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1.2. State of the art on physico-chemical modeling and reduction techniques . . . . 3

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Contents

Nomenclature xxi

1. Introduction 1

1.1. Challenges for future space missions . . . . 1

1.2. State of the art on physico-chemical modeling and reduction techniques . . . . 3

1.3. Objectives of this thesis . . . . 8

1.4. Manuscript Structure . . . . 9

1.5. Publications . . . . 10

2. Physico-chemical modeling of non-equilibrium plasma 13 2.1. Governing equations . . . . 13

2.2. Thermodynamic and transport properties . . . . 15

2.2.1. Kinetic mechanisms . . . . 15

2.2.2. Multi-Temperature thermodynamic properties . . . . 18

2.2.3. State-to-state thermodynamic properties . . . . 22

2.2.4. Multi-Temperature transport properties . . . . 24

2.3. Chemical production and energy transfer . . . . 26

2.3.1. Chemical mechanisms . . . . 26

2.3.2. Rate coefficients . . . . 30

2.3.3. Energy/chemistry exchange . . . . 33

3. Numerical calculation of normal shock waves 35 3.1. The Ordinary Differential Equations solver . . . . 35

3.2. Detailing the shock structure . . . . 38

3.3. Validation with UTIAS shock tube experiments . . . . 41

3.4. Influence of free-stream conditions . . . . 42

3.4.1. Pressure . . . . 42

3.4.2. Speed . . . . 44

3.5. Study of the electronically excited levels . . . . 44

3.5.1. Population distribution . . . . 44

3.5.2. Boltzmann excitation temperatures . . . . 45

3.6. Radiative properties . . . . 46

3.7. Influence of the various processes . . . . 46

3.7.1. Influence of the excitation and ionization reactions . . 46

3.8. Thermodynamic energy transfer compared to chemistry . . . 50

3.9. Study of the entropy . . . . 52

3.9.1. Entropy . . . . 52

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x Contents

3.9.2. Entropy production . . . . 54

3.9.3. Degree of disequilibrium . . . . 55

3.10. Normal shock waves in nitrogen . . . . 62

3.11. Conclusions on the dynamical study of normal shock waves . 64 4. A derived coarse grain model for argon 65 4.1. A modified reactive scheme . . . . 65

4.2. Governing equations in the shock frame . . . . 69

4.3. Entropy verification . . . . 73

4.3.1. Deriving the entropy production . . . . 73

4.3.2. Proof of the second law . . . . 77

4.4. Implementation of the model in the shock relaxation code . . 79

4.5. Conclusions on the novel coarse grain model for argon . . . . 79

5. Principal Component Analysis 81 5.1. Mathematical formulation and definition of PCA . . . . 81

5.2. Score-PCA . . . . 84

5.3. Manifold Generated Principal Component Analysis (MG-PCA) 85 5.3.1. Local MG-PCA . . . . 86

5.4. Pre-processing of the sample PCA data . . . . 87

5.4.1. Outlier detection and removal . . . . 87

5.4.2. Centering and scaling . . . . 88

5.5. Choosing a subset of principal variables . . . . 89

5.6. Rotation of principal components . . . . 90

5.7. Numerical implementation of PCA for CFD . . . . 91

6. Principal component analysis on a collisional-radiative argon model 95 6.1. Manifold Generated Principal component analysis . . . . 95

6.1.1. Pre-processing : centering and scaling . . . . 96

6.1.2. System reduction and principal variable extraction . . 96

6.1.3. A posteriori model evaluation . . . . 97

6.2. Local MG-PCA . . . 103

6.2.1. Pre-processing: centering and scaling . . . 104

6.2.2. Clustering algorithm for local MG-PCA . . . 104

6.2.3. A posteriori model evaluation . . . 105

6.3. Score principal component analysis . . . 108

6.3.1. A posteriori model evaluation . . . 108

6.4. Conclusions on the reduction of the argon CR model . . . 117

7. Reduction of the N2-N mechanism 121 7.1. State of the art . . . 121

7.2. Vibrational collisional model . . . 122

7.2.1. Reduction of the VC model . . . 123

7.3. Binning models . . . 127

7.3.1. The BRVC model . . . 127

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Contents xi

7.3.2. Reduction of the BRVC model . . . 129

7.3.3. The URVC model . . . 130

7.3.4. Reduction of the URVC model . . . 133

7.4. Conclusions on the reduction of the N-N

2

model . . . 138

8. Conclusion 141 8.1. Conclusions . . . 141

8.2. Suggestions for future work . . . 143

Appendices 145

A. Coefficients for transport properties 147

B. Degree of disequilibrium classification 149

Bibliography 159

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