC) method has been widely applied to study shockwaves, hypersonic reentry flows, and other nonequilibrium flow phenomena. Although there is currently active research on high-fidelity models based on ab-initio data, as presented through Chapters 4 to 6, the total collision energy (TCE) and Larsen-Borgnakke (LB) models remain the most often used chemistry and relaxation models in DSMC simulations, respectively. The conventional implementation of the discrete LB model, however, may not not satisfy detailed balance when recombination and exchange reactions play an important role in the flow energy balance. This issue can become even more critical in reacting mixtures involving polyatomic molecules, such as in combustion. In this chapter, this important shortcoming is addressed and two empirical strategies to specify the post-reaction energy states are proposed within the TCE framework. First, following Bird’s quantum-kinetic (QK) methodology for populating postreaction states, we propose a new TCE-based approach that involves two main steps: (i) the state-specific TCE reaction probabilities for a forward reaction are first precomputed from equilibrium 0-D simulations; (ii) these probabilities are then employed to populate the post-reaction vibrational states of the corresponding reverse reaction. This new approach is illustrated by application to exchange and recombination reactions relevant to H2 -O2 combustion processes. Second, we propose a simple change in the conventional DSMC implementation of the LB model for reacting collisions that satisfies detailed balance regardless the vibrational relaxation rates under consideration. While this strategy does not guarantee pre- and post-reaction molecular states are consistent with the reaction model,

12 as in the first approach, it does not require pre-calculation of the pre-reaction internal states. It means this approach can be readily implemented and efficiently used in any DSMC solver. The material presented in Secs. 2.1 to 2.4 was published in Ref. [29].

2.1

Motivation and Background In the DSMC modeling of chemical reaction processes, a correct calculation of

the reaction rates becomes as important as an accurate redistribution of the postreaction energy among the translational and internal modes of the product species. Both of these processes must be modeled in such a way that the principle of microscopic reversibility and consequently detailed balance are satisfied. Considering the AB+M=A+B+M reaction, microscopic reversibility means that the probability of molecule AB to dissociate from a specific energy state must equal the probability of species A+B to recombine to that same energy state. At thermochemical equilibrium, detailed balance implies that not only the probabilities of forward and reverse processes should match but also their corresponding rates [30]. DSMC studies have traditionally focused on high-enthalpy and high-Knudsen number reentry flows. In this class of problems, the chemistry is dominated by the dissociation of diatomic species into atoms and thus the requirement for accurate post-reaction energy redistribution models can be relaxed. For this reason, the LB model has been indistinguishably used to set post-collisional states in both nonreacting and reacting flows without major consequences. The LB model was developed to satisfy detailed balance and lead nonreacting gases to thermal equilibrium but its application in reacting flows dominated by recombination and exchange reactions is questionable. While continuous models describe the rotational energies with acceptable accuracy for non-cryogenic conditions, a better description of vibrational energies demands discrete models. Because of the large spacing between quantum vibrational energy levels, vibrational-translational (VT) relaxation processes are considerably slower than

13 rotational-translational (RT) energy exchanges and, therefore, have larger impact on thermochemical equilibration. The vibrational and rotational relaxation collision numbers, Zv and Zr , can be used to quantify the VT and RT relaxation rates, respectively. For a nonreacting collision pair, 1/Z determines the probability of a specific internal mode to undergo an energy relaxation during that collision. There is an extensive literature dealing with internal energy exchange for relaxation of nonreacting gases but only few DSMC studies have addressed the challenge of setting post-reaction states. Haas [31] proposed to redistribute the post-reaction energy proportionally to the degrees of freedom of the product species. Considering a constant vibrational relaxation collision number, Zv = 50, this approach successfully satisfied detailed balance for the thermochemical relaxation of diatomic molecules subject to dissociation-recombination reactions. This model, however, cannot be generalized for exchange reactions [32]. In a series of DSMC attempts to model the structure of a detonation wave in H2 -O2 mixtures, Bondar et al. [33] highlighted the importance of satisfying detailed balance in terms of pre- and post-reaction vibrational states. Assuming that all vibrational levels with energy below the dissociation threshold Ed were equally likely to occur in a recombined molecule, they prohibited dissociation-recombination events involving molecules with vibrational energy above Ed . To compensate this constraint, the corresponding reaction probabilities were corrected in such a way that detailed balance was satisfied. In contrast to the two aforementioned approaches, in which reaction rates were calculated using TCE-like models, Bird [34] presented a comprehensive discussion on setting the post-reaction vibrational levels in the framework of the quantumkinetic (QK) theory. Since the QK theory provides state-specific reaction rates, Bird derived an analytical expression to calculate the probability of a diatomic molecule to dissociate from a particular vibrational level i at equilibrium temperature T . This pre-reaction distribution was then used to set the post-reaction vibrational states of molecules recombining at that same temperature. It was shown that this QK

14 approach for populating post-reaction vibrational levels satisfies detailed balance for both recombination and exchange reactions of diatomic species. The popularity of QK and higher fidelity state-specific collision models that can be used in DSMC has increased over the last years. However, these latter models are usually based on extensive ab-initio calculation data that are not readily available for all possible collision pairs [35, 36]. For this reason, probably, the TCE model remains the most widely used chemistry model in DSMC. In this regard, the main goal of this chapter is to revisit the relevant aspects of the TCE-LB formulation and propose empirical approaches to better describe the post-reaction vibrational states. More specifically, we first propose a new methodology that extends Bird’s QK approach for populating post-reaction vibrational states to the TCE framework and later we propose a modified LB implementation for reacting collisions. It is well-known that both QK and TCE models involve a large degree of uncertainty for strongly nonequilibrium conditions. In this context, the new approaches for post-reaction energy redistribution are mainly targeted towards low-speed reacting flows dominated by recombination/exchange reactions and that do not significantly depart from thermal equilibrium. Such conditions can be found, for instance, in combustion [37, 38] and chemical vapor deposition (CVD) processes [39]. Nevertheless, even in rarefied hypersonic flows, where dissociation reactions play the major role in the flow energy balance, the production of vibrationally excited trace species via recombination/exchange reactions can strongly affect radiative emissions [40]. Relevant examples, in Earth’s atmosphere, include the formation of nitric oxide (NO) in the wakes of hypersonic vehicles [41] or ejecta spherules [9]. Likewise, the formation of carbon monoxide (CO) in Mars’ atmosphere entry flows [42, 43]. It is worthwhile to emphasize that, in the same manner TCE model can accurately reproduce only equilibrium rates, our newly proposed approaches for post-reaction energy redistribution are not intended to compete with high-fidelity models that are based, for example, on ab-initio data. Instead, they offer simple and robust strategies that ensure reacting mixtures will always relax towards full equilibrium. Under ther-

15 mal nonequilibrium conditions, however, these approaches cannot guarantee correct post-reaction energy distributions. The rest of the chapter is organized as follows. The collision models employed in this effort are summarized in Sec. 2.2. The conditions in which the LB model does not lead to thermochemical equilibrium are discussed in Sec 2.3. The first proposed strategy to solve this issue is illustrated by considering two reaction pairs in Sec. 2.4. Likewise, the second proposed approach is described and tested with a full H2 -O2 combustion mechanism in Sec. 2.5. Finally, the main findings and target applications in which the present study is relevant are briefly presented.

2.2

DSMC Collision Models

2.2.1

Collision Sampling and Internal Energy Modes

In this work, the collision frequency is calculated according to the no-time-counter (NTC) algorithm [4, 44] but other methodologies such as the majorant collision frequency [45] could also be considered. In NTC algorithm the number of candidate collision pairs Ncand =

N (N − 1)Wp (σT cr )max ∆t , 2 Vc

(2.1)

to be considered within a time step ∆t is a function of the number of simulated particles per collision cell N , cell volume Vc , number of real particles represented by each simulated one Wp , and the maximum value of total cross-section times the relative velocity (σT cr )max . The σT value is taken to be equal to the VHS/VSS crosssection. Based on its σT cr value, each of these Ncand candidate collision pairs is then accepted with probability Pc =

σT cr . (σT cr )max

(2.2)

Regarding the internal energy modes, the molecular rotational energy Er is assumed to follow a Boltzmann distribution at rotational temperature Tr ,  ζr /2−1   1 Er Er f (Er ) = exp − . Γ(ζr /2) kB Tr kB Tr

(2.3)

16 In this expression, kB is the Boltzmann constant and ζr is the number of rotational degrees of freedom. On the other hand, discrete energy levels are used to describe the vibrational energies. The vibrational modes are given according to the simple harmonic oscillator (SHO) model. Note, however, that the approach for post-reaction vibrational energy redistribution proposed in this chapter is readily applicable to other models. In SHO model, the energy of each vibrational mode m is a linear function of its vibrational quantum number i and characteristic vibrational temperature θvm , Evm = ikB θvm .

2.2.2

(2.4)

Chemical Reactions

The reaction rates are calculated using the TCE model [26], which is based on the collision theory for chemical reactions and is able to reproduce equilibrium rates expressed in the modified Arrhenius form, kr = Λ T η e−Eact /kB T .

(2.5)

The pre-exponential constants Λ and η, and the activation energy Eact can be obtained from experiments or ab-initio calculations. Bimolecular reactions can be expressed in two equivalent forms, dn1 = −n1 n2 kr = −n1 n2 σT12 cr12 PR , dt

(2.6)

where the number densities of reactant species 1 and 2 are denoted by n1 and n2 , and PR is the reaction probability, which can be further written as the ratio of the reaction to the total collision cross-sections, PR = σR12 /σT12 . While the right hand side of Equation 2.6 provides an exact result, within the TCE formulation, it is commonly approximated as n1 n2 σT12 cr12 PR ≈ Nc12 PR . From basic kinetic theory, the collision frequency Nc12 and average reaction probability PR can be obtained as the moment of the corresponding collisional quantities.

17 Assuming that the collision energy Ec and molecule velocity components c follow Boltzmann (equilibrium) distributions f0 , one can write: Z Nc12 = n1 n2 σT12 cr12 f0 (c1 )f0 (c2 )dc1 dc2

(2.7)

Z PR =

PR (Ec )f0 (Ec /kT )d(Ec /kT )

(2.8)

In order to be consistent with the NTC algorithm, σT is chosen again as the VHS/VSS cross-section such that Equation 2.7 can be explicitly solved. However, the general dependence of the reaction probability on the collision energy PR (Ec ) is usually unknown and the following functional form is assumed in the TCE model:   0 if Ec ≤ Eact PR (Ec ) = (2.9) C2 +ζc −1   C1 (Ec −Eactζ )−1 if E > E . c act E c c

The total collision energy includes the translational, rotational, and vibrational conP P tributions of both colliding molecules, Ec = Et + Er1 + Er2 + Ev1,m + Ev2,m . Similarly, (5 − 2ω12 ) + ζr1 + ζr2 + ζc = 2

P

ζv1,m +

P

ζv2,m

(2.10)

represents the average number of degrees of freedom contributing to Ec . The summation over vibrational modes accounts for polyatomic species and ω12 is the viscosity index as defined in the VHS/VSS models. The ζvm value is calculated according to the SHO model and the cell average vibrational temperature Tv , ζvm =

2 θvm /Tv . exp(θvm /Tv ) − 1

(2.11)

With these definitions, Equations 2.6 to 2.9 can be combined to analytically obtain constants C1 and C2 in terms of the Arrhenius and kinetic parameters. A similar analysis is also used to treat recombination reactions, which are ternary collision processes characterized by zero activation energy. Considering a third-body number density nM , the PR (Ec ) = nM C3 EcC4 functional form can be assumed. Note that here PR is still a bimolecular reaction probability and Ec does not include the third-body contribution.

18 It is worth pointing out that the use of discrete models to better describe the vibrational energy populations is inconsistent with the TCE formulation, which is built on the assumption that translational and internal energies have continuous distributions. As a consequence, TCE model can fail to reproduce a given Arrhenius rate when discrete vibrational energies are employed. This problem can be overcome by applying an empirical procedure [46] that consists in correcting constants Λ and η such that the DSMC rates match the original Arrhenius rates. The correction requires two 0-D DSMC simulations per equilibrium temperature; one to sample the rates obtained with the original Λ and η values and another to check that after adjusting these constants the DSMC results match the exact rates. This correction procedure can be similarly applied to other continuous reaction models. To illustrate the impact of not accounting for the discrete energy levels in TCE model, the H2 +H2 O H+H+H2 O equilibrium reaction rates are considered. The uncorrected Arrhenius coefficients and VHS model kinetic data used throughout next sections are listed in Tables 2.1 and 2.2, respectively. As showed in Fig. 2.1, without the modification of the Arrhenius coefficients, the DSMC rates can deviate from the exact values by more than one order of magnitude. Once the reaction event is accepted, the Ec value is updated by adding or subtracting the corresponding heat of reaction ∆HR , respectively, for exothermic or endothermic processes. For recombination reactions, the third-body energy should also be added. The next fundamental step in any DSMC collision model is to redistribute the post-collision energy Ec among the translational and internal modes. For reacting collisions, in particular, this redistribution has to account for the structure of the newly formed product species. In this context, the remaining of this section discusses the main concepts of the Larsen-Borgnakke (LB) model for nonreacting collisions while the next section is devoted to show the LB model limitations in describing some classes of reactive flows.

19

Table 2.1.: H2 -O2 combustion reaction mechanism [47]. Λ (mol-cm-K-s)

η (-)

Eact /kB (K)

1.700×1013

0.0

24044

OH+OH→H2 +O2

4.032×1010

0.317

14554

H+O2 →OH+O

1.987×1014

0.0

8456

OH+O→H+O2

8.930×1011

0.338

-118

H2 +OH→H2 O+H

1.024×108

1.6

1660

H2 O+H→H2 +OH

7.964×108

1.528

9300

H2 +O→OH+H

5.119×104

2.67

3163

OH+H→H2 +O

2.701×104

2.649

2240

OH+OH→H2 O+O

1.506×109

1.14

50

H2 O+O→OH+OH

2.220×1010

1.089

8613

H2 O+M→OH+H+M

8.936×1022

-1.835

59743

OH+H+M→H2 O+M

2.212×1022

-2.0

0

H2 +M→H+H+M

5.086×1016

-0.362

52105

H+H+M→H2 +M

9.791×1016

-0.6

0

H2 +O2 →OH+OH

Chemical species: H, O, OH, H2 , O2 , and H2 O. Third-body efficiencies are taken as 12 for M=H2 O and 2.5 for M=H2 .

Table 2.2.: VHS model kinetic parameters for Tref = 273 K. H2

H

O2

O

OH H2 O

Reference diameter, dref (˚ A) 2.92 2.33 4.07 3.00 3.50 4.50 Viscosity index, ω (-)

0.67 0.75 0.77 0.75 0.75 1.00

20 13

10

12

10

11

10

10

10

17

H2+H2O → H+H+H2O

H+H+H2O → H2+H2O Reaction Rate, k (cm-mol-s)

Reaction Rate, k (cm-mol-s)

10

109

10

10

8

10

7

10

6

10

5

Arrhenius DSMC - Uncorrected DSMC - Corrected

104 2000

3000

Figure 2.1.:

4000 T (K)

5000

6000

16

1015

Arrhenius DSMC - Uncorrected DSMC - Corrected 0

1000

2000

3000 T (K)

4000

5000

6000

Comparison of corrected and uncorrected TCE rates for the

H2 +H2 O=H+H+H2 O reaction pair.

2.2.3

Energy Redistribution in Nonreacting Collisions

Assuming that molecules are simple harmonic oscillators, Ev = ikB θv , and that only mono-quantum transitions can take place, ∆i = ±i, the rate of change in the population Ni of molecules at vibrational level i can be written as dNi = −ki→i+1 Ni + ki+1→i Ni+1 − ki→i−1 Ni + ki−1→i Ni−1 , dt

(2.12)

where, for instance, ki→i+1 represents the i → i + 1 vibrational transition rate coefficient. Further assuming that vibrational energies are given by Boltzmann distributions under equilibrium conditions and combining the detailed balance principle with the ki→i−1 = ik1→0 relation, obtained from quantum mechanics analysis for SHO model, Equation 2.12 can be rewritten in terms of the instantaneous vibrational energy Ev and its corresponding value when the system achieves equilibrium Eveq [2] E eq − Ev dEv = v . dt τv

(2.13)

21 The above expression is commonly known as the Landau-Teller (LT) equation and the vibrational relaxation time, τv =

1 , k1→0 [1 − exp (−θv /Tv )]

(2.14)

is the collision-specific parameter that defines the VT relaxation rate. From a continuum perspective, this rate can be related to the vibrational relaxation collision number, Zv = τv /τc , which is the vibrational relaxation time normalized by the corresponding mean collision time τc . This temperature-dependent parameter can be interpreted as the average number of collisions that take place within a τv interval. Likewise, all the previous concepts apply to RT energy exchanges and rotational relaxation numbers Zr can also be defined. Based on the aforementioned relaxation numbers, the DSMC-LB framework considers that only a fraction 1/Zv and 1/Zr of the total number of nonreacting collisions are vibrationally and rotationally inelastic, respectively. If multiple relaxation events are prohibited to take place in a single collision [48], i.e., only one internal energy mode can relax in a collision, this procedure ensures that the energy exchanges occur in accordance to the relaxation rates specified in terms of Z. For simplicity, however, a serial particle selection methodology [4, 49] is employed in this work such that the ro-vibrational relaxation rates might slightly deviate from the expected exact values. While this part of the LB implementation reproduces the relaxation rates from a macroscopic point of view, there is a complementary part that statistically accounts for the energy redistribution between translational and internal modes during individual collision events. A summary of the latter part, which seeks to satisfy detailed balance principle, is described as follows. Although the original LB model [25] considered continuous energy spectra for both rotational and vibrational modes, Bergemann and Boyd [50] showed that it can be extended to discrete vibrational energy levels. To illustrate the second part of the LB formulation, a nonreacting collision between molecules 1 and 2 that involves a total pre-collisional energy Ec and an average number of degrees of freedom ζc as defined in Equation 2.10 is first considered. It is important to highlight that in this event

22 there is no heat of reaction being released or absorbed and, therefore, Ec is conserved during the collision. For the general case, this total energy can be split into two arbitrary groups, a and b, such that Ec = Ea + Eb and ζc = ζa + ζb . Assuming that the post-collisional internal and relative translational energies follow local Boltzmann distributions, the LB model leads to  f

Ea Ec



 =f

Eb Ec



Γ(ζa + ζb ) = Γ(ζa ) + Γ(ζb )



Ea Ec

ζa −1 

Eb Ec

ζb −1 .

(2.15)

This relation provides the probability of finding a given energy redistribution, in terms of the Ea /Ec and Eb /Ec fractions, that is consistent with the number of degrees of freedom participating in each group. The ratio of this probability to its maximum value, f fmax

ζc − 2 = ζa − 1 



Ea Ec

ζa −1 

ζc − 2 ζb − 1

 ζb −1 Ea 1− , Ec

(2.16)

can then be used to sample Ea /Ec values with the acceptance-rejection method [4,44]. For the special case of sampling post-collision energies for a single internal mode with two degrees of freedom (ζa = 1), Equation 2.16 reduces to f fmax

 ζ −1 Ea b = 1− . Ec

(2.17)

As one can note, the probability of a given Ea /Ec value now depends only on the remaining average number of degrees of freedom participating in the energy exchange, ζb . The previous result can be directly applied to sample appropriate post-collisional Ev0 m values for each vibrational mode m in nonreacting inelastic collisions. Herein, primed variables indicate post-collision states. Thus, for VT exchanges, Ea = Evm , Eb = Et , ζa = 1 and ζb = 5/2 − ω12 such that f fmax

 =

E0 1 − vm Ec

3/2−ω12 .

(2.18)

In an analogous manner, Equations 2.16 and 2.17 can be used to populate the postcollisional rotational energies of polyatomic and diatomic species, respectively.

23 Figure 2.2 shows the adiabatic thermal relaxation of H2 O molecules via H2 O-H collisions for constant Zr = 1 and Zv = 10 values. It confirms that, under nonreacting conditions, LB model satisfies detailed balance even for polyatomic species. Based on the simple procedure proposed by Bird [44], which relies on the Landau-Teller equation, the difference between the average ro-translational and vibrational temperatures, (Tt + Tr )/2 − Tv , should fall to 1/e of its initial value at Zv mean collision times. Instead, the present particle selection methodology leads to a 9% faster VT relaxation rate. Nevertheless, it is important to keep in mind that such deviations only influence how fast the thermal equilibration is achieved and do not contribute to

00

detailed balance breakdown, which is the main issue being addressed in this chapter.

10

0

10

-1

10

-2

10

-3

Mode 1 (Sym. Stretching) Mode 2 (Bending) Mode 1 (Asym. Stretching)

0

10

00

20

5000/e

0

10

20

30 40 50 60 70 Mean Collision Times

TT TR TV (TT+TR)/2 - TV TV - Mode 1 TV - Mode 2 TV - Mode 3 80

90 100

Fraction (-)

00

T (K) 30 0

0

40

00

50

Sampling Vibrational Levels

Lines: Boltzmann Symbols: DSMC

10-4

10-5

0

5

10 Vibrational Level

15

20

Figure 2.2.: 0-D DSMC adiabatic ro-vibrational relaxation via H2 O-H collisions (left) and the corresponding equilibrium vibrational populations (right) for H2 O at n = 1020 molec/m3 .

24 2.3

Conventional LB Post-Reaction Energy Redistribution In principle, one can be attempted to extend the previous LB expressions to set

post-reaction vibrational levels by only specifying appropriate values to Ec and ζb in Equation 2.17. Following this route, because of the heat of reaction ∆HR , the post-reaction total energy should be written as Ec0 = Et + Er1 + Er2 +

X

Ev1,m +

X

Ev2,m + ∆HR + EM ,

(2.19)

where, for recombinations, EM stands for the relative translational and internal energies of the third-body interacting with species 1 and 2. Similarly, the number of remaining degrees of freedom ζb should account for all modes involved in the current energy exchange. For instance, considering a reacting collision that produces two diatomic (subscripts 1 and 2) and one atomic species and that LB redistribution is first applied to each vibrational mode, the procedure showed below is employed in this section. 1) Sample Ev0 1

   Ec = Ec0 with:   ζb = (5 − 2ω 0 ) + ζ 0 + ζ 0  /2 12 r1 r2

2) Sample Ev0 2

   Ec = Ec0 − Ev0 1 with:    ζb = (5 − 2ω 0 ) + ζ 0 + ζ 0  /2 r1 r2 12

3) Sample Er0 1

   Ec = Ec0 − Ev0 − Ev0 1 2 with:    ζb = (5 − 2ω 0 ) + ζ 0  /2 12 r2

4) Sample Er0 2 with:

   Ec = Ec0 − Ev0 − Ev0 − Er0 1 2 1   ζb = 5/2 − ω 0 12

The same sampling scheme is also applied to the recombination and exchange reactions investigated in the next sections. For such cases, the only difference is that Ev0 2 and Er0 2 are zero if one of the products is an atomic species. Finally, once all the

25 internal modes are sampled, the remaining energy in Ec is assigned to the relative translational modes and hence the product molecules are scattered according to the VHS/VSS model. This procedure is similar to what Dietrich [50, 51] proposed to sample post-recombination states. While exchange reactions were not discussed in his work, in order to satisfy detailed balance in the dissociation of diatomic species, 0 Dietrich considered that the dissociating particle contributes with 5 − 2ω12 internal

degrees of freedom to ζb . It means that, in contrast to the above expressions, the first   0 ) + ζr0 1 /2. After a series of tests (not shown for sampling step used ζb = 2(5 − 2ω12 brevity), the LB redistribution used in this section was found to better relax reacting mixtures to thermal relaxation than Dietrich’s approach. However, as illustrated in this section, even this improved LB scheme cannot lead reacting mixtures to complete thermal equilibrium under high Zv conditions. The thermochemical relaxation of a H-H2 -H2 O mixture in an adiabatic box is considered to evaluate the TCE-LB framework for reacting collisions. The mixture is initiated at 1 atm, 2000 K, and mole fractions XH =0.50 and XH2 =XH2 O =0.25. Particles in this system are allowed to react only via the H2 +H2 O H+H+H2 O reactions discussed in section 2.2. In this section, a constant Zr = 1 value is used but two different VT relaxation rates for nonreacting collisions are considered: constant Zv =1 and 104 . For each of these two scenarios, all types of collisions, H-H2 , H-H2 O, H2 -H2 , and so forth, are assumed to have the same Zv values, i.e., 1 or 104 . The DSMC results are compared with the numerical solution of the chemical kinetic equations obtained from the Chemked-I software package [52], which assumes thermal equilibrium. Figure 2.3 shows that although DSMC and Chemked results are in good agreement for Zv = 1, thermochemical equilibrium is not achieved for Zv = 104 . These large deviations can be attributed to the inadequacy of Equation 2.17, derived under constant Ec = Ec0 conditions, to reacting collisions. In practice, they can be explained as the combination of two factors. To explain the first factor, the AB(s1 )+M(s2 ) →A+B+M arbitrary reaction is again considered. In this endothermic reaction, the collision energy should be above

26 some activation threshold and thus the pre-reaction states, represented by s, usually do not follow equilibrium distributions. In the reverse reaction A+B+M→AB(s01 )+M(s02 ), however, LB model populates the post-reaction states s0 based on local equilibrium distributions. As one can see, even under chemical equilibrium, where forward and reverse reactions occur at the same rate, violation of detailed balance arises from the mismatch between s and s0 distributions [34]. The second factor is related to rate at which reacting and nonreacting collisions of a particular species take place. Although detailed balance is violated every time that post-reaction vibrational levels are populated with LB model, as presented in Fig. 2.3, this problem does not become apparent for Zv = 1. In this case, all nonreacting collisions are treated as inelastic and have their vibrational levels correctly assigned via the LB procedure for nonreacting collision. However, a different picture emerges with Zv = 104 . For the given flow conditions, the H+H→H2 reaction occurs about every 2.5×104 collisions, i.e., the reaction and VT relaxation rates are of the same order of magnitude. Clearly, under these conditions the errors in using LB model to populate post-reaction vibrational energies may become significant. Conversely,

36 34

T (K)

32 30 28

00 00

22 20

36

00

34

00

32

00 00

0 26 24

38

TT TR TV TV - H2 TV - H2O - Mode 1 TV - H2O - Mode 2 TV - H2O - Mode 3 Chemked

0

00 00 00

00 -8 18 10

10

-7

10

-6

10

-5

10

-4

t (s)

10

-3

T (K)

38

30 28

00 00 00 00 00 00

0 26 24 22 20

TT TR TV TV - H2 TV - H2O - Mode 1 TV - H2O - Mode 2 TV - H2O - Mode 3 Chemked

0

00 00 00

00 -8 18 10

10

-7

10

-6

10

-5

10

-4

10

-3

t (s)

Figure 2.3.: LB thermochemical relaxation via H2 +H2 O H+H+H2 O reactions with Zv = 1 (left) and Zv = 104 (right).

27 because the RT relaxation rates are much higher than the reactions and VT relaxation rates, the LB approach can be used to set post-reaction rotational modes without problems. While the chemistry in the shockwave region of hypersonic reentry flows is dominated by the dissociation of diatomic species, recombination and exchange reactions play an important role in combustion-like problems. Furthermore, the high temperatures (> 5000 K) observed in reentry flows lead to small Zv values such that the aforementioned second factor is not critical. Based on the approximated Zv (T ) expression proposed by Bird [44], Fig. 2.4 indicates that typical air species have Zv values of at least 104 for typical flame temperatures (∼ 3000 K). Therefore, although the LB model has been used to populate post-reaction vibrational levels in hypersonic flow problems without major consequences, another approach is needed to simulate combustion-like problems with DSMC. As the specific goal of the present chapter, the first alternative solution to this challenge is presented in next section.

9

10

8

10

7

10

6

10

5

Combustion

Reentry

ZV

10

10

4

10

3

10

2

10

1

10

0

N2-N2

O2-O2

0

2000

4000

6000

8000

10000

T (K)

Figure 2.4.: Vibrational relaxation number as a function of temperature for N2 -N2 and O2 -O2 collisions.

28 2.4

A TCE-Based Approach for Post-Reaction Vibrational Energy Redistribution Following the QK approach used by Bird [34] to redistribute energy in reacting

collisions, in this work it is proposed to use state-specific TCE reaction rates extracted from equilibrium 0-D DSMC simulations to specify the post-reaction vibrational levels. Considering the AB(i)+M(j)→A+B+M reaction, with pre-reaction vibrational levels i and j, the present approach can be summarized into two steps. 1) For a given reference equilibrium temperature TREF , sample the TCE probability of species AB and M to react from levels i and j, respectively. This task can be easily accomplished when the Arrhenius coefficients are corrected to account for discrete vibrational energies with the procedure discussed in section 2.2. Note that constant Zr = Zv = 1 should be used since Arrhenius rates intrinsically assume equilibrium. 2) Populate the post-reaction vibrational levels of the reverse reaction A+B+M →

AB(i0 )+M(j 0 ) by sampling them from the corresponding i and j pre-computed distributions. As the mixture approaches chemical equilibrium, this ensures that detailed balance is satisfied since the replenishment (recombination) rate of vibrational levels i0 and j 0 is enforced to match the depletion (dissociation) rate for these same levels.

The first step needs to be performed for different TREF values such that the entire temperature range of interest is covered. This is not a very restrictive requirement for combustion-like problems, which usually take place within well defined temperature ranges (300-5000 K). In this manner, reactions occurring within intermediate values of TREF can be populated by interpolating the corresponding i and j distributions. For nonequilibrium conditions, this interpolation is based on the translational temperature of the mixture and the respective two closest TREF values. To illustrate the use of this TCE-based post-reaction energy redistribution approach for exchange and recombination reactions, 0-D thermochemical relaxations

29 via the O2 +H OH+O and H2 +M H+H+M reactions are considered, respectively. Table 2.2 presents the corresponding Arrhenius coefficients, which were taken from a H2 -O2 combustion mechanism [47], and corrected to account for discrete vibrational levels. The following results were obtained with a highly modified DS1V solver [44].

2.4.1

Exchange Reactions

Because of the endothermic and exothermic nature of chemical reactions, the evolution of a system undergoing thermochemical relaxation strongly depends on its initial conditions. In order to evaluate the application of the proposed method for transient regimes dominated by either endothermic or exothermic processes, two cases are considered. Case 1 has initial mole fractions XO2 =XH =0.50 while case 2 has XOH =XO =0.50. Both of these mixture are initiated with n = 1020 molec/m3 and 3000 K. Assuming that only O2 +H OH+O forward and reverse reactions are allowed to occur, the solutions of the chemical kinetic equations obtained by Chemked are shown in Fig. 2.5. Based on these results and intervals of 200 K, nine TREF values ranging from 2200 to 3800 K are chosen. For each of these TREF values, equilibrium 0-D DSMC simulations are performed to sample the pre-reaction vibrational levels of both the forward, O2 (i)+H→OH+O, and reverse, OH(j)+O→O2 +H, reactions. For brevity, however, Fig. 2.6 presents only the vibrational populations associated with the O2 and OH species at 2400 K and 3600 K. As expected, when the vibrational levels of all O2 and OH molecules are sampled, the resulting distributions exactly matches the corresponding Boltzmann ones. On the other hand, nonequilibrium distributions are observed when only the O2 prereaction vibrational levels i are sampled. This overpopulation for higher i levels occurs because O2 (i)+H→OH+O is an endothermic process and only collisions above a specific energy barrier can react. Conversely, OH(j)+O→O2 +H has no energy barrier and hence j level closely follows Boltzmann distributions.

30

38 36

T (K)

34 32 3

O2+H = OH+O

TF=3650 K

00 00 00

T0

0 00

28 26 2

00

00 00

00 -8 22 10

TF=2420 K

Case 1 Case 2

0 40

10

-7

10

-6

10

-5

-4

10 t (s)

10

-3

10

-2

10

-1

10

0

Figure 2.5.: Solutions of the chemical kinetic equations for the thermochemical relaxation of O2 -O-H-OH mixtures. These results are used to guide the selection of TREF values.

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

Boltzmann All O2 Molecules All OH Molecules Only Pre-Reaction Level ( i ) Only Pre-Reaction Level ( j )

10

Fraction (-)

Fraction (-)

10

O2( i )+H → OH+O O2+H ← OH( j )+O

-6

0

5 10 Vibrational Level

15

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

Boltzmann All O2 Molecules All OH Molecules Only Pre-Reaction Level ( i ) Only Pre-Reaction Level ( j )

O2( i )+H → OH+O O2+H ← OH( j )+O

-6

0

5 10 Vibrational Level

15

Figure 2.6.: DSMC distribution functions for the O2 and OH vibrational levels in equilibrium mixtures at 2400 K (left) and 3600 K (right).

Once the pre-reaction vibrational distributions are known, DSMC calculations for the actual thermochemical relaxation can be performed using the proposed approach.

31 In order to be consistent with the temperature range under investigation, constant Zr = 5 and Zv = 104 values are used for nonreacting collisions. Figure 2.7 compares the results obtained with the LB redistribution discussed in section 2.3 and the proposed approach for case 1. As one can see, both approaches lead the mixture to similar translational TT , rotational TR , and vibrational TV temperatures at steady state. In spite of this reasonable agreement between the mixture T values, LB method clearly fails in relaxing O2 and OH vibrational temperatures to thermal equilibrium. The steady state (t >0.1 s) vibrational level distributions for these thermochemical relaxation cases are shown in Fig. 2.8. Although the LB method provides a surprisingly good match between pre- and post-reaction vibration populations, contradicting the results obtained by Bird [34] for the same case, only the distributions obtained with the new proposed approach match the values that correspond to thermal equilibrium (indicated by solid lines). According to Figs. 2.9 and 2.10, the same trends observed for case 1 also apply to case 2. These results show that the present approach can be used as an alternative to the LB vibrational energy redistribution, in particular, for mixtures dominated by exchange reactions.

2.4.2

Recombination Reactions

In this section the present approach is applied to populate the post-reaction vibrational levels of the H2 +M H+H+M reactions. Three different third-body species M are tested; namely, case 3 for M=H, case 4 for M=H2 , and case 5 for M=H2 O. Similarly to the cases investigated in section 2.3, the mixtures are initiated at 1 atm and 2000 K. Initial mole fractions of XH = XH2 =0.50 are considered for cases 3 and 4 while XH = XH2 =0.25 and XH2 O =0.50 are considered for case 5. Constant Zr = 5 and Zv = 104 are used again for all species and respective internal modes. The DSMC results using the new approach for H and H2 third-bodies are given in Fig. 2.11. For these cases, the corresponding solutions using the LB redistribution

32

32

T (K)

30 28 26 24

00

34

00

32

00

30

T (K)

34

00

TT TR TV TV - O2 TV - OH Chemked

00 00

00 -8 22 10

10

-7

10

-6

10

-5

28 26 24

-4

10 t (s)

10

-3

10

-2

10

-1

10

0

00 00 00 00

TT TR TV TV - O2 TV - OH Chemked

00 00

00 -8 22 10

10

-7

10

-6

10

-5

-4

10 t (s)

10

-3

10

-2

10

-1

10

0

Figure 2.7.: Thermochemical relaxation of the O2 +H OH+O system using LB model (left) and proposed approach (right) for case 1.

0

10

-1

10

-2

10

-3

10

-4

10

-5

4

10

Symbols: Thermochemical Relaxation (ZV=10 ) Lines: Equilibrium Mixture Pre-Reaction Levels (ZV=1)

i i’ j j’

Fraction (-)

Fraction (-)

10

O2( i )+H → OH( j’)+O

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

4

Symbols: Thermochemical Relaxation (ZV=10 ) Lines: Equilibrium Mixture Pre-Reaction Levels (ZV=1)

i i’ j j’

O2( i )+H → OH( j’)+O

O2( i’)+H ← OH( j )+O 10

O2( i’)+H ← OH( j )+O

-6

0

5 10 Vibrational Level

15

0

5 10 Vibrational Level

15

Figure 2.8.: Pre- and post-reaction vibrational level distributions of O2 and OH species using LB model (left) and proposed approach (right) for case 1.

also led to similar results and hence are omitted for brevity. However, as previously discussed in section 2.3 and showed in Fig. 2.3, the LB method does not lead the H-H2 -H2 O mixture to thermal equilibrium when Zv = 104 . As also presented in

33

50

T (K)

45 40 35 30

00

TT TR TV TV - O2 TV - OH Chemked

00

55 50

00

45

T (K)

55

00 00

35

00

00 -8 25 10

40

30

10

-7

10

-6

10

-5

-4

10 t (s)

10

-3

10

-2

10

-1

10

0

00

TT TR TV TV - O2 TV - OH Chemked

00 00 00 00 00

00 -8 25 10

10

-7

10

-6

10

-5

-4

10 t (s)

10

-3

10

-2

10

-1

10

0

Figure 2.9.: Thermochemical relaxation of the O2 +H OH+O system using LB model (left) and proposed approach (right) for case 2.

0

10

-1

10

-2

10

-3

10

-4

10

-5

4

10

Symbols: Thermochemical Relaxation (ZV=10 ) Lines: Equilibrium Mixture Pre-Reaction Levels (ZV=1)

i i’ j j’ Fraction (-)

Fraction (-)

10

O2( i )+H → OH( j’)+O

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

4

Symbols: Thermochemical Relaxation (ZV=10 ) Lines: Equilibrium Mixture Pre-Reaction Levels (ZV=1)

i i’ j j’

O2( i )+H → OH( j’)+O

O2( i’)+H ← OH( j )+O 10

O2( i’)+H ← OH( j )+O

-6

0

5 10 Vibrational Level

15

0

5 10 Vibrational Level

15

Figure 2.10.: Pre- and post-reaction vibrational level distributions of O2 and OH species using LB model (left) and proposed approach (right) for case 2.

Fig. 2.11, under the investigated conditions, the new approach provides a much better agreement between TT , TR , and Tvm values towards the steady state. This suggests that instead of the LB model the proposed approach can also be applied to reacting

34

36 34

T (K)

32 30 28 26 24 22 20

00

TT TR TV - H2 Chemked

00 00

38 36 34

00

32

00

T (K)

38

00 00

28 26

00

24

00

22

00

00 -8 18 10

30

20

10

-7

10

-6

10

-5

10

-4

10

-3

t (s)

38 36 34

T (K)

32 30 28

22 20

TT TR TV - H2 Chemked

00 00 00 00 00 00 00 00 00

00 -8 18 10

10

-7

10

-6

10

-5

10

-4

10

-3

t (s)

00 00 00 00 00 00

0 26 24

00

TT TR TV TV - H2 TV - H2O - Mode 1 TV - H2O - Mode 2 TV - H2O - Mode 3 Chemked

0

00 00 00

00 -8 18 10

10

-7

10

-6

10

-5

10

-4

10

-3

t (s)

Figure 2.11.: Thermochemical relaxation for H2 +M H+H+M reactions using proposed approach for H (left), H2 (right), and H2 O (bottom) third-bodies.

mixture having polyatomic molecules as major species. Such a situation is commonly encountered in the combustion of hydrocarbons.

35 2.5

A Modified LB Post-Reaction Energy Redistribution As discussed in Sec. 2.3, conventional DSMC implementation of the LB model to

study reacting mixtures may not lead the system to full thermochemical equilibrium under large ZV conditions. This problem can be overcome using the consistent TCEbased approach proposed and verified in Sec. 2.4, which relies on previous knowledge of the pre-reaction internal energy distributions. While this is a robust strategy that can be extended to other phenomenological and ab-initio chemistry models, tabulation of the pre-reaction internal state populations can become impractical for systems involving a large number of reaction pathways and wide range of TT , TR , and TV combinations. Keeping these issues in mind, we propose an alternative implementation of the LB method for post-reaction energy redistribution in this section. The main idea behind the present strategy is to perform a pre-redistribution of the post-collisional translational and internal energies before proceeding to the LB sampling scheme based on Equations 2.16 and 2.17. Considering a reacting collision that produces two diatomic species and following the nomenclature already introduced in Sec. 2.3, the new LB sampling scheme can be summarized as follows. I Pre-redistribution 1) Sample Et∗ proportionally to the translational degrees of freedom: 0 ) (5 − 2ω12 Ec 0 (5 − 2ω12 ) + ζv0 1 + ζv0 2 + ζr0 1 + ζr0 2 0    Ec = Ec0 − Et∗ ∗ 2) Sample Ev1 with:   ζb = ζ 0 + ζ 0 + ζ 0
/2 v2 r1 r2

Et∗ =

3) Sample Ev∗2

   Ec = Ec0 − Et∗ − Ev∗ 1 with:
  ζb = ζ 0 + ζ 0 /2 r1 r2

3) Sample Er∗1 with:

   Ec = Ec0 − Et∗ − Ev∗ − Ev∗ 1 2   ζb = ζ 0 /2 r2

36 4) Set Er∗2 as: Er∗2 = Ec0 − Et∗ − Ev∗1 − Ev∗2 − Er∗1 Having completed this preliminary energy redistribution, i.e., obtained the E ∗ values, we then simply perform the same LB redistribution employed for nonreacting collisions to sample the post-reaction energy states E 0 as shown below. I Final redistribution    Ec = Et∗ + Ev∗ 1 0 1a) Sample Ev1 with:   ζb = 5/2 − ω 0 12 1b) Update Et∗ as: Et∗ = Ec − Ev0 1    Ec = Et∗ + Ev∗ 2 0 2a) Sample Ev2 with:   ζb = 5/2 − ω 0 12 2b) Update Et∗ as: Et∗ = Ec − Ev0 2    Ec = Et∗ + Er∗ 1 0 3a) Sample Er1 with:   ζb = 5/2 − ω 0 12 3b) Update Et∗ as: Et∗ = Ec − Er0 1    Ec = Et∗ + Er∗ 2 0 4a) Sample Er2 with:   ζb = 5/2 − ω 0 12 4b) Set Et0 as: Et0 = Ec − Er0 2 Note that while Equation 2.16 is valid for molecules with ζr0 ≥ 2 and ζv0 ≥ 2, Equation 2.17 can only be used to sample internal states of diatomic species. For reacting collisions that produce three molecules, the corresponding E ∗ and E thirdbody energies should also be sampled and its degrees of freedom included in the pre-redistribution ζb calculation. In this case, note also that ζb should be divided by 3 instead of 2 in the pre-redistribution steps. As we demonstrate with the next calculations, this empirical modification in the LB procedures satisfies detailed balance regardless the VT and RT relaxation rates under consideration.

37 Aiming later DSMC simulations of multi-dimensional reacting flows, the postreaction energy redistribution methodology proposed in this section is implemented into the SPARTA (Stochastic PArallel Rarefied-gas Time-accurate Analyzer) opensource solver developed by Sandia National Laboratories [53]. SPARTA follows the same TCE procedures used in the original DS1V solver [44], which considers that only the translational and rotational modes contribute to total collision energy Ec . To be consistent with all the DSMC calculations presented in this dissertation, however, we further modified the TCE model implementation into the SPARTA-8Sep16 version to comply with the Ec definition given in Sec. 2.2.2. It means that all internal modes contribute to the total collision energy. Furthermore, as illustrated by Fig. 2.1, all the Arrhenius parameters used in our SPARTA simulations are also corrected to account for the use of discrete vibrational energy levels for diatomic species. In SPARTA, vibrational energy of polyatomic species are always based on continuum Boltzmann distributions.

2.5.1

Constant ZV and ZR Thermochemical Relaxation

The 0-D adiabatic thermochemical relaxation of an initial stoichiometric H2 -O2 mixture is considered to verify the modified LB implementation for reacting collisions. The mixture is initiated at 1 atm, 2000 K, and mole fractions XH2 =0.666 and XO2 =0.334. All the reaction pathways listed in Table 2.1 are allowed to take place. Similarly to previous sections, constant ZR = 1 value is used but two different VT relaxation rates for nonreacting collisions are considered: constant ZV =1 and 104 . For both scenarios, all types of collisions are assumed to have the same ZV values, i.e., 1 or 104 . The DSMC results are once again compared with the numerical solutions by Chemked, which assumes thermal equilibrium. As clearly shown by Fig. 2.12, both the original and modified LB models agree well with Chemked results for ZV = 1. Figure 2.13 shows, however, that only the new methodology leads all the species to the same vibrational temperature for ZV =

38

45

Temperature (K)

40 35 30 25 20 15 10

00

50

Chemked TT TR TV, H2 TV, O2 TV, OH TV, H2O

00 00 00 00

45 40

Temperature (K)

50

00 00 00

30 25 20 15

00

50

35

10

0

10-6

10-5

00 00 00 00 00

Chemked TT TR TV, H2 TV, O2 TV, OH TV, H2O

00 00 00

50

Squares: Rot. Temperatures

0 10-7

00

0

0 10-7

10-4

10-6

time (s) 100

H2 H 2O

O2

H 2O

O2 H Mole Fraction (-)

H Mole Fraction (-)

10-4

100

H2

10

10-5

time (s)

OH

-1

O

10

OH

-1

O

Lines: Chemked Symbols: DSMC 10-2 -7 10

10-6

10-5

time (s)

10-4

Lines: Chemked Symbols: DSMC 10-2 -7 10

10-6

10-5

10-4

time (s)

Figure 2.12.: Thermochemical relaxation of H2 -O2 combustion system using the conventional (left) and modified (right) LB implementaion for ZV = 1.

104 . The second set of plots also reveal that although the species-specific TV values differ considerably from one approach to another, the corresponding steady-state mole fractions are not so different.

39

45

Temperature (K)

40 35 30 25 20 15 10

00

50

00

45

00

40

00

Temperature (K)

50

00 00 00 00

30 25 20 15

00

50

35

10

0

00 00 00 00 00 00

00 00

50

0 10-7

10-6

10-5

Chemked TT TR TV, H2 TV, O2 TV, OH TV, H2O

00

0

0 10-7

10-4

10-6

time (s)

10-5

100

100

H2

H2 H 2O

H 2O

O2

O2 H Mole Fraction (-)

Mole Fraction (-)

H

10

10-4

time (s)

OH

-1

O

10

OH

-1

O

Lines: Chemked Symbols: DSMC

Lines: Chemked Symbols: DSMC 10-2 -7 10

10-6

10-5

10-4

10-2 -7 10

10-6

time (s)

10-5

10-4

time (s)

Figure 2.13.: Thermochemical relaxation of H2 -O2 combustion using the conventional (left) and modified (right) LB implementaion for ZV = 104 .

2.5.2

Realistic ZV (T) and ZR (T) for H2 -O2 Combustion

Previous thermochemical relaxation calculations were intended to evaluate how the different post-reaction energy redistribution schemes perform under the ZV = 1 and 104 limiting conditions. These values, however, were just reference collision numbers and do not necessarily represent actual H2 -O2 VT relaxation rates. Our next goal is, therefore, to test the modified LB model proposed in this section with more

40 realistic VT and RT relaxation rates. Before presenting these results, here, we first discuss some issues regarding the consistency between DSMC and continuum definition of the relaxation numbers Z. Finally, we present a brief survey on experimental based vibrational relaxation rates. Based on a continuum flow modeling convention, the vibrational relaxation number ZVC is the ratio of the vibrational relaxation and mean collision times, ZVC =

τv . τc

(2.20)

The mean collision time between species 1 and 2 depends on the corresponding number densities and the averaged cross-section data, τc12 =

1 n1 (σT cr )12

(2.21)

.

Considering collisions described by the VHS model, the above expression becomes, τc12 =

1 2 2π 1/2 dref n1 (T /Tref )1−ω12 (2kB Tref /mr )1/2

,

(2.22)

where mr is the reduced mass of the colliding pair, and the collision specific Tref and dref parameters are fit to comply with viscosity data [4]. The vibrational relaxation time of species 1 colliding with a heat bath gas of species 2, τv12 , is usually obtained from experiments or ab-initio calculations and rely on the assumption of isothermal relaxation processes. More details and examples on how to obtain τv12 values are given in Chapter 5. In the DSMC-LB implementations by Bird [4, 44], however, he considered adiabatic relaxation processes such that the probability of undergoing a VT exchange was taken to be 1/ZV . In other words, it considers that, on average, one out of ZV collisions experiences some degree of vibrational relaxation but it does not ensure the corresponding τv values are reproduced. The inconsistency between DSMC and continuum definitions of relaxation numbers, ZVD and ZVC , respectively, has been extensively discussed in the literature [49]. One of the ultimate approaches to link ZVD

41 and ZVC is given by Gimelshein et al. [48]. This correction has the following general form: ZVD = ZVC

ζT · (TT − T 0 ) , ζV (TT )TT − ζV (TV )TV

(2.23)

where ζT = 5−2ω is the collision translational degrees of freedom under the VHS/VSS models and ζV is the vibrational degrees of freedom calculated according to simple harmonic oscillator (SHO) model and the appropriate cell-based temperature T values. From energy conservation, the equilibrium post-collisional temperature T 0 can be numerically obtained by solving ζV (TV )TV + ζT (TV )TV = ζV (T 0 )T 0 + ζT (T 0 )T 0 .

(2.24)

Similarly, experimental-based rotational relaxation numbers should be corrected as ZRD = ZRC

ζT , ζT + ζR12

(2.25)

where ζR12 is the total number of rotational degrees of freedom involved in the collision [49]. For the particular case of vibrationally cold (TV < TT ) or nearly equilibrium (TV ≈ TT ) flows, Equation 2.23 can be approximated by ZVD = ZVC

ζT , ζT + A

(2.26)

with A being an explicit function of TT , A=

ζV2 (TT ) exp(θv /TT ) . 2

(2.27)

We also implemented this latter correction in SPARTA solver with a minor modification. Instead of using the instantaneous cell-averaged TT value in Equation 2.26, we use the quantized collision specific translational temperature TC = 2ET /(kB ζT ). Here the translational energy is calculated as ET = Ec ζT /(ζT + ζR12 + ζV12 ), where ζV12 is the collision total number of vibrational degrees of freedom. For clarity, unless stated otherwise, the superscript D used to denote the DSMC definition of Z is omitted in the rest of this chapter.

42 The SPARTA implementation of the ZV correction is compared against our modified DS1V solver results, which is has been previously validated [54]. Note that SPARTA and DS1V vibrational levels are based on SHO and AHO (anharmonic oscillator) models, respectively. Figure 2.14 presents the isothermal relaxation of an hypothetical mixture in which VT energy exchanges can occur only via O2 -O collisions. As one can see, the agreement between SPARTA and DS1V profiles verifies our implementation.

10

00

Temperature (K)

80

60

40

20

0

TRT TV, O2 (DS1V-AHO) TV, O2 (SPARTA-SHO)

00

00

00

00

ZV = 50 0 10-8

10-7

10-6

10-5

10-4

10-3

10-2

time (s)

Figure 2.14.: 0-D isothermal VT relaxation via O2 -O collisions for constant ZV = 50 and ZR = 1: SPARTA vs DS1V implementations.

Finally, in Fig. 2.15 we graphically present a survey of vibrational relaxation rates relevant to the H2 -O2 combustion system under investigation. These VT rates are expressed in terms of ZVC (T ), which are calculated according to Equation 2.22 and τv values obtained from literature [55–63]. Table 2.3 summarizes the ZVC (T ) values we arbitrarily chose for later DSMC calculations and the corresponding coefficients for the fitting function, ZVC (T ) = Z0 + exp A0 + A1 Θ + A2 Θ2 where Θ = T −1/3 .




(2.28)

10

7

106 10

5

10

4

10

3

10

2

Milikan-White (1963) Lutz (1966) Bird (1994) Nikitin (1994) Ibraguimova (2013)

+

+ + + + + + + +

+

+

++

+

O -O Collisions

2 2 ++ ++ ++ +++ +++ ++++ +

101 10

0

10

8

10

7

0

106 10

5

10

4

10

3

10

2

10

1

10

0

10

8

10

7

10

6

10

5

10

4

10

3

10

2

0

1000

2000

+ + + + ++ + + O2-Ar + ++ ++ + O2-He +++ +

++

++

1000

2000

++

+++

4000 3000 Temperature (K)

+

5000

6000

Camac (1961) Rao (1984) Losev (1970) Shatalov (1970) Nikitin (1994) Milikan-White (1963)

+

O2-H2

+++

++++

5000

++++

+

6000

Milikan-White (1963) Kiefer (1967) Breen (1975) Dushin (1991) MW-Park (1993) Sebastião (2017)

O2-O Collisions + ++ + ++ ++ ++

++

+++

101 10

4000 3000 Temperature (K)

++++

+++++

+++++++

++ ++ ++

0

0

1000

2000

4000 3000 Temperature (K)

5000

Vibrational Relaxation Number, ZVC (-)

8

Vibrational Relaxation Number, ZVC (-)

10

6000

Vibrational Relaxation Number, ZVC (-)

Vibrational Relaxation Number, ZVC (-)

Vibrational Relaxation Number, ZVC (-)

Vibrational Relaxation Number, ZVC (-)

43

10

8

10

7

Kiefer (1966) Kiefer (1966) Kiefer (1966) Dove (1974) Dove (1974) Dove (1974)

+

106 10

5

10

4

10

3

10

2

+ + + + H2-Ar + ++ + H2-He + ++ + H2-Ne +++ ++ ++ +

+++

+++

++++

+++++

+++++

H2-H2

101 10

0

10

8

10

7

0

1000

2000

4000 3000 Temperature (K)

5

10

4

10

3

10

2

10

1

10

0

10

8

10

7

10

6

10

5

10

4

10

3

10

2

OH(A) Quenching

0

++ ++ ++

OH(X)-H2O OH(X)-O2 OH(X)-O 2000

1000

OH-O2 OH-H2 OH-H OH-OH

4000 3000 Temperature (K)

+

5000

6000

Kung (1975) Zittel (1989) Zuev (1992) Kung (1975) Zuev (1992) Kung (1975) Kung (1975)

H2O-N2

101 10

+ + + ++ ++

6000

Lacousiere (2003) Tamura (1998) Tamura (1998) Tamura (1998) Tamura (1998) Tamura (1998) McCabe (2006) Atkinson (1992)

+

106 10

5000

H2O-Ar + + ++ ++

0

0

1000

H2O-He

H2O-H2O

++

++

+++

2000

+++ 4000 3000 Temperature (K)

5000

6000

Figure 2.15.: A survey of VT relaxation numbers relevant to H2 -O2 combustion.

44

Table 2.3.: Estimate for collision-specific ZVC (T ) based on Equation 2.28 fit. Collisiona

Z0

A0

A1

A2

Data Source

H2 -M

0

-2.281616

145.5725

-122.4369

Kiefer (1966)b

O2 -O2

0

-2.255037

127.0910

214.0946

Ibraguimova (2013) [62]

O2 -H2

0

0.6934441

84.97341

-131.5740

Milikan-White (1963) [55]b

O2 -O

0

3.700639

9.332155

-25.07215

Sebasti˜ao (2017) [54]

O2 -M

0

-2.018838

158.8461

38.37575

Losev (1970)b

OH-M

1000

0

0

0

Tamura (1998) [58]c

H2 O-H2 O

10

0

0

0

Kung (1975) [56]c

H2 O-M

500

0

0

0

Zittel (1989) [57]c,d

Most of these τv (T ) data were obtained from experiments covering temperatures below 3000 K. a

Collision pair order indicates: Relaxing Species - Heat Bath Species.

b

τv (T ) values extracted from Ibraguimova et al. (2012). [61]

c

approximated values.

d

τv (T ) values extracted from Capitelli (2013) [63].

For all collision pairs, the rotational relaxation numbers are calculated according to Parker’s expression [64], ZRC (T ) =

ZR∞ 1 + (π 1/2 /2) (T ∗ /TT )1/2 + (π + π 2 /4) (T ∗ /TT )

,

(2.29)

with ZR∞ = 20 and T ∗ = 100 K. Once again, in our SPARTA implementation, the quantized collision translational temperature TC is used instead of the cell-averaged TT value.

2.5.3

ZV (T) and ZR (T) Thermochemical Relaxation

Based on the VT and VR relaxation numbers presented in previous subsection, Fig. 2.16 shows the temperature and mole fraction evolutions for the same H2 -O2 mixture investigated in Sec. 2.5.1. For brevity, however, only the results obtained

45 with the modified LB methodology are shown. These temperature evolutions reveal that, except by the thermal nonequilibrium produced when H2 consumption begins, the mixture relaxation takes place essentially under equilibrium conditions. While this result suggests nonequilibrium effects are not critical when realistic ZV (T ) are used, it is important to point out these are all 0-D calculations and the impact of light species diffusion is not captured.

0 45

Temperature (K)

40 35 30 25

10

H2

0

H 2O O2

00

H

00 00 00

0 20 15

100

00

Chemked TT TR TV, H2 TV, O2 TV, OH TV, H2O

0

00 00

50

0

0 10-7

10-6

10-5

time (s)

10-4

Mole Fraction (-)

50

10

OH

-1

O

Lines: Chemked Symbols: DSMC 10-2 -7 10

10-6

10-5

10-4

time (s)

Figure 2.16.: H2 -O2 thermochemical relaxation system using the modified LB implementaion for ZV (T ) and ZR (T ).

2.6

Summary In DSMC simulations of reacting flows, the post-collisional energy must be par-

titioned among the translational and internal modes of the product species in such a way that detailed balance is satisfied. Otherwise, thermochemical equilibrium is not achieved as the mixture relaxes toward the steady state. Since in most DSMC studies the chemistry is dominated by dissociation of diatomic species into atoms, the LB model has been used to redistribute the collision energy of both nonreacting and reacting collisions without major consequences. However, as showed in this chap-

46 ter, the LB model does not lead reacting mixture to complete thermal equilibrium when VT relaxation and recombination/exchange reaction rates are of same order of magnitude. While LB model assumes equilibrium post-collisional distributions, experimental data support that post-reaction states have nonequilibrium distributions [65]. In this work, we propose two alternative solutions to sample post-reaction energy states within the TCE framework. The first proposed approach enforces that postreaction vibrational populations of a particular reaction match the corresponding pre-collisional vibrational levels of the reverse reaction. In doing so, the post-reaction vibrational energy redistribution is consistently linked to the reaction rates. The second proposed approach consists in performing a pre-redistribution of the collision energy before proceeding to the conventional LB sampling procedures. Keeping in mind that TCE model can accurately reproduce only equilibrium rates, the proposed framework is recommended for DSMC studies of low-speed reacting flows with moderate degree of nonequilibrium and where recombination and exchange reactions play an important role in the flow energy balance. Relevant applications of engineering interest include the formation of nitric oxide (NO) in the wakes behind hypersonic vehicles, which strongly impacts the after-body radiation, combustion of energetic materials such as aluminum nanoparticles, microcombustors, and materials synthesis with CVD-type process.

47

3. LOW-SPEED COMBUSTION FLOW MODELING Due to the large computational costs, practical atomistic-level modeling has been limited to study dilute gas flows via the direct simulation Monte Carlo (DSMC) method and material science, biophysical, and biochemistry problems, at reduced spatial/temporal scales, via molecular dynamics (MD) techniques. As high-performance computing (HPC) resources are evolving toward exascale platforms, however, problems that until now were considered intractable from a CPU-time perspective are becoming feasible. For instance, recent DSMC/SPARTA simulations have provided strong evidence that atomistic-level methods for gases can be used to study hydrodynamic instabilities [53, 66] and turbulent flows [67] quantitatively. Following this rationale, the main goal of the present work is to evaluate how DSMC performs in modeling atmospheric low-speed combustion problems. In particular, we employ a modified TCE-LB framework into SPARTA solver to model the 1-D laminar flame structure of H2 -O2 premixed systems. Although previous DSMC efforts have already addressed detonation waves [68], to the best of our knowledge, this is the first unsteady DSMC simulation of deflagration waves. Therefore, an empirical methodology to estimate the flame propagation speed is also proposed. The present DSMC results are compared with solutions by the well-established CHEMKIN/PREMIX [69] continuum solver and validated against experimental data available in the literature [70].

3.1

Motivation and Background Combustion occurs, in most cases, when a proper fuel-oxidant mixture reacts

in gas phase. This chemical process is characterized by a strong and irreversible heat release usually followed by a luminous flame [71–73]. A recent U.S. Energy Information Administration (EIA) report [74] predicts that although nuclear power

48 and renewable energy sources will be the fastest growing energy sources through 2040, combustion of fossil fuels – coal, oil, and natural gas – will still comprise around 80% of the world energy supply. Therefore, there exist a continuous demand to improve the efficiency of combustion based applications. While the overwhelming majority of combustion studies are based on the continuum fluid hypothesis, the underlying physics of combustion phenomena relies on molecular processes. Radical lifetimes and thermal relaxation between translational and internal molecular energy modes can vary considerably among the reacting species. As a result, both chemical and thermal nonequilibrium can take place in combustion. Plasma-assisted combustion [75,76], microcombustors [77,78], scramjets [79], and pulsed/rotating detonation engines (PDEs/RDEs) [80–82] are examples of potential applications where the flow molecular nature and nonequilibrium effects may become critical. Recent experimental and CFD efforts, mostly based on direct numerical simulations (DNS), addressed the effects of vibrational nonequilibrium on the combustion of H2 -air mixtures [83], detonation waves [84], turbulent mixing [85], and turbulent jets [86,87]. Although most combustion processes involve high concentrations of H2 O species, resulting in faster VT relaxation, these studies have shown that vibrational nonequilibrium can significantly alter ignition location in high-speed shock-containing flows. These simulations also suggest that VT relaxation has a major effect on the propagation of high-pressure waves and, consequently, the formation of pressure spikes behind the detonation front. The few DSMC combustion calculations available in literature have focused on H2 O2 . Unsteady 1-D and 2-D detonations in millimeter-sized channels were extensively investigated by Bondar et al. [33,68,88]. Temperature and mole fraction distributions of non-premixed combustion in a Y-shaped microchannel [37], estimation of ignition times [44,89], and preliminary studies of nonequilibrium effects on diffusion flames [38] have also been addressed with DSMC.

49 Taking into account the trade-off between computational costs and physical accuracy for full multi-scale flow simulations, conventional CFD approaches will be always preferable over DSMC. As the problem under consideration requires highfidelity models to capture the correct physics, however, DSMC and master equation approaches become necessary. Master equation solvers can be summarized as being CFD solvers in which one additional species-specific conservation equation has to be written and solved for each tracked discrete level of the internal molecular modes. Its 2 2 . × Nspecies computational costs should then nearly scale with Nlevels

Apart from the disadvantage of statistical noisy inherent of stochastic methods, complexity of DSMC algorithm is independent of Nlevels and Nspecies . While current computational resources prevent DSMC to address full multi-scale problems, using the proper collision models, its results can be used to study fundamental physics and interpret experimental data that cannot be reproduced by continuum fluid-flow solvers. For instance, high-fidelity DSMC simulations can serve as benchmark for code-code verification or provide data and insights that can be helpful in the development of CFD subgrid models. In this context, our current goal is to perform a preliminary assessment of DSMC capabilities and limitations in reproducing fundamental properties of low-speed combustion flows. Specifically, the laminar flame structure of H2 -O2 premixed systems and the corresponding deflagration wave speeds predicted by DSMC are compared against literature data. This configuration was chosen because of its simple 1-D nature and flexibility for using different burning conditions. The remaining of this chapter is organized as follows. Section 3.2 provides a brief summary of experimental techniques for measuring flame speeds. In Sec. 3.3, we compare different H2 -O2 reaction mechanisms and summarize the main features of PREMIX solver. The proposed DSMC framework to model the premixed flames is described in Sec. 3.4, where details on the verification and validation analysis are also given. In Sec. 3.5, we present the main findings and some concluding remarks.

50 3.2

Deflagration Waves Combustion is multi-disciplinary field that involves physical chemistry, fluid me-

chanics, and heat transfer knowledge. Numerical and theoretical combustion modeling has been extensively used to gain further insight into nature of processes that are not captured experimentally. Besides the temperature and mole fraction distributions across the flame, its shape and propagation velocity are relevant parameters used to evaluate combustion models. By definition, the premixed flame speed Sf lame is taken as the velocity at which a planar flame front travels with respect to the unburnt mixture, Sf lame = uf lame − ugas ,

(3.1)

where uf lame and ugas are the flame and unburnt gas velocities in the laboratory frame, respectively. Sometime, this is also called burning velocity as a metric of the rate at which reactants are consumed. A deflagration occurs when the flame propagates within the reactants under subsonic speeds. Conversely, a detonation takes place if the flame propagates under supersonic speeds as a shockwave. While in the former case pressure is nearly uniform across the flame, abrupt temperature and pressure changes take place across detonation waves. The most common experimental techniques for Sf lame calculation are summarized as follow. • In the Bunsen burner method, a stationary conical flame is anchored to the exit of a cylindrical tube. Sf lame is then estimated based on measurements of the flame geometry and the velocity of reactant gases into the tube. • In the spherical bomb method, the reactant gases are confined in a large constant-volume chamber with optical access. The mixture is then ignited, usually with an electrical discharge, creating an expanding spherical flame. Schlieren visualization is used to track the flame front evolution, and consequently, its velocity. Because of the flame curvature, however, effects of flame stretching must be taken into account.

51 • In counterflow methods, the main geometry consists of two counter jet flows or an impinging jet of premixed gases on stagnation surface. In any case, a flat flame disc establishes normal to the jet(s) such that the combustion products leave the reaction zone axially. Using velocimetry techniques, velocity of the unburnt gases near to the flame front can be obtained and, consequently, Sf lame . Because of the radial velocity components, stretching effects cannot be neglected. • In heat flux methods, temperature-controlled premixed gases are injected through a perforated, also temperture-controlled, flat plate. Parallel to this plate, a stable flame is established under adiabatic conditions, i.e., when the heat losses from the flame balances with the heat of reaction released by the incoming burning mixture. By measuring the radial temperature distribution, it is possible to determine Sf lame without corrections for flame stretching. A more comprehensive discussion pointing out the strengths and challenges of each technique is given by Alekseev [90]. As discussed in next sections, our DSMC simulations are somewhat based on the spherical bomb method but with the advantage of modeling unstretched flat flames.

3.3

Assessment of H2 -O2 Reaction Mechanisms Before proceeding to the DSMC implementation of a particular mechanism, we

compare the results obtained with the different H2 -O2 combustion mechanisms summarized in Table 3.1. As described in this table, the case referred to as “Reduced Shatalov” consists of a simplified version of the original mechanism. Two sets of calculations based on Chemked [52] and PREMIX [69] continuum solvers are considered, respectively. In both sets, initial stoichiometric H2 -O2 mixtures at 1 atm are considered. Figure 3.1 presents the 0-D thermochemical relaxations predicted by Chemked for initial temperatures, T0 , of 1000 and 2000 K. These evolutions are obtained by nu-

52

Table 3.1.: Investigated H2 -O2 reaction mechanisms. Reaction Mechanism #Species #Reversible Reactions

Reference

Davidenko (2006)

6a

7

[47]

` OConnaire (2004)

11b

21

[91]

Li (2004)

11b

16

[92]

Shatalov (2009)

11c

27

[93]

9d

14e

Present Work

Reduced Shatalov

a) H2 , H, O2 , O, OH, and H2 O. b) Same as in (a) plus HO2 , H2 O2 , N2 , Ar, and He. c) Same as in (a) plus HO2 , H2 O2 , N2 , Ar, and O3 . d) Same as in (a) plus HO2 , N2 , and Ar. e) Without reactions involving O3 and H2 O2 , and reaction O2 +H+M=HO2 +M.

45

Temperature (K)

40 35 30 25 20 15 10

00

ÒConaire (2004) Li (2004) Davidenko (2006) Shatalov (2009) Reduced Shatalov

00 00

50 45 40

00

Temperature (K)

50

00 00 00

30 25 20

00

15

00

0 5010-5

35

10

10-4

10-3

10-2

time (s)

10-1

100

101

00 00 00

ÒConaire (2004) Li (2004) Davidenko (2006) Shatalov (2009) Reduced Shatalov

00 00 00 00 00 00

0 5010-7

10-6

10-5

10-4

time (s)

Figure 3.1.: 0-D adiabatic Chemked simulations to compare different reaction mechanisms in predicting the self-ignition of a H2 -O2 stoichiometric mixture at 1 atm.

merically solving the set of differential equations that describe the principle of mass, species, and energy conservations. Defining the ignition time as the time required

53 to reach the equilibrium temperature, except by Shatalov-based results, all the investigated mechanisms lead to similar predictions. We obtained the present reduced version of Shatalov’s mechanism after analyzing the impact of different reaction pathways for the two considered initial conditions. Similarly, Table 3.2 compares the different kinetic mechanisms against the adiabatic flame temperatures predicted by the well-known NASA computer program CEA (Chemical Equilibrium with Applications), which relies on a minimization-offree-energy formulation [94]. As on can see, present Chemked and CEA-NASA results agree within 1%. The next set of comparisons is based on PREMIX calculations. This solver models steady, quasi-one-dimensional, isobaric premixed flames by solving the corresponding system of mass, species, and energy conservation equations [69]. Effective transport and thermodynamic properties are based on mixing rules and the rate of production of a given species depends on the reaction mechanisms provided by the user. It is important to mention that, in the solution of 1-D freely propagating flames, the mass flow rate across the flame is an eigenvalue. Since initial pressure and composition define densities far away from the flame, it means the gas velocity relative to the flame has to be determined as part of the solution.

Table 3.2.: Comparison of different reaction mechanismsa in predicting the adiabatic flame temperature of a H2 -O2 stoichiometric mixture at 1 atm. Thermochemical Equilibrium Temperature (K) ` Davidenko OConnaire Li Shatalov Red. Shatalov

T0 (K)

CEA

1000

3376

3373

3361

3377

3380

3412

2000

3363

3360

3346

3364

3364

3401

a

CEA-NASA does not account for kinetics but relies only on thermodynamic data. Other

values are obtained with Chemked.

54 The steady state 1-D flame structures calculated by PREMIX are shown in Fig. 3.2 in terms of temperature T and mole fraction X distributions. Taking the results based on Davidenko’s mechanism as the baseline, only two other mechanisms are compared. Except by the smaller temperature gradient across the flame observed in Davidenko’s case, the three mechanisms provide essentially the same (T, X) profiles. Surprisingly, the same degree of agreement is not observed for the gas velocity distributions shown in Fig. 3.3. The solution based on Davidenko’s mechanism predicts velocities that are about half of the other considered cases. Note that, since the flame is stationary in PREMIX solutions, the velocities shown in this plot are relative to the flame and therefore already represent the flame propagation speed. In summary, the present set of comparisons between reaction mechanisms suggests ` that Davidenko, OConnaire, and Li mechanisms all provide the same (T, X) flame structure. Even though Davidenko’s mechanism considerably underpredicts the flame speed for the investigated conditions, we will still used it for the subsequent DSMC simulations since its implementation has been already fully verified in Chapter 2 for

3500

1.0

Solid: Davidenko Mechanism Dashed: ÒConnaire Mechanism

3500

0.9

3000 0.8

H2

0.5 1500

0.4

O2

0.3

1000

H2

0.6

2000

0.5 1500

0.3

1000

OH

0.1

0.1

O 0

0.5

1

0.2

H 500

OH

0.4

O2

0.2

H

0.7

H 2O T (K)

0.6

2000

Mole Fraction (-)

0.7

H 2O

500

0.8

T 2500

Mole Fraction (-)

T

T (K)

0.9

3000

2500

0

1.0

Solid: Davidenko Mechanism Dashed: Li Mechanism

O 1.5

x (mm)

2

0.0

0

0

0.5

1

1.5

2

0.0

x (mm)

Figure 3.2.: 1-D PREMIX simulations of H2 -O2 laminar premixed flames based on different reaction mechanisms.

55 3500

120

3000

100

ÒConnaire 2500

80

T (K)

Li Mechanism 2000

60

1500

40

Davidenko

Sflame

1000

20

500 0

Gas Velocity (m/s)

T

0

Solid Lines: Temperature Dash-Dotted: Gas Velocity 0

0.5

1

1.5

2

-20

x (mm)

Figure 3.3.: Comparing temperatures and flame propagation speeds (Sf lame ) predicted by PREMIX for different reaction mechanisms.

0-D problems. In any case, the present evaluation of the reaction mechanisms might help explaining possible discrepancies in our later DSMC results.

3.4

DSMC Calculations of H2 -O2 Deflagration Waves

3.4.1

Flow and Numerical Conditions

As shown in Fig. 3.4, present DSMC/SPARTA premixed flame calculations consider a 2-D domain of lengths LX by LY . The boundary conditions consist of three specular walls and one constant pressure p = 1 atm and temperature T = 300 K boundary at the x-maximum contour. Details on the DSMC implementation of these boundaries conditions can be found in the literature [4, 95]. Velocities and internal states of initial gas particles are always based on Boltzmann (equilibrium) distributions and the corresponding pressure p0 = 1 atm and temperature T0 = 300 K values. Initial gas bulk velocity is zero for all cases. Since our goal is to model the flame propagation throughout this isobaric channel, we ignite the unburnt gas by setting an initial high-temperature, T1 = 3500 K,

56

Figure 3.4.: Schematic representation of the flow domain, initial, and boundary conditions.

stoichiometric mixture near the x-minimum boundary. The length δX of this zone is only 0.1 mm and represents no more than 5% of the considered LX values. Note that while the initial H2 /O2 molar ratio Y1 is 2 for all cases, different initial Y0 conditions are investigated. All subsequent calculations employ Davidenko’s H2 -O2 combustion mechanism as listed in Tables 2.1 and 3.1. It involves 6 (H2 , H, O2 , O, OH, and H2 O) species and 34 different reaction pathways. Here, we use a DSMC/SPARTA framework similar to that one described in Sec. 2.5, which includes: (1) modified LB post-reaction energy redistribution scheme; (2) TCE model implementation that assumes all modes contribute to the collision energy; (3) Arrhenius constants that are corrected to account for the discrete vibrational levels – SHO model – of diatomic species; (4) polyatomic species based on continuous Boltzmann distributions; and (5) constant ZV = 1 and ZR = 1 relaxation collision numbers in order to simulate equilibrition conditions. In order to minimize the computational costs, a very large aspect ratio LX /LY = 2 mm/1 µm = 20, 000 is considered for the baseline configuration referred to as “Standard”. Based on the p0 and T0 conditions, this LY value is on the order of one molecular mean free path λ. It means that, although we are carrying out 2-D calculations, our results can only be interpreted as 1-D. Future studies will focus on the 2-D/3-D nature of the flame front propagation. The following DSMC numerical parameters are

57 also employed in the standard configuration: constant time steps of 10 picoseconds; 100,000 collision/sampling cells in x-direction and 1 in y-direction; and an initial number of simulated particles of approximately 20×106 . Under these conditions, the local number of particles per cell is always within the Nparticles = [20, 200] range, where the first and second values correspond to burnt and unburnt gases, respectively. Likewise, the local mean free path to cell x-size ratio λ/∆X is always within the [80, 2.3] range.

3.4.2

DSMC Estimation of Deflagration Wave Speed

As demonstrated in Sec. 3.3, 1-D CFD calculation of isobaric laminar flame speeds is straightforward [69,96]. Once the solver iteratively finds the flow structure and mass flow rate that leads to a stationary flame, Sf lame will correspond to the gas velocity upstream to the flame. As shown in Fig. 3.3, this value becomes uniform throughout the unburned mixture. In 3-D CFD simulations, flame speeds calculations follow experimental methodologies [97]. On the other hand, there is not standard framework to obtain Sf lame from atomistic-level simulations. Recent molecular dynamics (MD) efforts have addressed flame propagation along solid materials [98] and supercritical gases [99]. None of these cases, however, involved velocity and pressure variations in the unburnt mixture and flame speeds were estimated based solely on flame position evolution, i.e., Sf lame = uf lame . Figure 3.5 illustrates a DSMC-predicted 1-D flame propagation toward a H2 -O2 stoichiometric mixture, 30 µs after ignition. The corresponding temperature profile and flame position evolution are given in Fig. 3.6. As shown later, for the premixed flame conditions investigated in this section, pressure and velocity changes take place in the entire domain according to our DSMC calculations. It means there is not an uniform constant ugas DSMC value that can be used in Equation 3.1. To overcome this issue, the following methodology is proposed to calculate Sf lame . First, a reference point that represents the flame position xf lame is defined at Tref = (Tmax + Tmin ) /2. An instantaneous gas velocity ugas (t) value is then obtained by spatially averaging

58

Figure 3.5.: Atomic-level 2-D representation of flame propagation toward a H2 -O2 premixed gas.

3500

Tmax

3000

T (K)

2500 2000

Reference Point

Tref 1500 1000 500 0

Flame Position, xflame (mm)

2.0 Region used to calculate the average upstream gas velocity (ugas)

1.5

xflame

1.0

0.5

Tmin 0

0.5

1

1.5

x (mm)

2

0.0

0

10

20 30 time (µs)

40

50

Figure 3.6.: Schematics of the proposed method to calculate deflagration wave speed under the DSMC framework. The reference point (left) is used to track the flame position evolution (right).

ugas within the [xref , xT min ] stations. Here, xT min is the first x-station where T = Tmin . The corresponding flame velocity, in the laboratory frame, is taken simply as uf lame (t) = dxref /dt. In this work, dxref /dt is approximated by standard secondorder finite difference schemes. With this approach, as shown later in the validation analysis, the time-averaged value of Sf lame (t) = uf lame (t) − ugas (t) reproduces well the available experimental data.

59 3.4.3

Verification and Validation Analysis

Before proceeding to the simulation of different flame conditions, we conduct a parametric study to ascertain the impact of numerical DSMC parameters on flame structure and average flame propagation speed results. Figure 3.7, for instance, compares the flame structure based on realistic temperature-dependent relaxation collision number, as given in Sec. 2.5.2, and our current ZV = ZR = 1 assumption. Clearly, both cases provide the same results. Calculations based on the previously described standard Ncells , Nparticles , and LX values are also compared with other configurations in which these parameters are independently modified. Unless stated otherwise, all subsequent verification cases consider an atmospheric stoichiometric H2 -O2 mixture as detailed in Sec. 3.4.1. Instantaneous macroscopic property profiles are dumped every 1 µs and correspond to 10-ns time-averaged values. The flame structure profiles based on the standard and modified cases are indicated, respectively, by lines and symbols in Figure 3.8. These profiles correspond to 30 µs after ignition. As one can see, same results are obtained when Ncells and

3500

1.0

Solid Lines: ZV = ZR= 1 Symbols: ZV(T) & ZR(T)

T

3500

ZV(T) & ZR(T)

0.9

3000

3000 0.8

H2

0.6 0.5 0.4

O2

0.3

1000

2000 1500

TRT TR, H2 TR, O2 TR, OH TR, H2O

1000

0.2

H 500

T (K)

T (K)

H 2O 2000 1500

2500

0.7

Mole Fraction (-)

2500

500

OH

0.1

O 0

0

0.5

1

1.5

x (mm)

2

0.0

0

0

0.5

1

1.5

2

x (mm)

Figure 3.7.: DSMC calculations to assess the impact of ZV and ZR values on the investigated 1-D flame structure.

60 Lines: Ncells,Nparticles Symbols: 2×Ncells

T(K)

1.0

00

Lines: Ncells,Nparticles Symbols: 2×Ncells

T

35

3500

0.9

p(atm) ugas(m/s) 1.4 200

00

3000

1500

0.4

O2

1.0

100

0.8

50

0.6

0

0.4

-50

25

00

150

00

0.5

1.2

20

0.6

00

T (K)

H 2O 2000

Mole Fraction (-)

0.7

15

2500

30

0.8

H2

0.3 0.2

H OH

0

0.1

50

500

10

00

1000

O 0

0.5

1

1.5

2

0.0

0

0

0

x (mm)

1.5

Lines: Ncells,Nparticles Symbols: 2×Nparticles

T(K)

1.0

00

Lines: Ncells,Nparticles Symbols: 2×Nparticles

T

1

2

x (mm)

35

3500

0.5

0.9

p(atm) ugas(m/s) 1.4 200

00

3000

1500

0.4

O2

1.0

100

0.8

50

0.6

0

0.4

-50

25

00

150

00

0.5

1.2

20

0.6

00

T (K)

H 2O 2000

Mole Fraction (-)

0.7

15

H2

2500

30

0.8

0.3 0.2

H OH

0

0.1

50

500

10

00

1000

O 0

0.5

1

1.5

2

0.0

0

0

0

x (mm)

1.5

Lines: Ncells,Nparticles Symbols: Extended LX

T(K)

1.0

00

Lines: Ncells,Nparticles Symbols: Extended LX

T

1

2

x (mm)

35

3500

0.5

0.9

p(atm) ugas(m/s) 1.4 200

00

3000

0.4

O2

150

1.0

100

0.8

50

0.6

0

0.4

-50

00

25

00

1.2

20

0.5

00

0.6

Mole Fraction (-)

T (K)

H 2O

15

0.7

2000 1500

30

0.8

H2

2500

0.3

0.1

0

OH

50

0.2

H 500

10

00

1000

O 0

0.5

1

x (mm)

1.5

2

0.0

0

0

0

0.5

1

1.5

2

x (mm)

Figure 3.8.: DSMC parametric study of the impact of Ncells (top), Nparticles (middle), and LX (bottom) on flame structure. Shown profiles correspond to t = 30 µs.

61 Nparticles are doubled. On the other hand, pressure and gas velocity profiles are considerably different between LX = 2 mm and LX = 20 mm configurations. This can be better explained by looking at the corresponding smoothened pressure and velocity evolutions shown in Fig. 3.9 and noting the domain x-maximum boundary condition enforces a constant pressure of 1 atm. As the combustion heat of reaction is released a pressure wave produced by the thermal expansion of the gases travels ahead and flame. In the right plot of Fig. 3.9 we can see a compression wave propagating toward the end of domain, where it reflects as an expansion wave. In general, these characteristic wavelengths are larger than 2 mm, which is the standard domain length. In other words, while these waves can fully develop and naturally propagate within the extended domain, LX = 20 mm, the constant pressure boundary produces an artificial dumping and all we can see for the standard LX case are uniform pressure profiles that oscillate with time. Such a behavior is clear in Figs. 3.10 and 3.11, which present complete temperature and pressure evolutions in terms of surface maps for the two considered LX cases.

ugas(m/s) 200

p (atm) ugas(m/s) 1.4 200

150

Lines: Gas Pressure (p)

1.2

150

1.0

100

0.8

50

0.6

0

p (atm) 1.4

1.2

Lines: Gas Pressure (p)

t1 t2 100

t3

50

0

0

0.5

1

x (mm)

1.5

1.0

0.8

0.6

Dots: Gas Velocity (ugas) -50

t2

t3

t1

Dots: Gas Velocity (ugas)

2

0.4

-50

0

5

10

15

0.4 20

x (mm)

Figure 3.9.: Impact of the DSMC domain size (LX ) on the propagation of pressure and gas velocity waves. Arrows indicate current direction of the wave propagation.

62

Figure 3.10.: DSMC predicted flame front evolution in terms of temperature (top) and pressure waves (bottom) for the standard length domain (LX = 2 mm) case.

At this point results have indicated DSMC-calculated temperature and mole fraction profiles are independent of the considered Ncells , Nparticles , and LX values. Velocity and pressure distributions, however, are strongly affected by the domain size.

63

Figure 3.11.: DSMC predicted flame front evolution in terms of temperature (top) and pressure waves (bottom) for the extended length domain (LX = 20 mm) case.

Our next goal is to evaluate how the average flame speed Sf lame depends on these parameters under the same stoichiometric conditions covered so far. Evolution of flame position for different DSMC parameters are given in Figure 3.12. Although the stan-

64 dard LX cases indicate some variations, on average, their ∆xf lame /∆t slope agrees well with the extended domain results. The slope for extended LX case is essentially constant until t ∼ 75µs, which is when the aforementioned reflected expansion pressure interacts with the propagating flame. Instantaneous flame and gas velocities – Sf lame , uf lame , and ugas – for standard and extended domain cases are presented in Fig. 3.13, where the red lines indicate the corresponding time-averaged Sf lame value. The two initial and five final sampling points are neglected in these averages. Note that even tough the previously discussed pressure-velocity coupling considerably affects uf lame and ugas evolutions, the average Sf lame value is essentially the same for both standard and extended LX cases. Table 3.3 and Fig. 3.14 summarize the DSMC verification study. All these comparisons confirm the standard configuration leads to grid and domain size independent Sf lame results. Finally, we compare our DSMC simulations against experimental data [100–103] and third-party CFD solutions [70] in Fig. 3.15. Present PREMIX calculations based on two different reaction mechanisms are also included. In general, the equilibrium

5

Ncells,Nparticles (Standard) 1/2×Ncells 2×Ncells 2×Nparticles Extended LX

1.5

Flame Position, xflame (mm)

Flame Position, xflame (mm)

2.0

1.0

0.5

Ncells,Nparticles (Standard) 10×LX,1/2×Nparticles

4

3

2

1

0.0

0

10

20 30 time (µs)

40

50

0

0

20

40 60 time (µs)

80

100

Figure 3.12.: DSMC parametric study of the impact of Ncells , Nparticles (left), and LX (right) on flame position evolution.

65

100

80

uflame ugas Sflame = uflame - ugas

90 80

70

70

60

60

Velocity (m/s)

Velocity (m/s)

100

uflame ugas Sflame = uflame - ugas

90

50 40 30 20

50 40 30 20



10 0

0

-10

-10

0

20

40 60 time (µs)



10

80

100

0

20

40 60 time (µs)

80

100

Figure 3.13.: Impact of the DSMC domain size (LX ) on the flame and gas velocities. Horizontal line indicates the time-averaged deflagration wave speed Sf lame .

Table 3.3.: Impact of DSMC numerical parameters on Sf lame and corresponding standard deviation σSf lame for under stoichiometric conditions. Case

Sf lame (m/s) σSf lame (m/s)

Standard

10.2128

6.1151

Ncells /2

11.1704

5.3482

2 Ncells

10.0678

5.7202

2 Nparticles

10.3516

5.0886

9.9120

9.7165

Extended LX

DSMC calculations agree well with the other data. These results demonstrate DSMC can reproduce both the laminar flame structure and propagation speed in premixed systems.

66 18

Calculations (Present): DSMC: Standard DSMC: 1/2×Ncells DSMC: 2×Ncells DSMC: 2×Nparticles DSMC: Extended LX

16 14

Sflame (m/s)

12 10 8 6 4 2 0

0

0.2

0.4 0.6 H2 Mole Fraction (-)

0.8

1

Figure 3.14.: Verification analysis of impact of the different DSMC numerical parameters on Sf lame .

18

Experimental Data: Jahn (1934) Senior (1961) Koroll (1988) Koroll (1993) Calculations (Present): DSMC: Standard PREMIX: Davidenko + PREMIX: ÒConnaire

16 14

Sflame (m/s)

12 10 8 6

+

+

+

4

+

2 0

++

+ 0

0.2

+

Gelfand (2012) Calculations

0.4 0.6 H2 Mole Fraction (-)

0.8

+ 1

Figure 3.15.: Comparison of present DSMC-based Sf lame calculations against thirdparty solutions and experimental data.

67 3.5

Summary This chapter presented a detailed verification and validation study to asses how

DSMC calculations of laminar premixed flames compare with available literature data. In general, DSMC is able to correctly capture the 1-D flame structure and its propagation speed. In order to properly reproduce gas pressure and velocity fields, however, the chosen DSMC domain should be long enough to allow full development of the thermal expansion wave. For the investigated conditions, preliminary DSMC calculations based on ZV = 1 and realistic ZV (T ) values suggest the 1-D flame structure is well described by assuming thermal equilibrium. Further research is still needed, however, to evaluate the impact of nonequilibrium effects on 2-D/3-D problems. These calculations can help interpret spectroscopy measurements and serve as background for subgrid modeling of reacting flows.

68

69

4. IMPLEMENTATION OF COMPACT QCT-BASED COLLISION MODELS While previous two chapters were related to low-speed reacting flows dominated by recombination/exchange reactions, the rest of the dissertation focuses on the DSMC implementation of high-fidelity collision models needed to accurately describe strong nonequilibrium flows, such as those occurring in shockwaves. These models build upon state-to-state internal energy exchange and reaction cross-sections obtained from quasi-classical trajectory (QCT) calculations. Present study data include only O2 +O and N2 +O ground-state systems but other binary collisions can be considered in the future. In this chapter we summarize the collision models proposed by Kulakhmetov [27, 104] for O2 -O and later extended by Luo [28, 105] for N2 -O. After describing these models we discuss their DSMC implementation.

4.1

Motivation and Background When perturbed from equilibrium, chemical reactions and molecular internal en-

ergy exchanges try to return the flow back to an equilibrium state. These processes, however, take place on finite timescales. Therefore, if thermal relaxation is slower than flow convection or reaction rates, regions of local nonequilibrium flow can be generated. Thermochemical nonequilibrium appears in many reacting flows of interest to engineering and scientific communities. Typical examples include shockwaves, hypersonic boundary layers, microscale and high-speed combustion processes [12, 78, 85, 106, 107]. Nonequilibrium rarefied flows can be modeled using the direct simulation Monte Carlo (DSMC) method [4], which is an atomistic approach for numerically solving the Boltzmann equation for problems of engineering interest. The reliability of DSMC results, however, directly depends on the accuracy of the

70 collision models employed. In this connection, high-fidelity internal energy relaxation and chemical reaction models become necessary to describe highly nonequilibrium flows. Due to the difficulty of measuring outcomes of high energy individual collisions experimentally, most relaxation and reaction models are phenomenological. These models have been developed based on computational convenience and are calibrated to reproduce available, often low temperature, measurements [4, 108–110]. For instance, the Larsen-Borgnakke (LB) [25] and total collision energy (TCE) [26] phenomenological models remain the most popular internal energy relaxation and chemistry models in the DSMC community, respectively. Once calibrated with near-equilibrium data, these models are typically extrapolated to high temperature and highly nonequilibrium conditions. Under these circumstances, however, they are not guaranteed to be accurate [111]. The aerothermodynamics community has been developing ab-initio relaxation and reaction models for both DSMC and computational fluid dynamics (CFD) approaches for the last two decades. Accurate potential energy surfaces are obtained from abinitio calculations, which are then used in quasi-classical trajectory (QCT) simulations [112]. These simulations can then be used to calculate either rates or statespecific cross-sections of molecular processes. Since typical air species have around 103 ro-vibrational levels, there are over 1012 possible state-to-state (STS) transition cross-sections for diatom-diatom interactions. Table lookup of pre-calculated crosssections have been used for diatom-atom collision pairs [35, 113]. However, due to computational memory limitations, large cross-section database lookup cannot be efficiently used in full flowfield solvers with diatom-diatom collision pairs. To reduce the size of state-specific cross-sectional data sets, it is possible to bin them into a reduced set at a smaller accuracy penalty [36,114,115] or use them to calibrate compact models that reproduce the QCT data. Figure 4.1 summarizes the typical methodology in constructing QCT-based models.

71

Figure 4.1.: Typical framework for the development of QCT-based models.

Recently, Kulakhmetov et al. [116] and Luo et al. [28] developed state-specific dissociation (SSD), state-specific exchange (SSE), and vibrational-translational (VT) internal energy relaxation models that can be calibrated to ab-initio calculations [105, 117] with a reduced set of parameters. The VT model, in particular, is based on maximum entropy (ME) considerations [30, 118–121] and for this reason it is referred to as ME-QCT-VT model. These compact models are intended to accurately reproduce QCT-calculated dissociation and VT relaxation rates and cross-sections for O2 -O and N2 -O collisions in electronic ground states. Here and in the following chapters we address the DSMC implementation and verification of the ME-QCT-VT [104,122] and SSD/SSE [123,124] models against the LB-TCE framework and master equation solutions.

4.2

State-Specific VT Relaxation and Reaction Models Ab-initio state-to-state relaxation and state-specific reaction cross-sections can be

obtained by running quasi-classical trajectory (QCT) calculations with an accurate potential energy surface (PES). Kulakhmetov et al. [117] performed QCT calculations 3 for an O2 (3 Σ− g ) + O( P) system based on the double many-body expansion (DMBE)

PES by Varandas and Pais [125]. Likewise, Luo et al. [105] performed N2 O(3A”) QCT calculations based on Gamallo et al.’s PESs [126]. The corresponding calculated cross-sections were then fit to the compact models discussed in this section.

72 4.2.1

ME-QCT-VT Model

The main idea of the present VT energy exchange model is to accurately and efficiently reproduce QCT-based cross-sectional data using a reduced number of parameters. Figure 4.2 illustrates O2 +O state-to-state cross-sections for VT transitions obtained from QCT calculations [27]. In these plots, v and J represent the pre-collision discrete vibrational and rotational levels and VR is the relative velocity between the O2 -O collision pair. From these results we can observe peaks in the cross-section distributions that correspond to vibrationally elastic collisions, i.e., collisions in which the vibrational level remains unchanged. These plots also indicate that the probability of a vibrational excitation decays first exponentially and then linearly towards higher post-collision vibrational levels. Instead of fitting the QCT data into an arbitrary functional form, Kulakhmetov et al. [116] and Luo et al. [28] proposed physically-based compact models to fit such data. In particular, the atom-diatom VT energy exchange model described in this

1

10

2

Vr=3km/s, J = 11 Vr=3km/s, J = 51 Vr=3km/s, J = 101 Vr=5km/s, J = 11 Vr=5km/s, J = 51 Vr=5km/s, J = 101 Vr=9km/s, J = 11 Vr=9km/s, J = 51 Vr=9km/s, J = 101

1

2

3

4

5

Final Vibrational Energy (eV) (a) Pre-collision levels: v = 0

6

10

0

-4

10

-3

10

-2

10

-1

10

0

10

Cross Section (A2)

1

10

-4

10

-3

10

-2

10

-1

10

0

10

Cross Section (A2)

10

2

Vr=3km/s, J = 11 Vr=3km/s, J = 51 Vr=3km/s, J = 101 Vr=5km/s, J = 11 Vr=5km/s, J = 51 Vr=5km/s, J = 101 Vr=9km/s, J = 11 Vr=9km/s, J = 51 Vr=9km/s, J = 101

0

1

2

3

4

5

6

Final Vibrational Energy (eV) (b) Pre-collision levels: v = 10

Figure 4.2.: O2 +O state-to-state vibrational cross-sections obtained from QCT calculations [27].

73 section is based on maximum entropy (ME) considerations. In summary, the ME theory considers that molecular collisions reproduce post-collision distributions for the internal states that maximize entropy under given constraints. In the original formulation, only two constraints were considered [23]. The first constraint refers to the normalization of the distribution function for the internal states and the second one imposes that the post-collision vibrational energy has a mean value Ev0 . Herein, primed variables stand for post-collision states. With these assumptions, the constrained optimization problem can be solved using Lagrange multipliers, λ0 and λ1 , to provide a post-collision distribution g(v 0 , J 0 , Ec ) exp(λ0 + λ1 Ev0 ) = P0 (v 0 , J 0 , Ec ) exp(S) 0, J 0, E ) g(v 0 0 c v ,J

P (v 0 , J 0 , Ec ) = P

(4.1)

that depends on the degeneracies g of the post-collision states v 0 and J 0 , Ev0 , and the total collision energy Ec . Note that in this expression P is the product of a prior distribution P0 and the exponential of the so-called surprisal function S. This function is consistent with the constraints under consideration and carries the information of how the actual post-collision distribution deviates from the initial estimate P0 . Initially, P0 was obtained by assuming that post-collision states follow bulk gas equilibrium distributions [23]. Later, P0 was replaced by the Larsen-Borgnakke (LB) distribution, which considers local equilibrium distributions according to the energy and degrees of freedom of the colliding particles [65]. While the LB distributions were more physically appropriate, the lack of data on state-resolved energy exchanges prevented the development of accurate functional forms for S. For this reason, in the past, S has been assumed simplified forms and fit to macroscopic observations or limited QCT calculations. Fortunately, recent accurate QCT calculations of O2 +O [117] with collision energies up to 20 eV make it possible to further develop the surprisal function. The ME considerations, also known as information theory models, were originally proposed by Levine and Bernstein [23] and then further extended by Procaccia and Levine [118]. Gonzales and Varghese [119, 127] applied the ME model in master

74 equation studies of hypersonic flows and Marriott and Harvey [128] and Gallis and Harvey [30, 129] then applied this model to DSMC studies. The present model, referred as ME-QCT-VT, is based on the original ME considerations but uses the newly extended surprisal function for vibrational-translational energy exchanges [116]. While Kulakhmetov et. al. have also developed a preliminary ME-QCT-VRT model for coupled vibrational-rotational-translational (VRT) energy exchanges, the evaluation of this model is a subject for future work. These models do not require creation of massive state-to-state tables and can be naturally implemented into DSMC codes. The ME-QCT-VT model is constructed such that microscopic reversibility and detailed balance, gEt σV Tv0 (Ec , v → v 0 ) = g 0 Et0 σV Tv (Ec , v 0 → v),

(4.2)

are satisfied. Here, g is the stage degeneracy, Et is the translational energy involved in the collision, and σV Tv0 (Ec , v → v 0 ) is the cross-section for the v to v 0 transition. By satisfying Eq. 4.2, the relaxation model will automatically achieve the correct equilibrium distribution [130]. In the ME-QCT-VT model, the degeneracy stays constant since only v → v 0 transitions are considered. For diatom-atom collisions, the ME-QCT-VT cross-sections are given by: 

σV Tv0 (Ec , v → v 0 ) = σref Etν

∗ −1

Ev0 Ec

1− P  v0 1 −

ν ∗

Ev0 Ec

0 ν ∗ exp [Sv (Ec , v, v )] ,

(4.3)

where the collision energy available for VT exchanges, Ec , includes only the translational and vibrational contributions, Ec = Et + Ev = Et0 + Ev0 . Here, σref and ν ∗ represent the reference cross-section and reference exponent, respectively. Note that Eq. 4.3 resembles the LB distribution with the additional surprisal function Sv , which for O2 -O collisions is given by,   Ev − Ev0 D Sv (Ec , v, v ) = A|Ev − Ev0 | + B exp − C     Ev 0 Ev + E exp −F −1 − 1 , (4.4) Ed Ed 0

75 where Ed is the dissociation energy and coefficients A through F are fit to comply with the QCT calculations. Specifically, coefficients A, B, and C are taken as linear functions of collision energy, i.e., A = A1 +A2 Ec , B = B1 +B2 Ec , and C = C1 +C2 Ec . These parameters were obtained by minimizing the model error at the following initial states: vibrational levels v = [0, 1, 5, 7, 10, 15, 20, 25, 30], rotational levels J = [11, 51, 101, 151], and collision relative velocities cr = [1, 3, 5, 7, 9, 11] km/s, which correspond to translational energies Et = [0.0553, 0.4975, 1.3819, 2.7085, 4.4774, 6.6884] eV. For collisions involving intermediate initial states, the model coefficients can be obtained by interpolation. In the 2,500-20,000 K temperature range, this 11parameter model reproduces the QCT-calculated state-specific relaxation rates within 30% at the optimized vibrational levels. Although the model was not optimized for vibrational levels 31 through 47, the populations of these levels are not expected to be significant unless there is a strong population inversion. In addition, previous master equation calculations based on the ME-QCT-VT model and a complete set of QCT-based state-to-state rates were in good agreement for temperatures up to 10,000 K [104, 116]. Such a comparison can be found in Sec. 5. Likewise, the surprisal function for N2 -O system is given by   Ev − Ev0 D Sv (Ec , v, v ) = A|Ev − Ev0 | + B exp − C F  1 (El − Ev ) (El − Ev0 ) , (4.5) +E 1eV 2 0

where El is the minimum between Ec and the N2 dissociation energy Ed , i.e., El = min(Ec , Ed ). The model coefficients were obtained by considering the following set of initial states: v = [0, 1, 5, 7, 10, 15, 20, 25, 30], rotational levels J = [0, 20, 50, 100, 150], and collision relative velocities cr = [1, 2, 3, 5, 7, 9, 11, 13] km/s, which correspond to translational energies Et = [0.0528, 0.2111, 0.4749, 1.3191, 2.5854, 4.2738, 6.3844, 8.917] eV. Preliminary values for A − F coefficients are available in the literature [28] but readers are encouraged to contact the authors for the newly optimized values used in this dissertation.

76 4.2.2

SSD and SSE Models

Similarly to the ME-QCT-VT model, and based on the same set of QCT calculations, the state-specific dissociation (SSD) and state-specific exchange (SSE) models assume the following functional form for the reaction cross-sections in O2 -O and N2 -O collisions:

σR (v, J, Ec ) =

   0     AR

Ec E∗

α1  α E∗ 2 1− Ec

for Ec ≤ E ∗

(4.6)

for Ec > E ∗ ,

where now the total collision energy includes both translational and ro-vibrational energy contributions, Ec = Et + Erv . In SSD model, the effective activation energy E ∗ is constant and equal to the dissociation energy Ed . In SSE model, however, it is taken as an additional state-specific parameter E ∗ = Ea (v, J). Note this σR functional form is somewhat based on the TCE model, which was modified to comply with the QCT observations. For O2 -O collisions, the SSD model parameters AR (v, J), α1 (v, J) and α2 (v, J) are fit for the following set of initial states: v = [0, 10, 20, 30, 40], J =[11, 51, 101, 151, 201, 231], and cR = [1, 3, 5, 7, 9, 11, 13, 17, 19] km/s. Note, for instance, that although v > 40 levels are not considered in the fitting procedure, such high levels have low equilibrium populations and therefore do not contribute significantly to equilibrium dissociation rates. For N2 -O dissociation, the model parameters are fit to translational energies ranging between 0.1 and 23 eV, v = [0, 1, 5, 7, 10, 15, 20, 25, 30, 50, 55], and J = [0, 20, 50, 100, 150]. In the SSE model, AR (v, J), α1 (v, J) and Ea (v, J) are fit to the same range of translational energies and initial rotational levels but v = [0, 1, 5, 7, 10, 20, 25, 30, 50]. In this model, the α2 parameter is calculated as  α2 (v, J) = α1 (v, J) 1 −

 Ed . Ea (v, J)

(4.7)

The optimized parameters and calibration details for both SSD and SSE models can be found in the literature for O2 -O [27, 104] and N2 -O [28, 105] collisions. These

77 3-parameter models reproduce the state-specific reaction rate coefficients calculated directly by QCT within 25% for temperatures above 5000 K. State-specific reaction rate coefficients can be calculated by integrating the dissociation cross-sections given by Eq. 4.6 over the translational energy distribution function: s kd (v, J, TT ) =

8kB TT πµ

Z 0

∞

  Et Et dEt σR (v, J, Et ) exp − , kB TT kB TT kB TT

(4.8)

where TT is the translational temperature and µ is the diatom-atom reduced mass. In the same manner, the nonequilibrium reaction rate coefficients are calculated by averaging Eq. 4.8 over the corresponding rotational and vibrational Boltzmann distribution functions: kd (TT , TV , TR ) =     vX max Jmax X(v)  Ev (v) Erv (v, J) − Ev (v) kd (v, J, T )(2J + 1)gN exp − exp − k T kB TR B V v=0 J=0 (4.9)     vX max Jmax X(v)  Ev (v) Erv (v, J) − Ev (v) (2J + 1)gN exp − exp − kB TV kB TR v=0 J=0 In this expression, TR and TV are the rotational and vibrational temperatures, and gN is the nuclear degeneracy, which for O2 is 1 for odd rotational levels and 0 for even rotational levels. The summations in Eq. 4.9 are up to the maximum rotational Jmax and vibrational vmax levels. Equilibrium dissociation rate coefficients are obtained when all three temperatures are the same, i.e., T = TT = TR = TV .

4.2.3

DSMC Implementation of State-Specific Models

Compared to standard implementations of LB and TCE models into DSMC [4], there is only one main difference in the current implementation of the ME-QCT-VT and SSD models. Instead of assuming that the total collision cross-section σT is equal to the scattering cross-section σV HS , we define   X X σT = max σV HS , σ Ri , σV Tv0 ,

(4.10)

78 where the total reaction cross-section

P

σRi is a summation over all possible reaction P pathways i, and the total VT relaxation cross-section σV Tv0 is a summation over all possible post-collision vibrational levels v 0 . Thus, the probability of undergoing a specific VT internal energy relaxation or reaction pathway is given by σV Tv0 /σT and σRi /σT , respectively. The VHS cross-sections are based on a phenomenological formulation that reproduces the viscosity temperature-dependence, µ ∝ T ω , of nonreacting gases [4]. This is accomplished by assuming that the scattering cross-section varies according to σV HS ∝ Et

1/2−ω

, where 0.5 < ω < 1. The VHS model, however, does not take

into account inelastic processes such as internal energy exchanges and reactions. As a result, at high temperatures, reaction cross-sections may become larger than the VHS cross-sections such that the standard DSMC definition, σT = σV HS , leads to reaction probabilities above unity [131]. As addressed in previous DSMC efforts [132–134], Eq. 4.10 fixes this physical inconsistency. In spite of this modification, the number of candidate collision pairs per cell and time step, Ncand , is still calculated according to the NTC scheme. It means that in each DSMC collision cell, Ncand remains proportional to the maximum value of total cross-section times the relative velocity (σT cr )max . This ensures that the collision frequency calculation is consistent with the present σT definition. Figure 4.3 summarizes the collision procedure employed for each one of the randomly chosen Ncand collision pairs. In this figure, R1 , R2 , R3 , and so forth, represent different random numbers uniformly distributed between 0 and 1. Note that while only the O2 dissociation is being considered here, in the general case, the post-reaction states also need to be specified. Since the LB model is still used for rotational relaxation, RT energy exchanges occur only if 1/ZR > R4 . The current VT and RT energy exchange approaches could be completely replaced by the ME-QCT-VRT model proposed by Kulakhmetov et al. [116], however, its implementation will be addressed in a future work. As the final step, the post-collision velocities are calculated according to the VHS/VSS models and the available collision translational energy.

79

Figure 4.3.: DSMC collision procedure for the ME-QCT-VT and SSD models.

It is also worthwhile to remember that although the O2 -O QCT calculations were based on a discrete and coupled ro-vibrational model [135], the present DSMC solver assumes a continuous energy spectrum for the rotational modes. Therefore, for simplicity, the discrete rotational levels required in the SSD model are calculated assuming a rigid rotator model, $

1 J= − + 2

r

% 1 Er + . 4 kB θr

(4.11)

Here, Er is the rotational energy, θr is the corresponding characteristic rotational temperature, and b.c denotes truncation.

80

81

5. COMPARING AB-INITIO AND PHENOMENOLOGICAL MODELS In this chapter we compare the DSMC results obtained with the QCT-based and the phenomenological TCE-LB models. In doing so, we first describe how the vibrational relaxation number Zv can be calibrated from DSMC simulations using the ME-QCTVT model. The model comparison test cases include 0-D thermal relaxations under isothermal (constant ro-translational temperature and variable vibrational temperature) conditions and 1-D nonreacting shockwaves. Later, we perform 0-D simulations to verify that DSMC implementation of the new reaction models matches the QCT-predicted equilibrium and nonequilibrium rates. These equilibrium rates are then used to calibrated the TCE model parameters. Finally, 0-D thermochemical relaxation and 1-D reacting shockwave problems are used to evaluate the differences between the models. Considering our current DSMC calculations concentrate on two similar diatom-atom systems, O2 -O and N2 -O, for clarity, their general aspects are first presented for O2 -O and further details for N2 -O are provided only when necessary. In order to consistently compare O2 -O collision models under different conditions, all the subsequent simulations consider a hypothetical O2 -O mixture in which the O2 dissociation and VT relaxation is prohibited via O2 -O2 collisions. In DSMC, this is achieved by setting Zv equal to 1012 for O2 -O2 collisions and Zr equal to unity for any O2 collision. This ensures translational and rotational temperatures remain in equilibrium (TRT = TT = TR ) and thermal relaxation can occur only via O2 -O VT exchanges. Unless stated otherwise, all the investigated cases consider mixture number densities of n = 1021 molec/m3 . In next chapter we focus on the application of the QCT-based models to real pure oxygen shockwave flows [62]. However, to

82 properly accomplish this task, the new models still need to calibrated for O2 -O2 collisions.

5.1

DSMC Numerical Models All the DSMC implementations and results presented in this chapter are based on

a highly modified DS1V code [44]. The collision frequency and elastic collisions are modeled according to the no-time-counter (NTC) scheme and variable hard sphere (VHS) model [4]. Table 5.1 shows the collision-specific VHS parameters used in the O2 -O simulations. These values were fit by Wysong el at. [136] to comply with viscosity predictions given by collision integrals within the 2,000-10,000 K range [137]. Likewise, Tables 5.2 lists the VHS parameters used in the N2 -O simulations. Note that for collisions involving different species, e.g. N2 +O, these parameters are calculated as the simple averages of the corresponding N2 and O values.

Table 5.1.: Collision-specific O2 -O system VHS model kinetic parameters for Tref = 273 K [136]. Collision pair

O-O O2 -O O2 -O2

Reference diameter, dref (˚ A) 3.458 3.442 3.985 Viscosity index, ω (-)

0.76

0.75

0.71

Table 5.2.: Air-system VHS model kinetic parameters for Tref = 273 K. Species

N2

O2

NO

N

O

Ar

Reference diameter, dref (˚ A) 4.17 3.98 4.20 3.00 3.46 4.17 Viscosity index, ω (-)

0.74 0.71 0.79 0.80 0.76 0.81

83 Regarding the O2 , N2 , and NO internal modes, continuous rotational and discrete vibrational energy spectra are considered. In order to be consistent with the QCT calculations used to calibrate the ME-QCT-VT and SSD models, the O2 vibrational energy levels, v, are those tabulated by Esposito et al. [138]. Similarly, the N2 and NO rotational energies are described by anharmonic vibrational levels that are consistent with the QCT data used to fit the state-specific models [28, 105, 139]. Unless stated otherwise, the vibrational temperature TV is numerically calculated based on the Boltzmann definition:   Ev (v) E (v) exp − v v kB TV   , P Ev (v) − exp v kB TV

P Ev (TV ) =

(5.1)

where Ev is the DSMC cell-wise average vibrational energy, Ev (v) is the vibrational energy of level v, summations are taken over all possible v values, and kB is the Boltzmann constant. The rotational-translational (RT) internal energy relaxation is described by the standard LB model and a particle selection methodology [49]. In this case, temperature dependent rotational relaxation numbers ZRC based on Parker’s expression [64] are used. To account for the inconsistency between the continuum and DSMC definitions of the relaxation collision numbers, the latter values are corrected according to ZRD /ZRC = ζT /(ζT +ζR ) [49] and the ZVD /ZVC correction by Gimelshein et al. [48]. Here, ζT and ζR stand for the number of translational and rotational degrees of freedom of the colliding particles. For simplicity, the superscript D used to denote the DSMC definitions of Z is omitted in the next sections.

5.2

O2 -O Vibrational Relaxation Time In the continuum framework, the vibrational energy relaxation is typically de-

scribed by the Landau-Teller (LT) equation [2, 140] dEv (t) E eq (t) − Ev (t) = v , dt τv

(5.2)

84 where Ev is the system vibrational energy, Eveq is the corresponding energy at equilibrium, t is time, and τv is the vibrational relaxation time. For a system in an isothermal heat bath, i.e., constant translational temperature TT , and assuming that τv is only a function of TT , LT equation predicts that Ev evolves exponentially in time. Under these conditions, solution of Eq. 5.2 can be written as   Eveq (t, TT ) − Ev (t) (t − t0 ) = exp − ≡ Λ(t). Eveq (t, TT ) − Ev (t0 ) τv

(5.3)

According to the e-folding approach employed by Park [141], τv can be defined as the time at which the vibrational energy difference falls to 1/e of its initial value, i.e., Λ(τv ) = 1/e. In order to calculate the O2 -O vibrational relaxation time τv,O2 −O as a function of TT , we performed a series of DSMC 0-D relaxation simulations using the ME-QCTVT model. In each of these simulations, shown in Fig. 5.1, the the ro-translational temperature TRT is kept constant and the initial vibrational temperature TV is set to 500 K. The only special case is for TT equal to 200 K, in which the initial TV is set to 1000 K to circumvent fluctuations in the TV calculation for very small Ev values. In all the cases, 50% O2 and 50% O mixtures are considered. In Fig. 5.3, the O2 -O vibrational relaxation times τv obtained from 0-D DSMC isothermal relaxation calculations based on the ME-QCT-VT model are compared to previous results available in the literature. In the first plot, the results are presented in terms of τv times the O2 partial pressure pO2 versus the ro-translational temperature TRT = TT = TR . As one can see, there is a remarkable agreement between the present DSMC and the master equation results by Kulakhmetov et al. [104] and Andrienko and Boyd [142]. The explanation for it is that all these studies rely on the same O2 -O PES [125] and e-folding [141] definition of τv . These results also follow the same trends observed in the QCT calculations by Esposito and Capitelli [143] and in the theoretical results – based on the statistical adiabatic channel model – by Quack and Troe [144]. In contrast to these calculations, the experimental-based MillikanWhite (MW) and Landau-Teller (LT) fits by Park [145] and Ibraguimova et al. [62],

Temperature (K) 80 100 120 140 160 180 200 00 00 00 00 00 00 00 00

85

20

00

40

00

60

TRT TV

0

10-7

10-6

10-5 10-4 Time (s)

10-3

10-2

Figure 5.1.: 0-D DSMC isothermal vibrational relaxations using the ME-QCT-VT

Λ (-)

0.6

τv

0.5 0.4

1/e

8. 0E -0 6 6

7. 0E -0

600 K 1000 K

6

0.7

TRT = 200 K

6. 0E -0

0.8

τc, O2-O (s)

TRT= 200 K 600 K 1000 K 1500 K 2000 K 4000 K 6000 K 8000 K 10000 K 15000 K 20000 K

0.9

5. 0E -0

1.0

6

model for O2 -O system.

1500 K

6

2000 K

4. 0E

-0

0.3

4000 K

0.2 6

6000 K

.0005

.001 Time (s)

.0015

6

3. 0E -0

0

10

2. 0E -0

0.1 0.0

10000 K

-7

10

-6

10

-5

10 Time (s)

8000 K

15000 K

-4

10

20000 K -3

10-2

Figure 5.2.: Vibrational relaxation time based on decay of the normalized energy difference (left) and corresponding O2 -O mean collision time τc (right).

  −1/3 respectively, obey the pτv ∝ exp TT dependence. Around 1,500 K, except by the LT fit, all the compared approaches agree well with the experimental measurements

86

10

-6

Theory (Quack,1975) QCT (Esposito,2007) Master Eq: QCT Rates (Andrienko,2015) Master Eq: QCT Rates (Kulakhmetov,2016) DSMC: ME-QCT-VT Model (Present)

-7

C Zv,O2-O (-)

pO2 τv,O2-O (atm-s)

10

Experimental (Breen,1973)

10

-8

0

2000

4000

6000 TRT (K)

10

4

10

3

O2-O Collisions 10

2

Millikan-White Fit (Park,1993)

10

1

8000

2000

Landau-Teller Fit (Ibraguimova,2013)

1000

0

MW (Milikan-White,1963) MW-P (Park,1993) Quasi-Classical SS (Gimelshein,1998) Landau-Teller (Ibraguimova,2013) ME-QCT-VT (Present)

4000

6000 TRT (K)

8000

1000

0

Figure 5.3.: Vibrational relaxation times for O2 -O collisions: results in terms of pτ (left) and the relaxation collision number ZVC (right).

by Breen et al. [146]. However, experimental data covering an wider temperature range are still needed to ultimately validate our τv (TT ) model predictions. Still focusing on Fig. 5.3, the second plot compares the vibrational relaxation numbers, ZVC = τv /τc , predicted by the ME-QCT-VT model to those compiled by Wysong et al [136]. In order to be consistent with Eq. 4.10, the present mean collision h i−1 time is calculated as τc = nO (σT cr )O2 −O and it is not based on VHS model as usual. For lower temperatures, the ZVC values based on the MW fits [55, 145] for pτv are about two orders of magnitude higher than the present results. While the LT-based [62] and ME-QCT-VT relaxation numbers agree well around 2,000 K, the former one predicts a much higher ZVC temperature dependence. Conversely, although the quasi-classical (QC) [147] and ME-QCT-VT results differ by a factor of approximately 6, both models predict that ZVC is essentially constant for TRT > 4, 000 K conditions.

87 5.2.1

0-D VT Relaxation

Figure 5.4 presents the 0-D isothermal VT relaxation via O2 -O collisions for a 50% O2 and 50% O mixture. In this case, TRT is kept constant at 10000 K and the initial TV is equal to 500 K. These plots reveal that using the previously described ZV calibration, both ME-QCT-VT and LB results lead to the same temperature and vibrational population evolutions. Furthermore, the corresponding master equation solutions by Kulakhmetov [27], using the complete set of state-to-state rates, are essentially indistinguishable from the DSMC results. Such a comparison serves as a validation of the present DSMC implementation of the ME-QCT-VT and LB models. As one can observe, under thermal nonequilibrium conditions, all these VT models predict that the evolution of the vibrational populations does not follow Boltzmann distributions. In order to evaluate the agreement between ME-QCT-VT and LB models for different TRT values, we carried out DSMC-LB simulations for all the TRT values showed in Fig. 5.1. For clarity, however, only two of these cases are shown in Fig. 5.5. The temperature profiles, normalized by the initial TRT − TV difference, indicate that ME-QCT-VT is slightly faster than the LB model at beginning of the relaxation. Surprisingly, the opposite trend is observed towards the end of the relaxation and both models achieve complete thermal equilibrium basically at the same time. The explanation for this behavior is still unclear for us and, in an attempt to answer it, further to compare the LB VT state-specific rates against ME-QCT-VT still need to be performed. The next investigated case is a vibrationally cold relaxation process in which TRT is equal to 4000 K and the initial TV is 250 K. These TRT and TV conditions resemble, respectively, typical downstream and upstream normal shockwave conditions observed in cruise missions at 40-km altitude. Similarly to the previous results, Fig. 5.6 shows that ME-QCT-VT and LB results are indistinguishable for TRT values below 10000 K. Keeping in mind that the QCT data used to calibrate these models do not cover TT

88 0

Lines: Boltzmann at Tv Gradients: Master Eq. with QCT Circles: DSMC with ME-QCT-VT Squares: DSMC with LB

00

0

10

0

Fraction (-)

t/τ = 1500

t/τ = 100

20

00

t/τ = 5

Temperature (K) 80 60 40 00 00 00

10

TRT TV - Master Equation TV - DSMC: ME-QCT TV - DSMC: LB

10

-8

10

-7

10

-6

-5

10 Time (s)

10

-4

10

-3

10

-2

(a) Temperature evolution.

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

t /τc = 5 t /τc = 100 t /τc = 1500

0

1

2 3 4 Vibrational Energy (eV)

5

6

(b) Vibrational population evolution.

Figure 5.4.: 0-D isothermal VT relaxation via O2 -O collisions; DSMC solutions with ME-QCT-VT and LB models, and master equation solution using the complete set of state-to-state rates [27].

1.0 ME-QCT-VT LB

0.9

1 - (TRT-TV) / (TRT-TV)0

0.8 0.7 0.6 TRT = 20000 K

0.5 0.4 0.3

TRT = 2000 K

0.2 0.1 0.0 10

-7

10

-6

10

-5

10 Time (s)

-4

10

-3

10

-2

Figure 5.5.: 0-D isothermal VT relaxation via O2 -O collisions for different TRT values; results are presented in a normalized form.

89

00

10

Fraction (-)

t /τc = 1000

t /τc = 50

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

Lines: Boltzmann at TV Circles: ME-QCT-VT Squares: LB t /τc = 0 t /τc = 1 t /τc = 50 t /τc = 1000

0

50

0

10

00

15

00

t /τc = 1

Temperature (K) 2 3 20 00 500 000

35

00

40

TRT TV - ME-QCT-VT TV - LB

0

10

-9

10

-8

10

-7

-6

10 10 Time (s)

-5

10

-4

10

-3

10

-2

0

(a) Temperature evolution.

1

2 3 4 Vibrational Energy (eV)

5

6

(b) Vibrational population evolution.

Figure 5.6.: 0-D isothermal VT relaxation via O2 -O collisions for a vibrationally cold mixture that resembles a Mach 8 normal shockwave.

t /τc = 3000

0

t /τc = 30

Fraction (-)

t /τc = 300

Temperature (K) 80 100 120 140 160 180 200 0 0 0 0 0 0 0

10

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

Lines: Boltzmann at TV Circles: ME-QCT-VT Squares: LB t /τc = 0 t /τc = 30 t /τc = 300 t /τc = 3000

0

20

0

40

0

60

TRT TV - ME-QCT-VT TV - LB

0

10

-8

10

-7

10

-6

-5

10 10 Time (s)

-4

10

-3

10

-2

10

-1

(a) Temperature evolution.

0

0.5

1 1.5 Vibrational Level

2

2.5

(b) Vibrational population evolution.

Figure 5.7.: 0-D isothermal VT relaxation via O2 -O collisions for a vibrationally hot mixture that resembles a nozzle expansion.

90 values below 500 K and that our Zv is extrapolated to lower temperatures, we also consider the isothermal relaxation of a vibrational hot mixture. A TRT equal to 200 K and an initial TV of 2000 K attempt to somewhat reproduce the temperature conditions in a nozzle expansion. The comparison given in Fig. 5.7 indicate a small and consistently difference between the two models. While we cannot ascertain the accuracy of these low TT relaxation rates, the results are still valuable to identify the ME-QCT-VT and LB features under different conditions.

5.2.2

1-D Nonreacting Shockwaves

In this section, 1-D nonreacting shockwaves with the following freestream conditions are investigated: gas composition of 50% O2 and 50% O, Mach number M1 = 8, overall equilibrium temperature T1 = 250 K, and n1 = 1021 molecules/m3 . The overall temperature T is defined as the weighted mean, T =

3TT + ζR TR + ζV TV . 3 + ζR + ζV

(5.4)

When there is, statistically, complete energy equipartition between translational and internal modes, T also represents the equilibrium temperature. In this effort, the Mach number is calculated using this temperature and the average number of degrees of freedom per molecule. The following numerical parameters are employed in the present shockwave simulations: 1000 sampling cells; 16 collision cells per sampling cell; and around 1.3×106 simulated molecules. Under steady state conditions, the mean collision separation δmcs to the mean free path λ local ratio becomes smaller than 0.15 everywhere in the domain. Similarly, the time step ∆t to the mean collision time τC local ratio is always smaller than 0.02. The stagnation streamline approach described in detailed by Bird [4,44] is used to generated a 1-D normal shockwave structure. With the aid of Fig. 5.8, this approach can be summarized as follows. Initially, the 1-D domain consists of a freestream and a solid specular wall boundaries at the minimum and maximum x-coordinates, respectively. In the initial time step t0 , the domain is populated with all particles

2. 3. 4. 5. 6. 7. 8. 0E 0E 0E 0E 0E 0E 0E +2 +2 +2 +2 +2 +2 +2 1 1 1 1 1 1 1

Freestream Mass Removal Region

M1, T1, n1 Steady State Shock Region

t2

t3

t4

t1

t0

0. 0E

+0 0

1. 0E

+2 1

3

n (molec/m )

91

-2.0

-1.5

-1.0 x (m)

-0.5

0.0

Figure 5.8.: Mach 8 shockwave stabilization using the stagnation streamline approach.

having the freestream conditions. In the next steps the flow then interacts with the solid wall and is reflected as a moving shockwave towards the inflow boundary. This situation is indicated as t1 . However, this moving shock is stronger than the desired one and we cannot easily sample unsteady flows in DSMC. Thus, based on the stagnation streamline technique, when the shock passes through the central xcoordinate we start to remove particles near the solid wall. The rate of mass removal is set to the match the freestream influx such that the number of particles in the domain eventually achieves steady state. In this process, the expansion wave generated in the mass removal region reaches the moving shock and after well damped oscillations the shock stabilizes. The corresponding steady state is indicate by t4 and the vertical lines within the (-1.1, -0.9) x-interval. All the subsequent DSMC results are obtained by sampling and averaging the data over this steady regime. The temperature profiles and respective vibrational populations predicted by MEQCT-VT and LB models are shown in Fig. 5.9. In Fig. 5.9(a), the vertical dash-dot

92

40

Temperature (K)

35 30 25 20

00 00

TT

TV

00 00

TT TR TV

0

00

50

Symbols: ME-QCT-VT Lines: LB

0

0 -1.10

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

Lines: Boltzmann at TV Circles: ME-QCT-VT Squares: LB

TR

00

0 15 10

10

00

Fraction (-)

45

-1.05

-1.00 x (m)

-0.95

-0.90

(a) Temperature distributions.

x = -1.10 m x = -1.05 m x = -1.02 m x = -0.90 m

0

1

2 3 Vibrational Energy (eV)

4

5

(b) Vibrational energy distributions.

Figure 5.9.: Temperature and vibrational population across a 1-D nonreacting shockwave with M1 =8.

lines indicate two out of the four x-stations we chose to sample and compare the vibrational populations. In agreement with the previous 0-D relaxation studies, both methods lead to very similar profiles for the investigated conditions. It suggests that once the phenomenological LB model is properly calibrated, it can accurately reproduce the results of state-specific models that rely on accurate ab-initio data. Figure 5.10(a) presents the complete 1-D shockwave structure. Since ME-QCTVT and LB profiles are almost indistinguishable, the latter ones are omitted for clarity. Some of these results are given in terms of Φ=

φ − φ1 , φ2 − φ1

(5.5)

where φ can be any of the macroscopic properties, and subscripts 1 and 2 stand for the corresponding values upstream and downstream the shock, respectively. In this plot, TTX and TTY are the translational temperatures based only on the xand y-components of the thermal molecular velocity. As one can see, across the present shockwave, nonequilibrium occurs even within the translational modes. In

93

Mach Number

7 6

1.4 1.2 1.0

5

0.8

4

0.6

3

1

Mach -1.05

-1.00 x (m)

-0.95

0

20

0

15

0

10

0

Dashed:Diffusion Velocity Solid: Mole Fraction

1.0 0.9 0.8

O2 - Diffusion

0.7 0.6

50

0.5

0

0.4

-5 0 -1 0

0

0.3

0.2

-1 5

0

0.2

0.0

-2 0

0

0.4

2

0 -1.10

1.6

Diffusion Velocity (m/s)

8

25

-0.90

(a) Shockwave structure in terms of normalized

O - Diffusion

50 -2-1.10

-1.05

-1.00 x (m)

LB Model -0.95

Mole Fraction (-)

1.8 TT TTx TTy p n

Normalized Property, Φ (-)

9

0.1 0.0 -0.90

(b) Diffusion velocities and mole fractions.

properties.

Figure 5.10.: 1-D nonreacting shockwave for a M1 =8 freestream.

Fig 5.10(b), it is also interesting to note the O2 -O separation that takes place in the shock region. This phenomenon has been previously reported in the literature [4, 148, 149]. According to Chapman and Cowling [5], the relative diffusion velocity between species 1 and 2 can be written as n2 C2 − C1 = D12 nn    1 2 n1 n2 (m2 − m1 ) DT ρ1 ρ2 n1 × ∇ + ∇(ln p) + ∇(ln T ) − (F1 − F2 ) . (5.6) n nρ D12 pρ Here D12 , m, ρ, DT , and F stand for diffusion coefficient, molecular mass, density, thermal diffusion coefficient, and any external force that acts only on a specific species. The first term in the square brackets describes the transport given by the well known Fick’s law. The species separation observed in Fig 5.10(b) is mainly assigned to the pressure diffusion given by the second term. The third term captures the thermal diffusion and the last one accounts for the effect of external fields. While DSMC naturally model all these process, only the first term is included in conventional Navier-Stokes solutions.

94 5.3

N2 -O Vibrational Relaxation Time Similarly to previous investigations, Fig. 5.11 presents a series of isothermal VT

relaxation cases for 50% N2 and 50% O hypothetical mixtures that can relax only via N2 -O collisions. The corresponding vibrational relaxation and mean collision times are shown in Fig. 5.12. Comparison of these values in terms of τv p and vibrational relaxation numbers are given in Fig. 5.13. Note that while there is no relaxation time measurements for high-temperature conditions, the τv (TT ) trend obtained wit ME-QCT-VT model deviate from available experiments as T decreases. As discussed by Luo [28], the possible explanation for such a problem is that the nature of low-temperature electronic transitions is not included in the present ME-QCT-VT calibration. As shown in the same plot, other model predictions available in literature poorly captures the experiments. For completeness, we suggest a curve fit that bridges high-temperature ME-QCT-VT and experimental results. Using VHS-based mean collision times, Eq. 5.7 presents the corresponding ZVC fit

Temperature (K) 10 12 14 16 8 00 000 000 000 000 000

ZVC (T ) = exp (22.3 − 1.917 log T ) .

(5.7)

20

00

40

00

60

TRT TV

0

10-8

10-7

10-6

10-5 Time (s)

10-4

10-3

10-2

Figure 5.11.: 0-D DSMC isothermal vibrational relaxations using the ME-QCT-VT model for N2 -O system.

95

7. 0E -0

6

1.0

6. 0E -0

0.1 0.0 -8 10

10-7

10-6

5. 0E -0 6

1500 K 2000 K

6

4000 K

6

τv

6000 K 10000 K

10-5 10-4 Time (s)

10-3

10-2

10-1

10-7

6

0.2

1/e

-0

0.3

τc, N2-O (s)

0.4

1000 K

4. 0E -0

0.5

600 K

3. 0E

200 K 600 K 1000 K 1500 K 2000 K 4000 K 6000 K 8000 K 10000 K 15000 K 20000 K

0.6

15000 K

2. 0E -0

0.7

6

TRT

1. 0E -0

0.8

Λ (-)

TRT = 200 K

6

0.9

10-6

10-5 10-4 Time (s)

8000 K 20000 K

10-3

10-2

Figure 5.12.: Vibrational relaxation time based on decay of the normalized energy

10

-1

10

-2

10

-3

10-4 10

-5

10

-6

10

-7

10

-8

10

Experiments (Breshears,1968) Experiments (Ekstrom,1973) QCT Cross-Sections (Luo, 2016) MEQCTVT Model (Luo, 2016)

Milikan-Whilte Fit (1961)

Suggested Fit (Present)

5

DSMC ME-QCT-VT Calculations Fitting Curve Suggested Fit

104 C Zv,N2-O (-)

p τv,N2-O (atm-s)

difference (left) and corresponding N2 -O mean collision time τc (right).

10

3

10

2

10

1

Theory (Fisher,1972)

DSMC: ME-QCT-VT Model (Present)

0

5000

0 1000 TRT (K)

1500

N2-O Collisions 0

2000

0

0

5000

0 1000 TRT (K)

1500

0

2000

0

Figure 5.13.: Vibrational relaxation times for N2 -O collisions: results in terms of pτ (left) and relaxation collision numbers ZVC (right).

While more accurate experimental data does not become available, this ZVC fit can serve as a reference expression for DSMC simulations involving N2 -O systems.

96 5.3.1

0-D VT Relaxation

Although Eq. 5.7 is believed to be more accurate than our current ME-QCT-VT relaxation rates, the present goal is to verify how our DSMC calculations compare with master equation results. Figures. 5.14 and 5.15 shows the normalized temperature evolutions for three isothermal relaxation cases. Once again, the same hypothetical flow conditions assumed for O2 -O are used here. In the first set of plots, we verify the DSMC implementation of ME-QCT-VT and calibrated-LB models agree well with master equations results based on the ME-QCT-VT cross-sections. In the second set of plots, third-party master equation results illustrate how the present ME-QCT-VT does not reproduce the actual set of QCT-based cross-section for TT < 10, 000 K. Note that in both cases, for brevity, only the vibrational distributions for TT = 10, 000 K are compared.

10

1.0 Master Eq: MEQCTVT DSMC: MEQCTVT DSMC: LB

0.9

0.7 0.6 0.5

Fraction (-)

(TV-TV,0) / (TRT-TV,0)

0.8

TRT = 15,000 K

0.4

TRT = 5,000 K TRT = 10,000 K

0.3

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

TRT = 10,000

t /τc ~ 1000

t /τc ~ 10

0.2 0.1

t /τc ~ 0 t /τc ~ 100

0.0 10

Lines: Boltzmann at TV Circles: ME-QCT-VT (DSMC) Squares: LB (DSMC)

-9

10

-8

10

-7

10

-6

-5

10 10 Time (s)

-4

10

-3

10

-2

(a) Temperature evolution.

Figure 5.14.:

10

-1

0

1

2 3 4 Vibrational Energy (eV)

5

6

(b) Vibrational population evolution.

0-D isothermal VT relaxation via N2 -O collisions; DSMC solutions

with ME-QCT-VT, calibrated LB, and third-party master equation solutions using the complete set of state-to-state [28].

97

10

1.0 Master Eq: QCT Rates Master Eq: MEQCTVT DSMC: MEQCTVT

0.9

0.7 0.6 0.5

TRT = 15,000 K

Fraction (-)

(TV-TV,0) / (TRT-TV,0)

0.8

TRT = 5,000 K

0.4 TRT = 10,000 K

0.3

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

TRT = 10,000

t /τc ~ 1000

t /τc ~ 10

0.2 0.1

Lines: Boltzmann at TV Gradients: QCT Rates (Master Eq) Crosses: MEQCTVT (Master Eq) Circles: MEQCTVT (DSMC)

t /τc ~ 0 t /τc ~ 100

0.0 10

-9

10

-8

10

-7

10

-6

-5

10 10 Time (s)

-4

10

-3

(a) Temperature evolution.

10

-2

10

-1

0

1

2 3 4 Vibrational Energy (eV)

5

6

(b) Vibrational population evolution.

Figure 5.15.: 0-D isothermal VT relaxation via N2 -O collisions; DSMC solutions with ME-QCT-VT and master equation solution using the complete set and ME-QCT-VT fit state-to-state rates [28].

5.4

O2 -O Dissociation Rates Having a clear quantitative picture of the ME-QCT-VT relaxation rates, we turn

our attention now to the O2 -O dissociation rate coefficients predicted by the statespecific dissociation (SSD) model, which is calibrated with QCT calculations. The SSD equilibrium rate coefficients are obtained by averaging the model cross-sections over Boltzmann distributions, as presented in Eq. 4.9. These equilibrium rate coefficients are compared to experimental data in Fig. 5.16. For the T < 11, 000 K range, the QCT calculations present a better overall agreement with the equilibrium rate coefficients given by Park [106]. While Baulch et al [150], Park [106], and Ibraguimova et al. [62] rate coefficients are based on experiments limited to T . 6, 000 K conditions, Ibraguimova et al. further provide rate coefficients for the 6, 000 − 11, 000 K range. In this plot, Tlow and Thigh indicate the approximate temperature ranges at which the experimental rate coefficients were obtained.

3

Reaction Rate Coefficient (m /s)

98 10

-15

10

-16

Tlow -range

Thigh -range

10-17 10

-18

10

-19

10

-20

10

-21

10-22 10

-23

O2 + O → 3O Baulch (1976): Tlow -range Park (1990): Tlow -range Ibraguimova (2013): Tlow -range Ibraguimova (2013): Thigh-range Kulakhmetov (2015): QCT Calc

-24

10 00 000 000 000 000 000 000 000 000 000 20 4 6 8 10 12 14 16 18 20

Temperature (K)

Figure 5.16.: Comparison of different O2 +O→3O equilibrium dissociation rate coefficients: QCT calculations versus experimental data.

A 0-D framework similar to that employed in the 0-D relaxation studies is used to calculate the reaction reaction rate coefficients predicted by the ab-initio and phenomenological TCE models. A single DSMC sampling/collision cell in which particles are not allowed to move is considered. Skipping the DSMC movement procedures saves computational time and is in agreement with the 0-D formulation, which intrinsically assumes a homogeneous mixture. For the equilibrium cases, the initial thermal state of the particles follow Boltzmann distributions appropriated to an unique temperature T = TT = TR = TV . Herein subscripts T , R, and V stand for translational, rotational, and vibrational modes, respectively. During the DSMC collision procedure, a cumulative counter Nr is incremented by one every time a reaction is accepted. Because here we are only interested on the rates of a specific reaction, there is no need to update the post-reaction states as in actual flow calculations. Thus, the structure of reacting particles remains unchanged. Such a strategy circumvents the changes in

99 the system temperature T and number densities of the reactant species. Under these conditions, the equilibrium reaction rate coefficient k0 (T ) be calculated as k0 (T ) =

Nr Wp , n1 n2 t˜s

(5.8)

where n1 and n2 are number densities of the reactant species, t˜s is the cumulative sampling time at which Nr is incremented, and Wp is the number of real particles represented by each simulated one. Likewise, for the nonequilibrium cases, the initial ro-translational and vibrational states follow Boltzmann distributions at TRT = TR = TT and TV values, respectively. While ZR is always set to unity in all simulations, in these nonequilibrium rate calculations, the VT energy exchanges are prohibited by using ZV = 1012 . These choices are used to keep a constant nonequilibrium degree during the calculation of the nonequilibrium rate coefficients k(TRT , TV ). In the first plot of Fig. 5.17, the QCT rate coefficients are compared to the corresponding SSD model, DSMC-TCE, and DSMC-SSD results. The last case, in particu-

10

-15

0.3

10

-16

0.2

10-17 10

-18

10

-19

10

-20

10

-21

10-22 10

-23

O2 + O → 3O

Kulakhmetov (2015): QCT Calc Kulakhmetov (2016): SSD Model DSMC: SSD Model DSMC: TCE Model

-24

10 00 000 000 000 000 000 000 000 000 000 20 4 6 8 10 12 14 16 18 20

Temperature (K)

Relative Error, k0 / k0,QCT -1 (-)

3

Reaction Rate Coefficient (m /s)

lar, relies on the procedures described in Sec. 4.2.3. In the second plot, the deviations

O2 + O → 3O

0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6

Kulakhmetov (2016): SSD Model DSMC: SSD Model DSMC: TCE Model

-0.7 00 000 000 000 000 000 000 000 000 000 20 4 6 8 10 12 14 16 18 20

Temperature (K)

Figure 5.17.: Comparison of different O2 +O→3O equilibrium dissociation rate coefficients: QCT calculations versus numerical models (left) and corresponding relative errors (right).

100 of model results to the QCT calculations are presented in terms of the relative error k0 /k0,QCT − 1, where k0 stands for the equilibrium reaction rate coefficient. Note that while SSD model predicts the QCT values within 25%, the DSMC-SSD approach consistently underpredicts the QCT results by approximately 30%. A possible explanation for this deviation is the fact that, in the current DSMC-SSD implementation, rotational levels are assumed to follow the rigid rotator model. Thereby, it is possible that we are erroneously using lower J values due to the absence of the ro-vibrational coupling energy. We plan to address this issue in a future work. In the DSMC-TCE cases, the QCT equilibrium rate coefficients are first fit into the modified Arrhenius form η

k0 (T ) = ΛT exp



−Ed kB T

 ,

(5.9)

and the corresponding Λ and η coefficients are further corrected to account for the discrete vibrational levels [46]. Moreover, the conventional TCE steric factor [4],     σR σV HS , is multiplied by to be consistent with Eq. 4.10. Using such an σV HS σT T CE

approach, the DSMC-TCE rate coefficients agree within 10% of the QCT values for the entire equilibrium temperature range. Similarly, Fig. 5.18 presents the O2 +O→3O nonequilibrium rate coefficients for three constant ro-translational temperatures TRT and varying vibrational temperatures TV . For comparison purposes, the Kuznetsov [151] and Macheret-Fridman (MF) [152] models are also calibrated to the SSD equilibrium rate coefficients and presented in the same figure. Both of these theoretical models are expressed in terms of the nonequilibrium coupling factor, Z(TT , TV ), such that the dissociation rate coefficient is given by k(TT , TV ) = k0 (T ) Z(TT , TV ).

(5.10)

For these models, the coupling factor can be rewritten in the general form [153]:    1 − exp(−θv /TV ) φ1 Ed 1 1 Z(TT , TV ) = (1 − L) exp − 1 − exp(−θv /TT ) kB TT T  V   φ1 Ed 1 1 + φ2 L exp − (5.11) kB TT Ta

101 where,  Ta = αTV + (1 − α)TT ,

L=

 p    9 π (1 − α) kB TT 1−η     64 Ed        5 (1 − α) k T B T   × 1+   2Ed  

α=

m1 m1 + m2

2 (5.12)

for molecule-atom collision (5.13)

    3/2−η   2 (1 − α) kB TT     π 2 α3/4 Ed    √   7 (1 − α) (1 + α) kB TT   for molecule-molecule collision  × 1+ 2Ed θv is the characteristic vibrational temperature of the dissociating species, and m1 and m2 stand for the mass of an atom in the dissociating molecule and colliding atom, respectively. In the Kuznetsov model, L = 0 and φ1 is approximately 0.7 for typical air species [134], which is the value employed in this work. In the MF model, 0 < L < 1 and φ1 = φ2 = 1. As one can see in Fig. 5.18, the DSMC-SSD nonequilibrium rate coefficients only slightly under predicts the QCT rate coefficients for TV < TT conditions. Even though the TCE model reproduces the QCT equilibrium rate coefficients within an error of 10%, as shown in Fig. 5.17, the present results reveal that it can deviate by more than one order of magnitude from the QCT values under nonequilibrium conditions. Apart from the fact that the dissociation rate coefficient approaches zero as TV → 0 in the Kuznetsov model, which is a nonrealistic feature also present in the well-known two-temperature model by Park [106], both the Kuznetsov and MF models capture the qualitative behavior predicted by the QCT data. Given the lack of experimental data for nonequilibrium conditions, the QCT-based results can be used to calibrate theoretical models. Fig. 5.19 illustrates how the aforementioned Kuznetsov and MF models can be empirically modified to comply with the O2 +O→3O QCT data under TV > TT conditions. In this preliminary example, the φ1 parameter in Eq. 5.11 was changed to be 0.78 in both models. Herein, these models are referred to as QCT-calibrated Kuznetsov and MF models. A more elabo-

-16

10

-17

10

-18

10

-19

10

-20

10

-21

10

-22

10

-23

10

-24

TRT = 5,000 K

3

10

-15

10

-16

10

-17

10

-18

10

-19

10

-20

10

-21

10

-22

10

-23

10

-24

0 2000

-15

10

-16

10

-17

10

-18

10

-19

10

-20

10

-21

10

-22

10

-23

10

-24

TRT = 10,000 K

QCT Calculation Kuznetsov Model Macheret-Fridman Model DSMC: SSD Model DSMC: TCE Model

0 0 5000 1000 1500 Vibrational Temperature (K)

0

2000

0

Equilibrium

0 0 5000 1000 1500 Vibrational Temperature (K)

Reaction Rate Coefficient (m /s)

0

QCT Calculation Kuznetsov Model Macheret-Fridman Model DSMC: SSD Model DSMC: TCE Model

10

Equilibrium

10

3

-15

Reaction Rate Coefficient (m /s)

10

Equilibrium

3

Reaction Rate Coefficient (m /s)

102

TRT = 20,000 K

0

QCT Calculation Kuznetsov Model Macheret-Fridman Model DSMC: SSD Model DSMC: TCE

0 0 5000 1000 1500 Vibrational Temperature (K)

2000

0

Figure 5.18.: Comparison of different O2 +O→3O nonequilibrium dissociation rate coefficients. Ro-translational temperatures TRT equal to 5,000 (left), 10,000 (right), and 20,000 K (bottom) are considered. Corresponding thermal equilibrium conditions (TRT = TV ) are denoted by the vertical lines.

rate calibration, for instance, involving also the L function in the MF model, could be done to match the model and QCT results within the entire range of temperatures. However, such a goal is beyond the scope of this work and it will be left for future study.

-16

10

-17

10

-18

10

-19

+ + + + + + + + + ++ + + + + T = 5,000 K + + RT

10

-20

10

-21

10

-22

10

-23

10

-24

0

+ + + + +

QCT Calculation Kuznetsov Model MF Model QCT-Calibrated Kuznetsov QCT-Calibrated MF

0 0 5000 1000 1500 Vibrational Temperature (K)

2000

0

10

-15

10

-16

10

-17

10

-18

10

-19

10

-20

10

-21

10

-22

10

-23

10

-24

+ + + + + + + + + ++ + ++ + + + + + Equilibrium

10

3

-15

Reaction Rate Coefficient (m /s)

10

Equilibrium

3

Reaction Rate Coefficient (m /s)

103

+ 0

TRT = 10,000 K

QCT Calculation Kuznetsov Model MF Model QCT-Calibrated Kuznetsov QCT-Calibrated MF

0 0 5000 1000 1500 Vibrational Temperature (K)

2000

0

Figure 5.19.: QCT-based calibration of Kuznetsov and Macheret-Fridman (MF) models for O2 +O→3O under TV > TT conditions. Ro-translational temperatures TRT equal to 5,000 (left), and 10,000 K (right) are considered. Corresponding thermal equilibrium conditions (TRT = TV ) are denoted by the vertical lines.

5.4.1

0-D Thermochemical Relaxation

Figure 5.20 presents 0-D adiabatic thermochemical relaxations via O2 -O collisions for three different initial temperature conditions. Specifically, the O2 -O mixtures are initiated at thermal equilibrium, vibrationally cold, and vibrationally hot conditions. For each case, the TRT and TV temperature evolutions along with the corresponding vibrational populations at different instants are shown for both the phenomenological, indicated as LB+TCE, and SS-QCT models. An initial mixture composition of 99% O2 and 1% O is considered for all these cases. In order to extract the vibrational energy populations with high time resolution and low statistical noise, the initial number of simulated particles and maximum value of the time step to the mean collision time ratio, ∆t/τc , are chosen to be 105 and 0.001, respectively. In the first case, top plots Fig. 5.20, the SSD model predicts TRT > TV in the beginning of the dissociation process. The opposite trend is observed for the TCE

104 Temperature (K) 9 10 8 7 6 5 4 3 2 00 000 000 000 000 000 000 000 000 000

0

Squares: LB+TCE Circles: SS-QCT

-1

SS-QCT

log ( Population Fraction )

t = 1×10-1 s

-5

t = 4×10 s

( t /τC ~ 30)

-2

LB+TCE

0

10

TRT TV

10

-6

10

-5

-3 -4 -5 t = 0s

-6 -7 -8 t = 0.1 s

-9

t = 4.0e-005 s

-10 -11

10

-4

10 Time (s)

-3

10

-2

10

-12

-1

0

1

2 3 4 Vibrational Energy (eV)

5

6

Temperature (K) 9 10 8 7 6 5 4 3 2 00 000 000 000 000 000 000 000 000 000

0

-2 log ( Population Fraction )

t = 1×10-2 s

-5

t = 3×10 s ( t /τC ~ 20)

SS-QCT

LB+TCE

TRT TV

10

-5 -6 -7

t = 0.01 s

-8 -9 t = 3.0e-005 s t = 0s

-4

10 Time (s)

10

-3

10

-3

Temperature (K) 9 10 8 7 6 5 4 3 2 00 000 000 000 000 000 000 000 000 000

10

10

10 Time (s)

-3

10

-2

10

4

Squares: LB+TCE Circles: SS-QCT

-3 -4 -5 -6

t = 0s

-7 -8 -9 -10

-4

2 3 Vibrational Energy (eV)

-2

t = 0.1 s

-11

LB+TCE -5

1

0

TV

-6

0

-1

SS-QCT

TRT

-12

-2

log ( Population Fraction )

-5

t = 7×10 s ( t /τC ~ 3000)

10

-2

-6

t = 1×10 s

0

-4

-10

10 0

-3

-11

10

10

Squares: LB+TCE Circles: SS-QCT

-1

-1

-12

t = 7.0e-003 s 0

1

2 3 4 Vibrational Energy (eV)

5

6

Figure 5.20.: 0-D adiabatic thermochemical relaxation via O2 -O collisions for mixtures initiated with TV = TRT (top), TV < TRT (middle), and TV > TRT (bottom).

105 results. The possible explanation for this behavior relies on the fact that the SSD model captures the vibrational favoring during the dissociation, as showed by the corresponding depopulation of vibrationally excited states (blue circles) at 4×10−5 s. On the other hand, in TCE model only the total energy is taken into account but not how it is distributed among the internal modes. As TRT approaches ∼5,000 K, the VT relaxation becomes faster than dissociation and the system reaches thermal equilibrium. In the second case, middle plots of Fig. 5.20, the SS-QCT model predicts a slower thermal relaxation for the the vibrationally cold mixture when compared to the LB+TCE solution. Keeping in mind that for nonreacting conditions both MEQCT-VT and LB models reproduce the same relaxation rate [104, 122], this behavior occurs because TCE model overpredicts the dissociation rates under TRT < TV conditions, as shown in Fig. 5.18. The same explanation also applies to the vibrationally hot thermochemical relaxation cases , bottom plots of Fig. 5.20, in which the SS-QCT relaxation is faster. Comparing the computational cost of the SS-QCT and standard TCE+LB frameworks, for the cases with TRT = TV , TRT > TV , and TRT < TV initial conditions, the new ab-initio models resulted in total wall-clock solution times that are, respectively, about 4x, 8x, and 2x longer than the corresponding TCE+LB ones. While the current SS-QCT solutions are considerably slower, note that these comparisons are not considering only the collision procedures but also the time required to sample, average, and write the output files. Since these output results are written at intervals of no more than τc /10, in order to obtain accurate time resolved vibrational populations, a significant portion of the computational time is spent to write the files instead of actually calculating the collision processes. Therefore, the aforementioned total solution time comparisons should serve only as preliminary estimates. Future efforts will focus on a rigorous evaluation of the computational cost of the SS-QCT models in terms of the average CPU time per simulated collision as usually encountered in the literature.

106 5.4.2

1-D Reacting Shockwaves

Comparisons of SS-QCT and LB+TCE approaches in modeling normal shockwaves are presented in Figures 5.21 and 5.22 for different freestream Mach numbers M1 . A 50% O2 and 50% O freestream at equilibrium temperature of 250 K is considered in all cases. The 1-D shockwave structure is generated according to Bird’s stagnation streamline technique [44]. In these simulations, the following numerical parameters are used: 500 sampling cells; 16 collision cells per sampling cell; and around 1.5×106 simulated molecules. Under steady state conditions, the mean collision separation to the mean free path, δmcs /λ, and the time step to the mean collision time, ∆t/τc , local ratios are always smaller than 0.15 and 0.02, respectively. Figure 5.21 shows the temperature and O2 mole fraction distributions along with the corresponding vibrational populations for the M1 = 8 case. Note that for this particular freestream condition, the O2 molecules essentially does not dissociate but only diffuse away from the shock region due to the strong temperature and pressure gradients [4,148,149]. As a consequence, the SS-QCT and LB+TCE solutions lead to very similar results [104]. For M1 = 12 and M1 = 16, however, Fig. 5.22 reveals that dissociation occurs and the downstream O2 mole fractions are about 25% and 2%, respectively. Following the same trends observed in the 0-D test cases, these last two plots indicate that TCE model can underpredict both the peak and the downstream shockwave temperatures.

107

10

00

15

00

TT TV XO2

50

0

M1 = 8 -1.25

-1.20 x (m)

0.6 0.5 0.4 0.3 0.2

Lines: Boltzmann at TV Squares: LB+TCE Circles: SS-QCT

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

x = -1.30 m x = -1.23 m x = -1.20 m x = -1.10 m

0.1 0.0 -1.10

-1.15

Fraction (-)

XO2

0

0.9

0.7

TV

0

10

0.8

TT

-1.30

1.0

O2 Mole Fraction (-)

Temperature (K) 4 4 3 2 3 20 00 500 000 500 000 500

Symbols: SS-QCT Lines: LB+TCE

0

1

2 3 Vibrational Energy (eV)

4

5

Figure 5.21.: Temperatures (left) and vibrational populations (right) across a M1 =8

-0.7

-0.6

-0.5 -0.4 x (m)

-0.3

-0.2

00

0.0 -0.1

0 00 Temperature (K) 10 80 60 00 00 00 0

0.3

-0.5

O2 Mole Fraction (-)

14 12

0.1

0

M1 = 12

0.6

0.4

TV

00

30 20

00

0.2

0.7

0.5

0.2

40

0.3

XO2

0.9 0.8

TT TV XO2

XO2

00

0.6

Symbols: SS-QCT Lines: LB+TCE

TT

20

0.7

0.4

00 0

0.8

0.5

TV

10 -0.8

0.9

O2 Mole Fraction (-)

TT TV XO2

1.0

0

1.0

Symbols: SS-QCT Lines: LB+TCE

TT

00

Temperature (K) 8 9 7 6 5 40 00 000 000 000 000 000

normal shockwave.

-0.4

-0.3 x (m)

M1 = 16 -0.2

0.1 0.0 -0.1

Figure 5.22.: Temperatures and mole fraction distributions across normal shockwaves with M1 =12 and M1 =16.

108 5.5

N2 -O Dissociation and Exchange Rates The N2 +O→NO+N exchange reaction has been extensively investigated in the lit-

erature. The corresponding NO product is a primary radiator for ultraviolet emissions of reentry vehicles [154]. In the flow around low Earth orbit vehicles, the existence of atomic oxygen and high energy collisions produce ro-vibrationally excited NO, which lead to infrared emissions. Such emissions, for instance, are directly related to the space shuttle glow [155]. Laboratory measurements have confirmed that formed NO has highly nonequilibrium state distributions [156]. This reaction rate has been measured in the 2000–4000 K range by shock tube and premixed oxygen-propane flames [157–159]. Walch and Jaffe [160], Gamallo et al. [126], and Lin et al. [161] have independently conducted ab-initio PES calculations for the ground N2 O(3 A00 ) state and first excited N2 O(3 A0 ) state. The QCT, quantum mechanical scattering, and quantum wave-packet methods have been used based on the PESs to obtain high temperature reaction rates, state-specific rates, and cross-sections [139, 161–165]. Although it is expected that early dissociation of O2 in hypersonic flows makes the N2 +O→2N+O reaction important, in contrast to the aforementioned exchange reaction, there are only few studies on the N2 +O dissociation [166]. The widely used N2 +O→2N+O rates [106, 150] are based on experimental measurements of N2 +N or N2 +N2 dissociation. In next sections we evaluate the N2 +O state-specific dissociation (SSD) and exchange (SSE) reaction models proposed by Luo et al. [105]. This material was published in Ref. [124]. Once again, for consistency purposes, the TCE reaction rate coefficients are obtained by first fitting the QCT-based equilibrium rates to the modified Arrhenius form k0 (T ) = ΛT η exp (−Ed /kB T ), where Ed is replaced by Ea for exchange reactions. The equilibrium rate coefficients used in this section are listed in Table 5.3. Note that, in particular, the O2 +O→3O Arrhenius parameters by Park [106] were to refit to be consistent with O2 dissociation energy (Ed ) used in the QCT calculations. It is also important to mention that the Λ and η parameters that best reproduce

109

Table 5.3.: Arrhenius parameters for the reaction rate coefficients used in the present TCE calculations. Reaction N2 +O→2N+O

Λ (mol-cm-K-s)

η (-)

Eact /kB (K)

Data Source

8.9330×1016 −0.3842

113, 950

Ab-initio (Luo [28])

N2 +O→NO+O

2.4411×1011

0.7071

37, 850

Ab-initio (Luo [28])

O2 +O→3O

2.5000×1018 −0.5650

60, 490

Ab-initio (Kulakhmetov [116])

60, 490

Experimental (Park [106])

O2 +O2 →2O+O2 1.9337×1022 −1.7334

the QCT equilibrium rates are further corrected to account for the use of discrete vibrational levels in DSMC-TCE calculations [46].

5.5.1

Dissociation Rate Coefficients

Figure 5.23 presents the N2 +O→2N+O equilibrium dissociation rate coefficients obtained from different models. To the best of authors’ knowledge, there is no experimental measurements for such a reaction. Park et al. [167] and Baulch et al. [150] estimated the rate coefficients based on similar, but different, N2 dissociation reactions. Moreover, only a limited temperature range (Texp−range ) was considered in their experiments. This range in indicated by the vertical lines in Fig. 5.23. Luo et al. [28, 139] obtained direct QCT-calculated rates (QCT Calc) and rates from integration of the SSD modeled cross-sections (SSD Integr). These baseline results are also presented in the plot. It is found that the DSMC implementation of both SSD and calibrated TCE models reproduce the QCT-based results. The experimentalbased empirical estimations are at most one order of magnitude higher than the QCT results. DSMC calculations are not conducted for T < 4, 000 K due to the lower probability of N2 dissociation under these conditions.

3

Reaction Rate Coefficient (m /s)

110 10

-16

10

-18

10

-20

Texp -range

10-22 10

-24

10

-26

10

-28

10

-30

N2 + O → 2N + O Baulch (1973): Texp-range Park (2001): Texp-range Luo (2016): QCT Calc Luo (2016): SSD Integr DSMC: SSD Model DSMC: TCE Model

10-32 10

-34

0

5000

0 0 1000 1500 Temperature (K)

2000

0

Figure 5.23.: Comparison of different N2 +O→2N+O equilibrium dissociation rate coefficients: QCT calculations versus experimental data and numerical models.

Figure 5.24 presents the nonequilibrium reaction factor, Z(TT , TV ) = k(TT = TR , TV )/k0 (TT ), for TT =5,000, 10,000, and 20,000 K with TV ranging from 1,000 to 20,000 K. DSMC calculations with SSD model predict the same Z(TT , TV ) values as the integration of SSD modeled cross-sections. Except for vibrationally cold condition at TT =5,000 K, a good agreement of the results with QCT calculations is observed. While TCE model overpredicts reaction rates at vibrationally cold conditions, the opposite behavior happens at vibrationally hot conditions. It also predicts an unphysical bump of Z(TT , TV ) at vibrationally cold conditions. This occurs because present TCE reaction probabilities are proportional to

¯ Γ(ζ+5/2−ω) ¯ Γ(ζ+η+3/2)

¯

× (1 − Ed /Ec )ζ ,

where ζ¯ = (ζR + ζV )/2 is the DSMC cell-wise average number of internal degrees

of freedom involved in the collision. In this product, as TV drops, the first term increases and the second term decreases. Hence, even under TV < TT , the nonequilibrium rates will increase at some point. Predictions by two models frequently used in the CFD community, Park’s two temperature model [106] and Macheret-Fridman (MF) model [152], are also presented in the same plot. The MF nonequilibrium factor differs from QCT-based results by up to two orders of magnitude, although both

6

104 10

2

10

0

TRT = 5,000 K

10

-2

10

-4

10

-6

0 0 5000 1000 1500 Vibrational Temperature (K)

Nonequilibrium Factor, Z (TT,TV)

0

Luo (2016): QCT Calc Luo (2016): SSD Integr MF Model Park Model DSMC: SSD Model DSMC: TCE Model

10

8

10

6

104 10

2

10

0

10

-2

10

-4

10

-6

2000

0

8

10

6

104 10

2

10

0

TRT = 10,000 K 10

-2

10

-4

10

-6

0 0 5000 1000 1500 Vibrational Temperature (K)

0

2000

0

Luo (2016): QCT Calc Luo (2016): SSD Integr MF Model Park Model DSMC: SSD Model DSMC: TCE Model

TRT = 15,000 K

0

Luo (2016): QCT Calc Luo (2016): SSD Integr MF Model Park Model DSMC: SSD Model DSMC: TCE Model

Equilibrium

10

10

Equilibrium

8

Nonequilibrium Factor, Z (TT,TV)

10

Equilibrium

Nonequilibrium Factor, Z (TT,TV)

111

0 0 5000 1000 1500 Vibrational Temperature (K)

2000

0

Figure 5.24.: Comparison of different N2 +O→2N+O reaction nonequilibrium factors Z(TT , TV ) = k(TT , TV )/k0 (TT ). Ro-translational temperatures TRT equal to 5,000 (left), 10,000 (right), and 15,000 K (bottom) are considered. Corresponding thermal equilibrium conditions (TRT = TV ) are denoted by the vertical lines.

approaches follow the same trend. On the other hand, Park’s model completely fails at extreme vibrationally cold conditions.

112 5.5.2

Exchange Rate Coefficients

For the N2 +O→NO+O exchange reaction, experimental measurements exist for temperatures ranging from 2,000 to 3,850 K. The first plot of Fig. 5.25 presents a comparison of Monat et al.’s shock tube measurements [158] and Livesey et al.’s study of NO formation in premixed oxygen-propane flames [157] against several models [145,159] and QCT-based calculations [139,162]. The QCT calculations match the experimental data and predict rate coefficients one order of magnitude higher than Park’s extrapolated Arrhenius rates. DSMC calculations using the SSE model and calibrated TCE model also reproduce the QCT-based rates, as shown in the second plot of Fig. 5.25. The corresponding nonequilibrium reaction factors are compared in Fig. 5.26. The DSMC calculations match the QCT-based results well for the temperature range considered. The DSMC-SSE results slightly underpredict the Z factor for TT =

10

-16

10

-16

10

-17

10

-17

10

-18

10

-18

3

10-19 10

-20

10

-21

10

-22

10

-23

N2 + O → NO + N

Livesey (1971): Exp Monat (1979): Exp Davidson (1958): Model Park (1994): Model Bose (1996): QCT Calc Luo (2016): QCT Calc

10-24 10

Reaction Rate Coefficient (m /s)

3

Reaction Rate Coefficient (m /s)

5,000 K and vibrationally hot conditions. Similar to the dissociation reaction, the

-25

0

5000

0 0 1500 1000 Temperature (K)

2000

0

10-19 10

-20

10

-21

10

-22

10

-23

N2 + O → NO + N

Bose (1996): QCT Calc Luo (2016): QCT Calc Luo (2016): SSE Integr DSMC: SSE Model DSMC: TCE Model

10-24 10

-25

0

5000

0 0 1500 1000 Temperature (K)

2000

0

Figure 5.25.: Comparison of different N2 +O→NO+O equilibrium exchange rate coefficients: QCT calculations versus experimental data (left) and numerical models (right).

2

10

1

10

0

10

-1

10

-2

10

-3

0 0 5000 1000 1500 Vibrational Temperature (K)

Nonequilibrium Factor, Z (TT,TV)

0

Luo (2016): QCT Calc Luo (2016): SSE Integr Bose-Candler Model Macheret Model Park Model DSMC: SSE Model DSMC: TCE Model

10

3

10

2

10

1

10

0

10

-1

10

-2

10

-3

2000

0

10

3

10

2

10

1

10

0

10

-1

10

-2

10

-3

TRT = 10,000 K

0 0 5000 1000 1500 Vibrational Temperature (K)

0

2000

0

Luo (2016): QCT Calc Luo (2016): SSE Integr Bose-Candler Model Macheret Model Park Model DSMC: SSE Model DSMC: TCE Model

TRT = 15,000 K

0

Luo (2016): QCT Calc Luo (2016): SSE Integr Bose-Candler Model Macheret Model Park Model DSMC: SSE Model DSMC: TCE Model

Equilibrium

10

TRT = 5,000 K

Equilibrium

3

Nonequilibrium Factor, Z (TT,TV)

10

Equilibrium

Nonequilibrium Factor, Z (TT,TV)

113

0 0 5000 1000 1500 Vibrational Temperature (K)

2000

0

Figure 5.26.: Comparison of different N2 +O→NO+O reaction nonequilibrium factors Z(TT , TV ) = k(TT , TV )/k0 (TT ). Ro-translational temperatures TRT equal to 5,000 (left), 10,000 (right), and 15,000 K (bottom) are considered. Corresponding thermal equilibrium conditions (TRT = TV ) are denoted by the vertical lines.

DSMC-TCE results overpredict the Z factor at vibrationally cold conditions, which may have a great influence on the prediction of NO generation for hypersonic vehicles. Bose and Candler’s three-temperature nonequilibrium model [168], based on their QCT calculations, is also compared in Fig. 5.26. The difference to our results is within one order of magnitude. Once again, Park’s two-temperature model fails for

114 vibrationally cold conditions. Macheret’s model [169] gives better estimations for moderate degrees of vibrational nonequilibrium.

5.5.3

0-D Thermochemical Relaxation

Figure 5.27 presents temperature, mole fraction, and vibrational energy population evolutions for the 0-D thermochemical relaxation of an initial 99%N2 +1%O mixture that can react only via the N2 +O→2N+O reaction pathway. In order to be better interpret the SSD model features, internal energy exchanges can only occur via N2 -O collisions. In this case, two limiting constant ZV values are considered: 102 and 1012 . For both cases, the LB+SSD model predicts vibrationally cold conditions, TV < TRT , during the entire relaxation process, which is in contrast to the LB+TCE model. The reason is that the SSD model inherently captures the vibrational favoring of N2 +O dissociation, i.e., N2 molecules with higher vibrational energy are more likely to dissociate. Thus, TV drops faster than TRT for the LB+SSD model. For the LB+TCE model, N2 molecules with either high translational or vibrational energies may have the same probability of dissociation since TCE model only accounts for the total collision energy but not how it is distributed among the different internal modes. The population of N2 molecules with high vibrational energy, however, is quite small. Hence, most N2 molecules with high translational energy are dissociated first and TRT drops faster than TV . This behavior can be observed in Fig. 5.27, which shows more depopulation of vibrationally excited molecules at t = 4 ms. Comparing the two cases, it can be found that VT relaxation becomes faster than chemical reaction after a time point and the system is equilibrated. Similarly, Fig. 5.28 presents the 0-D thermochemical relaxation of an initial 75%N2 , 20%O, and 5%NO mixture that can react only via the N2 +O→NO+O exchange. Energy states of formed NO species are set with the conventional LB approach [51] discussed in Sec. 2.3. In general, the LB+TCE model predicts a larger generation vibrationally hot NO particles in comparison to the LB+SSE model. A possible ex-

0 00 20 10

-1

10

Mole Fraction, X (-)

Mole Fraction, X (-)

N2 Z V = 102

0.4

N 0.3 0.2

10

-4

-3

10 Time (s)

10

-2

10

10

-5

10

-6

10

-1

Lines: LB+TCE Symbols: LB+SSD

0.7

N2

0.6

Z V = 1012

0.5 0.4

N 0.3

-1

10 10

Squares: LB+TCE Circles: LB+SSD

-3

-1

t = 1×10 s

t = 1×10 s Z V = 10 0

1

2

2

3 4 5 6 7 8 N2 Vibrational Energy (eV)

9

10

Population Fraction (-)

Population Fraction (-)

-5

t = 1×10-5s

-4

-2

O

0.0

0

10

10

0.1

O

0.0

-3

-3

10 Time (s)

0.2

0.1

10

-4

0.8

0.5

-2

10

0.9

0.6

10

-5

1.0

Lines: LB+TCE Symbols: LB+SSD

0.7

10

-1 -3

00 12 -2

0

10

Z V = 1012

00

-3

10 Time (s)

0.8

-1

LB+TCE

10

-4

0.9

10

TV

0

-3

t = 1×10-1 s

t = 1×10 s ( t /τC ~ 1000)

0 00 12 0

10

10

00

-5

LB+TCE

1.0

10

t = 1×10 s

Temperature (K) 18 16 14 00 00 00 0 0 0

TV

Z V = 102

10

LB+SSD TRT

Temperature (K) 18 16 14 00 00 00 0 0 0

LB+SSD

TRT

t = 4×10 s ( t /τC ~ 4000)

20

00

0

115

-5

10

-4

-3

10 Time (s)

10

-2

10

-1

0

10

-1

10

-2

Squares: LB+TCE Circles: LB+SSD

t = 1×10-5s 10

-3

10

-4

10

-5

10

-6

t = 1×10-1s t = 4×10-3s Z V = 10 0

1

2

12

3 4 5 6 7 8 N2 Vibrational Energy (eV)

9

10

Figure 5.27.: 0-D thermochemical relaxation for a 99%N2 and 1%O initial mixture. Only the N2 +O→2N+O reaction pathway is considered. Two constant ZV values are considered.

116

0 00 22 0 00

-2

-5

t = 4×10 s ( t /τC ~ 35)

t = 1×10 s

20 Temperature (K) 18 16 00 00 0 0 0

14 -7

1.0

-6

-5

10 Time (s)

Z V = 102

10

-4

10

-3

00 12 10

-2

10

-8

10

1.0

Lines: LB+TCE Symbols: LB+SSE

0.9

-7

LB+TCE 10

-6

-5

10 Time (s)

Z V = 1012

Mole Fraction, X (-)

0.6 0.5 0.4 0.3

O 0.2

LB+SSE -4

10

-3

0.4 0.3

O 0.1

10

10

-2

10

-3

10

-7

10

-6

-5

10 Time (s)

10

-4

10

-3

10

-2

10

t = 1×10-8s t = 1×10 s

-4

10

-5

10

10

Squares: LB+TCE Circles: LB+SSE

-2

10

-5

Z V = 102

t = 4×10 s

-6

0

NO

0.0 -8

0

-1

-2

0.5

N

NO

Population Fraction (-)

Population Fraction (-)

10

10

0.6

0.2

0.0 10

-2

0.7

N 0.1

10

N2

0.8

0.7

10

Lines: LB+TCE Symbols: LB+SSE

0.9

N2

0.8

Z V = 1012

0 10

0

10

LB+TCE

00

-8

LB+SSE

10

0

12

00

0

Z V = 102

10

00

10

Mole Fraction, X (-)

TRT TV, N2 TV, NO

00

-2

t = 1×10 s

14

00

-5

t = 4×10 s ( t /τC ~ 35)

Temperature (K) 18 16 00 00 0 0 0

20

00

0

22

00

0

TRT TV, N2 TV, NO

1

2

3 4 5 6 7 8 N2 Vibrational Energy (eV)

9

10

10

-7

10

-6

-5

10 Time (s)

10

-4

10

-3

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-8

Squares: LB+TCE Circles: LB+SSE

-8

t = 1×10 s t = 4×10-5s

Z V = 1012

t = 1×10-2s

-6

0

1

2

3 4 5 6 7 8 N2 Vibrational Energy (eV)

9

10

Figure 5.28.: 0-D thermochemical relaxation for a 75%N2 , 20%O, and 5%NO initial mixture. Only the N2 +O→NO+O reaction pathway is considered. Two constant ZV values are considered. Post-reaction NO states are based on LB model.

117 planation is that TCE model overpredicts the reaction rates at vibrationally cold conditions.

5.5.4

1-D Reacting Shockwaves

In this section we to compare the temperature and mole fraction distributions across 1-D reacting shockwaves based on the 4-reaction mechanism listed on Table 5.3. TCE model results are compared against a set of high-fidelity reaction models that includes: SSD model for O2 +O→3O and N2 +O→2N+O, SSE model for N2 +O2 →NO+N, and TCE+MF approach for O2 +O2 →2O+O2 . The latter case corresponds to an empirically modified TCE approach that reproduces the nonequilibrium rates given by the Macheret-Fridman (MF). Further details on TCE+MF implementation are given in Sec. 6.1.3. In order to isolate the influence of reaction models from the VT and RT energy relaxations, the LB model with constant ZR = 1 and ZV = 100 is used in for all collision types. Figure 5.29 presents the normal shockwave profiles obtained with the stagnation streamline approach [4, 44]. Two Mach numbers, M1 = 16 and 24, are considered for a freestream mixture of 78%N2 , 20%O2 , and 1%Ar at n = 1021 molec./m3 and T = 250 K. As aforementioned, two different sets of reaction models are compared for each M1 case. In these plots, results based on the high-fidelity reaction framework are indicated by symbols whereas the TCE-based results are shown by lines. In general, the high-fidelity models predict higher peak and post-shock temperature values than TCE model. This behavior is assigned to the fact TCE model leads to higher reaction rate coefficients under TV < TT conditions, as demonstrated in previous sections. This explanation is corroborated by the mole fraction distributions. Although the present simulations employ a hypothetical 4-reaction mechanism, which neglects many other reactions pathways that are relevant for hypersonic conditions, they are helpful in providing a quantitative evaluation of the difference between the

118

00

0

M1 = 24

35

Temperature (K) 80 100 120 140 160 00 00 00 00 00

M1 = 16

TT

00 10

-0.5

-0.4

-0.3

-0.2

Symbols: SSD / SSE / TCE+MF Lines: TCE

0

0

20 -0.6

TV

00

Symbols: SSD / SSE / TCE+MF Lines: TCE

00

40

TV

50

00

0

60

00

Temperature (K) 20 25 15 00 00 00 0 0 0

30

00

0

TT

-0.1

-0.6

-0.5

-0.4

x (m)

-0.1

00

0

M1 = 24

Vibrational Temperature (K) 30 20 25 15 10 00 00 00 00 00 0 0 0 0 0

N2

NO

40

00

O2

-0.5

-0.4

-0.3

-0.2

N2 O2

50

00

Symbols: SSD / SSE / TCE+MF Lines: TCE

0

0

20

00

Symbols: SSD / SSE / TCE+MF Lines: TCE

NO

-0.1

-0.6

-0.5

-0.4

x (m) M1 = 16

0.35 0.30

0.40

Symbols: SSD / SSE / TCE+MF Lines: TCE

0.05

Ar

O

+++++++

+++++++

0.00 -0.6

0.25 0.20 0.15 0.10 0.05 0.00

-0.5

-0.4

-0.3

-0.2

M1 = 24

-0.1

N NO

0.30

NO

O2

0.20+ + + + +++ N + + 0.15 + + + + 0.10 ++ ++ ++

-0.2

0.35

Thermal Nonequilibrium Region

0.25

-0.3 x (m)

Mole Fraction, X (-)

Mole Fraction, X (-)

-0.2

35

Vibrational Temperature (K) 14 16 12 10 60 8 00 000 000 000 000 000

M1 = 16

-0.6

0.40

-0.3 x (m)

-0.1

Thermal Nonequilibrium Region

O2

+ + + + + ++ + + + + + + O Symbols: SSD / SSE / TCE+MF + Lines: TCE Ar + ++ + + + + + + + + + + + + + + + + + +

-0.6

-0.5

-0.4

x (m)

-0.3

-0.2

-0.1

x (m)

Figure 5.29.: Temperature and mole fraction distributions across normal air shockwaves with M1 = 16 (left) and M1 = 24 (right). Only Table 5.3 reaction patways are considered. Vertical lines indicate mixture TT 6= TV regions.

119 high-fidelity and TCE frameworks. However, further studies considering a complete reaction mechanism and realistic VT and RT relaxation rates are still necessary.

5.6

Summary Compact ME-QCT-VT, SSD, and SSE models for O2 -O and N2 -O systems are

compared to available experimental data and phenomenological TCE-LB framework, when the latter one is calibrated to the corresponding QCT-calculated rates. Different test cases are considered: (i) 0-D VT thermal relaxations for constant compositions; (ii) 0-D adiabatic thermochemical relaxations; and (iii) 1-D nonreacting and reacting normal shockwaves. It was found that when QCT data is used to consistently calibrate the vibrational relaxation numbers, the phenomenological LB model can accurately reproduce the state-specific ME-QCT-VT and master equation results for nonreacting conditions. On the other hand, even when the TCE model is calibrated with the QCT equilibrium rate coefficients, the TCE nonequilibrium rates can deviate from the SSD and SSE results by more than three orders of magnitude. The 0-D calculations indicate SSD model reproduce vibrationally cold thermochemical relaxation whereas the TCE model predicts the opposite behavior. Shockwave calculations suggest the investigated state-specific N2 -O exchange rates will play a major role under low temperature nonequilibrium conditions.

120

121

6. DSMC SIMULATION OF O2 SHOCKWAVES BASED ON HIGH-FIDELITY MODELS Previous studies with the ME-QCT-VT and SSD models only considered hypothetical O2 -O mixtures. In these mixtures, the O2 VT relaxation and dissociation via O2 -O2 collisions were prohibited and the translational and rotational modes were assumed to be always in equilibrium. In this context, the main goal of this chapter is to validate these new models by simulating the O2 shock tube experiments by Ibraguimova et al. [62] with DSMC. A rigorous and consistent validation would require the use of the state-specific models for all types of collisions. However, this is still not possible due to our current lack of O2 -O2 ab-initio data. Therefore, we use an empirically modified TCE model to describe O2 -O2 dissociations. This modification allows the TCE model to reproduce nonequilibrium rate coefficients predicted by the MacheretFridman (MF) model [152]. The MF model is calibrated to experimentally-measured equilibrium dissociation rate coefficients and its nonequilibrium trends are calibrated to O2 -O ab-initio data. The O2 -O2 VT relaxation is described by the LB model, in which the relaxation collision numbers are calibrated with experimental data [62]. The final shockwave DSMC results, based on these high-fidelity models, are compared not only to experiments but also to other DSMC and master equation solutions available in the literature. In Sec. 6.1 we address the experimental-based LB calibration and DSMC implementation of the Macheret-Fridman dissociation model for O2 -O2 collisions. DSMC pure O2 shockwave calculations are presented in Sec. 6.2, where two different freestream conditions are considered. Concluding remarks are given in Sec. 6.3.

122 6.1

O2 -O2 VT Relaxation and Dissociation Models

6.1.1

VT Relaxation

Due to our current lack of QCT data, the O2 VT relaxation via O2 -O2 collisions is described with the LB phenomenological model. In contrast to the O2 -O system discussed in Chapter 5, there is a reasonable agreement between the O2 -O2 VT relaxation rates available in the literature. Figure 6.1 shows a compilation of different ZVC (TT ) predictions [55, 62, 145, 147] done by Wysong et al. [136]. In this work, however, two sets of ZVC results based on the recent LT fit by Ibraguimova et al. are considered. While one case assumes a mean collision time based on σT = σV HS , the other one follows Eq. 4.10. Clearly, the latter approach agrees well with the quasiclassical state-specific (SS) results for temperatures above 8, 000 K but the difference between these two approaches increases for lower temperatures. Besides the fact that the shockwaves investigated in Sec. 6.2 involve VT relaxation processes that take place under TT & 6, 000 K conditions, for comparison purposes, all the collision models employed in present simulations (NTC, LB, ME-QCT-VT, TCE, and SSD) need

10

6

10

5

MW (Milikan-White,1963) MW-P (Park,1993) Quasi-Classical SS (Gimelshein,1998) Landau-Teller (Ibraguimova,2013) + τC, VHS Landau-Teller (Ibraguimova,2013) + τ

C Zv,O2-O2 (-)

C, Total

104

O2-O2 Collisions 10

3

10

2

1

10 2000

4000

6000 TRT (K)

8000

1000

0

Figure 6.1.: Vibrational relaxation collision numbers for O2 -O2 collisions.

123 to be consistent with the same definition of total cross-section. For these reasons, the ZVC values based on τc,T otal are employed to simulated the O3 shockwaves.

6.1.2

Equilibrium Dissociation Rates

Figure 6.2 shows different experimental-based O2 +O2 →2O+O2 equilibrium rate coefficients. Based on this plot, for T < 11, 000 K, the rate coefficients predicted by Park [106] provide a good overall agreement with the low- and high-temperature rate coefficients given by Ibraguimova et al. [62,170]. For this reason, the equilibrium rate coefficients by Park are chosen to simulate the shockwave problems investigated in this work. Furthermore, we show that the corresponding DSMC-TCE implementation of these rate coefficients accurately matches the analytical values within the entire

3

Reaction Rate Coefficient (m /s)

temperature range. 10

-15

10

-16

Tlow -range

Thigh -range

10-17 10

-18

10

-19

10

-20

10

-21

10-22 10

-23

O2 + O2 → 2O + O2

Ibraguimova (1999): Tlow -range Ibraguimova (2013): Thigh-range Park (1990): Tlow -range DSMC: TCE

-24

10 00 000 000 000 000 000 000 000 000 000 20 4 6 8 10 12 14 16 18 20

Temperature (K)

Figure 6.2.: Comparison of different O2 +O2 → 2O+O2 equilibrium dissociation rate coefficients.

124 6.1.3

TCE+MF DSMC Implementation

Having illustrated the possibility of using QCT data to calibrate Kuznetsov and MF models, the actual implementation of such models into DSMC is the last topic of this section. To the best of the authors’ knowledge, the only DSMC implementation of dissociation models given in terms of the nonequilibrium coupling factor Z(TT , TV ) was done by Bondar et al. [134, 171]. They numerically found a set of vibrationally specific reaction cross-sections that satisfies the Kuznetsov form of Eqs. 5.10 and 5.11 when the translational, rotational, and vibrational states follow the corresponding Boltzmann distributions at TT , TR , and TV , respectively. Note that in this so-called Kuznetsov state-specific (KSS) model, TRT and TV can be different. In contrast to the KSS method, in this work, a simpler approach that extends the DSMC-TCE framework is proposed to reproduce the nonequilibrium dissociation rate coefficients given by Z(TT , TV ). First, based on the DSMC cell-wise values of TT and TV , the coupling factor is explicitly computed for a given pair of colliding species. Second, the original pre-exponential factor in Arrhenius expression is multiplied by Z(TT , TV ), i.e., Λ∗ = Λ × Z(TT , TV ). In doing so, it is expected that the use of

Λ∗ (TT , TV ) instead of Λ in the Arrhenius expression would satisfy the nonequilibrium

rate coefficient described by Eq. 5.10. However, this is not valid within the TCE formulation, which is derived under the assumption that the pre-exponential factor is a constant. As a result, the analytical rate coefficients given by Eq. 5.10 cannot be obtained by simply replacing Λ by Λ∗ in the standard DSMC-TCE implementation. To overcome this problem, we empirically introduced a calibration parameter, φ2 (TT ) , in the original MF model. This parameter is then calibrated such that the effective rate coefficients calculated with the TCE model, when it is using Λ∗ , nearly match the values predicted by Eq. 5.11 for TV < TT conditions. In the same fashion, for a given TT , the original Λ Arrhenius parameter is set to a constant value such that the TCE model also matches the nonequilibrium rate coefficients given by Eq. 5.11 for TV > TT conditions. This approach, herein referred to as TCE+MF, is somewhat similar

125 to those employed in the vibrationally dissociation favored (VFD) and generalized collision energy (GCE) models [172]. The O2 +O2 → 2O+O2 nonequilibrium dissociation rate coefficients based on the Arrhenius constants given by Park [106] and the φ1 value fit from O2 +O→3O QCT observations are shown in Fig. 6.3. In these plots, the TCE and TCE+MF nonequilibrium rate coefficients extracted from DSMC 0-D calculations are compared to the QCT-calibrated Kuznetsov and MF models. Three constant TRT and varying TV values are considered. While the TCE model underpredicts the QCT-calibrated MF rate coefficients by approximately a factor of 3 for TV > TT conditions, it overpredicts the same rate coefficients by more than 3 orders of magnitude for TV < TT . On the other hand, the present TCE+MF empirical approach agrees well with the analytical rates within the entire TV range under consideration. These results indicate that, when there is a lack of experimental data, the TCE+MF framework can be used to improve the DSMC solution of strongly nonequilibrium flows. For instance, in the shockwave problems that are investigated in the Sec. 6.2. Although there are other nonequilibrium models expressed in terms of Eq. 5.11 available in the literature [153], the majority of them require empirical fitting parameters such as the characteristic dissociation temperature U in the well-known MarroneTreanor model [173]. While the Kuznetsov model also does not rely on empirical parameters, as shown in the present nonequilibrium rate coefficient plots, it leads to excessively low dissociation rate coefficients as TV approaches zero. As aforementioned, this undesired feature is also observed in the widely used two-temperature model by Park. Therefore, the rationale behind adoption of the MF model in the present simulations relies on the following main points: (i) it does not depend on empirical parameters; and (ii) it also takes into account the contribution of low vibrational levels on the dissociation of weakly excited molecules, in which collision translational energy plays the major role. Even though comparisons of different multi-temperature models against QCT data have shown that the original MF model provides more accurate results for N2 dissociation via N2 -N collisions [174], as shown in Fig. 5.18, this

-16

10

-17

10

-18

10

-19

10

-20

10

-21

TRT = 5,000 K

-22

10

-23

10

-24

0 0 5000 1000 1500 Vibrational Temperature (K)

3

Reaction Rate Coefficient (m /s)

0

10

-15

10

-16

10

-17

10

-18

10

-19

10

-20

10

-21

10

-22

10

-23

10

-24

0 2000

-15

10

-16

10

-17

10

-18

10

-19

10

-20

10

-21

10

-22

10

-23

10

-24

TRT = 10,000 K

QCT-Calibrated Kuznetsov QCT-Calibrated MF DSMC: TCE DSMC: TCE+MF

0 0 5000 1000 1500 Vibrational Temperature (K)

0

2000

0

Equilibrium

10

QCT-Calibrated Kuznetsov QCT-Calibrated MF DSMC: TCE DSMC: TCE+MF

10

Equilibrium

10

3

-15

Reaction Rate Coefficient (m /s)

10

Equilibrium

3

Reaction Rate Coefficient (m /s)

126

TRT = 20,000 K

0

QCT-Calibrated Kuznetsov QCT-Calibrated MF DSMC: TCE DSMC: TCE+MF

0 0 5000 1000 1500 Vibrational Temperature (K)

2000

0

Figure 6.3.: Comparison of different O2 +O2 → 2O+O2 nonequilibrium dissociation rate coefficients. Ro-translational temperatures TRT equal to 5,000 (left), 10,000 (right), and 20,000 K (bottom) are considered. Corresponding thermal equilibrium conditions (TRT = TV ) are denoted by the vertical lines.

feature cannot be directly extended to the O2 -O system given the higher complexity of its PESs.

127 6.2

DSMC Simulation of O2 Shockwaves In this section, we evaluate the use of high-fidelity VT relaxation and dissociation

models in simulating the shock tube experiments by Ibraguimova et al [62]. As listed in Table 6.1, two different pure O2 freestream conditions are considered. For the investigated conditions, it has been shown that recombination reactions have only a minor impact on the downstream properties [136]. Therefore, since the experimental data is available only near the shock front, for comparison purposes, recombination reactions are neglected in this work. Table 6.1.: O2 freestream shockwave conditions. Case Mach (-) Pressure (torr) Temperature (K) 1

9.3

2.0

295

2

13.4

0.8

295

The DSMC normal shockwave structures are generated using the stagnation streamline approach [4, 44, 104], which can summarized as follows. The 1-D domain is first initialized with the freestream conditions and inflow and solid specular wall boundary conditions (BCs) at the minimum and maximum x-coordinates, respectively. The freestream flow, which is initially directed towards the solid wall, is then reflected as a moving shock towards the inflow boundary. When this moving shock passes at a pre-defined arbitrary distance from the wall, δ, particles lying within an arbitrary distance δ ∗ from the wall start to be uniformly removed from the domain. This artificial process takes place such that the removal mass rate is equal to influx mass rate specified at the left boundary condition. As a result, the moving shock starts to oscillate within the 1-D domain until, eventually, a steady state is achieved. Under this condition, the final DSMC results are obtained by sampling and averaging the microscopic properties.

128 The following numerical parameters are used in the present DSMC 1-D simulations: a 30-mm domain length; 500 sampling cells; 32 collision cells per sampling cell; and around 1.3 × 106 simulated molecules. Under steady state conditions, the mean collision separation to the mean free path and the DSMC time step to the mean collision time local ratios are always smaller than 0.01 and 0.02, respectively. The maximum allowed time step is 0.2 ns. Table 6.2 summarizes the four different sets of numerical models that are evaluated in this work. These sets differ only on how the nonequilibrium dissociation rate coefficients are calculated. Before proceeding to the comparison between the different models, we conduct a simple parametric study to evaluate the impact of the removal of particles performed in the stagnation streamline approach, the total number of simulated particles Npart , and the number of sampling cells Ncell on the steady state results. In the first case, this is accomplished by testing two different δ values for the M = 13.4 case and model 3 framework. As clearly shown in Fig. 6.4, the TT and TV distributions near to the front shock are independent of the chosen δ value. It indicates that with

Table 6.2.: Summary of the DSMC modeling frameworks employed in this work. VT Relaxation

Dissociation

Model

O2 -O

O2 -O2

O2 -O

O2 -O2

1

ME-QCT-VT

LBa

TCEb

TCEc

2

ME-QCT-VT

LBa

TCEb

TCE+MFc,d

3

ME-QCT-VT

LBa

SSD

TCEc

4

ME-QCT-VT

LBa

SSD

TCE+MFc,d

a

ZVC based on τv by Ibraguimova et al [62] and τc assuming σT as in Eq. 4.10.

b

Arrhenius coefficients are fit to the QCT rate coefficients by Kulakhmetov et

al. [117]. c

Equilibrium rate coefficients by Park [106].

d

Using φ1 = 0.78, which is calibrated based on O2 -O QCT calculations.

129 M = 13.4

Dashed Lines: δ = 10 mm Solid Lines: δ = 20 mm Circles: Doubled Npart Crosses: Doubled Npart and Ncell

10

00

0

12

00

0

+

20

00

40

00

Temperature (K) 60 80 00 00

TT

0

+ ++ -2 -1

+ + +

Mass Removal Region (δ*= 1 mm)

+ + + + + ++ ++ + TV + ++ + 0

1

2

3

+

4

5

+

++ +

6

7

Mass Removal Region (δ*= 5 mm)

+ ++

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 x (mm)

Figure 6.4.: Parametric study of the impact of δ, Npart , and Ncell on the steady state temperature distributions.

δ = 10 mm, the stagnation streamline approach can be used to accurately describe the shock structure within a distance of at least 10 mm downstream from the shock front. This plot also indicates the length of the mass removal region δ ∗ , which is is set to 1 and 5 mm in the cases with δ = 10 and 20 mm, respectively. Based on the standard Npart (u 1.3 × 106 ) and Ncell (=500) values used in this section, Fig. 6.4 further shows that essentially the same results are obtained when the value of these DSMC parameters are independently doubled. The same type of analysis is also done for the M = 9.3 case but, for brevity, it is not shown.

6.2.1

Comparing M=9.3 Cases

For the M = 9.3 case, only the vibrational temperature measurements are available. These results are shown as black circles in Fig. 6.5. In the first plot, the present DSMC solutions for TT and TV are given by solid and dashed lines, respectively. The atomic oxygen mole fraction distributions XO are given by dashed-dotted lines. Note that, for comparison purposes, all the property distributions are shifted in the

130

-1

0

1

2

3

4 5 x (mm)

6

7

8

9

Model 4 LB,TCE (Wysong,2014) QC,KSS (Wysong,2014) Master Eq (Andrienko,2016) Master Eq (Kustova,2016) Experimental

10

M = 9.3 .00 10

0

0

10

XO

00 60

00 20 00

.05

Vibrational Temperature, TV (K) 20 30 40 50 00 00 00 00

.15

.10

TV

M = 9.3

00

Temperature (K) 30 40 00 00

50

00

TT

.20 Atomic Oxygen Mole Fraction, XO (-)

60

00

Model 1 Model 2 Model 3 Model 4 TV - Experimental

-1

0

1

2

3

4 5 x (mm)

Figure 6.5.: Temperature and atomic oxygen mole fraction (

TT ,

6

7

TV ,

8

9

10

XO )

distributions across the M = 9.3 shockwave (left) and comparison of model 4 against third-party solutions for the corresponding TV distributions.

x−direction such that the TV profiles overlap at TV = 1, 000 K. Since at this Mach number the oxygen dissociation is not significant and the atomic oxygen mole fraction remains below 0.07 within the nonequilibrium region, the shock profile is more sensitive to the O2 -O2 dissociation models than O2 -O models. As a result, models 1 and 3, both employing O2 -O2 TCE rate coefficients, produce almost identical results. The same scenario is also observed between models 2 and 4, which employ the TCE+MF approach for O2 -O2 collisions. Calculations with the TCE+MF O2 -O2 dissociation models preformed better than with just the TCE model. However, based on the experimental data, all models underpredict vibrational temperature for x < 3 mm and overpredict vibrational temperature for x > 3 mm. This fact suggests that our O2 -O2 vibrational relaxation and dissociation rates are slower than those observed in experiments. This assumption will need to be evaluated with additional calculations based on O2 -O2 ab-initio data. The solution that is believed to be the most accurate one, model 4, is also compared to the DSMC results by Wysong et al. [136] and master equation calculations

131 by Andrienko et al. [175] and Kustova et al. [176]. While one of Wysong’s cases used the LB model calibrated with the VT relaxation rates by Millikan and White [55], the other one used the state-specific quasi-classical (QC) VT rates, as shown in Sec. 5.1. Regarding the O2 dissociation, both their TCE and Kuznetsov state-specific (KSS) implementations were calibrated with the equilibrium rate coefficients by Ibraguimova et al. [62]. In Andrienko’s calculations, similarly to the present DSMC efforts, the O2 -O collisions were modeled with state-specific rate coefficients obtained from QCT calculations. However, their O2 -O2 VT relaxation and dissociation rate coefficients were described by the state-specific forced harmonic oscillator (FHO) and MarroneTreanor (MT) models, respectively. Kustova et al. employed a modified MT dissociation model and VT relaxation rates also given by the FHO model for all types of collisions. Although the calculations by Andrienko better captures the TV peak, as all the other showed results, it does not reproduce the correct post shock TV decay. In general, all these compared approaches provide a similar degree of agreement with the experimental data for M = 9.3.

6.2.2

Comparing M=13.4 Cases

For the M = 13.4 case, experimental data is available not just for the translational and vibrational temperature profiles but also for the mole fraction distribution of atomic oxygen across the shock. In Fig. 6.6, the experimental TT and TV values are indicated by symbols whereas the present DSMC results are shown by lines. In contrast to the M = 9.3 case, there is a significant atomic oxygen concentration at M = 13.4 such that the O2 -O dissociation and relaxation models also play an important role on the shock profile. The model results are compared such that the differences in employing the SSD model over the standard TCE model for O2 -O dissociation are better highlighted. In the first plot, comparison of models 1 and 3 indicates that both frameworks significantly underpredict the TV peak value. Model 3 better reproduces the experimental post shock temperature distributions. By com-

0 14

00

TT - Model 1

M = 13.4

TT - Model 4

00

TV - Model 3 TT - Experimental

TV - Model 4

TT

TT - Experimental TV - Experimental

40

40

00

TV - Experimental

00

Temperature (K) 10 60 80 00 00 00 0

TV - Model 2

12

12

00

TT - Model 3

TT

TT - Model 2

M = 13.4

0

0

TV - Model 1

Temperature (K) 10 60 80 00 00 00 0

14

00

0

132

00

TV

-1

0

1

2

3

4 5 x (mm)

6

7

8

9

Figure 6.6.: Temperature distributions (

0

20

0

20

00

TV

-1

TT ,

0

1

2

3

4 5 x (mm)

6

7

8

9

TV ) across the M = 13.4 shock-

wave: comparison of models 1 and 3 (left) and models 2 and 4 (right) against experimental data.

paring models 2 and 4, the second plot reveals that, in general, the TV peak is better predicted when the O2 -O dissociation is described by the SSD model. Model 4, which combines SSD model for O2 -O and TCE+MF model for O2 -O2 collisions, reproduces both the TT decay and TV experimental distribution across the shock. It is also worth mentioning that, apart from the value of TT peak, the shock front TT gradient predicted by these four models is almost indistinguishable. This is easily seen when all the models are compared in the same plot (not shown for brevity). It occurs because just upstream the shock front O2 -O2 collisions are dominant and all models rely on the same VT relaxation rates for such a type of collision. The first plot of Fig. 6.7 compares the vibrational temperature profiles obtained with models 3 and 4 to the same third-party solutions presented for the M = 9.3 case. The master equation calculation by Andrienko et al., once again, captures well the measured maximum TV value. However, it slightly overpredicts the TV gradient in the shock front, which is better captured by the present models and Kustova’s solution. The largest deviations from the experiments are observed for the calcula-

0

10

00 -1

0. 5

Model 3 Model 4 LB,TCE (Wysong,2014) QC,KSS (Wysong,2014) Master Eq (Andrienko,2016) Master Eq (Kustova,2016) Experimental

0

1

2

3

4 5 x (mm)

6

7

8

9

M = 13.4

Atomic Oxygen Mole Fraction, XO (-) 0. 0. 0. 0. 4 3 2 1

Vibrational Temperature, TV (K) 50 60 30 40 20 00 00 00 00 00

70

M = 13.4

0

00

133

-1

Model 3 Model 4 Master Eq (Andrienko,2016) Master Eq (Kustova,2016) Experimental

0

1

2

3

4 5 x (mm)

6

7

8

9

Figure 6.7.: Comparison of models 3 and 4 against third-party solutions for the corresponding TV distributions (left) and distribution of the atomic oxygen mole fraction (right) across the M = 13.4 shockwave.

tion relying on the standard LB and TCE models. It occurs because, as shown in Sec. 5.1, the TCE model overpredicts the nonequilibrium dissociation rate coefficients for vibrationally cold conditions. The atomic oxygen mole fraction XO distributions obtained by different approaches are shown in the second plot. It indicates that the O2 dissociation predicted by model 3 is faster than the other models under consideration. It is also interesting to note that, although model 3 and Kustova’s results for the TV distributions are very similar, the O2 dissociation is considerably slower in the latter case. Among these approaches, model 4 provides a better overall match with the XO measurements. Model 3 and Andrienko’s XO profiles are also similar to the experimental data but they are shifted in the x-coordinate. Based on these results, one can infer that once a significant concentration of atomic oxygen is generated in the flow, it will have a significant impact on the distribution of macroscopic parameters. Ab-initio models for atomic oxygen are critical for such conditions. Finally, Fig. 6.8 presents the vibrational energy populations at different x−coordinates across the shockwave predicted by each model. In these plots, the solid lines represent

134

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

10

-8

10

-9

x = 1 mm

Lines: Boltzmann at TV Symbols: Model 1 x = -1 mm x = 0 mm x = 1 mm x = 2 mm

x = 0 mm

x = 2 mm

x = -1 mm 0

1

2 3 4 Vibrational Energy (eV)

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

10

-8

10

-9

5

6

Lines: Boltzmann at TV Symbols: Model 3 x = -1 mm x = 0 mm x = 1 mm x = 1 mm x = 2 mm

x = 0 mm

x = 2 mm x = -1 mm 0

1

10

Population Fraction (-)

10

10

Population Fraction (-)

0

2 3 4 Vibrational Energy (eV)

5

6

0

10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

10

-8

10

-9

x = 1 mm

Lines: Boltzmann at TV Symbols: Model 2 x = -1 mm x = 0 mm x = 1 mm x = 2 mm

x = 0 mm

x = 2 mm

x = -1 mm 0

1

2 3 4 Vibrational Energy (eV)

0

5

6

10

-1

10

-2

10

-3

Lines: Boltzmann at TV Symbols: Model 4 x = -1 mm x = 0 mm x = 1 mm x = 2 mm

10

-4

x = 1 mm

10

-5

10

-6

10

-7

10

-8

10

-9

10

Population Fraction (-)

Population Fraction (-)

10

x = 0 mm x = 2 mm

x = -1 mm 0

1

2 3 4 Vibrational Energy (eV)

5

6

Figure 6.8.: Vibrational energy populations at different x−coordinates across the M = 13.4 shockwave.

the Boltzmann distributions corresponding to the local TV value. From a qualitative point of view, all models demonstrate the same trends. At x = −1 mm, the flow is essentially in thermal equilibrium such that the DSMC and Boltzmann populations agree well. Deviation from such a behavior can be observed only for population fractions below 10−6 . This is clearly seen for models 3 and 4, where some highly vibrationally excited molecules can diffuse in the upstream direction before dissocia-

135 tion. A possible explanation for this fact is that, under TV < TT conditions, models 1 and 2 predict higher O2 -O2 dissociation rate coefficients than models 3 and 4. As shown in Fig. 6.6, the highest thermal nonequilibrium degree occurs at x ∼ 0 mm and this is clearly seen in the corresponding vibrational energy populations, which significantly deviate from the respective Boltzmann distribution. Instead, the populations nearly follow bimodal distributions that can be approximated as a linear combination of the Boltzmann distributions based on the freestream TV and local equilibrium temperatures [104]. At x = 1 mm and x = 2 mm, the DSMC calculations show that lower vibrational energy levels follow Boltzmann distributions. However, a non-Boltzmann behavior takes place for highly vibrationally excited states, which are more likely to dissociate. It can be further observed that the degree of depopulation of higher vibrational levels is considerably larger for models 1 and 2 due to fact they predict, as aforementioned, higher dissociation rate coefficients when TV < TT . Although all these observations are already well documented in the literature, the present ME-QCT-VT and SSD results can serve as reference data for comparison with future calculations based on different models or flow conditions.

6.3

Summary Previous DSMC efforts have focused on the implementation and verification of the

recently proposed ab-initio ME-QCT-VT and SSD models for O2 -O collisions [104, 122,123]. In order to validate the ability of these ab-initio models to predict real flow conditions, in this chapter, we simulated O2 experimental shock conditions [62] with the DSMC method. Since ab-initio data exists only for the O2 -O system, we further propose DSMC strategies for building O2 -O2 nonequilibrium dissociation models that are consistent with the O2 +O QCT observations. More specifically, we empirically modify the standard TCE implementation to reproduce the rate coefficients given by the Macheret-Fridman (MF) model. The combination of ME-QCT-VT and SSD models for O2 -O and LB and TCE+MF models for O2 -O2 provides a remarkable

136 agreement with the M = 13.4 shockwave measurements. These DSMC calculations suggest ab-initio models are preferred when available and that the proposed TCE+MF approach improves the standard TCE results in predicting the macroscopic properties of flows with moderate degree of thermochemical nonequilibrium. However, it is worth emphasizing that even multi-temperature high-fidelity models may be inaccurate under strong nonequilibrium conditions due to the impossibility of capturing the non-Boltzmann nature of highly excited vibrational states [121]. In these cases, state-specific models are still necessary.

137

7. CONCLUSIONS 7.1

Concluding Remarks Although there is currently active research on high-fidelity models based on ab-

initio data, the standard DSMC framework still relies on phenomenological collision models. In this work, we revisit the relevant aspects of the standard TCE-LB formulation and propose two new empirical approaches to consistently sample post-reaction internal energy states. In the first approach, we populate post-reaction states based on the corresponding pre-reaction distribution. In the second approach, before performing the conventional LB procedure, we perform a pre-redistribution of the collision energy among the translation and internal mode. Both strategies are illustrated by application to combustion reactions and demonstrate they satisfy detailed balance regardless the mixture internal energy relaxation rates. These strategies are relevant to the DSMC modeling of reacting flows dominated by exchange/recombinations and where the production of nascent excited species is critical. As high-performance computing resources are evolving toward exascale platforms, DSMC modeling of problems that until recently were considered intractable from a CPU-time perspective are becoming feasible. Among others, one can list turbulent flows and low-speed combustion. In this context, in the second part of this dissertation, we implement one of the proposed post-reaction energy redistribution strategies into the SPARTA/DSMC solver and model the 1-D laminar flame structure of H2 -O2 premixed systems. To the best of our knowledge, this is the first unsteady DSMC simulation of deflagration waves. Present results indicate the acoustic nature of the flame is strongly affect by the numerical boundary conditions. On the other hand, for nearly isobaric flows, the flame structure and its propagation speed are insensitive to the pressure-velocity coupling. Based on the proposed methodology to estimate

138 the flame propagation speed in atomistic-level simulations, DSMC results reproduced available experimental data in good agreement. The third and last part of the dissertation concentrates on the DSMC implementation and validation of compact state-specific ab-initio models for O2 -O and N2 -O systems. Before applying these VT internal energy exchange and reaction models to real gas flow conditions, we consistently compare their results against the standard TCE-LB framework and available experimental data. Present results show that when ab-initio data is used to consistently calibrate the vibrational relaxation numbers, phenomenological LB model can accurately reproduce the state-specific ME-QCTVT and master equation results for nonreacting conditions. On the other hand, even when the TCE model is calibrated with the QCT-based equilibrium rate coefficients, its nonequilibrium rates can deviate from the SSD and SSE results by more than three orders of magnitude. We also show that the proposed TCE+MF formulation can be used to improve the standard TCE model results when ab-initio data is not available or limited. Compared against different sets of models, combination of present QCT-based and TCE+MF models provided the best degree of agreement with pure O2 shockwave measurements. This confirms that, when available, efficient ab-initio models are preferred over phenomenological ones.

7.2

Future Work As aforementioned, the main contributions of the present dissertation can be sum-

marized as: (1) development of robust post-reaction energy redistribution strategies; (2) assessments of DSMC capabilities in simulating atmospheric low-speed combustion flows; and (3) consistent implementation and evaluation of ab-initio VT energy relaxation, dissociation, and exchange reaction models. In despite of the fact these studies have employed reasonable physical assumptions, there is a significant number of improvements that can, and should, be implemented in the future as well as

139 open questions that demand further investigation. Among others, we encourage the following extensions of the present work: • Develop ab-initio, state-specific models for post-reaction energy redistribution. For instance, the QCT data obtained by Kulakhmetov [27] and Luo [28] can be fit to compact models that describe the initial states of nascent species. Since post-reaction states are empirically specified in current DSMC models, QCTinformed modeling might have major impact on DSMC-based predictions of electromagnetic radiation of high-enthalpy flows. • Apply DSMC method to investigate other fundamental combustion properties such as flame strain rates and/or more complex 2-D/3-D applications in which nonequilibrium effects cannot be neglected, e.g., in microcombustion flows [78]. • Implement present ME-QCT-VT, SSD, and SSE models into a high-performance DSMC solver such as SPARTA and evaluate their computational efficiency and physical accuracy for 3-D realistic flow simulations.

REFERENCES

140

REFERENCES

[1] Richard Phillips Feynman, Robert B Leighton, and Matthew Sands. Feynman lectures on physics. vol. 1: Mainly mechanics, radiation and heat. AddisonWesley, 1963. [2] Walter Guido Vincenti and Charles H Kruger. Introduction to physical gas dynamics, volume 1. Wiley, New York, 1965. [3] G. A. Bird. Molecular Gas Dynamics. Clarendon Press, Oxford, 1976. [4] G. A. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford University Press, New York, 1994. [5] Sydney Chapman and Thomas George Cowling. The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge university press, 1970. [6] Mohamed Gad-el Hak. MEMS: Introduction and Fundamentals. CRC press, 2005. [7] G. A. Bird. Approach to translational equilibrium in a rigid sphere gas. Physics of Fluids (1958-1988), 6(10):1518–1519, 1963. [8] G. A. Bird. Shock wave structure in a rigid sphere gas. In J. H. de Leeuw, editor, Rarefied gas dynamics, volume 1, pages 216–222. Academic Press, New York, 1965. [9] Devon Parkos, Alina Alexeenko, Marat Kulakhmetov, Brandon C. Johnson, and H. Jay Melosh. NOx production and rainout from Chicxulub impact ejecta reentry. Journal of Geophysical Research: Planets, 120(12):2152–2168, 2015. [10] GN Markelov, AV Kashkovsky, and MS Ivanov. Space station Mir aerodynamics along the descent trajectory. Journal of Spacecraft and Rockets, 38(1):43–50, 2001. [11] Israel Borges Sebasti˜ao and Wilson FN Santos. Gas–surface interaction effects on the flowfield structure of a high speed microchannel flow. Applied Thermal Engineering, 52(2):566–575, 2013. [12] Israel Borges Sebasti˜ao and Wilson FN Santos. Gas–surface interaction impact on heat transfer and pressure distributions of a high speed microchannel flow. Applied Thermal Engineering, 62(1):58–68, 2014. [13] Israel Borges Sebasti˜ao and Wilson FN Santos. Numerical simulation of heat transfer and pressure distributions in micronozzles with surface discontinuities on the divergent contour. Computers & Fluids, 92:125–137, 2014.

141 [14] Israel Borges Sebasti˜ao and Wilson FN Santos. Impact of surface discontinuities on flowfield structure of a micronozzle array. Nanoscale and Microscale Thermophysical Engineering, 18(1):54–79, 2014. [15] Andrew Strongrich, Aaron Pikus, Israel Borges Sebasti˜ao, and Alina Alexeenko. Microscale in-plane knudsen radiometric actuator: Design, characterization, and performance modeling. Journal of Microelectromechanical Systems, 26(3):528–538, 2017. [16] Ayyaswamy Venkattraman and Alina A Alexeenko. Direct simulation Monte Carlo modeling of metal vapor flows in application to thin film deposition. Vacuum, 86(11):1748–1758, 2012. [17] Alina A. Alexeenko and Sergey F. Gimelshein. Numerical Simulation Methods: Direct Simulation Monte Carlo. In Richard W. Johnson, editor, Handbook of Fluid Dynamics, chapter 39, page 1560. CRC Press, 2 edition, 2016. [18] Andrew Brian Weaver. Assessment of high-fidelity collision models in the direct simulation Monte Carlo method. PhD thesis, Purdue University, 2015. [19] Tianfeng Lu and Chung K Law. Toward accommodating realistic fuel chemistry in large-scale computations. Progress in Energy and Combustion Science, 35(2):192–215, 2009. [20] Morton Mitchner and Charles H Kruger. Partially ionized gases. Wiley New York, 1973. [21] Evgeni˘ı Nikitin. Theory of elementary atomic and molecular processes in gases. Oxford, Clarendon Press, 1974. [22] Evgeni˘ı Nikitin and Stanislav Ya Umanskii. Theory of slow atomic collisions, volume 30. Springer Science & Business Media, 1984. [23] Raphael D Levine and Richard B Bernstein. Molecular Reaction Dynamics and Chemical Reactivity. Oxford University Press, 1987. [24] Katsuhisa Koura and Hiroaki Matsumoto. Variable soft sphere molecular model for inverse-power-law or Lennard Jones potential. Physics of Fluids A, 3(10):2459–2465, 1991. [25] C. Borgnakke and P. Larsen. Statistical collision model for Monte Carlo simulation of polyatomic gas mixture. Journal of Computational Physics, 18:405–420, 1975. [26] G. A. Bird. Monte-Carlo simulation in an engineering context. In S. S. Fisher, editor, Rarefied gas dynamics, volume 74, pages 239–255, New York, NY, 1981. American Inst. of Physics. [27] Marat Kulakhmetov. Upscaling ab-initio chemistry models to non-equilibrium flow simulations. PhD thesis, Purdue University, 2016. [28] Han Luo. Ab-initio based state-specific modeling of N2 +O system. Master’s thesis, Purdue University, 2016.

142 [29] Israel Borges Sebasti˜ao and Alina Alexeenko. Consistent post-reaction vibrational energy redistribution in DSMC simulations using TCE model. Physics of Fluids, 28(10):107103, 2016. [30] Michael A. Gallis and John K. Harvey. The modeling of chemical reactions and thermochemical nonequilibrium in particle simulation computations. Physics of Fluids, 10(6):1344–1358, 1998. [31] B. L. Haas. Models of energy-exchange mechanics applicable to a particle simulation of reactive flow. Journal of Thermophysics and Heat Transfer, 6(2):200– 207, 1992. [32] N. E. Gimelshein, D. A. Levin, and S. F. Gimelshein. Hydroxyl formation mechanisms and models in high-altitude hypersonic flows. AIAA Journal, 41(7):1323–1331, 2003. [33] Yevgeniy A. Bondar, Kaoru Maruta, and Mikhail S. Ivanov. Hydrogenoxygen detonation study by the DSMC method. AIP Conference Proceedings, 1333(1):1209–1214, 2011. [34] G. A. Bird. Setting the post-reaction internal energies in direct simulation Monte Carlo chemistry simulations. Physics of Fluids, 24(12):127104 1–11, 2012. [35] Jae Gang Kim and Iain D Boyd. Monte Carlo simulation of nitrogen dissociation based on state-resolved cross sections. Physics of Fluids, 26(1):012006, 2014. [36] Neal Parsons, Deborah A Levin, Adri CT van Duin, and Tong Zhu. Modeling of molecular nitrogen collisions and dissociation processes for direct simulation Monte Carlo. The Journal of Chemical Physics, 141(23):234307, 2014. [37] Hailong Zhang and Maozhao Xie. DSMC Simulation of Non-Premixed Combustion of H2/O2 in a Y-shaped Microchannel. Nanoscale and Microscale Thermophysical Engineering, 19(1):31–62, 2015. [38] Israel Borges Sebasti˜ao and Alina Alexeenko. DSMC investigation of nonequilibrium effects in a H2-O2 unstretched diffusion flame. In 45th AIAA Thermophysics Conference, page 3372, 2015. [39] Andrea Pfl¨ uger, Bernd Schr¨oder, and Hans-J¨org Bart. Monte Carlo simulations on large-area deposition of amorphous silicon by hot-wire CVD. Thin Solid Films, 430(1):73–77, 2003. [40] Iain D. Boyd, Graham V. Candler, and Deborah A. Levin. Dissociation modeling in low density hypersonic flows of air. Physics of Fluids, 7(7):1757–1763, 1995. [41] Deepak Bose and Graham V Candler. Simulation of hypersonic flows using a detailed nitric oxide formation model. Physics of Fluids (1994-present), 9(4):1171– 1181, 1997. [42] David B Hash and HA Hassan. Monte Carlo simulation of entry in the Martian atmosphere. Journal of thermophysics and heat transfer, 7(2):228–232, 1993.

143 [43] Michael J Wright, Chun Y Tang, Karl T Edquist, Brian R Hollis, Paul Krasa, and Charles A Campbell. A review of aerothermal modeling for Mars entry missions. AIAA paper, 443:4–7, 2010. [44] G. A. Bird. The DSMC Method. CreateSpace Independent Publishing Platform, 2013. [45] MS Ivanov, GN Markelov, and SF Gimelshein. Statistical simulation of reactive rarefied flows: numerical approach and applications. 1998. [46] S. F. Gimelshein, N. E. Gimelshein, D. A. Levin, M. S. Ivanov, and I. J. Wysong. On the use of chemical reaction rates with discrete internal energies in the direct simulation Monte Carlo method. Physics of Fluids, 16(7):2442–2451, 2004. [47] D. Davidenko, I. G¨okalp, E. Dufour, and P. Magre. Systematic numerical study of the supersonic combustion in an experimental combustion chamber. AIAA 2006-7913, 2006. [48] NE Gimelshein, SF Gimelshein, and DA Levin. Vibrational relaxation rates in the direct simulation Monte Carlo method. Physics of Fluids, 14:4452–4455, 2002. [49] Brian L Haas, David B Hash, Graeme A Bird, Forrest E Lumpkin III, and Hassan A Hassan. Rates of thermal relaxation in direct simulation Monte Carlo methods. Physics of Fluids (1994-present), 6(6):2191–2201, 1994. [50] F. Bergemann and I. D. Boyd. New discrete vibrational energy model for the direct simulation Monte Carlo method. In David P. Weaver and Bernie D. Shizgal, editors, Rarefied Gas Dynamics: Experimental Techniques and Physical Systems, pages 174–183. AIAA, 1994. [51] S. Dietrich. Direct simulation Monte Carlo modeling of near-continuum flows with chemical reactions. In David P. Weaver and Bernie D. Shizgal, editors, Rarefied Gas Dynamics: Theory and Simulations, pages 208–217. AIAA, 1994. [52] M. Jelezniak and I. Jelezniak. Chemked–a program for chemical kinetics of gas phase reactions. 2013. Available at www.chemked.com. [53] Michail A Gallis, Timothy P Koehler, John R Torczynski, and Steven J Plimpton. Direct simulation Monte Carlo investigation of the Richtmyer-Meshkov instability. Physics of Fluids, 27(8):084105, 2015. [54] Israel Borges Sebasti˜ao, Marat Kulakhmetov, and Alina Alexeenko. DSMC study of oxygen shockwaves based on high-fidelity vibrational relaxation and dissociation models. Physics of Fluids, 29(1):017102, 2017. [55] Roger C Millikan and Donald R White. Systematics of vibrational relaxation. The Journal of Chemical Physics, 39(12):3209–3213, 1963. [56] RTV Kung and RE Center. High temperature vibrational relaxation of H2O by H2O, He, Ar, and N2. The Journal of Chemical Physics, 62(6):2187–2194, 1975. [57] PF Zittel and DE Masturzo. Vibrational relaxation of H2O from 295 to 1020 K. The Journal of chemical physics, 90(2):977–989, 1989.

144 [58] Masayuki Tamura, Pamela A Berg, Joel E Harrington, Jorge Luque, Jay B Jeffries, Gregory P Smith, and David R Crosley. Collisional quenching of CH(A), OH(A), and NO(A) in low pressure hydrocarbon flames. Combustion and Flame, 114(3):502–514, 1998. [59] Jean Lacoursiere, Mark J Dyer, and Richard A Copeland. Temperature dependence of the collisional energy transfer of OH (v=10) between 220 and 310 k. The Journal of chemical physics, 118(4):1661–1666, 2003. [60] DC McCabe, B Rajakumar, P Marshall, IWM Smith, and AR Ravishankara. The relaxation of oh(v= 1) and od(v= 1) by h2o and d2o at temperatures from 251 to 390 k. Physical Chemistry Chemical Physics, 8(39):4563–4574, 2006. [61] L Ibraguimova and O Shatalov. Non-equilibrium kinetics behind shock waves experimental aspects. In High Temperature Phenomena in Shock Waves, pages 99–147. Springer, 2012. [62] LB Ibraguimova, AL Sergievskaya, V Yu Levashov, OP Shatalov, Yu V Tunik, and IE Zabelinskii. Investigation of oxygen dissociation and vibrational relaxation at temperatures 4000–10,800 K. J. Chem. Phys., 139(3):034317, 2013. [63] Mario Capitelli, Carlos M Ferreira, Boris F Gordiets, and Alexey I Osipov. Plasma kinetics in atmospheric gases, volume 31. Springer Science & Business Media, 2013. [64] JG Parker. Rotational and vibrational relaxation in diatomic gases. Physics of Fluids (1958-1988), 2(4):449–462, 1959. [65] Michael Gallis and John Harvey. New ionization model for the direct simulation Monte Carlo method. In Thermophysics Conference. AIAA, 1996. [66] M. A. Gallis, T. P. Koehler, J. R. Torczynski, and S. J. Plimpton. Direct simulation Monte Carlo investigation of the Rayleigh-Taylor instability. Phys. Rev. Fluids, 1:043403, 2016. [67] MA Gallis, NP Bitter, TP Koehler, JR Torczynski, SJ Plimpton, and G Papadakis. Molecular-level simulations of turbulence and its decay. Physical Review Letters, 118(6):064501, 2017. [68] Ye A Bondar, K Maruta, and MS Ivanov. Numerical study of gas detonation at the molecular kinetic level. In 28th International Symposium on Shock Waves, pages 313–318. Springer, 2012. [69] Robert J Kee, Joseph F Grcar, Mitchell D Smooke, JA Miller, and E Meeks. PREMIX: a fortran program for modeling steady laminar one-dimensional premixed flames. Report SAND85-8249, Sandia National Laboratories, 1985. [70] Boris E Gelfand, Mikhail V Silnikov, Sergey P Medvedev, and Sergey V Khomik. Laminar and cellular combustion of hydrogenous gaseous mixtures. In Thermo-Gas Dynamics of Hydrogen Combustion and Explosion, pages 15–51. Springer, 2012. [71] F. A. Williams. Combustion Theory. The Benjamin-Cummings Publishing Company, Inc., New York, 2 edition, 1985.

145 [72] S. R. Turns. An Introduction to Combustion: Concepts and Applications. McGraw-Hill Science, New York, 2 edition, 2000. [73] Chung K Law. Combustion Physics. Cambridge University Press, 2006. [74] Adam Sieminski. International energy outlook. Energy Information Administration (EIA), 2013. [75] SM Starikovskaia. Plasma assisted ignition and combustion. Journal of Physics D: Applied Physics, 39(16):R265, 2006. [76] N. A. Popov. The effect of nonequilibrium excitation on the ignition of hydrogen-oxygen mixtures. High Temperature, 45(2):261–279, 2007. [77] Christopher M Spadaccini, Xin Zhang, Christopher P Cadou, Norihisa Miki, and Ian A Waitz. Preliminary development of a hydrocarbon-fueled catalytic micro-combustor. Sensors and Actuators A: Physical, 103(1):219–224, 2003. [78] Kaoru Maruta. Micro and mesoscale combustion. Proceedings of the Combustion Institute, 33(1):125–150, 2011. [79] Romain Fievet, Stephen Voelkel, Heeseok Koo, Venkat Raman, and Philip L Varghese. Effect of thermal nonequilibrium on ignition in scramjet combustors. Proceedings of the Combustion Institute, 36(2):2901–2910, 2017. [80] Sergio Escobar. Rotating Detonation Combustion: A Computational Study for Stationary Power Generation. PhD thesis, West Virginia University, 2014. [81] Rui Zhou, Dan Wu, and Jianping Wang. Progress of continuously rotating detonation engines. Chinese Journal of Aeronautics, 29(1):15–29, 2016. [82] James Braun, Bayindir H Saracoglu, and Guillermo Paniagua. Unsteady performance of rotating detonation engines with different exhaust nozzles. Journal of Propulsion and Power, pages 1–10, 2016. [83] S Voelkel, V Raman, and PL Varghese. Effect of thermal nonequilibrium on reactions in hydrogen combustion. Shock Waves, 26(5):539–549, 2016. [84] Stephen Voelkel, Damien Masselot, Philip L Varghese, Venkat Raman, Andrew Ketsdever, and Henning Struchtrup. Analysis of hydrogen-air detonation waves with vibrational nonequilibrium. In AIP Conference Proceedings, volume 1786, page 070015. AIP Publishing, 2016. [85] Heath H Reising, KC Utsav, Stephen Voelkel, Noel Clemens, Venkatramanan Raman, Philip Varghese, and Heeseok Koo. Vibrational non-equilibrium effects in supersonic jet mixing. In 52nd Aerospace Sciences Meeting, page 0231, 2014. [86] Heath H Reising, Timothy W Haller, Noel T Clemens, Philip L Varghese, Romain Fi´evet, and Venkatramanan Raman. Spontaneous Raman scattering temperature measurements and large eddy simulations of vibrational nonequilibrium in high-speed jet flames. In 32nd AIAA Aerodynamic Measurement Technology and Ground Testing Conference, page 3550, 2016. [87] Romain Fievet, Stephen J Voelkel, Venkatramanan Raman, and Philip L Varghese. Numerical investigation of vibrational relaxation coupling with turbulent mixing. In 55th AIAA Aerospace Sciences Meeting, page 0663, 2017.

146 [88] Yevgeniy Bondar, Anatoly Trotsyuk, and Mikhail Ivanov. DSMC modeling of the detonation wave structure in narrow channels. In 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, page 1568, 2009. [89] Chao Yang and Quanhua Sun. Investigation of spontaneous combustion of hydrogen-oxygen mixture using DSMC simulation. 1628:1261–1267, 2014. [90] Vladimir Alekseev. Laminar burning velocity of hydrogen and flame structure of related fuels for detailed kinetic model validation. PhD thesis, Tryckeriet i E-huset, Lunds universitet, 2015. ´ [91] Marcus OConaire, Henry J Curran, John M Simmie, William J Pitz, and Charles K Westbrook. A comprehensive modeling study of hydrogen oxidation. International journal of chemical kinetics, 36(11):603–622, 2004. [92] Juan Li, Zhenwei Zhao, Andrei Kazakov, and Frederick L Dryer. An updated comprehensive kinetic model of hydrogen combustion. International journal of chemical kinetics, 36(10):566–575, 2004. [93] OP Shatalov, LB Ibraguimova, VA Pavlov, GD Smekhov, and Yu V Tunik. Analysis of the kinetic data described oxygen-hydrogen mixtures combustion. In 4th European Combustion Meeting, page 222. Citeseer, 2009. [94] Sanford Gordon and Bonnie J McBride. Computer program for calculation of complex chemical equilibrium compositions and applications. part 1: Analysis. Technical Report Reference Publication 1311, NASA, 1996. [95] MA Gallis and SJ Plimpton. SPARTA direct simulation Monte Carlo (DSMC) simulator, 2014. Available at http://sparta.sandia.gov/index.html. [96] Henning Bonart. Implementation and validation of a solver for direct numerical simulations of turbulent reacting flows in OpenFOAM. Master’s thesis, Karlsruhe Institute of Technology, 2012. [97] Feichi Zhang, Henning Bonart, Thorsten Zirwes, Peter Habisreuther, Henning Bockhorn, and Nikolaos Zarzalis. Direct numerical simulation of chemically reacting flows with the public domain code OpenFOAM. In High Performance Computing in Science and Engineering 14, pages 221–236. Springer, 2015. [98] S Jain, G Mo, and L Qiao. Molecular dynamics simulations of flame propagation along a monopropellant PETN coupled with multi-walled carbon nanotubes. Journal of Applied Physics, 121(5):054902, 2017. [99] Chowdhury Ashraf, Abhishek Jain, Yuan Xuan, and Adri CT van Duin. ReaxFF based molecular dynamics simulations of ignition front propagation in hydrocarbon/oxygen mixtures under high temperature and pressure conditions. Physical Chemistry Chemical Physics, 19(7):5004–5017, 2017. [100] B Lewis and G Von Elbe. Combustion, flames and explosions of gases. Academic Press Inc., Orlando, FL, 3rd edition, 1987. [101] DA Senior. Burning velocities of hydrogen-air and hydrogen-oxygen mixtures: Determination by burner method with schlieren photography. Combustion and Flame, 5:7–10, 1961.

147 [102] GW Koroll, RK Kumar, and CK Chan. Combustion behaviour in the moderator cover gas. 1988. [103] GW Koroll, RK Kumar, and EM Bowles. Burning velocities of hydrogen-air mixtures. Combustion and Flame, 94(3):330–340, 1993. [104] M Kulakhmetov, Isreal B Sebastiao, and Alina Alexeenko. Adapting vibrational relaxation models in DSMC and CFD to ab-initio calculations. In 46th AIAA Thermophysics Conference, volume AIAA 2016-3844, 2016. [105] Han Luo, Marat Kulakhmetov, and Alina Alexeenko. Ab initio state-specific N2+O dissociation and exchange modeling for molecular simulations. The Journal of Chemical Physics, 146(7):074303, 2017. [106] Chul Park. Nonequilibrium Hypersonic Aerothermodynamics. New York: Wiley, 1990. [107] Ross Wagnild, Graham V Candler, Pramod Subbareddy, and Heath Johnson. Vibrational relaxation effects on acoustic disturbances in a hypersonic boundary layer over a cone. In 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, page 922, 2012. [108] GA Bird. The QK model for gas-phase chemical reaction rates. Physics of Fluids, 23(10):106101, 2011. [109] Brian L Haas and Iain D Boyd. Models for direct Monte Carlo simulation of coupled vibration–dissociation. Physics of Fluids A: Fluid Dynamics, 5(2):478– 489, 1993. [110] Dean C Wadsworth and Ingrid J Wysong. Vibrational favoring effect in DSMC dissociation models. Physics of Fluids, 9(12):3873–3884, 1997. [111] Ingrid Wysong, Sergey Gimelshein, Natalia Gimelshein, William McKeon, and Fabrizio Esposito. Reaction cross sections for two direct simulation Monte Carlo models: Accuracy and sensitivity analysis. Physics of Fluids, 24(4):042002, 2012. [112] DG Truhlar, JT Muckerman, and RB Bernstein. Atom-Molecule Collision Theory: A Guide for the Experimentalist. Plenum, New York, 1979. [113] Jae Gang Kim and Iain D Boyd. State-resolved master equation analysis of thermochemical nonequilibrium of nitrogen. Chemical Physics, 415:237–246, 2013. [114] M Panesi, A Munafo, TE Magin, and RL Jaffe. Nonequilibrium shock-heated nitrogen flows using a rovibrational state-to-state method. Physical Review E, 90(1):013009, 2014. [115] Neal Parsons, Tong Zhu, Deborah A Levin, and Adri CT van Duin. Development of DSMC chemistry models for nitrogen collisions using accurate theoretical calculations. In 52nd AIAA Aerospace Sciences Meeting, volume AIAA 2014-1213, 2014.

148 [116] Marat Kulakhmetov, Michael Gallis, and Alina Alexeenko. Ab initio-informed maximum entropy modeling of rovibrational relaxation and state-specific dissociation with application to the O2 +O system. The Journal of Chemical Physics, 144(17), 2016. [117] Marat Kulakhmetov, Michael Gallis, and Alina Alexeenko. Effect of O2 -O abinitio and Morse additive pairwise potentials on dissociation and relaxation rates for nonequilibrium flow calculations. Physics of Fluids, 27(8):087104, 2015. [118] I Procaccia and RD Levine. Vibrational energy transfer in molecular collisions: An information theoretic analysis and synthesis. The Journal of Chemical Physics, 63(10):4261–4279, 1975. [119] David A Gonzales and Philip L Varghese. Evaluation of simple rate expressions for vibration-dissociation coupling. Journal of Thermophysics and Heat Transfer, 8(2):236–243, 1994. [120] Yen Liu, Marco Panesi, Amal Sahai, and Marcel Vinokur. General multi-group macroscopic modeling for thermo-chemical non-equilibrium gas mixtures. The Journal of chemical physics, 142(13):134109, 2015. [121] A Munaf`o, Y Liu, and M Panesi. Modeling of dissociation and energy transfer in shock-heated nitrogen flows. Physics of Fluids (1994-present), 27(12):127101, 2015. [122] M Kulakhmetov, I Sebastiao, A Alexeenko, and Michael Gallis. Maximum entropy modeling of vibrational-translational energy exchange in O2 +O collisions. In 54th AIAA Aerospace Sciences Meeting, volume AIAA 2016-0504, 2016. [123] Isreal Borges Sebasti˜ao, M Kulakhmetov, and Alina Alexeenko. Comparison between phenomenological and ab-initio reaction and relaxation models in DSMC. In 30th International Symposium on Rarefied Gas Dynamics. AIP Publishing, 2016. [124] Israel Borges Sebasti˜ao, Han Luo, Marat F Kulakhmetov, and Alina Alexeenko. DSMC implementation of compact state-specific N2+O dissociation and exchange models. In 55th AIAA Aerospace Sciences Meeting, page 1842, 2017. [125] AJC Varandas and AACC Pais. A realistic double many-body expansion DMBE potential energy surface for ground-state O3 from a multiproperty fit to ab-initio calculations, and to experimental spectroscopic, inelastic scattering, and kinetic isotope thermal rate data. Mol. Phys., 65(4):843–860, 1988. [126] P. Gamallo, Miguel Gonz´alez, and R. Say´os. Ab initio derived analytical fits of the two lowest triplet potential energy surfaces and theoretical rate constants for the N(4S) + NO(X2II) system. J. Chem. Phys., 119(5):2545–2556, 2003. [127] David A Gonzales and Philip L Varghese. A simple model for state-specific diatomic dissociation. The Journal of Physical Chemistry, 97(29):7612–7622, 1993. [128] PM Marriott and JK Harvey. New approach for modelling energy exchange and chemical reactions in the direct simulation Monte Carlo method. In Rarefied Gas Dynamics, volume 1, pages 784–791, 1991.

149 [129] Michael A Gallis and John K Harvey. Maximum entropy analysis of chemical reaction energy dependence. J. Thermophys. Heat Transfer, 10(2):217–223, 1996. [130] Chonglin Zhang and Thomas E Schwartzentruber. Consistent implementation of state-to-state collision models for direct simulation Monte Carlo. In 52nd Aerospace Sciences Meeting, AIAA Paper 2014-0866, 2014. [131] Hao Deng, T Ozawa, and DA Levin. Analysis of chemistry models for DSMC simulations of the atmosphere of Io. Journal of Thermophysics and Heat Transfer, 26(1):36–46, 2012. [132] NE Gimelshein, SF Gimelshein, DA Levin, MS Ivanov, and IJ Wysong. Reconsideration of DSMC models for internal energy transfer and chemical reactions. In AIP Conference Proceedings, pages 349–357, 2003. [133] Ye Bondar, N Gimelshein, S Gimelshein, M Ivanov, and I Wysong. On the accuracy of DSMC modeling of rarefied flows with real gas effects. In 24th International Symposium on Rarefied Gas Dynamics, volume 762, pages 607– 613. AIP Publishing, 2005. [134] Ye A Bondar, AA Shevyrin, YS Chen, AN Shumakova, AV Kashkovsky, and MS Ivanov. Direct Monte Carlo simulation of high-temperature chemical reactions in air. Thermophysics and Aeromechanics, 20(5):553–564, 2013. [135] Normand M Laurendeau. Statistical Thermodynamics: Fundamentals and Applications. Cambridge University Press, 2005. [136] Ingrid Wysong, Sergey Gimelshein, Yevgeniy Bondar, and Mikhail Ivanov. Comparison of direct simulation Monte Carlo chemistry and vibrational models applied to oxygen shock measurements. Physics of Fluids (1994-present), 26(4):043101, 2014. [137] M Capitelli, C Gorse, S Longo, and D Giordano. Collision integrals of high-temperature air species. Journal of Thermophysics and Heat Transfer, 14(2):259–268, 2000. [138] Fabrizio Esposito, Iole Armenise, Giulia Capitta, and Mario Capitelli. O-O2 state-to-state vibrational relaxation and dissociation rates based on quasiclassical calculations. Chem. Phys., 351(1):91–98, 2008. [139] Han Luo, Marat Kulakhmetov, and Alina Alexeenko. Calculation of stateresolved cross sections for the n2(x1 σ)+o(3 p) system. AIP Conference Proceedings, 1786(1), 2016. [140] L Landau and E Teller. Zur theorie der schalldispersion. Phys. Z. Sowjetunion, 10(1):34, 1936. [141] Chul Park. Rotational relaxation of N2 behind a strong shock wave. Journal of Thermophysics and Heat Transfer, 18(4):527–533, 2004. [142] Daniil Andrienko and Iain D Boyd. Investigation of oxygen vibrational relaxation by quasi-classical trajectory method. Chemical Physics, 459:1–13, 2015.

150 [143] Fabrizio Esposito and Mario Capitelli. The relaxation of vibrationally excited O2 molecules by atomic oxygen. Chemical Physics Letters, 443(4):222–226, 2007. [144] M Quack and J Troe. Complex formation in reactive and inelastic scattering: Statistical adiabatic channel model of unimolecular processes III. Berichte der Bunsengesellschaft f¨ ur physikalische Chemie, 79(2):170–183, 1975. [145] Chul Park. Review of chemical-kinetic problems of future NASA missions. I Earth entries. Journal of Thermophysics and Heat transfer, 7(3):385–398, 1993. [146] JE Breen, RB Quy, and GP Glass. Vibrational relaxation of O2 in the presence of atomic oxygen. The Journal of Chemical Physics, 59(1):556–557, 1973. [147] SF Gimelshein, MS Ivanov, GN Markelov, and Yu E Gorbachev. Statistical simulation of nonequilibrium rarefied flows with quasiclassical vibrational energy transfer models. Journal of Thermophysics and Heat Transfer, 12(4):489–495, 1998. [148] Vladimir Riabov. Numerical analysis of compression and expansion binary gasmixture flows. AIAA Paper, pages 96–0109, 1996. [149] Vladimir Riabov. Numerical simulations of nonequilibrium and diffusive effects in spherical shock waves. In 28th International Symposium on Shock Waves, pages 209–214. Springer, 2012. [150] DL Baulch, DD Drysdale, J Duxbury, and S Grant. Evaluated Kinetic Data for High-Temperature Reactions, volume 3. London: Butterworths, 1976. [151] N. M. Kuznetsov. Kinetics for the dissociation of molecules in a molecular gas. Theoretical and Experimental Chemistry, 7:17–26, 1973. [152] SO Macheret, A Fridman, IV Adamovich, JWb Rich, and Treanor CE. Mechanisms of nonequilibrium dissociation of diatomic molecules. In AIAA and ASME, Joint Thermophysics and Heat Transfer Conference, 6 th, Colorado Springs, CO, volume AIAA 1994-1984, 1994. [153] A. Losev, B. V. Potapkin, S. O. Macheret, and G. G. Chernyi. Physical and Chemical Processes in Gas Dynamics: Physical and Chemical Kinetics and Thermodynamics of Gases and Plasmas, volume II. Progress in Astronautics and Aeronautics, 2004. [154] Peter W Erdman, Edward C Zipf, Patrick Espy, Carl L Howlett, Deborah A Levin, Robert J Collins, and Graham V Candler. Measurements of ultraviolet radiation from a 5-km/s bow shock. JTHT, 8(3):441–446, 1994. [155] M Ahmadjian, D E Jennings, M J Mumma, F Espenak, C. J. Rice, R. W. Russell, and B. D. Green. Infrared spectral measurement of space shuttle glow. Geophysical Research Letters, 19(10):989–992, 1992. [156] D B Oakes, D M Sonnenfroh, G E Caledonia, and W. A. M. Blumberg. Velocity dependent O atom IR excitation cross sections: Connections with flight data. Journal of Geophysical Research, 99(A12):23249, 1994.

151 [157] JB Livesey, AL Roberts, and Alan Williams. The formation of oxides of nitrogen in some oxy-propane flames. Combustion Science and Technology, 4(1):9–15, 1971. [158] JP Monat, RK Hanson, and CH Kruger. Shock tube determination of the rate coefficient for the reaction N2 + O → N O + N . In Symp. Int. Combust. Proc., volume 17, 1979. [159] N. Davidson. Selected reactions involving nitrogen and oxygen, report no. 32. Technical report, Avco Research Lab, 1958. [160] Stephen P. Walch and Richard L. Jaffe. Calculated potential surfaces for the reactions: O+N2→NO+N and N+O2→NO+O. J. Chem. Phys., 86(12):6946, 1987. [161] Wei Lin, Rub´en Meana-Pa˜ neda, Zoltan Varga, and Donald G. Truhlar. A quasiclassical trajectory study of the N2 (X1 Σ)+O(3 P)→NO(X2 Π)+N(4 S) reaction. The Journal of Chemical Physics, 144(23):234314, 2016. [162] Deepak Bose and Graham V Candler. Thermal rate constants of the N2 + O → N O + N reaction using abinitio 3 A and 3 A0 potential energy surfaces. J. Chem. Phys., 104(8):2825–2833, 1996. [163] S. Losev, S. Kanne, O. Knab, H.-H. Fruehauf, M. Pogosbekyan, S. Kanne, O. Knab, H.-H. Fruehauf, M. Pogosbekyan, and S. Losev. Calibration of the CVCV-model against quasiclassical trajectory calculations. 32nd Thermophysics Conference, 1997. [164] M. V. Ivanov, R. Schinke, and G. C. Mcbane. Theoretical investigation of vi1 + 3 brational relaxation of NO(2 Π ), O2 (3 Σ− g ), and N2 ( Σg )in collisions with O( P). Molecular Physics, 105(9):1183–1191, 2007. [165] Sinan Akpinar, Iole Armenise, Paolo Defazio, Fabrizio Esposito, Pablo Gamallo, Carlo Petrongolo, and Ram´on Say´os. Quantum mechanical and quasiclassical Born-Oppenheimer dynamics of the reaction N2 on the N2O and surfaces. Chem. Phys., 398:81–89, 2012. [166] A Lagana, A Lombardi, F Pirani, P Gamallo, R Say´os, I Armenise, M Cacciatore, F Esposito, and M Rutigliano. Molecular Physics of Elementary Processes Relevant to Hypersonics : Atom-Molecule, Molecule-Molecule and Atoms-Surface Processes. The Open Plasma Physics Journal, 7:48–59, 2014. [167] Chul Park, Richard L. Jaffe, and Harry Partridge. Chemical-Kinetic Parameters of Hyperbolic Earth Entry. JTHT, 15(1):76–90, 2001. [168] Deepak Bose and Graham V. Candler. Kinetics of the N2 + O yields N O + N reaction under thermodynamic nonequilibrium. JTHT, 10(1):148–154, jan 1996. [169] GG Chernyi, SA Losev, SO Macheret, and BV Potapkin. Physical and chemical processes in gas dynamics: cross sections and rate constants. volume i. Progress in Astronautics and Aeronautics, 196:311, 2002. [170] L. B. Ibragimova, G. D. Smekhov, and O. P. Shatalov. Dissociation rate constants of diatomic molecules under thermal equilibrium conditions. Fluid Dynamics, 34(1):153–157, 1999.

152 [171] Yevgeniy A Bondar and Mikhail S Ivanov. DSMC dissociation model based on two-temperature chemical rate constant. AIAA Paper, pages 2007–614, 2007. [172] Iain D Boyd. Computation of hypersonic flows using the direct simulation Monte Carlo method. Journal of Spacecraft and Rockets, 52(1):38–53, 2014. [173] Paul V Marrone and Charles E Treanor. Chemical relaxation with preferential dissociation from excited vibrational levels. Physics of Fluids (1958-1988), 6(9):1215–1221, 1963. [174] M Lino Da Silva, V Guerra, and J Loureiro. Two-temperature models for nitrogen dissociation. Chemical Physics, 342(1):275–287, 2007. [175] Daniil Andrienko and Iain D Boyd. Vibrational relaxation and dissociation in O2 -O mixtures. page 4021, 2016. [176] E Kustova, E Nagnibeda, G Oblapenko, A Savelev, and I Sharafutdinov. Advanced models for vibrational-chemical coupling in multi-temperature flows. Chemical Physics, 464:1–13, 2016.

VITA

153

VITA

Israel Borges Sebasti˜ ao Contact Information Purdue University School of Aeronautics & Astronautics Neil Armstrong Hall of Engineering, ARMS 2203 701 W. Stadium Ave., West Lafayette, IN 47907

office: (765) 494-7867 e-mail: [email protected]

Research Interests Rarefied gas dynamics, nonequilibrium reacting flows, and direct simulation Monte Carlo (DSMC) method; High-altitude aerothermodynamics and MEMS gas flow modeling; Atmospheric and vacuum spray freeze-drying technology.

Education PhD – Aeronautics & Astronautics (GPA: 3.8/4.0) Jan 2013 – Aug 2017 Purdue University, IN, USA Thesis: “Ab-initio collision models for DSMC and their applications to reacting flows” Advisor: Alina A. Alexeenko MSc – Aerospace Engineering and Technology (GPA: 4.0/4.2) Mar 2009 – Sep 2011 National Institute for Space Research, SP, Brazil Thesis: “Numerical simulation of MEMS-based cold gas micronozzle flows” Advisor: Wilson F. N. Santos BSc – Mechanical Engineering (GPA: 8.6/10.0) Jan 2004 – Jul 2008 University of Mogi das Cruzes, SP, Brazil Final Project: “Design and fabrication of a didactic bench for heat transfer study” (in Portuguese) Advisor: Jeronimo S. Travelho

154

Honors and Awards • Best poster Dr. Thomas Jenkins Student Award, Midwest Chapter Meeting, International Society of Lyophilization/Freeze-Drying (ISLFD) Conference, IL, USA, 2015 • 2nd Best Poster Award at Nanotechnology Student Advisory Council (NSAC) Undergraduate Student Research Symposium, Birck Nanotechnology Center, Purdue Discovery Park, IN, USA, 2015. Mentored the first author, Aaron Pikus. • 2013-2017 CNPq PhD Fellowship, National Council for Scientific and Technological Development (CNPq-Brazil), Grant GDE/201444/2012-7 • Outstanding Undergraduate Engineering Award for ranking first in class, Regional Engineering, Architecture and Agronomy Council (CREA), SP, Brazil, 2008 • Outstanding Undergraduate Engineering Award for ranking first in class, University of Mogi das Cruzes, SP, Brazil, 2008 • Outstanding Toolmaker Award for ranking first in class, National Service for Industrial Training (SENAI), SP, Brazil 2003

Professional Experience Purdue University, W. Lafayette, IN, USA Professor Alexeenko’s Research Team Jan 2013 – current Graduate Student: Research activities on rarefied gas flows and spray freeze-drying; Mentoring undergraduate and masters students; Grader and teaching assistant for Fall-2016 Prof. Alexeenko’s AAE 590 course - Molecular Gas Dynamics. National Institute for Space Research (INPE), SP, Brazil Center for Weather Forecasts and Climate Studies (CPTEC) Mar 2011 – Dec 2011 Programmer Analyst: Improvement of the computational performance of the regional weather prediction model ETA/CPTEC. Combustion and Propulsion Laboratory (LCP) Mar 2009 – Mar 2011 Graduate Student: DSMC investigation of convergent-divergent micronozzle flows with discontinuities in the divergent contour. CNPq Project No. 580249/2008-5. Computing and Applied Mathematics Laboratory (LAC) Aug 2007 – Jul 2008 Undergraduate Student: DSMC investigation of the impact of accommodation coefficients on supersonic microchannel flows. CNPq undergraduate research project. University of Mogi das Cruzes, SP, Brazil Vocational/Technical School Aug 2007 – Mar 2009 Lecturer: Thermal and Hydraulic Machines, and Materials Technology. SEW Eurodrive, SP, Brazil Investment Engineering Department Jan 2007 – Jul 2007 Engineer Trainee: Computer-aided mechanical design (AutoCad and SolidWorks), quotation, purchase, and tryout of new equipments and facilities.

155 Orbital - Industrial Automation, SP, Brazil Mechanical Project Department Nov 2003 – Dec 2006 Engineer Trainee: Computer-aided mechanical design of industrial automatic equipments, computer-aided manufacturing, and customer support for automotive and chemical industries. Villares Metals S.A., SP, Brazil Main Machining Shop Mar 2000 – Mar 2003 Machinist Apprentice: General and high precision machining, and maintenance activities.

Vocational/Technical Courses Toolmaker – National Service for Industrial Training, SP/Brazil Mechanical Design Technician – State Technical School, SP/Brazil Mechanical Technician – State Technical School, SP/Brazil Machinist – National Service for Industrial Training, SP/Brazil

Jan Jul Jan Jan

2003 2002 2001 2000

– – – –

Dec Dec Jun Dec

2003 2003 2002 2001

Computer/Programming Skills Linux, Shell, Python, Fortran, C++, MATLAB, OpenMP, MPI, SolidWorks, and LATEX 2ε .

Publications Journal Articles: J1. I.B. Sebasti˜ ao and W.F.N. Santos, “Gas-surface interaction effects on the flowfield structure of a high speed microchannel flow”, Applied Thermal Engineering, v. 52, p. 566-575, 2013. J2. I.B. Sebasti˜ ao and W.F.N. Santos, “Gas-surface interaction impact on heat transfer and pressure distributions of a high speed microchannel flow”, Applied Thermal Engineering, v. 62, p. 58-68, 2014. J3. I.B. Sebasti˜ ao and W.F.N. Santos, “Impact of surface discontinuities on flowfield structure of a micronozzle array”, Nanoscale and Microscale Thermophysical Engineering, v. 18, p. 54-79, 2014. J4. I.B. Sebasti˜ ao and W.F.N. Santos, “Numerical simulation of heat transfer and pressure distributions in micronozzles with surface discontinuities on the divergent contour”, Computers & Fluids, v. 92, p. 125-137, 2014. J5. I.B. Sebasti˜ ao and A. Alexeenko, “Consistent post-reaction vibrational energy redistribution in DSMC simulations using TCE model”, Physics of Fluids, v. 28(10), p. 107103, 2016. J6. I.B. Sebasti˜ ao, M.F. Kulakhmetov, and A. Alexeenko, “DSMC study of oxygen shockwaves based on high-fidelity vibrational relaxation and dissociation models”, Physics of Fluids, v. 29(1), p. 017102, 2017. J7. D. Strongrich, A. Pikus, I.B. Sebasti˜ ao, and A. Alexeenko, “Microscale In-Plane Knudsen Radiometric Actuator: Design, Characterization, and Performance Modeling”, Journal of Microelectromechanical Systems, v. 26(3), p. 528-538, 2017.

156 J8. I.B. Sebasti˜ ao, T.D. Robinson, and A. Alexeenko, “Atmospheric Spray FreezeDrying: Numerical Modeling and Comparison With Experimental Measurements”, Journal of Pharmaceutical Sciences, v. 106(1), p. 183-192, 2017 (selected for inclusion in the Peter York’s dedicated issue). Conference Papers: C1. I.B. Sebasti˜ ao and J.S. Travelho, “Experimental study of natural convection in flat heated surfaces under ordinary environments” (in Portuguese), 6th National Congress of Mechanical Engineering (CONEM), 2010, Campina Grande, PB, Brazil. C2. I.B. Sebasti˜ ao and W.F.N. Santos, “Gas-surface interaction effects on the flowfield structure of high speed microchannel flow”, 13th Brazilian Congress of Thermal Engineering and Sciences (ENCIT), 2010, Uberlˆ andia, MG, Brazil. C3. I.B. Sebasti˜ ao and W.F.N. Santos, “Gas-surface interaction impact on heat transfer and pressure distribution of high speed microchannel flow”, 4th Southern Conference on Computational Modeling (MCSUL), 2010, Rio Grande, RS, Brazil. C4. I.B. Sebasti˜ ao and W.F.N. Santos, “Surface smoothness on flowfield structure in a micronozzle array”, 32nd Iberian Latin American Congress on Computational Methods in Engineering (CILAMCE), 2011, Ouro Preto, MG, Brazil. C5. I.B. Sebasti˜ ao and W.F.N. Santos, “Surface curvature influence on flowfield structure in a micronozzle array”, 21st International Congress of Mechanical Engineering (COBEM), 2011, Natal, RN, Brazil. C6. I.B. Sebasti˜ ao and A. Alexeenko, I. B. Sebasti˜ ao, A. Alexeenko, “DSMC Investigation of Nonequilibrium Effects in a H2 -O2 Unstretched Diffusion Flame”, 45th AIAA Thermophysics Conference, Dallas, TX, June 22-25 AIAA Paper 2015-3372. C7. M. Kulakhmetov, I.B. Sebasti˜ ao, M. Gallis, and A. Alexeenko, “Maximum entropy modeling of vibrational-translational energy exchange in O2 +O collisions”, 54th AIAA Aerospace Sciences Meeting, San Diego, CA, January 3-8, 2016, Paper 2016-0504. C8. A. Strongrich, A. Pikus, I.B. Sebasti˜ ao, D. Peroulis, and A. Alexeenko, “Lowpressure gas sensor exploiting the Knudsen thermal force: DSMC modeling and experimental validation”, IEEE 29th International Conference on MicroElectroMechanical Systems (MEMS), Shanghai, China, pp. 828-831, 2016. C9. M. Kulakhmetov, I.B. Sebasti˜ ao, and A. Alexeenko, “Adapting vibrational relaxation models in DSMC and CFD to ab-initio calculations”, 46th AIAA Thermophysics Conference, AIAA Aviation Forum, AIAA Paper 2016-3844. C10. A. Alexeenko, A. Strongrich, A. Cofer, A. Pikus, I.B. Sebasti˜ ao, S. Tholeti, and G. Shivkumar, “Microdevices enabled by rarefied flow phenomena”, 30th International Symposium on Rarefied Gas Dynamics: RGD 30, AIP Conference Proceedings, Vol. 1786 (1), article 080001. C11. A. Pikus, I.B. Sebasti˜ ao, A. Strongrich, A. Alexeenko, “DSMC simulation of microstructure actuation by Knudsen thermal forces including binary mixtures”, 30th International Symposium on Rarefied Gas Dynamics: RGD 30, AIP Conference Proceedings,

157 Vol. 1786 (1), article 080003. C12. I.B. Sebasti˜ ao, M. Kulakhmetov, A. Alexeenko,“Comparison between phenomenological and ab-initio reaction and relaxation models in DSMC”, 30th International Symposium on Rarefied Gas Dynamics: RGD 30, AIP Conference Proceedings, Vol. 1786 (1), article 150015. C13. I.B. Sebasti˜ ao, H. Luo, M. Kulakhmetov, and A. Alexeenko, “DSMC implementation of compact state-specific N2 +O dissociation and exchange models”, 55th AIAA Aerospace Sciences Meeting, Grapevine, TX, January 9-13, 2017, Paper 2017-1842. Technical Reports: R1. I.B. Sebasti˜ ao and W.F.N. Santos, “Performance analysis of MEMS-based micronozzles for satellite attitude control” (in Portuguese), Research report for the National Council for Scientific and Technological Development (CNPq), Project No. 580249/2008-5, SP, Brazil, 2011. R2. I.B. Sebasti˜ ao and J.S. Travelho, “Study of the influence of the accommodation coefficient in microchannel flows”(in Portuguese), Undergraduate research project (PIBIC/INPE/CNPq), SP, Brazil, 2008. Posters, Workshop & Seminar Presentations: P1. A. Pikus, I. Sebasti˜ ao, A. Strongrich, A. Alexeenko, “DSMC Simulation of Microstructure Actuation by Knudsen Thermal Force”, 68th Annual Meeting of the American Physical Society Division of Fluid Dynamics, 2015 Boston, MA. P2. I.B. Sebasti˜ ao, M. Kulakhmetov, and A. Alexeenko, “Post-Reaction Vibrational Energy Redistribution in TCE Model”, DSMC 2015 Conference, 2015, Kauai, HI. P3. I.B. Sebasti˜ ao, T.D. Robinson, and A. Alexeenko, “Atmospheric Spray Freeze Drying, a New Approach to Drying Solutions: Modeling and Measurements”, 7th Meeting of the Midwest Chapter of the International Society of Lyophilization/Freeze-Drying (ISLFD) Conference, 2015, Chicago, IL. P4. I.B. Sebasti˜ ao, T.D. Robinson, and A. Alexeenko, “Atmospheric Spray Freeze Drying: Process Modeling and Measurements”, American Association of Pharmaceutical Scientists (AAPS) Annual Meeting and Exposition, 2015, Orlando, FL. P5. I.B. Sebasti˜ ao, “The DSMC Method: A Brief Introduction and Applications”, Onehour guest lecture for Prof. Hu’s MA 692 course - Introduction to Kinetic Theory, Purdue University, Spring 2016, W. Lafayette, IN. P6. I.B. Sebasti˜ ao, “The DSMC Method: A Brief Introduction and Applications”, Twohour guest lecture for Prof. Alexeenko’s AAE 590 course - Molecular Gas Dynamics, Purdue University, September, Fall 2016, W. Lafayette, IN.

158

Other Skills and Activities Serving as peer reviewer to: • Physics of Fluids

• Journal of Vacuum Science and Technology A Research mentoring: • Aaron Pikus, Purdue AAE undergraduate student, 2015-current. • Aizhan Ibrayeva, Purdue AAE MSc student, 2016-current.