Introduction to Reactive Gas Dynamics...CONTENTS vii 4.2 Gas mixtures with elastic collisions 100 4.2.1 Chapman-Enskog method 100 4.2.2 Transport terms: Navier-Stokes equations 103

Introduction to Reactive Gas Dynamics

Raymond Brun

OXPORD UNIVERSITY PRESS

Contents

Introduction xiii

General Notations xvii

Part I Fundamental Statistical Aspects 1

Notations to Part I 3

1 Statistical Description and Evolution of Reactive Gas Systems 5 1.1 Introduction 5

1.2 Statistical description 6

1.2.1 State parameters 7

1.2.2 Transport parameters 9

1.3 Evolution of gas systems 11

1.3.1 Boltzmann equation 11

1.3.2 General properties 12

1.3.3 Macroscopic balance equations 12

1.4 General properties of collisions 14

1.4.1 Elastic collisions 14

1.4.2 Inelastic collisions 17

1.4.3 Reactive collisions 18

1.5 Properties of collisional terms 18

1.5.1 Collisional term expressions 18

1.5.2 Characteristic times: collision frequencies 21

Appendix 1.1 Elements of tensorial algebra 22

Appendix 1.2 Elements of molecular physics 25

Appendix 1.3 Mechanics of collisions 31

2 Equilibrium and Non-Equil ibrium Collisional Regimes 36

2.1 Introduction 36

2.2 Collisional regimes: generalities 37

2.3 Pure gases: equilibrium regimes 38

2.3.1 Monatomic gases 39

2.3.2 Diatomic gases 41

vi CONTENTS

2.4 Pure diatomic gases: general non-equilibrium regime 43

2.5 Pure diatomic gases: specific non-equilibrium regimes 46

2.5.1 Dominant TV collisions 47

2.5.2 Dominant VV collisions 47

2.5.3 Dominant resonant collisions 49

2.5.4 Physical applications of the results 50

2.6 Gas mixtures: equilibrium regimes 50

2.6.1 Mixtures of monatomic gases 50

2.6.2 Mixtures of diatomic gases 51

2.7 Mixtures of diatomic gases in vibrational non-equilibrium 52

2.8 Mixtures of reactive gases 53

2.8.1 Reactive gases without internal modes 53

2.8.2 Reactive gases with internal modes 55

Appendix 2.1 The H theorem 56

Appendix 2.2 Properties of the Maxwellian distribution 57

Appendix 2.3 Models for internal modes 59

Appendix 2.4 General vibrational relaxation equation 60

Appendix 2.5 Specific vibrational relaxation equations 62

Appendix 2.6 Properties of the Eulerian integrals 65

Transport and Relaxation in Quasi-Equilibrium Regimes: Pure Gases 66 3.1 Introduction 66

3.2 Expansion of the distribution function 66

3.2.1 Definition of flow regimes 66

3.2.2 Classification of flow regimes 68

3.3 First-order solutions 69

3.3.1 Pure gases with elastic collisions: monatomic gases 70

3.3.2 Pure diatomic gases with one internal mode 75

3.3.3 Pure diatomic gases with two internal modes 82

Appendix 3.1 Orthogonal bases 87

Appendix 3.2 Systems of equations for a, b, d coefficients 91

Appendix 3.3 Expressions of the collisional integrals 92

Appendix 3.4 Influence of the collisional model on the transport terms 95

Appendix 3.5 Linearization of the relaxation equation 96

Appendix 3.6 Vibrational non-equilibrium distribution 98

Transport and Relaxation in Quasi-Equilibrium Regimes: Gas Mixtures 100 4.1 Introduction 100

CONTENTS vii

4.2 Gas mixtures with elastic collisions 100

4.2.1 Chapman-Enskog method 100

4.2.2 Transport terms: Navier-Stokes equations 103

4.3 Binary mixtures of diatomic gases 106

4.3.1 One internal mode 106

4.3.2 Two internal modes 109

4.4 Mixtures of reactive gases 112

Appendix 4.1 Systems of equations for a, b, I, d coefficients 113

Appendix 4.2 Collisional integrals and simplifications 117

Appendix 4.3 Simplified transport coefficients 122

Appendix 4.4 Alternative technique: Gross-Jackson method 124

Appendix 4.5 Alternative technique: method of moments 128

Transport and Relaxation in Non-Equilibrium Regimes 131 5.1 Introduction 131

5.2 Vibrational non-equilibrium gases: SNE case 131

5.2.1 Pure diatomic gases 131

5.2.2 Mixtures of diatomic gases 135

5.2.3 Usual approximations: SNE case 137

5.3 Mixtures of reactive gases: (SNE)c case 138

5.3.1 (SNE)c + (WNE)l/case 138

5.3.2 (SNE)C+(SNEV case 144

Appendix 5.1 Pure gases in vibrational non-equilibrium 147

Appendix 5.2 First-order expression of the vibrational relaxation equation 149

Appendix 5.3 Gas mixtures in vibrational non-equilibrium 150

Appendix 5.4 Expressions of g coefficients and relaxation pressure 154

Appendix 5.5 Vibration-dissociation-recombination interaction 156

Generalized Chapman-Enskog Method 160 6.1 Introduction 160

6.2 General method 160

6.3 Vibrational^ excited pure gases 162

6.3.1 Transport terms 164

6.3.2 Approximate expressions of heat fluxes 165

6.4 Extension to mixtures of vibrational non-equilibrium gases 166

6.5 Reactive gases 167

6.6 Conclusions on non-equilibrium flows 169

Appendix 6.1 Vibrational^ excited pure gases 169

Appendix 6.2 Transport terms in non-dissociated media 171

viii CONTENTS

Appendix 6.3 Example of gases with dominant W collisions 173

Appendix 6.4 A simplified technique: BGK method 175

Appendix 6.5 Boundary conditions for the Boltzmann equation 178

Appendix 6.6 Free molecular regime 181

Appendix 6.7 Direct simulation Monte Carlo methods 183

Appendix 6.8 Hypersonic flow regimes 186

Part II Macroscopic Aspects and Applications 189

Notations to Part II 191

7 General Aspects of Gas Flows 195 7.1 Introduction 195

7.2 General equations: macroscopic aspects and review 195

7.2.1 Comments on the transport terms 196

7.2.2 Particular forms of balance equations 197

7.2.3 Entropy balance 199

7.2.4 Boundary conditions 200

7.3 Physical aspects of the general equations 201

7.3.1 Characteristic quantities 201

7.3.2 Dimensionless conservation equations 202

7.3.3 Dimensionless numbers: flow classification 204

7.4 Characteristic general flows 207

7.4.1 Steady flows 207

7.4.2 Unsteady flows 209

7.4.3 Simplified flow models 210

7.4.4 Stability of the flows: turbulent flows 211

Appendix 7.1 General equations: review 212

Appendix 7.2 Unsteady heat flux at a gas-solid interface 216

Appendix 7.3 Gas-liquid interfaces 217

Appendix 7.4 Dimensional analysis 219

Appendix 7.5 Generalities on total balances 220

Appendix 7.6 Elements of magnetohydrodynamics 221

8 Elements of Gas Dynamics 224 8.1 Introduction 224

8.2 Ideal gas model: consequences 224

8.3 Isentropic flows 226

8.3.1 One-dimensional steady flows 226

CONTENTS ix

8.3.2 Multidimensional steady flows 226

8.3.3 One-dimensional unsteady flows 227

8.4 Shock waves and flow discontinuities 229

8.4.1 Straight shock wave: Rankine-Hugoniot relations 229

8.4.2 Ideal gas model 230

8.5 Dissipative flows 231

8.5.1 Domain of influence: boundary layer 231

8.5.2 General equations: two-dimensional flows 233

Appendix 8.1 Method of characteristics 236

Appendix 8.2 Fundamentals of supersonic nozzles 237

Appendix 8.3 Shock waves: configuration and kinematics 239

Appendix 8.4 Generalities on the boundary layer 242

Appendix 8.5 Simple boundary layers: typical cases 247

Appendix 8.6 The turbulent boundary layer 252

Appendix 8.7 Flow separation and drag in MHD 255

Reactive Flows 259 9.1 Introduction 259

9.2 Generalities on chemical reactions 259

9.3 Equilibrium flows 260

9.3.1 Law of mass action: chemical equilibrium constant 260

9.3.2 Examples of reactions 261

9.3.3 Examples of equilibrium flows 264

9.4 Non-equilibrium flows 266

9.4.1 Chemical kinetics 266

9.4.2 Vibrational kinetics 268

9.4.3 General kinetics 271

9.5 Typical cases of Eulerian non-equilibrium flows 271

9.5.1 Flow behind a straight shock wave 271

9.5.2 Flow in a supersonic nozzle 278

9.5.3 Flow around a body 282

Appendix 9.1 Evolution of vibrational populations behind a shock wave 283

9.1.1 Evolution without dissociation 284

9.1.2 Evolution with dissociation 285

Appendix 9.2 Air chemistry at high temperature 286

9.2.1 Air chemistry in equilibrium conditions 286

9.2.2 Ionization phenomena 287

Appendix 9.3 Reaction-rate constants 290

Appendix 9.4 Nozzle flows 292

x CONTENTS

10 Reactive Flows in the Dissipative Regime 294 10.1 Introduction 294

10.2 Boundary layers in chemical equilibrium 295

10.2.1 The flat plate 295

10.2.2 The stagnation point 296

10.2.3 Reactive boundary layer and wall catalycity 298

10.2.4 Boundary layer along a body 300

10.3 Boundary layers in vibrational non-equilibrium 300

10.3.1 Example 1: boundary layer behind a moving shock wave 300

10.3.2 Example 2: boundary layer in a supersonic nozzle 301

10.3.3 Example 3: boundary layer behind a reflected shock wave 303

10.4 Two-dimensional flows 305

10.4.1 Hypersonic flow in a nozzle 305

10.4.2 Hypersonic flow around a body 308

10.4.3 Mixtures of supersonic reactive jets 311

Appendix 10.1 Catalycity in the vibrational non-equilibrium regime 313

Appendix 10.2 Generalized Rankine-Hugoniot relations 315

Appendix 10.3 Unsteady boundary layers 316

Appendix 10.4 CO2/N2 gas-dynamic lasers 317

Appendix 10.5 Transport terms in the non-equilibrium regime 320

Appendix 10.6 Numerical method for solving the Navier-Stokes equations 323

11 Facilities and Experimental Methods 326 11.1 Introduction 326

11.2 The shock tube 327

11.2.1 Simple shock tube theory 327

11.2.2 Disturbing effects 330

11.2.3 Reflected Shockwaves 335

11.2.4 General techniques: configurations and operation 337

11.2.5 General methods of measurement 341

11.3 The hypersonic tunnel 347

11.3.1 Generalities 347

11.3.2 The hypersonic shock tunnel 347

Appendix 11.1 Experiments in real flight 350

Appendix 11.2 Optimum flow duration in a shock tube 352

Appendix 11.3 Heat flux measurements in a shock tube 353

Appendix 11.4 Shock-interface interactions 355

Appendix 11.5 Operation of a free-piston shock tunnel 356

Appendix 11.6 Source flow in hypersonic nozzles 358

CONTENTS xi

12 Relaxation and Kinetics in Shock Tubes and Shock Tunnels 360 12.1 Introduction 360

12.2 Vibrational relaxation 361

12.2.1 Relaxation times: general methods 361

12.2.2 Vibrational populations 366

12.2.3 Vibrational catalycity 372

12.3 Chemical kinetics 374

12.3.1 Dissociation-rate constants 374

12.3.2 Time-resolved spectroscopic methods 376

12.3.3 Chemical catalycity 382

12.3.4 Hypersonic flow around bodies 383

Appendix 12.1 Generalities on IR emission 385

Appendix 12.2 Models for vibration relaxation times 386

Appendix 12.3 Simulation of emission spectra 387

Appendix 12.4 Precursor radiation in shock tubes 391

Appendix 12.5 Examples of kinetic models 394

References 397

Index 405