High Temperature Phenomena in Shock Waves

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Shock Wave Science and Technology Reference Library

The new Springer collection, Shock Wave Science and Technology Reference Library, conceived in the style of the famous Handbuch der Physik has as its principal motivation to assemble authoritative, state-of-the-art, archival reference articles by leading scientists and engineers in the field of shock wave research and its applications. A numbered and bounded collection, this reference library will consist of specifically commissioned volumes with internationally renowned experts as editors and contributing authors. Each volume consists of a small collection of extensive, topical and independent surveys and reviews. Typical articles start at an elementary level that is accessible to non-specialists and beginners. The main part of the articles deals with the most recent advances in the field with focus on experiment, instrumentation, theory, and modeling. Finally, prospects and opportunities for new developments are examined. Last but not least, the authors offer expert advice and cautions that are valuable for both the novice and the well-seasoned specialist.

Shock Wave Science and Technology Reference Library

Collection Editors Hans Grönig Hans Grönig is Professor emeritus at the Shock Wave Laboratory of RWTH Aachen University, Germany. He obtained his Dr. rer. nat. degree in Mechanical Engineering and then worked as postdoctoral fellow at GALCIT, Pasadena, for one year. For more than 50 years he has been engaged in many aspects of mainly experimental shock wave research including hypersonics, gaseous and dust detonations. For about 10 years he was Editorin-Chief of the journal Shock Waves.

Yasuyuki Horie Professor Yasuyuki (Yuki) Horie is internationally recognized for his contributions in high-pressure shock compression of solids and energetic materials modeling. He is a co-chief editor of the Springer series on Shock Wave and High Pressure Phenomena and the Shock Wave Science and Technology Reference Library, and a Liaison editor of the journal Shock Waves. He is a Fellow of the American Physical Society, and Secretary of the International Institute of Shock Wave Research. His current interests include fundamental understanding of (a) the impact sensitivity of energetic solids and its relation to microstructure attributes such as particle size distribution and interface morphology, and (b) heterogeneous and nonequilibrium effects in shock compression of solids at the mesoscale.

Kazuyoshi Takayama Professor Kazuyoshi Takayama obtained his doctoral degree from Tohoku University in 1970 and was then appointed lecturer at the Institute of High Speed Mechanics, Tohoku University, promoted to associate professor in 1975 and to professor in 1986. He was appointed director of the Shock Wave Research Center at the Institute of High Speed Mechanics in 1988. The Institute of High Speed Mechanics was restructured as the Institute of Fluid Science in 1989. He retired in 2004 and became emeritus professor of Tohoku University. In 1990 he launched Shock Waves, an international journal, taking on the role of managing editor and in 2002 became editorinchief. He was elected president of the Japan Society for Aeronautical and Space Sciences for one year in 2000 and was chairman of the Japanese Society of Shock Wave Research in 2000. He was appointed president of the International Shock Wave Institute in 2005. His research interests range from fundamental shock wave studies to the interdisciplinary application of shock wave research.

R. Brun (Ed.)

High Temperature Phenomena in Shock Waves With 179 Figures, 48 in Color, and 41 Tables

ABC

Prof. Raymond Brun Université de Provence 42 Chemin des Petits Cadeneaux 13170 Les Pennes Mirabeau France E-mail: [email protected]

ISBN: 978-3-642-25118-4

e-ISBN: 978-3-642-25119-1

DOI 10.1007/978-3-642-25119-1 Library of Congress Control Number: 2011940964 © 2012 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com

Contents

General Introduction.................................................................................................. 1 R. Brun References .................................................................................................................... 9 Chapter 1: Thermodynamic Properties of Gases behind Shock Waves .............. 11 M. Capitelli, D. Bruno, G. Colonna, G. d’Ammando, A. d’Angola, D. Giordano, C. Gorse, A. Laricchiuta, S. Longo 1 Introduction ...................................................................................................... 11 2 Partition Functions for Atomic Levels. General and Few-Level Approaches ........................................................................................................ 13 3 Partition Functions for Diatomic Molecules..................................................... 15 4 Transport Cross Sections and Collision Integrals............................................. 18 4.1 Phenomenological Approaches for Unknown Colliding Systems ........... 19 4.2 Resonant Charge Transfer........................................................................ 20 4.3 Neutral-Neutral Interactions.................................................................... 22 4.4 Neutral-Ion Interactions ........................................................................... 25 4.5 Charge-Charge Interactions ..................................................................... 26 4.6 Electron-Neutral Interactions ................................................................... 27 5 Thermodynamic Properties of Thermal Plasmas: The Cut-Off Issue............... 28 5.1 Cut-Off Criteria........................................................................................ 29 5.1.1 The Ground State Method ............................................................. 29 5.1.2 Debye-Hückel Based Criteria........................................................ 29 5.1.3 The Fermi Criterion....................................................................... 30 5.2 The Cut-Off Criteria Based on the Solution of Schrödinger Equation..... 30 5.3 Case Study: Air Plasma............................................................................ 34 6 Transport of Two-Temperature Plasmas .......................................................... 41 7 Transport Properties of Plasmas: The Role of Electronically Excited States ... 43 8 Conclusions ...................................................................................................... 47 Acknowledgments.. ............................................................................................... 47 Appendix: Thermodynamic and Transport Properties of Air Plasmas .................. 48 References .................................................................................................................. 54 Chapter 2: Non-equilibrium Kinetics and Transport Properties behind Shock Waves .......................................................................................... 59 E.V. Kustova, E.A. Nagnibeda 1 Introduction ...................................................................................................... 59 2 State-to-State Approach.................................................................................... 60

VI

Contents

2.1 Distribution Functions and Macroscopic Parameters............................... 60 2.2 Governing Equations................................................................................ 61 2.3 First-Order Approximation. ..................................................................... 63 3 Quasi-stationary Approaches............................................................................ 71 3.1 Vibrational Distributions: Governing Equations...................................... 71 3.2 Transport Terms....................................................................................... 75 3.3 Production Terms..................................................................................... 77 4 Non-equilibrium Processes behind Shock Waves in Air Components and CO2 Mixtures ............................................................................................. 79 4.1 Non-equilibrium Kinetics and Transport Properties in Diatomic Gas Mixtures............................................................................. 79 4.1.1 Governing Equations and Flow Parameters ................................... 79 4.1.2 Transport Properties ....................................................................... 83 4.1.3 Electronic Excitation and Radiation............................................... 84 4.2 Non-equilibrium Kinetics and Transport Processes in Air Mixture......... 87 4.3 Mixtures Containing CO2 Molecules ....................................................... 90 References .................................................................................................................. 96 Chapter 3: Non-equilibrium Kinetics behind Shock Waves Experimental Aspects .................................................................................................... 99 L. Ibraguimova, O. Shatalov 1 Introduction ...................................................................................................... 99 2 Vibrational Relaxation of Diatomic Molecules .............................................. 100 2.1 Vibrational Relaxation of O2, N2, CO, NO ............................................ 100 2.2 Vibrational Relaxation of Diatomic Molecules in Collisions with Potentially Reactive Atoms .................................................................... 104 2.3 Vibrational Relaxation of H2 and D2 ...................................................... 106 2.4 Vibrational Relaxation of Halides and Hydrogen Halides ..................... 107 3 V-T Relaxation of Three- and Multi-atomic Molecules ................................. 111 3.1 CO2 V-T Relaxation............................................................................... 113 3.2 N2O V-T Relaxation............................................................................... 115 3.3 H2O V-T Relaxation............................................................................... 117 3.4 NO2 V-T Relaxation............................................................................... 118 4 Chemical Reactions ........................................................................................ 119 4.1 Chemical Reactions in the System C-O ................................................. 120 4.2 Chemical Reactions in the System N-C-O ............................................. 128 5 Non-equilibrium Radiation............................................................................. 135 References ................................................................................................................ 141 Chapter 4: Ionization Phenomena behind Shock Waves .................................... 149 W.M. Huo, M. Panesi, T.E. Magin 1 Introduction .................................................................................................... 149 2 Production, Reaction and Removal of Charged Species................................. 151 2.1 Electrons ................................................................................................ 151 2.1.1 Production of Electrons............................................................... 153

Contents

VII

2.1.2 Reactions Involving Electrons .................................................... 159 2.1.3 Electron Recombination.............................................................. 164 2.1.4 Interparticle Interactions and Free Electron Number Density..... 164 2.2 Ions......................................................................................................... 165 3 Modeling Collisional and Radiative Processes in a Weakly Ionized Plasma ............................................................................................................ 165 3.1 The Collisional-Radiative Model ........................................................... 165 3.1.1 Transport Equations .................................................................... 166 3.1.2 Reaction Source Terms ............................................................... 168 3.1.3 Radiative Processes and Radiative Transport.............................. 173 3.2 Results.................................................................................................... 179 3.2.1 Fire II Flight Experiment ............................................................ 179 3.2.2 Chemistry and Flow Field Energy Distribution .......................... 180 3.2.3 Radiative Transport and Interaction between Radiation and Matter................................................................................... 182 3.2.4 Quasi-steady State Distribution................................................... 184 3.2.5 Comparison with Experimental Data .......................................... 186 4 Conclusions .................................................................................................... 188 Acknowledgments. .............................................................................................. 188 References ................................................................................................................ 189 Chapter 5: Radiation Phenomena behind Shock Waves..................................... 193 M.Y. Perrin, Ph. Rivière, A. Soufiani 1 Introduction .................................................................................................... 193 2 Radiative Mechanisms and Radiative Properties............................................ 195 2.1 Bound-Bound Transitions....................................................................... 195 2.1.1 General Formulation .................................................................... 195 2.1.2 Atomic Line Spectra .................................................................... 198 2.1.3 Diatomic Line Spectra.................................................................. 200 2.2 Bound-Free Transitions.......................................................................... 204 2.2.1 General Formulation ................................................................... 204 2.2.2 Atomic Photoionization .............................................................. 205 2.2.3 Molecular Photodissociation ....................................................... 206 2.2.4 Molecular Photoionization .......................................................... 206 2.2.5 Photodetachment ......................................................................... 207 2.3 Free-Free Transitions ............................................................................. 207 3 Example of Application.................................................................................. 208 4 Radiative Transfer Modeling.......................................................................... 212 4.1 The Escape Factor Approach ................................................................. 213 4.2 Spectral Models...................................................................................... 214 4.2.1 Statistical Narrow Band Models ................................................. 214 4.2.2 Global Models............................................................................. 218 4.3 Geometrical Treatment of Radiative Transfer........................................ 219 4.3.1 The Discrete Ordinate Method (DOM) ....................................... 220 4.3.2 Spherical Harmonics (PN) and Related Methods........................ 220

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Contents

4.3.3 The Tangent Slab Approximation ............................................... 222 4.4 The Monte Carlo Method....................................................................... 223 5 Radiation and Flow-Field Coupling ............................................................... 225 6 Conclusion and Perspectives .......................................................................... 227 References ................................................................................................................ 227 Chapter 6: Structure of Shock Waves .................................................................. 231 A.A. Raines, F.G. Tcheremissine 1 Introduction .................................................................................................... 231 2 Methodology of Computations ....................................................................... 233 2.1 Solution of the Boltzmann Equation for a Pure Monatomic Gas ........... 233 2.2 Solution of the Generalized Boltzmann Equation .................................. 237 2.3 Two Levels Kinetic Model of RT Relaxation ........................................ 239 2.4 Solution of the Boltzmann Equation for a Gas Mixture......................... 241 2.5 Statement of the Boundary Problem and Presentation of the Computed Data ....................................................................................... 244 3 Shock Wave Structure in a Pure Monatomic Gas........................................... 245 3.1 Gas of Hard Sphere Molecules .............................................................. 245 3.2 The Lennard-Jones Gas.......................................................................... 250 4 Shock Wave Structure in a Polyatomic Gas ................................................... 253 4.1 Shock Wave Structure with Frozen Vibrational Levels ......................... 253 4.2 Shock Wave Structure with Excited Rotational and Vibrational Levels .................................................................................. 259 5 Shock Wave Structure in a Mixture of Monatomic Gases.............................. 261 6 Conclusion...................................................................................................... 266 References ................................................................................................................ 266 Chapter 7: Shock Waves in Hypersonic Rarefied Flows .................................... 271 V. Lago, A. Chpoun, B. Chanetz 1 Introduction .................................................................................................... 271 2 General Phenomena in Rarefied Flows .......................................................... 274 2.1 Flow Regime Classification ................................................................... 274 2.2 Shock Waves Thickness and Stand-Off Distance .................................. 275 2.2.1 Mixing Reynolds Number ........................................................... 275 2.2.2 Shock-Wave Stand-Off Distance Ahead of Blunt Bodies........... 276 2.2.3 Shock Wave Thickness Ahead of Blunt Bodies.......................... 276 2.2.4 Impact of Flat-Faced Leading-Edge Effects on Shock Stand-Off Distance and Shock Wave Thickness ......................... 277 2.2.5 Characterisation of Shock Waves in Rarefied Regime over a Flat Plate .......................................................................... 278 2.3 Heat Flux in Rarefied Conditions........................................................... 279 2.4 Leading Edge Flow and Viscous Interaction in Supersonic Rarefied Flow ......................................................................................... 280 2.5 Wall Pressure in Free Molecular Flow Regime ..................................... 281

Contents

IX

3 Experimental Approach.................................................................................. 281 3.1 Hypersonic Rarefied Wind Tunnel ........................................................ 281 3.2 Shock Wave-Boundary Layer Interactions in Low Density Flow ......... 283 3.3 Shock-Shock Interferences in Low Density........................................... 286 3.4 Pressure Measurements in Rarefied Flow Regimes ............................... 288 3.4.1 Pressure Measurements and Orifice Diameter Effects ................ 288 3.4.2 Pitot Pressure Measurement ........................................................ 288 3.4.3 Static Pressure Measurement ..................................................... 290 3.4.4 Pressure Transducers................................................................... 291 3.5 Heat Flux Measurements........................................................................ 291 3.6 Shock Wave Control .............................................................................. 293 References ................................................................................................................ 297 Chapter 8: High Enthalpy Non-equilibrium Shock Layer Flows: Selected Practical Applications ......................................................................... 299 S. Karl, J. Martinez Schramm, K. Hannemann 1 Introduction .................................................................................................... 299 2 Chemical Relaxation in High Enthalpy Cylinder Shock Layer Flow ............. 300 2.1 High Enthalpy Shock Tunnel Göttingen (HEG) .................................... 300 2.2 Phase Step Holographic Interferometry ................................................. 305 2.3 CFD Code .............................................................................................. 307 2.4 Experimental Setup and Results............................................................. 308 2.5 Summary and Conclusions..................................................................... 313 3 CFD Modeling of Radiation Phenomena in Shock Layers............................. 314 3.1 Introduction, Definitions and Nomenclature.......................................... 314 3.2 The Radiative Transfer Equation in Participating Media....................... 316 3.3 One-Dimendional Approximations for the Solution of the Radiative Transfer Equation ................................................................. 317 3.3.1 The Infinite Slab Model .............................................................. 317 3.3.2 Infinite Cylinder .......................................................................... 319 3.4 Approximate Solution Methods of the Radiative Transfer Equation in Three Dimensions ............................................................... 320 3.4.1 The Discrete Transfer Model ...................................................... 320 3.4.2 Solution of the Radiative Transfer Equation Using a Monte Carlo Method ...................................................... 321 3.4.3 Isothermal Cylinder..................................................................... 324 3.5 Huygens Entry Peak Heating Prediction ................................................ 327 4 Summary and Conclusions ............................................................................. 334 References ................................................................................................................ 335 Author Index........................................................................................................... 337

General Introduction R. Brun Université de Provence, Marseille, France

The production of high temperatures in gases constitutes one major feature of shock waves which represent one of the best means to transform kinetic energy into thermal energy. One of the most important consequences lies in the physical and chemical phenomena which may arise from these high temperatures such as rotational and vibrational excitation of molecules, dissociation, ionization and various chemical reactions, as well as associated radiation. Another fundamental feature proceeds from the fact that, in collisional regime (continuum), the shock wave may be considered as a discontinuity, so that the temperature rise is quasi-instantaneous. Thus, as the characteristic time required for the development of the physical and chemical phenomena is non-negligible, since it is related to collisions between elementary particles, the gaseous medium behind shock waves is in a non-equilibrium thermodynamic and chemical state. These essential points have thus contributed to the fundamental knowledge of the gaseous reactive flows at high temperature and, in particular, to the kinetics of the above phenomena from an experimental and theoretical point of view. Moreover, many applications of these properties, related to the development of aero-spatial flights, thermonuclear fusion or combustion, may thus be found in hypersonic flow, plasma generation or propulsion. After crossing a shock wave, the fluid particles, in absence of other perturbations, tend to a physical and chemical equilibrium state corresponding to the pressure and temperature conditions determined by flow conservation equations (Euler or NavierStokes) and boundary conditions, using adequate expressions for internal energy (or enthalpy). This variable, in equilibrium conditions, may be calculated a priori as a function of pressure and temperature by methods of statistical mechanics involving the computation of partition functions and transport cross-sections, keys of the determination of thermodynamic and transport properties in high temperature gases. Details on these computations are given in Chapter 1, with a particular emphasis on air plasmas obtained behind very strong shock waves. A simple introductory example is given by a straight shock wave for which the downstream equilibrium flow quantities result from the Rankine-Hugoniot relations: thus, in Fig. 1, equilibrium temperature and density ratios across a straight shock wave propagating in pure oxygen [1], i.e. T2 T1 and ρ 2 ρ1 are represented as functions of shock Mach number Ms . In the same way, as an example of chemical reactions taking place at high temperatures, the equilibrium composition of air as a function of temperature [2,3] is represented in Fig. 2.

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High Temperature Phenomena in Shock Waves

Fig. 1. Equilibrium temperature and density ratios across a straight shock wave : Temperature, : Density (Oxygen, T1 = 300K , p1 = 10 3 Pa )

Fig. 2. Equilibrium air composition (Mass concentrations, p = 10 7 Pa )

Before reaching equilibrium, the gas “just” behind the shock wave may be considered in a “frozen” state corresponding to its chemical state in front of the shock. Then, various reactions start with, generally, different characteristic times. Detailed kinetics of chemical reactions and vibrational populations is given in Chapter 2, in which the “state to state” approach as well as more global relaxation models are presented, including the computation of transport properties in non-equilibrium conditions. Examples of the kinetics of air and carbon dioxide are also given.

General Introduction

3

Many experimental aspects of vibrational relaxation and chemical kinetics behind shock waves are proposed in Chapter 3, owing to the numerous measurements of relaxation times and chemical rate constants carried out in shock tube [1,4,5] by means of various optical and spectroscopic diagnostic methods. As introductory examples, Fig. 3 represents the experimental evolution of vibrational populations of carbon monoxide behind a shock wave [6] and Fig. 4 show experimental values of dissociation rate constants of oxygen as functions of temperature [4].

Fig. 3. Evolution of the relative population of the 3rd and the 6th vibrational level behind a straight shock wave (CO ; M s = 5, 60; T1 = 293 K ; p1 = 196 Pa )

Fig. 4. Measured values of dissociation rate constant (Oxygen)

4

High Temperature Phenomena in Shock Waves

The interaction of these various processes with the flow parameters behind shock waves results in a variation of all macroscopic quantities in the non-equilibrium region. Thus in Fig. 5, an example of the variation of translation-rotation temperature, vibrational temperature and density behind a straight shock wave in nitrogen is presented [6].

Fig. 5. Evolution of temperatures and density ratio behind a straight shock wave (Nitrogen, M s = 6,12, pi = 3947 Pa , Ti = 295 K )

In the same way, the variation of temperatures and species concentrations behind a strong straight shock wave in air [6] is represented in Figs. 6 and 7 respectively.

Fig. 6. Spatial variation of temperatures behind a straight shock wave in air ( M s = 25, p1 = 8, 5 Pa, T1 = 205 K )

General Introduction

5

Fig. 7. Spatial variation of concentrations behind a straight shock wave in air (Conditions of Fig.6)

The coupling between the chemical processes themselves also may be important, as illustrated by Fig. 8, in which the evolution of the dissociation rate constant of nitrogen is represented, with (curve A) and without (curve B) neglecting the influence of the vibrational relaxation [6]. It may be also noted that the interaction between dissipative processes and non-equilibrium phenomena may lead to complex, indeed “anomalous” situations [7].

Fig. 8. Dissociation rate constants of nitrogen behind a straight shock wave in air ( M s = 25, p1 = 8, 5 Pa, T1 = 205 K ) A:

kD

Arrhenius, B:

kD

with vibrational interaction

6

High Temperature Phenomena in Shock Waves

When the Mach number becomes sufficiently high, the collisions between elementary particles are more intense and ionization phenomena can become important. Thus, the aim of Chapter 4 is to describe the various collision processes contributing to create ions and electrons as well as the reactions involving charged species. Applications to air plasmas in shock tube are given including radiative processes. An introductory example of the spatial variation of ionized species behind a straight shock wave in air is presented in Fig. 9.

Fig. 9. Spatial variations of ionized species concentrations behind a straight shock wave in air (Conditions of Fig.6)

It is of course impossible to dissociate the physical and chemical processes described in Chapters 1-4 from the radiation inherently connected to these processes. It is the aim of Chapter 5 to describe various radiative mechanisms in hot gases, particularly in non-equilibrium conditions. Examples are given, showing the importance of radiative fluxes in hypersonic flight and emphasizing the coupling between radiation and aerothermodynamics. An introductory example is given in Fig.10 which represents the experimental evolution of the spontaneous emission of the Δv = 0 band of the electronic transition B2Σ+ ↔ X2Σ+ of CN behind a strong shock wave in the Titan simulated atmosphere [8] ( 92%N 2 , 3%CH 4 , 5%Ar ). It is thus possible to deduce from Fig.10 the intensity profiles of the lines as functions of the wavelength (Fig. 11) at different instants. In the same way, the time evolution of the intensity of different lines may be determined, as represented in Fig. 12 where a strong overshoot of non-equilibrium radiation is clearly visible. From this type of results, vibrational populations, species concentrations and (or) temperatures can be deduced.

General Introduction

7

Fig. 10. Example of streak image behind a straight shock wave ( Δv = 0 , CN band of the Titan mixture CH 4 / N 2 / Ar ) U s = 5560m / s, p1 = 220 Pa

Fig. 11. Experimental spectra of the (Conditions of Fig.10)

Δv = 0 band of CN at two instants behind the shock

8

High Temperature Phenomena in Shock Waves

Fig. 12. Time evolution of the 0-0, 1-1, 2-2 vibrational bands (Conditions of Fig.10)

As stated above, in continuum regime, a shock wave can be represented as a discontinuity, but, in reality, the passage from an upstream state to a downstream one (frozen state) through a shock wave requires a few collisions between elementary particles, so that the “thickness” of the shock wave is of the order of several mean free paths: in this zone, the gas can be considered in strong translational and rotational non-equilibrium. Thus, Chapter 6 deals with methods used for the study of the shock wave itself, in which Euler or Navier-Stokes equations are invalid. These methods essentially consist in directly solving the Boltzmann equation. Distribution functions and profiles of macroscopic quantities can then be obtained inside the shock wave; solutions are presented for pure monatomic and diatomic gases, as well as for gas mixtures, and non-monotonous temperature profiles can be found. An example of temperature profiles obtained for a mixture He/Ar with the Direct Simulation Monte-Carlo Method [9] (DSMC) is presented in Fig.13. When the gaseous medium becomes “rarefied”, the mean free path is lengthening, so that the shock wave can no longer be considered as a discontinuity and presents a non-negligible thickness; then, non-equilibrium phenomena take particular aspects which are examined in Chapter 7: thus, different flow regimes may be defined between the continuum regime and the “free molecular” one, which are likely to be analysed by modified Navier-Stokes equations (slip flow) or purely numerical methods (DSMC); in the same way, specific diagnostic experimental methods are presented in this chapter. In the last chapter (Chapter 8), concrete examples of specific non-equilibrium flows are presented: thus, results of measurements of static pressure, heat flux, standoff distance, phase shift in the dissociated air flow around a cylinder placed in the test section of a free piston shock tunnel (HEG Göttingen) are presented. Moreover, various models for the computation of radiative heat flux in non-equilibrium conditions are presented, completed by a calculation of the peak heat flux undergone by the Huygens probe during its entry into the Titan atmosphere.

General Introduction

(

9

)

Fig. 13. Temperature profiles across a shock wave ⎡ M s = 8, nAr = 0.1⎤ , nHe a ⎣⎢ ⎦⎥ : TAr , a: upstream, b: downstream :T , : THe ,

Finally, we might consider that combustion phenomena induced by shock waves should be included in the present book; the particular specificity of this topics however, as well as the important developments required for its treatment, would be beyond the framework of one single chapter and should be the subject of one another book. To sum up, the main aspects of phenomena related to high temperatures prevailing behind shock waves are presented in the present book, at least the actual knowledge on the matter, which of course remains an active research field.

References 1. Gaydon, A.G., Hurle, I.R.: The Shock Tube in High Temperature Research. Chapman and Hall, London (1963) 2. Park, C.: Nonequilibrium Hypersonic Aerothermodynamics. J.Wiley, New-York (1990) 3. Vincenti, W.G., Krüger, C.H.: Introduction to Physical Gas Dynamics. R.G.Krieger, Florida (1965) 4. Stupochenko, Y.V., Losev, S.A., Osipov, A.I.: Relaxation in Shock Waves. Springer, Berlin (1967) 5. Oertel, H.: Stossrohre. Springer, Wien (1966) 6. Brun, R.: Introduction to Reactive Gas Dynamics. Oxford Univ. Press, Oxford (2009) 7. Belouaggadia, N., Armenise, I., Capitelli, M., Esposito, F., Brun, R.: J. Therm. Heat Transf. 24(4), 684 (2010) 8. Ramjaun, D., Dumitrescu, M.P., Brun, R.: J. Therm. Heat Transf. 13(2), 219 (1999) 9. Bird, G.A.: Rarefied Gas Dynamics, vol. 175. Tokyo Univ. Press, Tokyo (1984)

Chapter 1

Thermodynamic Properties of Gases behind Shock Waves M. Capitelli1,2, D. Bruno2, G. Colonna2, G. D’Ammando1, A. D’Angola3, D. Giordano4, C. Gorse1,2, A. Laricchiuta2, and S. Longo1,2 1

Department of Chemistry, University of Bari (Italy) CNR Institute of Inorganic Methodologies and Plasmas (IMIP) Bari (Italy) 3 Università della Basilicata, Potenza (Italy) 4 ESA ESTEC Aerothermodynamics Section Noordwijk (The Netherlands)

2

1 Introduction The research on high-energy shock wave is a field of large interest including nuclear explosion, hypersonic flows as well as laser forming plasmas. During the relevant interaction a high-temperature, high-pressure plasma is formed, which in some cases can be ascribed to the family of thermal plasmas, characterized by equilibrium between the different degrees of freedom, including chemical and internal ones. Thermal plasmas can be described by equilibrium chemical thermodynamics, in particular statistical thermodynamics is used in this field to get information about the input data (entropy, enthalpy and specific heat of single species). Thermal plasmas are usually characterized by a single temperature for all species, including the vibrationally, rotationally and electronically distributions among the excited states, while dissociation and ionization (Saha) equilibria characterize them. On the other hand thermal plasmas with different temperatures are still accepted in this kind of literature, the different temperatures characterizing the corresponding reservoirs of energy. The internal distributions are still Boltzmann at a given (different) temperature; chemical equilibrium thermodynamics again characterizes the plasma properties even though caution must be exercised in using it. Typical conditions for thermal plasmas are temperatures in the range of 5 000-50 000 K, pressure in the 10-2103 atm. range and ionization degree larger than 10-5. Characterization of thermal plasma flows is obtained by using NS (Navier-Stokes) fluid-dynamic codes; thermodynamics in this case provides to the CFD (computational fluid dynamics) community important input data for the different species as well as the properties of the mixture if the hypothesis of local equilibrium holds in the flow. Moreover NS equations need the transport properties (thermal conductivity, diffusion coefficients, viscosity and electrical conductivity) of the plasma components as well as of the mixture, these quantities determining the heat flux from plasma to solid samples that can be heated during the plasma-material interaction.

12

High Temperature Phenomena in Shock Waves

The experimental determination of both thermodynamic and transport properties of ionized gases is very difficult to be achieved so that one demands to the theory the calculation of these quantities. Statistical thermodynamics is used, as anticipated, to get information about the thermodynamic properties of the high temperature components, while statistical mechanics is used for getting information of transport properties through the Chapman-Enskog solution of the Boltzmann equation[1,2]. The key point in this characterization is the calculation of partition functions of atomic and molecular species as well as of transport cross sections (collision integrals) for the relevant interactions. The knowledge of the partition function, in fact, is the basis for calculating the thermodynamic properties of single species as well as of the mixture, while the transport cross sections are essential ingredients to calculate the transport properties of the system. In both cases we present in this chapter simplified and accurate methods to calculate these two quantities. Concerning the partition function of atomic species we will present essentially two methods, the first one based on the inclusion in the partition function of a complete set of energy levels, subjected to an appropriate cut-off criterion to avoid the divergence. This approach involves the insertion of thousands of electronic energy levels, representing a computational problem when these partition functions must be calculated in the mathematical grid used by CFD. An alternative approach is based on particular grouping of levels such to reproduce the thermodynamic behavior of the multilevel system. As an example the nitrogen atom is reduced to a three-level system composed by the ground state (4S), a second level, which coalesces, with appropriate energy and multiplicity, the two low-lying excited states 2P and 2D, and a third level, which accounts for the huge number of electronically excited states. Two levels of accuracy are also used to characterize the viscosity-type collision integrals of the different atom-atom, atom-ion interactions. In this case in fact accurate transport cross sections can be obtained by averaging the different contributions coming from the numerous potentials arising in the interaction. As an example two nitrogen atoms in the ground state (4S) can interact along four potentials 1,3,5,7 Σ and the transport cross sections are obtained by averaging the different contributions with suitable statistical weights. This procedure becomes prohibitive when the interaction occurs through unknown potentials, a situation met in the interaction between electronically excited states. An alternative is the use of an average potential and in this direction the phenomenological potential, which is actually an improvement of the Lennard-Jones, is a good candidate to get accurate values for transport cross sections. The situation is more complicated when dealing with atom-parent-ion interactions, the diffusion-type transport cross section being governed by the resonant chargeexchange process. In turn these cross sections can be obtained by using the huge number of gerade-ungerade (g-u) potential pairs arising in the interaction (e.g. for N(4S)-N+(3P) the g-u electronic pairs 2,4,6Σgu, 2,4,6Πgu of the molecular ion should be considered). The asymptotic theory could also be alternatively used for getting these data, avoiding the quantum mechanical derivation of the relevant potentials. Partition functions and transport cross sections are the ingredients for the calculation of plasma properties. However problems are met in the derivation of thermodynamic and transport properties of plasmas. In particular we refer to the dependence of thermodynamic properties of thermal plasmas on the cut-off criterion

Thermodynamic Properties of Gases behind Shock Waves

13

used in the calculation of self-consistent partition function. A similar problem arises in the calculation of transport coefficients i.e. the role of electronically excited states in affecting the transport properties of the plasmas. All these concepts will be analyzed in this chapter, which is divided in different sections. Section 2 is devoted to the calculation of partition function of atomic species, either using the complete set of levels or by using a three-level approach. Section 3 deals with the calculation of partition function of diatomic species. Section 4 focuses on the calculation of transport cross sections for interactions involving atomic species by using both multi-potential and phenomenological approaches, also discussing the estimation of inelastic corrections to diffusion-type collision integrals in ion-parent-atom collisions. Section 5 reports examples of thermodynamic properties of thermal plasmas, emphasizing the role of cut-off criteria in affecting the results. Section 6 deals with the treatment of thermodynamics and transport for two-temperature plasmas. Finally Section 7 discusses the role of electronically excited states in affecting transport properties of thermal plasmas.

2 Partition Functions for Atomic Levels. General and Few-Level Approaches The partition function of an atomic system is the product of translational and internal contributions

Qa = Q tr ⋅ Q int

(2.1)

The translational partition function is given by the closed form ⎡ 2π mkT ⎤ , Qtr = ⎢ 2 ⎥⎦ V ⎣ h 32

(2.2)

whereas the internal partition function is expressed as Qint = ∑ g n e − En

kT

,

(2.3)

n

where gn and En represent, in the order, the degeneracy and the energy of the nth level. In the case of atomic hydrogen we sum over the principal quantum number, keeping in mind that 1⎤ ⎡ En = I H ⎢1 − 2 ⎥ ⎣ n ⎦

and

g n = 2n 2

(2.4)

These equations lead to the so-called divergence of partition function, in fact, once the exponential factor converge to e − I H kT , the factor gn diverges as n2. A suitable cutoff criterion is then necessary for the truncation of electronic partition function of atoms. This problem will be discussed in details in Section 5.

14

High Temperature Phenomena in Shock Waves

In general the sum in equation (2.3) includes thousands of levels so that, in practical calculations, few-level approaches have been developed, based on grouping criteria for electronic energy levels. These approaches will be demonstrated in the following for benchmark atomic systems. For atomic hydrogen, in the frame of the two-level approach, the ground state is characterized by EH,0=0 and gH,0=2, while the large number of electronically excited states is reduced to one lumped level, having the degeneracy equal to the sum of degeneracies and the energy equal to the mean value in the excited manifold nHm

nHm

n =2

n=2

EH ,1 = ( g H ,1 ) ∑ g H , n EH , n and g H ,1 = ∑ g H ,n

(2.5) m

The summation is performed up to a maximum number of levels nN . It is straightforward that both the value of energy and its degeneracy factor depend on the m choice of nN . Atomic nitrogen is the case study for the three-level system. The ground state configuration is 4S3/2 having a statistical weight gn = 4 . There are other two low-lying levels, having the same 2s22p3 electronic configuration of the ground state, i.e. 2D5/2,3/2 (EN=2.3839 eV, gN = 10 ) and 2P3/2,1/2 (EN=3.5756 eV, g N = 6 ), grouped to form the first excited level in the three-level model, with energy at EN,1=2.8308 eV and statistical weight g N ,1 = 16 . All other levels are grouped to form the third level, whose energy and degeneracy have been calculated applying equations (2.5), starting the summation from n = 3 to the selected maximum number n Nm . Excited state energies can be calculated in the hydrogen-like approximation or extending available data following the Ritz-Rydberg series. In Fig.1, the internal specific heat of atomic nitrogen considered as a three-level system, for different values of the maximum number of levels actually included, is reported. Note that the curve labeled with nNm = 2 corresponds to neglect the third level and exhibits a well-defined maximum at T~15 000 K. Including an increasing number of excited states in the definition of the higher lumped level, transient bimodal behaviors are found leading, for very high degeneracy of the third level, to the disappearance of the maximum due to the first levels. Note that the results reported in Fig.1 have been obtained by using the following expressions for the energy and degeneracy of electronically excited states En ≅ I N −

Ry n2

and

g n = g core ⋅ 2n 2 = 9 ⋅ 2n 2 ,

(2.6)

Thermodynamic Properties of Gases behind Shock Waves

15

The accuracy of two- and three-level models has been validated by comparison with partition functions including thousands of energy levels, as shown in Refs[3,4].

Fig. 1. Internal specific heat of atomic nitrogen as a function of temperature for different number of levels included in the highest lumped level.

3 Partition Functions for Diatomic Molecules Closed form for the vibrational and rotational partition functions can be obtained separating the different degrees of freedom. In doing so we should be aware that, summing over the vibrational and rotational ladders, we are considering levels with energy exceeding the dissociation limit. In the most general case, each electronic state is described by its own potential energy curve that can be approximated by an harmonic oscillator only in a small region close to the minimum. As a consequence, the momentum of inertia depends on vibrational state. Moreover, the potential energy curve should be corrected by adding the contribution of the centrifugal force, which depends on the rotational state. In this way the vibrational and rotational states are strictly related and we cannot consider them separately. For this reason the relative motion of molecular nuclei is described by ro–vibrational levels. In this picture, the energy of internal levels of the sth diatomic molecule depend on the electronic state n, vibrational v and rotational j quantum numbers, writing the partition function as

Q

int

=

1

m nsm vsm js ( nv )

∑ σ ∑∑ n = 0 v = 0 j =0

g s ,n ( 2 j + 1) e

−

ε s ,nvj kT

,

(3.1)

16

High Temperature Phenomena in Shock Waves

where we can understand that the vibrational energy depends not only on the vibrational quantum number but also on the electronic state, as well as the rotational energy depends on the electronic and vibrational state. The relevant sums are extended on the available electronic states and the limiting values vsm , jsm represent the maximum vibrational and rotational quantum numbers such that the maximum total energy is below the dissociation limit of the corresponding electronic state. Using this approach, it is not possible to obtain closed form for the partition function and thermodynamic quantities and the sum over the energy levels must be calculated directly, as for the atomic species. Energy of ro–vibrational levels is calculated by semi–empirical formula, with coefficients determined from molecular spectra [5]. The treatment for diatomic molecules follows the method developed by Drellishak et al. [6,7] and by Stupochenko et al.[8] In this method the energy of a molecular state is split into three contributions: the electronic excitation, the vibrational and the rotational energy rot ε s,nvj = ε sel,n + ε svib , nv + ε s , nvj

(3.2)

The vibrational energy associated with the vth vibrational level of the nth electronic state of a diatomic molecule, referred to the ground electronic state, is expressed in analytical form as

ε svib,v hc

= ω0 v − ω0 x0 v 2 + ω0 y0 v 3 + ω0 z0 v 4 + ω0 k0 v 5

(3.3)

The rotational energy for a non–rigid rotor, associated to each vibrational level in a given electronic state, is given by

ε srot, nvj hc

= Bs , nv j ( j + 1) − Ds , nv j 2 ( j + 1)

2

(3.4)

The spectroscopic constants entering equations (3.2)-(3.4) can be found in Refs[9,10,11,12]. Equation (3.3) can be used for calculating the maximum vibrational quantum number of each electronic state, while a more elaborate approach can be used to calculate the maximum rotational quantum number for each vibrational level[9,10,11,12]. The method has been extensively applied to many diatomic molecules existing in planetary atmospheres[9,10,11,12]. A sample of results for N2, N2+ and O2, O2+ has been presented in Fig.2 (a) and (b). In each figure we report the temperature dependence of the internal partition function, Qint, of the reduced internal energy, Eint/RT, and of the reduced specific heat C int p / R . The last two quantities can be written as a function of the logarithmic derivatives of the internal partition functions as follows Eint d ln Qint = RT d ln T

(3.5)

Thermodynamic Properties of Gases behind Shock Waves

2 int ⎡ d ln Q int ⎤ ⎤ 2 ⎡ d ln Q = 2⎢ ⎥ +T ⎢ ⎥ 2 R ⎣ d ln T ⎦ ⎣ dT ⎦

C int p

17

(3.6)

very low temperature (activation of rotational degree of freedom), reaching asymptotically the value 2 (activation of rotational and vibrational degrees of freedom), that is kept constant from a given high temperature. Quantitatively the same numbers occur for the reduced internal specific heat.

Fig. 2. Partition function, first logarithmic derivative and specific heat as a function of temperature for diatomic species at 1 atm pressure. (left) nitrogen (right) oxygen.

We qualitatively recover these numbers by inspection of the results obtained by the more complicated approach. Looking at the trend of Eint/RT for N2 reported in Fig.2 (a) we can observe that the reduced internal energy presents a value of 1 at very low temperature asymptotically reaching a value of 2 at high temperature. For T>10 000 K the internal energy starts increasing reaching a maximum, after which it starts decreasing up to zero (at very high temperature not reported in the figure). The same trend is observed for the reduced internal specific heat, rapidly converging to zero after a strong maximum. We therefore note that the behavior of Eint/RT and for low and intermediate temperatures (i.e. in the temperature range important C int p / R for the activation of rotational and vibrational degrees of freedom) can be explained by the well-known ho-rr approximation. On the other hand for T>10 000 K strong

18

High Temperature Phenomena in Shock Waves

the internal partition function, neglected in the ho-rr approximation as well as the insertion of finite number of rotational, vibrational and electronic states in our partition function, differently from that obtained in the frame of ho-rr approximation, including an infinite number of rotational and vibrational states. The results for N2+ and O2, O2+ systems, also reported in Fig.2 (a) and (b), present a similar qualitative trend, strong differences appearing in the maximum of internal energy and specific heat due to the different number of electronic states inserted in the relevant partition functions [9,10,11,12]. In the given references the quality of the present results can be appreciated, looking at their comparison with existing accurate calculations as well as similar results for diatomic and polyatomic molecules for high temperature planetary atmospheres (Earth, Mars and Jupiter).

4 Transport Cross Sections and Collision Integrals The heart of the Chapman-Enskog theory lies on some hypotheses about the dynamics of collisions at a microscopic level, that are assumed to be binary, elastic, with isotropic interaction potentials. The dynamical information is contained in the socalled collision integrals which can be classically obtained by performing a threefold integral [2], i.e. integration over inter-particle distance r, leading to the deflection angle ϑ, over the impact parameter b for transport cross sections QA and, finally, over reduced energy γ2=E/kT ∞

⎡

ϑ (b, E ) = π − 2b ∫ ⎢1 − rc

⎣

b2 ϕ ( r ) ⎤ − ⎥ r2 E ⎦

−1/ 2

r −2 dr

(4.1)

∞

Q A ( E ) = 2π ∫ [1 − cos A (ϑ )] b db

(4.2)

0

Ω( A , s ) (T ) =

kT 2πμ

∞

∫Q

A

e − γ γ 2 s +3 d γ 2

(4.3)

0

where rc is the distance of closest approach and ϕ ( r ) is the spherically symmetric interaction potential. (A,s) represents the collision integral order§, related to the momentum, A, of transport cross section. A number of model potentials (inverse power[13], polarization potential, exponential repulsive[14,15], Morse potential[16], Lennard-Jones[17], modified Buckingham[18], Hulburt-Hirschfelder[19], Tang&Toennies[20]) have been proposed in literature, whose parameters could be estimated theoretically or experimentally, including generally a

§

Traditionally orders (1,1) and (2,2) are associated to so-called diffusion- and viscosity-type collision integrals

Thermodynamic Properties of Gases behind Shock Waves

19

short-range repulsive term and long-range attractive term, and an attractive well, whose depth is related to the strength of the chemical bond. In general for these model potentials dimensionless reduced collision integrals are reported, physically representing the deviation of actual values from the rigid-sphere case Ω( A , s )∗ =

∞

2 Ω( A , s ) kT = Q l e − γ γ 2 s +3 d γ ( A,s ) ∫ Ω rs 2πμ 0

(4.4)

Actually interactions between open-shell chemical species rarely can be described by means of a single potential, due to the large number of molecular states, bound and repulsive, arising in the approaching of colliding species in a defined quantum state, predicted through the rules of momentum addition in different coupling schemes. The collision integral results from a weighted average of the contributions of each state, the statistical weight being the product of spin multiplicity (2s+1) and of the multiplicity due to the axial projection of the orbital angular momentum which, in the case of electronic terms for diatomic molecules, is 1 for Σ states and 2 for all other symmetries (Π, Δ, Φ, …) Ω(avA ,s )∗ =

∑

ωn Ω(nl ,s )∗ ∑ n ωn n

(4.5)

4.1 Phenomenological Approaches for Unknown Colliding Systems Despite the incredible advancement of theoretical chemistry, accurate knowledge of interaction potentials is still a challenging problem. For this reason phenomenological potentials are being reconsidered as a tool to describe in a satisfactory way the average interaction. Large interest in this direction is devoted to the potential energy function developed by Pirani et al.[21,22], which can be considered as an improvement of the Lennard-Jones potential. This potential is able to predict intermolecular interactions in a variety of systems (neutral-neutral and neutral-ion). Fundamental interaction features, i.e. binding energy and equilibrium distance, enter in relevant equations as parameters and their values are determined on the base of correlation formulas of the physical properties of colliding partners (polarizability, charge, number of electrons effective in polarization)[23,24,25,26]. The proposed full-range phenomenological potential, simulating the average interaction, could allow direct evaluation of internally consistent complete sets of collision integrals for different atmospheres. The validity of this approach was demonstrated[27] by comparing, for some benchmark systems, results obtained using the model potential with those calculated with more accurate methods[28,29,30,31]. The interaction potential is modeled with the function m ⎡ m ⎛ re ⎞n n ⎛ re ⎞ ⎤ − ⎜ ⎟ ⎜ ⎟ ⎥ ⎣⎢ n − m ⎝ r ⎠ n − m ⎝ r ⎠ ⎦⎥

ϕ ( r ) = ϕ0 ⎢

(4.6)

20

High Temperature Phenomena in Shock Waves

where n = β + 4 ( r re ) . The parameter m assumes different values depending on the nature of the interaction, i.e. 4 for neutral-ion and 6 for neutral-neutral interactions. The value of β parameter, ranging from 6 to 10 depending on the hardness of interacting electronic distribution densities, could be estimated through the following empirical formula[32] 2

β =6+

5 s1 + s2

(4.7)

where the subscripts 1 and 2 identify the colliding partners. The softness s, entering in equation (4.7), is defined as the cubic root of the polarizability. For open-shell atoms and ions a multiplicative factor, which is the ground state spin multiplicity, should be also considered. Useful bi-dimensional fitting relations have been derived, depending on both temperature and β parameter, allowing the estimation of classical reduced collision integrals up to order (4,4) for any colliding pair[32]. Another approach has been proposed[33], modeling the average interaction with the Lennard-Jones potential, estimating the parameters( σ , ϕ0 ) for the asymmetrical interactions through simplified mixing rules involving well-known parameters for symmetric colliding pairs, i.e. arithmetic mean of collision diameter and a geometric mean of the potential well depth

σ ij =

1 (σ ii + σ jj ) 2

(ϕ0 )ij = ⎡⎣(ϕ0 )ii (ϕ0 ) jj ⎤⎦

1

2

(4.8)

4.2 Resonant Charge Transfer The above considerations completely neglect inelastic channels, in fact cross sections for inelastic processes (internal energy transfer, chemical processes, …) are usually too small to be effective, but in low temperature range[34,35]. However the assumption is not acceptable for resonant processes, i.e. charge-transfer in neutral— parent-ion interactions and excitation-transfer in interaction involving identical atoms in excited states, characterized by high-value cross sections. The simplest theoretical treatment of resonant charge transfer processes relies on the two-state approximation, the atom and its parent ion interacting along two possible molecular states of different symmetry with respect to the interchange of nuclei, i.e. gerade or ungerade parity. If a higher number of (g-u) electronic states arise in the interaction the cross section results from the weighted average of various pairs. A quantum mechanical treatment leads to expression for the charge transfer cross section in terms of phase shifts for gerade and ungerade electronic terms

σ ex ( E ) =

π ∑ ( 2n + 1) sin 2 (ηng −ηnu ) κ2 n

(4.9)

Thermodynamic Properties of Gases behind Shock Waves

21

with diffusion cross section defined as Q( ) = 1

4π

κ2

∑ ( n + 1) sin (η even

2

n

g n +1

− ηnu ) +

4π

κ2

∑ ( n + 1) sin (η odd

2

n

u n +1

− η ng )

(4.10)

Alternatively a very powerful theoretical tool is represented by the asymptotic approach, formulated by Firsov[36] and developed by Nikitin & Smirnov[37]. In the frame of the semi-classical impact parameter method the cross section for resonant charge transfer process can be written as ∞

∞

+∞

0

0

−∞

σ ex = 2π ∫ Pex ( b ) bdb = 2π ∫ bdb sin 2

∫

Δ ( R) 2h

dt

(4.11)

where Pex represents the charge exchange probability and Δ(R) is the exchange interaction potential, that is the g-u energy splitting Δ ( R ) = ϕu − ϕ g

(4.12)

The exchange interaction potential, in the frame of the asymptotic approach, is expressed in terms of the parameter describing the asymptotic behavior of the radial wave function of the valence electron undergoing the transition between the two ionic cores R (r ) = Ar (1/γ −1) e −γ r

(4.13)

−γ 2 2 representing the electron binding energy and A is the normalization factor,

evaluated tailoring the long-range asymptotic representation of electron wavefunction with accurate results obtained by ab-initio Hartree-Fock approach (see Refs.[36,37] for details). The g-u splitting decreases exponentially at large inter-particle distances, as molecular states become degenerate correlating with the same dissociation limit, thus the integral in equation (4.12) can be divided in two parts, the region of impact parameters b less than a critical value b*, and of high values of the phase ξ, where the probability value rapidly oscillates between 0 and 1 and may be replaced by its average value 1/2, while in the second region, at large b values, the probability falls off to zero ∞

σ ex = π ( b∗ ) + 2π ∫ Pex ( b ) bdb ≈ π ( b∗ ) 1 2

2

b∗

1 2

2

(4.14)

The calculation of cross sections thus reduced to the estimation of the critical impact parameter. The transport cross section is affected by elastic and inelastic scattering, and it can be demonstrated that resonant charge transfer processes do not affect even orders[38]. It can be also shown[38] that, neglecting the elastic contribution, the simplified expression holds

22

High Temperature Phenomena in Shock Waves

Qin = 2σ ex ,

(4.15)

and assuming a form for the dependence of the charge-transfer cross section on the relative velocity the inelastic contribution to the odd-order collision integrals have a closed form[39]. Finally the effective diffusion-type collision integral[40] can be defined as Ω(eff1,1)∗ =

4.3

( Ω ( ) ) + ( Ω( ) ) 1,1 ∗ in

2

1,1 ∗ el

2

(4.16)

Neutral-Neutral Interactions

A wide literature does exist on the exact multi-potential treatment of ground-state interactions relevant to hydrogen and air plasmas. The O(3P)-O(3P) can be a suitable example. The momentum-coupling of atomic electronic terms originates 18 molecular states (2 1,5 Σ +g , 1,5 Σu− , 2 3 Σu+ ,3 Σ −g ,1,3,5 Π gu ,1,5 Δ g , 3 Δ u ) and the collision integral is defined as the weighted average of contributions from each term, equation (4.5). The results by Yun & Mason[41], dated 1962, were based on accurate force laws and the advancement in the theoretical study of electronic structure of O2 molecule has been followed by an increasing accuracy in the calculation of corresponding collision integrals. The commonly adopted approach is the fitting of ab-initio data using model potentials, though quantum mechanical approaches can be also found in literature[42]. It is possible to trace the improvements either moving from a rigid classification of Morse and exponential decaying potentials, for bound and repulsive states respectively[43,44] to functional forms able to accommodate peculiar potential features, as with Hulburt-Hirschfelder potential[33,45] or using re-evaluated ab-initio potential energy curves. Table 1. Diffusion and viscosity-type collision integrals [Å2] for O(3P)-O(3P) interaction. (results of Ref.[33] have been obtained by using the fitting formula given in the corresponding paper)

σ 2rs Ω (1,1)∗

T[K] 2 000 4 000 6 000 8 000 10 000 12 000 14 000 16 000 18 000 20 000

σ 2rs Ω (2,2)∗

41

43

42

33

44

43

42

33

44

5.27 4.39 3.90 3.58 3.34 3.15 3.00

4.69 3.98 3.58 3.30 3.09 2.93 2.79 2.68 2.58 2.49

4.84 4.00 3.57 3.27 3.05 2.87 2.72 2.59 2.48 2.38

4.81 4.07 3.63 3.33 3.11 2.94 2.79 2.67 2.56 2.47

6.01 4.88 4.26 3.84 3.53 3.28 3.08 2.91 2.77 2.64

5.45 4.66 4.21 3.90 3.67 3.49 3.34 3.21 3.10 3.00

5.58 4.67 4.20 3.88 3.64 3.44 3.28 3.14 3.02 2.91

5.57 4.74 4.26 3.94 3.69 3.50 3.34 3.20 3.08 2.98

6.97 5.74 5.06 4.59 4.24 3.97 3.74 3.55 3.38 3.24

Thermodynamic Properties of Gases behind Shock Waves

23

In Table 1, theoretical diffusion and viscosity-type collision integrals for groundstate oxygen-oxygen interaction by different authors[33,41-44] are reported, considering the temperature interval relevant to the existence of atomic oxygen in an LTE plasma. An excellent agreement is found among the results of Refs.[33,43], both based on analytical fits and the ones by Levin[42] based on quantum mechanically derived potential energy surfaces, with deviations below 4% in the whole temperature range. Instead discrepancies within 20% are observed in the low-temperature region with collision integrals by Ref.[44] obtained following the same approach used in Ref.[43] with up-dated interaction potentials. The relative difference decreases with temperature, reaching about 10% at 20 000 K. It should be noted that different sets show quite similar values (10% at T=2 000 K) for the relative distance from the old results by Yun & Mason[41], this agreement being also due to compensation effects between the contribution coming from the 18 potential curves. The case of O(3P)O(3P) interaction clarifies the critical point of the traditional approach, i.e. the availability of reliable curves for the ensemble of electronic terms. Recently a phenomenological approach has been proposed, overcoming this difficulty by considering a modified Lennard-Jones potential, describing the average interaction. The investigation on the applicability of the proposed methodology can proceed through the analysis of benchmark systems, such as N(4S)-N(4S). According to the Withmer-Wigner rules the interaction occurs along four different potential curves corresponding to 1 Σ +g , 3 Σ u+ , 5 Σ +g , 7 Σ u+ electronic terms. Following the pair-valence theory we can rationalize the increase of the unbound character of the state with spin multiplicity. So while the singlet state is characterized by a strong chemical bond, the septet exhibits a repulsive potential, as can be appreciated in Fig.3 (a), where relevant potential energy curves are reported.

Fig. 3. The interaction potential energy in the N2 system. (left) Potential energy curves for the electronic states correlating with N(4S)-N(4S), (right) detail of averaged potential (full line) and of phenomenological potential (dotted line).

24

High Temperature Phenomena in Shock Waves

In the same figure the curve <ϕ> is shown, resulting from the statistical average of the four potentials. This kind of averaging emphasizes the role of the repulsive states in smoothing the attractive parts of chemical bonds. The curve <ϕ> for the N2 system is compared, in Fig.3 (b), with the one obtained with the phenomenological procedure. The comparison shows that the wells are quite similar, with a depth three orders of magnitude lower than that of the ground singlet state, and located approximately in the same inter-nuclear-distance range. Table 2. Diffusion-type collision integrals [Å2] for N(4S)-N(4S) interaction. 2

σ rs Ω

T[K] 27

500 1 000 2 000 4 000 5 000 6 000 8 000 10 000 15 000 20 000

7.34 6.30 5.42 4.64 4.40 4.21 3.93 3.72 3.36 3.13

<ϕ>

5.54 4.82 4.25 3.74 3.58 3.45 3.26 3.11 2.84 2.66

(1,1) ∗

46

42

7.76 6.79 5.25 4.50 4.27 4.09 3.79 3.55 3.12 2.82

7.03 5.96 5.15 4.39 4.14 3.94 3.61 3.37 2.92 2.62

In Table 2, diffusion-type collision integrals, obtained integrating the classical deflection angle on the averaged and phenomenological potentials, are reported. In the same table a comparison with results from literature[42,46], obtained with the standard procedure, i.e. adiabatically averaging the contributions coming from the four different states, is also performed. In particular, collision integrals by Capitelli et al.[46] result from a Morse fitting of experimental potential curves for the bound states and an exponential-repulsive function reproducing an Heitler-London calculation of septet state, while in the low temperature region (T<1 000 K) a Lennard-Jones potential has been used. Levin et al.[42] results are derived on the base of accurate ab-initio calculations. We note a substantial agreement between data sets in literature. A better agreement is found when data in literature are compared with the collision integrals obtained by using the phenomenological potential. In this case the differences increase with temperature, not exceeding 20% below 15 000 K. Such behavior indicates that the phenomenological approach describes accurately the potential well which plays the major role in the low temperature region. The models discussed above have been developed for isotropic interaction between colliding partners. This approach seems to be valid only for atom-atom and atomatomic ion encounters, in fact atom-diatom dynamics develops on multi-dimensional surfaces and the interaction potential should take into account all the involved channels. Some authors have used classical trajectory calculation on ab-initio surfaces for collisions involving molecules[34,35], considering anisotropic potentials. However for small molecules, rapidly rotating, an isotropic potential can be assumed, as demonstrated by Stallcop et al.[30], who performed ab-initio multi-reference configuration-interaction calculations and derived an effective angle-averaged

Thermodynamic Properties of Gases behind Shock Waves

25

potential for the estimation of transport cross sections of O(3P)-O2(X 3Σg-) and N(4S)N2(X 1Σg+), thus reducing to a mono-dimensional problem. A good agreement is found in these cases between accurate results of Ref.[30] and those obtained modeling the average interaction with the phenomenological potential [32] with larger deviations in the high-temperature region (T>2 000 K). 4.4 Neutral-Ion Interactions The procedure for estimation of elastic collision integrals in the case of neutral-ion interactions is the same already outlined for neutral-neutral collisions, thus characterized by the same drawbacks. Additionally in atom—parent-ion collisions the contribution coming from the resonant charge-transfer channel to odd-order terms should be estimated. Also for this class of interactions the phenomenological approach has been validated for the derivation of viscosity-type and of elastic contribution to diffusion-type collision integrals, considering benchmark systems[27], for example the N(4S)-N+(3P) system interacting along the 12 related electronic states 2,4,6 Σ gu ,2,4,6 Π gu. Table 3. Viscosity-type collision integrals [Å2] for atom-parent ion interactions O(3P)-O+(4S)

(2,2) ∗

2 (2,2) ∗ σ rs Ω

2 σ rs

T[K] 47

500 1 000 2 000 4 000 5 000 6 000 8 000 10 000 15 000 20 000

N(4S)-N+(3P)

9.32 8.64 7.67 6.99 5.91 5.25

Ω

29

46

27

29

46

27

16.41 13.27 10.50 8.33 7.74 7.26 6.48 5.84 4.66 3.87

13.25 11.32 9.55 7.85 7.32 6.90 6.26 5.79 4.99 4.46

18.54 11.65 7.88 6.09 5.72 5.45 5.09 4.83 4.41 4.13

14.78 11.14 8.72 6.94 6.39 5.95 5.26 4.75 3.92 3.41

10.19 8.73 7.40 6.09 5.65 5.29 4.73 4.31 3.61 3.18

15.22 9.58 6.50 5.05 4.74 4.53 4.23 4.02 3.67 3.45

In Table 3, the viscosity-type collision integrals, not affected by the charge transfer process, calculated with the phenomenological potential27 are reported together with data in Refs.[29,46,47]. Collision integrals calculated according to the phenomenological potential show a reasonable agreement with Stallcop et al.[29] and Capitelli et al.[46] results, especially in the temperature range (5 000-20 000 K) in which N and N+ are the major species. The results in Ref.[47] are, on the contrary, higher with maximum relative difference of about 35%. The behavior, in the considered temperature range, of the absolute error of data obtained with the phenomenological approach with respect to the accurate calculations, based on ab-initio potentials for each interaction channel, is the same displayed in Figure 1 of Ref.[31] Levin used the effective potential in the Tang & Toennies form, which is actually a more complex function than the phenomenological potential. However, it should be noted that in Ref.[31] the binding energy and the equilibrium distance, the two basic potential parameters, have been obtained using the

26

High Temperature Phenomena in Shock Waves

methodology outlined above. Same considerations apply to the case of O(3P)-O+(4S) collision, viscosity-type collision integrals being also presented in Table 3. A comparison with data in literature, still gives a satisfactory agreement, especially when compared with data in Refs.[29,46] in the temperature range of interest (5 00020 000 K) and confirming that an effective potential, not directly connected with details of the interacting system in different electronic states, can be used for transport cross section prediction. The inelastic contribution dominates the effective odd-order collision integral, mainly in the high-temperature region, and the estimation proceeds through the knowledge of corresponding resonant charge transfer cross sections. Due to its relevance in affecting the transport properties resonant process has been investigated, both theoretically and experimentally, for a number of systems. The analysis here is limited to the case of N(4S)-N+(3P) interaction. Theoretical cross sections for resonant charge transfer have been obtained by Stallcop & Partridge[29] by a phase-shift approach, based on accurate ab-initio potential energy curves for the molecular ion N2+ calculated at CASSCF level. Two sets of data refer to calculations performed in the framework of the asymptotic theory, the paper by Eletskii et al.[48]also extending to the treatment of highly excited states, and the re-evaluation recently done by Kosarim et al.[49] that critically selects the proper scheme of coupling momenta on the basis of hierarchy of interactions, including also the low-lying excited states. It should be emphasized the satisfactory agreement among different approaches also confirmed by experimental measurements by Belyaev[50]. This reflects on the corresponding inelastic contribution to collision integrals reported in Table 4. Deviations from the accurate quantum results of Ref.[29] of collision integrals based on asymptotic cross sections of Ref.[48] are within 15% and a little larger 18% for results by Kosarim et al[51]. In the same table, collision integrals by Capitelli et al.[46,52,53] have been also reported. The agreement is ∗ were derived by satisfactory but in the case of the oldest Refs.[52,53] where σ rs2 Ω(1,1) in integration of cross sections obtained by using the g-u splitting of the relevant states of the molecular ion. The observed discrepancies could be rationalized considering that the g-u splitting for different pairs were estimated in a range of atom-ion internuclear separations too close to the equilibrium distance of the molecular ion, far from the region relevant for charge-exchange process, thus underestimating the cross sections. 4.5 Charge-Charge Interactions The interaction between charged particles, i.e. ion-ion and electron-ion collision pairs, are modeled with the screened Coulomb potential

ϕ (r) +

zi z j r

e − r λD

(4.17)

zi and zj being the charge of i and j ions, e the electron charge and λD the Debye length.

Thermodynamic Properties of Gases behind Shock Waves

27

Collision integrals for this potential do exist either in analytical[54] or in tabular form[55], and recently accurate collision integrals by Mason[56] have been fitted by the following equation[57]

( )

6

ln ⎡⎣Ω( l , s )∗ ⎤⎦ = ∑ c j ln j T ∗ j =0

(4.18)

Table 4. Inelastic diffusion-type collision integrals [Å2] for N(4S)-N+(3P) interaction. 2

52

500 1 000 2 000 3 000 4 000 5 000 6 000 7 000 8 000 9 000 10 000 12 000 14 000 15 000 16 000 18 000 20 000 30 000 40 000 50 000

(1,1)∗

σ sr Ω in

T[K]

14.5 14.2 13.9 13.8 13.6 13.4

12.5

53

16.3 16.0 15.8 15.7 15.6 15.3 15.2

29 38.17 34.27 31.39 29.95 29.00 28.30 27.73 27.26 26.86 26.50 26.19 25.65 25.20 25.00 24.82 24.48 24.18 23.05 22.28 21.70

46

40.6 37.5 34.5 32.8 31.6 30.7 30.0 29.4 28.9 28.5 28.1 27.4 26.8 26.5 26.3 25.9 25.5 24.0 23.0 22.3

48

38.88 36.52 34.24 32.94 32.03 31.34 30.78 30.31 29.90 29.55 29.23 28.69 28.23 28.03 27.84 27.50 27.19 26.04 25.23 24.61

49

40.61 38.17 35.81 34.46 33.52 32.80 32.22 31.73 31.31 30.94 30.61 30.05 29.58 29.37 28.82 28.50 28.50 27.30 26.46 25.82

4.6 Electron-Neutral Interactions Collision integrals for electron-neutral interactions are usually calculated by integration of theoretical or experimental differential elastic electron-scattering cross sections, so as to include quantum effects. An illustrative example is represented by the electron elastic scattering by atomic argon, exhibiting a low-energy Ramsauer minimum. This system has been deeply studied and, combining theoretical results in the low-energy region[58] and measured elastic differential cross sections[59,60,61], a wide energy range can be explored, allowing accurate estimation of transport cross sections and high-order collision integrals, displayed in Fig.4 (a). However the critical point still remains the knowledge of the differential cross sections. In fact experimental data are usually available for few collision energy values, also missing extreme values of the scattering angle, and accurate theoretical results are also difficult to be retrieved for all the interactions. On the contrary the integral transport cross sections, elastic term Q(0) and momentum transfer Q(1) are found in literature for a number of systems, allowing the straightforward derivation of

28

High Temperature Phenomena in Shock Waves

diffusion-type collision integrals, but again for the higher order viscosity-type collision integrals the Q(2) is not always readily available. These difficulties could be overcome by using different techniques to estimate this last from basic models or additional information. In Ref.[62] the ratio Q(2)/Q(1) has been determined from the known Q(1)/Q(0) assuming a model angular dependence of the differential cross section. The approach is demonstrated for the e-CO2 interaction, taking the elastic and momentum transfer cross section from Ref.[63], giving a satisfactory agreement with recommended data in literature[64] as shown in Fig.4 (b).

Fig. 4. Diffusion (continuous line) and viscosity-type (dotted lines) collision integrals for electron-neutral interactions, compared with recommended data in Ref.64 (left) e-Ar; (right)e-CO2.

5 Thermodynamic Properties of Thermal Plasmas: The Cut-Off Issue In this chapter we will show the importance of electronic excitation in deriving partition functions and their first and second logarithmic derivatives as well as thermodynamic properties of single atomic species and of plasma mixture. Recent results obtained by using different cut-off criteria are discussed and compared with the so ground state method i.e. by inserting in the electronic partition function only the ground electronic state of the atomic species. A rich literature does exist on the subject, indicating the existence of compensation effects in the calculation of the thermodynamic properties of thermal plasmas. These compensations hide in some cases the role of electronic excitation of atomic species in affecting the thermodynamic properties of thermal plasmas. Results for a case study, i.e. air plasma in a wide range of temperature (500-100 000 K) and pressure (1-1000 bar§), are reported and can be considered representative of many other systems. §

1 atm=1.01325 bar.

Thermodynamic Properties of Gases behind Shock Waves

29

5.1 Cut-Off Criteria It has been already pointed out the necessity of introducing a suitable cut-off criterion to prevent the divergence of electronic partition functions of atomic (neutral and ionic) species. This section will be focused on the following criteria (a) the groundstate method (GS) (b) the Debye Hückel criteria (c) the Fermi criterion (F). Strong differences are expected especially when a complete set (observed and missing) of electronic levels is used in the calculation. 5.1.1 The Ground State Method The partition function includes only the ground state i.e.

Qej = g e0 ,

(5.1)

involving that the first and second logarithmic derivatives are zero. (5.2) In this case the electronic excitation is completely disregarded. 5.1.2 Debye-Hückel Based Criteria In this case we have two types of approaches, one due to Griem (G)[65] and the other one to Margenau and Lewis (ML)[66]. According to Griem we write the electronic partition function of the jth species as Qej =

Enj max

∑g

nj

e

− Enj kT

,

(5.3)

0

where Enj and gnj represent in the order the energy and the statistical weight of the nth level of the jth species. The sum includes all levels up to a maximum value given by

Enj max = I j − ΔI j , j +1

(5.4)

In turn, the lowering of the ionization potential ΔIj,j+1 is given by e3 ⎛ π ⎞ πζ 3 2 ⎜⎝ kT ⎟⎠

12

ΔI j , j +1 =

(∑

n 2 i =1 i i

z n

)

1

2

(z

j

+ 1)

(5.5)

Note the nth level in this case does not refer to the principal quantum number. Following Margenau and Lewis the electronic partition function is written as nmax

Qej = ∑ g nj e n

− Enj kT

,

(5.6)

30

High Temperature Phenomena in Shock Waves

where nmax is the maximum principal quantum number to be inserted in the partition function. In turn nmax is obtained by assuming that the classical Bohr radius does not exceed the Debye length λD. 2 a0 nmax = λD , Z eff

(5.7)

where Z eff = z + 1 is the effective charge seen by the electronic excited state (z is the charge of the atom/ion, z=0 for neutral and so on) and a0 is the Bohr radius. The two formulations coincide when use is made of hydrogen-like levels, presenting however large differences when the dependence of energy levels on the angular and spin momenta and their coupling are considered. In this last case the partition functions and related properties calculated according to ML method exceed the corresponding G values (see Refs.[67,68]). This point should be taken into account when comparing the well-known Drellishak et al.[69] partition functions based on the ML theory and the corresponding values obtained by the G method. 5.1.3 The Fermi Criterion According to the Fermi criterion[70] an electronic state is considered still bound and therefore to be included in the partition function if his classical Bohr’s radius does not exceed the inter-particle distance. One can therefore write 2 Z eff a0 nmax 1 = 1 3 ⇒ nmax = Z eff n a0 n1 3

,

(5.8)

where n, not to be confused with the principal quantum number, is the particle density [cm-3] linked to the pressure by

p = NkT

(5.9)

Therefore the electronic partition function depends on pressure, in particular a decrease of nmax is to be expected with the increase of pressure. 5.2 The Cut-Off Criteria Based on the Solution of Schrödinger Equation The results reported in the previous sections can be rationalized by solving the Schrödinger equation for atomic systems in particular for hydrogen. The energy levels of the hydrogen atom as well as the degeneracy can be obtained by solving the radial Schrödinger equation −

d ⎛ 2 dR ⎞ ⎡ h 2 l ( l + 1) ⎤ ⎥ R = ER ⎜r ⎟ + ⎢V ( r ) + 2 8π μ r dr ⎝ dr ⎠ ⎣ 8π μ r 2 ⎦ h2 2

2

(5.10)

Thermodynamic Properties of Gases behind Shock Waves

31

E is the energy, h is the Planck constant, A (0, 1, 2, 3 …) is the azimuthal quantum number, r is the radial coordinate, V(r) is the potential energy and µ is the reduced mass for the electron-proton system. Energy levels from Eq. (2.4) are eigenvalues of Eq. (5.11) for a Coulomb potential V (r ) = −

e2 r

(5.11)

In this section we re-examine the problem by considering not an isolated atom but an atom closed in a spherical box of radius δ, i.e. we numerically solve the radial part of the Schrödinger equation for atomic hydrogen by considering the following boundary conditions[71] R (r = δ ) = 0

(5.12)

This boundary condition is completely different from the one appearing in the analytical solution of the Schrödinger equation i.e. R(r=∞)=0. In Ref.[71] results are reported for δ a0 = 103 and δ a0 = 10 4 values by imposing A=0, i.e. ns levels. More in details Fig.5 reports the non-dimensional energy level values

α=

En IH

(5.13)

obtained with δ a0 = 103 as a function of the number of grid points. In the same figure we have also reported the analytical reduced energy levels i.e. α=En/IH=-1/n2 (called Bohr), which show the well-known asymptotic trend of energy levels to α=0. The numerical results present values which closely follow the analytical ones suddenly becoming positive from n=28 on, clearly showing the existence of two types of energy levels, the negative ones, assimilated to the bound states, and the positive ones, representing the discretized continuum. These last levels strongly increase their energy with n asymptotically going to the analytical energy levels obtained by the particle in cell model, described by the following equations En =

αn =

h2 n2 8meδ 2

En ⎛ π n ⎞ =⎜ ⎟ I H ⎝ δ a0 ⎠

(5.14) 2

,

(5.15)

32

High Temperature Phenomena in Shock Waves

where me is the electron mass (see Fig.5). Similar results have been obtained for δ a0 = 10 4 where the numerical results reproduce the analytical ones up to n=89, suddenly becoming positive for n>89. Again the positive levels asymptotically go toward the corresponding particle in the box values. Different interesting points can be derived from these calculations. The first one is linked to the fact that the partition function of atomic hydrogen, including bound and positive levels, converges since the positive levels present energies increasing with n2. Thus the solution of the Schrödinger equation for hydrogen confined in a box can be considered as a natural cut-off criterion for the partition function. Moreover the principal quantum number at which occurs the sudden onset of the positive energy levels is in satisfactory agreement with the corresponding value obtained by applying the Fermi cut-off, which gives for the conditions above studied (i.e. δ/a0=103 and δ/a0=104) the values of nmax 40 and 120 (see Ref.[71] for details). Finally we want to mention that the equilibrium between bound and continuum levels can be used in the so-called physical picture to recover the well-known Saha’s equation in the chemical picture. All these effects have been obtained by using ns levels. Going beyond this approximation, it can be done by calculating the energy levels with different A values. The energy levels as a function of the azimuthal quantum number for the δ/a0=103 case start to be affected by A only for n>15, the dependence on A becoming dramatic when we consider very small δ/a0 values.

Fig. 5. Reduced energy levels calculated according to numerical δ/a0=103 and analytical Bohr atom solution of the Schrödinger equation. In the same figure are also reported the particle in the box energy levels calculated numerically and analytically.

As far we have presented results of the Schrödinger equation in the box considering a Coulomb potential. Now we solve the same problem accounting for a Debye potential[72,73,74], i.e.

Thermodynamic Properties of Gases behind Shock Waves

V (r ) = −

e − r λD e r2

33

(5.16)

Fig.6 reports the energy levels for two values of the Debye length, i.e. λD/a0=102 3 and λD/a0=108 for a box δ a0 = 10 . In the same figure, we also report the Bohr’s results as well as the particle in the box values. Inspection of the figure shows again the transition from bound to continuum levels occurring respectively at nmax=11 for λD/a0=102 and nmax=29 for λD/a0=108. The last value coincides with the corresponding value in the presence of Coulomb potential i.e. the Debye length is too high for affecting the results. In both cases the positive levels asymptotically go toward the particle in the box, while the bound levels closely follow the Bohr results. On the other hand the energy levels are strongly affected by the Debye length for λD/a0<102. Note also that the transition between negative and positive energy values at λD/a0=10 and δ/a0=103 occurs nmax=3 i.e. the Debye potential dominates the action of box confinement. It should be noted that the ML cut-off criterion[66] for the reported three conditions give respectively nmax values of 3, 10 and 104 compared to 3, 11 and 29 from numerical results. The excellent agreement between ML results and numerical ones for the first two cases is not surprising because the ML results were based on the energy levels calculated by Ecker & Weizel solving the Schrödinger equation in the presence of the Debye potential. The disagreement for the third case, i.e. for λD/a0=108, is the consequence of the loss of the role of Debye potential in affecting the energy levels for this large value of Debye length.

Fig. 6. Influence of Coulomb-potential screening and confinement on the hydrogen-atom energy levels; A = 0; δ/a0=103.

34

High Temperature Phenomena in Shock Waves

5.3 Case Study: Air Plasma We consider an air plasma composed by O2, O, O+, O2+, O3+, O4+, N2, N, N+, N2+, N3+, N4+ and electrons plus minority species represented by N2+, O2+, O2-, NO, NO+ and O-. A set of equilibrium constants with relations for the electro-neutrality condition and the Dalton’s law for the total pressure can be written. The equilibrium constants, calculated by using the statistical thermodynamics, depend on the relevant partition functions, which, in the case of atomic species, depend on electron and ionic species densities and temperature (Griem criterion) and on the pressure and temperature in the case of Fermi criterion. On the other hand the partition functions of ground state method depend only on the degeneracy of ground state. Different iterative steps are necessary when applying the cut-off criteria to reach convergence. Note also that in the case of atomic species we use a complete set of energy levels including observed and missing ones. Semi-empirical methods based on Ritz and Ritz-Rydberg equations are used to this end. Once obtained the number densities ni (or better Ni=ni/ρ particles g-1) of the different species, by solving the equilibrium problem, we can calculate the different thermodynamic properties. To understand the role of electronic excitation we report the equations for the entropy, enthalpy and specific heats (frozen and total) of the mixture. ⎡ ⎛Q ⎞ ⎛ ∂A ⎞ ⎛ ∂G ⎞ ⎛ ∂ ln Qi ⎞ ⎤ S = −⎜ = −⎜ = k ∑ ⎢ N i ln ⎜ i ⎟ + 1 + ⎜ ⎟ ⎟ ⎟ ⎥ ⎝ ∂T ⎠V , Ni ⎝ ∂T ⎠ p , Ni ⎝ ∂ ln T ⎠V , Ni ⎥⎦ i ⎢ ⎝ Ni ⎠ ⎣ ⎡ ⎛Q = k ∑ ⎢ N i ln ⎜ i i ⎣ ⎢ ⎝ Ni

⎞ ⎛ ∂ ln Qi ⎞ ⎤ tr int ⎟+⎜ ⎟ ⎥ = ∑ N i Si = ∑ N i S i + ∑ N i S i ln T ∂ ⎝ ⎠ i i ⎥ i p , Ni ⎦ ⎠

(5.17)

where the entropy has been divided in the translational and internal contributions. ⎡ ⎛ ∂ ln Qi ⎞ ⎤ H = G + TS = kT ∑ ⎢ N i ⎜ ⎟ ⎥ + ∑ N iε i i ⎣ ⎢ ⎝ ∂ ln T ⎠ p , Ni ⎦⎥ i ⎡ ⎛ ∂ ln Q int ⎞ ⎤ 5 i = ∑ N i H i = kT ∑ N i + ∑ ⎢ N i ⎜ ⎟ ⎥ + ∑ N iε i 2 ln T ∂ i i i ⎢ ⎝ ⎠ p , Ni ⎥⎦ i ⎣

⎡ ⎛ ∂ ln Q ⎛ ∂H ⎞ i = k ∑ ⎢ Ni ⎜ c pf = ⎜ ⎟ ∂ T ln ⎝ ∂T ⎠ p , Ni ⎝ i ⎢ ⎣

⎛ ∂ 2 ln Qi ⎞ ⎤ ⎞ ⎥ = ∑ Ni c pi ⎟ + ∑ Ni ⎜ ∂ 2 ⎟ ⎠ i ⎝ ln T ⎠ p , Ni ⎦⎥ i

(5.18)

(5.19)

Thermodynamic Properties of Gases behind Shock Waves

⎡⎛ ∂ ln Qi ⎞ ⎛ ∂Ni ⎞ ⎛ ∂H ⎞ ⎛ ∂N ⎞ ⎤ + ∑ ε i ⎜ i ⎟ ⎥ = c pf + c pr cp = ⎜ ⎟ = c pf + kT ∑ ⎢⎜ ⎟ ⎜ ⎟ ⎝ ∂T ⎠ p i ⎢ ⎣⎝ ∂ ln T ⎠ p , Ni ⎝ ∂T ⎠ p i ⎝ ∂T ⎠ p ⎥⎦

35

(5.20)

In turn the frozen specific heat can be split into two contributions one due to the translational contribution and the other one due to internal one i.e. ⎡ ⎛ ∂ ln Qint ⎞ ⎛ ∂ 2 ln Qiint ⎞ ⎤ 5 ⎛ ∂H ⎞ i ⎢ = + c pf = ⎜ k N N ∑i ⎢ i ⎜ ∂ ln T ⎟ ∑i i ⎜ ∂ ln T 2 ⎟ ⎥⎥ + 2 k ∑i Ni = cintp + ctrp (5.21) ⎟ ⎝ ∂T ⎠ p, Ni ⎠ p, Ni ⎝ ⎠ p, Ni ⎦ ⎣ ⎝

These equations will be used to better understand the results reported in this section. We start examining the properties of selected atomic single species obtained by using the G and F cut-off criteria. These data should be compared with the ground state values that in the case of O, O+ and O2+ species and for T>2000 K are Q[O( 3 P)] = 9 Q[O + ( 4 S)] = 4 Q[O2+ (2 P)] = 6

Qint ′ [O( 3 P)] = Qint ′′ [O(3 P)] = 0 Qint ′ [O + ( 4 S)] = Qint ′′ [O + ( 4 S)] = 0 2+ 2 Qint ′ [O ( P)] = Qint ′′ [O2+ (2 P)] = 0

Values for ground state nitrogen species can be found in Ref.[75]. In Fig.7 and Fig.8 electronic partition functions, their first and second logarithmic derivatives and internal specific heats, calculated according to G and F cut-off criteria, are reported as a function of temperature for the three species of oxygen (O, O+ and O2+) and of nitrogen (N, N+ and N2+) at different pressures. In both cases the two cut-off criteria give different values of partition function, these differences propagating on the first and second logarithmic derivatives as well as in the specific heat. In particular the F criterion introduces more levels in the partition function as compared with the G criterion with the consequence of increasing the partition function. Note also that, due to the energy range of electronic levels, the partition function of the different species present the sudden increase in well-defined and not overlapping temperature ranges. This aspect is better evidenced in the first and second logarithmic derivatives, which present well distinct maxima. Note that in the second logarithmic derivative the values calculated according to the Fermi criterion overcome the corresponding Griem values up to the maximum, the opposite occurring in the decreasing region. This behavior is reflected on the specific heat of the single species which in any case presents the trend characteristic of a system containing a finite number of excited levels i.e. the internal specific heat after the maximum asymptotically reaches a zero value (see Section 1). Moreover the large influence of electronic excitation on the specific heat can be understood by reminding that the corresponding values for the ground state are zero independently of temperature and the reduced translational contribution to the specific heat is 5/2.

36

High Temperature Phenomena in Shock Waves

Fig. 7. Partition function, first and second logarithmic derivatives and specific heat as a function of temperature for oxygen species, (left) p=1 bar, (right) p=100 bar

Fig. 8. Partition function, first and second logarithmic derivatives and specific heat as a function of temperature for nitrogen species, (left) p=1 bar, (right) p=100 bar

Thermodynamic Properties of Gases behind Shock Waves

37

Fig.9 reports the entropy of the oxygen (O and O+) species, as a function of temperature at different pressures, calculated according to the ground state method (translational contribution) and to G and F cut-off criteria (translational and electronic excitation contribution). The differences between the three methods reflect the corresponding trend of the electronic partition function and of the corresponding first logarithmic derivative. The contribution of the electronic states is well evident in the different plots when G and F values start deviating from the corresponding values calculated from the ground state method. In any case the trend of the entropy for the different species monotonically increase passing from ground state to Griem and Fermi methods following the corresponding increase of electronic contribution. Before examining the dependence of total thermodynamic properties on the cut-off criterion we report the corresponding dependence of the molar fractions of the major species of the air plasma. In Fig.10 in particular the temperature dependence of molar fractions of selected species at different pressures is displayed, showing a small dependence although not negligible of the molar fractions on the adopted cut-off criterion.

Fig. 9. Entropy for oxygen species as a function of temperature for different pressures, (left) oxygen atom, (right) oxygen ion.

38

High Temperature Phenomena in Shock Waves

Fig. 10. Molar fractions for air species as a function of temperature for different pressures and different cut-off criteria, (left) oxygen, (right) nitrogen.

Fig. 11. Total entropy for air plasma as a function of temperature for different pressures with ground-state cut-off. The right axis reports percentage relative difference of results obtained with different criteria with respect to the ground-state cut-off.

Thermodynamic Properties of Gases behind Shock Waves

39

Fig. 12. Specific heat, (left) frozen, (right) total, for air plasma as a function of temperature for different pressures and different cut-off criteria.

Let us now examine the behavior of the thermodynamic properties of air plasma mixture, starting with the total entropy, reported in Fig.11 as a function of temperature at different pressures, calculated with the different methods. In general the Fermi criterion presents larger entropy values compared to G and GS methods, the differences not exceeding 6%. The behavior of the frozen and total specific heats with temperature, for the different pressures, is reported in Fig.12. In this case the differences between the three methods can reach, at high pressure, a factor larger than 2, the values calculated by using the Fermi criterion overcoming in any case the values obtained by G and GS methods. It should be again reminded that the ground state method includes only the translational degree of freedom. More complicated is the situation for the reactive contribution. The dissociation regime is not affected by the cut-off of electronic partition function as confirmed by the results of Fig.10. The ionization regimes are strongly affected by the chosen cut-off criterion. The compensation between F, G and GS methods occurs only in the dissociation and first ionization regimes, while large differences are observed for the second, third and fourth ionization reactions, these differences increasing with pressure. At 1000 bar we observe the largest deviations in the three methods. The differences between G and F values are reduced in the total specific heat due to the partial compensation between frozen and reactive specific heats. On the other hand this compensation tends to disappear when comparing these values with the corresponding ground state values. Only at 1 bar the ground state values are in good agreement with the other two methods, while the differences strongly increase at high pressure.

40

High Temperature Phenomena in Shock Waves

Finally Fig.13 reports the frozen and the total isentropic coefficients for the air plasmas calculated according to the three methods. Once more the effects of the electronic excitation are well evident on the frozen coefficient, being in any case appreciable for the total isentropic coefficient. Note that the frozen isentropic coefficient for the ground state method reaches the constant value of 1.67 in the ionization regime (see Ref.[76]). Results for hydrogen and argon-hydrogen plasmas, recently reported by Sing et al.[77], confirm the above observations.

Fig. 13. Isentropic coefficients, (left) frozen, (right) total, for air plasma as a function of temperature for different pressures and different cut-off criteria.

The results reported in this section do not exhaust the numerous methods used in the literature for the calculation of the electronic partition functions of atomic (neutral and ionized) species. It is worth noting that many researchers calculate the partition function inserting in it only the observed levels[78,79], avoiding in this case any cutoff criterion. Of course this method or other similar methods, which insert in the partition only few levels above the ground state, dramatically underestimate the electronic contribution to the thermodynamic properties of thermal plasmas being not so far from the corresponding values obtained with the ground state method. Note that the famous Gurvich’s tables[80] include in the partition function only the electronic levels coming from the rearrangement of valence electrons i.e. only the low-lying excited states. As an example the partition function of an oxygen atom is obtained by inserting the ground state 3P and the 1D and 1S metastable excited states. On the other hand the well-known JANAF tables, as well as the pioneering calculations of Gordon

Thermodynamic Properties of Gases behind Shock Waves

41

& McBride[81], include those levels whose energy is lower than I-kT, where I is the ionization potential of the species. Also in this case the electronic partition function is strongly underestimated.

6 Transport of Two-Temperature Plasmas The general equations of multi-temperature plasmas have been discussed in different papers. Simplified formulations are nowadays used to calculate the relevant transport coefficients[82,83,84]. The results depend not only on the transport equations but also on the thermodynamic model used to calculate composition and thermodynamic properties of multi-temperature plasmas. We can obtain in fact a multiplicity of Saha's equations depending on the equilibrium criterion adopted (minimization of Gibbs potential, maximization of entropy) as well as the definition of the different temperature existing in the plasma[85,86]. Let us consider hydrogen plasmas characterized by the dissociation and ionization reactions H 2 R 2H

(6.1)

H R H+ +e

The ionization constants are derived by minimization of Gibbs potential and maximization of entropy subjected to different constraints. •

minimization of Gibbs potential with the constraint Th = Tel ≠ Te ϑ

⎡ nH + ⎤ ( 2π me kTe ) ⎢ ⎥ ne = h3 ⎣ nH ⎦

•

ϑ

32

kTe

(6.2)

ϑ

⎡ Q + (Te ) ⎤ − lH Qe ⎢ H ⎥ e ⎣⎢ QH (Te ) ⎦⎥

kTe

(6.3)

maximization of entropy with the constraint Th ≠ Tel = Te ⎡ nH + ⎤ ( 2π me kTe ) ⎢ ⎥ ne = h3 ⎣ nH ⎦

32

•

ϑ

⎡ Q + (Th ) ⎤ − lH Qe ⎢ H ⎥ e ⎢⎣ QH (Th ) ⎥⎦

minimization of Gibbs potential with the constraint Th ≠ Tel = Te ⎡ nH + ⎤ ( 2π me kTe ) ⎢ ⎥ ne = n h3 ⎣ H ⎦

•

32

ϑ

⎡ Q + (Te ) ⎤ −lH Qe ⎢ H ⎥ e ⎣⎢ QH (Te ) ⎦⎥

kTe

(6.4)

maximization of entropy with the constraint Th = Tel ≠ Te ϑ

⎡ nH + ⎤ ( 2π me kTe ) ⎢ ⎥ ne = h3 ⎣ nH ⎦

32

ϑ

⎡ Q + (Th ) ⎤ − lH Qe ⎢ H ⎥ e ⎣⎢ QH (Th ) ⎦⎥

kTh

(6.5)

42

High Temperature Phenomena in Shock Waves

These equations can be further simplified in the specific case by putting Qe = 2 and

QH + (Th ) = QH + (Te ) = 1 . Comparison of the different equilibrium equations for our case study (i.e. H2 plasmas) shows differences in both the exponential ϑ= Th/Te factor disappearing in the equations which derive from the maximization of entropy as well as on the different temperatures appearing in the partition function and in the exponential term. Again we note that all exponential terms contain the electron temperature with the exception of equation (6.5) which contains the heavy particle temperature. This difference should have strong consequences in the relevant results. It should be noted that equation (6.4) is nowadays the most used equation for the two temperature plasmas based on the kinetic idea that electrons are responsible of the ionization equilibrium as well as of the excitation one, this idea i.e. mixing thermodynamic and kinetic concepts being a little contradictory. The presentation of results taken from Ref.[87] is made either as a function of Th (in the range 2 50010 000 K) keeping constant Te=10 000 K or as a function of Th (in the range 8 00030 000 K) at constant Th=8 000 K, thus meaning that the different plots are made at different 1/ϑ values rather than fixing it. In the first case only the electron density is strongly affected by the choice of the different equations while atomic and molecular hydrogen densities scarcely depend on this choice. In any case electron and ions densities keep values well below the corresponding values for atomic and molecular hydrogen (at Th=Te=10 000 K the electron density is a factor 100 less than atom density). As a consequence only the transport coefficients which depend on the electron density (i.e. total thermal conductivity λ and electrical conductivity σ) will be affected by the choice of the equilibrium constants.

Fig. 14. (left) Electrical conductivity, as a function of gas translation temperature at Te=10 000 K, (right) total thermal conductivity, as a function of the electron temperature at Th=8 000 K, of H2 plasma, corresponding to different Saha equations. (solid line)- Eq. (6.2), (dashed line)-Eq. (6.3), (dashed-dotted line)-Eq. (6.4), (dotted line)-Eq. (6.5)

Thermodynamic Properties of Gases behind Shock Waves

43

This is indeed the case, as can be appreciated by looking at Fig.14 (a) where we have reported t versus Th at Te=10 000 K for an atmospheric hydrogen plasma calculated inserting in the transport equations the compositions coming from equations (6.2)-(6.5). No appreciable change is observed by using equations (6.2) and (6.3), while the use of equation (6.4) strongly increases the electrical conductivity. On the other hand use of equation (6.5) is such to strongly underestimate the electrical conductivity as a consequence of the exponential factor calculated at Th (see equation (6.5)). All the curves converge to the same values for the one temperature case i.e. Th= Te=10 000 K. Let us consider now the results obtained as a function of Te for Th=8 000 K. In this case the influence of Te can play an important role only when Te ≥ 2Th i.e. from 15 000 K on. This is indeed the case as shown in Fig.14 (b), where we have reported the total thermal conductivity calculated by inserting in the transport equations the different compositions coming from equations (6.2)-(6.5). Inspection of the results shows that equations (6.2) and (6.3) give practically the same results, while an appreciable change is observed when using the maximization of entropy in the form of equation (6.4). Again we note that use of equation (6.5) does not allow the onset of the ionization reaction yielding a total thermal conductivity basically given by the atomic hydrogen contribution. To conclude this section we want to emphasize that transport coefficients of two temperature plasmas strongly depend on the used Saha's equation, a problem still open to discussion despite the numerous researchers contributing to the field[88,89,90,91].

7 Transport Properties of Plasmas: The Role of Electronically Excited States The excited state issue is still debated, being the dependence of collision integrals on the quantum state of chemical species largely unknown. However it has been shown[92,93] the role of electronically excited states in affecting transport properties of equilibrium atomic hydrogen plasmas. The plasma is composed by H(n), H+ and electrons, where n is the principal quantum number of excited atomic hydrogen, i.e. each electronic excited state is considered as an independent species with its own transport cross section. Electronically excited state cross sections dramatically increase with n. In Table 5 the approaches adopted in the derivation of collision integrals for the different kind of interactions occurring in the plasma and involving excited atoms, H(n), with n≤12, are summarized. The diffusion-type collision integrals for the H(n)-H+ interaction are dominated by the inelastic contribution, due to resonant charge-transfer process, that is characterized by a strong increase with n (between n3 and n4)[94], while for H(n)-H(n) and e-H(n) interactions the dependence on n is weaker. In considering collisions between excited atoms, in asymmetric collisional schemes (n≠m), the inelastic channel due to resonant excitation transfer should be properly account in the estimation of diffusion-type collision integrals[92].

44

High Temperature Phenomena in Shock Waves Table 5. Collision integrals for interaction involving excited states in hydrogen plasma.

interaction

H(n)-H+

collision integrals ( 1,1) ∗

σ Ω el 2

σ Ω 2

( 2 , 2 )∗ ( 1,1 ) ∗

σ Ω in 2

H(n)-e H(n)-H(n)

2

σ Ω

σ 2Ω (1,1)∗ σ 2Ω (2,2)∗ ( 1,1) ∗

σ Ω el 2

H(n)-H(m)

(1,1)∗

σ 2Ω (2,2)∗ σ

2

(1,1) ∗ Ω in

adopted approach polarizability model extrapolation formulas of accurate resonant chargeexchange collision integrals for n≤5[94] integration of momentum transfer cross sections[95] extrapolation formulas of accurate collision integrals for n≤5[96] averaging of symmetric interactions scaling of accurate resonant excitation-exchange collision integrals n≤3[92]

The effect on the different contributions of the total thermal conductivity is reported in Fig.15, where it is shown the ratio between the relevant contributions calculated with this kind of cross sections (called abnormal) and with the ground state approximation (called usual). In the last case excited state collision integrals are set equal to the ground state ones. Fig.15(a) reports the ratio between the translational thermal conductivity values calculated with the abnormal cross sections (λha) and the corresponding results calculated with the usual cross sections (λhu) as a function of temperature for different pressures. For T<104 K the presence of excited states does not alter the results. Inspection of Fig.15(a) shows that the differences in the ratio λha/λhu strongly increase with the pressure. By defining a relative error as err=100 |λha/λhu|/λha, the maximum error in the curves is 7%, 40% and 240% for 1, 10 and 100 atm. respectively. It should be noted that the small effect observed at 1 atm is due to a compensation effect between diagonal and off-diagonal terms in the determinant expression of translational thermal conductivity of the heavy components, this compensation disappearing when considering only the diagonal terms[97]. The compensation between diagonal and non-diagonal terms disappears at high pressure as a result of the shifting of the ionization equilibrium towards higher temperatures where excited states are more easily populated. The results of Fig.15(a) have been obtained by considering in all cases twelve excited states. However the number of excited states has been decreased up to n=7 at 100 atm. As expected the differences in the ratio strongly decrease (see the dotted line in Fig.15 (a)). The relative error can reach in this case a value of 42%.

Thermodynamic Properties of Gases behind Shock Waves

45

Fig. 15. Ratio between transport coefficients calculated by using abnormal (a) and usual (u) collision integrals, as a function of temperature, at different pressures and for different number of atomic levels. (a) Translational thermal conductivity of heavy particles; (b) translational thermal conductivity of electrons; (c) reactive thermal conductivity; (d) total thermal conductivity.

The electronically excited states affect the translational thermal conductivity of free electrons only through the interaction of electrons with H(n). Fig.15 (b) reports the abnormal to usual ratio, showing the increase of excited state influence with pressure. The contribution due to the reactive thermal conductivity has been extensively analyzed in Ref.[97] and the behavior of the ratio is displayed in Fig.15 (c). The main conclusions follow the trend already illustrated for translational components. Finally Fig.15 (d) reports the ratio for the total thermal conductivity. The viscosity and the electrical conductivity behave respectively like the heavy-particle translational thermal conductivity and like electron translational thermal conductivity[93]. The results have been obtained by parametrizing the number of excited states, however this condition can be overcome by self-consistently calculating the equilibrium composition of LTE plasma, obtaining results that, while confirming the qualitative nature of the presented calculations, show the strong dependence of the transport coefficients on the adopted cut-off criterion for truncating the electronic partition function of atomic hydrogen[98,99]. It should be noted that a

46

High Temperature Phenomena in Shock Waves

different formalism is adopted for transport coefficients, allowing a separated estimation of internal and reactive contributions and the interplay between them and the resulting compensation effects[98]. Air Plasmas A collision-integral database for interactions involving electronically excited atoms, relevant to LTE atomic air plasma, is far from being complete. Reference should be done to the pioneering work in this field by different authors[43,53,100,101], attempting the accurate calculation of diffusion and viscosity-type collision integrals for interactions involving the ground and the so-called low-lying excited states of nitrogen and oxygen atoms [N(2P,2D), O(1D,1S)] and ions [N+(1D,1S), O+(2P,2D)], i.e. low-energy states characterized by the same electronic configuration of the ground state. Recently a complete revision of old results for oxygen system has been performed[44], based on accurate ab-initio interaction potentials for valence states. The inelastic contribution to odd-order collision integrals due to resonant chargeexchange processes in atom-parent-ion collisions have been evaluated from the corresponding cross sections recalculated in the framework of the asymptotic theory[48,49,51] considering different momentum coupling schemes. An attempt to study the effects of excited species in an LTE nitrogen plasma can be found in recent papers by Sourd et al.[102], however restricted to dissociative regime, i.e. neutralatom interactions. In the cited references collision integrals have been obtained in the traditional multi-potential approach, thus requiring the accurate knowledge of a huge number of electronic terms increasing the quantum state of colliding partners. In the case of N2 and N2+ excited systems only few electronic terms are available from ab-initio calculations or from spectroscopic data models, reflecting the theoretical and experimental limits in studying excited states. In this context the phenomenological approach could represent a valuable tool, allowing the derivation of complete and internally consistent set of collision integrals[103] . The potential function adopted for low-lying excited states is the same used for describing the interaction between ground-state species, these states being characterized by physical properties quite similar to the ground state and by small energy separation, while for interactions involving higher excited species a different functional form has been proposed[103]. In comparing viscosity-type collision integrals, obtained by using the phenomenological approach, for atomic oxygen low-lying excited-state interactions with results available in literature[43,44], based on a complete (62 valence states) multi-potential traditional approach, a general satisfactory agreement is found, observed deviations giving an indication of the accuracy of the phenomenological procedure. Cross sections for resonant charge-transfer processes, affecting diffusion-type collision integrals, for atom-ion interactions involving nitrogen and oxygen atoms in highly-excited states (n>2)[48] have been calculated with the asymptotic approach and results show a n4-n5 dependence on the principal quantum number of the atomic valence shell. Results for transport coefficients of air plasmas, limiting the analysis to the role of low-lying electronically excited states[75], demonstrate a large effect on the internal thermal conductivity value in the whole temperature range explored, thus confirming

Thermodynamic Properties of Gases behind Shock Waves

47

the importance of the inclusion of higher excited states in transport calculations due to the enormous increase of transport cross sections on n. Future work in this direction should develop a complete set of state-specific collision integrals for electronically excited states, analogously to the database built for hydrogen.

8 Conclusions Simulation of shock wave entails the reliable estimation of thermodynamics and transport properties characterizing the plasma. Fundamental quantities are represented by translational and internal partition functions for thermodynamic properties, including Saha equilibrium constants fully determining the equilibrium plasma composition, and transport cross sections for the description of microscopic collisional dynamics of chemical species. Accurate theoretical methods are presented in this chapter together with alternative approximate approaches, representing valuable tools when a rigorous treatment is hindered by a high computational load or by lack in the knowledge of the fundamental ab-initio structural information about chemical systems. This is the case of few-level approaches for internal partition function of atoms, reducing the huge number of levels to be included in the summation by adopting a suitable grouping criterion, or the use of phenomenological potential for the average interaction in the calculation of collision integrals of colliding systems, whose multi-surface interaction potential is unknown. The central point in the development of accurate theoretical thermodynamic methods is the definition of physically sounded criteria for truncation of the internal partition function for atomic species, affected by divergence. For thermal plasmas actually accurate values of thermodynamic and transport coefficients, obtained by using sophisticated approaches, are nowadays available and in Appendix A recommended values for air plasmas[104] are tabulated. These data for atmospheric air plasmas are in good in agreement with corresponding ones reported by Boulos et al.[105], even though this comparison suffers to some extent of the compensation effects already pointed out in this chapter. Improvements of transport properties of air plasmas can be achieved from a more complete treatment of transport cross sections of electronically excited states, as done for atomic hydrogen plasma. Insertion of electronically excited states represents an important issue also in the calculation of the transport properties of multi-temperature air plasmas[106]. Finally we want to stress that the bulk of present results has been obtained in the assumption of ideal plasmas, whose validity can be questioned at very high pressure and temperature regimes[107,108,109]. Acknowledgments. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement n° 242311.

48

High Temperature Phenomena in Shock Waves

Appendix: Thermodynamic and Transport Properties of Air Plasmas Table A.1. Air plasma density, ρ [kg m-3], as a function of temperature for different pressures T [K] 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

pressure [atm]

10-2

10-1

1

10

102

0.03465120 0.01732560 0.01155040 0.00866281 0.00693025 0.00577521 0.00495018 0.00433140 0.00385014 0.00346512 0.00231006 0.00173001 0.00134653 0.00101631 0.00083214 0.00071898 0.00062014 0.00051192 0.00031352 0.00024800 0.00021184 0.00017757 0.00014036 0.00010536 0.00008212 0.00007001 0.00006312 0.00005830 0.00005440 0.00005104 0.00004792 0.00004477 0.00004127 0.00003329 0.00002692 0.00002318 0.00002093 0.00001922 0.00001478 0.00001131 0.00000954 0.00000792

0.34651200 0.17325600 0.11550400 0.08662810 0.06930250 0.05775200 0.04950180 0.04331400 0.03850140 0.03465120 0.02310080 0.01731750 0.01372700 0.01086230 0.00859941 0.00729257 0.00637863 0.00555825 0.00380173 0.00260883 0.00216573 0.00187718 0.00160975 0.00133530 0.00106728 0.00084792 0.00070257 0.00061502 0.00055905 0.00051886 0.00048632 0.00045856 0.00043317 0.00038244 0.00032463 0.00026839 0.00022758 0.00020155 0.00016333 0.00012852 0.00010172 0.00008657

3.46512000 1.73256000 1.15504000 0.86628100 0.69302500 0.57752000 0.49501800 0.43314000 0.38501400 0.34651200 0.23100800 0.17323100 0.13817400 0.11295300 0.09180140 0.07569730 0.06527960 0.05761850 0.04419050 0.03105880 0.02295250 0.01933550 0.01696560 0.01489580 0.01288570 0.01091190 0.00908662 0.00757016 0.00644765 0.00567209 0.00513651 0.00474947 0.00444922 0.00397611 0.00358058 0.00317687 0.00274892 0.00235249 0.00173932 0.00143504 0.00117183 0.00094912

34.6512000 17.3256000 11.5504000 8.66281000 6.93025000 5.77520000 4.95018000 4.33140000 3.85014000 3.46512000 2.31008000 1.73248000 1.38468000 1.14655000 0.96185800 0.80757500 0.68592900 0.59777100 0.47461300 0.37068700 0.27559000 0.21114100 0.17719800 0.15616600 0.13942700 0.12427700 0.10986700 0.09613230 0.08339210 0.07220580 0.06293000 0.05560990 0.04999950 0.04238760 0.03750020 0.03387140 0.03078790 0.02779030 0.02052750 0.01575240 0.01306980 0.01103000

346.512000 173.256000 115.504000 86.6281000 69.3025000 57.7520000 49.5018000 43.3140000 38.5014000 34.6512000 23.1008000 17.3254000 13.8562000 11.5231000 9.80484000 8.43599000 7.29747000 6.35892000 5.01336000 4.08181000 3.29544000 2.59484000 2.04911000 1.69664000 1.47733000 1.32233000 1.19611000 1.08400000 0.98021800 0.88296600 0.79208000 0.70894000 0.63478700 0.51540300 0.43155300 0.37389700 0.33286300 0.30155000 0.24163100 0.18981200 0.15005200 0.12422300

Thermodynamic Properties of Gases behind Shock Waves

49

Table A.2. Air plasma enthalpy, H [kJ kg-1], as a function of temperature for different pressures T [K]

pressure [atm]

10-2 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

100.958 201.916 303.240 404.808 508.248 612.579 720.163 829.321 941.415 1055.425 1654.368 2323.130 3498.021 6070.441 7660.881 8698.956 10577.950 15049.494 33236.326 40446.037 44739.726 53430.686 72082.461 103389.677 135571.933 154883.548 164365.813 169871.874 174122.732 178361.867 183959.618 193464.910 210984.575 279812.636 362905.544 419073.559 449704.188 473912.266 628974.590 849979.015 975976.574 1200753.306

10-1 100.958 201.916 303.240 404.808 508.248 612.579 720.163 829.321 941.415 1055.425 1654.259 2305.327 3152.698 4804.604 7023.943 8340.692 9510.274 11475.684 22160.691 36908.484 42579.756 47098.151 54698.867 68354.953 90261.128 117569.949 142041.377 158386.225 168220.181 174593.747 179537.520 184168.540 189574.536 209290.177 254714.026 324232.491 390543.029 436641.372 509759.647 666287.352 863540.060 994645.516

1 100.958 201.916 303.240 404.808 508.248 612.579 720.163 829.321 941.415 1055.425 1654.221 2299.679 3036.460 4094.762 5789.172 7620.688 8929.649 10175.724 14960.316 26338.112 38411.294 44274.573 48741.888 54759.995 64040.455 77801.211 96334.081 117805.248 138488.040 155176.206 167288.618 175923.474 182447.346 193424.717 207299.354 232796.547 276868.060 334847.704 456654.276 533201.540 662213.596 840160.969

10 100.958 201.916 303.240 404.808 508.248 612.579 720.163 829.321 941.415 1055.425 1654.213 2297.896 2998.996 3829.926 4945.197 6422.216 7983.307 9328.274 12156.206 17495.214 27273.108 38054.914 44957.858 49658.797 54531.395 60800.710 69155.888 80237.510 93874.755 109816.833 126631.859 142712.940 156853.857 178182.253 192937.792 205756.099 221105.745 243944.321 345876.572 456846.869 537790.273 634877.978

102 100.958 201.916 303.240 404.808 508.248 612.579 720.163 829.321 941.415 1055.425 1654.209 2297.331 2987.081 3742.003 4608.160 5646.193 6863.672 8174.183 10731.689 13731.197 18496.571 25983.993 35217.812 43303.235 49186.526 53985.855 58847.149 64370.794 71103.168 79256.961 88659.677 99586.806 111679.541 137192.225 161050.705 181025.090 197503.079 212326.863 257609.617 335320.393 431910.288 518237.917

50

High Temperature Phenomena in Shock Waves

Table A.3. Air plasma entropy, S [J K-1 kg-1], as a function of temperature for different pressures T [K]

pressure [atm]

10-2 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

7319.489 7819.896 8230.788 8522.692 8753.887 8943.382 9109.891 9254.838 9387.601 9506.925 9991.966 10374.514 10890.202 11823.314 12319.994 12596.448 13035.602 13970.891 17279.175 18411.453 18981.905 19997.245 21949.676 24924.062 27728.255 29280.176 29985.485 30366.400 30640.886 30897.830 31217.116 31729.496 32625.472 35888.245 39505.807 41761.739 42900.465 43734.894 48435.833 54368.947 57340.319 62046.283

10-1 6654.542 7154.948 7565.841 7857.744 8088.939 8278.434 8444.943 8589.890 8722.654 8841.977 9326.934 9700.062 10075.241 10670.855 11357.156 11710.856 11985.636 12397.366 14313.497 16602.881 17367.726 17897.942 18693.978 19988.411 21886.790 24066.857 25880.286 27009.256 27644.687 28032.175 28314.784 28565.615 28842.949 29775.056 31739.000 34514.011 36971.663 38564.824 40818.914 44955.346 49613.580 52378.961

1 5989.594 6490.001 6900.893 7192.797 7423.992 7613.486 7779.995 7924.943 8057.706 8177.030 8661.945 9032.100 9359.926 9742.725 10262.768 10753.000 11062.363 11324.373 12183.588 13923.622 15545.002 16240.095 16709.937 17280.850 18084.003 19178.810 20543.958 22017.125 23346.350 24354.951 25045.605 25512.308 25847.377 26369.722 26971.198 27984.194 29607.500 31599.371 35371.510 37411.486 40423.931 44166.135

10 5324.647 5825.053 6235.946 6527.849 6759.044 6948.539 7115.048 7259.995 7392.758 7512.082 7996.997 8366.231 8678.776 8980.393 9322.957 9716.390 10084.536 10368.275 10880.865 11696.161 12995.618 14268.489 14999.085 15447.199 15869.437 16367.960 16981.410 17738.425 18607.395 19562.656 20512.724 21372.442 22090.143 23099.790 23738.277 24248.271 24813.824 25594.955 28690.005 31642.243 33539.827 35565.736

102 4659.699 5160.105 5570.998 5862.902 6094.097 6283.591 6450.100 6595.047 6727.811 6847.135 7332.050 7700.949 8008.679 8283.207 8549.864 8826.319 9112.905 9388.899 9855.392 10315.689 10948.398 11827.040 12799.466 13571.512 14083.893 14466.776 14824.119 15199.842 15627.022 16111.393 16634.408 17209.590 17813.327 18995.512 20006.749 20788.006 21386.844 21889.511 23256.962 25282.578 27513.263 29303.288

Thermodynamic Properties of Gases behind Shock Waves

51

Table A.4. Air plasma specific heat, cp [J K-1 kg-1], as a function of temperature for different pressures T [K] 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

pressure [atm]

10-2

10-1

1

10

102

1014.520 1024.986 1035.496 1046.007 1056.632 1067.303 1078.075 1089.129 1100.460 1112.283 1197.811 1576.336 3852.955 4780.056 2224.724 2521.479 5381.352 13324.183 14199.251 3986.872 5566.082 12436.343 25858.943 35022.360 25917.821 12890.191 6447.673 4484.200 4539.567 5836.254 8580.667 13520.980 21472.278 41635.856 38583.471 19648.378 11861.544 14464.231 49705.790 29666.227 30732.499 55626.641

1007.379 1019.193 1031.067 1042.975 1055.002 1067.108 1079.354 1091.868 1104.679 1117.918 1201.673 1414.532 2310.936 4396.216 3534.237 2304.186 2819.383 5163.011 16723.042 9129.795 4161.324 5550.852 9990.580 17849.586 26178.366 27101.316 19688.604 11760.530 7201.355 5381.051 5132.901 5895.724 7616.617 15119.992 28861.573 37528.527 28695.069 17113.209 18594.428 42697.574 31126.417 26315.412

999.692 1012.905 1026.177 1039.520 1052.987 1066.540 1080.253 1094.242 1108.478 1123.082 1207.825 1353.060 1732.113 2736.026 3782.774 3108.601 2493.950 2899.424 7450.140 14323.441 8416.463 4507.565 4871.495 7112.580 11059.886 16452.059 21274.323 22257.378 18702.803 13514.762 9384.078 6976.605 5932.350 6368.178 9512.442 16242.392 25924.269 31322.438 16157.240 18627.342 33526.227 33745.157

991.135 1005.712 1020.358 1035.057 1049.905 1064.828 1079.881 1095.180 1110.624 1126.371 1212.330 1326.512 1523.123 1926.976 2627.059 3128.177 2877.939 2605.873 3634.367 7355.384 11594.896 9193.429 5453.854 4465.663 5095.517 6677.107 9130.361 12337.200 15649.028 17795.560 17691.110 15504.099 12482.605 7927.448 6462.797 7012.757 9149.809 13188.905 25600.297 17641.543 16552.424 24246.007

980.921 996.771 1012.707 1028.671 1044.764 1060.917 1077.178 1093.577 1110.069 1126.802 1214.302 1314.418 1443.388 1633.423 1925.484 2293.618 2541.829 2543.750 2651.560 3674.722 5921.948 8779.046 9341.867 7153.383 5214.909 4555.560 4774.278 5541.464 6758.608 8396.067 10338.294 12268.423 13669.132 13373.046 10341.591 7963.109 7124.192 7395.310 11542.517 19040.416 18912.212 15715.859

52

High Temperature Phenomena in Shock Waves

Table A.5. Air plasma viscosity µ, 104 [kg m-1 s-1], as a function of temperature for different pressure T [K]

pressure [atm]

10-2 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

0.070478 0.130294 0.180187 0.224078 0.263933 0.300875 0.335592 0.368557 0.400092 0.430435 0.569453 0.695858 0.836173 1.037112 1.198476 1.330414 1.466503 1.624842 1.879189 2.062503 2.146783 1.849861 1.204350 0.606252 0.260733 0.118569 0.070457 0.056345 0.053812 0.056137 0.059585 0.060515 0.055158 0.033235 0.019890 0.014835 0.013904 0.014255 0.011272 0.007566 0.007388 0.006370

10-1 0.070478 0.130294 0.180187 0.224078 0.263933 0.300875 0.335592 0.368557 0.400092 0.430435 0.569453 0.695156 0.820403 0.981896 1.171143 1.319642 1.450967 1.587976 1.903264 2.106463 2.280353 2.346231 2.115625 1.591983 1.015144 0.574555 0.312060 0.182016 0.124785 0.103038 0.095507 0.096062 0.098504 0.095813 0.073339 0.048236 0.033162 0.026430 0.024102 0.018756 0.013457 0.012461

1 0.070478 0.130294 0.180187 0.224078 0.263933 0.300875 0.335592 0.368557 0.400092 0.430435 0.569447 0.694933 0.814916 0.946145 1.111411 1.285015 1.431124 1.563677 1.849121 2.153168 2.343793 2.508764 2.603900 2.508539 2.167383 1.675900 1.187453 0.794995 0.523175 0.355312 0.260472 0.211367 0.188537 0.174879 0.175703 0.159413 0.125436 0.091633 0.051352 0.044750 0.036925 0.027489

10 0.070478 0.130294 0.180187 0.224078 0.263933 0.300875 0.335592 0.368557 0.400092 0.430435 0.569447 0.694864 0.813126 0.931844 1.064314 1.218951 1.379692 1.526672 1.796698 2.092230 2.390949 2.595010 2.756634 2.891139 2.935977 2.828907 2.553361 2.161841 1.736515 1.349495 1.033251 0.794597 0.625603 0.438867 0.367527 0.345225 0.337274 0.313844 0.194299 0.121025 0.097066 0.083487

102 0.070478 0.130294 0.180187 0.224078 0.263933 0.300875 0.335592 0.368557 0.400092 0.430435 0.569447 0.694843 0.812549 0.926965 1.043886 1.170230 1.308917 1.454556 1.735612 2.007099 2.301107 2.604812 2.850717 3.029418 3.182606 3.309411 3.380286 3.363595 3.241351 3.018712 2.720730 2.390710 2.066136 1.519453 1.151920 0.936084 0.821281 0.763227 0.676374 0.503380 0.347650 0.264329

Thermodynamic Properties of Gases behind Shock Waves

53

Table A.6. Air plasma total thermal conductivity, λ [W m-1 K-1], as a function of temperature for different pressures T [K]

pressure [atm]

10-2 100 200 300 400 500 600 700 800 900 1 000 1 500 2 000 2 500 3 000 3 500 4 000 4 500 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 22 000 24 000 26 000 28 000 30 000 35 000 40 000 45 000 50 000

0.010347 0.019126 0.026511 0.033244 0.039759 0.046247 0.052733 0.059184 0.065562 0.071851 0.101662 0.169482 0.621638 0.774534 0.347090 0.550411 1.475929 3.575126 3.343822 0.890515 1.040140 1.719839 2.775433 3.267816 2.388875 1.473512 1.119406 1.062241 1.116666 1.224217 1.357653 1.504522 1.654833 1.934224 2.165996 2.429127 2.774950 3.173694 4.115690 5.065981 6.350769 7.559955

10-1 0.010347 0.019126 0.026511 0.033244 0.039759 0.046247 0.052733 0.059184 0.065562 0.071851 0.101453 0.145464 0.334716 0.805333 0.594779 0.418192 0.694294 1.543614 4.654813 2.361559 1.039848 1.196090 1.717283 2.454675 3.107835 3.109148 2.517998 1.983151 1.742015 1.713820 1.788025 1.929433 2.104051 2.504460 2.914643 3.282035 3.610524 3.987431 5.267572 6.546212 7.859838 9.532709

1

10

102

0.010347 0.019126 0.026511 0.033244 0.039759 0.046247 0.052733 0.059184 0.065562 0.071851 0.101387 0.137798 0.222197 0.483524 0.750142 0.583697 0.504607 0.750090 2.467025 4.356947 2.423929 1.317651 1.364111 1.750623 2.289554 2.889072 3.360348 3.486108 3.283431 3.000136 2.829727 2.812017 2.910924 3.306078 3.859325 4.459614 5.037119 5.548666 6.849608 8.650775 10.488277 12.282686

0.010347 0.019126 0.026511 0.033244 0.039759 0.046247 0.052733 0.059184 0.065562 0.071851 0.101366 0.135367 0.184226 0.299898 0.511206 0.667127 0.618947 0.584469 1.141813 2.681341 4.022729 2.990622 1.847529 1.657301 1.927490 2.389902 2.943731 3.519161 4.029105 4.380553 4.549753 4.598192 4.628028 4.875871 5.430368 6.210712 7.137552 8.092531 9.994419 11.869134 14.389841 17.134915

0.010347 0.019126 0.026511 0.033244 0.039759 0.046247 0.052733 0.059184 0.065562 0.071851 0.101360 0.134598 0.171972 0.231259 0.332557 0.469773 0.581421 0.620108 0.723585 1.297034 2.424558 3.608618 3.620807 2.797966 2.313369 2.369820 2.771441 3.368757 4.064296 4.783103 5.466364 6.068133 6.582626 7.447319 8.318966 9.371899 10.663627 12.161004 15.865971 18.572724 21.436432 25.063154

54

High Temperature Phenomena in Shock Waves

Table A.7. Air plasma electrical conductivity, σ [S m-1], as a function of temperature for different pressures T [K]

pressure [atm]

10-2 500 1 000 2 000 3 000 4 000 5 000 6 000 7 000 8 000 9 000 10 000 11 000 12 000 13 000 14 000 15 000 16 000 17 000 18 000 19 000 20 000 21 000 22 000 23 000 24 000 25 000 26 000 27 000 28 000 29 000 30 000 32 000 34 000 36 000 38 000 40 000 42 000 44 000 46 000 48 000 50 000

10-1

1

102

10

0

0

0

0

1.0632·10-18 1.0016·10-5 0.17 8.43 68.54 350.45 1010.99 1631.16 2199.05 2741.77 3241.13 3676.99 4067.29 4441.00 4812.06 5182.68 5542.91 5860.13 6063.66 6076.90 5930.40 5772.09 5708.86 5753.60 5879.75 6061.33 6277.30 6510.88 6747.01 6969.80 7302.48 7418.15 7446.47 7573.16 7828.16 8162.27 8508.67 8798.67 8990.36 9113.16

2.9968⋅10-19 3.1574⋅10-6 0.064 4.75 41.46 175.07 681.92 1493.24 2260.52 2986.86 3685.75 4345.01 4940.96 5468.36 5949.25 6408.06 6857.69 7301.67 7729.61 8115.27 8408.03 8545.59 8507.50 8360.64 8215.79 8149.32 8179.25 8293.27 8471.19 8695.07 9210.38 9719.56 10104.20 10296.50 10370.90 10481.60 10704.70 11028.70 11405.00 11776.20

5.4226⋅10-20 9.7366⋅10-7 0.021 2.35 24.39 101.61 318.76 999.82 1992.66 2977.39 3934.26 4869.85 5772.02 6620.13 7394.90 8091.58 8725.14 9317.21 9883.90 10433.60 10966.20 11466.30 11905.10 12241.80 12437.00 12475.30 12391.80 12258.80 12148.30 12104.40 12255.60 12660.70 13211.00 13789.90 14301.60 14670.50 14880.00 15013.50 15176.30 15424.70

6.1694⋅10-21 2.5908⋅10-7 0.0063 0.88 12.66 59.44 171.88 443.79 1182.84 2360.71 3618.40 4896.25 6175.89 7441.91 8671.96 9846.78 10941.10 11945.90 12863.70 13707.20 14492.80 15235.00 15942.50 16616.10 17248.10 17821.60 18310.10 18686.40 18930.50 19044.60 18984.40 18836.10 18871.70 19155.90 19649.30 20245.30 20870.90 21448.80 21920.20 22265.60

0 6.2623⋅10-22 4.2388⋅10-8 0.0015 0.25 4.73 28.88 94.12 225.25 496.76 1134.63 2309.08 3821.74 5454.82 7155.50 8903.00 10668.50 12421.60 14141.20 15790.10 17347.00 18802.70 20153.50 21404.90 22566.60 23650.10 24665.60 25619.20 26514.50 27347.40 28108.60 29367.60 30181.30 30571.90 30683.90 30715.80 30838.00 31125.30 31575.30 32149.40 32795.80

References 1. Ferziger, J.H., Kaper, H.G.: Mathematical theory of transport processes in gases. NorthHolland, Amsterdam (1972) 2. Hirschfelder, J., Curtiss, C., Bird, R.: Molecular theory of gases and liquids, p. 525. Wiley, New York (1964) 3. Colonna, G., Capitelli, M.: Spectrochimica Acta B 64, 863 (2009)

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4. D’Ammando, G., Colonna, G., Pietanza, L.D., Capitelli, M.: Spectrochimica Acta B 65, 603 (2010) 5. http://webbook.nist.gov/ 6. Drellishak, K.S., Aeschliman, D.P., Cambel, A.B.: Tables of thermodynamic properties of argon, nitrogen and oxygen plasmas. Aedc-tdr 64-12 (1964) 7. Drellishak, K.S., Aeschliman, D.P., Cambel, A.B.: Physics of Fluids 8, 1590 (1965) 8. Stupochenko, E.V., Stakhenov, I.P., Samuilov, E.V., Pleshanov, A.S., Rozhdestvenskii, I.B.: American Rocket Society Journal Supplement 30, 98 (1960) 9. Capitelli, M., Colonna, G., Giordano, D., Marraffa, L., Casavola, A., Minelli, P., Pagano, D., Pietanza, L.D., Taccogna, F.: Tables of internal partition functions and thermodynamic properties of high-temperature Mars atmosphere species from 50 to 50 000 K. In: Giordano, D., Warmbein, B. (eds.) ESA STR–246. ESA Publication Division (2005) 10. Capitelli, M., Colonna, G., Giordano, D., Marraffa, L., Casavola, A., Minelli, P., Pagano, D., Pietanza, L.D., Taccogna, F.: Journal of Spacecrafts and Rockets 42, 980 (2005) 11. Pagano, D., Casavola, A., Pietanza, L.D., Capitelli, M., Colonna, G., Giordano, D., Marraffa, L.: Internal partition functions and thermodynamic properties of hightemperature Jupiter-atmosphere species from 50 to 50 000 K. In: Giordano, D., Fletcher, K. (eds.) ESA STR–257. ESA Communication Production Office (2009) 12. Pagano, D., Casavola, A., Pietanza, L.D., Colonna, G., Giordano, D., Capitelli, M.: Journal of Thermophysics and Heat Transfer 22, 434 (2008) 13. Kihara, T., Taylor, M.H., Hirschfelder, J.O.: Physics of Fluids 3, 715 (1960) 14. Monchik, L.: Physics of Fluids 2, 695 (1959) 15. Kalinin, A.P., Dubrovitskii, D.Y.: High Temperature 38, 848 (2000) 16. Smith, F.J., Munn, R.J.: Journal of Chemical Physics 41, 3560 (1964) 17. Neufeld, P.D., Janzen, A.R., Aziz, R.A.: Journal of Chemical Physics 57, 1100 (1972) 18. Mason, E.A.: Journal of Chemical Physics 22, 169 (1954) 19. Rainwater, J., Holland, P., Biolsi, L.: Journal of Chemical Physics 77, 434 (1982) 20. Tang, K.T., Toennies, J.P.: Journal of Chemical Physics 118, 4976 (2003) 21. Pirani, F., Maciel, G., Cappelletti, D., Aquilanti, V.: International Reviews in Physical Chemistry 25, 165 (2006) 22. Pirani, F., Albertí, M., Castro, A., Teixidor, M.M., Cappelletti, D.: Chemical Physics Letters 394, 37 (2004) 23. Liuti, G., Pirani, F.: Chemical Physics Letters 122, 245 (1985) 24. Cambi, R., Cappelletti, D., Liuti, G., Pirani, F.: Journal Chemical Physics 95, 1852 (1991) 25. Cappelletti, D., Liuti, G., Pirani, F.: Chemical Physics Letters 183, 297 (1991) 26. Aquilanti, V., Cappelletti, D., Piran, F.: Chemical Physics 209, 299 (1996) 27. Capitelli, M., Cappelletti, D., Colonna, G., Gorse, C., Laricchiuta, A., Liuti, G., Longo, S., Pirani, F.: Chemical Physics 338, 62 (2007) 28. Stallcop, J., Partridge, H., Pradhan, A., Levin, E.: Journal of Thermophysics and Heat Transfer 14, 480 (2000) 29. Stallcop, J., Partridge, H., Levin, E.: Journal of Chemical Physics 95, 6429 (1991) 30. Stallcop, J., Partridge, H., Levin, E.: Physical Review A 64, 0427221 (2001) 31. Levin, E., Wright, M.: Journal of Thermophysics and Heat Transfer 18, 143 (2004) 32. Laricchiuta, A., Colonna, G., Bruno, D., Celiberto, R., Gorse, C., Pirani, F., Capitelli, M.: Chemical Physics Letters 445, 133 (2007) 33. André, P., Bussiére, W., Rochette, D.: Plasma Chemistry and Plasma Processing 27, 381 (2007)

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34. Viehland, L.A., Dickinson, A.S., Maclagan, R.G.A.R.: Chemical Physics 211, 1 (1996) 35. Maclagan, R.G.A.R., Viehland, L.A., Dickinson, A.S.: Journal of Physics B 32, 4947 (1999) 36. Firsov, O.B.: Journal of Experimental and Theoretical Physics 21, 1001 (1951) (in Russian) 37. Nikitin, E.E., Smirnov, B.M.: Soviet Physics Uspekhi 21, 95 (1978) 38. Mason, E.A., Vanderslice, J.T., Yos, J.M.: Physics of Fluids 2, 688 (1959) 39. Devoto, R.S.: Physics of Fluids 10, 354 (1967) 40. Murphy, A.B.: Plasma Chemistry and Plasma Processing 15, 279 (1995) 41. Yun, K.S., Mason, E.A.: Physics of Fluids 5, 380 (1962) 42. Levin, E., Partridge, H., Stallcop, J.R.: Journal of Thermophysics and Heat Transfer 4, 469 (1990) 43. Capitelli, M., Ficocelli, E.: Journal of Physics B 5, 2066 (1972) 44. Laricchiuta, A., Bruno, D., Capitelli, M., Celiberto, R., Gorse, G., Pintus, G.: Chemical Physics 344, 13 (2008) 45. Sourd, B., Aubreton, J., Elchinger, M.F., Labrot, M., Michon, U.: Journal of Physics D 39, 1105 (2006) 46. Capitelli, M., Gorse, C., Longo, S., Giordano, D.: Journal of Thermophysics and Heat Transfer 14, 259 (2000) 47. Gupta, R.N., Yos, J.M., Thompson, R.A., Lee, K.P.: NASA Report RP-1232 (1990) 48. Eletskii, A.V., Capitelli, M., Celiberto, R., Laricchiuta, A.: Physical Review A 69, 042718 (2004) 49. Kosarim, A., Smirnov, B., Capitelli, M., Celiberto, R., Laricchiuta, A.: Physical Review A 74, 0627071 (2006) 50. Belyaev, Y.N., Brezhnev, B.G., Erastov, E.M.: Soviet Physics JEPT 27, 924 (1968) 51. Kosarim, A., Smirnov, B.: Journal of Experimental and Theoretical Physics 101, 611 (2005) 52. Capitelli, M., Devoto, R.S.: Physics of Fluids 16, 1835 (1973) 53. Capitelli, M.: Journal of Plasma Physics 14, 365 (1975) 54. Liboff, R.L.: Physics of Fluids 2, 40 (1959) 55. Hahn, H.S., Mason, E.A., Smith, F.J.: Physics of Fluids 14, 278 (1971) 56. Mason, E.A., Munn, R.J., Smith, F.J.: Physics of Fluids 10, 1827 (1967) 57. D’Angola, A., Colonna, G., Gorse, C., Capitelli, M.: European Physical Journal D 46, 129 (2008) 58. Bell, K.L., Scott, N.S., Lennon, M.A.: Journal of Physics B 17, 4757 (1984) 59. Gibson, J.C., Gulley, R.J., Sullivan, J.P., Buckman, S.J., Chan, V., Burrow, P.D.: Journal of Physics B 29, 3177 (1996) 60. Panajotovic, R., Filipovic, D., Marinkovic, B., Pejcev, V., Kurepa, M., Vuskovic, L.: Journal of Physics B 30, 5877 (1997) 61. Nahar, S.N., Wadehra, J.M.: Physical Review A 35, 2051 (1987) 62. Bruno, D., Capitelli, M., Catalfamo, C., Celiberto, R., Colonna, G., Diomede, P., Gorse, C., Laricchiuta, A., Longo, S., Pagano, D., Pirani, F.: Transport Properties of HighTemperature Mars Atmosphere Components, ESA STR 256. In: Giordano, D., Fletcher, K. (eds.). ESA Communication Production Office (2008) 63. Itikawa, Y.: J. Phys. Chem. Ref. Data 31, 749 (2002) 64. Wright, M.J., Bose, D., Palmer, G.E., Levin, E.: AIAA Journal 43, 2558 (2005) 65. Griem, H.R.: Physical Review 128, 1280 (1962) 66. Margenau, H., Lewis, M.: Rev. Modern Physics 31, 594 (1959) 67. Capitelli, M., Ficocelli, E.V.: Zeitschrift für Naturforschung A 25, 977 (1970)

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68. Capitelli, M., Molinari, E.: Journal of Plasma Physics 4, 335 (1970) 69. Drellishak, K.S., Knopp, C.F., Cambel, A.B.: Physics of Fluids 6, 1280 (1963) 70. Drellishak, K.S., Knopp, C.F., Cambel, A.B.: Partition functions and thermodynamic properties of argon plasmas, Arnold Engineering Development Center, Tullahome, Tennessee, report TDR 63-146 (1963) 71. Fermi, E., für, Z.: Physik 26, 54 (1924) 72. Capitelli, M., Giordano, D.: Physical Review A 80, 32113 (2009) 73. Ecker, G., Weizel, W.: Annalen der Physik 17, 126 (1956) 74. Ecker, G., Kroll, W.: Zeitschrift für Naturforschung 21A, 2012 (1966) 75. Roussel, K., O’Connell, R.: Physical Review A 9, 52 (1974) 76. Giordano, D., Capitelli, M.: Unpublished results 77. Capitelli, M., Bruno, D., Colonna, G., Catalfamo, C., Laricchiuta, A.: Journal of Physics D 42, 194005 (2009) 78. Capitelli, M., Giordano, D., Colonna, G.: Physics of Plasmas 15, 082115 (2008) 79. Sing, K., Sing, G., Sharma, R.: Physics of Plasmas 17, 72309 (2010) 80. Moore, C.E.: Atomic Energy Levels NBS Circular N 467, 1949 (1958) 81. http://physics.nist.gov/PhysRefData/ASD/levels_form.html 82. Gurvich, L.V., Veyts, I.V., Alcock, C.B.: Thermodynamic Properties of Individual Substances. Hemisphere Publishing Corporation, New York (1989) 83. Gordon, S., McBride, B.J.: Thermodynamic data to 20 000 K for monatomic gases. NASA/TP-1999-208523 (1999) 84. Aubreton, J., Elchinger, M.F., Fauchais, P.: Plasma Chemistry and Plasma Processing 18, 1 (1998) 85. Rat, V., André, P., Aubreton, J., Elchinger, M.F., Fauchais, P., Lefort, A.: Physical Review E 64, 026409 (2004) 86. Rat, V., Murphy, A.B., Aubreton, J., Elchinger, M.F., Fauchais, P.: Journal of Physics D 41, 183001 (2008) 87. Giordano, D., Capitelli, M.: Physical Review E 65, 16401 (2001) 88. Capitelli, M., Giordano, D.: Journal of Thermophysics and Heat Transfer 16, 283–285 (2002) 89. Capitelli, M., Colonna, G., Gorse, C., Minelli, P., Pagano, D., Giordano, D.: AIAA paper 2001-3018 (2001) 90. Potapov, A.: High Temperature 4, 48 (1966) 91. Chen, X., Han, P.: Journal of Physics D 32, 1711 (1999) 92. Van de Sanden, M.C.M., Schram, P.P.J.M., Peeters, A.G., van der Mullen, J.A.M., Kroesen, G.M.W.: Physical Review A 40, 5273 (1989) 93. Morro, A., Romeo, M.: Journal of Non-Equilibrium Thermodynamics 13, 339 (1988) 94. Capitelli, M., Colonna, G., Gorse, C., Minelli, P., Pagano, D., Giordano, D.: Journal of Thermophysics and Heat Transfer 16, 469 (2002) 95. Capitelli, M., Celiberto, R., Gorse, C., Laricchiuta, A., Pagano, D., Traversa, P.: Physical Review E 69, 26412 (2004) 96. Capitelli, M., Lamanna, U.: Journal of Plasma Physics 12, 71 (1974) 97. Ignjatovic, L., Mihajlov, A.A.: Contributions to Plasma Physics 37, 309 (1997) 98. Celiberto, R., Lamanna, U.T., Capitelli, M.: Physical Review A 58, 2106 (1998) 99. Capitelli, M., Celiberto, R., Gorse, C., Laricchiuta, A., Minelli, P., Pagano, D.: Physical Review E 66, 16403 (2002) 100. Bruno, D., Capitelli, M., Catalfamo, C., Laricchiuta, A.: Physics of Plasmas 14, 072308 (2007)

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High Temperature Phenomena in Shock Waves

101. Bruno, D., Laricchiuta, A., Capitelli, M., Catalfamo, C.: Physics of Plasmas 14, 022303 (2007) 102. Capitelli, M., Lamanna, U.T., Guidotti, C., Arrighini, G.P.: Chemical Physics 19, 269 (1977) 103. Nyeland, C., Mason, E.A.: Physics of Fluids 10, 985 (1967) 104. Sourd, B., André, P., Aubreton, J., Elchinger, M.F.: Plasma Chemistry and Plasma Processing 27, 35 (2007); ibidem 27, 225 (2007) 105. Laricchiuta, A., Pirani, F., Colonna, G., Bruno, D., Gorse, C., Celiberto, R., Capitelli, M.: Journal of Physical Chemistry A 113, 15250 (2009) 106. D’Angola, A., Colonna, G., Gorse, C., Capitelli, M.: European Physical Journal D 46, 129 (2008) 107. Boulos, M.I., Fauchais, P., Pfender, E.: Thermal plasmas: fundamentals and applications. Plenum Press, New York (1994) 108. Ghorui, S., Heberlein, J.V.R., Pfender, E.: Plasma Chemistry and Plasma Processing 28, 553 (2008) 109. Kremp, D., Schlanges, M., Kraeft, W.: Quantum statistics of non-ideal plasmas. Atomic, Molecular and Plasma Physics Series, vol. 25. Springer, Heidelberg (2005) 110. Zivny, O.: European Physical Journal D 54, 349 (2009) 111. Zaghoul, M.R.: Physics of Plasmas 17, 062701 (2010)

Chapter 2

Non-equilibrium Kinetics and Transport Properties behind Shock Waves E.V. Kustova and E.A. Nagnibeda Saint Petersburg State University, Saint Petersburg, Russia

1 Introduction In high-temperature and hypersonic flows of gas mixtures, the energy exchange between translational and internal degrees of freedom, chemical reactions, ionization and radiation may result in noticeable violation of thermodynamic equilibrium when the characteristic times of kinetic and gas-dynamic processes are comparable. Therefore the non-equilibrium effects become important and for a correct prediction of gas flow parameters, non-equilibrium kinetics and gas dynamics should be considered jointly. In shock waves occurring in hypersonic flows, the rapid gas compression within a thin shock front results in a temperature jump which leads to excitation of internal degrees of freedom of molecular species and chemical reactions. Experimental data show the significant difference in relaxation times of various kinetic processes. The theoretical models adequately describing physical-chemical kinetics behind shock waves depend on relations between relaxation times of various kinetic processes. At the high temperature conditions which are typical just behind the shock front, the equilibrium between the translational and rotational degrees of freedom is established in a substantially shorter time than that of vibrational relaxation and chemical reactions, and therefore the following relation takes place[1]: ~

~ .

(1)

Here, , , , and are, respectively, the relaxation times for the translational, rotational and vibrational degrees of freedom, and the characteristic time for chemical reactions; is the mean time of macroscopic parameters variation. In this case it is usually supposed that translational and rotational relaxation occurs in a thin shock front (with a characteristic length of about several mean free paths of molecules) essentially without variation in the mixture composition and distributions over the vibrational energies. Then, in the relaxation zone (with a length of many tens or even hundreds mean free paths) behind the shock front, the excitation of vibrational degrees of freedom and chemical reactions take place, while equilibrium or weakly non-equilibrium distributions over translation and rotational energies established in the shock front are maintained. For the description of the non-equilibrium flow under the

60

High Temperature Phenomena in Shock Waves

condition (1) it is necessary to consider the equations of the state-to-state vibrational and chemical kinetics coupled to the gas dynamic equations. It is the most detailed description of the non-equilibrium flow. More simple models are based on quasi-stationary multi-temperature or onetemperature vibrational distributions. In the vibrationally excited gas at moderate temperatures, the near-resonant vibrational energy exchanges between molecules of the same chemical species occur much more frequently compared to the non-resonant transitions between different molecules as well as transfers of vibrational energy to the translational and rotational ones and chemical reactions[1,2]: ~

~ .

(2)

Here , , are, respectively, the mean times for the VV1 vibrational energy exchange between molecules of the same species, VV2 vibrational transitions between molecules of different species and TRV transitions of the vibrational energy into other , quasi-stationary (multimodes. Under the condition (2), during the time temperature) distributions over the vibrational levels establish, which then maintain in non-equilibrium chemical reactions (model of non-equilibrium multi-temperature kinetics). For tempered reaction regime, with the chemical reaction rate considerably lower than that for the internal energy relaxation, the following characteristic time relation takes place: ~ ,

(3)

where is the mean time for the internal energy relaxation. Under this condition, the non-equilibrium chemical kinetics can be considered on the basis of the maintaining thermal-equilibrium one-temperature Boltzmann distributions over the internal energy levels of molecular species (the model of one-temperature chemical kinetics [3], often used in applications). The most accurate theoretical description of non-equilibrium flows behind shock waves can be given using the kinetic theory methods. The kinetic theory makes it possible to develop mathematical models of a flow under different non-equilibrium conditions, i.e. to obtain closed systems of the non-equilibrium flow equations and to elaborate calculation procedures for transport and relaxation properties.

2 State-to-State Approach 2.1 Distribution Functions and Macroscopic Parameters In reacting mixtures with rapid and slow physical-chemical processes, the kinetic , , over chemical species c, vibrational equations for the distribution functions i and rotational j energy levels in the phase space of the velocity coordinates and time may be written in the form [4,5]: ·

.

(4)

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

Under the conditions (4), the integral operator of rapid processes

61

describes elastic

collisions and rotational energy exchange whereas the operator of slow processes describes the vibrational energy exchange and chemical reactions: ,

.

(5)

The parameter in Eqs.(4) represents the ratio of the characteristic times: ⁄ , ~ , ~ . The integral operators (5) are given in [4,6]. Modification of the Chapman–Enskog method for the solution of the kinetic equations (4), (5) [7,8], makes it possible to derive governing equations of the flow, expressions for the dissipative and relaxation terms in these equations and algorithms for the calculation of transport and reaction rate coefficients. The solution of the kinetic equations in the zero-order approximation 0

(6)

is specified by the independent collision invariants of the most frequent collisions. These invariants include the momentum and particle total energy which are conserved at any collision, and additional invariants for the most probable collisions which are given by any value independent of the velocity and rotational level j and depending arbitrarily on the vibrational level i and chemical species c. The additional invariants appear because vibrational energy exchange and chemical reactions are supposed to be frozen in rapid processes. Based on the above set of the collision invariants, the zero-order solution of Eqs.(4) takes the form exp

.

(7)

is the population of vibrational level i of species c, , is the Here, macroscopic velocity, the rotational energy of the molecule at jth rotational and ith vibrational levels, is the gas temperature, the molecular mass, k the Botzmann the rotational partition constant, the rotational statistical weight, functions. For the rigid rotator model, , , is the moment of inertia, h is the Planck constant, is the symmetry factor. The distribution functions (7) are specified by the macroscopic gas parameters , ( 1, , , 0,1, , , is the number of chemical species, is the number of excited vibrational levels in species c), , , and , which correspond to the set of the collision invariants of rapid processes. 2.2 Governing Equations The closed set of equations for the macroscopic quantities , , , , and , follows from the kinetic equations and includes the conservation equations of

62

High Temperature Phenomena in Shock Waves

momentum and total energy coupled to the equations of detailed state-to-state vibrational and chemical kinetics [4]: ·

·

,

1,

·

, ,

0,1,

0,

·

:

,

,

(8)

(9) 0.

(10)

diffusion velocities of Here is the pressure tensor, the total energy flux, molecules at different vibrational states and the total energy per unit mass ∑

∑

(11)

,

is the rotational energy per unit mass, the vibrational energy of a molecule of species con the i-th vibrational level and the energy of formation of the particle of species c. The source terms in the equations (8) are expressed via the integral operators of slow processes ∑

,

(12)

and characterize the variation of the vibrational level populations and atomic number densities caused by different vibrational energy exchanges and chemical reactions. For this approach, the vibrational level populations are included to the set of main macroscopic parameters, and the equations for their calculation are coupled to the equations of gas dynamics. Particles of various chemical species in different vibrational states represent the mixture components, and the corresponding equations contain the diffusion velocities of molecules at different vibrational states. In the zero-order approximation of Chapman-Enskog method ,

0,

0

, ,

(13)

and the set of governing equations takes the form ·

,

1,

, ,

0, ·

0.

0,1,

,

(14) (15) (16)

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

63

contain the microscopic rate coefficients for vibrational The expressions for energy exchanges and chemical reactions averaged with the Maxwell–Boltzmann distribution over the velocity and rotational energy levels and depending on the vibrational states and chemical species of interacting particles. The equations (14)–(16) describe detailed state-to-state vibrational and chemical kinetics in an inviscid non-conductive gas mixture flow in the Euler approximation. Taking into account the first-order approximation makes it possible to consider dissipative properties in a non-equilibrium viscous gas. 2.3 First-Order Approximation. The first-order distribution functions can be written in the following structural form [4]: · ln

∑

·

:

·

.

(17)

depend on the derivatives of all macroscopic The distribution functions parameters: temperature , velocity , and vibrational level populations via the diffusive driving forces ln .

(18)

, , , and depend on the peculiar velocity and the The functions flow parameters, and satisfy the linear integral equations with linearized operator for rapid processes. The transport kinetic theory in the state-to-state approximation was developed, for the first time, in [8] and also given in [4]. The expressions for the transport terms in the equations (8)–(10) in the first order approximation are derived on the basis of the distribution functions (17). The viscous stress tensor is described by the expression: 2

·

.

(19)

is the relaxation pressure, and are the coefficients of shear and bulk Here, viscosity. The additional terms connected to the bulk viscosity and relaxation pressure appear in the diagonal terms of the stress tensor in this case due to rapid inelastic TR exchange between the translational and rotational energies. The existence of the relaxation pressure is caused also by slow processes of vibrational and chemical 0. relaxation. If all slow relaxation processes in a system disappear, then The diffusion velocity of molecular components c at the vibrational level i is specified in the state-to-state approach by the expression [4,8]: ∑

ln ,

(20)

64

High Temperature Phenomena in Shock Waves

where and are the multi-component diffusion and thermal diffusion coefficients for each chemical and vibrational species. The total energy flux in the first-order approximation has the form: ∑

∑

,

(21)

where is the thermal conductivity coefficient, is the mean and are responsible for the energy transfer rotational energy. The coefficients associated with the most probable processes which, in the present case, are the elastic collisions and inelastic TR- and RR rotational energy exchanges. In the state-to-state approach, the transport of the vibrational energy is described by the diffusion of vibrationally excited molecules rather than by the thermal conductivity. In particular, the diffusion of the vibrational energy is simulated by introducing independent diffusion coefficients for each vibrational state. It should be noted that all transport coefficients are specified by the cross sections of rapid processes except the relaxation pressure depending also on the cross sections of slow processes of vibrational relaxation and chemical reactions. From the expressions (20), (21), and (18), it is seen that the energy flux and diffusion velocities include along with the gradients of temperature and atomic number densities also the gradients of all vibrational level populations. This constitutes the main feature of the heat transfer and diffusion in the state-to-state and and the diffusion approach and the fundamental difference between velocities and heat flux obtained on the basis of one-temperature, multi-temperature or weakly non-equilibrium approaches. The transport coefficients in the expressions (19)–(21) can be written in terms of , , , and : functions 10

,

,

, ,

,

,

, ,

, ,

,

.

(22)

Here , are the bracket integrals associated with the linearized operator of rapid processes. They were introduced in [4] for strongly non-equilibrium reacting mixtures similarly to those defined in [9] for a non-reacting gas mixture under the conditions for weak deviations from the equilibrium. For the transport coefficients calculation, the functions , , , and are expanded into the Sonine polynomials in the reduced peculiar velocity and those of Waldmann-Trübenbacher in the dimensionless rotational energy. For the coefficients of these expansions, the linear transport systems are derived, and the transport coefficients are expressed in terms of the solutions of these systems. in Eqs. (8) describe slow processes of vibrational relaxation The source terms and chemical reactions. These terms can be written as follows ,

,

,

,

,

(23)

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

(24)

, ,

,

,

and dissociation-recombination ,

,

(25)

,

(26)

They contain the rate coefficients for the energy transitions ,

,

,

,

,

,

65

, ,

, exchange reactions

respectively

,

. (27)

,

The expressions for the zero-order rate coefficients for binary reactions have the following form , ,

∑

exp

,

exp

,

,

.

(28)

is the reduced mass, the relative velocity, the integral cross section , of the collisions resulting in a binary reaction. The expressions for rate coefficients for the remaining processes have a similar form. In the first order approximation, the rate coefficient, contrarily to the zero-order coefficients, depend not only on the temperature but also on the vibrational level populations and includethe term proportional to the velocity divergence · . The procedure for calculation of the zero-order and first-order rate coefficients is given in [4]. It should be noted, that in the practical simulations of dynamics of viscous conducting gases, in the equations of non-equilibrium kinetics (in the state-to-state, multi-temperature or one-temperature approaches), the reaction rate coefficients are calculated using the zero-order distribution function. Up to now, no reliable calculations for the first order state-depending reaction rate coefficients are available. Such estimations were proposed only for the multi-temperature model of coupled dissociation and vibrational relaxation in [10], and for a one-temperature approach in [11]. In the literature, a number of theoretical and experimental estimates for the zeroorder rate coefficients for vibrational energy transitions in different temperature intervals are available (see for example [12]). Up to the recent time, the most commonly used are the formulas of the Schwartz, Slawsky and Herzfeld theory (known as the SSH-theory), developed for the harmonic oscillator model in [13] and later generalized for anharmonic oscillators in [2,14], as well as the Landau-Teller theory (for the VT exchange) with various semi-empirical expressions for the vibrational relaxation time [15,1]. In addition, semi-empirical formulas for the rate coefficients of vibrational energy transitions written in a form similar to the

66

High Temperature Phenomena in Shock Waves

expressions of the SSH-theory are often used; they provide a satisfactory consistency with the experimental results due to introduction of some additional empirical parameters. More accurate results are based on the quantum-mechanical and semiclassical techniques applied to the calculation of the cross sections for inelastic collision and probabilities for vibrational and rotational energy transitions in various gases [16,17], as well as on the trajectory calculations [18,19,20]. In particular, in [21] it is shown that at low temperatures, the SSH-theory does not provide a satisfactory accuracy for the evaluation of the atoms efficiency in VT energy transitions. Furthermore, at high temperatures, the SSH-theory overestimates the probabilities for VT transitions from high vibrational states (compared to those obtained in [16,17]). However, practical implementation of the quantum-mechanical methods and trajectory calculations is restricted by the computational costs of the calculation of the cross section for each specific transition. Among the up-to-date analytical models for the vibrational transition probabilities, we can recommend the semi-classical model of forced harmonic oscillator (FHO) [22,23] which makes it possible to obtain correct values for the rate coefficients of VV and VT transitions (including multi-quantum jumps which are particularly important if the partner in the collision is an atom [16] at high temperatures. A model proposed in [24] based on the information theory can also be appreciated. The analytical approximations useful for practical calculations of the probabilities of different vibrational energy transitions in air components are obtained in [25,26] by interpolation of accurate numerical results presented by G. Billing in [16,17]. These approximate formulas are valid for temperatures below 12000 K. Comparison of the results obtained using the SSH formulas and the expressions proposed in [25,26] , on the show [4] that, while the dependences of the coefficients , and , vibrational quantum number are similar to those given by the formulas of the generalized SSH theory, for the coefficient , , an essential difference can be noticed. Figure 1 presents the dependence of the rate coefficients for VT transitions 1 1 (b) on the (a), and vibrational level i at T = 6000 K. One can observe a qualitative agreement of the VT rate coefficients for the transition in a collision with a molecule. Quantitatively, the generalized SSH theory gives considerably higher rates of VT transitions for upper vibrational levels ( 20). On the other hand, for the transition proceeding through a collision with an atom, theoretical models and quasi-classical trajectory (QCT)calculations give a different dependence of the rate coefficient on the vibrational state. According to QCT results, it increases almost linearly with whereas analytical models provide non-linear rising of the rate coefficient. Therefore using the SSH and FHO models for the description of non-equilibrium vibrational kinetics in mixtures with high atomic concentrations can lead to a certain error. Nevertheless, for investigation of kinetics in strong shock waves, this transition does not play a crucial role because the vibrational distributions are established at a rather short distance close to the shock front, where atomic concentrations remain low.

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

67

Fig. 1. Rate coefficients for VT transitions in N2 in a collision with N2 molecule (a) and N atom (b) respectively. T=6000 K. 1: SSH model for anharmonic oscillators [2]; 2: SSH model for harmonic oscillators; 3: FHO model [22]; 4: formulas [25, 26].

Fig. 2. Vibrational distributions behind a shock wave. 1: SSH model for anharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHO model. Curves 1-3 in graph correspond to x=0.01cm; 1'-3’ to x=2cm. (The conditions in the free stream: T=293 K, p=100 Pa, Mach number M=15).

In Figs.2 and 3, vibrational distributions and gas temperature calculated in [27] behind a shock wave using different models for the VV and VT rate coefficients are presented. One can notice that the SSH model for anharmonic oscillators provides higher population of the upper vibrational levels and lower values of temperature. The SSH model for harmonic oscillators gives a slower excitation of high vibrational levels, and, as a result, lower rate of temperature decrease. The discrepancy between the results obtained on the basis of the FHO model and the SSH model for anharmonic oscillators is small: it does not exceed 2% for the temperature. Since the

68

High Temperature Phenomena in Shock Waves

FHO model can be considered as the most accurate one in the high temperature conditions, one can conclude that the SSH model for harmonic oscillators leads to a noticeable error in the predicted values of gas dynamic parameters whereas its generalization for anharmonic oscillators works rather well in shock heated gases.

Fig. 3. Temperature behind a shock wave. 1: SSH model for anharmonic oscillators; 2: SSH model for harmonic oscillators; 3: FHO model.

The important role of multi-quantum transitions in a collision of N2 molecules with atoms is discussed in [25,26]. Solution of equations for the vibrational level populations in N2-N mixture [28] shows that multi-quantum transitions in collisions with atoms influence significantly vibrational distributions and macroscopic flow parameters reducing the vibrational relaxation time. Multi-quantum transitions in molecule-molecule collisions play a weaker role and can be neglected. The rate coefficients for dissociation from different vibrational levels have been studied much less widely than those for vibrational energy transitions. Two models are commonly used in the literature: the ladder-climbing model assuming dissociation only from the last vibrational level (see, for instance [29,25,26]), and that of Treanor and Marrone [30] allowing for dissociation from any vibrational state. Originally this model was proposed for the two-temperature approximation; its modification for the state-to-state approach [4,31] makes it possible to present the rate coefficient for dissociation of a molecule on the vibrational level iin the form: ,

where

,

,

is the state-dependent non-equilibrium factor

(29)

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

,

exp

,

69

(30)

is the thermal equilibrium dissociation rate coefficient, the , equilibrium vibrational partition function, and the parameter of the model. For , the empirical Arrhenius law can be applied: , ,

exp

(31)

,

where is the dissociation energy; the coefficients and are generally obtained from a best fit of experimental data. The tables of the coefficients in the Arrhenius formula for various chemical reactions can be found in [1,3,12,32,33]. For a practical implementation of the Treanor–Marrone model, it is important to select the parameter so that a good consistency is reached for the dissociation rate coefficient with experimental data or the results of calculations based on more accurate models. The following approximations for are commonly used: ∞, ⁄6 , and 3 . For ∞, dissociation is assumed to be equi-probable for each vibrational level, whereas the other values of the parameter describe preferential dissociation from high vibrational states. In [34], the dissociation rate coefficients calculated within the framework of the Treanor–Marrone model are compared , with those obtained from trajectory calculations [20]. The Figs.4 present the for temperature dependence of the state-dependent dissociation rate coefficient , 0 (a) and 20 (b) obtained in [20] and using the Treanor-Marrone model with different U values.

Fig. 4. Temperature dependence of the dissociation rate coefficient 20 (b). Curve 1: results of Ref. [20], curves 2–4 correspond to ∞

,

for ⁄6 ,

0 (a) and 3 , and

70

High Temperature Phenomena in Shock Waves

It can be noticed that the choice ∞results in significant overestimation for at low vibrational levels. With the increase of the vibrational quantum number , a better agreement with the results of accurate trajectory calculations is found. It confirms the common assumption of the preferential dissociation from high ⁄6 and vibrational states. The values 3 provide good consistency for at intermediate levels. It is also shown that using the same value of the , parameter for any i and T can result in considerable error in the calculation of statedependent dissociation rate coefficients. The choice of the parameter should be specified by the conditions of a particular problem. In some studies [35,25,26], a possibility for dissociation from any vibrational state is suggested within the framework of the ladder-climbing model. To this end, it is supposed that a transition to the continuum occurs as a result of multi-quantum vibrational energy transfers. From this point of view, the authors of [35] conclude that dissociation from low vibrational levels is preferential for high gas temperatures. A similar effect is also mentioned in [36]. However, this conclusion has not been justified either by accurate trajectory calculations or by experiments. The influence of the dissociation model on the vibrational level populations behind the shock wave is shown in Figure 5, where the population of the tenth level versus the distance from the shock front is presented for different values for the parameters U, A, and n, as well as for the ladder-climbing model. Three values for U are ⁄6 , and considered: ∞, 3 , the parameters in the Arrhenius formula are taken from [37,32]. We can see that the vibrational distributions calculated with the ladder-climbing dissociation model are significantly different from those obtained using the Treanor–Marrone model. The rate coefficients for bimolecular exchange reactions depending on the vibrational states of reagents and products have been less thoroughly studied than those for dissociation processes. Theoretical and experimental studies for the influence of the vibrational excitation of reagents on reaction rates were started by J. Polanyi [38]; some experimental results were also obtained in [39]. The accurate theoretical approach to this problem primarily requires a calculation for the statedependent differential cross sections for collisions resulting in chemical reactions, and their subsequent averaging over the velocity distributions. In the recent years, the dynamics of atmospheric reactions has been studied, and quasi-classical trajectory calculations for the cross sections and state-dependent rate coefficients for the and have been carried out by reactions several authors. The reactions of NO formation are considered in [40]; the effect of translational, rotational, and vibrational energy of reagents on the reaction is discussed in [41,42]. For the application of the existing results to the problems of non-equilibrium fluid dynamics, the analytical expressions for the dependence of the reaction rate coefficients on the vibrational states of molecules participating in the reactions are needed. Two kinds of such expressions are available in the literature. The first kind includes analytical approximations for numerical results obtained for particular reactions (see [43,12,44]). These expressions are sufficiently accurate and convenient for practical use; however, their application is

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

71

restricted by the considered temperature range. Another approach is based on the generalizations of the Treanor–Marrone model to exchange reactions suggested in [45,46].

⁄6 (the parameters A, n are taken Fig .5. 10th level population as a function of x. 1: ⁄6 ;3: from Ref. [37]); 2: 3 ; 4: ∞ (for 2–4A, n are from Ref. [32]); 5: ladderclimbing model. (The conditions in the free stream: T=293 K, p=100 Pa, Mach number M=15).

These models can be used for more general cases, but the theoretical expressions for the rate coefficients contain additional parameters, which should be validated using experimental data. A lack of data for these parameters restricts the implementation of the above semi-empirical models. Therefore, the development of justified theoretical models for cross sections of reactive collisions and statedependent rate coefficients for exchange reactions remains a very important problem of the non-equilibrium kinetics.

3 Quasi-stationary Approaches 3.1 Vibrational Distributions: Governing Equations Practical implementation of the state-to-state kinetic model leads to serious difficulties. The important problem encountered in the realization of the state-to-state model is its computational cost. Indeed, the solution of the fluid dynamics equations coupled to the equations of the state-to-state vibrational and chemical kinetics in multi-component mixtures requires numerical simulation of a great number of equations for the vibrational level populations of all molecular species. Moreover, in

72

High Temperature Phenomena in Shock Waves

the viscous gas approximation, numerical simulations require the calculation of a large number of transport coefficients, particularly, diffusion coefficients in each space cell and at each time step, which significantly complicates the study of specific flows. Simplifications proposed in [47] make it possible to reduce the number of state-dependent transport coefficients, but even after that, the state-to-state model for multi-component reacting flows remains time consuming and numerically expensive. Therefore simpler models based on quasi-stationary vibrational distributions are rather attractive for practical applications. In quasi-stationary approaches, the vibrational level populations are expressed in terms of a few macroscopic parameters; consequently, non-equilibrium kinetics can be described by a considerably reduced set of governing equations. Commonly used models are based on the Boltzmann distribution with the vibrational temperature different from the gas temperature. However, such a distribution is valid solely for the harmonic oscillator model, which describes adequately only the low vibrational states. The more accurate quasistationary model is based on the Treanor two-temperature vibrational distribution for anharmonic oscillators. The state-to-state kinetic model of non-equilibrium flow may be reduced to multitemperature description under the condition (2). In this case, the integral operator of the most frequent collisions in the kinetic equations (4) includes the operator of VV1 vibrational energy transitions between molecules of the same species along with the operators of elastic collisions and collisions with rotational energy exchanges; the operator of slow processes consists of the operator of VV2 vibrational transitions between molecules of different species, as well as the operators describing the transfer of vibrational energy into rotational and translational modes and chemical reactions ,

.

(32)

The zero-order distribution functions are specified by the invariants of the most frequent collisions. In addition to the invariants which are conserved in any collision, under the condition (2) there are additional independent invariants of rapid processes: the number of the vibrational quanta in each molecular species c, and an arbitrary value independent of the velocity, vibrational i and rotational j quantum numbers and depending arbitrarily on the particle chemical species c. Conservation of vibrational quantum for VV transitions in a single-component gas was found for the first time in [48] and the non-equilibrium two-temperature distribution was derived, now it is called the Treanor distribution. In a gas mixture, conservation of vibrational quanta in each species during VV1 transitions takes place. The existence of the other additional invariants is explained by the fact that under the condition (2), slow chemical reactions remain frozen in the most rapid process. Taking into account the system of collision invariants one can obtain the zero-order distribution functions and the following expressions for vibrational level populations: ,

exp

,

(33)

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

73

, the non-equilibrium is the number density of cth species, Where partition function of vibrational degrees of freedom and the temperature the first vibrational level for each molecular species c. The distributions (33) generalize the Treanor distributions [48] for a multicomponent reacting mixture. Similarly to a single-component gas, the distribution (33) describes adequately only the populations of low vibrational levels , . This is explained by the where corresponds to the minimum of the function fact that the conservation of vibrational quantum takes place only at low levels . However, in the high temperature gas when , the level appears to be close 2 to the last vibrational level . Therefore the Treanor distribution may be used for all vibrational levels in the relaxation zone behind shock waves. On condition that the anharmonic effects can be neglected, the distribution (33) is reduced to the non-equilibrium Boltzmann distribution with the vibrational different from T. In the case of the temperature of molecular components local thermal equilibrium, the vibrational temperatures of all molecular species are equal to the gas temperature , and the Treanor distribution (33) is reduced to the one-temperature Boltzmann distribution. The zero-order distribution functions depend on the macroscopic parameters , , , , , , and , . In the present case, in contrast to the state-tostate model, the number of main macroscopic parameters is reduced, and instead of it includes the vibrational the level populations of all vibrational states temperatures and number densities of chemical species. The governing , , , , , , and , . are equations for the macroscopic quantities derived in [7,49]. A closed system of reacting multi-component mixture dynamics consists of the equations of the multi-temperature chemical kinetics for the species number densities, conservation equations for the momentum and the total energy, and additional relaxation equations for molecular species: ·

·

, · ·

·

,

1,

, ,

(34)

0, :

(35) 0, ·

(36) ,

1,

,

,

(37)

where are diffusion velocities of different chemical species, is a specific number of vibrational quanta in molecules of c species and , the flux of vibrational quanta of c molecular species:

74

High Temperature Phenomena in Shock Waves

,

∑

.

(38)

In the multi-temperature approach, the total energy is a function of , , , in contrast to the state-to-state model where it depends on all level populations and gas temperature. The source terms in Eqs. (34), (37) have the form: ∑

, ,

∑

(39)

,

,

(40)

.

The equations (34)–(37) form a closed system of equations for the macroscopic parameters of a reacting gas mixture flow in the multi-temperature approach. It is obvious that the system (34)–(37) is considerably simpler than the corresponding system (8)–(10) in the state-to-state approach, as it contains much fewer equations. Thus, instead of ∑ equations for the vibrational level populations, one should equations for the numbers of quanta and equations for the number solve densities of the chemical components ( is the number of vibrational levels in the is the number of the molecular species in a mixture). molecular species c, Consequently, for a two-component mixture containing nitrogen molecules and and should be solved instead of 46 equations for atoms, two equations for the level populations of N2 molecules. While studying the (important for practical applications) five-component air mixture N2, O2, NO, N, O in the state-to-state 114 equations for the vibrational approach, one should solve level populations. In the multi-temperature approach, they are reduced to six equations: three for the molecular number densities , , and three for , , and . vibrational temperatures In a system of harmonic oscillators, the relaxation equations (37) are transformed into those for the specific vibrational energy which is defined by non-equilibrium Boltzmann distribution with vibrational temperature : ,

·

,

,

,

·

,

1,

,

, (41)

with ,

,

,

(42)

.

In the zero-order approximation of the Chapman–Enskog method, the transport terms are as follows ,

,

0,

0

,

(43)

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

75

and the system (34)–(37) takes the form typical for inviscid non-conductive flows ·

,

1,

, ,

(44)

0,

·

(45)

0,

(46)

,

1,

,

(47)

.

The production terms in equations (44), (47) are given by the formulas ∑

∑

,

(48)

,

and contain the zero-order operators of VV2 and TRV vibrational energy exchanges and chemical reactions. 3.2 Transport Terms In the multi-temperature approach, the first-order distribution functions have the following form [7,4]: 1 ∑

1

· ln

·

:

· ln ·

.

(49)

The coefficients , , , , and are functions of the peculiar velocity and macroscopic parameters and satisfy the linear integral equations with linearized operator of rapid processes of VV2, VT vibrational transitions and chemical reactions. First-order distribution functions (49) define transport terms in the equations (34)– (37). The pressure tensor has the form (19) where the relaxation pressure and bulk viscosity coefficient are presented as sums of two terms: ,

,

(50)

where the first term is due to inelastic RT rotational energy exchange, whereas the second is connected to the VV1 transitions in each vibrational mode.

76

High Temperature Phenomena in Shock Waves

The diffusion velocity takes the form ∑

ln ,

(51)

and are the diffusion and thermal diffusion coefficients . The total energy flux and the fluxes of vibrational quanta depend on the gradients of the gas temperature T, temperatures of the first vibrational level , and molar fractions of chemical species ⁄ : ∑

∑

∑

,

.

∑

,

(52) (53)

In Eqs. (52), (53), , , , are thermal conductivity coefficients and is the specific enthalpy of c particles. Transport coefficients are defined by bracket integrals, depending, in the multitemperature approximation, on the cross sections of elastic collisions and collisions resulting from the RT and non-resonant VV1 energy exchanges. The expressions for shear and bulk viscosity and relaxation pressure via bracket integrals are the same as in the state-to-state approximation (see Eq. (22)). However, bracket integrals in these two approaches are different because they are defined with cross sections of various rapid processes [4]. In the expressions (51)–(53), the diffusion, thermal diffusion and heat conductivity coefficients have the forms ,

, ,

, ,

,

,

, ,

, ,

.

(54)

The coefficient describes the transport of the translational, rotational energy and a small part of the vibrational energy, which is transferred to the translational mode as a result of the non-resonant VV1 transitions between molecules simulated by anharmonic oscillators and is presented as a sum of three corresponding terms: . The coefficients are associated with the transport of vibrational quanta in each molecular species and thus describe the transport of the . The cross coefficients , are specified by main part of vibrational energy both the transport of vibrational quanta and the vibrational energy loss (or gain) as a ⁄ , the result of non-resonant VV1 transitions. For low values of the ratio , , and are much smaller than , and for the harmonic coefficients 0 since VV1 transitions appear to be strictly oscillator model and disappear in a system resonant. For the same reason, the coefficients of harmonic oscillators.

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

77

The number of independent diffusion coefficients in the multi-temperature model is considerably smaller than that in the approach accounting for the detailed vibrational kinetics. Therefore the use of the quasi-stationary vibrational distributions noticeably facilitates the heat fluxes calculation in a multi-component reacting gas mixture. The proposed kinetic theory was applied in [49] for the simulation of gasdynamic parameters, transport coefficients and heat fluxes in non-equilibrium reacting air flows behind strong shock waves. In the one-temperature approach based on the thermal equilibrium Boltzmann vibrational distributions, the closed set of governing equations includes equations for , , which have the form (34), (35), (36). However, one should keep in mind that transport and relaxation terms in these equations differ from those obtained in the multi-temperature approach because they are defined by different collision processes: in the one-temperature approximation rapid processes include along with elastic collisions all internal energy transitions while slow processes are specified by only chemical reactions. The total heat flux is described by the expression ∑

∑

,

(55)

where , is the vibrational thermal conductivity coefficient. In this approach the bulk viscosity and relaxation pressure in the stress , . tensor are Possibility of limit transition from the state-to-state heat transfer description to the quasi-stationary models is discussed in [4]. 3.3 Production Terms in the equations (34) describes the variation of The chemical production term the particle c number density due to chemical reactions, whereas the term in in Eq.(37) characterizes the variation of the specific number of vibrational quanta molecular species c due to both slow vibrational energy exchanges and chemical reactions. The term describes exchange reactions, dissociation and recombination and can be written in the form (56)

, where ∑ ∑

,

,

,

.

(57)

(58)

78

High Temperature Phenomena in Shock Waves

Here, is the multi-temperature rate coefficient of the exchange reaction (during a ). The coefficients , , collision of two molecules or a molecule and an atom are the rate coefficients of dissociation and recombination reactions. , In the quasi-stationary approaches the zero-order reaction rate coefficients are expressed in terms of state-dependent rate coefficients ,

∑

,

,

∑

,

(59)

,

,

(60)

, where denotes quasi-stationary distributions and , are state, , dependent rate coefficients of bimolecular reactions (28) and dissociation. For the generalized Treanor distribution, the rate coefficients of exchange reactions occurring in the collision of two molecules have the form

,

,

,

∑

, ,

exp ,

,

(61)

and depend on the gas temperature and vibrational temperatures of the first levels of reagents. Rate coefficients for dissociation depend on two temperatures

,

,

,

∑

exp

(62)

.

,

Neglecting the anharmonic effects, we can find the reaction rate coefficients averaged over the non-equilibrium Boltzmann distribution ,

∑

,

,

∑

,

,

exp

exp

,

,

(64)

.

,

(63)

In a thermal equilibrium gas mixture, the reaction rate coefficients depend only on the gas temperature and are specified by the expressions ∑

exp

, ,

,

(65)

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

∑

,

exp

and can be described by the Arrhenius law. The total recombination rate coefficient specific rate coefficients as follows ,

∑

,

,

(66)

is defined in terms of the state-

,

,

79

,

(67)

and depends on the gas temperature only. The superscript ”0” in the notations for the state-to-state rate coefficients indicates that they are calculated by averaging the corresponding inelastic collision cross sections with the Maxwell-Boltzmann distribution over the velocity and rotational energy. in the relaxation equations for can be expressed in The production term terms of macroscopic parameters substituting the zero-order or the first-order distribution functions into the formulas (40). In the zero-order approximation, includes the vibrational distributions (33) and the state-to-state rate coefficients of VV2, VT vibrational energy transitions and chemical reactions [4]. The expressions can be also simplified if non-equilibrium or thermal equilibrium Boltzmann for distributions are used instead of Treanor distributions.

4 Non-equilibrium Processes behind Shock Waves in Air Components and CO2 Mixtures 4.1 Non-equilibrium Kinetics and Transport Properties in Diatomic Gas Mixtures In this section the results of applications of the state-to-state, multi-temperature and one-temperature models for evaluation of non-equilibrium kinetics and transport properties behind shock waves in the air components are presented. 4.1.1 Governing Equations and Flow Parameters The gas state in the unperturbed flow before the shock front is supposed to be in equilibrium, and vibrational and chemical kinetics in the relaxation zone behind the shock front is studied in the Euler approximation for an inviscid non-conducting gas flow. If the flow is assumed to be one-dimensional and steady-state, the governing equations are substantially simplified. Thus, the flow of a binary mixture of molecules A2 and atoms A behind a plane shock wave with dissociation, recombination, TV and VV vibrational energy transitions is described by the following set of equations for , atomic number densities , macroscopic gas the vibrational level populations velocity , and temperature ,

1,

, ,

(68)

80

High Temperature Phenomena in Shock Waves

2∑

(69)

,

(70)

,

(71)

,

(72)

,

where is the distance from the shock front. The subscript ”0” denotes the parameters in the free stream and the specific enthalpy h is equal to (73)

, ,

are the mass fractions of molecules and atoms, ,

,

∑

.

(74)

The right-hand sides of Eqs.(68), (69) include the state-to-state coefficients for vibrational energy transitions, dissociation, and recombination. The distribution in the free stream is usually assumed to be the Boltzmann one with a given temperature T0. The results obtained from the numerical solution of the system (68)-(72) for the N2/N mixture under the following conditions in the free stream: T0 =293 K, p0 =100 Pa, M0 =15 are presented below. The vibrational energy is simulated by the Morse anharmonic oscillator, the rate coefficients for vibrational energy transitions are calculated on the basis of the SSH-theory generalized for anharmonic oscillators [2,14], dissociation is described using the generalized Treanor–Marrone model with different values for the parameter U and parameters A, n in the Arrhenius law (31). The vibrational level populations and macroscopic flow parameters are computed in the state-to-state, two-temperature, and one-temperature approaches. The quasistationary vibrational distributions for different distances from the shock front are calculated using the obtained values for the macroscopic parameters. In Figure 6, the dimensionless vibrational level populations / calculated with the above three models are given as functions of i for various distances from the shock front. We can see a substantial discrepancy between the vibrational distributions obtained with different models close to the shock front, where the quasistationary distributions have not been yet established. The two-temperature and onetemperature approaches overestimate the vibrational level populations in a thin layer immediately behind the shock front. This discrepancy decreases with the distance from the shock front.

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

81

Fig. 6. Vibrational level populations. 1, 2: state-to-state model; 1’, 2’: two-temperature approach; 1’’, 2’’: one-temperature model. Solid curves correspond to x = 0.03, dashed to x = 0.8 cm.

Fig. 7. Gas temperature T behind the shock as a function of x. The curves 1, 2, 3 represent the state-to-state, two-temperature, and one-temperature approaches, respectively.

Figure 7 presents the variation of the gas temperature, calculated with the three approaches, as a function of the distance from the shock front. With the onetemperature and two-temperature models, the temperature is underestimated, since these approaches assume the existence of quasi-stationary distributions immediately behind the shock front and do not take into account the process of vibrational excitation in the very beginning of the relaxation zone.

82

High Temperature Phenomena in Shock Waves

Fig. 8. Atomic molar fraction behind the shock front as a function of x. The curves 1, 2, 3 represent the state-to-state, two-temperature, and one-temperature approaches, respectively.

Fig. 9. Averaged dissociation rate coefficient as a function of x. The curves 1, 2, 3 represent the state-to-state, two-temperature, and one-temperature approaches, respectively.

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

83

In Fig.8, the number densities of nitrogen atoms calculated in the three approaches are compared. It can be noticed that the one-temperature model does not describe the dissociation delay immediately behind the shock front. Both quasi-stationary models overestimate the dissociation degree in the vicinity of the front. From the vibrational distributions found behind the shock wave for the state-tostate and two-temperature approaches, the averaged dissociation rate coefficient can be calculated. For this purpose, the vibrational level populations, molecular , and temperature T obtained in these approaches for various points number density behind the shock are substituted into the formula (60). In the one-temperature approach, the dissociation rate coefficient is calculated using the Arrhenius formula (31). The results are presented in Fig.9. It is seen that the one-temperature model describes the behavior of the dissociation rate coefficient inadequately, particularly close to the shock front. The two-temperature approach provides more realistic values in comparison to for the dissociation rate coefficient, overestimating however the state-to-state approximation at x <0.5 cm. 4.1.2 Transport Properties In order to evaluate transport properties in the state-to-state approach, it is necessary to solve numerically the system (8)–(10) for the macroscopic parameters , , and in the first-order approximation of the Chapman–Enskog method. In the framework of the rigorous formalism, the state-dependent transport coefficients in the equations (8)–(10) should be calculated at each step of the numerical solution. Such a technique appears to be extremely time-consuming. In [50], an approximate approach for evaluation of gas dissipative properties was suggested. First, the vibrational level populations, molar fractions of atoms, and temperature are found from the governing equations in the zero-order approximation (68)–(72). Then the obtained non-equilibrium distributions are used to calculate the transport coefficients, diffusion velocities, and heat flux with the accurate formulae of the kinetic theory. The results obtained with this approach in the framework of the state-to-state, two-temperature and one-temperature models are presented in Fig.10, where the variation of the total energy flux in the relaxation zone behind the shock front is given. The one-temperature and two-temperature approaches substantially underestimate the absolute values of the heat flux in the very beginning of the relaxation zone, where the process of vibrational excitation is essential. The more rigorous state-to-state approach should be applied close to the shock front (x <0.3cm, or about twenty mean free path lengths of the unperturbed flow), in the domain of simultaneous vibrational relaxation and chemical reactions.

84

High Temperature Phenomena in Shock Waves

Fig. 10. Heat flux q as a function of x. The curves 1, 2, 3 represent the state-to-state, two-temperature, and one-temperature approaches, respectively.

4.1.3 Electronic Excitation and Radiation In high-temperature flows, it is necessary to take into account the excitation of electronic degrees of freedom and radiation from non-equilibrium flow regions. Thus the following processes involving electronic states should be taken into account in addition to the kinetic scheme described above (ionization is still neglected in the present contribution): AB

M M

M

AB

M

2

AB AB M

M

ET transitions,

(75)

VE transitions,

(76)

Induced emission and absorption, (77) Spontaneous emission,

(78)

Dissociation from excited states,

(79)

The rate coefficients for ET and VE transitions for high-temperature air components are given, for instance, in [51]. The radiative transitions are characterized by the Einstein coefficients[52]. The rate coefficients for dissociation from the excited states can be calculated using the generalization of the Treanor-Marrone model presented in[53]. The non-equilibrium factor , depending on the electronic state is given by the expression

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

,

exp

∑

exp

,

85

(80)

is the electronic partition function, is the electronic statistical weight, where and are the vibrational energy and vibrational partition function associated to is the electronic energy. If excited electronic states are the electronic state , neglected, Eq.(80) reduces to Eq.(30).

Fig. 11. Rate coefficients of dissociation from different electronic levels. Curves 1, 2, 3 correspond to the electronic states Σ, Π, Π, 1' corresponds to the rate coefficient calculated taking into account only ground electronic state Σ

Fig.11 presents the temperature dependence of the state-to-state rate coefficient for CO dissociation from the first vibrational level of various electronic states in the temperature range 5000–25000 K for the case of preferential dissociation (U = 3T) [53]. The coefficient of dissociation from the ground electronic state is calculated using both expressions (80) and (30). One can see that all rate coefficients increase with temperature, the rate of dissociation from excited electronic states exceeds noticeably the corresponding rate for lower states. Taking into account several excited electronic levels leads to a considerably lower rate of dissociation from the ground electronic state; this is explained by the repartition of internal energy between electronic states. The detailed state-to-state kinetics of a CO flow behind the shock wave taking into account electronic excitation, dissociation from higher electronic states, and radiation 5200 m/s, = 300 K, was studied in [53,54]. The free stream conditions are 500Pa. Three electronic states of CO were considered: Σ, Π, Π. It is

86

High Temperature Phenomena in Shock Waves

found that the contribution of ET and VE processes (75), (76) to the formation of vibrational distributions in a shock heated CO is rather weak compared to the role of these transitions in low temperature CO systems [55], where a strong depletion of the selected vibrational levels of the ground electronic state due to near-resonant VE transitions was observed. In the high temperature case, the rate of dissociation from the upper vibrational states appears to be much higher than the rate of VE transitions, and thermal decomposition of CO molecules becomes dominant compared to VE transitions from the ground electronic state. However VE transitions provide a source of electronically excited molecules and thus influence significantly the dissociation process and the UV emission intensity. The mixture composition behind the shock front calculated taking into account all considered processes is given in Fig.12. One can see a weak degree of dissociation as well as low concentrations of electronically excited molecules. Nevertheless, with increasing distance from the shock front, even low populations of excited electronic states provide a higher ultra-violet (UV) radiation intensity compared to the infra-red (IR) intensity (see Fig.13).

Fig. 12. Mass fractions of atoms and CO electronic levels as functions of x. = 300 K, 500 Pa.

5200 m/s,

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

Fig. 13. IR and UV intensities as functions of x. Solid lines: dissociation from Σ state (Eq.(30)) states; dashed lines: dissociation only from

Σ,

87

Π,

Π

The intensity of UV emission is found to be lower for the case when dissociation from the excited electronic levels is neglected, especially close to the shock front (see Fig.13). On the other hand, the IR radiation remains practically identical in both cases, because it is emitted essentially by the first vibrational levels of the Σ electronic state which are not affected by the dissociation model and VE transitions. 4.2 Non-equilibrium Kinetics and Transport Processes in Air Mixture In the simulation of vibrational and chemical relaxation in air, a five-component mixture (N2, O2, NO, N, O) is usually considered, and the following reactions are taken into account: N

M

N

N

M,

(81)

O

M

O

O

M,

(82)

N

O

NO

N,

(83)

O

N

NO

O,

(84)

NO

M

N

M.

(85)

O

In order to study the state-to-state vibrational and chemical kinetics in such a mixture, 114 equations for the populations of the vibrational levels of N2, O2, and NO, as well as the equations for the atomic number densities are to be solved. To reduce

88

High Temperature Phenomena in Shock Waves

the number of the equations, it is often supposed[26] that NO molecules forming as a result of the reactions (83)–(85) are in the ground vibrational state. This makes it possible to consider only a single vibrational level of NO instead of 36. In the quasistationary multi-temperature approach, the number of kinetic equations is substantially reduced: instead of equations for the populations of vibrational levels , and , we should solve the equations for the molecular number densities , , and , and effective temperatures of the first vibrational levels , , and . In addition, the distribution of NO molecules over the vibrational states is , since the vibrational usually supposed to be in thermal-equilibrium with relaxation time for NO molecules is appreciably smaller than that for N2 and O2[1]. However, it should be noted that in the recent years, some information about a deviation of the NO vibrational temperature from the gas temperature was revealed (see, for instance[56]). In the harmonic oscillator approach, the equations for the , are reduced to those for temperatures of the first vibrational levels , . vibrational temperatures The results obtained in [49] for the kinetics, dynamics, and transport processes behind strong shock waves propagating in air on the basis of the reaction scheme (81)–(85) in three quasi-stationary approaches: 1) the multi-temperature (generalized Treanor) approach; 2) the multi-temperature Boltzmann approach for harmonic oscillators, and 3) the one-temperature thermal equilibrium approach are presented in Figs. 14, 15, 16. A numerical solution for the equations for the chemical species number densities , , , , and , velocity , and temperatures ( , , , in the first case; , , and in the second case, and in the third case), and yielded the macroscopic flow parameters behind the shock front. The conditions in the free stream are as follows: M0 = 15, T0 = 271 K, p0 = 100 Pa, / = 0.79, / = 0.21. Figure 14 presents the variation of the gas temperature as well as of the temperatures , , , and behind the shock wave. We can see that the onetemperature model substantially underestimates the gas temperature in the relaxation zone. The comparison of the results obtained within the second and third models makes it possible to estimate the influence of anharmonicity of molecular vibrations: while it essentially does not affect the gas temperature T, the temperatures , , obtained for harmonic calculated for anharmonic oscillators differ from and . oscillators. The maximum discrepancy is found for The dependence of the molecular molar fractions on the distance x is presented in Fig.15. Since the one-temperature approach does not account for the dissociation delay caused by the finite time of the excitation of the vibrational degrees of freedom, this approach overestimates the dissociation and exchange reaction rates. As a result, the rate of decomposition of molecules N2 and O2, as well as the rate of formation of NO appear to be substantially higher than that with the multi-temperature approaches. The influence of the anharmonicity on the concentrations of chemical species behind the shock wave is rather weak.

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

89

Fig. 14.Temperatures T, , and as functions of x. The curve 1 displays T calculated in the one-temperature approach; 2, 2’: T in the multi-temperature approaches for anharmonic and harmonic oscillators, respectively; 3: ; 3’: ; 4: ; 4’: .

Fig. 15. Molecular molar fractions / (1, 1’, 1’’), / (2, 2’,2’’), and / (3, 3’,3’’) as functions ofx. The curves 1–3 represent the one-temperature approach; 1’–3’: the multitemperature approach (anharmonic oscillator); 1’’–3’’: the multi-temperature approach (harmonic oscillator).

90

High Temperature Phenomena in Shock Waves

The energy flux calculated with three approaches using the results obtained for the macroscopic parameters behind the shock front in the Euler approximation and rigorous kinetic algorithms for transport properties is plotted in Fig.16. It is important to emphasize that the one-temperature approach yields an inadequately high heat flux for x <0.2 cm, which can be explained by the overestimated role of the diffusion processes in the beginning of the relaxation zone. As mentioned above, the onetemperature approach does not describe the dissociation delay, and, consequently, results in the overrated gradients of species concentrations in the vicinity of the shock front. The influence of anharmonicity on the heat flux under the considered conditions appears to be of minor importance.

Fig. 16. Heat flux q as a function of x. The curves 1–3 represent the one-temperature, multitemperature (anharmonic oscillator), and multi-temperature (harmonic oscillator) approaches, respectively.

4.3 Mixtures Containing CO2 Molecules Modeling of non-equilibrium flows containing CO2 molecules is very important for the prediction of the parameters near a spacecraft entering the Mars atmosphere. Recent advances in the design of the Mars Sample Return Orbiter (MSRO) have triggered a renewed interest in this problem. The important problem is the development of adequate kinetic models suitable for applications and their implementation into the computational fluid dynamics (CFD) schemes. A linear tri-atomic CO2 molecule in the ground electronic state has three vibrational modes[57]. The first mode is the symmetric stretching mode with the frequency , the second is the doubly degenerated bending mode ( ), and the third mode is the asymmetric stretching mode ( ). Different vibrational energy exchanges of these modes should be taken into account in modeling of CO2 kinetics. However,

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

91

up to the present time, in numerical simulations of supersonic CO2 flows, basically, only simplified two-temperature or one-temperature models are used for the description of CO2 kinetics. These models cannot correctly describe the complex vibrational kinetics of CO2 molecules. Thus, the one-temperature model is valid only for thermal equilibrium flows while the two-temperature models are commonly developed introducing a single vibrational temperature for all three CO2 modes[58,59,60]. More rigorous models based on the multi-temperature distributions are developed in[61-66] for the vibrational kinetics, and in[64-66] for the transport properties. A detailed state-to-state kinetic theory approach for the modeling of flows containing CO2 is proposed in[67]. These models take into account the real structure of carbon dioxide molecules, anharmonic vibrations, and include to the kinetic scheme different vibrational energy transitions: intra- and inter-mode exchanges (VV and VV’) as well as VT transitions. Based on the rigorous kinetic theory algorithms proposed in[64,67], gas dynamic parameters, state-to-state distributions and transport properties in dissociating CO2 flows behind shock waves were evaluated for some particular cases[66]; however, in these studies, the estimates were performed for the flow parameters found previously from the simplified fluid dynamic equations. The accurate models developed in[64,67] are too complicated and time consuming to be inserted into the equations of non-equilibrium viscous flows, especially for the evaluation of the transport terms. Actually, in the state-to-state approximation[67], the tremendous number of the transport coefficients should be calculated at each step of the CFD code. Moreover, even simpler multi-temperature models[64,66] remain unused in the numerical simulations of viscous non-equilibrium flows due to their complexity, if the vibrational temperatures are introduced for each mode of the anharmonic CO2 vibrations and if different rates of energy transitions within and between modes are taken into account. A self-consistent three-temperature description of the three-component CO2/CO/O and five-component CO2/CO/O2/C/O mixtures was proposed on the basis of the kinetic theory methods[68,69]. The model takes into account various mechanisms of the CO2 vibrational relaxation and gives the expressions for the transport coefficients suitable for practical applications. The channels of vibration relaxation in mixtures containing CO2 are: intra-mode VV and VT transitions, inter-mode and inter-molecular VV’ transitions. The analysis of the existing data on the rate coefficients of these processes[70,71] shows that, for a wide variety of high-temperature flows, the following relation between the characteristic times holds ~

~

~

~

~ ,

1, 2, 3.

(86)

Here are the times of the intra-mode VV exchanges in the mth mode; is the correspond to the inter-mode time of the VT transitions in the bending mode, transitions. This condition corresponds to rapid translational and rotational relaxation, VVm vibrational energy exchanges within all kinds of vibrations and VV’1-2 exchange between symmetric and bending CO2 modes. In this case, the vibrational CO2 of the combined distributions depend on the vibrational temperatures

92

High Temperature Phenomena in Shock Waves

(symmetric+bending) mode and of the asymmetric mode [68]. The vibrational distributions of CO and O2 are supposed to be close to the thermal equilibrium; vibrational spectra are simulated using the harmonic oscillator model. From the kinetic equations for distribution functions one can derive the set of governing equations of a non-equilibrium flow. Under condition (86), the set of equations consists of conservation equations of mass, momentum and total energy coupled to the equations of non-equilibrium chemical and vibrational kinetics written in the following form[69]

· ·

·

0,

(87)

·

0,

(88)

:

·

0, ,

(89) ,

·

,

, , ,

·

·

(90)

,

·

(91)

.

(92)

, are the specific vibrational energies of non-equilibrium CO2 Here, modes, , are the production terms due to slow processes of CO2 vibrational , are the fluxes of vibrational energy in the combined and relaxation; asymmetric modes, respectively. Note that for harmonic oscillators, vibrational energies of CO2 modes are uncoupled and depend only on the corresponding and depend also on vibrational temperature while for anharmonic oscillators, the gas temperature [64]. The expressions for transport terms are given in[69]. The expressions for diffusion velocity and stress tensor are similar to those given in Section 3. All transport coefficients are defined by rapid processes. In the present case, in the pressure tensor, the bulk viscosity coefficient is defined by rotational energy transitions of all molecular species and VT vibrational energy transfer in CO and O2 molecules and can be written as follows , , . For harmonic oscillators, rapid inelastic VV and VV’1−2 exchanges are resonant, and therefore do not give contribution to the coefficient . The relaxation pressure is basically supposed to be small compared to , and usually is neglected. The heat flux is given by the formula ,

,

∑

∑

,

(93)

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

93

is the thermal conductivity coefficient of all degrees of where freedom which deviate weakly from local thermal equilibrium. They include the translational and rotational modes as well as CO and O2 vibrational degrees of freedom. Therefore , , . The coefficients , , , correspond to the thermal conductivity of strongly non-equilibrium modes: combined (symmetric+bending) and asymmetric ones. The fluxes of vibrational energy in the combined and asymmetric CO2 modes in the harmonic oscillator approach depend only on the gradient of corresponding vibrational temperature: ,

,

,

.

(94)

The transport algorithms derived in[68] were generalized to the case of the fivecomponent CO2/CO/O2/C/O mixture and implemented directly into the 2D viscous flow solver for the numerical investigation of the equations (87)–(92) in the shock layer[69]. The source terms in the equations (90)–(92) describe the slow processes, in our case they are CO2 vibrational energy transitions and chemical reactions. The rates of vibrational energy transitions are expressed in terms of the corresponding relaxation times. The rate coefficients for non-equilibrium CO2 dissociation were calculated using the expressions proposed in[68] as an extension of the Treanor-Marrone model[30] for three-atomic molecules. For the recombination rate coefficients, the detailed balance principle is used. For the rate coefficients of exchange reactions and dissociation of diatomic molecules, the Arrhenius formulas are applied. Equations (87)–(92) with rigorous kinetic schemes for transport coefficients described above were solved numerically for a flow in a 2D viscous shock layer near the blunt body imitating the form of the space craft MRSRO (Mars Sample Return Orbiter) for conditions typical of the re-entering regime[69]. Two test cases are = 5223 m/s, =2.9310−4 kg/m3, = 140 K (TC1) and =5687 m/s, studied: −5 3 =3.141·10 kg/m , = 140 K (TC2). The free stream is supposed to be constituted of pure CO2, the body surface is assumed to be either non-catalytic or fully catalytic. The results of this study are presented below and show the behavior of flow parameters in a shock layer. Fig.17 depicts the profiles of gas temperature and , along the stagnation line and demonstrates a length of vibrational temperatures a non-equilibrium zone in the shock layer. The system of governing equations has been solved not only in the threetemperature approximation described above but also using two simplified models: the two-temperature and the one-temperature approaches. In the first case, a single vibrational temperature for all three CO2 modes is introduced[58]; and in the second model, CO2 distributions are supposed to be close to thermal . In these approaches, the transport equilibrium with the gas temperature coefficients are calculated using approximate formulas given in[72-74]: for the twotemperature CO2 dissociation rates, the Park model[75] is applied.

94

High Temperature Phenomena in Shock Waves

Fig. 17. Temperature profiles along the stagnation line. (a) Test case 1; (b) Test case 2

Fig. 18. Gas temperature along the stagnation line. Test case 1. Curve 1: one-temperature model; curve 2: three-temperature model; curve 3: two-temperature model [58].

Fig.18 presents the comparison of the gas temperatures obtained in the frame of the one-temperature approach for weak deviations from thermal equilibrium (curves 1) and using rigorous three-temperature (curves 2) and simplified two-temperature (curves 3) models for vibrational non-equilibrium flows. One can see that the profiles of gas temperature corresponding to the two- and three-temperature models differ rather weakly; the discrepancy does not exceed 5%. On the other hand, the difference between the results obtained for one-temperature and multi-temperature flows occurs more essential. Non-equilibrium vibrational CO2 excitation leads to a slight increase of a distance between the shock wave and body surface, noticeable rising (up to 30%) of the gas temperature near the shock front and weakly affects the gas temperature near the surface. Vibrational non-equilibrium of CO2 molecules also leads to the increase of the surface heat flux up to 10%.

Non-equilibrium Kinetics and Transport Properties behind Shock Waves

95

The effect of bulk viscosity in a shock layer is demonstrated in Figs.19 and 20. Bulk and shear viscosity coefficients in a shock layer are given in Fig.19 along the stagnation line. Close to the shock front, the bulk viscosity coefficient is approximately twice higher than the shear viscosity one; the difference decreases coming to the surface. The heat fluxes to the body surface calculated taking into account the bulk viscosity coefficient in the flow equations and neglecting it are presented in Fig.20. Including the bulk viscosity coefficient to the fluid dynamic equations increases the heat flux up to 10%.

Fig. 19. Bulk and shear viscosity coefficients along the stagnation line.

Fig. 20. Heat flux along the body surface taking into account and neglecting bulk viscosity.

96

High Temperature Phenomena in Shock Waves

It may be noted that if the diatomic species CO and O2 are considered to be out of the thermal equilibrium, the VV vibrational energy exchanges in these molecules are included to the group of rapid processes while TV transitions define slow processes. For these conditions, five-temperature kinetic model including the equations for vibrational temperatures for CO and O2 molecules are proposed in[76].

References 1. Stupochenko, Y., Losev, S., Osipov, A.: Relaxation in Shock Waves, Nauka, Moscow (1965); Engl.Transl. Springer, Heidelberg (1967) 2. Gordiets, B., Osipov, A., Shelepin, L.: Kinetic Processes in Gases and Molecular Lasers, Nauka, Moscow (1980); Engl. Transl., Gordon and Breach Science Publishers, Amsterdam (1988) 3. Kondratiev, V., Nikitin, E.: Kinetics and Mechanism of Gas Phase Reactions, Nauka, Moscow (1974) 4. Nagnibeda, E., Kustova, E.: Nonequilibrium Reacting Gas Flows. Kinetic Theory of Transport and Relaxation Processes. Springer, Heidelberg (2009) 5. Brun, R.: Introduction to Reactive Gas Dynamics. Oxford University Press (2009) 6. Ern, A., Giovangigli, V.: Multicomponent Transport Algorithms. Lect. Notes Phys., Series Monographs, vol. 24. Springer, Heidelberg (1994) 7. Chikhaoui, A., Dudon, J., Kustova, E., Nagnibeda, E.: Physica A 247 (1-4), 526 (1997) 8. Kustova, E., Nagnibeda, E.: Chem. Phys. 233, 57 (1998) 9. Ferziger, J., Kaper, H.: Mathematical Theory of Transport Processes in Gases. NorthHolland, Amsterdam (1972) 10. Belouaggadia, N., Brun, R.: J. Thermophys. Heat Transfer 12(4), 482 (1998) 11. Kustova, E.V.: Rarefied Gas Dynamics. In: Abe, Takashi (eds.) Proc. 26th Int. Symp., Series: AIP Conference Proceedings, Kyoto, Japan, July 2008, vol. 1084, p. 807 (2009) 12. Chernyi, G., Losev, S. (eds.): Physical-Chemical Processes in Gas Dynamics, vol. 1,2. Moscow University Press, Moscow (1995) 13. Schwartz, R., Slawsky, Z., Herzfeld, K.: J. Chem. Phys. 20, 1591 (1952) 14. Gordiets, B., Zhdanok, S.: In: Capitelli, M. (ed.) Nonequilibrium Vibrational Kinetics, vol. 43. Springer, Heidelberg (1986) 15. Millikan, R.C., White, D.R.: J. Chem. Phys., 39, 3209 (1963) 16. Billing, G., Fisher, E.: Chem. Phys. 43, 395 (1979) 17. Billing, G., Kolesnick, R.: Chem. Phys. Lett. 200(4), 382 (1992) 18. Laganà, A., Garcia, E.: J. Chem. Phys. 98, 502 (1994) 19. Laganà, A., Riganelli, A., de Aspuru, G., Garcia, E., Martinez, M.: In: Capitelli, M. (ed.) Molecular Physics and Hypersonic Flows, pp. 35–42. Kluwer Acad. Publishers, Netherlands (1996) 20. Esposito, F., Capitelli, M., Gorse, C.: Chem. Phys. 257, 193 (2000) 21. Armenise, I., Capitelli, M., Celiberto, R., Colonna, G., Gorse, C., Laganà, A.: Chem. Phys. Lett. 227, 157 (1994) 22. Adamovich, I., Macheret, S., Rich, J., Treanor, C.: J. Thermophys. Heat Transfer 12(1), 57 (1998) 23. Adamovich, I., Rich, J.: J. Chem. Phys. 109 (18), 7711 (1998) 24. Gonzales, D., Varghese, P.: J. Thermophys. Heat Transfer 8(2), 236 (1994) 25. Armenise, I., Capitelli, M., Colonna, G., Gorse, C.: J. Thermophys. Heat Transfer 10(3), 397 (1996)

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26. Capitelli, M., Armenise, I., Gorse, C.: J. Thermophys. Heat Transfer 11(4), 570 (1997) 27. Losev, S., Pogosbekian, M., Sergievskaya, A., Kustova, E., Nagnibeda, E.: In: Capitelli, M. (ed.) Rarefied Gas Dynamics, AIP Conference Proceedings, vol. 762, p. 1049 (2005) 28. Nagnibeda, E., Novikov, K.: In: Ivanov, M., Rebrov, A. (eds.) 25th International Symposium on Rarefied Gas Dynamics, Novosibirsk, vol. 971 (2007) 29. Osipov, A.: Teor. Exp. Khim. 2(11), 649 (1966) 30. Marrone, P., Treanor, C.: Phys. Fluids 6(9), 1215 (1963) 31. Lordet, F., Meolans, J., Chauvin, A., Brun, R.: Shock Waves 4, 299 (1995) 32. Gardiner, W. (ed.): Combustion Chemistry. Springer, New York (1984) 33. Park, C.: Nonequilibrium Hypersonic Aerothermodynamics. J.Wiley and Sons, Chichester (1990) 34. Esposito, F., Capitelli, M., Kustova, E., Nagnibeda, E.: Chem. Phys. Lett. 330, 207 (2000) 35. Candler, G., Olejniczak, J., Harrold, B.: Phys. Fluids 9(7), 2108 (1997) 36. Varghese, P., Gonzales, D.: In: Capitelli, M. (ed.) Molecular Physics and Hypersonic Flows, p. 105. Kluwer Acad. Publishers, Netherlands (1996) 37. Kovach, E., Losev, S., Sergievskaya, A.: Chem. Phys. Rep. 14, 1353 (1995) 38. Polanyi, J.: Acc. Chem. Res. 5, 161 (1972) 39. Birely, J., Lyman, J.: J. Photochem. 4, 269 (1975) 40. Gilibert, M., Aguilar, A., Gonzales, M., Sayos, R.: Chem. Phys. 178, 287 (1993) 41. Gilibert, M., Aguilar, A., Gonzales, M., Mota, F., Sayos, R.: J. Chem. Phys. 97, 5542 (1992) 42. Gilibert, M., Aguilar, A., Gonzales, M., Sayos, R.: J. Chem. Phys. 99, 1719 (1993) 43. Bose, D., Candler, G.: J. Chem. Phys. 104(8), 2825 (1996) 44. Capitelli, M., Ferreira, C., Gordiets, B., Osipov, A.: Plasma Kinetics in Atmospheric Gases. Series on atomic, optical and plasma physics, vol. 31. Springer, Berlin (2000) 45. Knab, O., Frühauf, H., Messerschmid, E.: J. Thermophys. Heat Transfer 9(2), 219 (1995) 46. Aliat, A.: Physica A 387, 4163 (2008) 47. Kustova, E.: Chem. Phys. 270(1), 177 (2001) 48. Treanor, C., Rich, J., Rehm, R.: J. Chem. Phys. 48, 1798 (1968) 49. Chikhaoui, A., Dudon, J., Genieys, S., Kustova, E., Nagnibeda, E.: Phys. Fluids 12(1), 220 (2000) 50. Kustova, E., Nagnibeda, E.: In: Brun, R., et al. (eds.) Rarefied Gas Dynamics, CEPADUES, 21th edn., Toulouse, vol. 1, p. 231 (1999) 51. Losev, S., Yarygina, V.: Russ. J. Phys. Chem. B 3, 641 (2009) 52. Heaps, H., Herzberg, G.: Z. Physik 133, 49 (1953) 53. Aliat, A., Kustova, E., Chikhaoui, A.: Chem. Phys. 314, 37 (2005) 54. Aliat, A., Kustova, E., Chikhaoui, A.: Phys. Rev. E 68, 056306 (2003) 55. Deleon, R., Rich, J.: Chem. Phys 107, 283 (1986) 56. Takahashi, T., Yamada, T., Inatani, Y.: In: 20th International Symposium on Space Technology and Science, Gifu, Japan (1996) 57. Herzberg, G.: Infrared and Raman Spectra of Polyatomic Molecules. D. Van Nostrand Company, Inc., New York (1951) 58. Taylor, R., Bitterman, S.: Rev. Mod. Phys. 41(1), 26 (1969) 59. Thomson, R.: J. Phys. D: Appl. Phys. 11, 2509 (1978) 60. Brun, R.: AIAA Paper 88-2655 (1988) 61. Likalter, A.: Prikl. Mekh. Tekn. Fiz., 4, 3 (1976) (in Russian) 62. Anderson, J.: Gasdynamic Lasers: An Introduction. Academic Press, New York (1976) 63. Cenian, A.: Chem. Phys., 132, 41 (1989)

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64. Kustova, E., Nagnibeda, E.: In: Brun, R., et al. (eds.) Rarefied Gas Dynamics, CEPADUES, 2nd edn., Toulouse, France, pp. 289–296 (1999) 65. Chikhaoui, A., Kustova, E.: Chem. Phys. 216, 297 (1997) 66. Kustova, E., Nagnibeda, E., Chikhaoui, A.: In: Munz, E., Ketsdever, A. (eds.) Rarefied Gas Dynamics, AIP Conference Proceedings, vol. 663, p. 100 (2003) 67. Kustova, E., Nagnibeda, E.: In: Bartel, T., Gallis, M. (eds.) Rarefied Gas Dynamics, AIP Conference Proceedings, vol. 585, p. 620 (2001) 68. Kustova, E., Nagnibeda, E.: Chem. Phys. 321, 293 (2006) 69. Kustova, E., Nagnibeda, E.: In: Abe, T. (ed.) Rarefied Gas Dynamics, AIP Conference Proceedings, vol. 1084, p. 801 (2009) 70. Makarov, V., Losev, S.: Khim. Phizika, 16 (5), 29 (1997) (in Russian) 71. Losev, S., Kozlov, P., Kuznezova, L., Makarov, V., Romanenko, Y., Surzhikov, S., Zalogin, G.: In: Harris, R. (ed.) Proc. of the 3rd European Symp. on Aerothermodynamics for Space Vehicles, ESTEC, Noordwijk, The Netherlands, vol. 426, p. 437. ESA Publication Division, ESA (1998) 72. Wilke, C.: J. Chem. Phys. 18, 517 (1950) 73. Mason, E., Saxena, S.: Phys. Fluids 1, 361 (1958) 74. Armaly, B., Sutton, K.: AIAA Paper 82-0469 (1982) 75. Park, C., Howe, J., Jaffe, R., Candler, G.: J. Thermophys. Heat Transfer 8(1), 9 (1994) 76. Kustova, E., Nagnibeda, E., Shevelev, Y., Syzranova, N.: In: Proc. 27th Int. Symp. on Rarefied Gas Dynamics (2010) (accepted for publication)

Chapter 3

Non-equilibrium Kinetics behind Shock Waves Experimental Aspects L. Ibraguimova and O. Shatalov Institute of Mechanics of Lomonosov Moscow State University, Moscow, Russia

1 Introduction The investigation of physical-chemical kinetics in high-temperature gases is motivated by its importance in the aerodynamics of hypersonic flight, in upper atmosphere phenomena, in numerous practical applications as well as by the academic interest. The use of shock tubes for the experimental study of physical and chemical processes in gases started several decades ago and is still in progress. This is due to improvements made in the classical experimental methods (interferometry, pressure measurements, laser-schlieren method, emission and absorption spectroscopy including atomic resonance absorption spectroscopy (ARAS)) and also to the use of relatively new techniques, such as coherent anti-Stokes Raman scattering (CARS) and laser induced incandescence (LII). Numerous reviews have been published concerning the chemistry of high-temperature gases, combustion chemistry, and environmental problems, for example[1-9]. Partially, this information can be found in monographs[10-15]. Due to the possibility of practically instantaneous and uniform heating in cross-sections, a wide range of relaxation processes occurring in gases was studied in shock tubes. These include the establishment of translational and rotational equilibrium in the shock front, the processes of vibrational relaxation of diatomic and polyatomic molecules, as well as direct and reverse chemical reactions, processes involving mutual influence of chemical, vibrational-rotational and electron kinetics, radiation processes, the cluster formation, etc. It is hardly possible to describe all these results in one chapter. Only a few aspects of these studies will be presented below, in particular, the vibrational relaxation of di and tri-atomic molecules, including relaxation in non-adiabatic collisions and the results of investigation of chemical reactions in the C-O and N-C-O atom systems. The kinetic coefficients (rate coefficients or rate constants) characterizing the rate of chemical reaction processes are measured in the region of gas flow behind the shock front, where the chemical equilibrium has not yet been attained. The problem of non-equilibrium radiation connected with non-equilibrium population of radiating excited electronic states of molecules and atoms is also shortly described. It is shown that different versions of non-equilibrium are possible depending on the participation of different types of internal states of molecules (and

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atoms) in high temperature relaxation process taking into account spontaneous emission, energy-change processes, exothermic reactions, and so on. The overall balance of these processes depends on the properties of molecules and atoms under consideration and on the ambient conditions.

2 Vibrational Relaxation of Diatomic Molecules The experimental investigation of reaction rates, in particular, vibrational relaxation rates, makes possible a direct comparison of the experimental and theoretical data and provides the basis for theory evolution and numerical modeling of real gas flows. Generally, the characteristic vibrational relaxation time τ was determined from the experimental data (pressure, density, absorption etc), associated with the energy relaxation equation d ε v / dt = (ε veq − ε v ) / τ

Here,

(1)

εv is the instantaneous vibrational energy of the gas, and ε veq is the

instantaneous equilibrium vibrational energy, which is a function of the translational temperature. In turn, the temperature dependence of τ is usually approximated according to Landau-Teller theory[16] as

τ = A ⋅ exp( BT −1/3 )

(2)

This is justified for adiabatic conditions (Massey parameter ωτcoll>>1, where ω is the vibration frequency and τcoll the collision time) and for two-level vibrators or onequantum transitions of a harmonic oscillator at not too high temperatures. At present, there is a voluminous literature containing experimental data on vibrational relaxation of different molecules in collision with various species[13,17]. One of the most cited publications is that of Millikan and White[18] which contains a simple empirical correlation of a large number of data on relaxation times of diatomic molecules with a variety of colliding partners. However the systematics[18] deviate from the form of the theoretical function developed by the original Landau-Teller theory[16] even under adiabatic collisional conditions. This can be due not only to too high temperatures but also to some other factors, such as the influence of the attractive branch of the interaction potential, (as in collisions CO2(0001) + N2(0)), the influence of molecular rotations, (as in collisions of hydrogen-containing molecules), the participation of electronic degrees of freedom, (as for NO relaxation) or in the case of non-adiabatic collisions of heavy molecules. Examples of classical LandauTeller type vibrational relaxation as well as the deviations from this dependence, are presented below. 2.1 Vibrational Relaxation of O2, N2, CO, NO O2-O2. Vibrational relaxation of oxygen in collisions O2-O2 was studied behind the front of shock waves by different methods. Ultraviolet absorption methods, (λ=210-240 nm), were used for the study of relaxation for T=1200-7000K[19,20] and T=600-2000K[21],

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laser schlieren technique for T=1000-3700K[22], interferometry for T=600-2600K[23]. These methods are based on gas density measurements or time evolution of oxygen absorption behind the shock wave front owing to excitation of vibrational states of molecules. In experiments[24,25] the deactivation of O2 vibrations was studied by O2 absorption in a nozzle behind a reflected shock wave for T=500-2050K. In papers[26,27] the results of shock tube measurements of O2-O2 vibrational relaxation are presented for T=2500-10200K. For T>6000-7000K, these data are weakly dependent on the temperature; they are represented in the systematic[18]. However, in later publications[19,20], the same authors suggest investigated temperature range not exceeding 7000K. For T=500-7000K, all the measured data are described within ±50% by expression (2) with parameters A, B taken from[22]. They are presented in Table 1. O2-Ar. To analyze the oxygen vibrational relaxation data in mixtures (such as O2-Ar) where φ is the oxygen molar fraction, the linear mixing rule is used

1 / τ mix = ϕ / τ O2 −O2 + (1 − ϕ ) / τ O2 −Ar . Measurement of the oxygen vibrational relaxation time in O2-Ar collisions was performed in[28] by registration of vacuum ultraviolet absorption (VUV, λ=147 nm) behind the front of an incident shock wave for temperatures T=1200-7000K. The obtained result is presented in Table 1. The results[29] obtained by the interferometric method are in agreement with those of[28] for T=400-1600K. The values of τ (O2-Ar), determined in[30] (Table 1) behind the reflected shock wave for T=1000-1600K with the α-Lyman absorption method (121.6 nm), are slightly higher than those of[28] in the temperature region where they overlap. Absorption measurements[31]at seven wavelengths (164-224.5 nm) in O2-Ar mixtures for T=3000-10000K behind the shock wave show a certain deceleration of vibrational relaxation in comparison with the previous data, as the temperature grows (2.3 times at 7000K), see Table 1. Table 1.

T, K Collision A (atm·sec) B (K)1/3 -10 O2-O2 2.92·10 ±50% 126 500-7000 O2-Ar 8.3·10-11 163±30% 1200-7000 O2-Ar 2.0·10-10±25% 156 1000-1600 O2-Ar (1.6·10-14±30%)·T 173.13 3000-10000 O2-He 5.3·10-9 60 400-8600 O2-H2 1.0·10-8 36 400-2000 O2-D2 4.5·10-9 62.9 300-850 O2-O 7.48·10-10 47.7 1500-3000

Ref. 22 28 30 31 29,33 34 35 67

O2-He, O2-H2. The efficiency of O2 collisions with light species (H2, He) is usually very high, i.e. vibrational relaxation is much faster, than the efficiency of collisions with heavy species.

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This follows from the Landau-Teller expression for the probability of V-T transition P(1→0) depending on the reduced mass of the collision pair, verified by numerous experimental data. Relaxation of oxygen molecules in collisions with He behind the shock wave front was observed for T=400-1600K with the interferometric method of density variation registration[29] and with UV absorption spectroscopy for T~3600K[32] and for T=5600-8600K [33]. The O2 vibrational relaxation in collision with molecular hydrogen and deuterium was studied by interferometric methods[35,34]. CO-CO. (Table 2). Vibrational relaxation of carbon monoxide was measured in shock waves by using interferometry[36] for T=2200-4900K, by detecting CO infrared emission[37-40], and by recording the pressure evolution for 2200-6000K at the end wall of a shock tube[41]. The measurements[36,41] practically coincide with the data[37]. The possibility of extrapolating the data[37] up to 5000K was experimentally verified[42]. However, in a later work by the same author, a slight increase in the relaxation time for T>5000K (~20% at 6000K) was recorded[43]. Table 2.

Collision A (atm·sec) B (K)1/3 -11 CO-CO 7.1·10 174.57 CO-CO 5.74·10-11 181 CO-Ar 3.55·10-12 232.6 CO-Kr 2.07·10-10 187 CO-Ne 6.86·10-10 142 CO-He 5.0·10-9 87 CO-H2 3.0·10-9 67 CO-O (6.8±50%)·10-10 54 CO-O (0.6±0.1)·10-7 0 CO-H (1.4±0.3)·10-8 3±2 CO-H 2.3·10-16 188.9

T, K 2200-6000 1500-3000 1100-2500 2100-7000 1400-3000 580-1500 580-2900 1800-4000 2800-3900 840-2680 1400-3000

Ref. 36,37,41 38 44 45 45 45 37 68 69 71 72,73

The latest data[38] obtained for T=1500-3000K by detecting the global IR emission of CO, also exceeds the measured data[37] by a factor 1.4 at 1500K, and by a factor 1.16 at 5000K(Table 2). The results[39] of registration of CO first overtone emission 2.34 μm and fundamental band 4.66 μm at 1500-3000K behind the shock wave front practically coincide with the results[38]. CO-Ar, Ne, Kr. Vibrational relaxation of carbon monoxide in collisions CO-Ar was studied by IR emission registration behind incident shock waves in 5%CO+95%Ar mixtures[37] for 1100-2500K, and in expansion waves behind reflected shocks in 0.5%CO+95.5%Ar mixtures[44]. Both results are practically coincident (data in Table 2 are taken from[44]). The data on CO vibrational relaxation in collision with Ne and Kr were determined from the infrared emission behind incident shock waves in [45] and also presented in Table 2.

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CO-He, CO-H2. The rate of vibrational excitation of CO in collisions with He and H2 was measured from the infrared emission intensity behind incident shock waves[37,45]. As for the oxygen relaxation, CO relaxation in collisions with He and H2 is very fast. N2-N2. (Table 3). Earlier results of interferometric measurements of nitrogen vibrational relaxation for the approximate temperature range 2000-5500K in N2-N2 collisions were summarized in [18]. Later, vibrational relaxation of nitrogen was studied with a vacuum-ultraviolet light-absorption technique at 117.6 nm for temperatures 3000-9000K[46], with time-resolved measurements of pressure on the end wall of a shock tube for the temperature range 3300-7000K[47], with interferometeric density variation measurements for T=1900-2800K[48], and for 1500-5000K[49], and with a laser schlieren technique for T=3000-4500K[50]. Nitrogen vibrational relaxation was also studied with a broad-band method of coherent anti-Stokes scattering (CARS) for 1200-3000K in the expansion flow of a reflected shock tunnel[51] by measuring bow shock stand-off distances49. All these studies present a good agreement, in the overlapping temperature range, with the average dependence of the relaxation time on temperature18. Relaxation N2-N2 was also measured for T=8000-15500K behind an incident shock wave[52] using nitrogen absorption of VUV emission of discharge impulse argon source, λ=127 nm. The results of this work display a deceleration of temperature dependence of vibrational relaxation at high temperatures, in comparison with the previous low-temperature data. Table 3.

Collision N2-N2 N2-N2 N2-H2O N2-H2O N2-H2 N2-He N2-O N2-O N2-O

A (atm·sec) 6.0·10-12 (4.6·10-15±30%)T 0.26·10-6 0.18·10-6 0.24·10-8 3.8·10-10 (2÷0.4)·10-6 (2-0.8)·10-6±50% (0.36±0.1)·10-6

B (K)1/3 234.9 230 21 11.33 80.6 123 0 0 0

T, K Ref. 1200-9000 18 8000-15500 52 1600-3100 55 1300-3100 56 1630-2370 57 1940-3100 58 3000-4500 50 1200-3000 74 2800-3900 69

N2-H2O. The study of vibrational relaxation of N2 with H2O was stimulated in order to estimate the potential performance of the high-temperature CO2 gasdynamic laser with N2 as the main storage source and H2O as the lower-level deactivant. The earlier studies of this process[53,54] presented an uncertainty as for the order of magnitude for the rate constant at temperatures near 2000K. Later, vibrational relaxation of N2 with H2O was measured behind incident shock waves in the temperature range 16003100K[55]. The vibrational relaxation was monitored by interferometric measurements of the gas density and by infrared emission measurements of the CO

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fundamental band using CO as a tracer species to detect the N2 vibrational energy. Taking into account the available low-temperature data, the result[55] is presented in Table 3. This result confirms the ealier results [54]. In experiments [56] the vibrational relaxation of N2 by H2O was studied behind incident shock waves in the temperature range from 1300 to 3100K using the laserschlieren technique. The relaxation time was described by the formula also presented in Table 3. N2-He, N2-H2. Optical interferometric method was used [57,58] to measure the rate of vibrational relaxation of nitrogen in collision with He and H2. As the majority of other experimental results on vibrational relaxation, these results were presented as functions of the average temperature of the relaxation zone (Table 3). NO-NO, NO-Ar. Relaxation times of nitric oxide in collisions with NO and Ar were measured in shock tube experiments using the laser schlieren technique [59] at 7302700K, IR emission registration [60] at 900-2700K, and UV absorption for the temperature range T=900-2700K [61] and for 1500-6800K [62]. Table 4.

Collision A (atm·sec) NO-Ar 1.4·10-6 NO-NO 7.9·10-8 NO-O 1.7·10-8 NO-Cl 1.7·10-8

B (K)1/3 33.85 14.05 0 0

T, K 730-6800 730-6800 2700 1700

Ref. 59-62 59-62 61 61

All these experiments show a very high rate of these processes. The reason for such high rate is that NO molecules exist in orbital-degenerated electronic Π-state, and NO-NO collisions generate several surfaces of potential energy which, in turn, cause an extremally high rate of NO-NO vibrational relaxation. The expressions of nitric oxide relaxation times in collisions with NO and Ar presented in Table 4 are taken from the work [60]. They correlate fairly well with the experimental data of other investigations. 2.2 Vibrational Relaxation of Diatomic Molecules in Collisions with Potentially Reactive Atoms Vibrational relaxation of diatomic molecules in collisions with potentially reactive atoms is extremely efficient. Many studies of vibrational relaxation of diatomic molecules in systems O2-O, CO-H, CO-Fe, NO-O, NO-Cl have been undertaken using shock tubes, however the nature of this efficiency is still poorly understood. One explanation consists in the strong influence of long-range attractive interaction in collisions involving reactive atoms. Another possible explanation[63] could be the contribution to the relaxation rate of the electronically non-adiabatic transitions, when one of colliding partners possesses nonzero electronic angular momentum. A third mechanism that has been postulated involves the formation of bond intermediate

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collision complexes, with energy transfer occurring during the complex lifetime[64]. Another mechanism would involve relaxation via reactive collisions leading to chemical exchange. Some of reactions described later can be explained by two or three above-listed mechanisms, for example O2-O; however, this problem has not yet been solved definitively. O2-O. Vibrational relaxation in O2-O collisions was studied using ozone decomposition as a source of oxygen atoms [65], or discharge-flow shock tube system, where the test section of shock tube forms a part of discharge flow [66]. The data of work [65] may be fitted as follows:

pτ O 2 −O = 4.35 ⋅ 10 −8 − 7.75 ⋅ 10 −12 T , atm ⋅ s. The fitting of data[65] presented in work[67] is given in Table 1. Laser schlieren detector was used in experiments[66] to measure the rate of vibrational relaxation O2-O, the concentration of O atoms being monitored by air afterglow produced when adding a small measured amount of NO to the gas downstream of the discharge. For T=1000-3400K, the result[66] is as follows: pτ O 2 − O = (3.06 ± 0.19 ) ⋅ 10 −8 − (2.18 ± 8 .34 ) ⋅ 10 −13 T , atm ⋅ s.

Deactivation of oxygen vibrations was measured using UV absorption (210 and 230 nm) in the flow of partially dissociated oxygen in a wedge-shaped nozzle mounted at the end of a shock tube[25]. Numerical simulation of supersonic gas flow with test values of vibrational relaxation time made it possible to estimate the vibrational relaxation time in O2-O collisions for T=1000-3000K: p τ O 2 − O [± 35 % ] = 3 ⋅ 10 − 8 + 4 .5 ⋅ 10 − 15 T exp (110 T − 1 / 3 ), atm ⋅ s.

The data[25,65,66] on pτ (O2-O) differ from one another by a factor of 2 to 3 at T=1000K, probably because of inadequate interpretation and processing of the measurement, and almost coincide at T~3000K. At the same time, the measured times of vibrational relaxation in O2-O mixture are 2-3 orders of magnitude shorter than in O2-O2 at these temperatures. CO-O. Extremely high efficiency of O and H atoms was found in deactivation of CO vibrations. The values of pτ (CO-O) were measured for 1800-4000K in experiments [68] behind incident shock waves from the infrared emission of the CO fundamental vibration. The atomic oxygen was produced by rapid thermal decomposition of ozone. Relaxation of CO in collisions with O atoms was also studied in the work [69] (cited in the article [70]) for T=2800-3900K. Results are presented in Table 2. CO-H. The rate of vibrational relaxation in CO-H collisions was measured for T=840-2680K with a discharge-flow shock tube[71]. The upper vibrational levels of carbon monoxide were monitored by infrared emission spectroscopy, and the concentration of H-atoms prior to shock heating was determined by chemiluminescent

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titration reaction. Vibrational relaxation CO-H was also studied in the rapid nonequilibrium expansion of a shock tunnel[72]. In this work a small amount of hydrogen was added in the shock tube and the vibrational temperature of CO was determined from the emission intensity of the first overtone 2.3 μm, using a calibrated infrared detection system. The measured vibrational relaxation time for T=1400-2800K was smaller than in the work [71]. Practically the same result was obtained in the supersonic flow of a shock tunnel for T=1800-3000K by means of the CARSspectroscopy method [73]. In the author’s[73] opinion, the uncertainty of this result, equal to a factor 2, is due, first of all, to the uncertainty of CO vibrational temperature measurements. Results[71-73] are given in Table 2. N2-O. Vibrational relaxation of nitrogen in collisions with oxygen atoms behind shock waves was measured with the laser schlieren method for T=3000-4500K[50]. In these experiments O atoms were produced by ozone decomposition behind the shock wave front. In investigation[74] a CO tracer technique was used to study N2-O vibrational relaxation for T=1200-3000K. In this temperature range, the measured relaxation times pτ (N2-O) vary from 2 to 0.8 μs.atm with an uncertainty of ±50%. Finally, a laser schlieren method was used [69,70] for the study of vibrational relaxation in N2-O collisions for T=2800-3900K. O atoms were generated behind the front in rapid dissociation of N2O, and monitored by CO-tracer method (registration of absolute intensity of CO-O recombination radiation). The measured value pτ (N2-O) coincides with previous results. All these results are nearly two orders of magnitude smaller than those for pure nitrogen and are presented in Table 3. NO-O, NO-Cl. Vibrational relaxation of NO in collisions with O and Cl atoms was studied behind incident and reflected shock waves by monitoring NO absorption in the γ-band [61] (λ=226, 236 and 247 nm). Oxygen and chlorine atoms were produced in fast thermal dissociation of NO2 and N2O at T~2700K and of ClNO at T~1700K. The shock-heated mixtures contained 0.2%-4.4% NO in Ar. The high efficiency of O and Cl atoms in NO relaxation (see Table 4) suggests a chemical nature of this process involving bond (NO2) and (ClNO) collision complexes. 2.3 Vibrational Relaxation of H2 and D2

Vibrational relaxation of hydrogen is of interest both from a theoretical point of view, (H2 being a molecule with a maximum rotational quantum number), and from a practical one, as a means for isotope separation and as an important component in combustion chemistry. Shock tube interferometry was used to observe the vibrational relaxation of H2 and D2 molecules[35]. However the method of conventional interferometry used in that study appeared to be inadequate to resolve the too rapid H2 and D2 relaxation. The results are pτ (H2-H2) <2·10-6 atm·s and pτ (D2-D2)<6·10-6 atm·s at T=1400K. Later, the vibrational relaxation of H2 and D2 and their mixtures with other gases was studied using narrow-beam [75-77] and broad-beam[78] laser-schlierenphotomultiplier technique. The results are presented in Table 5 in the usual Landau-Teller form: τp =A exp(BT-1/3) atm.s.

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Table 5.

Collision H2-H2 H2-H2 H2-Ar H2-Ar H2-He H2-Ne H2-Kr D2-Kr D2-D2 D2-D2 D2-Ar D2-Ar

A (atm·sec) (3.9±0.8)·10-10 (2.06±0.06)·10-9 (16.0±3.3)·10-10 (1.1±0.1)·10-9 (1.05±0.15)·10-9 (8.05±0.65)·10-9 (1.27±0.34)·10-8 1.3·10-9 (2.7±0.3)·10-10 0.79·10-10 (1.0±0.7)·10-9 3.8·10-10

B (K)1/3 100.0±2.6 80.0±0.4 100.0±2.6 103.8±1.3 95.2±1.84 65.5±1.1 84.0±3.66 125 110.5±1.5 125 118.0±10 125

T, K Ref. 1100-2700 76 1350-3000 76 1500-2700 76 1350-3000 77 1350-3000 77 1350-3000 77 1350-3000 77 1200-2300 78 1100-3000 75 1200-2300 78 1600-3000 75 1200-2300 78

2.4 Vibrational Relaxation of Halides and Hydrogen Halides HI-Ar. Hydrogen halides being polar molecules with large rotational velocities and strong intermolecular attractive forces, provide a sensitive test for theories of vibrational energy transfer. In the work[79] the vibrational relaxation of HI was studied in a mixture of 10% HI in Ar behind a shock wave for 1400-2300K by measuring light absorption from 4 vibrational states v=0-3 of ground electronic state (λ=249,8-413,0 nm). It was shown that all the states relax with the same rate constant (for HI-Ar collision k≈1.0·10-13 cc/(molec·s), i.e. pτ≈3.4·10-6 atm·s at 2000K). This rate is 2-3 orders smaller than the value predicted by the SSH theory[80] (Table 6). The rate of HI-HI relaxation is 2-3 times smaller than in collisions HI-Ar. HI-HI. Relaxation HI-HI was studied later in[81] by time resolved measurements of the post-shock density gradient for 800-1800K using the laser schlieren technique developed by Kiefer and Lutz[75,82]. The measured pτ values were generally in agreement with those found in study[79]. DI-DI. The same densitometric method was used for the determination of the vibrational relaxation of DI-DI for 700-2000K[83]. The fact that the deuterated species relaxes slower than hydrogen analog was interpreted as the evidence that the R-V energy transfer can play a significant role in the vibrational relaxation of hydrogen halides. The results of investigations[81,83,84] are summarized in the work [83] as a modified Landau-Teller function in the temperature range 700-2000K:

pτ = A ⋅ exp( BT −1/3 − CT −1 ), atm.s. The parameters A, B and C are given in Table 6.

(3)

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High Temperature Phenomena in Shock Waves

HCl, DCl and HBr, DB Relaxation. Densitometric laser-beam-deflection method was used for the study of the vibrational relaxation of HCl and DCl behind incident shock waves [84], and for the relaxation of HBr and DBr [81,83]. As in HI, DI relaxation studies, the results of these investigations are in serious disagreement with the predictions of the SSH theory [80] and in good agreement with the theories [85,86], suggesting that the R-V energy transfer is important in hydrogen halides relaxation. Experiments [87] conducted at temperatures 1100-2100K in HCl-Ar mixtures by monitoring IR emission 3.4 μ were in approximate agreement with the data [84] for collisions HCl-HCl. The vibrational relaxation time for HCl-Ar [87] is such that pτ (HCl-Ar) ≈ 45pτ (HCl-HCl), see Table 6. Later, the measurements were made of the vibrational relaxation time for HCl in mixtures with He, Ne, Ar and Kr [88] to study a simple mass effect on the vibrational relaxation process. The experiments were carried out behind incident shock waves in the temperature range 800-4100K. The measured relative efficiencies of colliding partners in HCl vibrational excitation were as follows:

HCl:He:Ne:Ar=1:(6±20%):(16±30%):(86±50%-70%). It was noted that Kr is a less efficient collision partner than Ar. Table 6.

Collision HI-Ar HI-HI DI-DI HI-N2 DI-N2 HI-CO HCl-HCl DCl-DCl HCl-CO DCl-CO HCl-Ar HBr-DBr DBr-DBr HBr-CO

A (atm·sec) 3.4·10-6 1.55·10-9 2.04·10-10 1.32·10-8 1.08·10-8 (3.5±1.5)·10-9 8.39·10-13 8.77·10-12 (3.7±2.7)·10-10 (1.05±0.45)·10-10 3.4·10-11 4.01·10-12 2.75·10-9 (5.05±1.15)·10-9

B (K)1/3 0 68.1 107 60.7 52.6 66.8±4.6 184 151 109.6±9.2 104.5±5.8 184 166 59.8 73.2±2.3

C (K) 0 646 1594 0 0 0 4729 3080 0 0 4729 4070 -277 0

T, K 1400-2300 800-1800 700-2000 1000-2700 1200-2000 1400-2000 700-2000 700-2000 1200-2000 1350-1850 800-4100 700-2000 700-2000 1200-2000

Ref. 79 81 83 89 89 90 84 84 90 90 87,88 81,83 81,83 90

HI-N2, DI-N2. Time resolved laser schlieren method [75] was also used for the analysis of the vibrational energy transfer in shock-heated mixtures N2-HI (T=10002700K) and N2-DI (T=1200-2000K) [89]. From an oscillogram analysis it was deduced that in both systems, vibrational relaxation of N2 is dominated by rapid V-V energy transfer from the halide gas. The results concerning the effect of N2 on relaxation of HI and DI confirm the previous suggestions that R-V energy transfer is the predominant mechanism for vibrational relaxation of hydrogen halides (Table 6).

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

109

HX-CO, X=Cl, Br, I. The vibrational relaxation times of shock-heated mixtures of hydrogen halides in collisions with carbon monoxide [90] were measured for the overtone emission from shock-heated CO to which a small amount of hydrogen halides was added. From the decrease of the experimental relaxation time with concentration of hydrogen halide HX, the rate constants for the reactions HX(v=1)+CO→HX(v=0)+CO were determined. The results, in atm·sec units, is presented in Table 6. HF, DF Relaxation. Vibrational relaxation of hydrogen and deuterium fluorides HF and DF at high temperatures was studied by monitoring infrared emission of HF (2.5-2.7 μ) or 3.5 μ of DF behind incident shock waves [91-94]. The measured vibrational relaxation times are presented as an usual Landau-Teller function pτ = A ⋅ exp ( B ⋅ T − 1/3 ) , atm .s in Tables 7 and 8. As in the case of other hydrogen halides, neither systematic [18] nor SSH theory [80] describe vibrational relaxation of HF and DF. According to an analysis [91], the theory[86,87], suggesting R-V energy transfer in hydrogen halides relaxation, has shown more promise in explaining the experimental data. A reasonable fit to experimental data was obtained in works [91,92] using the modified theory [86]. In this case the dominant role of the attractive potential in energy transfer was demonstrated especially at low temperatures. At the same time, in work [92] it was noted that it is only the short-range repulsive intermolecular potential that needs to be considered in the high-temperature limit, and that the effects of attractive forces can be neglected. Nevertheless, it seems clear that, at the present time, the mechanism of vibrational relaxation in the hydrogen halide molecule remains unclear. Table 7.

Table 8. 1/3

Collision A (atm·sec) B (K) HF-HF 1.02·10-8 34.39 HF-HF 0.6·10-9 64±4 HF-Ar 1.62·10-9 111.97 -10 HF-He 1.52·10 133.3 HF-F 3.33·10-11 64 HF-Cl <1.10-10 64

T (K) Ref. 1350-4000 92 1400-4100 93 1350-4000 92 1350-4000 92 1400-4100 93 3000-4100 94

Collision A (atm·sec) B (K)1/3 DF-DF (1.4±0.15)·10-9 63.7±3 DF-DF (2.5±1.6)·10-9 56±3 DF-N2 (1.26±3.2)·10-10 127±12 -9 DF-Ar (7.1±1)·10 128.6±6 DF-H2 1.9·10-8 35 DF-F (0.5-1.5)·10-8 0

T (K) Ref. 2000-4000 91 1500-4000 94 1500-4000 94 1500-4000 91 800-4100 91 2000-3000 91

F2, Cl2, ClF Relaxation. Vibrational relaxation of Cl2 was measured behind a shock front at four temperatures from 578 to 1466K with an optical interferometer [95]. These results exhibit a considerable scatter. Later, vibrational relaxation of F2, Cl2 and ClF was studied by recording the density change behind the shock waves using the above-mentioned laser-beam deflection technique [96-98]. Relaxation of fluorine was studied in mixtures 10% and 20% F2 in Ar in the work [96], and in mixtures 10%He+70%Ar+20%F2 in the work [97]. Relaxation of chlorine was studied in pure gases and in mixtures of 1.0, 2.0 and 5% HCl in Cl2, in mixtures 5 and 10.1% DCl in Cl2, as well as in mixtures 10% CO in Cl2[98]. The measurements of vibrational relaxation time in collisions Cl2-Cl2[98] agrees fairly well with results[95]. It was noted that the effects of HCl and DCl on Cl2 relaxation are large, particulary at low temperatures, e.g., HCl is about 100 times more effective than Cl2 at 400K. Molecule

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High Temperature Phenomena in Shock Waves

HCl is about 2-4 times more efficient than DCl, depending on the temperature. In the work[98] these relative efficiencies are interpreted as a strong indication of the rotational effect. The same laser schlieren technique was used to measure the vibrational relaxation time of chlorine monofluoride ClF behind shock waves in the temperature range between 500 and 1000K[99]and for mixtures 10% ClF+90% Ar and 20% ClF+80% Ar. The vibrational relaxation times measured in investigations[96-99] are presented as a modified Landau-Teller function (3.3) with parameters A, B and C given in Table 9. Table 9.

Collision F2-F2 F2-Ar F2-He Cl2-Cl2 Cl2-CO Cl2-HCl Cl2-DCl ClF-ClF ClF-Ar

A (atm·sec) (12.6±8.4)·10-10 (2.49±1.7)·10-10 (6.85±1.8)·10-10 (4.75±2.42)·10-10 (2.57±1.67)·10-9 (4.77±1.20)·10-8 (7.89±2.00)·10-9 (7.64±6.97)·10-9 (1.58±1.03)·10-9

B (K)1/3 65.2±7.02 96.97±7.73 47.21±2.33 73.8±6.8 40.6±5.3 -3.64±2.06 20.9±2.1 50.57±13.26 68.44±6.78

C (K) 0 0 0 (537±174) 0 0 0 0 0

T (K) 500-1300 500-1300 500-1050 400-1400 400-1100 400-1100 400-1100 500-1000 500-1000

Ref. 96 96 97 98 98 98 98 99 99

Br2, I2. The above-described examples of Cl2 and F2 vibrational relaxation were studied in shock waves at temperatures not exceeding 1500K. The characteristic vibrational temperatures θ=ћω/k of these molecules are equal to 797.61K for Cl2 and 1286.46K for F2. Under these conditions, the rate of the relative motion of colliding particles v~(2kT/μ)1/2 is slow, as compared with the rate of vibrations, i.e. the collision time τcol~1/αv is greater than the oscillation period (α is the exponential interaction potential parameter, α~(4-6)·108 cm-1). In these cases, the Massey parameter, responsible for the probability of vibrational transition, is such that ωτcol>>1, and the vibrational transition probability P1-0 is low, increasing exponentially with the temperature[16,80]. As a result, vibrational relaxation of Cl2 and F2 obeys the Landau-Teller temperature dependence. Therefore, of particular interest is to study the vibrational relaxation of molecules Br2 and I2, having minimal vibrational quantum energies of 465K and 306.9K, respectively. Vibrational relaxation of I2 was investigated behind incident shock waves in vapors of pure I2 and in mixtures of I2 with Ar, He and N2[100]. The study was conducted by recording the absorption of I2 molecules in the visible spectrum for T=800-3500K. The results indicate that when T≤1000K (ωτcoll>1) relaxation of molecular iodine is described by the usual Landau-Teller temperature dependence. At higher temperatures, the growth of the deactivation probability P1-0 decelerates, so that it passes through a maximum and then decreases. In mixtures of I2 with Ar, He and N2 the investigation was performed at ωτcoll<1, and in all experiments the values P1-0 decreased with increasing temperature. This was particularly evident in I2-He mixtures, due to minimal reduced masses of the colliding particles and, correspondingly, minimal values of τcoll and ωτcoll. A result of this investigation is presented in Fig.1 taken from[100].

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

Fig. 1. Dependence100 of deactivation probability P1-0 versus the adiabatic parameter ωτcoll in collisions of: 1. I2-I2; 2. I2-I2; 3. I2-He; 4. I2-Ar; 5. I2-N2

111

Fig. 2. Dependence of Br2 vibrational relaxation time on T at P=1 atm. 1 - Data [101]; 2 - data of ultra-acoustic measurements (taken from work [101])

A similar study was carried out behind the shock wave front in vapors of pure bromine for T=580-3300K using the absorption of radiation 414.0 nm and 565,0 nm[101], and in its mixtures with He, Ar, Ne and Xe at T=500-2250K (by registration of absorption of 414 nm and 490 nm)[102]. Registration of absorption made it possible to measure the time evolution of the vibrational temperature Tv. Thereupon, the vibrational energy, the gas density, and the translational temperature were calculated using the conservation laws, and, finally, the vibrational relaxation time was determined τ=(εveq-εv)/(dεv/dt). An example of the measured relaxation time τ (Br2-Br2) is shown in Fig.2. Clearly, at T~1000K τ presents a minimum corresponding to ωτcol~1, then, with increase in T, τ begins to increase. The study[102] of bromine vibrational relaxation in mixtures with He, Ar, Ne and Xe confirmed the observed dependence. These results proved that a maximum probability of vibrational excitation takes place when the duration of molecule collision with a colliding partner is close to the molecular oscillation period.

3 V-T Relaxation of Three- and Multi-atomic Molecules Vibrational relaxation of polyatomic molecules, primarily CO2, was studied in many works. This is essentially due to the development of laser technologies in the 60's and 70's of last century. Partially, the results of these investigations are systematized in a few reviews and books; see, for example[13,15,103,104]. In order to determine the vibrational relaxation time τ, the simple relaxation equation (1) is often used. However, the process of vibrational relaxation of three- or more atomic molecules is more complicated due to the fact that such molecules are characterized by several types (modes) of vibration. An analysis of vibrational relaxation is simplified if

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High Temperature Phenomena in Shock Waves

intramode VV and intermode VV' exchange is faster than VT relaxation of the mode having the lowest vibrational frequency. In this case, the relaxation process can be described by the same relaxation equation (1), where

τ = τ 1 ( ∑ cvi ) cv1

(4)

Here, cvi is the vibrational heat capacity of the ith vibrational mode, τ is the vibrational relaxation time, and τ1 is the vibrational relaxation time of the mode having the lowest vibrational frequency. In some cases, to describe vibrational relaxation of molecules A in collision with a M

partner M, the rate constant of VT exchange W is introduced. In this case the equation of vibrational relaxation can be rewritten as follows[12]: d ε / dt = ( p / kT )( ε 0 − ε ) ∑ W M γ M

(5)

where p is the pressure, and γ M is the molar fraction of component M in the mixture. In equation (5) the dimension of the rate constant is cm3/s. This rate constant is related to the usual vibrational relaxation time as follows: pτ (atm. s) = 1.36 ⋅10−22 T [W M (cm3 /s)(1 − exp(−θ / T ))]

In the vibrational kinetics the (atm·s)-1 and (atm·μs)-1 dimensions of the rate constant for vibrational relaxation are also used. They are usually denoted as kM. In this case Eq.(5) can be written as follows:

d ε / dt = p ( ε 0 − ε ) ∑ k Mγ M

(6)

In different publications, the equations describing VT relaxation are used either in form (1) or in forms (5), (6). Thus, when using the quantities τ, WM or kM, it is necessary to indicate how these values were determined and to use them only in the relevant equations. Table 10.

M CO2 CS2 N2O NO2 H2O SO2

υ1 1388 658 1285 1320 3657 1151

υ2 667 397 589 750 1595 518

υ3 2349 1532 2224 1618 3756 1362

The data on vibrational relaxation of triatomic molecules, given in Table 10, are taken first of all from review[104] and monograph[13] with additional data issued after these publications or those not mentioned in[13,104]. For these molecules the lowest frequency corresponds to the bending vibrations υ2 (Table 10). Usually, VT relaxation of the other modes can be neglected, in view of a rapid intermode exchange in these molecules.

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

113

3.1 CO2 V-T Relaxation CO2(0110)+M→CO2(0000)+M CO2-CO2. The main results obtained in shock tube experiments after the review[103] agree well with the high-temperature data given in review [103]. In CO2 intermode VV' exchange is faster than the VT relaxation. Therefore, the use of Eq. (4) to determine τ(CO2-CO2) is justified. In papers [105-107] a Mach-Zehnder interferometer and the laser-schlieren method were used to measure τ (CO2-CO2) for temperatures up to 3000K. It was shown that relaxation time measured by these two methods agree well. This result is in agreement with the data [108] obtained in a shock tube for 380-1950K using the laser-schlieren method. The vibrational relaxation of shock heated CO2 was also studied with a tuned CO2 laser absorption technique in the temperature range from 500 to 2000K [109-110]. An absorption method was used for different rotational states of bending and symmetric-stretching modes. In this case, the measured bending-mode relaxation was shorter than that predicted from interferometry results using the ratio of specific heats (4). In work [111] the CO2 deactivation time in the nozzle mounted behind the reflected shock wave was measured using the registration of CO2 laser absorption (10.6 μm) for T=600-1400K. It was shown that τdeexit/τexit=1.1±0.4. As noted in review [104], the best agreement with experiment both at low and at high temperatures is given by the approximation of the constant kM proposed in the works [108,112]. In Table 11, the approximation used in [112] is given. CO2-N2. The data on VT relaxation of CO2 in collisions with nitrogen at high temperatures are scarce. In experiments [113] the data on the vibrational relaxation of CO2-N2 at T=800-2500K were obtained by recording the infrared radiation of CO2 molecules behind the shock wave front. In a shock tube experiments [107] the schlieren technique was used for T=360-1500K and the laser-schlieren method was used in works [105,114] for T=380-1950K and for T=350-1400K. respectively The best temperature approximations of τCO2-N2 in a wide range of temperatures are given in studies [112,115] (see Table 11). It should be noted that the dependence τCO2-N2, recommended in review[103] and often used in calculations, is wrong. CO2-O2. In review [103] it was assumed that the CO2 relaxation rates are the same in collisions with nitrogen and oxygen. Later, experimental data were obtained, indicating that these rates are really close in a wide temperature range. Shock tube measurements [114] made using laser-schlieren method at temperatures between 350 and 1500K suggest that O2 is slightly more efficient than N2 in deactivating the bending mode of CO2. However, later shock tube experiments [108] using the same method for T=380-1950K gave smaller values of the rate constant, (equal to half of data [114] at 1000K). The authors of study [108] suggest that the discrepancy between the results can be due to the use in work [114] of a simplified model of vibrational energy exchange in the mixture CO2-O2. An analysis, performed in review [104], makes it possible to recommend an approximation of the rate constant kCO2-O2 for temperatures from 200 to 2000K taking into account the low-temperature data [116] (Table 11).

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High Temperature Phenomena in Shock Waves

CO2-H2O. Relaxation of CO2 containing trace amounts of H2O has been extensively studied, mainly due to a high rate of deactivation of mode υ2 by water vapor, which greatly affects the population inversion in carbon dioxide laser. The widest temperature range in determining kCO2-H2O was investigated in the work [117]. In that paper, the influence of water vapor on the vibrational relaxation of CO2 was studied by the laser-schlieren method behind an incident shock wave in the temperature range 320-1540K. The relaxation time was measured at fixed temperatures depending on humidity, which varied from 0 to 6%. The relaxation rate constant of υ2 mode was determined in the framework of a model taking account the V-T and V-V' processes. A constant error was estimated as ±10%. These data agree well with the results of the investigation [118] performed in a shock tube and also using the laser-schlieren method. The approximation of the rate constant measured in work [117] is given in Table 11. This approximation also describes the available low-temperature data satisfactorily. CO2-H2. Shock tube study of vibrational relaxation of CO2 in collisions with hydrogen molecules was made using a Mach-Zehnder interferometer for temperatures 350-1200K [119] and using the laser-schlieren method for T=350-1500K [107]. In these studies it was shown that the efficiency of hydrogen in the excitation of the mode υ2 is high and close to the effectiveness of H2O molecules. An approximately similar dependence of the CO2-H2 relaxation rate constants recommended in the works [112,115] in the range T=150-1500K was found (Table 11). The temperature dependence τCO2-H2O(T) and τCO2-H2(T) considerably differ from that predicted by the Landau-Teller theory. CO2-CO. The data concerning the efficiency of CO molecules in the relaxation of CO2 at high temperatures are almost lacking. An estimation of this quantity is made in work [113] for T= 800-2500K. In Table 11, the rate constant proposed in work[112] is given. This approximation is mainly based on the low-temperature data, and is given only for completeness. CO2-NO. Efficiency of NO in the vibrational-translational exchange with CO2 was studied in work [59] in a shock tube for T=320-900K using the laser-schlieren method. The relaxation rate constant obtained in that investigation was determined using a model that takes into account both VT and VV' exchanges of vibrational energy. This constant practically does not depend on the temperature and at T≈300K is 50 times higher than the constant with M=CO2. Approximation of the rate constant is given in Table 11. CO2-Ar, Kr, He, O. VT vibrational relaxation of CO2 in collisions with Ar was studied in shock tubes using the laser-schlieren method in the temperature ranges 360-3000K[106] and 360-1500K[107]. CO2-Ar relaxation was also studied in work[120] by monitoring the emission at 4.3 μ of the asymmetric stretching mode for T=700-2000K and by monitoring the emission at 4.3 μ and 2.7 μ for T=2000-4000K. in work[121].

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

115

Vibrational relaxation of CO2 in collisions with O atoms was studied in work[65]. It was shown that all modes of CO2 exhibit the same relaxation time, so that the measured data could be directly related to the relaxation rate for the translationvibration excitation of the bending mode. The experimental results indicate that O-atoms are approximately an order of magnitude more efficient than Ar in the relaxation of the bending mode. Vibrational relaxation of CO2 in collisions with Kr atoms was studied in work[122] using the laser-schlieren method for temperatures 1377-6478K. The effectiveness of krypton in the temperature range 1377-3000K was half the efficiency of argon, but with increasing temperature above 4000K this efficiency decreased down to a quarter of the argon efficiency. Relaxation of CO2 in collisions with atoms He was studied using the conventional interferometry[95,119] and the laser-schlieren method[107] at temperatures from 350 to 1500K. It was shown[119] that at T> 1000K the efficiency of light gases (H2, He, D2) in the excitation of CO2 vibrations are comparable and considerably higher than the efficiency of CO2-CO2 collisions. Approximations of the rate constants for M=Ar, He, Kr and O are presented in Table 11. Here θ=1-exp(-960/T). Table 11.

M CO2 N2 O2 CO NO H2O H2 He Ar Kr O

log10[θ -1 kCO2-M] (atm·s)-1 10.327 - 57.31T-1/3 + 156.7T-2/3 11.013 - 68.78T-1/3 + 188.5T-2/3 3.4578 +35.957T-1/3 - 17.079T-2/3 9.96 - 51.65T-1/3 + 121.8T-2/3 6.86 - 5.87T-1/3 5.97 + 17.53T-1/3 - log10θ 8.903 - 28.46T-1/3 + 144T-2/3 8.711 - 14.75T-1/3 10.011 - 49.4T-1/3 + 77.3T-2/3 9.71 - 49.4T-1/3 + 77.3T-2/3 11.011 - 49.4T-1/3 + 77.3T-2/3

3.2 N2O V-T Relaxation N2O(0110) + M → N2O(0000) + M

Vibrational relaxation of molecules N2O at high temperatures was studied in shock tubes using the classical interferometry and the registration of infrared radiation, but mainly using the laser-schlieren method, as in the study of CO2 relaxation. These investigations were analyzed in detail in studies[13,104], where the references to the original experimental studies are given. Thus, here only the main results taken from [13,104] are presented. N2O-N2O. The vibrational relaxation data for collisions N2O-N2O were obtained at T ≤ 2060K. The corresponding approximation of the rate constant was recommended

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High Temperature Phenomena in Shock Waves

in review[104] for the range 300-2000K with an accuracy of ±15% at high temperatures, see Table 12. N2O-N2. Relaxation of the N2O bending mode in collisions with N2 was investigated both by the laser-schlieren method and by recording N2O infra-red emission at temperatures up to 2000K. Corresponding rate constant recommended in review[104] agrees well with the results of low-temperature measurements and at T ~ 1000K has an uncertainty of 20-30%. N2O-O2. In the case of M=O2 the vibrational relaxation of N2O was studied at temperatures not higher than 2000K. As in the case of M=N2, the V-V' exchange between N2O and O2 complicates the interpretation of the experimental results, because the channels of V-V' exchange have not been reliably established. Therefore, according to the review [104] the approximation of the rate constants of V-T relaxation of N2O in collisions with O2, given in Table 12, at high temperatures can have a systematic error of ±30%. N2O-NO. The relaxation rate of N2O in collisions with NO molecules is high and weakly dependent on the temperature, due to electronically degenerate states of NO. This leads to the E-V channel of vibrational relaxation, as was observed in collisions of NO with other molecules. An approximation of the vibrational relaxation rate in N2O-NO collisions, measured at temperatures up to 1800K, is recommended in review[104] with an accuracy of ±30% at high temperatures. N2O-H2O. The study of the processes with participation of the H2O vapor is complicated because of the difficulty of monitoring their concentration in the mixture. Therefore, the scatter of the experimental data in different investigations is often greater than an order of magnitude. The relaxation rate in collisions with H2O molecules is large owing to the V-R mechanism of interaction. In Table 13 the approximation of the experimental data obtained in a shock tube using the laserschlieren technique for T=300-1200K is given. The processing of experiments was carried out taking into account both V-T and V-V' exchanges. Table 12.

M N2O N2 O2 CO NO H2O H2 Ar He

log10[kN2O-M] (atm·s)-1 8.964 - 32.88T-1/3 + 88.954T-2/3 8.380 - 16.70T-1/3 8.455 - 17.75T-1/3 11.399 - 65.345T-1/3 + 196.47T-2/3 7.08 - 1.6T-1/3 8.85 - 2.04(T/1000) + 0.892(T/1000)2 7.846 7.584 - 8.619T-1/3 - 47.62T-2/3 7.85 - 5.75T-1/3

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

117

N2O-H2. In the shock-tube studies of N2O relaxation in mixtures with H2 the contributions of the para- and ortho- hydrogen were separated. It was noted that in the case of para-hydrogen the relaxation of N2O occurs via V-T exchanges, whereas in the case of ortho-hydrogen via a resonance V-R process. Measured up to T=1000K, the relaxation rate of N2O in collisions with H2 is high (an order of magnitude higher than in collisions with H2O) and is almost independent of the temperature. The rate constant for relaxation of para-hydrogen, recalculated taking into account the relation (3.4), is given in Table 12. N2O-Ar, He. Relaxation of the mode υ2 of N2O in mixtures with Ar at temperatures up to 2000K and with He up to T=1000K was studied in shock-tube experiments taking the V-T and V-V' exchanges into account. The approximations of the rate constants are presented in Table 12. 3.3 H2O V-T Relaxation H2O(0110)+M→H2O(0000)+M

An important part of experimental and theoretical data on vibrational relaxation of water vapors are presented in works[13,104]. However, at high temperatures, this process was studied only in the investigation[123], which is essentially due to the difficulty of monitoring the water vapor concentration and to a high rate of vibrational relaxation. In this work, the vibrational relaxation of H2O with H2O, He, Ar and N2 was studied behind incident shock waves at temperatures between 1800 and 4100K. The relaxation processes were monitored by registering infrared emission of the water vapor at 6.3 μ (bending mode) and 2.7 μ (asymmetric mode). In H2O-Ar mixtures the over-all υ3 relaxation was studied between 1900 and 3800K with H2O vapor pressure ranging from 5 to 50 μ Hg, and the total pressure ranging from 0.5 to 3.0 Torr. A value of 1/54 was obtained for the relative efficiency of Ar as compared with H2OH2O collisions. Owing to the scatter (±50%) in the data, no temperature dependence was obtained. The υ2 relaxation was studied at initial pressures pH2O ~ 20 μ Hg and pAr ~ 0.25 Torr, and in the temperature range 1800-4100K. Within experimental uncertainties, no detectable differences were observed in the relaxation times of the two modes, nor any evident temperature dependence. In the mixtures H2O-He the initial pressure ranges covered for H2O and He were 6 μ-19 μ Hg and 0.25-1 Torr, respectively. The temperature varied between 1800 and 2500K. The relative efficiency of He, as compared with H2O, turned out to be equal to 1/35. The agreement between the data for the two modes confirms the same relaxation time and the absence of a temperature dependence on the covered temperature range. The H2O-N2 data were obtained at temperatures 2000-2700K. As in the cases of Ar and He, no temperature dependence was observed, and the relaxation times of υ3 and υ2 modes presented similar values. The results of this investigation involve rapid intramolecular energy transfer between the various modes of H2O, and a very weak temperature dependence of V-T deexitation rate constants in the studied temperature ranges. The υ2 V-T deexitation rate constants obtained are presented in Table 13.

118

High Temperature Phenomena in Shock Waves Table 13.

Collision partner

T, K

H2O-H2O H2O-Ar H2O-He H2O-N2

1800-4100 1800-4100 1850-2500 2000-2700

k20, (cm3/sec), at T=2500K 1.9·10-10 2.9·10-12 6.1·10-12 2.0·10-12

3.4 NO2 V-T Relaxation NO2(010)+M→NO2(000)+M NO2-NO2. Vibrational relaxation in collisions NO2-NO2 was studied with the laserschlieren method in the temperature range 400-2000K[124,125]. The measured relaxation rate was very high and weakly temperature dependent. This indicates that the relaxation mechanism is non-adiabatic and cannot be described by the SSH theory. In review[104] it was noted that the observed nature of NO2 relaxation is probably caused by the formation of N2O4 dimer as an intermediate stage of relaxation. NO2-N2. The vibrational relaxation of NO2 in collisions with N2 was investigated with the laser-schlieren technique for temperatures 590-1890KP[126]. In processing the experimental data it was assumed that the V-T relaxation of NO2 takes place when equilibrium between all the three modes of vibration has been attained. The relaxation rate increases rapidly with the temperature and can be satisfactorily described by the SSH theory. NO2-O2. The rate constant of V-T relaxation of NO2 in collision with oxygen molecules was measured using both the laser-schlieren method and infrared diagnostics in the temperature range 360-2400K[127]. Experiments were interpreted taking both V-V' and V-T processes in the gas into account. It was found that the relaxation rates of NO2 in the mixtures with O2 and N2 are similar in value. NO2-CO, Ar. V-T relaxation in a mixture NO2-CO was studied for T=6901180K[126]. In that study the infrared radiation of CO molecules and of the υ2 mode of NO2 molecules was registered in addition to the measurement of the density gradient behind the shock wave front. The nature of the temperature dependence of the relaxation rate constant in collisions NO2-CO was close to that observed in NO2N2 collisions. A similar study of the relaxation of NO2 in the mixture with Ar was performed in work[125] for 450-2400K. The approximate values of the relaxation rate constants of NO2 molecules obtained in works[124-127] are presented in Table 14. The accuracy of these approximations is estimated by their authors as ±30%. However, this error reflects only the random experimental errors and does not take into account the imperfections of the models used in processing the experiments.

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

119

Table 14.

M NO2 N2 O2 CO Ar

log10 kNO2-M, (atm·sec)-1 0.727 + 3.944T-1/3 9.48 - 28.33T-1/3 9.587 - 28.18T-1/3 10.43 - 28.33T-1/3 8.947 - 29T-1/3

T, K 400-2000 590-1890 360-2400 690-1180 450-2400

No data on the high-temperature relaxation of CS2 and SO2 are available. In conclusion, several other aspects of vibrational relaxation studies performed in shock tubes should be noted. A good summary of the experimental data concerning the V-V and V-V' exchange of CO2 is given in monograph[15] and, in a more detailed way, in review[128]. In addition, there are a number of publications devoted to the study of vibrational relaxation of polyatomic molecules, first of all, relaxation of hydrocarbons, see, for example[129-131]. However, these studies are beyond the scope of the present review, as well as the studies of vibrational relaxation during thermally non-equilibrium chemical reactions (Tv≠T); see, for example[67,132,133].

4 Chemical Reactions A large number of data on chemical kinetics has been obtained in shock-tube experiments. The conventional and most widely used methods of investigation are emission and absorption spectroscopy. In recent years, the accuracy of the results has been considerably improved thanks to the application of advanced measurement techniques. In essence, these methods often represent a refined version of the absorption technique. In particular, this is the case for the atomic resonance absorption spectroscopy (ARAS) method which makes it possible to monitor the atom concentration from light absorption of their resonance lines. At present, the resonance emission sources have been developed on the O, C, H, N, S, Si and other atoms. Another method, namely that of laser absorption, consists of the use of a laser as a light source in a very narrow spectral region; it makes it possible to selectively measure the absorption of the molecular gas under study and to eliminate the superimposition of the spectra of accompanying objects. Widely used is also the technique of measuring absorption in wider spectrum bands emitted on tunable wavelengths (dye lasers). At present, these are the most accurate methods for determining atomic and molecular concentrations ensuring high reliability of the measurements. This is also favored by the use of multi-channel recording for simultaneously monitoring both reactants and reaction products in experiments. With these methods, many kinetic coefficients, in particular, those of dissociation of many diatomic and triatomic molecules, for example, O2, N2, CO, NO, C2, CO2, C2N2, etc., were refined.

120

High Temperature Phenomena in Shock Waves

In this Section the results of investigation of reactions in the C-O and N-C-O atom systems are presented. These systems include sets of reactions the study of which is stimulated by the development of space engineering and the flights in the atmospheres of Earth, Mars, and Venus, as well as by the problems related with combustion in various media. In many cases, the same reaction was investigated in different studies, at different temperatures and pressures. In these cases the references of overviews are given, in which an analysis of these studies is made and recommendations for kinetic coefficients are presented, together with an estimation of their errors. These data make it possible to estimate the completeness of reaction investigation and the degree of reliability of the values recommended. 4.1 Chemical Reactions in the System C-O Reaction CO2+M→CO+O+M (R1) Dissociation of carbon dioxide has been investigated in shock tubes behind incident and reflected shock waves. Pure CO2 and CO2 mixtures with Ar, N2, Kr were studied. The measured rate constants of CO2 decomposition relate mainly to the low pressure limit. Absorption and emission spectroscopy technique in IR and UV spectra, laser absorption, atomic resonance absorption spectroscopy (ARAS) and density measurement using the laser schlieren method were applied in these studies. The results on dissociation of CO2 relate mainly to moderate temperatures, T<6000K. In the temperature range from 2500 to 5000K, the main contribution to dissociation of CO2 is made by the decay via the spin-forbidden channel from excited electronic term 3B2 associated with the formation of the products CO(1Σ)+O(3P). The decay of CO2 was studied under vibrational non-equillibrium conditions [134,135]. As the temperature increases, it is required to take into account other channels of the CO2 decay. It was shown[136,137] that, already for T=3000-4000K, the 1B2 term is populated during the CO2 dissociation, and the radiation from this state was detected in experiments. The 1B2 excitation was realized in conditions of incompleteness of vibration-electronic relaxation and its population temperature was found to be lower than the translational one. As the temperature increases, the probability of the CO2 dissociation via the channel from the ground electronic state X1Σ increases. In works[138-140] the electron-excited O(1D) atoms were detected behind the shock wave front that was evidence for additional channels of molecular dissociation directly from high vibrational levels of the ground term X1Σ and from the excited term 1B2. In these studies a partial contribution of the spin-forbidden and allowed channels of decay was analyzed at different temperatures. An incubation period prior to decomposition of CO2 behind shock waves was observed in study[141] for the first time. The thermal decomposition of carbon dioxide was investigated behind reflected shock waves at temperatures 3200-4600K and pressures 337.5-750 Torr. Ultraviolet laser absorption at λ=216.5 and 244 nm was used to monitor the CO2 concentration with microsecond time resolution, allowing observation of a pronounced incubation period prior to steady CO2 dissociation. Mixtures of 1% and 2% CO2 diluted in argon were used in experiments. In Fig.3, an incubation time on the CO2 absorbtion signal is observable prior to the onset of steady dissociation.

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

121

Fig. 3. CO2 absorbance (ln(I0/I)) at 216.5 nm. Initial mixture 2% CO2/Ar[141]. Initial (prior to decomposition) vibrationally equilibrated reflected shock conditions: 3838K, 623 Torr

Fig. 4. Rate constant of CO2 dissociation in Ar. Experimental data: 1 - Davies, 1964[145]; 2 Davies, 1965[142]; 3 - Fishburne et al., 1966[147]; 4 – Dean 1973144, 5 - Zabelinsky et al., 1986[146] , 6 - Fujii et al.,1989[143]; 7 - Burmeister and Roth, 1990[148]; 8 - Hardy et al., 1974[149]; 9 - Oehlschlaeger et al., 2005[141]. Recommendation: 10 - Ibraguimova, 2000[150]

122

High Temperature Phenomena in Shock Waves

The passage of the incident shock wave causes a schlieren spike in the signal. The vibrational relaxation time behind the reflected shock wave in these conditions (Fig.3) is estimated to be 0.7 μs. The incubation time was treated in work[141] as the time of activation of molecules undergoing collisions with other gas species. The rate constants of CO2 dissociation in Ar measured in works[141-148] are shown in Fig.4. The ARAS method was used for the C atom concentration measurements[148]. The rate constants k1 [141] are in excellent agreement with the results of studies[146,149]. The preferred rate constant obtained on the basis of expert-statistical method in review[150] for M=Ar in the temperature range 230011000K is also shown. In deriving the preferred value, the results of experimental studies analyzed in reviews[148,150,151] were used. In Table 15 the rate constants k1 [148,150] are presented. CO2 dissociation was studied in shock tubes both in pure CO2 and in mixtures of CO2 with N2, Kr, Ne[147,152-156]. The efficiencies of N2 and Ar as collissional partners in dissociation of CO2 differ only slightly from that of CO2. For T>4000K, the efficiency of CO2 exceeds that of Ar by a factor of 3 (see Table 15). The extrapolation of the CO2 dissociation rate constants to wider temperature ranges must inevitably be based on an analysis of the interaction between dissociation and vibrational relaxation. The presence of three vibrational modes and several lowlying “embedded” electronic excited states of different symmetry, determining their significant role in the processes of excitation and decay, requires considering energy exchange processes between them. Reaction CO+O+M→CO2+M (R2) Reaction (R2) was studied in shock tubes with gas mixtures diluted in Ar for temperatures 1200-3500K[10,157-162]. Methods of detecting CO2 radiation in IR and UV spectrum, and measuring C atom concentration by titration, atomic fluorescence, and resonance absorption were applied. The presence of impurities in the test gases has an influence on the results of measurements. Careful experiments were performed in works[159,160]. In work[159] the level of impurities was about 10 ppm (CH4 and H2O). A small controllable amount of H2 was added to O2/CO/Ar gas mixture[160]. Reactions involving H2 and H were included in the chemical model (27 reactions). The optimal composition of mixture H2/O2/CO/Ar was chosen in calculations to reduce the influence of “error of model” on the determination of k2 . Experimental data on recombination process CO+O+M→CO2+M were analyzed in detail in reviews[150,151]. In addition to the results of shock-tube studies, the data obtained in flames, and discharges at low temperatures were also considered. For the collisional partner M=Ar the preferred rate constant was determined in the temperature range 250-11000K (Table 15). The efficiencies of species N2, CO, CO2 in recombination CO+O+M were given in review[150] in which the experimental data obtained at temperatures not above 800K were analyzed (Table 15).

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

123

Table 15. Preferred rate constants k = A ⋅ T n exp (− E a / T ) , cm3·mole-1·s-1

Reaction

M

T, 103 K

A

n

Ea, K

0 52525 CO2+M→CO+O+M Ar 2400-4400 3.65·10 Ar 2300-11000 1.7·1031 -4.22 64800 14

N2 2900-11000 2.8·1013

0

±Δlgk; %

Ref.

±15%

148

0.1

150

40320 0.3 (T<6500K)

150

0.4 (T>6500K) CO2 2500-11000 7.9·10

14

0

52290 0.38 (T<7000K)

150

0.5 (T>7000K) -2.97 3830 CO+O+M→CO2+M Ar 250-11000 4·10 15 6 -2 -1 0 2185 (k, cm ·mol ·s ) CO 300-800 2.3·10 24

0.2

150

0.4

151

N2

300-800

1.5·1015

0

2185

0.4

CO2

300-800

4.7·1015

0

2185

0.4

151

±50%

173

±30%

176

CO+M→C+O+M

Ar 5500-9000 4.3·10

C2+M→C+C+M

Ar 2580-4650 1.5·1016

27

-3.1 129000 0

71650

151

The rate constant k2 has a maximum at T~1500K. This indicates that the recombination mechanism includes an energy barrier overcoming stage. The activation energy in the expression for k2 (Table 15) is equal to about 0.3 eV, which approximately corresponds to the barrier on the potential surface of the 3B2 term of the CO2 molecule formed from the products in ground states CO(1Σ)+O(3P). The existence of such a barrier on the potential surface of the 3B2 follows from a number of studies in which chemiluminescence was studied under the CO+O recombination[163,164]. Dissociation of molecules CO2 and recombination CO+O was also studied under the “high pressure limit” conditions and in the transitional region[165-167]. It was found that at T=300K the region of low pressures for recombination CO+O is limited to a pressure of about 2 atm[165]. Reaction CO+M→C+O+M (R3) The CO dissociation was studied in shock tubes both in undiluted CO [168,169] and in CO diluted in argon[170-173]. In the experiments the disappearance of CO molecules was observed during dissociation process by detecting emission in the IRspectrum or absorption in the vacuum-ultraviolet range. In work[173] the occurrence of O and C atoms formed as a result of dissociation was detected using the ARASmethod. In some studies [170,171], dissociation of CO accompanied by C2 and C radiation was also observed. In studying CO dissociation in mixtures with argon, a delay of CO dissociation was observed. The delay time was considerably greater than vibrational relaxation time of CO molecules (with the same conditions in the gas). In study [173] the delay time was observed as a delay in the appearance of the C and O atoms. As the temperature and gas pressure increase in shock wave, the delay time decreases. In undiluted CO, no delay was observed.

124

High Temperature Phenomena in Shock Waves

Fig. 5. The rate constant of dissociation CO inAr. Experimental data: 1 - Mick et al., 1993[173]; 2 - Appleton et al., 1970[172]; 3 - Davies, 1964 [170]. Recommendation: 4 - Baulch et al., 1976[3]

A decrease of the concentration of CO molecules was accompanied by appearance of C2 molecule. The delay was attributed to an accumulation of active intermediates, either CO molecules in excited electronic states or C2 and O2 molecules dissociated into C and O[171,172]. The following reaction chains occurring simultaneously with CO dissociation were suggested[171]: CO + C → C 2 + O, C 2 + M → 2C + M , CO + O → O2 + C , O2 + M → 2O + M

These reactions can play an important role in the dissociation of pure CO. The first recommendation3 of the rate constant for reaction (R4) was given on the temperature range 7000-15000K for collision partners Ar, CO, on the basis of the experimental data obtained up to 1974. In Fig.5, the results[170,172,173] with the recommendation from review3 are given. In Table 15, the rate constant k3 [173] is presented. Reaction C2+M→2C+M (R4) Estimations of value k4 were made in a few studies devoted to CO and CN dissociation in different gas mixtures CO/Ar171, C2N2/Ar and BrCN/Ar[174], and pure

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

125

CO [169]. In these studies, the reaction R4 was one of numerous processes in the observable chemical mechanism. The first measurement of the rate constant was performed in study[175], where k4 was determined by considering the time history of C2 emission formed during acetylene pyrolysis behind the shock front. These data were often used in reaction mechanisms in other works.

Fig. 6. Rate constants of the C2 molecule dissociation: 1 - Hanson, 1974[169]; 2 - Beck and Mackie, 1975[175]; 3 - Kruze and Roth, 1997176;4 - The data were derived from Slack, 1976[174]

The rate constant of reaction R4 (Table 15) was reliably measured in the temperature range 2580-4650K[176], where acetylene pyrolysis in shock wave was studied for a gas mixture with small acetylene content in argon: 5-50 ppm. In that study the quantitative results for the time history of the C atoms, and C2 and C3 molecule concentrations were obtained. The C atom concentrations were measured by applying atomic resonance absorption spectroscopy. Quantitative C2 detection was performed by ring dye laser absorption spectroscopy, and C3 radicals were monitored by their emission using a combination of a spectrograph and ICCD camera system. Detection of three species made it possible to study also reaction C2+C2→C3+C which, together with dissociation C2+M→2C+M, is an effective channel of C2 molecule removal. The results of works[175,176], together with estimations[169] and the value k4 derived using the rate constant of back reaction[174] are shown in Fig.6. In review[4] the rate constant (Table 15).

k4 from work[176] is given as a preferred value

126

High Temperature Phenomena in Shock Waves

Reactions CO+O2→CO2+O (R5), CO2+O→CO+O2 (R(-5)) Reaction R5 was studied in shock tubes at temperatures 1500-4500K in mixtures CO/O2/Ar[177,178], CO/O2/CH4/Ar[179], CO/NO/Ar[180]. The IR emission of CO (λ=5.07μ) and CO2 (λ=4.25μ), and UV emission of OH (λ=306.4 nm) and CO2 (λ=445.7 nm) were detected[177] behind the front of shock wave. The value of the rate constant was determined from the measurements of the delay time between the passage of the front and the start of the CO2 emission at λ=445.7 nm. High-purity gases previously frozen to eliminate water vapors were used. Similar methods were used in studies[179-181]. The concentrations of atoms O and H were measured[178] using the ARAS technique. In processing the experiments, chemical reactions involving small admixtures (H2, CH4, N2O) were usually taken into account since they have an effect on CO oxidation. The studies concerning reactions R5 and R(-5) are reviewed[182]. In Fig.7 the results of the above-mentioned studies are shown. For comparison, the preferred value of k5 from review3 obtained up to 1973 is given. In Table 16 the results[178] and the preferred values of k5 proposed for the temperature range 1500-5000K are given. Table 16. Rate constants of exchange reactions k = A ⋅ T n exp (− Ea / T ) , cm3·mole-1·s-1

Reaction CO+O2→CO2+O CO2+O→CO+O2 C+O2→CO+O (3P,1D)

T, K 1500-5000 1700-5000 1500-4200 298-4000

A 5·1013 2.7·1014 1.2·1014 6·1012

n 0 0 0 0

E a, K 31800 33750 2010 320

C2+C2→C3+C

2580-4650

3.2·1014

0

0

±Δlgk; % 0.2 0.3 ±50% 0.15 (298K) 0.5 (4000K) ±12.5%

Ref. 178,182 182 186 4

176

Reaction R(-5) is very important in the CO2 decomposition since it accompanies CO2 dissociation. The experimental results[183-185] for k − 5 were obtained at T>3000K. Recommendation of k − 5 at T=1500-3000K was derived in review3. These data are shown in Fig.8. On the plot the value of k − 5 calculated using k5 [178] and the equilibrium constant is also presented as curve 6. The data[182] are presented in Table 16. Reactions C+O2→CO+O(3P, 1D) (R6) Reaction R6 was studied[186] behind shock waves in mixture C3O2/O2/Ar at T=15252540K. The C atoms were formed during the pyrolysis of C3O2 which was realized using an ArF laser. The C concentration was measured by ARAS. The rate constant is given in Table 16. Using the results[186] and low-temperature data, the preferred rate constants4 in the temperature range 298-4000K were proposed (Table 16). The dominant channel of reaction R6 is that of atom O(1D) formation.

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

127

Reaction C2+C2→C3+C (R7) The kinetics of C2 radical reactions during the first stage of acetylene hightemperature pyrolysis was studied[176] by monitoring C, C2 and C3 radicals. Quantitative C2 detection was performed by ring dye laser absorption spectroscopy; C atoms were measured by applying atomic resonance absorption spectroscopy (ARAS), and C3 radicals were monitored by their emission using a combination of a spectrograph and ICCD camera system. The experiments were performed behind reflected shock waves and cover the temperature range of 2580-4650 K. Initial mixtures containing Ar with 5-50 ppm C2H2 were used. In this very low concentration range, the rate coefficients k7 and k4 were determined. The k7 value is presented in Table 16. In the very low concentration range of 50 ppm C2H2 or less, a simplified description of the measured C2 profiles could be given on the basis of four elementary reactions, for which the rate coefficients were evaluated. In addition to reactions R7 and R4 (C2 dissociation), these are the reactions C2+H2→C2H+H and C2H+H2→C2H2+H.

Fig. 7. Rate constants of reaction CO+O2→CO2+O. Experimental data: 1 Sulzmann et al., 1965[177]; 2 - Sulzmann et al., 1971[180]; 3 - Dean and Kistiakowsky, 1971[179]; 4 - Roth and Thielen, 1983[178]. Recommendation: 5 - Baulch et al., 19763; 6 - Ibraguopimova, 1991[182].

Fig. 8. Rate constants of reaction CO2+O→CO+O2. Experimental data: 1 Bartle and Myers, 1969 [183]; 2 - Baber and Dean, 1974[184]; 3 - Korovkina, 1976 [185]. Recommendations: 4 - Baulch et al., 19763; 5 - Ibraguimova, 1991 [182]. Curve 6 is the

value of

k− 5

derived from work [178].

128

High Temperature Phenomena in Shock Waves

4.2 Chemical Reactions in the System N-C-O Reaction CN+M→C+N+M (R8) The CN dissociation was studied174 behind reflected shock waves in lean mixtures BrCN (0.1%)/Ar, C2N2 (0.2%)/Ar, C2N2 (0.2%)/N2 (4%)/Ar at total pressure 1-20 Torr. In the experiments the CN (λ=390, 419 nm) and C2 (λ=514 nm) radiation was monitored. The total range of temperature was 4400-13000 K. A complex mechanism of the CN decomposition earlier proposed187 was confirmed. Dissociation itself (reaction R8) is accompanied by secondary reactions CN+C=C2+N (R15), CN+N=N2+C (R(-14)), C2+M=2C+M (R4), N2+M=2N+M. At T<5000 K and high concentrations of CN, it was proposed to consider also another channel of CN removal via the reaction CN+CN=C2+N2 (R17). The rate constant k8 estimated in

that study was in good agreement with the data187 at T=8000 K. The use of gas mixtures of different compositions made it possible to estimate the rate constants of reactions R15 and R(-14). The values of k8 [174] were used in many studies concerned with simulations in high-temperature gases. Recommendation for k8 (M=Ar) was given in reviews[188,189], where the data obtained up to 1976 were considered.

Fig. 9. Rate constant of CN dissociation Experimental data: 1 - Mozzhukhin et al.,1989[190] 2 Slack, 1976[174]; 3 - Ibraguimova, 2000[192]. Recommendation: 4 - Baulch et al., 1981[188]

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

129

In experiments[190,191] the C and N atom concentrations were measured behind the front of reflected shock waves by the ARAS method in highly diluted mixtures C2N2/Ar in the temperature range 4060-6060 K. The rate constants were determined from the initial slope of the atom concentration time history. In Table 17 the values k8 [188,190] are given. The rate constant was evaluated[192] in the temperature range 4000-10000 K. The radiation of molecules CN (λ=421.6 nm) and C2 (λ=516.5 nm), and of atoms C (λ=247.8 nm) was studied behind the front of shock waves in mixtures CO(CO2)/N2/Ar. At T=4000-6000 K the rate constant values obtained in gas mixtures CO:N2:Ar=1:2:7; 2:3:5 exceed the data[190,191] by 40%. With increase in the temperature the deceleration of the rate constant growth was observed and attributed to thermally non-equilibrium dissociation processes. In Fig.9 the data[192] were shown as black points. Table 17. Preferred rate constants: k = A ⋅ T n exp (− E a / T ), cm3·mole-1·s-1

Reactions CN+M→C+N+M

M T, 103K Ar 500012000

Ar NCO+M→CO+N+M Ar C2N2+M→CN+CN+M Ar Ar Ar

40606060 20003100 20004000 19002650 19003450

2.5·1014

n Ea, K ±Δlgk; % Ref. 188 0 75000 ±60% (5000K) ±30% (12000K) 190,191 0 71000 0.2

2.2·1014

0 27200

0.2

4

3.2·1016

0 47500

0.3

188

1.8·1017

0 53665

±30%

200

1.07·1034 - 65420 4.32

±30%

200

A 2·1014

Reaction NCO+M→CO+N+M (R9) Reaction R9 was studied in works[194-197]. The gas mixtures C2N2/N2O/O2/Ar at temperatures 2150-2400K were studied[195] in shock waves. The CN concentration was measured using absorption in band (0-0) of violet system (λ=388 nm). The coincidence of infrared line of the CO spectrum with line of CO-laser was used to monitor the CO concentration in absorption. The CO-laser was tuned to an NO(v=0) absorption line (Λ-doublet) for measuring NO concentration. The ratio of the rate constants of reactions R9 and NCO+O→CO+NO was determined in the study. The rate constant k9 was determined directly from the time history of the NCO molecules concentration[194]. The diagnostic was narrow-line of NCO at 440.479 nm using a

130

High Temperature Phenomena in Shock Waves

remotely located ring dye laser source. Recommendations for k9 [189,193] were based on the results[194,196]. However, the rate constant k9 [197] appeared to be considerably lower than in study[194]. A mixture HNCO diluted in argon was studied[197] and the same method was applied for determining the NCO concentration. The reason of such a discrepancy between the k9 values lies in large errors of reaction rate constants related to the NCO formation in work[194] that resulted in large errors in the determination of the rate constant for reaction R9. In study[197] new refined values of these coefficients were used. In addition, new values of the NCO absorption coefficient were obtained. The value of k9 [197] is given in Table 17 and in review[4] as the preferred value. In accordance with the estimations[189] made on the basis of the RRKM theory, dissociation of molecules NCO at pressures 10-300 atm and temperatures 10002500K was found to proceed in the low pressure limit. Reactions C2N2+Ar→CN+CN+Ar (R10), CN+CN+Ar→C2N2+Ar (R(-10))

Fig. 10. Dissociation rate constant of C2N2 Experimental data: 1 - Fueno et al., 1973[202]; 2 Colket, 1984[198]; 3 - Schekely et al., 1984[199]; 4 - Natarajan et al,. 1986[200]. Recommendation: 5 - Baulch et al., 1981[188]

The CN absorption was studied[198,199] at λ=388 nm in gas mixtures C2N2/Ar heated by shock wave. The ARAS method was applied[200] to measure the

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

131

concentration of O and H atoms, where the thermal decomposition of cyanogen was studied for temperatures 1900-2650 K behind reflected shock waves in mixtures C2N2/H2 and C2N2/O2 highly diluted in argon. Owing to fast secondary reaction of CN dissociation, the measured concentrations of O and H atoms strongly depended on the cyanogen decomposition rate. The measured rate constant k10 is presented in Table 17. One can see in Fig.10 that results[199-201] are close; therefore, the value k10 was determined in the entire temperature range 1900-3450 K[200]. In Table 17 the preferred value k10 [188] is also presented. It was obtained using the results of studies carried out up to 1975. Recombination reaction R(-10) was studied in a few works (see review[188]) but interpretations of the experiments were contradictory. The main reason was the impossibility to exclude the influence of accompanying reactions in the experiments. That is why, in review[188] it was proposed to derive the rate constant k −10 from k10 using

the

equilibrium

constant

on

the

temperature

k−10 = 3.8 ⋅ 10 exp(15000 / T ), cm ·mole ·s . 14

6

-2

range

2000-4000K:

-1

Reactions CN+O→CO+N(4S) (R11a), CN+O→CO+N(2D) (R11b) This reaction is very effective in the CN molecule removal[188] because already at

T=298 K the rate constant k11 = 10 , cm3mole-1c-1. Reaction R11 was studied in several works carried out at a temperature T=298 K and in shock-tube investigations. The time history of CN absorption was monitored at 388 nm[195] and at 388.44 nm (absorption line of laser line)[204]. The O and N atom concentrations were measured[203] using the ARAS method. Gas mixtures diluted in argon contained C2N2 with additions of O2[195], CO2[203] and N2O [204]. The rate constant k11 was determined by fitting the measured O- and N-atom concentrations at later reaction times[203]. In review[4] the preferred value k11 was based on the data obtained at T=298 K and the results[195,203,204] obtained at high temperatures (Table 18). In some studies it is assumed that reaction R11 proceeds mainly by channel (b) (about 80%). 13

Reaction CO+N→CN+O (R12) The rate constant k12 was measured in work[205], where CN absorption was studied in the band (0-0) of the violet system (λ=378.6-387.6 nm) behind the front of an incident shock wave in gas mixtures CO/N2/Ar=1/2/7 and 2/3/5. The time history of absorption near the shock front was shown to be related with the rate of the CN formation via reaction R12, since it was dominant for CN production in the experimental conditions. The rate constant k12 was obtained for temperatures 45007600K (Table 18).

132

High Temperature Phenomena in Shock Waves

Reactions NO+C→CO+N (R13a), NO+C→CN+O (R13b) Reaction R13 was studied[206] behind reflected shock waves for temperatures 15504050K in gas mixture C3O2/NO/Ar. The C atoms were formed by pyrolysis or photolysis of C3O2 (laser photolysis at λ=193 nm). Detection of C atoms was carried out using atomic resonance absorption (ARAS) at 156.1 nm. In the presence of an excess NO, the C atoms were rapidly removed by reaction R13. The branching ratios of the product channels (a) and (b) were determined by measuring the formation of product species, CN, N atoms and O atoms, using laser absorption or ARAS. It was obtained from computer simulation that the CN time history was very sensitive to the branching ratio of channels. The preferred value k13 [4] and its uncertainty were

presented (Table 18), together with the ratio of the rate constants of channels (a) and (b). Reactions N2+C→CN+N (R14), CN+N→N2+C (R(-14)) The rate constant k14 was determined in shock-tube experiments[201,207,208]. Gas mixtures C2N2/N2/Ar and CH4/N2/Ar with small content of C2H2 and CH4 (5-20 ppm), highly diluted in argon were studied in work[201]. The time histories of the C atom concentration were measured behind the shock front. Mixtures C2H6/N2/Ar and CH4/N2/Ar were used in study[208] where the C concentration was also measured. The C atoms were formed under C3O2 pyrolysis[207] and then they disappeared very quickly due to reaction R14 in excess of N2. In all the studies the C and N atom concentrations were measured by ARAS (λ=156.1 and 119.9 nm, respectively). Using mainly these data the authors4 proposed a preferred rate constant for temperatures 2000-5000 K (Table 18). The back reaction R(-14) was investigated in several studies[174,190,203,209,210]. The results[174,190,203,210] had large data scatter in the temperatures range 3000-8000 K (about 20 times), with no clear indication of the temperature dependence. This reaction was studied in mixture C2N2/NO/Ar behind reflected shock waves[209]. The N atom concentration was monitored by ARAS. The data[209] and the results obtained at low temperatures were used for deriving the recommended value4 for k−14 (Table 18). Reaction CN+C→C2+N (R15) Reaction R15 is very important for the C2 molecule formation in the high-temperature gas CO2/N2. Reaction R15, together with reaction R(-14), was investigated[190] as secondary reactions under CN dissociation. In mixture C2N2/Ar heated by shock wave, the N and C atom concentrations were measured by ARAS. The rate constant of CN dissociation was determined from the initial slope of the measured concentrations, whereas the rate constants of secondary reactions were obtained by fitting the experimental and simulated results. In Table 18 the value of k15 is

presented at temperatures 4060-6060 K. The data[174] obtained earlier in a shocktube study for higher temperatures 5000-8000 K exceed the rate constant[190] by a factor of 2.5 at T=6000 K.

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

133

Table 18. Preferred rate constants: k = A ⋅ T n exp (− E a / T ), cm3·mole-1·s-1*) The uncertainties are given for overall reaction (a)+(b).**) The uncertainties are given for overall reaction (a)+(b)+(c). Reactions CN+OĺCO+N(4S) (a) CN+OĺCO+N(2D) (b) CO+N→CN+O NO+CĺCO+N (a) NO+CĺCN+O (b)

ΔT, 103K 295-4500 295-4500 4500-7600 290-4050 290-4050

N2+C→CN+N CN+NĺN2+C CN+C→C2+N CN+O2→NCO+O (a) CN+O2→CO+NO (b) CN+O2→N+CO2 (c) C2+N2→CN+CN CN+NOĺNCO+N CN+CO2→NCO+CO CO2+N→CO+NO

A 3·1013 for k=ka+kb

n 0

Ea, K 200

±Δlgk; % 0.5*)

Ref.

1.5·1015 4.8Â1013 for k=ka+kb ka/k=0.6 kb/k=0.4

0 0

39300 0

205

2000-5000 300-3000

5.2·1013 5.9·1014

0 -0.4

22600 0

0.2 0.3 *) Δk a Δk b = = ±0.3 k k (T=1500-4050K) 0.15 0.3 (T=300K)

4

4060-6060 200-4500 200-4500 200-4500

5·1013 7.2·1012 for k=ka+kb+kc ka/k=1 (T>1000K) kb/k=0.25 (T=298K)

0 0

13000 -210

2900-3420 2480-3160 2510-3510 2510-3510

1.5·1013 9.6·1013 4·1014 8.6·1011

0 0 0 0

4

0.5 (T=3000K) 0.3 0.1 (T=200K)

**)

4

4

190 4

0.3 (T=4500K)**) 21000 21200 19200 1110

±50% ±50% ±40% -

216 210 203 203

Reactions CN+O2→NCO+O (R16a), CN+O2→CO+NO (R16b), CN+O2→N+CO2 (R16c) Reaction R16 was studied in many works in shock tubes and at low temperatures. In studies[211,212] carried out at low temperatures (T=13-760 K) it was clearly shown that the rate constant k16 increases with decrease in the temperature. In several

studies the ratio of reaction channels (a, b, c) was investigated. The channel (a) of reaction R16 was studied in the experiments[201,204,213] carried out in shock tubes, with the gas mixture C2N2/O2/Ar. In studies[201,204] the C and N atom concentrations were measured by ARAS. Using pulsed radiation from an ArF excimer laser at 193 nm, a fraction of the C2N2 was photolyzed to produce CN. The presence of N atoms in the high-temperature gas was explained by the fast decomposition of the NCO molecules (formed via reaction R16) into CO and N for T>2000 K. In experiments[213], mixtures C2N2/O2/Ar and BrCN/O2/Ar were used, and the concentrations of N and O were monitored by ARAS. The preferred rate constant k16 was obtained in review[4] (Table 18) but without regard for the results[201,204,213]. For temperatures 2000-3000 K, the preferred value and the rate constant[201] are close, and the difference between them and the rate constants[204,213] does not exceed 30%. The channel (a) of reaction R16 was

134

High Temperature Phenomena in Shock Waves

recommended[4] as the principal channel at T>1000 K. When the temperature decreases, the contribution of channel (b) increases. The channel (c) gives a slight contribution to the overall reaction R16. Reactions CN+CN→C2+N2 (R17), C2+N2→CN+CN (R(-17)) Reaction R17 was studied in works[214,215]. Gas mixtures C2N2/Ar and C2N2/N2 were used[214]. Absorption of CN and emission of C2 were detected behind the shock wave at temperatures not higher than 4800 K. A consequence of the observation of C2 radiation was the assumption that CN decomposition occurred mainly via reaction R17. The possibility of other scenarios via reactions R8 and R15 was also pointed out:

CN+M→C+N+M (R8), CN+C→C2+N (R15) Radiation of CN and C2 was detected[215] behind the shock front in mixture BrCN/Ar. The authors also assumed that reaction R17 was dominant and that the CN and C2 concentrations were directly characterized by this reaction. However, the experiments carried out in lean mixtures showed that the CN molecules could be removed via dissociation itself. Its role becomes more important when the temperature increases. Later, the authors of studies[174,215] performed at higher temperatures concluded that it is necessary to consider also reactions R4, R(-14) and N2 dissociation with CN decomposition: CN+N→N2+C (R(-14)), C2+M→C+C+M (R4), N2+M→N+N+M There are no direct measurements of the k17 values. In works[190,201] the rate constant

k17 was

obtained

by

computer

simulation

(T=4060-6060

K):

k17 = 6.3 ⋅ 10 , cm3mole-1s-1. 11

The backward reaction R(-17) was studied[216] behind the reflected shock waves using time dependent absorption and emission measurements. Mixtures of fullerene C60 highly diluted in argon were shock heated and used as C2 sources. Perturbation of the reaction system by addition of N2 results in changes of C2 absorption and causes strong CN emission. The C2 concentration was quantitatively monitored by ring dye laser absorption spectroscopy at ν=19355.6 cm-1. Simultaneously, the temporal behavior of spectrally-resolved light emitted from the shock-heated mixtures was recorded by an intensified CCD camera in the wavelength range 315-570 nm. By comparing the integrated C2 and CN emission signals for Δv=0 progressions, the CN concentration can be determined on the basis of the known C2 concentration and the relative line strengths. The experiments were performed in the temperature range 2896-3420 K. The evaluation of C2 and CN concentration measurements leads to the determination of the rate constant of the reaction R(-17) with an uncertainty of ±50% (Table 18). Reaction CN+NO→NCO+N (R18) Products of the reaction R18 were observed in shock-tube experiments[210]. Gas mixture C2N2/NO/Ar was heated by a shock wave for temperatures higher than 2000 K. Time histories of the O and N atom concentrations were detected by ARAS. The

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

135

measured atom concentrations were compared with computed values based on a kinetic mechanism of 12 reactions. Both atom concentrations were very sensitive to the rate of reaction R18. The rate constant k18 obtained in that study is presented in Table 18. Reaction CN+CO2 →NCO+CO (R19) Reaction R19 was studied[201.203] behind reflected shock waves for temperatures 2510-3510 K. Mixtures C2N2 and CO2 highly diluted in argon were used as initial reactants, in which C2N2 served as a thermal CN source. Its reaction with CO2 was followed by N- and O-atom concentration measurements monitored with ARAS. Because of the very fast decomposition of NCO into N and CO, the rate coefficient of the reaction CN+CO2 is very sensitive to the measured N-atom concentration. A rate coefficient k19 was determined by fitting the computed N-atom concentrations to the

measured N-atom profiles at early reaction times. This value of k19 is presented in Table 18. Reaction CO2+N→CO+NO (R20) Reaction R20 was studied in work[203] (see description of reaction R19). The rate coefficient k 20 was determined using the fitting procedure for the measured N- and

O- atom concentrations at later reaction times. The result is presented in Table 18. The backward reaction has not been studied but the value k 20 was obtained[198] by processing the experiments performed in a shock tube in the temperature range 400015000K, i.e. 1 ⋅ 103 exp(−21000 / T ) , cm3·mole-1·c-1.

5 Non-equilibrium Radiation Non-equilibrium radiation was observed for high velocity shock waves (V>9.5 km/s) and at low pressures (P1 ≤ 1 Torr) both in shock tubes and in full-scale experiments. It manifested itself as depletion of the population of excited states of atoms and molecules. In work[217] the results of a few experiments on ionization and air radiation behind an incident shock wave were analyzed. The gas at the end of a relaxation region was observed to reach a stationary state with constant parameters different from the thermodynamic ones. In this case, the radiation intensity of continuous spectra and atomic lines was lower than at equilibrium. This effect is attributable to low gas density and, correspondingly, to a lower rate of excited state population in collisions, as compared with the rate of spontaneous emission. The same effect was also observed for electron concentration[217], and attributed to the influence of depleted population of atomic excited states on the degree of ionization. This effect practically disappears at higher pressures. In Fig.11 the distribution of electron concentration Ne (curve 1) measured behind a shock front in air at

136

High Temperature Ph henomena in Shock Waves

V=9.2 km/s and P1=0.5 Torr T is presented. The values of Ne calculated using the method described in the sam me study (curve 2) and the equilibrium values of Ne (cuurve 3) are also presented. Thee radiation profile of an oxygen line (transition 5s - 5p, λ0=777.3 nm) presented in Fig.12 was obtained in an experiment at V=12 km/s and ditions, the equilibrium intensity of this line (not shownn in P1=0.2 Torr. In these cond the figure) is higher than the measured intensity by a factor of more than 40.

Fig. 11. The electron concen ntration Ne in air Fig. 12. Radiation intensity of oxygen line behind the shock front at V=9.2 km/s and (W/cm3·sr) λ0=777.3 nm in air behind the P1=0.5 Torr[217] shock front (V=12 km/s, P1=0.2 Torr[217]])

A similar effect of depleetion of the radiating state population of C2 molecules w was observed in study[218], an nd attributed to a weak efficiency of the electronic teerm excitation in collisions with h argon, as compared with that of its depopulation duee to spontaneous emission, rath her than to a low gas pressure. CN and C2 radiation w were observed[218] behind the front of shock wave in a gas mixture CO-N2 dilutedd in argon (90-99%). The radiaation intensity was measured in a stationary region behhind the front where the chemicaal equilibrium was already attained and the gas temperatture was thermodynamic (Teq). For F a gas mixture CO-N2 highly diluted in argon (95-999%), the radiation intensity in Sw wan band (0-0) of C2 was found to be considerably bellow the equilibrium intensity att temperature Teq. With increasing proportion of molecuular species in the mixture (fro om 1% to 10%), its intensity approached the equilibriium value. At the same time, th he CN radiation in violet band (0-1) corresponded to the equilibrium, independently y of the argon content. The temperature and the total gas density in the experiments were w almost constant: Teq=6950K, neq=1.8·1018 cm-3. For adequately interpretting the experimental data[218], the main conclusionss of reviews[219,220] were used. There concerned intramolecular radiationlless Ts) between mixed equal-energy vibronic states pertainning nonadiabatic transitions (NT to different electronic term ms of diatomic molecules. NTs represent a very effecttive mechanism of intramolecullar energy transfer in molecules CN, C2, CO, N2, N2+. T The coupling of rovibronic states s occurs because of intramolecular non-adiabatic

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

137

interaction in isolated molecules or in collisions of molecules with particles of the ambient gas. In both cases, the mixed states are generated in each adjacent term, and NTs take place between them. The authors of study[221] draw special attention to the collisional coupling of pairs of vibronic levels in neighboring electronic states and to the phenomenon of intersystem collisional transfer of excitation as a dominant mechanism controlling the distribution of vibronic level population under discharge conditions. In particular, it is found that there occur many pairs of vibronic levels in adjacent electronic states in which the pair partners attain essentially identical population through the agency of collisional coupling. Undoubtedly, this mechanism occurs also in gases heated by shock waves. The cross-sections of NTs depend exponentially on the energy gap ΔE between the initial and final states. For the cases in which the band origins of adjacent electronic states are below 500-600 cm-1 the accidental resonance among the individual rotational-vibrational-electronic levels in neighboring electronic states is highly probable such that the energy gaps may fall below the average kinetic energy of the molecules. Such resonance regions were named superchannels[221] of very large cross-section for coupling of a few favored rotational levels.

Fig. 13. Effective rate coefficient (cm3c-1) for quenching of the d3Πg(v′=0) versus summary concentration S of atoms N, C, and O[218]

The salient features and properties of NTs are presented in review[219]. The rate constants of these processes amount to 10-11 - 10-9 cm3·s-1. The large efficiency of NTs is characterized by the participation of a great number of vibronic levels of adjacent electronic states in energy transfer.

138

High Temperature Phenomena in Shock Waves

The depletion of population C2 (d3Πg, υ'=0) behind the shock front for gas mixtures highly diluted in argon was attributed, in study[222], to the existence of limiting stages in NTs between singlet and triplet states of molecule C2 under excitation of the d3Πg state from the ground one. In collisions with Ar, the limiting stages have small cross-sections, and, as a consequence, the population of electronic state (as well as radiation intensity in Swan band (0-0)) appeared to be lower than the equilibrium value in the stationary region behind the shock front. It was found[218] that the eff

effective rate coefficient k−1 for deactivation state C2(d3Πg, v′=0) depends on the total concentration S of atoms N, C, O in the stationary region behind the shock front dependent on the content of CO and N2 in the gas mixture. It strongly increases at S>1.5·1017 cm-3 (Fig.13) demonstrating the appearance of some additive processes of effective electronic state quenching. The same is true for the effective rate constant of excitation of the term C2(d3Πg). The mechanism of electronic excitation of diatomic molecules from ground electronic state, taking NTs into account in relaxation processes, was formulated for a gas heated by shock wave[222]. The proposed mechanism was based on the following conception: Excitation of higher energy levels of a molecule immediately behind the shock front occurs primarily owing to rotational-vibrational relaxation in ground electronic state in collisions with other particles of the gas. The excited electronic states are populated mainly because of collisional NTs in regions of close approach to neighboring electronic states including the ground state. Through the agency of NTs, the energy redistribution over rotational-vibrational-electronic states continues during vibrational relaxation in the ground state. Keeping in view the CN, N2, C2, O2, CO molecules and gas temperatures not higher than 10,000K, the vibrational relaxation time was assumed to be much greater than the times of translational and rotational relaxation, and to be much less than the time of chemical reactions, especially, reactions of dissociation. This assumption was confirmed by a comparative analysis of the relaxation times[222]. Thus, near the front of a shock wave, the populations of excited states reach its equilibrium values during the time of vibrational relaxation. At next moments, when chemical reactions start to proceed and that the gas temperature changes, the distribution over all internal degrees of freedom is in equilibrium at the local gas temperature. The proposed mechanism of electronic state excitation is mainly collisional in nature, and, for this reason, is universal. At the same time, the participation of other processes of electronic excitation is not excluded, for example, any resonance energyexchange processes and exothermic chemical reactions. This mechanism was used in processing the experimental data[223,224] where radiation of CN, C2 and atoms C was studied behind the front of a shock wave in gas mixtures CO/N2/Ar at temperatures T=4000-9500K. The kinetics of the CN, C2, C radiation in heated gas region behind the shock front was simulated by nonequilibrium chemical reactions with the assumption of equilibrium distribution over all internal degrees of freedom of molecules at the local gas temperature. The experimental time histories of radiation obtained in experiments were satisfactorily described by this mechanism (Fig.14).

Non-equilibriu um Kinetics behind Shock Waves. Experimental Aspects

139

Fig. 14. Experimental (black points) and simulated (solid curves) profiles of CN, C2 annd C radiation intensity (in relative units) behind a shock front in gas mixture 10% CO/20% N2/770% Ar at V=3.07 km/s, P1=8.1 Torrr[224]

The processes selectively y populating the vibrational levels of the excited termss of molecules may complicate a total picture of relaxation dependent on selecting souurce power. Such an example waas presented in study[225], where the CN radiation spectrra in bands of Δv=0 sequence (traansition B2Σ→X2Σ) were investigated behind a shock fronnt in Titan atmosphere (92% N2-3 3% CH4-5% Ar). The experiments were carried out for two cases: with an initial pressurre P1=1.5 Torr and a shock wave velocity V=5.56 km/s (ccase A) and P1=8.2 Torr, and V= =5.13 km/s (case B). An imaging spectroscopy technique w was used to supply simultaneou us wavelength-intensity-time information behind the shhock front. The observation of tim me-resolved spectra of the band of CN showed evidencee of non-Boltzmann distribution ns in the vibrational populations. In both cases, an overpopulation of the level v=6 was found. With distance from the shock front, this ore rapidly in case A. At the higher initial pressure (case B), overpopulation decreased mo the population for higher leveels (v=6 and higher) deviated from the Boltzmann distribution and this took place during th he entire recording. The studies of the CN spectra in CO2-N2 mixture[226] showed that vib brational relaxation of CN reached the Boltzmann distribuution very rapidly. The results obtaained in N2-CH4 mixture may be explained by the availabiility of any resonance energy-ch hange processes or even exothermic chemical reactions w with production of CN molecules mainly in highly vibrational levels of the excited state. In this e for the higher initial pressure in case B (that is, forr the case, the more pronounced effect greater rate of chemical reacttion) becomes comprehensible. The conditions in heaated gases corresponding to the entry into planettary atmospheres with velocitiess higher than 10 km/s and pressures lower than 1 Torr w were modeled[227-229]. The stu udies were carried out in a free-piston double-diaphraagm shock tube. A spatial variattion of air radiation spectra behind strong shock waves w was obtained by spatially resolv ved imaging spectroscopy. Two peaks of radiation behhind the shock front were observ ved[227] when detecting the profiles of integral radiationn in the range 300-445 nm. Th he radiation of molecular species (N2, N2+) gave a m main contribution to the first peak immediately after the shock front, whereas the atom mic me intense shortly after it. Spatial profiles of temperatuures line spectra (O+, N) becam were obtained[228,229] forr a shock velocity 11.9 km/s and an ambient pressure off 0.3 Torr ahead of the shock wave, w using pure nitrogen as a test gas. In Fig.15 the tiime histories of rotational, vibraational and electronic temperatures behind the shock frront are shown and compared with the calculated ones obtained using Park’s model[14].

140

High Temperature Phenomena in Shock Waves

a

b

Fig. 15 (a, b). Distributions of temperature (rotational (a) and vibrational (b)). Shock velocity is 11.9 km/s, ambient pressure is 0.3 Torr[228]

The estimated electronic excitation temperature of N was found to be approximately 4000K at a 5-mm distance behind the shock front and to increase very gradually with the distance from the shock. The vibrational temperature of N2 was close to this temperature, whereas that of N2+ was two times higher and quickly increased with the distance from the shock front. The rotational temperatures of N2 and N2+ were turned out to be below 6500 and 23000K within 3-mm distance from the shock front, respectively, that is, much lower than the translational temperatures simulated by two-temperature Park’s model. Thus, these results indicated that the region immediately behind the shock front was in strong non-equilibrium, even for the rotational mode (Fig.15). The rotational and vibrational temperatures obtained[228] with a higher initial pressure and a lower velocity (P1=2.1 Torr, V=8.14 km/s) showed good agreement in equilibrium region with simulations using the two-temperature model. In many studies, the shock-heated gas in the conditions of vehicle entry into the atmospheres of Mars (CO2-N2) and Titan (N2-CH4) was investigated[230-235]. The authors of these studies were faced with unsatisfactory modeling description of results of their experiments in conditions of low pressure and high shock speed (Fig.16). In studies[230,231] the necessity of developing more sophisticated models adequately describing collisional-radiative processes and excited state formation in nonequilibrium zones was noticed. Many questions remain on chemical reaction modeling, especially on dissociation and, more especially, on carbon dioxide dissociation. This is not a simple problem because reliable data on the rate coefficients of dissociation at temperatures higher than 10,000 K are lacking and experimental studies are difficult because of the strongly non-equilibrium nature of dissociation.

Non-equilibrium Kinetics behind Shock Waves. Experimental Aspects

141

Fig. 16. Comparison between experimental and numerical spectra[230]

With increase of the temperature, the chemical reaction rates strongly increase, and near the shock front all the processes take place simultaneously and proceed in the absence of equilibrium with respect to internal degrees of freedom. In addition, for molecules such as CN, C2, N2, N2+, rotational and vibrational relaxations are essentially non-adiabatic processes, especially in excited electronic states, and often, apparently, they are inseparable from relaxation over electronic states221. Thus, there exists a diversity of situations for non-equilibrium radiation in shock waves depending on molecule and atom properties as well as on the gas conditions which have an influence on population and deactivation of excited states, namely gas composition, pressure, temperature, and species concentrations, including electrons and chemically active species.

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144 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129.

High Temperature Phenomena in Shock Waves Moor, C.W.: J. Chem. Phys. 43, 2979 (1965) Shin, K.H.: Chem. Phys. Lett. 6, 494 (1970); J. Phys. Chem. 75, 1079 (1971) Bowman, C.T., Seery, D.J.: J. Chem. Phys. 50, 1904 (1969) Smiley, E.F., Winkler, E.H.: J. Chem. Phys. 22, 2018 (1954) Breshears, W.D., Bird, P.F.: J. Chem. Phys. 54, 2968 (1971) Borrell, P.M., Borrell, P., Gutteridge, R.: J. Chem. Soc., Farad. Trans. 2(71), 571 (1975) Bott, J.F., Cohen, N.: J. Chem. Phys. 58, 934 (1973) Bott, J.F., Cohen, N.: J. Chem. Phys. 53, 3698 (1971) Solomon, W.C., Blauer, J.A., Jaye, F.C., Hnat, J.G.: Int. J. Chem. Kinetics 3, 215 (1971) Blauer, J.A., Solomon, W.C., Owens, T.W.: Int. J. Chem. Kinetics 4, 293 (1972) Smiley, E.F., Winkler, E.H.: J. Chem. Phys. 22, 2018 (1954) Diebold, G.J., Santoro, R.J., Goldsmith, G.J.: J. Chem. Phys. 60, 4170 (1974) Diebold, G., Santoro, R., Goldsmith, G.: J. Chem. Phys. 62, 296 (1975) Breshears, W.D., Bird, P.F.: J. Chem. Phys. 51, 3660 (1969) Santoro, R.J., Diebold, G.J.: J. Chem. Phys. 69, 1787 (1978) Generalov, N.A., Kosinkin, B.D.: Sov. Phys. Doklady 12 (1967) Generalov, N.A., Maximenko, V.A.: Sov. Phys. Doklady 14 (1969) Generalov, N.A., Maximenko, V.A.: J. Exp. Theor. Phys. 58, 420 (1970) (in Russian) Taylor, R.L., Bitterman, S.: Rev. Mod. Phys. 41, 26 (1969) Zuev, A.P., Losev, S.A., Osipov, A.I., Starik, A.M.: Himicheskaja Physika 11, 4 (1992) (in Russian) Simpson, C.J.S.M., Bridgman, K.B., Chandler, T.R.D.: J. Chem. Phys. 49, 513 (1968) Simpson, C.J.S.M., Chandler, T.R.D., Strawson, A.C.: J. Chem. Phys. 51, 2214 (1969) Simpson, C.J.S.M.: Proc. Roy. Soc., London A317, 265 (1970) Zuev, A.P., Tkachenko, B.K.: Himicheskaja Physika 5, 1307 (1986) (in Russian) Eckstrom, D.J., Bershader, D.: J. Chem. Phys. 53, 2978 (1970) Eckstrom, D.J., Bershader, D.: J. Chem. Phys. 57, 632 (1972) Britan, A.B., Losev, S.A., Makarov, V.N., Pavlov, V.A., Shatalov, O.P.: Fluid Dynamics 2, 204 (1976) Achasov, O.V., Ragosin, D.S.: Preprint No. 16. Minsk. ITMO AN BSSR (1986) (in Russian) Sato, Y., Tsuchiya, S.:J. Phys. Soc., Japan 33, 1120 (1972) Simpson, C.J.S.M., Gait, P.D., Simmie, J.M.: Chem. Phys. Lett. 47, 133 (1977) Blauer, J.A., Nickerson, G.R.: AIAA - paper 74 (1974) Taine, J., Wichman-Jones, C.T., Simpson, C.J.S.M.: Chem. Phys. Lett. 115, 60 (1985) Zuev, A.P., Negodiaev, S.S.:In: Proc. Moscow Physical-Technical Institute, Moscow (1992) (in Russian) Buchwald, M.I., Bauer, S.H.: J. Phys. Chem. 76, 310 (1972) Rees, T., Bhangu, J.K.: J. Fluid Mech. 39, 601 (1969) Kamimoto, G., Matsui, H.: J. Chem. Phys. 53, 3990 (1970) Center, R.E.: J. Chem. Phys. 59, 3523 (1973) Saxena, S., Kiefer, J.H., Tranter, R.S.: J. Phys. Chem. A 111, 3884 (2007) Kung, R.T.V., Center, R.E.: J. Chem. Phys. 62, 2187 (1975) Zuev, A.P., Starikovskii, A.Y.: Khimicheskaja Physika 7, 1431 (1988) (in Russian) Zuev, A.P., Tkachenko, B.K.: Khimicheskaja Physika 9, 180 (1990) (in Russian) Zuev, A.P., Tkachenko, B.K.: Khimicheskaja Physika 9, 1427 (1990) (in Russian) Zuev, A.P., Starikovskii, A.Y.: Khimicheskaja Physika 9, 877 (1990) (in Russian) Losev, S.A.: Combustion, Explosion, Shock Waves 12, 141 (1976) Richards, L.W., Sigafoos, D.H.: J. Chem. Phys. 43, 492 (1965)

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130. Kiefer, J.H., Buzyna, L.L., Dib, A., Sundaram, S.: J. Chem. Phys. 113, 48 (2000) 131. Davis, M.J., Kiefer, J.H.: J. Chem. Phys. 116, 7814 (2002) 132. Ibraguimova, L.B., Bykova, N.G., Zabelinskii, I.E., Shatalov, O.P.: In: West-East High Speed Flow Field Conference, CD Proceedings, Section 2, 9, Moscow, Russia, November 19 - 22 (2007) 133. Zabelinskii, I.E., Ibraguimova, L.B., Shatalov, O.P.: Fluid Dynamics 45, 485 (2010) 134. Eremin, V., Shumova, V.V.: In: Abstracts 21th Symp. Rarefied Gas Dynamics, Marseille, vol.1, p. 306 (1998) 135. Eremin, A.V., Shumova, V.V., Ziborov, V.S., Roth, P.: In: 21st Int. Symp. Shock Waves, p. 2180 (1997) 136. Eremin, A.V., Ziborov, V.S.: Sov. J. Chem. Phys. 8, 475 (1989) 137. Eremin, A.V., Ziborov, V.S.: Shock waves 3, 11 (1993) 138. Eremin, A.V., Ziborov, V.S., Shumova, V.V.: Kinetics and Catalysis 38, 1 (1997) 139. Eremin, A.V., Zaslonko, I.S., Shumova, V.V.: Kinetics and Catalysis 37, 455 (1996) 140. Eremin, A.V., Roth, P., Woiki, D.: Shock Waves 6, 79 (1996) 141. Oehlschlaeger, M., Davidson, D.F., Jeffries, J.B., Hanson, R.K.: Z. Phys. Chem. 219, 555 (2005) 142. Davies, W.O.: J. Chem. Phys. 43, 2809 (1965) 143. Fujii, N., Sagawai, S., Sato, T., Nosaka, Y., Miyama, H.: J. Phys. Chem. 93, 5474 (1989) 144. Dean, A.M.: J. Chem. Phys. 58, 5202 (1973) 145. Davies, W.O.: J. Chem. Phys. 41, 1846 (1964) 146. Zabelinsky, E., Ibraguimova, L.B., Krivonosova, O.E., Shatalov, O.P.: Physical-chemical kinetics in gasdynamics, p. 126. Moscow State University, Moscow (1986) (in Russian) 147. Fishburne, E.S., Belwakesh, K.R., Edse, R.: J. Chem. Phys. 45, 160 (1966) 148. Burmeister, M., Roth, P.: AIAA J. 28, 402 (1990) 149. Hardy, W.A., Vasatko, H., Wagner, H.G., Zabel, F.: Ber. Bunsenges. Phys. Chem. 78, 76 (1974) 150. Ibraguimova, L.B.: Mathematical Modeling 12, 3 (2000) (in Russian) 151. Ibraguimova, L.B.: Sov. J. Chem. Phys. 9, 785 (1990) (in Russian) 152. Michel, K.W., Olschewsky, H.A., Richetering, H., Wagner, H.G.: Z. Physik, N.F. 39, 129 (1963); 44, 60 (1965) 153. Losev, S.A., Generalov, N.A., Maksimenko, V.A.: Doklady Akademii Nauk SSSR 150, 839 (1963) (in Russian) 154. Galaktionov, I.I., Korovkina, T.D.: Teplofizika vysokih temperatur 7, 1211 (1969) (in Russian) 155. Kiefer, J.H.: J. Chem. Phys. 61, 244 (1974) 156. Saxena, S., Kiefer, J.H., Tranter, R.S.: J. Phys. Chem. A 111, 3884 (2007) 157. Starikovsky, A.Y.:Doctoral Dissertation, Moscow Physico-Technical Institute, p. 307 (1991) (in Russian) 158. Meyer, I., Olschewsky, H.A., Schecke, W.G., et al.: Inst. Fur Physik. Chem. Universitat Gottingen, AD 706, S.898, (1970) (cited by3) 159. Dean, A.M., Steiner, D.C.: J. Chem. Phys. 66, 598 (1977) 160. Hardy, J.E., Gardiner, W.C., Burcat, A.: Int. J. Chem. Kinet. 10, 503 (1978) 161. Zaslonko, I.S.: Doctoral Dissertation, Moscow, Inst. of Chem. Phys., SSSR, p. 502 (1980) (in Russian) 162. Wagner, H.G.: In: Proc. 8th Int. Symp. Shock Waves, London, p. 4 (1971) 163. Pravilov, A.M., Smirnova, L.G.: Kinetika i Kataliz 22, 107 (1981) (in Russian) 164. Pravilov, A.M., Smirnova, L.G.: Kinetika i Kataliz 22, 559 (1981) (in Russian) 165. Wagner, H.G., Zabel, F.: Ber. Bunsenges. Phys. Chem. 78, 705 (1974)

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166. Troe, J.: In: Proc. 15th Int Symp. on Combustion, Pittsburgh, The Combust. Inst., p. 667 (1975) 167. Simonatis, R., Heicklen, J.: J. Chem. Phys. 56, 2004 (1972) 168. Chackerian Jr., C.: In: Proc. Eighth Intern. Shock Tube Symp., p. 40. Chapman Hall, London (1971) 169. Hanson, R.K.: J. Chem. Phys. 60, 4970 (1974) 170. Davies, W.O.: National Aeronautics and Space Administration, CR-58574 (1964) (cited by 3) 171. Fairbairn, A.R.: Proc. Roy. Soc. A312, 1509, 207 (1969) 172. Appleton, J.P., Steinberg, M., Liquornik, J.: J. Chem. Phys. 52, 2205 (1970) 173. Mick, H.J., Burmeister, M., Roth, P.: AIAA J. 31, 671 (1993) 174. Slack, M.W.: J. Chem. Phys. 64, 228 (1976) 175. Beck, W.H., Mackie, J.C.: J. Chem. Soc. Far. Tr.I. 71, 1363 (1975) 176. Kruze, T., Roth, P.: J. Phys. Chem. A 101, 2138 (1997) 177. Sulzmann, K.G.P., Myers, B.F., Bartle, E.R.: J. Chem. Phys. 42, 3969 (1965) 178. Roth, P., Thielen, K.: In: Proc. 14th Int. Symp. Shock Waves, Australia, p. 624 (1983) 179. Dean, A.M., Kistiakowsky, G.B.: J. Chem. Phys. 54, 1718 (1971) 180. Sulzmann, K.G.P., Leibowitz, L., Penner, S.:In: Proc. 13th Int. Symp. Combust., The Combust. Inst., p. 137 (1971) 181. Dean, A.M., Kistiakowsky, G.B.: J. Chem. Phys. 53, 830 (1970) 182. Ibraguimova, L.B.: Sov. J. Chem. Phys. 10, 456 (1992) 183. Bartle, E.R., Myers, B.F.: Amer. Chem. Soc., Divis. of Phys. Chem. Abstracts (1969); Abstract No. 152 184. Baber, S.C., Dean, A.M.: J. Chem. Phys. 60, 307 (1974) 185. Korovkina, T.D.: Khimiya vysokih energuii 10, 87 (1976) 186. Dean, A.J., Davidson, D.F., Hanson, R.K.: J. Phys. Chem. 95, 183 (1991) 187. Fairbairn, A.R.: J. Chem. Phys. 51, 972 (1969) 188. Baulch, D.L., Duxbury, J., Grant, S.J., Montague, D.C.: J. Phys. Chem. Ref. Data 10, 576 (1981) 189. Tsang, W.: J. Phys. Chem. Ref. Data 21, 753 (1992) 190. Mozzhukhin, E., Burmeister, M., Roth, P.: Ber. Bunsenges. Phys. Chem. 93, 70 (1989) 191. Mick, H.J., Roth, P.: In: 18th Int. Symp. Shock Waves, Book of Abstracts, Japan, Sendai, F31 (1991) 192. Ibraguimova, L.B., Smekhov, G.D., Dikovskaya, G.S.: Chem. Phys. Reports 19, 57 (2000) 193. Baulch, D.L., Cobos, C.J., Cox, R.A., et al.: J. Phys. Chem. Ref. Data 21, 411 (1992) 194. Louge, M.Y., Hanson, R.K.: Combust. Flame 58, 291 (1984) 195. Louge, M.Y., Hanson, R.K.: Int. J. Chem. Kin. 16, 231 (1984) 196. Higashihara, T., Saito, K., Murakami, I.: J. Phys. Chem. 87, 3707 (1983) 197. Mertens, J.D., Hanson, R.K.: In: International Symposium on Combustion, vol. 26, p. 551 (1996) 198. Colket III, M.B.: Int. J. Chem. Kinet. 16, 353 (1984) 199. Szekely, A., Hanson, R.K., Bowman, C.T.: J. Chem. Phys. 80, 4982 (1984) 200. Natarajan, K., Thielen, K., Hermans, H.D., Roth, P.: Ber. Bunsenges. Phys. Chem. 90, 533 (1986) 201. Burmeister, M.: Untersuchungen zur kinetic Homogener C-, CN-, und CH- radical reactionen bei hohen temperaturen, PhD Thesis, Universitat Duisburg, p. 170 (1991) 202. Fueno, T., Tabayashi, K., Kajimoto, O.: J. Phys. Chem. 77, 575 (1973) 203. Lindackers, D., Burmeister, M., Roth, P.: Combust. Flame, 81, 251 (1990)

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204. Davidson, D.F., Dean, A.J., DiRosa, M.D., Hanson, K.: Int. J. Chem. Kin. 23, 1035 (1991) 205. Ibraguimova, L.B., Kuznetsova, L.A.: Chem. Phys. Reports 23, 82 (2004) 206. Dean, A.J., Hanson, R.K., Bowman, C.T.: J. Phys. Chem. 95, 3180 (1991) 207. Dean, A.J., Hanson, R.K., Bowman, C.T.: In: International Symposium on Combustion, vol. 23, p. 259 (1990) 208. Lindeckers, D., Burmeister, M., Roth, P.:In: International Symposium on Combustion, vol. 23, p. 251 (1990) 209. Natarajan, K., Woiki, D., Roth, P.: Int. J. of Chem. Kinet. 29, 35 (1997) 210. Natarayan, K., Roth, P.: International Symposium on Combustion, vol. 21, p. 729 (1988) 211. Sims, J.R., Queffelec, J.L., Defrance, A., et al.: J. Chem. Phys. 100, 4229 (1994) 212. Sims, J.R., Smith, W.M.: Chem. Phys, Lett. 151, 481 (1988) 213. Burmeister, M., Gulati, S.K., Natarayan, K., et al.: In: Int. Symp. on Combustion, vol. 22, p. 1083 (1989) 214. Patterson, W.L., Green, E.F.: J. Chem. Phys. 36, 1146 (1962) 215. Faibairn, A.R.: J. Chem. Phys. 51, 972 (1969) 216. Sommer, T., Kruse, T., Roth, P., Hippler, H.: J. Phys. Chem. A 101, 3720 (1997) 217. Zaloguin, G.N., Lunev, V.V., Plastinin, Y.A.: Fluid Dynamics 15, 85 (1980) 218. Ibraguimova, L.B.: Zhurnal Prikladnoi Spectroskopii. J. of Applied Spectrosc. 28, 612 (1978) (in Russian) 219. Ibraguimova, L.B.: Chem. Phys. Reports 15, 939 (1996) 220. Dvoraynkin, A.N., Ibraguimova, L.B., Kulagin, Y.A., Shelepin, L.A.: Review of Plasma Chemistry, Consultants Bureau, NY, p. 1 (1991) 221. Benesch, W., Fraedrich, D.: J. Chem. Phys. 81, 5367 (1984) 222. Ibraguimova, L.B.: Chem Phys. Reports 15, 959 (1996) 223. Dushin, V.K., Ibraguimova, L.B.: Fluid Dynamics 16, 253 (1981) 224. Ibraguimova, L.B., Losev, S.A.: Kinetika i Kataliz 24, 263 (1983) (in Russian) 225. Ramjaun, D.H., Dumitrescu, M.P., Brun, R.: In: Proc. 21th Int. Symp. Rarefied Gas Dynamics, vol. 2, p. 361 (1999) 226. Dumitrescu, M.P., Ramjaun, D.H., Chaix, A., et al.: In: Proc. 20th Int. Symp. Shock Waves (1997) 227. Morioka, T., Sakurai, N., Maeno, K., Honma, H.: In: Proc. 21th Int. Symp. Rarefied Gas Dynamics, vol. 2, p. 345 (1999) 228. Fujita, K., Sato, S., Ebinuma, Y., et al.: In: Proc. 21th Int. Symp. Rarefied Gas Dynamics, vol. 2, p. 353 (1999) 229. Fujita, K., Sato, S., Abe, T.: J. of Thermophysics, Heat Transfer 16, 77 (2002) 230. Rond, C., Boubert, P., Felio, J.-M., Chikhaoui, A.: Chemical Physics 340, 93 (2007) 231. Boubert, P., Rond, C.: J. of Thermophysics, Heat Transfer 24, 40 (2010) 232. Grinstead, J.H., Wright, M.J., Bogdanov, D.W., Alen, G.A.: J. Thermophysics, Heat Transfer, 23, 249 (2009) 233. Lee, E., Park, C., Chang, K.: J. Thermophysics, Heat Transfer 21, 50 (2007) 234. Lee, E., Park, C., Chang, K.: J. Thermophysics, Heat Transfer 23, 226 (2009) 235. Brandis, A.M., Morgan, R.G., McIntyre, T.J., Jacobs, P.A.: J. Thermophysics, Heat Transfer 24, 291 (2010)

Chapter 4 Ionization Phenomena behind Shock Waves W.M. Huo1,*, M. Panesi2, and T.E. Magin3 1

2

NASA Ames Research Center, Moffett Field, U.S.A. Institute for Computational Engineering and Sciences, University of Texas, Austin U.S.A. 3 Aeronautics and Aerospace Department, Von Karman Institute Rhode-S-Genèse, Belgium

1 Introduction During the hypersonic entry of a space vehicle into a planetary or lunar atmosphere, the flow field becomes partially ionized. The percentage of ionization depends on the entry speed and the vehicle size. The electrons and atomic/molecular ions produced by the ionization introduce new reaction mechanisms that significantly influence the radiative and convective heat loads. Thus a simulation of the flow field behind the shock wave needs to incorporate the production of ionic species, chemical reactions due to charge-neutral and charge-charge interactions, and the removal of the charged species by recombination. In this regime, electron collision provides an efficient means of producing electronic excited states of the atoms and molecules in the flow field. Thus it plays an important role in determining the internal energy and state distribution of the gaseous particles. The excited states in turn are the source of radiation observed during a hypersonic entry. Electron-atom/molecule collisions differ from heavy particle (atom-atom, atom-molecule and molecule-molecule) collisions in two aspects. First, the mass of an electron is more than four orders of magnitude smaller than the reduced mass of N2. Thus its average speed, and hence its average collision frequency, is more than 100 times larger. Even in the slightly ionized regime with only 1% electrons, the frequency of electronatom/molecule collisions is equal to or larger than that of heavy particle collisions, an important consideration in the low density part of the atmosphere where the reaction probability is frequently controlled by the collision frequency. Second, the interaction potential between a charged particle (electron) and a neutral particle is longer range than neutral-neutral interactions. Hence electron-atom/molecule collision cross sections tend to be larger. A characteristic of electron collisions is that it generally produces a variety of excited states whereas heavy particle collisions tend to produce specific excited states. Also, low-energy electron collisions can be effective in spin changing excitations. Recombination of electron and ions removes the charge particles from the flow field. Radiative recombination produces continuum radiation extending from the VUV to the far infrared, providing a radiation source for photoionization downstream. Due to their heavier mass, ions are not as effective in generating excited states of atoms and molecules as the free electron, but charge transfer between ions and neural *

Ames Associate.

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atoms or molecules provides a means of generating new ionic species. Beside the radiation emitted upon recombination with electrons, excited states of ions also are a source of radiation from bound-bound transitions. Their spectra, however, are often imbedded under the more intense spectra of the neutral species. Modeling electron and ion collisions in nonequilibrium gas dynamics requires data to simulate their production and removal in the flow field. In the non-equilibrium regime their number densities, temperature, and reaction rate coefficients are part of the input data needed to determine the populations of the radiative species. Using a combination of experimental data and a variety of approximate formulas, a number of databases have been developed in the entry physics community. The data sets by Park[1,2], Losev[3] and Bird’s TCE[4,5] models are well-established examples of this approach. The NEQAIR package[6] that simulates nonequilibrium radiation in an entry flow employs Gryzinski’s classical formula [7] for electron-impact excitation of atoms whereas for molecules experimental data are used, sometimes by extrapolation or by analogy. More recent models have incorporated improved databases, based on new experimental data and/or theoretical calculations. The collisional radiative model by Bultel et al.[8] includes many updates. Similarly SPRADIAN07[9] incorporated new, improved data into the NEQAIR model. This chapter reviews the electron and ion collision processes relevant to hypersonic entry. Most of the examples given are for air. Since rare gases are also used in shock tube experiments, selected references to rare gas data are provided. In the ionized regime, molecules are mostly dissociated. Thus the discussions mainly concern the atomic species. Emphasis is on the collision data obtained either by using a quantum mechanical method or recent experimental data. A one-dimensional flow solver coupled to the gas kinetics based on a collisionalradiative (CR) model[8] and to the radiative transfer equation is used to illustrate how the ionization process influences the flow properties and its effect on the radiative and convective heat load. In the application considered here, where entry speeds exceed 9-10 km/s, the primary contributor to the radiative processes are atomic species (mainly nitrogen atoms), which account for about 90% of the overall radiation output. Note that a realistic representation of the ionization and radiative processes, occurring in shock heated air, can only be achieved through the explicit calculation of the population of the atomic electronic states using a state-to-state description of the gas kinetics, i.e., by treating the quantum states of atoms as separate pseudospecies[74,75]. Often in the literature the calculation of the radiation field is decoupled from the solution of the flow-field quantities (species densities, temperatures etc.) and escape factors are used in the flow equations to model the effects of the radiative processes on the population of the excited states[74,75,76,9]. The analysis of ionizing flow adopting this simplified treatment of radiation clearly shows the strong dependence of the result on the assumptions made when selecting the escape factors[74,75,77] (e.g. thin or thick assumptions). By substituting escape factors with the source terms resulting from the solution the radiative transfer equation, a fully consistent treatment of the radiation processes is employed in this chapter. The influence of the radiation processes on the population of the excited states as well as the cooling effects is thus correctly modeled. Modeling examples using this approach are presented using the conditions of FIRE II flight experiment and EAST shock tube facility.

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2 Production, Reaction and Removal of Charged Species 2.1 Electrons Due to the light electron mass, in general e-atom/molecule collisions should be treated using quantum mechanics instead of classical mechanics. The Schrödinger equation for an e-atom/molecule system is given by ( H − E ) Ψ (τ 1

τ N +1 , R 1

RM ) = 0

(1)

Here H is the Hamiltonian of the e + target system, where the target is an atom or molecule, E the total energy, Ψ the corresponding wave function, τi the spatial (ri) and spin (si) coordinates of the ith electron and RK the spatial coordinate of the Kth nucleus. The indices i and j label the bound electrons, and N+1 labels the free electron. Due the large difference between electron and nuclear mass, it is common to use the center of mass of the target as the center of mass of the colliding system. The total Hamiltonian consists of the target Hamiltonian, HA, the kinetic energy operator of the free electron, Te, and the Coulomb potential V between the free electron and the target. H = H A +T e + V , HA = −

N 1 N 2 1 M 2 ∇i − ∑ ∇ K + ∑ ∑ 2 i =1 2 K =1 j >i

N

∑r i =1

M N M M 1 ZK ZK ZL − ∑∑ +∑∑ − r R R − ri K =1 i =1 i L > K K =1 K L − RK j

1 2

T e = − ∇ 2N +1 N

V =∑ i

,

M 1 ZK −∑ rN +1 − ri K =1 rN +1 − R K

(2)

,

(3)

(4)

(5)

Here ZK is the charge of the Kth nucleus. The origin of the coordinate system is chosen to be at the center of the mass of the target. Note that HA in Eq. (3) is the nonrelativistic Hamiltonian of the target. Even for light atoms such as N and O and molecules consisting of light atoms, relativistic corrections are generally required to obtain accurate energy levels. The solution to Eq. (1) are obtained by first solving the target Schrödinger equation ( H A − E A ) Φ (τ 1

τ N , R1

RM ) = 0

(6)

The solution of Eq. (6) is then used as a building block for solving Eq. (1). In the range of incident electron energy of interest to hypersonic flow modeling, from threshold to ≈100 eV, the close coupling method[10] provides the most reliable solution of Eq. (1)

152

High Temperature Phenomena in Shock Waves

Ψ(τ 1

τ N +1,R1

∞

R M ) = ∑ A {f m (τ N +1 )Φ m (τ1 m =1

τ N ,R1

R M )}.

(7)

The antisymmetrizer A permutes the free electron with the bound electrons to account for the fact that electrons are indistinguishable and must satisfy Fermi statistics. The summation is over all possible states of the target, including the continuum. The summation in Eq. (7) becomes integration in the continuum region. The excitation/ionization cross section is obtained by analyzing the asymptotic behavior of the function fm. Because there are an infinite number of discrete and continuum target states, for practical reasons the summation is necessarily truncated. The success of a close coupling calculation depends on the quality of the target wave functions used and the number of terms included in the summation. Two of the most successful closed coupling approaches used in e-atom collisions are the convergent close-coupling (CCC)[11-13] method and the R-matrix with pseudo states (RMPS)[14, 15]. The CCC method expands target states using squareintegrable functions. The convergence of such representation is tested by successively increasing the size of the basis. While the CCC method has demonstrated excellent results, so far its application is limited to atoms/ions with one or two electrons outside a closed shell core, e.g., the alkali and alkaline earth atoms. The current version is not applicable to N, O, and their ions that are important in the Earth entry environment. The RMPS method is an extension of the R-matrix method[16]. In the R-matrix approach the scattering problem is separated into two regions. Inside the R-matrix hypersphere of electronic radius r = ae, the full N+1-electron Schrödinger equation is used with the proper boundary condition. Outside ae, the collisional system is regarded as consisting of a well separated atom and the scattering electron. Analogous of the close coupling method, the quality of the calculation depends on the number of terms used in the expansion in Eq. (7). Insufficient number of terms in the expansion may lead to pseudo resonances, i.e., a false trapping of the electron. The RMPS method corrects this problem by constructing target functions using both physical wave functions determined from the solution of Eq. (6) and pseudo states. The latter approximately represent the high-lying bound states and continuum states. This method is applicable to atoms/ions of arbitrary structure. A more recent version of the R-matrix method, the B-spline RMPS method, employs nonorthogonal one-electron orbitals that lead to a more compact calculation. Simulation of nonequilibrium gas dynamics requires a complete set of cross sections involving all possible initial and final states of the atom/molecule. The NIST database[17] lists 381 levels for N atom and many high-lying Rydberg states are known to be missing in this database. Thus a very large set of cross-section data are needed. As discussed in the above, modern quantum mechanical calculations can provide reliable cross section data but the accurate calculations are limited to a small set of low-lying states for a given atom, or applied only to a particular class of atoms. Therefore it is necessary to employ more approximate methods. At present the best practice is to use experimental or more accurate quantal treatment when available, and supplement the data set using more approximate treatments.

Ionization Phenomena behind Shock Waves

153

2.1.1 Production of Electrons In the neutral atmosphere immediately behind the shock, molecules are first dissociated to form atoms. The initial production of electrons is achieved by associative ionization of two atoms. In the Earth’s atmosphere, these reactions are N + O → NO + + e,

(8)

N + N →N2+ + e,

(9)

O+ O →O2+ + e.

(10)

In Eqs. (8) – (10), the internal states of the reactants and products are not specified. This notation implies that a number of state-to-state reactions of the same type are possible. For a specific state-to-state reaction, the internal states are explicitly written out. See, for example, Eqs. (15) – (17). This convention is used throughout the chapter. Reaction (8) has the lowest threshold among the three reactions and it dominates the initialization of electron production in air. At high entry velocity, as the electron number density increases electron-impact ionization becomes dominant. N + e → N + + 2e,

(11)

O+ e →O+ + 2e.

(12)

The VUV radiation emitted upstream can also photoionize the neutral species downstream.

N + hν →N + + e.

(13)

O + hν →O+ + e.

(14)

Precursor photoionization has been reported previously[18]. (a) Associative Ionization Associative ionization (AI) is a resonant process involving both electronic and nuclear motions. In the simplest case (called the direct process), the potential energy curve for the relative motion of the two atoms crosses with the potential curve of the ion. Simultaneously at the crossing region the electronic wave functions of the two atoms are at resonance with a compound state of the diatomic ion and an electron. Autoionization can occur, producing a free electron. Thus Eq. (8) can be more explicitly written as

N + O ⇔ NO** → NO+ + e.

(8a)

Figure 1, taken from Vejby-Christensen et al.[19], illustrates this process. The ground states of the atoms, N (4S) and O (3P), can follow the A′ 2Σ+ curve and cross the NO+ (X1Σ+) ground state at the right limb of its potential curve. Autoionization at the proximity of the crossing point produces the ground state NO+ and an electron.

154

High Temperature Ph henomena in Shock Waves

The potential curves for ex xcited states of the two atoms can also cross the ion currve. For example, N (2D) and O (3P) can follow the B 2Π, B′ 2Δ and L 2Π curves and crross the NO+ ground state. An indirect i process, involving first crossing with the potenntial curve of a Rydberg state of molecule that converges to the ion state, can also occurr. mental and theoretical studies are devoted to the inveerse The majority of experim process, dissociative recom mbination (DR). The AI rate coefficient is then obtaiined using detailed balancing. In I the past two decades, storage ring experiments hhave greatly advanced the knowlledge on DR, including those for atmospheric ions. VejjbyChristensen et al.[19] and Hellberg H et al.[20] reported the DR cross sections and the branching ratios for the production of N (4So) + O (3P), N (4So) + O (1D) and N (2Do) + O (3P) from v = 0 of NO+. NO+ (X 1Σ+ ,vv = 0) + e →N( 4 S o ) + O( 3 P) + 2.7eV, NO+ (X 1Σ+ ,vv = 0) + e →N( 4 S o ) + O(1 D) + 0.80eV, +

1 +

((15)

NO (X Σ ,vv = 0) + e →N( D ) + O( P) + 0.38eV . 2

o

3

Hellberg et al. found 95% of the product to be N (2Do) + O (3P), 5% N (4So) + O ( P), and the production of N (4So) + O (1D) is negligible. Employing the Multichannel Quantum Deefect method[21] and potential curves determined ussing R-matrix[22] calculations adjusted to match experiment, Motapon et al.[[23] calculated the rate coefficieents for DR covering v = 0 – 14 of NO+ that can be usedd in plasma modeling. It should d be pointed out the application of detailed balancingg to determine AI rate coefficiients requires the reactants and products to be uniquuely defined. The DR rate coeffiicients tabulated in Ref.[23], on the other hand, correspoond to a mixed product of N (4So) + O (3P) and N (2Do) + O (3P). Their branching ratioo for v > 0 of NO+ have not beeen reported. Thus additional data is required for a uniique determination of the AI ratee coefficients. 3

Fig. 1. Reaction pathways for the associative ionization of N + O. The solid curves are forr the neutral N + O system whereas the dotted curve is for NO+. This figure is taken from Ref. [199].

Ionization Phenomena behind Shock Waves

155

As seen in Eq. (15), the electron production via AI is a competition between the reaction of the more abundant N (4So) with O (3P), but with a smaller cross section and higher threshold kinetic energy, versus the less abundant N (2Do) but with a larger cross section and lower threshold kinetic energy. Since N (4So) is expected to be the major dissociation product of N2 immediately behind the shock, the AI rate in this region is relatively small. For N2+ Peterson et al.[24] measured the DR rate coefficients for electron energy 10 meV – 30 eV. They found the DR rate coefficients to be weakly dependent on the N2+ vibrational level. At zero electron energy and v = 0 level of the ion, they determined the branching ratio of the dissociation products N 2 + ( X 2Σ g + , v = 0) + e → N ( 4 S o ) + N ( 4 S o ) + 5.82eV , N 2 + ( X 2Σ g + , v = 0) + e → N ( 4 S o ) + N (2 D o ) + 3.44eV , N 2 + ( X 2Σ g + , v = 0) + e → N ( 4 S o ) + N ( 2 P o ) + 2.25eV ,

(16)

N 2 + ( X 2Σ g + , v = 0) + e → N ( 2 D o ) + N ( 2 D o ) + 1.06eV .

to be 0:0.37:0.11:0.52. Even with the weak dependence of the DR rate coefficient on v, a unique determination of AI rate coefficients from the DR data is still not possible because the branching ratio is determined only at a single electron energy. The DR of O2+ has been measured by Peverall et al.[25] at electron energy 1meV – 3eV. O2+ (X 2Πg+ ,v = 0) + e →O( 3P) + O( 3 P) + 6.65eV, O2+ (X 2Πg+ ,v = 0) + e →O( 3P) + O(1 D) + 4.99eV, O2+ (X 2Πg+ ,v = 0) + e →O(1 D) + O(1 D) + 3.02eV, +

+

(17)

O2 (X Πg ,v = 0) + e →O( P) + O( S) + 2.77eV, 2

3

1

O2+ (X 2Πg+ ,v = 0) + e →O(1D) + O(1S) + 0.80eV,

The branching ratio for the formation of O (3P) + O (3P), O (3P) + O (1D) and O ( D) + O (1D) is 0.20:0.45:0.30 at zero electron energy. The branching ratio of O (3P) + O (1S) is negligible and for O (1D) + O (1S) it is less than 0.06. 1

(b) Electron-Impact Ionization There are two mechanisms in electron-impact ionization. The first is direct ionization where the colliding electron directly detaches a bound electron from the atom or molecule, as shown in Eqs. (11) and (12). The second mechanism is via autoionization. It is an indirect process where the atom is first excited to a metastable bound electronic state that lies in the continuum. As example consider the N atom,

N + e →N * + e →N + + 2e.

(18)

156

High Temperature Phenomena in Shock Waves

The lowest state of N+ is the 2s22p2 3P state. The next state of the ion is the 2s22p2 D state that is 15,316.2 cm-1 above the ground 3P state. Thus the series of neutral atomic states with the configuration 2s22p2(1D)nl converging to the 2s22p2 1D ionization limit include states that lie above the 2s22p2 3P ion state. One state of N atom that lies above the first ionization limit is the 2s2p4 2D state. It is 3974.8 cm-1 above the ground state of N+. Since these states lie above the first ionization limit they are metastable states. It can decay by radiation to a neutral state that is below the first ionization limit, or it can autoionize and emitted an electron. In Fig. 2 the initial state 2s22p3 2Do of N atom is first excited to the metastable state 2s2p4 2D by electron impact. The metastable state subsequently emitted an electron and produces the 2s22p3 3P state of N+. 1

Fig. 2. Schematic diagram of the indirect ionization process

The total electron-impact ionization cross section is the sum of the contributions from direct ionization and autoionization. σ I = σ DI + σ Auto

(19)

The cross term between direct and autoionization has been neglected. Generally experimental measurements of electron impact ionization are available for the ground state of the atom or low-lying metastable states, but not high-lying excited states. Quantum theory, on the other hand, has a demonstrated record of successful calculations of the total ionization cross sections by electron impact[26] and can be applied to excited states as well as the ground state. Different methods are used to calculate direct ionization and autoionization cross sections. The improved Binary-Encounter-Dipole (iBED) model[27] expresses the direct ionization cross section in two terms. σ DI = σ BinaryEncounter + σ BornDipole

(20)

Ionization Phenomena behind Shock Waves

157

The Binary-Encounter cross section σBinaryEncounter describes the close collision between the free electron and bound electron and the dipole Born cross section accounts for the long-range interaction between the free electron and the target. The autoionization cross section through a specific metastable state m is given by the product of the electron-impact excitation cross section σim from the initial state i to the metastable state m and the ionization probability of the metastable state, PmI.

σ Auto = PmI σ im

(21)

The ionization probability is given by PmI =

k I ,m k I ,m + k R ,m

,

(22)

with kI,m and kR,m the ionization and radiative rate coefficients of state m. Generally, autoionization can occur through several metastable states. In that case the total autoionization cross section is expressed as a sum of these processes. While direct ionization applies to any state of the atom, autoionization is possible only if electron impact can excite the initial state to an autoionizing state. Figure 3 presents σI of the 4So and 2Do states of N atom. For 2Do the two components of σI, σDI and σAuto, are also presented. σDI is calculated using the iBED method[28] whereas σAuto with 2s2p4 2D as the intermediate is from the calculation of Kim and Desclaux[29]. For the 4So state, the autoionization probability is expected to be small. Thus the σI of 4So includes only σDI. Figure 3 shows σDI of 2Do larger than that of 4So, mainly due to the lower ionization threshold. Note σDI and σAuto of 2Do have significantly different electron energy dependence. While σDI consistently increases with energy between threshold to 50 eV, σAuto reaches a plateau and then remains almost constant with energy. Also, σAuto is consistently smaller than σDI except near the threshold. However, under most entry conditions only the high-energy tail of the electron energy distribution reaches the ionization threshold. Thus the larger σAuto near the threshold makes it an important path for ionization. Figure 4 presents the ionization rate coefficients for 10 states of the N atom with the outermost electron at n=2 and 3[30,31]. The states are 2s22p3 4So, 2s22p3 2Do, 2s22p3 2Po, 2s22p2(3P)3s 4P, 2s22p2(3P)3p 4Do, 2s22p2(3P)3p 4Po, 2s22p2(3P)3d 4F, 2s22p2(3P)3d 4D, 2s22p2(3P)3s 2P, 2s22p2(1D)3s 2D. It is seen that the rate coefficients separate into two groups. The three lowest states with the outermost electron at n=2 are more tightly bound and have smaller ionization rate coefficients than the seven states that has one electron in the n=3 shell. The difference is particularly striking at low electron temperatures. The large ionization rates of the upper states lends to the possibility that the upper states will first reach Saha equilibrium with the free electrons and ions before they reach Boltzmann equilibrium with the lower states.

158

High Temperature Phenomena in Shock Waves

Fig. 3. Ionization cross sections for the ground 4So and first excited 2Do state of N atom.

Fig. 4. Ionization rate coefficients for 10 states of N atom as a function of electron temperature.

Electron-impact ionization cross sections for the ground states of O and C atoms have been calculated by Kim and Desclaux[29]. Straub et al.[32] measured the partial and total ionization cross sections of Ar.

Ionization Phenomena behind Shock Waves

159

(c) Photoionization In analogy to electron-impact ionization, photoionization consists of direct ionization and autoionization,

N + hν →N + + e,

(23)

N + hν → N * → N + + e.

(24)

The TOPBase data from the Opacity Project[33] tabulates the photoionization cross sections for a large collection of atoms as a function of their electronic states and photon frequency, including the data for N, O, and C of interest to Earth entry. It should be noted, however, that the TOPBase tabulation does not provide information on the final ion state. Detailed discussions on photoionization are given in Chapter 5. 2.1.2 Reactions Involving Electrons Electron collisions provide an efficient means for electronic excitation/de-excitation of atoms and molecules. In the ionized regime, this is a major source of radiative species. In addition, an electron can be attached to an atom or molecule to form a negative ion and molecules can be vibrationally excited/de-excited and dissociated by electron-impact. (a) Electronic Excitation by Electron-Impact Modeling the distribution of electronic excited states of atoms in a plasma requires a complete set of electron-impact excitation cross sections. However, both experimental and theoretical cross-section data cover only a small number of initial and final states. Experimental data on e - N, O collisions are sparse. The review of Laher and Gilmore[34] on O atom covers data prior to 1990. Landolt-Börnstein[35] covers the experimental database up to 1990’s. Also, the International Atomic Energy Agency (IAEA) Atomic Molecular Data Services[36] includes electron collisions in their database. For molecules a recent review is by Brunger and Buckman[37]. Recent theoretical calculations using improved quantal treatments produce cross section data with accuracy comparable to experiment. These include the B-spline RMPS calculations of N38, O39, and Ar40. Figure 5, taken from Tayal and Zatsarinny[38], presents the electron-impact excitation cross section for the 4So – 2Do transition in the N atom. It compares the experimental data by Yang and Doering[41] with the B-spline RMPS calculation by Tayal and Zatsarinny using 24 spectroscopic bound and autoionizing states together with 15 pseudo states in the close coupling calculation. The pseudo states are determined by the requirement that the polarizability for the ground 2s22p3 configuration is approximately accounted for. Earlier R-matrix calculations[38,42,43,44] are also included in the comparison. The oscillatory structures in the older calculations are due to pseudo resonances. These pseudo resonances have been removed in the latest calculation[38] by increasing the size of the calculation and the use of the pseudo states. The sharp structures between 10 – 13 eV in the RMPS curve are real resonance structures due to the transient formations of the N- ion [38]. Theoretical calculations are in general agreement with experiment[41]. Unlike the close coupling approach, perturbation theory can be used to treat excitation to high-lying electronic states without the corresponding problem of

160

High Temperature Phenomena in Shock Waves

increasing the size of the calculation. The Born approximation is a first order perburbation treatment.[45] Due to the simplicity in its calculation, it is frequently used in the database for plasma modeling. However, while the Born approximation describes long-range interactions such as dipole and quadrupole interactions well, it does not account for electron exchange, the polarization of the target electrons by the free electron, and the distortion of the free electron by the target. These features are described by higher order terms in the perturbation series. Thus the Born approximation is generally applicable only at high electron energies. The energy regime of interest in modeling hypersonic flow is too low for it to be valid. The BE scaling method by Kim[46] incorporates the high order effects approximately into the Born cross section by the use of energy scaling. This method has been applied successfully in treating electron collisions with neutral atoms and ions[46,47]. It should be noted, however, that this method does not include resonance effects, that is, the enhancement of the collision cross section due to the formation of a transient compound state. In this approach resonances should be treated separately. The distorted wave approximation is another perturbation approach. Here the incoming and outgoing electrons are described by elastically scattered waves and their coupling is treated by the first order Born approximation. This approximation is employed by the HULLAC[104] code in the calculation of electron-impact excitation cross sections.

Fig. 5. Electron-impact excitation cross section (10-22 m2) for the 4So – 2Do transition in N atom as a function of electron energy (eV). Solid curve: Tayal and Zatsarinny[38] 39-state B-spline RMPS result; long-dashed curve: Tayal and Zatsarinny[38] 21-state R-matrix result; shortdashed curve: 8-state R-matrix calculation of Berrington et al.[42]; dashed-dotted curve: 7-state R-matrix calculation of Ramsbottom and Bell[43]; dotted curve: 11-state R-matrix calculation of Tayal and Beatty[44]; diamonds: experimental cross section of Yang and Doering[41]. This figure is from Ref. [38].

Ionization Phenomena behind Shock Waves

161

Figure 6 presents the rate coefficients for the electron-impact excitation of 4So – 2s 2p2(3P)3s 4P of N atom. The three theoretical curves are calculated using the B-spline RMPS method[38], a 33-state R-matrix method[48] and the BE scaling method[31]. The experimental curve labeled as Stone and Zipf are calculated from their cross-section data[49] as recalibrated by Doering and Goembel[50]. The crosssection measurement of Doering and Goembel[50] does not cover the region below 30 eV electron energy and hence cannot be used to deduce rate coefficients. Figure 6 also include a single data point from the arc chamber measurement of Frost et al.[48] at 52,220 K. The B-spline RMPS and BE scaling curves are in good agreement whereas the R-matrix curve of Frost et al. is consistently larger. The experimental curve from Stone and Zipf are also higher than the B-spline RMPS and BE scaling curves, where the arc chamber measurement of Frost et al. is lower. The arc chamber data at higher electron temperatures, not shown in Fig. 6, are also lower than the B-spline RMPS/BE scaling curves. The above examples indicate that the optimal approach to build an electron-impact excitation data set is to use a combination of quantal methods with guidance from available experimental data. 2

(b) Vibrational Excitation by Electron-Impact Since the ionization potentials of air molecules are higher than their dissociation energies, electron production in the shock layer occurs after molecular dissociation has initiated. Thus e-molecule collisions do not play as important a role as e-atom collisions. Note, however, there is a significant resonance enhancement in the vibration excitation cross sections of N2 and O2 by electron impact[51,37]. For N2 the resonance enhancement extends to high vibrational levels[52,53]. As a result of this resonance, the e + N2 vibrational excitation rate coefficients are two orders of magnitude larger than the corresponding N + N2 vibrational excitation rates. Figures 7 and 8 compare the two sets of rate coefficients at initial v = 2 and 10, respectively, and both translational temperature (T) and electronic temperature (Te) at 10,000 K. The N + N2 vibrational excitation rate coefficient[54] is obtained from a weighed sum of rovibrational rate coefficients, K v →v ′ (T) = ∑ PvJ (T)K vJ →v ′J ′ (T), JJ ′

(25)

with PvJ(T) the statistical weight of the initial (v, J) level, KvJ→v’J’(T) the rovibration excitation rate coefficient and Kv→v’(T) the vibrational excitation rate coefficient. The rotational temperature is assumed to be the same as the translational temperature. The electron-impact vibrational excitation rate coefficient is calculated at J = 50[55] (26) The magnitude of this e + N2 vibrational excitation rate coefficient implies the e-N2 vibrational excitation/de-excitation may play a role in determining electron temperature in the nonequilibrium regime in the shock layer. The e-N2 vibrational rate coefficient have been tabulated by Huo et al.[55] up to v = 12 and J = 50.

162

High Temperature Phenomena in Shock Waves

Fig. 6. Electron-impact excitation rate coefficient for the 4So – 2s22p2(3P)3s 4P transition in N atom as a function of electron temperature. Theoretical curves are from the B-spline RMPS calculation of Tayal and Zatsarinny[38], 33-state R-matrix calculation of Frost et al.[48] and the BE-scaling method[31]. The experimental curve of Stone and Zipf[49] are calculated from their cross-section data as recalibrated by Doering and Goembel[50]. A single data point from the arc chamber measurement by Frost et al.[48] is also included.

1010 v = 2, T/T = 10,000K

108

3

Rate coefficient (m mole

-1

-1

s )

e

10

9

107 106 105

N+N 2 e+N 2

104

0

5

v'

10

15

Fig. 7. Comparison of e + N2 and N + N2 vibrational excitation rate coefficients at initial v = 2 and T = Te = 10,000 K. The e+ N2 data are from Ref. [55] and the N + N2 data are from Ref. [54].

Ionization Phenomena behind Shock Waves

163

Fig. 8. Comparison of e + N2 and N + N2 vibrational excitation rate coefficients at initial v = 10 and T = Te = 10,000 K. The e+ N2 data are from Ref. [55] and the N + N2 data are from Ref. [54].

(c) Electron-Impact Dissociation Electron-impact excitation of a molecule to a dissociative state leads to dissociation. The process is written as N 2 + e → N 2* + e → N + N + e.

(27)

N2*

Here denotes an electronic excited state of N2 that is dissociative. Cosby[56,57] measured the electron-impact dissociation cross sections of N2 and O2. His paper also reviewed older data for this process. (d) Electron Attachment For atom and molecules with positive electron affinity, electron collision can lead to the formation of a stable ion with the release of the excess energy. For atoms the energy is released by emitting a photon. The electron attachment of O atom is written as O + e →O− + hν .

(28)

The reverse process, photo detachment, has been studied both experimentally [58,59,60] and theoretically[61,62,63,64]. The cross section for electron attachment can be deduced using microscopic reversibility. For molecules the excess energy in the attachment can be transformed into vibrational energy. For example, the Bloch-Bradbury mechanism for O2¯ production

164

High Temperature Phenomena in Shock Waves

first forms a vibrational excited ion O2¯ *. The vibrational excitation is subsequently quenched by a second collision. O2 + e →O2−*, O2−* + M →O2− + M.

(29)

Electron attachment of O2 has been reviewed by Hatano and Shimamori[65]. The nitrogen species, N and N2, have negative electron affinity and no stable negative ions exist. Using a high-resolution electron beam experiment of the dissociative attachment of N2, Mazeau et al.[66] determined that the ground (3P) state of N¯ is located at 0.07 ± 0.02 eV above the ground (4So) state of N atom. Theoretical calculations of Thomas and Nesbet[67] give N¯ (3P) 0.1 eV above N (4So). Thus unlike the O atom, electron attachment of N and the reverse reaction, photo detachment of N¯ , are not possible. Instead, the presence of N¯ is seen as resonance structures in e - N collision cross sections. Similarly, the ground 2Πg state of N2¯ is seen as prominent resonance structures in the elastic and vibrational excitation cross sections in e - N2 collisions between 2 – 5 eV electron energy. See discussion in Sec. 2.1.2b. The e - N2 data have been reviewed by Brunger and Buckman[37]. 2.1.3 Electron Recombination Recombination of electrons and ion is the reverse of the ionization process discussed in Sec. 2.1.1. Thus radiative recombination is the reverse of direct photoionization in Eq.(23) and dielectronic recombination is the reverse of autoionization in Eq. (24). Both processes remove the charge species through the formation of a neutral atom and emit a photon. They can produce significant radiative heat load during the vehicle entry and detailed discussions are given in Chapter 5. It is worth noting that the TOPBase data[33] currently available do not distinguish transitions to different final states. This prevents the use of microscopic reversibility to deduce the recombination cross sections from the photoionization cross sections. The AMDPP (Atomic and Molecular Diagnostic Processes in Plasmas) database[68], on the other hand, provides state-to-state partial radiative recombination rate coefficients and partial dielectronic recombination rate coefficients that may be used for modeling purposes[28,30,31]. Dissociative recombination is an important recombination pathway for molecular ions. This process is discussed in Sec. 2.1.1. 2.1.4 Interparticle Interactions and Free Electron Number Density The atomic and molecular data discussed so far are either calculated for an isolated atom/molecule or from measurements extrapolated to zero pressure. The presence of neighboring atoms or molecules in a real plasma means the data must be modified to account for the effect of interparticle interactions. Griem[69] uses Debye shielding to describe the influence of all the charged species surrounding an atom and deduces an approximate expression for the lowering of ionization potential of the atom in a plasma. However, neutral-neutral interactions are ignored. A more general approach is the occupation probability formalism of Hummer and Mihalas[70] where the probability of finding a particle occupying states i is calculated directly from a physical description of interparticle interactions. This approach has been applied to astrophysical plasma with some success, but has not yet been applied to entry plasma.

Ionization Phenomena behind Shock Waves

165

2.2 Ions

The production of ions and electrons are in pairs. Thus the discussion in Sec. 2.1.1 for electron production also applies to ion production. Similarly the removal of ions by recombination is also paired with electrons and Sec. 2.1.3 also applies to ion removal. However, the reactions of ions with neutral particles are different from electrons. The heavier ion mass means the collision frequency between ions and neutral are two orders of magnitude smaller than electron-neutral collision frequency. The heavier mass of the ion also mean that ion-atom/molecule collisions may be amenable to semi-classical or quasi-classical calculations. Charge transfer between ion and neutral does not have an analogy in electron collisions. In nitrogen discharges with collision energy under 10 eV, fast N atoms are produced mainly by the charge exchange mechanism[71]. Freysinger et al.[72] reported measurements of the charge transfer reaction, N + + N 2 →N + N 2+ ,

(30)

from thermal to 100 eV. Excited states of ions can be produced by electron-impact and bound-bound transitions from these excited states provide another source of radiation. The R-matrix method, RMPS, and the BE-scaling method have all been applied to e-ion collisions.

3 Modeling Collisional and Radiative Processes in a Weakly Ionized Plasma 3.1 The Collisional-Radiative Model

The modeling of the kinetic and radiative processes and the analysis of available experimental and in-flight data put forward the importance of ionization processes in shocked heated air[76]. The radiative and convective heat loads generated during Earth entry are also shown to be strongly dependent on the degree of ionization in the gas. In ionizing air, the formation of the first electrons is due to the association of N and O atoms into NO+. This is the first step in a two-step ionization process. It is favored by the relatively low activation energy[73]. Also the reaction does not require the presence of charged particles. Thus it is well suited as the initialization step. In a second step, when the number of electrons is sufficiently large, the high-speed electrons ionize the neutral atoms, rapidly increasing the electron density. The accurate modeling of this two-step process requires the correct modeling of the thermo-chemical relaxation leading to the formation of the atomic species (e.g. dissociation), as well as a detailed modeling of the excitation and ionization of the atomic species, contributing to the production of electrons in the hypersonic regimes. One important aspect in the present discussion is the explicit coupling of the kinetic equations with radiation by solving of the radiative transfer equation, instead of employing an escape factor to approximately account for the effect of the radiative processes on the on the population of the excited states. This results in a consistent modeling of the radiative processes.

166

High Temperature Phenomena in Shock Waves

Two important issues for re-entry applications into the earth’s atmosphere are considered in this section: (1) the treatment of the time-dependent chemistry of an ionizing air plasma for different re-entry conditions; (2) the importance of a consistent treatment of kinetics and radiation, via the numerical solution of the radiative transport equation coupled with the kinetic set of equations. 3.1.1 Transport Equations In this section, consideration is given to the governing equations used to describe the dynamics of multi-component, multi-temperature, chemically reacting, relaxing and emitting flows. The detailed derivation of the equations used in this work is omitted, and instead the discussion is limited to a brief analysis of the fundamental assumptions underlying the mathematical construct. Further information concerning the derivation can be found in Refs.[78,79,80]. The air mixture used in this work comprises 95 chemical components, including the electronic energy levels of atomic nitrogen and oxygen. The populations of the vibrational energy levels of the molecules (N2, NO, O2, N2+, NO+, O2+) are assumed to follow Boltzmann distributions and share a common vibrational temperature TV. The rotational energy level populations are assumed to follow Boltzmann distributions at the translational temperature T of the gas. The CR model provides the electronic state populations of the N and O atoms. (a) Thermodynamics In this study, air is considered as a mixture of nitrogen and oxygen and their products. It is composed of neutral species (N2, O2, NO, N(1-46), and O(1-40)) and charged species (N2+ , O2+ , NO+, N+, O+, and e+). Forty-six electronic energy levels for N and 40 levels for O are employed[74,75]. The levels used are a combination of physically real states and lumped states, obtained by averaging energies and by summing the statistical weights of the states that are lumped together. The final reduced atomic model obtained allows one to accurately calculate: 1) ionization of the N and O atoms by electron impact and 2) the net population of the excited states resulting from the collisional and radiative processes. Furthermore, the coupling of the atom electronic energy levels through the different elementary processes considered in the following section allows for explicit determination of their excitation and consequently the radiative signature of the plasma without using any assumption on their populations a priori. The number of electronic levels used to compute the energy of the ions and molecules is tuned to yield the best matching agreement between values of the computed energies and the reference tables of Gurvich et al.[81]. Molecular energy is computed assuming the rigid rotor and harmonic oscillator approximations. Spectroscopic constants are taken from Ref.[81]. Electronic-specific data have been used for the vibrational and rotational constants of the molecules. In general, such simplified thermodynamic models for the rotation and vibration of molecules are not good approximations in high-temperature flow conditions. However, in this work the fraction of bound molecules is very low; thus our results are relatively insensitive to the molecular model chosen. Although the negative ions (e.g. O-2 and O-) can also be formed, their contribution to chemistry can be considered to be negligible, as a result of the high temperatures

Ionization Phenomena behind Shock Waves

167

reached behind the shock wave, and due to the high rate coefficient for detachment processes. On the other hand, these processes have to be accounted for, when considering radiation processes, due to the formation of background continuum radiation, which often characterizes the photo-detachment process. (b) Shock Tube Flow Solver: Mass, Momentum and Energy Equations We have developed a one-dimensional flow solver, SHOCKING to simulate air plasmas obtained in shock-tube facilities, based on the model presented in Ref.[82]. This model has been modified to simulate re-entries at speeds higher than 10 km/s. First, a radiative source term QRad has been added in the equation that expresses conservation of the total energy. The inclusion of this term is important as radiative transitions tend to deplete/replenish the flow energy for an optical thin/thick medium. Second, a separate source term in the species continuity equation has been added to account for the effect of radiation on the population of electronic states. Post-shock conditions are derived from the jump relations (Ranking-Hugoniot equations) assuming frozen-gas composition and vibrational and electronic energy modes, and the rotational mode is in equilibrium with the translational mode. It is important to stress that the Ranking-Hugoniot equations tend to overestimate the jump of the flow quantities across the shock as they do not account for the dissipative effects, which are not negligible within the shock, owing to the strong gradients. A better approximation of the physical phenomena can be obtained by using the shock slip conditions as suggested in Refs.[76,78]. The downstream flow-field is determined by solving a set of continuity equations for each chemical component, including its electronic structure in the case of the atoms, complemented by the Euler system of equations, namely mass, momentum and energy conservation, which allow one to retrieve the remaining characteristic flow quantities, such as pressure, temperature and flow speed. Also, the relaxation of the vibrational and free-electron energy is modeled using separate conservation equations accounting for the energy exchanges with the other modes (i.e. translation) and chemistry. Finally, the characterization of the radiative field is modeled solving the radiative transfer equation, discussed in details in Section 3.1.3. The physicalmathematical structure of the model is summarized as follows:

• Euler equations: conservation of mass for species i, momentum and total energy

⎛ ρiu ∂ ⎜ 2 ⎜ ρu + ∂x ⎜ ρ uH ⎝

⎛ ⎞ ⎞ ⎜ miωi + miωir ⎟ ⎟ ⎟ ⎜ p⎟ = ⎜ 0 ⎟ ⎟ ⎜ ⎟ ∂ ⎠ ⎜ − Qrad ⎟ ⎝ ⎠ ∂x

(31)

• MultiT models: additional energy conservation eqs., e.g. vibrational energy for the mth molecule in the mixture eq.

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High Temperature Phenomena in Shock Waves

∂ ∂x

( ρ ue ) = Ω

m

(32)

m

• Radiation transport models:

μ ∂I λ κ λ ∂x

+ Iλ =

ηλ κλ

( x) = Sλ ( x) ,

(33)

where i stands for the set of indices of the mixture species (including species and pseudo-species), ρi the mass density of the ith species; mi the mass of the species;

ωi and ω ir the mass production source terms due to chemical and radiative processes, respectively. The velocity is indicated with u, while p stands for the static pressure and H stands for the total enthalpy and accounts for the kinetic and mixture enthalpy. The radiation losses resulting from the emission of radiation are indicated with Qrad In Eq.(32), em refers to the energy of the internal energy modes. In the particular case, the kinetic energy of the free electrons and all the internal energy modes with the exception of the rotational structure and the internal energy of the atomic species are included in em . The energy exchanged by the lumped energy modes em with chemistry and the translational energy modes are indicated with Ω m . In Eq.(33), I λ is the spectral intensity, μ is the cosine of the angle

ϑ between

ˆ and the axis x, κ indicates spectral absorption coefficient and η the direction Ω λ λ the spectral emission coefficients. The ratio between emission and absorption coefficients is indicated with Sλ (x) and it is referred to as source function.

(c) Internal Energy Modes of Gaseous Particles For atoms the only internal energy mode is the electronic energy. Molecules have three internal energy modes: electronic, vibrational and rotational energies. 3.1.2 Reaction Source Terms The electronic energy relaxation is accounted for by solving electronic master equations based on the kinetic processes previously discussed. In particular, the rate for the production of an atom s, at the electronic level i, via excitation or ionization by electron-impact or heavy-particle impact, can be written as ⎡

⎤

ωi = ∑ k eji N s j N e +∑ k lji N s j N l +N e Ni + ⎢ β i e ,b N e + ∑ β il ,b Nl + α iRRκ iRR + α iDRκ iDR ⎥ j∈A

j∈A l∈H

⎢ ⎣⎢

j∈A l∈H

⎡ ⎤ − N s i ⎢∑ kije N e + ∑ kijl N l + βi e , f N e + ∑ βi l , f Nl ⎥ ⎢ j∈A ⎥ j∈A j∈A l∈H l∈H ⎣⎢ ⎦⎥

⎥ ⎦⎥

(34)

Ionization Phenomena behind Shock Waves

169

where the symbol A stands for the set of indices for the electronic energy levels of the N and O atoms. The set of indices for companion electronic levels is denoted by Ai. This set is N for an electronic level of the nitrogen atom and O for oxygen, with A = N ∪ O . Symbol H stands for the set of indices for the heavy particles. Symbol Ni stands for the molar number density of the species or pseudo species “i”. The reaction rate coefficient for excitation “kei,j”, ionization “β fi”, and recombination “βbi” depends on the collision partner: electron-impact interactions are denoted by the superscript e, and heavy-particle impact interactions are denoted by the superscript l. For radiative and dielectronic recombinations, the recombination rate coefficients are given by κ iDR and κ iRR , respectively. For these two recombination processes, an escape factor α i is introduced in Eq. (34) because the corresponding photoionization processes are not included in our radiative transport model. Here α iDR is the escape factor for dielectronic recombination, and α iRR for radiative recombination. Note that escape factors are only used for these two processes. All other kinetic processes are properly coupled with radiation. (a) Kinetic Processes Included in the Model The closure of the complex non-linear set of equations previously discussed requires the knowledge of rate parameters governing the dynamics of the particles and photons. Recently, Bultel et al.[8] compiled an electronic specific kinetic mechanism for air, which was applied to the study of compressing and expanding flow situations using a 0D model. In Bultel’s model, also referred to as ABBA model, 13 species in their ground state and numerous electronic excited states were taken into account. Although similar models have already been proposed by Teulet et al.[83,84] and Sarrette et al.[85], none are valid for pressures between 1 kPa and the atmospheric pressure. Furthermore, numerous recent experiments and ab-initio calculations have been carried out to improve the accuracy of excitation cross sections by electron impact, dissociative recombination rates of NO+, O2+ and N2+ and their branching fractions as well as vibrational processes. The more recent data are incorporated in our database. Atomic Processes The inelastic collisions between the species lead to chemical changes. The N and O atoms are efficiently excited and ionized by electron-impact reactions. Due to their small mass and the long-range charge-neutral interaction potential, free electrons are effective in exciting the atomic electrons to a higher state and they provide the major source of excited populations of the atoms. A number of models exist for the related cross sections and rate coefficients. Whenever possible, our calculation makes use of recent data from ab initio calculations. For N atom, the excitation rate coefficients from the R-matrix calculation of Frost et al.[48] cover the transitions from the ground and first two metastable states to the first twenty levels, and have been included in the present model. Furthermore, the cross sections from the RMPS calculation by Tayal et al.[38] have been tabulated and included in the model as well. When the two sources provide different data for the same transitions, Tayal’s rate coefficients have been preferred to the data of Frost. Finally, all the data provided by Huo[30,31] have

170

High Temperature Phenomena in Shock Waves

been used to model the electron-induced ionization and excitation of atomic species. Additional calculations have been performed using the BE scaling method for transition among the following states of atomic nitrogen: 4S0(2s2.2p3)4Pe(2s2.2p2.3s); 4Pe(2s2.2p2.3s)-4D0(2s2.2p2.3p); 2Pe(2s2.2p2.3s)-2S0(2s2.2p2.3p); 4Pe(2s2.2p4)-4D0(2s2.2p2.3p). Drawin’s expressions [86, 87] of electron-impact excitation and ionization, based on a simplified model originally developed for atom-atom collisional excitations, provide an efficient method for this type of calculations and thus are adopted for excitation and ionization involving higher states. The corresponding rate coefficients have been expressed in analytical form obtained through the integration of Drawin’s cross-sections over a Maxwell-Boltzmann distribution at the electron Te. For an electronic transition from i to j level, where j > i, the rate coefficient kf is a function of the secondary quantum number l of each level involved. For an optically allowed transition (li ≠ lj): 2

⎛ E ⎞ k = 4π v a α ⎜ H ⎟ Σ1 ( ε ) ⎝ kBTe ⎠ e ij

2 e 0

(35)

where quantity ve = ⎡⎣8RTe / (π me ) ⎤⎦ is the electron thermal speed, R is the universal gas constant, me is the electron molar mass, a0 is the first Bohr radius, EH = 13.6 eV is the ionization energy of the hydrogen atom, α=0.05, and Σ1 ( ε ) = 0.63255ε −1.6454 exp(−ε ) with the reduced energy ε = ( E j − Ei ) / k BTe . 1/2

For an optically forbidden transition (li = lj): 2

⎛ E j − Ei ⎞ kijε = 4π ve a02α ⎜ ⎟ Σ 2 (ε ) ⎝ k BTe ⎠

(36)

with Σ2 ( ε ) = 0.23933ε −1.4933 exp( −ε ) . For ionization of an atom under electron impact, β ie , f , Eq.35 is used with α=1 and a reduced energy ⎛ E Ion − Ei ⎞ , a=⎜ i ⎟ ⎝ k BTe ⎠

(37)

where EiIon is the energy of the ground state of the ion related to that of the ground state of the atom.

Ionization Phenomena behind Shock Waves

171

The rate coefficients for the excitation of atoms from the ground state into the metastable states due to atom-atom collisions are taken from Capitelli et al.[88], while excitation of the remaining excited levels by heavy particle impact have been neglected. Molecular Processes Throughout this investigation, the population of the molecular excited states will be supposed to be in Maxwell-Boltzmann equilibrium at the kinetic temperature of the free-electrons, thus reducing the complexity of the model. In fact, based on the analyses presented in Ref.[76,89], the electronic states of the major molecular species are very likely to follow a Maxwell-Boltzmann distribution, making the state-specific treatment unnecessary. The kinetic mechanism comprises different types of forward and backward reactions involving molecular and atomic species: 1) dissociation of N2, O2, and NO by atomic or molecular impact/recombination; 2) dissociation of N2 by electron impact/recombination; 3) associative ionization/dissociative recombination; 4) radical reactions (including Zel’dovich reactions); and 5) charge exchange. Dissociation processes, as well as their coupling with vibration, are of key importance for hypersonic reentry applications, as they significantly affect aerodynamics, the radiative and convective heat fluxes, and the spectral signatures of vehicles flying at suborbital to super-orbital velocities in rarefied atmospheres. To determine the population of the internal energy states, it would be more accurate to consider the vibronic states as pseudo-species, then treat inelastic collisions as chemical reactions, and finally compute averaged quantities for the VT, VV, and VVT processes. This type of model requires a very large number of data in terms of transition rate coefficients. Theoretical calculations have become possible only recently [90, 91], but an exhaustive database for air is still lacking. In this Chapter, Park’s model is used to describe the influence of the vibration on the dissociation of the molecular species and to account for the influence of the chemistry on the vibrational energy. The T -Tv Park’s model[1] is the most widely spread and used model in the aerospace community, mainly due to its simplicity. The geometrical average temperature is used in the Arrhenius law for the rate coefficients. This model is purely heuristic and based on the analysis of experimental data coming from the study of the post shock radiative signature in shock-tube facilities. The dissociative recombination of molecular ions is known to play an important role in the case of recombining plasmas. The inverse process, associative ionization, allows for formation of the first electrons in many cases, such as in shock tubes and reentry problems. Consequently, it allows for many ionizing situations to be explained. Since N2, O2, and NO are present in the molecular air plasma described here, dissociative recombination has to be considered[8]. Zel’dovitch reactions are known to greatly influence the distribution of nitrogen and oxygen between atomic and molecular systems and to contribute to the destruction of O2 and N2 and the formation of NO. For these processes, we have used the rate coefficients estimated by Bose and Candler [92,93], which are based on quasi-classical trajectory study performed on ab-initio potential energy surfaces. Further details about the model are given in Ref. [8].

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High Temperature Phenomena in Shock Waves

(b) Chemistry and Flow Field Energy Distribution Chemistry and energy exchange source terms are the key parameters in the set of equations governing the behavior of high enthalpy gases, allowing the changes in the mixture due to chemical reactions as well as accounting for the exchanges of energy among the internal energy modes. Also, chemistry and internal excitation are closely correlated and are known to influence each other. Non-equilibrium models are distinguished by the way they describe the interaction among the excitation of the internal energy modes and the macroscopic kinetic mechanisms employed. The model proposed is hybrid in the sense that it combines a state-to-state treatment of the atomic electronic levels with a multi-temperature approach for the remaining modes, thus requiring the modeling of the energy exchanges among the different energy modes. In the case of two temperature models the relaxation terms to be considered are: Ω = Ω VT + Ω CV + Ω ET + Ω E + Ω I m

(38)

Vibrational Translational exchanges ( ΩVT ): The rate of vibrational-translational energy transfer follows a Landau-Teller formula, where the species relaxation time is based on the Millikan–White formula, including Park’s high-temperature correction[48]. Exchanges between chemistry and vibration ( Ω CV ): The model employed was proposed by Candler [94]. Elastic exchanges among electrons and heavy species ( Ω ET ): The form of the source term is very similar to VT (Landau-Teller form). Note, however, due to the disparity of masses of an electron and a heavy particle, kinetic energy transfer between them is inefficient, resulting from the simultaneous requirement of energy and momentum conservation. Energy losses due to excitation/ionization ( Ω I / Ω E ): At high speeds, it is important to account for the energy lost by the free electrons during ionization and excitation of the atoms and molecules, as already stressed in Ref. [82]. If neglected, electron-impact ionization reactions (and in general, all the reactions involving free electrons) produce a large amount of free electrons without depleting their kinetic energy, thus enhancing their production. This phenomenon may lead to an “avalanche” ionization, which introduces related numerical problems, especially for high-speed conditions. The expression for the related source terms for electron-impact ionization and excitation reactions is: I

ΩI =

∑ω

e ,r

U

r

r

,

E

ΩE =

∑ω r

e,r

U

r

(39)

Ionization Phenomena behind Shock Waves

173

where Ur is the reaction enthalpy of the r reaction, ω e,r is the electron chemical production term of the r reaction, RI denotes the set of indices of the electron-impact ionization reactions, RE is the set of indices of electron-impact excitation reactions, Ω I accounts for the energy removed by electron-impact ionization reactions, and Ω E accounts for the energy removal by electron-impact excitation reactions. 3.1.3 Radiative Processes and Radiative Transport The system under study is composed of molecules, atoms, ions and electrons interacting with each other and with the radiation field. This system is modeled with the photons regarded simply as another species of particles. This simplifying assumption is appropriate for the description of the chaotic light characterizing the reentry environment. Under these assumptions, the kinetic theory of particles and photons is thus reduced to the set of kinetic equations for the material particles (e.g. Navier-Stokes system of equations, complemented by conservation equations for energy and the various chemical components) and the well-known Radiative Transfer Equation (RTE). The theory governing the dynamics of such systems in nonequilibrium is well established and is discussed in details in Ref.[95]. (a) Atomic and Molecular Spectral Properties When considering atomic systems, all the transitions characterized by absorption and emission of light is subdivided into three types: free-free transitions; bound-free transitions; and bound-bound transitions. The present analysis is carried out considering only the bound-bound radiation, which is the result of an electronic transition among bound atomic or bound molecular states. This type of radiation is also referred to as line radiation, owing to its discrete nature. When modeling atomic line radiation, three mechanisms have to be considered: spontaneous emission, absorption, and stimulated emission. These processes require the knowledge of one of the three transition probabilities also known as Einstein coefficients: Aji, Bij and Bji. The Einstein coefficients are not independent of each other and they must satisfy the Einstein relations:

B ji =

c2

8π hν 3 g j B ji = g i Bij

A ji

,

(40)

where c indicates the speed of light, h is the Planck’s constant and ν the frequency of the radiation. The upper states, in the electronic transition are indicated by j, while the lower states by i. The degeneracy of the states is indicated with g. Note that different definitions of B can be found in literature. A comprehensive list of atomic line probabilities, in terms of Einstein coefficients, for nitrogen and oxygen is provided by the NIST atomic line database[17], which has been used in this work. When the gas cannot be considered optically thin, an accurate modeling of the spectral line shape is also important. In this work we account for Doppler, natural and collisional broadening. Among the pressure broadening

174

High Temperature Phenomena in Shock Waves

mechanisms considered, the broadening due to collisions with charged particles (electron and ions), known as Stark broadening, has to be accounted for. Three different models have been included: Johnston's fits of experimental data[76], Cowley's and Arnold's curve fits[96] and experimental values taken from Griem[97]. (b) Energy Levels The atomic model used in radiation and the thermodynamic model used in the calculation of the flow properties differ significantly. In fact, the number of atomic levels considered on the radiation side is considerably larger than the corresponding number of pseudo-species, used in the solution of the flow field. This is due to the fact that while radiation is very sensitive to the accuracy of the atomic model, the kinetic model is relatively insensitive number of atomic levels employed. That is to say that the kinetics of the gas can be faithfully represented with a reduced number of levels, while a large number of levels is required in order to have a good representation of the radiative signature of the gas. The reduction in the number of the atomic levels in the flow case is obtained by lumping the electronic states into groups, assuming a uniform distribution within each bin [74, 75]. When computing the radiation, the population of each lumped level is distributed among the levels included in the considered group according to a Maxwell-Boltzmann distribution at a local temperature Te. Thus, the population of the ith state of the un-grouped system is obtained as: ⎛ ΔE ⎞ gi exp ⎜ − i ⎟ ⎝ kbTe ⎠ ni = nk Qˆ

(41)

k

where ni indicates the population of the ungrouped level, nk refers to the grouped level and the partition function of the grouped level is defined as: ⎛ ΔE ⎞ Qˆ k = ∑ gi exp ⎜ − i ⎟ i∈I k ⎝ kbTe ⎠

(42)

The definition of the partition function of each single bin is based upon the definition of ΔEi for each single level, which reads: ΔEi = Ek − Ei

where Ek is the energy of the grouped level Ek = ∑ i∈I k

gi Ei ∑ gj j∈I k

and I k is the set of indices of the states belonging to the group k.

(43)

Ionization Phenomena behind Shock Waves

175

A special treatment is reequired for the auto-ionizing states, which are assumedd to be in Saha equilibrium (cheemical equilibrium) with the free-state and their populattion is obtained as follows I ne ni = nIon

⎛ Ei − E Ion ⎞ gi 3 λ exp ⎜− ⎟ e kbTe ⎠ 2QIon ⎝

((44)

where λe = hP / (2π me kBTe ) is the thermal De Broglie wavelength of the free electrrons and Q Ion is the partition fun nction of the ion. In Fig.9 the distribution of the electronic states of the nitrogen atoms is shown. T The black part of the curve reprresents what is actually computed by the kinetic model and the red is the part, which is added in the calculation of the radiative properties. IIt is worth mentioning that any attempt to lump the auto-ionizing states in a unique grooup, as done for the bound stattes, resulted in a distortion of the tail of the distributtion function (red part in Fig.9), which is reflected in the spectrum. The inconsistent groupiing strategy adopted for kinetics, based on the unifoorm averaging, and radiation relying r on a Boltzmann distribution, is the main souurce inaccuracies in the spectru um and the calculation of the optical properties of the plasma. 1/2

Fig. 9. Electronic energy disttribution function for N. Equilibrium mixture at 10000 K and 1 atm

176

High Temperature Phenomena in Shock Waves

(c) Fluid Description of a Gas with Radiative Interaction This section considers the transport of radiation in a participating medium (a medium that absorbs and emits radiation). Thus, the output from the flow calculation in terms of flow quantities, i.e. pressure, temperature, electron density and the population of the electronic states of the atomic species is used to compute the spectral dependent emission and adsorption coefficients (see Eq.33). The radiative transfer problem behind the normal shock is modeled using the tangent slab approximation, thus assuming that the radiative properties, in terms of absorption and emission coefficients, of a plane layer of gas of finite thickness vary only in direction perpendicular to the surface (or the shock wave). This assumption is consistent with the assumption of one-dimensionality used for the flow-field calculations, and it is often successfully employed in literature for three-dimensional calculations to describe the radiation field in the shock-layer of blunt bodies, where the flow is almost one-dimensional. In the following, the formal solutions for the net radiative heat-flux, the divergence of the radiative heat-flux and the incident radiation for a plane-parallel medium are presented. (See Refs. [98,99] for the full derivation). The knowledge of the radiative intensity allows for the estimation of the energy source terms, thus accounting for the radiative power emitted or adsorbed by the gas. This term is simply obtained by computing the divergence of the radiative heat-flux. This can be obtained by: Incident intensity: ⎡ I + (τ b ) E (τ ) + I − (τ s ) E (τ s − τ ) + τ S (τ ) E (τ − τˆ ) dτˆ ⎤ 2 λ λ λ λ ∫τb λ λ 1 λ λ λ ⎥ ⎢ λ λ 2 λ Gλ (τ λ ) = 2π ⎢ ⎥ τs ⎢⎣ + ∫τ S λ (τ λ ) E1 (τˆλ − τ λ ) dτˆλ ⎥⎦

(45)

where En (τ λ ) is the exponential integral of nth kind. τ λ indicates thee optical depth, defined as τ λ = κ λ dx . The superscript “b” and “s” indicates the body and the shock

∫

location respectively, whereas the “+” and “-“ sign corresponds respectively to the positive or negative cosine of the angle ϑ . All the expressions above have been calculated by numerically integrating the product of the exponential integral and the source function. The solution of those integrals is easily obtained by assuming a piece-wise constant representation of the spatial evolution of the source function, given the existence of an analytical solution for the exponential functions. Divergence of Radiative Heating:

dqλ (τ λ ) = 4π Sλ (τ λ ) − G (τ λ ) dτ λ

(46)

Ionization Phenomena behind Shock Waves

177

Radiative Heating:

( )

( ) (

)

⎡ I + τ b E (τ ) − I − τ s E τ s −τ + τ S (τ ) E (τ −τˆ ) dτˆ ⎤ 3 λ λ λ λ ∫τb λ λ 2 λ λ λ ⎥ ⎢λ λ 3 λ qλR (τ λ ) = 2π ⎢ ⎥ τs ⎢⎣+∫τ Sλ (τλ ) E2 (τˆλ −τλ ) dτˆλ ⎥⎦

(47)

The other output from the radiation calculation are the rates of change of the number densities of the chemical component s in the quantum electronic state i. γ ⎡ ⎤ λmax ⎢ ⎥ i, j ⎢ Aj ,i ns , j − ( Bi , j ni − B j ,i n j ) ∫ Gλ ( x ) Φ λ d λ ⎥ λmin ⎢ ⎥ ⎣ ⎦ ωir = ∑ γ ⎡ ⎤ j >i λmax ⎢ ⎥ i, j −∑ ⎢ Ai , j ns ,i − ( B j ,i n j − Bi , j ni ) ∫ Gλ ( x ) Φ λ d λ ⎥ i> j ⎢ λmin ⎥ ⎣ ⎦

(48)

(a) Numerical Solution of Non-equilibrium Flows with Radiative Interactions The description of the non-equilibrium kinetic processes taking place behind a strong normal shock requires the solution of the system of stiff ordinary differential equations discussed, in Section 3.1.1. The reasons of the stiffness in the system of equations originate from differences in the time scales at which the different state variables relax towards their equilibrium value. In the case under analysis, the dynamics of the high lying excited states of the atomic species is extremely fast and tends to quickly equilibrate, reaching a state of quasi-equilibrium, whereas the lower states relax according to slower kinetics. The presence of different time scales suggested the development of simplified methods based on the solution of asymptotic analytical methods, also known as quasisteady state (QSS) approaches (or statistical equilibrium approaches (SE) in the astrophysical community). The use of QSS methods reduces the system of ODEs to a hybrid system of equations, since the population of the high lying states can be determined via the solution of a non-linear algebraic system of equations. Within the aerospace community, the application of the QSS methods is often used to determine the internal distribution of the excited states, when departures from the MaxwellBoltzmann distribution are expected to occur. In general however, the a priori use of these methods is highly discouraged, since the validity of the underlying assumptions strongly depends on the case to be investigated. An example of the inappropriate

178

High Temperature Phenomena in Shock Waves

application of the QSS method is discussed in Ref. [100], where Magin et al. demonstrate its inability to properly estimate the radiative signature of the hot gases surrounding the Huygens probe, by comparing QSS predictions and the results obtained using a time dependent model. Also, Ref. [82] clearly show that the low lying metastable states of nitrogen and oxygen do not reach steady-state conditions at the same time as the upper states, making any asymptotic approach inapplicable. For these reasons the direct numerical solution of the non-linear set of ODE is preferred to the asymptotic analytical methods. The introduction of the radiation coupling drastically changes the mathematical structure of the system of equations from ODE to integral-differential equations, causing the properties of very distant points in a medium to be non-locally coupled through the radiation field. In this work, the numerical solution of this problem is thus obtained through a semi-implicit iteration scheme, as follows:

∂y n = f ⎡⎣ y n , Γ( n −1,n ) Ω slx ⎤⎦ ∂x

( )

Γ where

( n −1, n )

(Ω ) sl x

∂ n −1 ⎤ ⎡ = ⎢ωir ( n−1, n ) , 0, 0, − Qrad ⎥ ∂x ⎣ ⎦

T

(49)

y n is the state vector, and it includes mass fractions for the chemical

components and the atomic excited states, velocity and temperatures. The Γ ( n −1,n ) is the vector of the coupling source terms, composed of the source terms in the species conservation equations and the divergence of the radiative heating for the total energy equation. The spatial domain is indicated using the greek letter Ω slx . It is written explicitly to stress the non-local nature of the coupling source term. The index n refers to the iterations needed to obtain convergence. The loosely coupled approach employed requires the independent solution of the kinetic and the radiative transfer problem. When solving the kinetic problem, the radiation source term used corresponds to the value at the previous step. Once the state vector y is updated the new source term can be computed and used in the following kinetic iteration. It is important to note that in order to improve the stability of the algorithm, part of G( Ω slx ) is treated implicitly.

ω r ±( n −1,n ) = ± nun Aul ∓ ( nln Blu − nun Bul ) ϒ( n −1) i

where

ϒ ( n −1) =

λmax

∫ λ

Gλ( n −1) ( x )Φ iλ, j d λ

(50) (51)

min

The only part of the source terms to be treated explicitly is ϒ {n−1} , which improves the stability and convergence of the numerical algorithm.

Ionization Phenomena behind Shock Waves

179

3.2 Results

In this section a hybrid CR model, modified from the ABBA model [8], is used to analyze the behavior of the electronically excited states of atomic species behind a strong shock. Sec. 3.2.1 is devoted to the analysis of the non-equilibrium ionization process occurring in the post-shock region of the FIRE II capsule, a well-known flight experiment from the 1960s. The dynamics of the electronic level populations of N and O is discussed. Comparison of the non-equilibrium distribution with the Boltzmann populations is made in order to quantify its departure from equilibrium. Furthermore, the effect of a self-consistent treatment of radiation coupling on the ionization process and on the internal electronic distribution is discussed. Sec. 3.2.2 studies the flow field energy distribution and Sec. 2.2.3 the interaction between radiation and matter. Sec. 3.2.4 is devoted to the analysis of the QSS regime of high-lying excited electronic states. In particular the limits of its applicability are discussed. Sec. 3.2.5 compares the modeling result with experimental measurement in the EAST shock tube facility. 3.2.1 Fire II Flight Experiment One of the primary objectives of the Fire project was to define the radiative heating environment associated with the re-entry of a large-scale Apollo vehicle at a velocity of 11.4 km/s. During this re-entry, a large portion of the overall wall heat flux was due to radiation. Most of the radiation (approximately 90%) came from atomic lines, and thus an accurate prediction of the populations of excited electronic states of the atoms is crucial. The aim of the present work is to test the CR model for different physicochemical conditions, from electronic energy level populations in strong nonequilibrium to populations following Boltzmann distributions. The shock-tube operating conditions corresponding to the trajectory point investigated here is presented in Table 1. Free stream characteristic quantities are denoted by the subscript 1, post-shock characteristic quantities by the subscript 2. Symbols U represents the shock velocity. The mole fractions of nitrogen and oxygen are assumed to be constant through the shock (xN2 = 0.79 and xO2 = 0.21). We recall that, after the shock, the rotational temperature is equal to the post-shock gas temperature T2, whereas the vibrational and electron/electronic temperatures are still equal to the free stream gas temperature T1. Table 1. Shock-tube operating conditions used in the simulation

Time P1 [Pa] T1 [K] U1 [m/s] P2 [Pa] T2 [K] U2 [m/s]

1634 s 2.0 195 11360 3827 62337 1899

180

High Temperature Ph henomena in Shock Waves

The numerical solution of o the governing equations requires iterations between the system of equation describiing the dynamics of the particles and the radiative transfer equation, solved within thee tangent slab approximation. Monitoring the relative errror in the coupling term betweeen the nth iteration and the previous (n-1)th iteration alloows one to assess the convergen nce of the numerical algorithm. Figure 10 shows the noorm of the error as a function off the iteration number. The relative error is steadily reduuced by ten orders of magnitud de in only 24 iterations. The stability of the algorithm m is ±{ n −1, n} ω achieved through the semi--implicit treatment of the term i . Every attemppt to treat this term explicitly, as done for the divergence of the radiative heating, has faiiled completely, due to the onseet of un-dumped oscillations in the coupling source terrms, which lead to the divergencce of the solution.

Fig. 10. Infinity no orm of the relative error on the coupling source terms

3.2.2 Chemistry and Flow w Field Energy Distribution The characterization of thee physical-chemical state of the plasma in the shock laayer requires the knowledge of its i chemical composition as well as the internal energies of the particles. To this aim m, Fig.11 shows the evolution of the rotational and translational temperature an nd the vibrational/free-electron temperature. After a juump in the translational temperaature across the shock (located on the left at x=0), the gas redistributes energy throug gh activation of the internal energy modes as well as the onset of chemical reactio ons, until the flow eventually reaches its post-shhock equilibrium state. The inteernal temperature profile, indicated by Tv, shows a raapid initial increase due to the excitation e of the vibrational energy states, but then flatttens out due to the eventual onseet of chemical reactions (see Fig.12). The extent of therm mal non-equilibrium (i.e. T ≠T TV) is dictated by the coupling of the internal energy and

Ionization Phenomena behind Shock Waves

181

chemistry. In the particular case considered here, the extent of thermal nonequilibrium strongly depends on the dissipation of free-electron energy by the excitation and ionization of atomic species. This relaxation process is modeled by Eqs.(39). The kinetic processes, acting as energy sinks, tend to slow down further reactions, thus reducing the rate at which the thermal energy stored in the form of translational energy is converted into chemical energy. As a result, a higher average kinetic temperature can be observed throughout the shock layer. The use of an electronic specific model allows for an accurate estimation of the energy losses to be obtained, without the use of ad-hoc parameters calibrated on the basis of experimental data. Most of the losses are due to excitation of the electronic structure of the atomic species (and, to a lesser extent, to the direct ionization). In this picture, radiative processes play a fundamental role, since they influence the dynamics of the thermal and chemical relaxation. The influence of the optical thickness of the gas on the chemistry is clear when observing the electron density profiles, shown in Fig. 12. When the medium is assumed optically thick, all of the radiation emitted is immediately self-absorbed and the electron density quickly reaches its equilibrium value. In the case of optically thin gas case, the ionization rate is considerably slower; this is due to the fact that radiative processes (when only emissive processes are considered) tend to deplete the excited states, resulting in a delay of the excitation as well as the ionization processes. In Fig. 12, the density plot exhibits a maximum and is monotonically decreasing due to the radiative energy losses, which work to consume energy, thereby preventing the gas from reaching the desired equilibrium electron density. The result of the coupled calculation is in close agreement with the optically thick case. This is the first time a rigorous modeling of the radiative processes and their interaction with the surrounding matter is used to test the escape factor approximation. 70000

T TV

Temperature [K]

60000 50000 40000 30000 20000 10000 0 0

0.02

0.04

0.06

0.08

Distance from the shock [m] Fig. 11. Post shock temperature profile

0.1

182

High Temperature Phenomena in Shock Waves

21

10 10 -3

21

a

Number Density [m ]

3

Electron Number Density [m ]

5.0×10

4.0×10

21

3.0×10

21

2.0×10

Coupled Model Optically Thick Optically Thin

21

1.0×10

0.0 0

0.02

0.04

0.06

0.08

Distance from the shock [m]

10 10

10 10 10 10

10

b

N

N2 O

21

N

+

O2

20 19

+

O

NO

18

+

NO

-

e

17

16

10 0.1

23

22

N2+

O+2

15

14

10

-4

10

-3

10

-2

10

-1

Distance from the shock [m]

Fig. 12. a) Electron density as a function of the optical thickness. b) density plots.

3.2.3 Radiative Transport and Interaction between Radiation and Matter At high shock speed, associative ionization reactions are responsible for the production of the prime electrons, thus enabling the ionization by electron collision with the atomic species. A comparison of the electron density profiles with the densities of NO+, N+ and O+ in Fig.12b shows this phenomenon very clearly: NO+ is responsible for the creation of free electrons only in the shock area (x<1mm), while after 2-3 mm from the shock most of the electrons are produced by N and O atoms. At high speeds, dissociation tends to be nearly complete before the onset of ionization. In this case, the ionization process becomes very similar to the ionization in the atomic gases as it is dominated by electron-impact processes, which can easily excite and ionize the nitrogen and oxygen atoms. In the electron temperature regime under consideration, the average electron energy is insufficient to ionize the N and O atom in their ground state and low-lying metastable states. The ionization of the atomic species follows a ladder climbing dynamic, in that it requires the excitation of the atomic species before ionizing. Thus, an atom needs to “climb” the electronic ladder before ionizing. Hence, in order to characterize non-equilibrium ionization processes, it is necessary to characterize the distribution of the electronic states of atomic nitrogen and oxygen. To this aim, the Boltzmann distribution (Te) in the post-shock area is compared with the distribution of the atomic species at 1 cm from the shock front (Figure 13(a,b)). These results demonstrate how the dynamics of upper and lower excited states are significantly different: while the excitation from the lower to the upper states as well as the ionization processes take place at relatively slow rates for the lower energy levels (e.g. ground and metastable), the higher levels exhibit opposite behavior, as the rates for these processes are very rapid. As a consequence the lower states are very likely to follow a Maxwell-Boltzmann distribution, while the higher states are more likely to be in Saha equilibrium with the free electrons, (and their population can be approximated by Eq.(44), as shown in Figure 13 a) and b). In Ref. [101], Biberman and Ul’yanov suggest the possibility of lumping all the excited states in a unique group, assumed to be in Saha equilibrium. The present analysis of the distribution functions of the atomic species seems to support such an idea. A similar approach,

Ionization Phenomena behind Shock Waves

183

discussed in Ref. [89], is based on the same concept of lumping states into macroscopic bins. Furthermore, rather than using one single group to represent the dynamics of the high-lying electronic states, the distribution is “discretized” by several groups, each of which is assumed to be populated according to a MaxwellBoltzmann distribution. Note that in Figure 13, a coupled model refers to the self-consistent model, which correctly models radiation and its interaction with the material particles. A strong depletion in the population of the excited states, (previously discussed in Ref. [82]), is shown here. Furthermore, the population of the low-lying Rydberg states seems to be strongly influenced by the assumptions made regarding the optical thickness. At this particular location and for these flight conditions, the results delivered by the selfconsistent model seem to agree well with the predictions obtained when assuming an optically thick medium. To further investigate the effects of the radiation coupling, Figure 14 presents the evolution of the population of a few electronic states of N as a function of the distance from the shock. The results obtained with the coupled model strongly contrast with the calculations obtained using escape factors. Only the coupled results, correctly estimate the concentration in the near post shock area. The radiation coming from the equilibrium part of the shock layer, by interacting with the non-equilibrium part of the gas, promotes the excitation of its internal structure populating the upper electronic states. This effect cannot be accounted for by using escape factors. Differences can be found also in the remainder part of the shock layer as the optically thick model over-predicts and the optically thin model under-predicts the population in comparison with the coupled model. 22

10

21

20

-3

Coupled Model Optically Thick Optically Thin

10 10

Normalized Population [m ]

-3

Normalized Population [m ]

10

19

10

18

10

17

10 10

16 15

10 10

14 13

10

a

Coupled Model Optically Thick Optically Thin

10

19

10

18

10

17

10

16

10

15

10

14

10

13

10

b

12

12

10 0

21

20

2

4

6

8

10

Energy level [eV]

12

14

10 0

2

4

6

8

10

12

14

Energy level [eV]

Fig. 13. a) Electronic distribution of nitrogen; b) Electronic distribution of oxygen

Finally, in order to quantify the extent of this non-equilibrium effect, we compared the spectrum obtained by means of the collisional-radiative model with the one based on Boltzmann distributions in Figure 15. The large differences observed in the radiative signature of the gas are due to the large overestimation of the population densities of the excited states when assuming Maxwell-Boltzmann equilibrium distribution.

184

High Temperature Phenomena in Shock Waves

10

Coupled Model Optically Thick Optically Thin

17

4

2

2

N P(2s 2p 3s)

-3

Population [m ]

10

18

2

2

2

N P(2s 2p 3s)

16

10

15

10

14

10

4

4

N P(2s 2p )

13

10

10

-4

10

-3

-2

10

10

-1

Distance from the shock [m] Fig. 14. Evolution of the population of the 3 lowest Rydberg states of atomic nitrogen 5

10

Intensity [W/cm -μm-sr]

4

10

4

2

10

5

2

Intensity [W/cm -μm-sr]

10

10

10

3

2

1

10 0

3

10

10

2

1

500

1000

Wavelength [nm]

1500

2000

10 0

500

1000

1500

2000

Wavelength [nm]

Fig. 15. Atomic spectrum at 1 cm from the shock front. On the left, the equilibrium results. On the right the non-equilibrium calculations are shown.

3.2.4 Quasi-steady State Distribution The analysis to be presented addresses the validity of the QSS assumption and it is presented in greater details in Ref [82]. The model employed here relies on Drawin’s cross sections for the modeling of the atomic excitation and ionization processes by electron impact and it describes the optical thickness of the medium by using escape factors. Thus this calculation uses a different dataset in comparison with the calculation described in Section 3.2.3. The conclusions drawn here are therefore conditioned to the validity of the foregoing assumptions. Future investigations based on the more physically consistent model presented in Section 3.1 should be carried out. Nevertheless, it is expected that the main conclusion on the applicability of QSS for upper electronic states to remain valid.

Ionization Phenomena behind Shock Waves

185

Quasi-steady-state models are often used in the literature [76,9] to study electronic energy level populations in strong non-equilibrium conditions, under which the hypothesis of a Boltzmann distribution does not hold. They constitute a valid alternative to the time-dependent CR model presented in this paper when the characteristic time of the excited state processes is very short with respect to the characteristic time of the flow. However, it is well known that the regime of validity of the QSS assumption is strongly influenced by a sudden change in the plasma conditions, such as after a strong shock, when the electron density is very scarce. Owing to the fact that the collisional processes are responsible for the equilibration of the internal energy states (in particular, the processes involving electrons as collision partners), a lack of electrons contributes to the failure of the QSS assumption. This section addresses the validity of the QSS assumption based on the same set of reaction rates at several locations in the flow. For this purpose, we compute the electronic energy population of atomic nitrogen by means of the full CR model. From the calculation, we extract the profiles of the flow characteristic quantities (pressure, temperatures, and composition). Then the QSS populations of excited electronic states are computed. This approach is called the simplified CR model in the following. In the first model (simplified CR metastable), no assumption is made for the population of the metastable, whereas in the second model (simplified CR), the metastable states are considered in Boltzmann equilibrium with the ground state. The latter hypothesis is justified by the small difference in terms of electronic energy among the ground state and the two metastable states, which enhances the excitation due to impacts with light and heavy particles promoting thermalization. The flow is investigated at three locations: 0.3, 0.5, and 1 cm from the shock. The electronic energy level populations for atomic nitrogen are shown in Figure 16. At 0.3 cm, the populations obtained by means of the simplified CR model are higher than the populations obtained by means of the full CR model; the QSS assumption is not valid in the near-shock region. This difference of populations is much more pronounced for the metastable states that follow a Boltzmann distribution when the QSS assumption is used. At 0.5 cm, all the excited states practically satisfy the QSS condition. After 1 cm, no difference is noticed between the results obtained by means of the two models. An explanation is found by examining the characteristic time for the atomic excitation and ionization processes. For instance, it is observed that this characteristic time for the first metastable state is of the same order of magnitude as the characteristic time of the flow 5 10-6 s computed at 1 cm from the shock. The results of the analysis suggest that one may consider the atomic metastable states as separate species for this trajectory point and compute the upper electronic states by means of a QSS model, in order to reduce the computational cost in multidimensional flow simulations. The CR model could be used as a tool to derive effective reaction rates for the simplified mechanism associated with this mixture of species and the expressions for the free-electron energy loss term for electron-impact ionization and ionization reactions. The work briefly presented is discussed in greater details in Ref. [82].

186

High Temperature Phenomena in Shock Waves

Fig. 16. Plots showing electronic energy level populations for atomic nitrogen (m3)

3.2.5 Comparison with Experimental Data The present analysis is committed to the partial validation of the electronically specific collisional-radiative, against the recent measurements performed in the EAST shock tube facility and reported in Ref. [102]. The work briefly presented hereafter is discussed in greater details in Ref. [103]. The EAST facility at NASA Ames Research Center was developed to simulate high-enthalpy, real gas phenomena encountered by hypersonic vehicles entering planetary atmospheres. It has the capability of producing super-orbital shock speeds using an electric arc driver with a tube diameter of 10.16 cm. The facility was built in the late 1960s to support research in aero-thermo-chemistry of hypervelocity flight through Earth and planetary atmospheres. The heated test gas behind the shock front simulates conditions behind the bow shock on a re-entry vehicle. This enables the experiments to be performed with flow parameters, such as velocity, static pressure, and atmospheric composition close to actual flight conditions. As the tests are performed at real flight pressures, full capsule geometry must be used if it is intended to test aerodynamic features of the flow. Due to the small region of test gas in a nonreflected shock tube, they are therefore not suitable for reproducing the aerodynamic flow-field, just a small region of the flow where it is radiating. The region of valid test gas lies between the shock front and the contact surface that separates the driver and driven gases.

Ionization Phenomena behind Shock Waves

187

The shock layer thermo-chemistry predicted by our model has been used to compute the shock layer radiance. This shock layer radiance prediction was then compared to spectrographic data acquired during the shots. In particular, during each shot under consideration two imaging spectrographs were utilized to collect spectrally and spatially resolved shock layer radiance as the shock travels though a test section.

Fig. 17. Comparison between experimental and simulated radiation traces for Boltzmann and non-Boltzmann solvers. Vs = 9.165 km/s; ps = 0.1 Torr.

The kinetic model used in this case is based on the baseline Abba model presented in Ref. [82] complemented by additional rate constants for excitation provided by Frost et al in Ref. [48]. The comparison between the simulated and the predicted intensities are shown in Figures 17-18, for two different shock-tube runs, characterized by different shock speed. Two models are employed in the prediction: a model based on the equilibrium distribution of the electronic energy levels and the CR model labeled Abba. In general both models well predict the shape of the experimental curve, however, while the non-equilibrium treatment of the electronic states shows and excellent agreement in terms of magnitude of the radiative intensity, the Boltzmann results strongly overestimate the intensity at the peak. The peaked shape present in the radiation traces, shown in Figure 17, tends to disappear at higher shock velocities (see Figure 18) as discussed in Ref. [103]. The reason of the overshoot is to be sought in the different path followed by ionization in the two cases. While in the lower speed case, associative ionization plays an important role in the production of free electrons at higher speeds its role is limited to the generation of the “prime electrons” which enable the onset of direct ionization by electron collision. Also, at lower speeds, nitrogen molecules, often vibrationally excited beyond their equilibrium values in the post shock region, tend to dissociate less, thus heating the free electrons beyond the equilibrium values, via e-V energy exchanges, which significantly contribute to the over-shoot in the radiation trace.

188

High Temperature Phenomena in Shock Waves

Fig. 18. Comparison between experimental and simulated radiation traces for Boltzmann and non-Boltzmann solvers. Vs = 9.980 km/s; ps = 0.1 Torr.

4 Conclusions This chapter delineates the importance of ionization processes in a shocked heated gas. The radiative and convective heat loads generated during Earth entry are shown to be strongly dependent on the degree of ionization in the flow. Modeling the entry physics in this regime requires a reliable database for the physical processes involved, including the production of the charged particles, chemical reactions involving charge-neutral and charge-charge interactions, and the removal of charged particles by recombination. Current status of the database is reviewed. Whenever possible, atomic and molecular data calculated using a quantum mechanical method or obtained from recent experiments are recommended. Modeling of the partially ionized flow is carried out using a one-dimensional flow solver coupled with kinetic processes described by the collisional-radiative model and the radiative transfer equation. Explicit calculations of the population of the atomic electronic states are carried out using state-to-state gas kinetics and treating the quantum states of the atoms as separate pseudo-species. Another important factor in a realistic simulation of the ionized flow is the coupling of the kinetic equations with the radiative transfer equation, resulting in a fully consistent treatment of the radiation processes. Examples are presented using the condition of FIRE II flight experiment. Differences are found between the fully coupled result and uncoupled calculations using escape factors based on optically thick or optically thin models. Generally, the coupled result is closer to the optically thick model. Calculations are also made to compare with experimental measurements from EAST shock tube facility. Results from the non-Boltzmann solver using the CR model are in significant better agreement with experiment than the Boltzmann result.

Ionization Phenomena behind Shock Waves

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Acknowledgments. Discussions with Prof. Anne Bourdon, Prof. Arnaud Bultel and Dr. Yen Liu are gratefully acknowledged.

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Mazeau, J., Gresteau, F., Hall, R.I., Huetz, A.: J. Phys. B 11, 557 (1978) Thomas, L.D., Nesbet, R.K.: Phys. Rev. A 12, 2369 (1975) http://amdpp.phys.strath.ac.uk/tamoc/DATA/ Griem, H.R.: Plasma Spectroscopy. McGraw-Hill, New York (1964) Hummer, D.G., Mihalas, D.: Astrophys. J. 331, 704 (1988) Gylys, V.T., Jelenkovic, B.M., Phelps, A.V.: J. Appl. Phys. 65, 3369 (1989) Freysinger, W., Khan, F.A., Armentrout, P.B., Tosi, P., Dmitriev, O., Bassi, D.: J. Chem, Phys. 101, 3688 (1994) Zeldovich, Y., Raizer, Y.: Physics of Shock Waves and High Temperature Hydrodynamic Phenomena. Academic Press Inc., Berkeley Square House (1966) Bourdon, A., Vervisch, P.: Phys. Rev. E 55 (4), 4634 Bourdon, A., Teresiak, Y., Vervisch, P.: Phys. Rev. E 57(4), 4684–4692 Johnston, C.O.: Non-equilibrium Shock-Layer Radiative Heating for Earth and Titan Entry, PhD thesis, Virginia Polytechnic Institute and State University, Virginia (2006) Panesi, M., Magin, E.T., Huo, W.: Nonequilibrium ionization phenomena behind shock waves. In: 27th International Symposium on Rarefied Gas Dynamics (2010) Brun, R.: Introduction to reactive gas dynamics. Toulouse, France, Cepadues (2006) Giordano, D.: Hypersonic-flow governing equations with electromagnetic fields. RTOEN-AVT-162 VKI lecture series 1(1) (2008) Lee, J.H.: AIAA Paper 84-1729 (1984) Gurvich, L., Veyts, I., Alcock, C.: Thermodynamic properties of individual substances, p. 1. Hemisphere Publishing Corporation (1989) Panesi, M., Magin, T., Bourdon, A., Bultel, A., Chazot, O.: J. Thermophysics and Heat Transfer 23, 236 (2009) Teulet, P., Sarrette, J.-P., Gomes, A.-M.: J. Quantitative Spectroscopy and Radiative Transfer 1, 69 (1999) Teulet, P., Sarrette, J.-P., Gomes, A.-M.: J. Quantitative Spectroscopy and Radiative Transfer 1, 70 (2001) Sarrette, J.-P., Gomes, J., Bacri, A.-M., Laux, C., Kruger, C.: J. Quantitative Spectroscopy and Radiative Transfer 1, 53 (2001) Drawin, H.: Atomic Cross Sections for Inelastic Electronic Collisions, Association Euratom-CEA. Rept. EUR-CEA-FC 236, Cadarache, France (1963) Drawin, H.: Zeitschrift fur Physik D:Atoms, Molecules and Clusters 211, 3, 404 (1968) Capitelli, M., Ferreira, C., Gordiets, B., Osipov, A.: Plasma Kinetics in Atmospheric Gases. Springer, Berlin (2000) Panesi, M., Magin, T., Bourdon, A., Bultel, A., Chazot, O.: J. Thermophysics and Heat Transfer (2011) Chaban, G., Jaffe, R., Schwenke, D., Huo, W.: AIAA Paper 2008-1209 (2008) Jaffe, R., Schwenke, D., Chaban, G., Huo, W.: AIAA Paper 2008-1208 (2008) Bose, D., Candler, G.: J. Chem. Phys. 104, 8, 2825 (1996) Bose, D., Candler, G.: J. Chem. Phys. 107, 16, 6136 (1997) Candler, G., MacCormack: J. Thermophysics and Heat Transfer 5(11), 266 (1991) Oxenius, J.: Kinetic Theory of Particles and Photons: Theoretical Foundations of Non-Lte Plasma Spectroscopy, p. 20. Springer, Heidelberg (1986) Cowley, C.R.: An Approximate Stark Broadening Formula for Use in Spectrum Synthesis. The Observatory 91, 139 (1971) Griem, H.R.: Spectral Line Broadening by Plasmas. Academic Press, New York (1974) Özisik, M.N.: Radiative Transfer and Interactions with Conduction and Convection. Wiley-Interscience Publication (1973)

192 99. 100. 101. 102.

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Siegel, R., Howell, J.R.: Thermal Radiation Heat Transfer. Taylor & Francis (2010) Magin, T.E., Bourdon, A., Laux, C.O.: J. Geophysical Research 111, E07S12 (2006) Biberman, L.M., Ulyanov, K.N.: Optical Spectroscopy (U.S.S.R.) 16, 216 (1964) Grinstead, J.H., Olejniczak, J., Wilder, M.C., Bogdanoff, M.W., Allen, G.A., Lilliar, R.: Shock-heated Air Radiation Measurements at Lunar Return Conditions: Phase I EAST Test Report, NASA EG-CAP-07-142 (2007) 103. Panesi, M., Babou, Y., Chazot, O.: AIAA Paper 2008-3812 (2008) 104. Klapisch, M., Busquet, M., Bar-Shalom, A.: In: Gillaspy, J.D., Curry, J.J., Wiese, W.L. (eds.) 15th International Conference on Atomic Processes in Plasmas, AIP, vol. 206 (2007)

Chapter 5

Radiation Phenomena behind Shock Waves M.Y. Perrin, Ph. Rivière, and A. Soufiani Laboratoire EM2C, CNRS UPR288, Ecole Centrale Paris, 92295 Chatenay-Malabry, France

1 Introduction Shock waves produce hot gases, which radiate. Radiation is a full partner of the many physical and chemical processes which have to be taken into account in physical gasdynamics in hot gases [1-3]. As the emitted radiation is linked to the thermochemical state of the media, it has been widely used as a non disturbing tool to characterize the state of media behind shock waves. The emitted radiation may also contribute to the heat flux suffered by an obstacle. This contribution will be important for a vehicle entering an atmosphere [4] at very high speed such those experienced in aerocapture entry or lunar return for example. For vehicles entering the Earth’s atmosphere at velocity higher than 10km/s, the role of radiative heating in the total flux balance becomes essential. For Galileo entry into Jovian atmosphere the contribution of radiation was dominant for most of the entry trajectory. Accurate predictions of the non-equilibrium radiation in shock layers are thus required for efficient design of thermal protection systems. Radiation may also modify the gas dynamics. The emitted photons can either leave the flow, giving rise to the so-called radiative cooling, or can be re-absorbed, contributing to the transport of energy. Under some conditions, the processes of emission and absorption of photons have to be included in the equations describing the evolution of atomic and molecular internal states. As the emission and absorption coefficients depend on the internal state of gases, the radiation field and the internal state of gases must be determined selfconsistently. The role of radiation is particularly important in the so-called radiative shocks [1,2], which are present in a wide range of astronomical objects and which can be generated in the laboratory using high-power lasers. In these high Mach number shocks, the radiation energy density, flux and stress tensor have to be included in the set of conservative equations; furthermore the medium may be photoionized ahead of the shock front giving rise to precursors which modify the shock jump relations. In these cases, the radiation drives the flow. In the present paper, we will mainly be concerned by radiation in hypersonic flows encountered in atmospheric entries. The incident probe velocities range from about 5 km/s for a low-speed Mars or Titan entry to almost 60 km/s for a polar probe to Jupiter. At these speeds, a strong shock wave forms in front of the entering probe that dissociates and, for the highest velocities, ionizes the gas.

194

High Temperature Phenomena in Shock Waves

The radiative intensity Iσ (u) (W.m-2.sr-1.(m-1)-1) which characterizes the radiation field can be obtained at each point for each propagation direction u , and each wavenumber σ (m-1) by solving the radiative transfer equation (RTE) :

dIσ (u) = ησ − κσ Iσ (u) , ds

(1)

where s is the optical path along u . κσ is the absorption coefficient (m-1), and ησ is the emission coefficient (W.m-3.sr-1.(m-1)-1). Equation (1) is written assuming that scattering by solid particles is negligible, otherwise an additional term has to be included and the intensities along different propagation directions become coupled. The media will be called optically thin if the second term, i.e. the absorption, can be neglected in (1). The radiative flux (W.m-2) on a surface is calculated from the following intensity ∞

qR = ∫ dσ ∫ Iσ (u) u. n du , 4π

0

(2)

where n is the normal to the considered surface. The amount of power exchanged between the matter and the radiation field per unit volume (W.m-3) is given locally by ∞

PR = ∫ dσ 0

∫ (ησ − κσ Iσ (u) ) du

(3)

4π

κσ and ησ are local quantities which depend on the thermochemical states of the emitting and absorbing species. If the medium is at local thermodynamic equilibrium (LTE) at temperature T, κσ and ησ are linked through the Kirchhoff’s law:

ησ = κσ Iσ0 (T) ,

(4)

where Iσ0 (T) is the equilibrium intensity given by the Planck’s law : Iσ0 (T) =

2hc 2σ 3 , ⎛ hcσ ⎞ −1 exp ⎝ kT ⎠

(5)

h, k and c being respectively the Planck and Boltzmann constants and speed of light. For medium, which is not in equilibrium, both κσ and ησ have to be determined. The different mechanisms, which contribute to the emission and absorption coefficients, are presented in section 2. A few illustrations are given in section 3 for air flows whose spectra show a rich structure. The modeling of spectral and directional aspects of radiative transfer is discussed in section 4. We briefly review in section 5 the different approaches used to couple radiation and other phenomena in shock wave flows. The emission and absorption coefficients for the different radiative mechanisms, which can be observed behind shock waves for a flow, depend on the thermochemical state of the plasma. We will consider in the following that the translation energy distributions of electrons and heavy particles can be represented by

Radiation Phenomena behind Shock Waves

195

Boltzmann distributions at Te and Tt respectively. For the internal degrees of freedom, the thermodynamic state is characterized by the energy level populations. We will also consider a frequently used two-temperature model, which introduces Ttr for the translational motion of heavy particles and rotation of molecules, and Tve for molecular vibration, electronic excitation and free electron translation.

2 Radiative Mechanisms and Radiative Properties The selection of the radiative mechanisms and associated spectroscopic data is a critical issue. All the mechanisms, which may contribute to emission and absorption, have to be considered. Due to the wide temperature range that can be encountered (up to 60000K for entry applications), the spectral range has to cover Infrared (IR) to Vacuum Ultra-Violet (VUV). Moreover the fine structure of bound-bound atomic and molecular spectra has to be taken accurately into account to predict radiative transfer with absorption. Several computer codes and spectral databases have been developed in the last decades for radiation analysis: NEQUAIR [5], LORAN [6], SPRADIAN [7], MONSTER [8], SPECAIR [9], PARADE [10], HARA [11], GPRD [12]. The simulations, which are presented here, have been made using the HTGR [13] database developed by the authors. The selection of the data, their accuracy and completeness is detailed in Refs.14,15 for air and in Ref. 16 for CO2. The general methodology consists in selecting the most precise data available and generating the missing ones. The derived expressions of emission and absorption coefficients rely on spectroscopic concepts, which are not all detailed. The interested reader is invited to refer to specialized books on spectroscopy for a deeper insight [17-20]. 2.1 Bound-Bound Transitions 2.1.1 General Formulation Bound-bound transitions correspond to transitions between bound energy levels of atoms or molecules; they give rise to line spectra. The three important mechanisms leading to transitions between bound energy levels u (upper) and l (lower) of species A are described by the following reaction schemes:

A(u) → A(l) + h c σ ul A(l) + h c σ ul → A(u)

A(u) + h c σ ul → A(l) + 2 h c σ ul

emission

(6)

absorption

(7)

stimulated emission

(8)

σul , the wavenumber of the emitted or absorbed photon, satisfies the relation

h c σ ul = E u − E l

,

(9)

where E u and E l are the energies of the upper and lower levels respectively. The three radiative mechanisms are characterized by the Einstein coefficients Aul, Blu and Bul which allow to express the monochromatic emission and absorption coefficients as

196

High Temperature Phenomena in Shock Waves

ησ = ∑ ul

κσ =

Aul hcσ ul N u f ul (σ − σ ul ) , 4π

∑ (N B l

− N u B ul )h σ ul f ul (σ − σ ul ) ,

lu

(10)

(11)

ul

where Nu and Nl designate the population of the upper and lower transition levels. f ul (σ − σ ul ) is the spectral line shape of the transition accounting for Doppler, natural and collisional broadening. As can be seen from the detailed balance of the processes at thermodynamic equilibrium, the Einstein coefficients satisfy the relations:

Aul = 8π hcσ ul3 Bul ,

(12)

gu Bul = gl Blu ,

(13)

where gu and gl are the degeneracies of levels u and l. To predict bound-bound emission or absorption spectra, we need thus to know for each transition the transition strength, characterized by Blu for instance, the line position σul, the thermodynamic state of the medium, i.e. the Nu and Nl values, and the line shape f ul (σ − σ ul ) . Einstein coefficients are intrinsic quantities of the radiating particle. For electric dipolar transitions, which are the main contributors, the Einstein absorption coefficient may be expressed as

Blu =

8π 3 1 1 R , 3h 2c 4πε 0 gl ul

(14)

where ε 0 is the permittivity of free space. The transition dipole moment Rul is obtained form the dipolar moment operator μ of the radiating particle according to

Rul =

∑

l,ml μ u,mu

2

.

(15)

ml mu

The summation on m l , resp. m u , extends over a complete basis of the state space associated to the level l, resp. u. A non zero transition dipole moment means that the changes in quantum numbers satisfy the electric dipolar selection rules: pu ≠ pl ΔJ = Ju − Jl = 0, ± 1 J l = 0 ↔ Ju = 0 is not allowed

(16)

where pu and pl are the level parities, J u and J l are the total angular momentum quantum numbers.

Radiation Phenomena behind Shock Waves

197

The main mechanisms contributing to lineshapes in the present application are the broadening due to thermal motions of the radiating particle, namely the Doppler broadening, and the collisional broadening. The Doppler profile associated to a Maxwellian velocity distribution at Tt for the radiating particles is given by f D (σ − σ ul ) =

2 ⎡ ⎛ σ − σ ul − δ ulD ⎞ ⎤ exp − ln 2 ⎢ ⎟ ⎥. ⎜ π γ ulD γ ulD ⎠ ⎥⎦ ⎝ ⎣⎢

ln 2 1

(17)

δ ulD is the Doppler shift of the line center due to the hydrodynamic velocity. The halfwidth at half-maximum (HWHM) γ ulD is expressed as γ ulD = σ ul

2kTt ln2 , mrc 2

(18)

where m r is the mass of the radiating particle. The collisional broadening is usually described by a Lorentzian profile of HWHM γ ulL :

f L (σ − σ ul ) =

γ ulL 1 2 π ( γ ulL ) + (σ − σ ul − δ ulL )2

(19)

More complex formulations have to be considered for hydrogen plasma [21]. The convolution of the Doppler and Lorentz line shapes leads to a Voigt line shape. If the shifts δ ulD and δ ulL are neglected, the Voigt profile is given by

fV (σ − σ ul ) =

a

πγ ulD

e −y dy, π ∫−∞ a2 + (b − y) 2

ln2

+∞

2

(20)

with

γ ulL ln2 , γ ulD σ − σ ul b = ln2 . γ ulD a=

(21)

The specificities related to atomic line spectra and diatomic line spectra are detailed in the following sections.

198

High Temperature Phenomena in Shock Waves

2.1.2 Atomic Line Spectra The atomic line spectra correspond to transitions between atomic electronic levels. In a level-by-level description, the different electronic populations are determined and expressions (10) and (11) can be directly used. If a Boltzmann distribution of electronic atomic states at Tve is assumed, the populations of electronic levels are given by:

Nu = Nat

gu exp(−E u /kTve ) , Qat

(22)

where N at and Qat are the total population and the internal partition function of the atomic species. An electronic level of a light atom (Z<30) is represented in the LS coupling scheme by 2S +1 LJ ; L, S and J are the total orbital angular momentum (the numerical values of L=0,1,2,… are replaced by letters S,P,D,…), total electronic spin angular momentum and total angular momentum respectively (the nuclear spin angular momentum is not considered). J takes the values J = L − S, L − S +1, ..., L + S . Each level has a degeneracy equal to 2J+1. The electric dipolar transitions satisfy the general selection rules given by Eqs.16 and, in case of pure LS coupling scheme, the additional selection rules on the changes in S and L: ΔS = 0 ΔL = 0,± 1

(23)

L = 0 ↔ L = 0 is not allowed

Atomic line data can be found in the NIST atomic spectra database, which provides critically evaluated data on atomic energy levels, wavelengths, and transition probabilities that are reasonably up-to-date [22]. For example the database records 898 lines for allowed transitions in O from 200 cm-1 up to 160 000 cm-1. Atomic line data can also be found in the most complete TOPBASE database, which has been developed to estimate stellar opacities [23]. The data have been computed quantummechanically with advanced methods but can be less accurate than the selected NIST data; moreover the fine structure due to spin-orbit interaction has not been considered. The atomic data are thus related to the terms 2S +1 L . A good compromise consists in using the NIST database, keeping the accuracy and the fine structure for the strongest lines, and complementing with the TOPBASE multiplet lines, which will be in most cases individually optically thin but whose contributions can be non negligible [11] if the strong lines are strongly self-absorbed. As many atomic lines cannot be assumed optically thin (in particular those involving the ground energy level), the line shapes have to be carefully determined. As it is not possible to find all the data, systematic calculations have to be performed to account for the different collisional broadening mechanisms [21,24], which are contributing.

Radiation Phenomena behind Shock Waves

199

Collisional broadening by neutral species is generally accounted for using the impact approximation, which leads to a Lorentzian profile with a van der Waals contribution and a resonance contribution for a perturber of the same type as the radiating particle. The Van der Waals line width is given by

γ ulvdw =

⎛ 9π 5 Δ r 2 ⎜ ⎜ 16 m3 E 2 e p ⎜ ⎝

1 ∑ N p v 3/5 2c p

⎞ ⎟ ⎟ ⎟ ⎠

2/5

, (24)

where N p and E p are the population and the energy of the first excited state of the neutral perturber. Δr 2 = r 2 − r 2 is the difference between the mean square radii of the u

l

radiating particle in the upper and lower transitions levels – for a perturber identical to the radiator, only the level of the same parity as the ground state is taken into account when evaluating Δr 2 . The brackets designate an average on the distribution of the relative velocity of colliding particles. Mean square radii can be evaluated from the Bates-Dammgard approximation [18] and the quantum defect values listed in TOPBASE [23]. The resonance contribution occurs only for perturber of the same type as the radiator and is expressed according to

γ ulres =

3qe2 16π ε 0 m e c 2 2

∑n j

j

⎛ g j f ju ⎜ g σ + ⎝ u uj

g j f jl ⎞ , gl σ lj ⎟⎠

(25)

where qe is the electron charge, f ij is the oscillator strength of the transition i → j

which is defined by Bij ( E j − Ei ) / π rec where re is the electron classical radius. The oscillator strengths f ij and the transition wavenumbers σ lj can be found in TOPBASE [23]. Collisional broadening by impact of charged particles, known as Stark broadening can be modeled in the impact approximation for weakly ionized media, at medium pressure and for non hydrogenoid species. Relatively simple expressions, similar to those given above for neutral contribution, can be obtained using a semi-classical adiabatic approach [24]. For example, the Stark line width for a neutral radiator can be expressed as a sum of a contribution due to dipolar polarization interaction potential

γ uldip = N p

1 ⎛π ⎞ 2πc ⎝ 2 ⎠

5/3

Γ

2/3 ⎛ 1⎞ ΔC 4 v1/ 3 , ⎝ 3⎠

(26)

and a contribution due to the quadripolar polarization interaction potential

γ ulquad = N p

π 2c

ΔC3 ,

(27)

200

High Temperature Phenomena in Shock Waves

with

ΔC4 = C4 u − C 4 l , I H2 ⎛ a0qp ⎞ C4 j = πme hc2 ⎜⎝ qe ⎟⎠ ΔC 3 =

2

f jl

∑σ l

(B r ) + (B r ) − B r r 2 u u

2

2

2

l l

(28)

2 2 ul u l

(29)

2 jl

2I H q p me qe

(30)

In the above expressions, I H is the Rydberg constant, a0 the Bohr radius and q p the perturber charge. B j and Bul constants may be crudely estimated from the total electronic orbital angular momentum quantum numbers Lu and Ll of the radiator according to

Bj =

2L j +1 ⎛ L j ⎜ 15 ⎝ 0

⎧L u Bul = −2Bu Bl ⎨ ⎩ Ll

2 L j⎞ L j ⎟ (−) , 0 0⎠

Ll Lu

1⎫ ⎬, 2⎭

(31)

(32)

where the symbols ( ) and {} represent the Wigner 3-j and 6-j symbols. More sophisticated semi-classical approaches can be used [21,25]. They enable to achieve better accuracy but are more tedious to use. 2.1.3 Diatomic Line Spectra The diatomic line spectra correspond to transitions between rovibronic states, which are characterized by:

n: the electronic state v: the vibrational number J: the total angular momentum without nuclear spin. i: the spin multiplet component which takes 2S+1 values p: the parity which characterizes the symmetry of the total wavefunction (without nuclear spin) with respect to the inversion symmetry operator I. The energy of the rovibronic states can be expressed as the sum of an electronic contribution, a vibrational contribution and a rotational contribution:

Ev,in (J, p) = Eel (n) + Evib(n,v) + Erot(n,v,J,i, p) .

(33)

Radiation Phenomena behind Shock Waves

201

In the framework of the two-temperature model, the population of an energy level is expressed as

N nvJip = N r gsnJip

⎛ E (n) + E vib (n,v) E rot (n,v,J,i, p) ⎞ 2J + 1 exp⎜ − el − ⎟, Q(Tve,Ttr ) kTve kTtr ⎝ ⎠

(34)

where N r is the total population of the radiating molecule and Q(Tve,Ttr ) is the twotemperature partition function [26]. gsnJip is the nuclear spin factor which applies only for homonuclear molecules. Similarly to atoms, there is a coupling between electronic spin, orbital angular momenta and moreover the angular momentum R associated with the rotation of the molecular nuclei. For molecules with light nuclei, this coupling is weak and we can define the projection of the total electronic orbital momentum L on the internuclear axis. The absolute value of this projection is noted Λ. An electronic state of a diatomic molecule is characterized by the label 2S+1Λ where 2S + 1 is the spin multiplicity. The numerical values 0,1,2,. . . of Λ are replaced by Σ, Π, Δ, . . . For Λ > 0 there is a double orbital degeneracy called Λ-doubling which will be lifted in the rotating molecule. The multiplet structure of energy levels will depend on how the electronic angular momenta couple to the molecular rotation. Several limiting coupling cases are defined [19]. We are, in the present study, concerned by the Hund’s coupling cases a and b. In Hund's case a, the interaction between the molecular rotation and the electrons motion is weak, L and S are closely coupled to the internuclear axis, their projections being Λ and Σ. In Hund's case b, S does not couple to the internuclear axis and recouples to N = L+R. The total angular momentum quantum number J ranges from N+S to ⎢N-S ⎢ by units step. The radiative transitions considered for air can be represented by intermediate a/b coupling. The level energies are then obtained as the roots of a determinant calculated from the matrix of a model Hamiltonian whose parameters can be adjusted on experimental 1 data [27,28]. The simplest expression of level energies is obtained for Σ state which can be expressed 2 3 E1Σ v (J) = Tv + Bv x − Dv x + Hv x + ...

(35)

with x = J(J + 1) . A rotational line will correspond to the transition between two rotational levels characterized by (n’, v’, J’, i’, p’) and (n’’, v’’, J’’, i’’, p’’). Such transitions obey to the general selection rules given by Eq. (5.16) and to additional selection rules for Hund's cases a and b which lead to complex branch structure [28]. The set of transitions between rovibrational levels of the electronic states A and B gives rise to the A-B system. The strength of the different transitions will depend on the transition dipole moment Rul which can be written as the product of two terms assuming that centrifugal distortion can be neglected:

(

Rul = Rev 'v ''

)

2

S Jv ''vJ''''

(36)

202

High Temperature Phenomena in Shock Waves

(R )

v ' v '' 2 e

is the square of the electronic vibrational moment, and S Jv'' Jv '''' is the HönlLondon factor [28]. The prime quantities refer to the upper level of the transition, the double prime quantities to the lower level. The sum rule of Whiting et al [29] is adopted:

∑S J'

v ' v '' J ' J ''

= ( 2 − δ 0,Λ 'δ 0,Λ '' ) ( 2 S + 1)( 2 J "+ 1)

(37)

The electronic-vibrational part of the line intensity is given by

(R )

v ' v '' 2 e

2

∞ = ⎡ ∫ Ψ v ' (r ) Re ( r ) Ψ v '' ( r ) dr ⎤ , ⎣⎢ 0 ⎦⎥

(38)

where r is the internuclear distance, Ψv' (r) and Ψv'' (r) are the radial rotationaless vibrational wavefunctions of the upper and lower vibrational states, Re is the electronic transition moment function (ETMF).

We need to know ( Rev 'v '' ) for all the contributing bands. This information can be extracted from spectroscopic measurements but the latter are not exhaustive. To remedy this lack of experimental data, the electronic-vibrational part of the line intensities can be calculated systematically. The method comprises three steps [14]. In a first step, the potential energy curves for all the electronic states are reconstructed via the Rydberg-Klein-Rees (RKR) procedure from the vibrational energy term and the rotational constants. In a second step, the radial Schrödinger equation is solved to get the vibrational wavefunctions Ψv (r) . Finally expression (38) is evaluated by using electronic transition moment functions Re selected in the literature. Many transient molecular species are highly radiative, meaning that even trace concentrations of these species may have a significant impact on the radiation field. Table 1 gives the molecular systems considered for air plasma. Rotational lines have been calculated up to the last rotational quantum number below the centrifugal dissociation of the upper and lower vibrational levels, i.e. widely beyond the validity range of the selected spectroscopic constants. Such extrapolations are required insofar as this database is devoted to predict radiative transfer inside very high rotational temperature plasmas. We have ensured that no unphysical line positions occur due to these extrapolations. The contribution of the different systems as a percentage of the total emission coefficient integrated over the range 1000-150000 cm-1 is given in Fig.1 as a function of temperature for an atmospheric air mixture at equilibrium. The thick lines separate the contributions of the different molecules. As expected at low temperature, the emission occurs mainly in the infrared. At higher temperature many systems are contributing. If the gases are not in equilibrium behind the shock wave, the contribution of a given system may be decreased if the electronic upper state is less populated or increased if the electronic upper state is more populated. Moreover the contribution percentages for the intensity escaping from a flow column may be different due to absorption. 2

Radiation Phenomena behind Shock Waves

203

Table 1. Diatomic electronic systems considered for air [14] Radiating molecule N2

+

N2

NO

O2

System name

Upper state – lower state

Calculated bands

First-Positive

B3Π g − A 3 Σ +u

(0 : 21; 0 : 16)

Second Positive

C Πu − B Πg

(0 : 4; 0 : 21)

Birge-Hopfield 1

b1Π u − X1Σ +g

(0 : 19; 0 : 15)

Birge-Hopfield 2

b Πu − X Σ

+ g

(0 : 28; 0 : 15)

Carroll-Yoshino

c'14 Πu − X1Σ +g

(0 : 8; 0 : 15)

Worley-Jenkins

c Πu − X Σ

+ g

(0 : 4; 0 : 15)

Worley

o13Πu − X1Σ +g

(0 : 4; 0 : 15)

Meinel

A2Πu − X 2 Σ +g

(0 : 27; 0 : 21)

3

3

'1

1

1

1 3

+ g

(0 : 8; 0 : 21)

First-Negative

B Σ −X Σ

Second-Negative

C Σ −X Σ

+ g

(0 : 6; 0 : 21)

γ δ δ ε γ’ β’

A2Σ + − X 2Πr

(0 : 8; 0 : 22)

B Πr − X Πr

(0 : 37; 0 : 22)

C Πr − X Πr

(0 : 9; 0 : 22)

D Σ + − X Πr

(0 : 5; 0 : 22)

E Σ + − X Πr

(0 : 4; 0 : 22)

B Δ − X Πr

(0 : 6; 0 : 22)

2

2

+ u

2

+ u

2

2

2

2

2

2

2

2

2

'2

2

+

+

11000 A

D Σ −A Σ

infrared

X Πr − X Πr

Schumann-Runge

B Σ −X Σ

3

2

3

2

2

− u

3

− g

(0 : 5; 0 : 8) (0 : 22; 0 : 22) (0 : 19; 0 : 21)

Fig. 1. Relative contributions of the different molecular systems to emission versus temperature for an air plasma at LTE and atmospheric pressure. The emission is integrated over the spectral range 1000–150 000 cm−1(adapted from Ref.14).

204

High Temperature Phenomena in Shock Waves

Collisional broadening of diatomic rovibrational lines in the infrared has been the subject of many studies [30]. However, very few data can be found on collisional broadening of diatomic rovibronic lines. When the data are not available, pragmatic collisionnal linewidths can be used [31]: γ = 0.1

⎛ 273⎞ ⎝ T ⎠

cm-1 atm-1

(39)

Additional Lorentzian line broadening due to predissociation has to be taken into account for several systems ( N2 VUV, NO, and O2 systems [14],…) 2.2 Bound-Free Transitions 2.2.1 General Formulation The bound-free transitions correspond to exchange between the internal states of the atom or molecule and kinetic energy; they lead to continuous spectra. Three types of mechanism can be involved:

A(k) + hcσ ↔ A + (i) + e

photoionization/radiative recombination

(40)

AB(k) + hcσ ↔ A(i) + B( j)

photodissociation/radiative recombination

(41)

A − (k) + hcσ ↔ A(i) + e

photodetachment/radiative electron attachment (42)

They can be represented formally by:

AB(k) + hcσ → A(i) + B( j) .

(43)

The energy conservation leads to:

1 hcσ = μg 2 + E iA + E Bj − E kAB , 2

(44)

where μ is the reduced mass of the products (A,B), g is the relative velocity of the products, E lX is the energy of level l of species X. For each mechanism, level-to-level cross sections S characterize spontaneous emission, induced emission, and absorption. Similarly to bound-bound transitions, the three cross sections are linked by the Einstein-Milne relations [2]. Assuming a Boltzmann distribution at Trel for the A/B relative velocity distribution enables to write the emission and absorption coefficients as [32]:

ησ , fb = 2hc 2σ 3 ∑ ijk

⎛ E A + E Bj − E kAB − hcσ ⎞ gk abs hσ Sk,ij (σ )exp⎜ i ⎟, ξ (μ,Trel ) gi g j kTrel ⎠ ⎝ N iA N Bj

(45)

Radiation Phenomena behind Shock Waves

⎡

AB κσ , fb = hσ ∑ S kabs ,ij ( σ ) N k ⎢1 −

⎣⎢

ijk

N iA N Bj

ξ ( μ , Trel ) n

AB k

⎛ EiA + E Bj − EkAB − hcσ gk exp ⎜ ⎜ gi g j kTrel ⎝

205

⎞⎤ ⎟⎟ ⎥ , (46) ⎠ ⎦⎥

where ξ (μ,Trel ) is the volumetric translational partition function of the particle of mass μ at temperature Trel defined by:

ξ(μ,Trel ) =

⎛ 2πμkTrel ⎞ ⎝ h2 ⎠

3/2

.

(47)

2.2.2 Atomic Photoionization In the framework of the two-temperature model, Eqs.45 and 46 allow to express the emission and absorption coefficients corresponding to the mechanism (40) as: ⎛ hcσ ⎞ N at neq ⎛ E ⎞ χ ∑ gk exp⎜ − k ⎟ Sk (σ ) , ⎟ ⎝ kTve ⎠ Qat ⎝ kTve ⎠ k

(48)

⎤⎡ ⎛ Ek ⎞ ⎛ hcσ ⎞ ⎤ N at ⎡ neq ⎢ ∑ g exp⎜ − ⎟ Sk (σ ) ⎥ ⎢1− χ exp ⎜ − ⎟⎥, Qat ⎣ k k ⎝ kTve ⎠ ⎝ kTve ⎠ ⎦ ⎦⎣

(49)

⎛E ⎞ N ion N e Qat exp⎜ ion ⎟ , N at 2Qionξ (me ,Tve ) ⎝ kTve ⎠

(50)

ησ = 2 hc 2σ 3 exp⎜ −

κσ =

χ neq =

where N at , N ion and N e are the total populations of the atom, produced ion and electron. Qat and Qion are the atom and ion internal partition functions. m e is the electron mass and E ion is the ionization energy corrected from the Debye ionization lowering. The factor χ neq characterizes the deviation from equilibrium and consequently from Kirchoff’s law. S k (σ ) is the total cross section of photoionization from level k: abs Sk (σ ) = hσ ∑ Sk,ij . i, j

(51)

TOPBASE [23] provides spectrally resolved absorption cross section S k (σ ) from levels up to a principal quantum number equal to 10 or 11. However these data cannot be directly used if electron translation and ion electronic excitation cannot be abs represented by the same temperature. In order to calculate S k,ij , simpler models for bound-free transitions can be developed from first principles assuming hydrogenoïd atoms1 or using more complex theories (quantum defect method, Thomas-Fermi method) [18].

206

High Temperature Phenomena in Shock Waves

2.2.3 Molecular Photodissociation The O2 photodissociation corresponding to Eq.(41) and leading to the SchumannRunge continuum is known to be an important mechanism above 50,000 cm-1. Only equilibrium absorption cross sections are available. By assuming that the oxygen atoms are only produced in the 3P and 1D terms of the oxygen ground configuration, which corresponds to the asymptote state of the O2 B 3Σ −u electronic state, the thermal cross section S LTE can be written as: S

LTE

⎛ E kO 2 ⎞ abs hσ (σ ,T ) = ∑ g exp ⎜ − kT ⎟ S k,i0 j0 (σ ) , QO 2 (T ) k k ⎝ ⎠

(52)

where i0 and j 0 represent the 3P and 1D terms of atomic oxygen, and QO2 is the internal partition function of O2. This equation leads to the expressions of the spontaneous and induced emission coefficients in the two-temperature model:

⎛ hcσ ⎝ kTtr

ησ = 2hc 2σ 3 exp ⎜ −

⎞ LTE neq ⎟ S (σ , Ttr ) N O2 χ ⎠ ⎛ hcσ ⎞ ⎟ ⎝ kTtr ⎠

κ σi = χ neq N O S LTE (σ ,Ttr )exp⎜ − 2

χ neq =

⎛ E diss + E 10D E 10D ⎞ QO 2 (Ttr ) N O2 , exp ⎜ − 2 N O 2 QO (Tve )ξ (m O ,Ttr ) kTtr kTve ⎟⎠ ⎝

(53)

(54)

(55)

where N O and NO2 are the atomic and molecular oxygen concentrations. E diss is the 0 dissociation energy of the ground electronic state of O2 and E 1 D is the energy of the 1 D term of oxygen ground configuration. The absorption coefficient cannot be determined from S LTE . It was approximated in Ref. 32 by:

κσabs = NO S LTE (σ , Tve )

(56)

2

2.2.4 Molecular Photoionization Most of the available data for this type of mechanisms correspond to room temperature and to global absorption cross section Σ . Fortunately the contribution remains small and a pragmatic estimation can be adopted:

⎛ hc ⎞ σ⎟ ⎝ kTe ⎠

ησ = 2hc 2σ 3 N mol χ neq Σ exp⎜ −

(57)

Radiation Phenomena behind Shock Waves

⎛ hc ⎞ ⎤ σ⎟ ⎥ ⎝ kTe ⎠ ⎦

⎡

κσ = Nmol Σ ⎢1− χ neq exp⎜ − ⎣

207

(58)

where the expression of χ neq follows the one given in (50) at the temperature Te. 2.2.5 Photodetachment The photodetachment process, which corresponds to Eq.(42), is far to be fully understood. However the inverse process, i.e. radiative electron attachment, was put forward to explain emission spectra of oxygen and nitrogen plasmas for temperatures around 10000K and atmospheric pressure [33]. According to the literature, the major contributions correspond to the photodetachment from the 2P term for O-, and from the 3P term of the ground configuration of N- (whose existence is under debate). The concentrations of negative ions are generally not calculated in fluid dynamics. The concentrations are thus estimated through the Saha equation from neutral atom and free electron populations at the temperature Tve for the two-temperature model. The absorption coefficient can be expressed as: ⎡

⎛ hc ⎞ ⎤ σ⎟ ⎥ , ⎝ kTve ⎠ ⎦

κσ = N i Σ i (σ ) ⎢1− exp⎜ − ⎣

(59)

where the absorption cross section Σ i (σ ) can be found in the literature [13]. The associated emission coefficients can be approximated using the Kirchhoff’s law at Tve . 2.3 Free-Free Transitions

The free-free emission (Bremsstrahlung emission) is observed when an electron is slowed down by a collision with an ion, atom or molecule. The energy of the emitted photon is equal to the loss of kinetic energy. The cross section of the process depends on the relative velocity of colliders, which can be taken equal to the electron velocity. As the electron velocities are assumed to follow a Maxwell-Boltzmann distribution at Te, the Bremsstrahlung emission coefficient and the inverse Bremsstrahlung absorption coefficient are linked by the Kirchhoff's law (Eq. (4)) with a Planck's function (Eq. (5)) at Te. The free-free emission coefficient in the field of ions are generally expressed by the classical Kramers formula corrected by a Gaunt factor g(σ ,T) which includes the quantum-mechanical correction [21]: 8 ⎛ 2π ⎞ ησ = g(σ ,T) ⎜ ⎟ 3 ⎝ 3kTe m e ⎠

1/ 2

α 2e 6 mec 3

⎛ hc ⎞ exp⎜ − ⎟ N e N ion ⎝ kTe ⎠

(60)

208

High Temperature Phenomena in Shock Waves

Where Ne and Nion are the densities of electrons and ions respectively, me is the electron mass, α is the ionization degree and . Cross sections can be found in the literature for free-free transitions involving atoms and molecules[15]. The contribution of these processes remains small. Fig.2 gives the contributions of the different bound-free and free-free processes to the absorption coefficient for an atmospheric plasma at LTE at two temperatures. At low temperature, the contribution is due to molecules. The absorption is important above 50,000 cm-1. At 1200K the main contribution is due to neutral atoms and ions. The contribution of N- photodetachment is clearly seen between 10,000 and 70,000 cm-1

Fig. 2. Continuum absorption coefficient for an atmospheric air plasma at LTE at 2000K (left) and 15000K (right)

3 Example of Application The spectra presented in the following have been obtained using the HTGR data base [13], which includes for air the radiative processes listed in Table 2. Table 2. Radiative procsses taken into account for air [15] Bound-bound transitions: - atomic species - diatomic species

N, N+, N++, O, O+, O++ N2, O2, NO, N2+

Bound-free transitions: - atomic photoionization - atomic photodetachment - molecular photoionization - molecular photodissociation

N, N+, O, O+ O-, NN2, O2, NO O2 Schumann-Runge

Free-free transitions

N, N+, N++, O, O+, O++ , N2, O2, NO

Radiation Phenomena behind Shock Waves

209

Figs.3 and 4 give the spectral emission and absorption coefficient for an atmospheric air plasma at equilibrium at two temperatures, 2000K and 12000K, respectively. These spectra have been calculated by a line-by-line approach using a sufficiently fine spectral grid to keep the spectral dynamics. A linearly variable spectral increment, varying from 0.01cm-1 at 1000 cm-1 to 0.12 cm-1 at 150,000 cm-1, has been chosen. The main contributions are given in these figures.

Fig. 3. Spectral emission and absorption coefficients an atmospheric air plasma at LTE at 2000K

At 2000K, the molecular contributions are predominant. The emission occurs mainly in the infrared with the NO infrared system. Including a small amount of CO2, which is a highly radiating molecule, would increase the emission [16]. The absorption spectrum shows that the gas is mostly transparent to the radiation up to 50,000 cm-1. The absorption is important above. Absorption of cold layers, close to the wall for example in re-entry problems, has to be taken into account. At 12000 K the diatomic molecules are dissociated. As a result, line spectra and bound-free continua from atomic species become more significant than other processes. We can observe that the emission intensity moves toward the highest wave numbers. The ultra-violet (UV) and vacuum-ultra-violet (VUV) radiation should be treated carefully at high temperatures. The absorption spectrum shows that many atomic lines are optically thick as well as the photoionization continuum for column length higher than 1 mm.

210

High Temperature Phenomena in Shock Waves

It should be noticed that the spectral range which needs to be considered for total radiative transfer is wider than the experimentally observed one in shock tubes, which usually covers the UV-Visible-near Infrared with recent extension towards near VUV [34,35].

Fig. 4. Spectral emission and absorption coefficients an atmospheric air plasma at LTE at 12000K

We discuss in the following an application to FireII, a reentry experiment, which was conducted in the framework of the Apollo program. The capsule was equipped with a total radiometer, a spectral radiometer and a calorimeter [36]. Figure 5 gives the aerothermal fields on the stagnation line, which have been calculated for the 1642.66s flight time [37]; this point corresponds to heating peak. These aerothermal fields have been obtained without taking into account the radiation. We have used them to generate local and spectral emission and absorption coefficients on the stagnation line.

Radiation Phenomena behind Shock Waves

211

Fig. 5. Temperature (left) and concentration (right) fields along the stagnation line for the flight time 1642.66s as calculated in Ref.37

The radiative transfer equation (1) has been solved along the stagnation line. Figure 6 gives the spectral incident intensity at the stagnation wall point on the 1000200,000 cm-1 spectral range. The main contributing processes are given in the different spectral ranges. The figure shows also the cumulated distribution of the incident intensity. The VUV contribution (over 50,000 cm-1) represents 58% of the total intensity. The contributions of molecular systems, atomic lines and continua are all roughly equal to one third for this flight time. The cumulated distribution of incident intensity up to 50,000 cm-1 is equal to 62.6 W. cm-2.sr-1, in good agreement with the total radiometer measurement of 63 W. cm-2.sr-1. This agreement should be considered with caution due to uncertainties in the aerothermal fields.

Fig. 6. Spectral incident intensity and cumulated distribution of the incident intensity on the stagnation wall point for the flight time 1642.66s (from Ref.38)

212

High Temperature Phenomena in Shock Waves

The left part of Fig.7 illustrates the influence of the chemical non-equilibrium on continua emission corresponding to inverse-photodissociation and inversephotoionization. Assuming chemical equilibrium leads to an overestimation of the incident intensity. Assuming equilibrium for the photodissociation leads to a higher intensity; the flow is less dissociated than the equilibrium flow. The opposite is observed for the photoionization. The right part of Fig.7 illustrates the importance of absorption due to VUV atomic lines, N2 VUV systems and molecular and atomic continua and to a lesser extent to IR lines. Neglecting the absorption overestimates the total incident intensity by a factor of 36.

Fig. 7. Effect of chemical non-equilibrium (left) and absorption (right) on the incident intensity on the stagnation wall point for the flight time 1642.66s (from Ref.38)

4 Radiative Transfer Modeling The practical prediction of radiative fluxes and radiative source terms in the energy balance equations, or in the master equations for level densities, requires generally recourse to approximate models, the complexity of which depends firstly on the optical thickness of the medium for a given radiation mechanism. When the medium can be considered as optically thin and surrounded by a relatively cold environment, local absorption can be neglected in comparison with emission. Radiation becomes then a local phenomenon that just removes energy from the material system through emission. The radiative source term in the total energy balance equation is then simply given by: ∞

S R = −PR = −4π ∫ ησ dσ .

(61)

0

In a similar manner, the radiative source term in the equation governing the density Ni of an atom level i reduces schematically to ⎛ dN i ⎞ ⎝ dt ⎠

rad

= ∑ N j Aij + N + N e A fi − N i ∑ Aik , j >i

k
(62)

Radiation Phenomena behind Shock Waves

213

where Aij designates the Einstein spontaneous emission coefficient from level i to level j, Afi the radiative recombination rate, Ne the electron number density and N+ the ion density. When the medium cannot be considered as optically thin, the absorption of radiation leads to non-local phenomena and several issues must be addressed to enable practical prediction of radiative quantities. On the one hand, spectrally resolved calculations of radiation transport require the use of a huge number of discrete wavenumbers. Approximate radiative properties are generally required. On the other hand, spatial and directional aspects of radiation transport involve also important computational times and simplified models are whished. Some simple models such as the escape factor approach can handle both aspects of spectral and geometrical complexities. This concept is discussed in the first part of this section. We discuss in the second part some approximate treatments of the spectral integration problem and, in the third part, some geometrical approximations and methods to solve the radiative transport equation. We conclude this section by a brief description of the Monte Carlo method, especially when applied to nonequilibrium radiation. 4.1 The Escape Factor Approach

The main idea in this approach consists in still using Eqs.(61) or (62) but with a correction to take into account absorption. For the total energy, equation (61) is transformed to [39] ∞

SR = −4π ∫ ησ e −κ σ R dσ , O

(63)

where R is a characteristic length of the medium. This expression is rigorous at the center of a uniform spherical medium of radius R if one neglects the radiation incoming on the sphere. It remains however very approximate for shock waves and the choice of the parameter R is not intuitive for 3D geometries. On the other hand, this approximation leads everywhere to a negative source term and cannot predict the net absorption in the cold regions. For the level number density equation, self-absorption is very important close to resonance atomic lines and it was early recognized that a specific treatment is required to account for the “imprisonment” of radiation in the medium [40,41]. Equation (62) is generally written with effective Einstein emission coefficients Ajkeff=Λjk Ajk and effective cross sections for radiative recombination where the escape factor Λjk is introduced to take into account the trapping of radiation in the medium [42]. Λjk =1 corresponds to an optically thin medium and Λjk =0 to the optically thick case where the effects of radiation vanish. The escape factor is often taken as an adjustable parameter and the results are presented in the limiting thin and thick cases (see e.g. Ref.43). The rigorous calculation of Λ requires to solve the full radiative transfer equation. Let us consider for instance the radiative de-excitation of an atom from level i to a lower level j. The effective rate due to the transition i-j can be written as N i Aijeff = N i Aij − ∫

∞

0

κ σ ij I dΩ dσ , hcσ ∫4 π σ

(64)

214

High Temperature Phenomena in Shock Waves

where κσij is the absorption coefficient due to transition i-j. Using the definition of the absorption coefficient and the relations between Einstein spontaneous emission, absorption and induced emission coefficients, the escape factor is given by Λ ij = 1− ∫

∞

0

Iσ f (σ )

1 ⎛ gi N j ⎞ − 1 dσ , 2hc 2σ 3 ⎜⎝ g j N i ⎟⎠

(65) 1

where Iσ is the directionally averaged incoming intensity, Iσ = ∫4π Iσ dΩ . Under 4π LTE conditions, and assuming that the wavenumber σ remains close to (Ei-Ej)/hc under the line profile, the last expression reduces to Λ ij = 1− ∫

∞

0

Iσ f (σ )dσ , Iσ0

(66)

which shows clearly that Λij vanishes for very thick lines since we have, locally, Iσ ≈ Iσ0 close to line center. Several other definitions of the escape factor, and other ways to use it either for radiative transfer or for population density calculations, exist in the literature. These definitions are summarized in Ref.44 where some approximate expressions may be found. Such approximate expressions are generally welcome since they avoid the solution of the radiation transport equation for several millions of wavenumbers, especially when radiation must be coupled to aero-thermal phenomena. However, the correct and self-consistent treatment of radiation transport in collisional-radiative models has been undertaken in some simple (small number of transitions) or with the Quasi-Steady-State (QSS) approximation [45-47]. 4.2 Spectral Models 4.2.1 Statistical Narrow Band Models Statistical Narrow-Band (SNB) models were first developed and widely used to study infrared molecular radiation in atmospheric and in combustion applications [48-49]. They are particularly suited to spectra dominated by a dense structure of molecular (or atomic) lines for which the statistical treatment inside a narrow band (but containing a great number of lines) is an efficient way to summarize the whole spectroscopic properties of the lines belonging to the band. We briefly recall here the principles of SNB models and discuss their possible use for nonequilibrium radiation. Let us consider first a uniform gaseous column of length l, at equilibrium at temperature T, and a spectral band Δσ containing a large number N of lines but sufficiently narrow to consider that the Planck function may be considered constant inside Δσ. Under the assumption that line centers may be considered as randomly located inside Δσ, the column transmissivity, averaged over Δσ may be written

Radiation Phenomena behind Shock Waves

τσ =

1 Δσ

⎛ W⎞ ⎟, ⎠

∫ σ exp ( −κσ l ) dσ = exp ⎜⎝ − δ Δ

215

(67)

where δ is the mean distance between two consecutive line centers δ = Δσ / N and is the mean value of the black equivalent line width given by W =

1 N

N

∞ ; Wi = ∫ (1 − exp( −κ σ ,i l ) ) d σ ,

∑W i =1

i

0

W

(68)

where κσ,i is the contribution of the ith line to the absorption coefficient. The expression of W depends in practice on the line broadening regime (Doppler, Lorentz or Voigt line shapes) and on the statistical distribution of line intensities characterized by a probability density function P(S). Here, the intensity S for a transition u-l is defined by S=

(N l Blu − N uBul ) hσ ul , p

(69)

where p is the partial pressure of the emitting/absorbing species. Figure 8 shows a typical behavior of the curve of growth − ln( τ) = W / δ as function of the column length l for CO2 lines at 2000K and 100 Pa. A linear growth is first observed in the optically thin limit, followed by a Doppler dominated growth, then a transition regime, and finally a Lorentz regime characterized by a root square power law l1/2 resulting from the Lorentz shape which dominates line wings at high optical thicknesses. There are no general analytical expressions for WV in the general case of Voigt line shapes. Several approximations exist in the literature and the following one, due to Ludwig et al [50], yields generally accurate results WV

δ

=

plS

δ

1 − Ω−1/ 2 , −2

(70) −2

⎡ ⎛ W / δ ⎞2 ⎤ ⎡ ⎛ W / δ ⎞2 ⎤ L D Ω = ⎢1 − ⎜ ⎟ ⎥ + ⎢1 − ⎜ ⎟ ⎥ −1 , ⎢⎣ ⎝ plS / δ ⎠ ⎥⎦ ⎢⎣ ⎝ plS / δ ⎠ ⎥⎦

where

S=

1 N ∑S N i=1 i

(71)

is the mean line intensity and W L and W D are the mean equivalent

line black widths in the pure Lorentz and Doppler regimes, respectively. Comparison between different line intensity distribution functions show that the closest agreement with line by line calculations is obtained with the inverseexponential tailed distribution for the Lorentz profile, and the exponential distribution for Doppler profile. These distributions lead to

216

High Temperature Phenomena in Shock Waves

WL

δ

WD

δ

=

=

⎞ 2γ L ⎛ ( S / δ ) pl ⎜ 1+ − 1⎟ , ⎜ ⎟ δ ⎝ γL /δ ⎠

γ D ⎛ ( S / δ ) pl ⎞ E⎜ ⎟ with δ ⎝ γD /δ ⎠

E ( y) =

1

π

(72)

+∞

y e −ξ

−∞

1 + y e −ξ

∫

2

2

dξ ,

(73)

where γ L and γ D are the mean Lorentz and Doppler line half-widths respectively, and

δ is an effective mean distance between two consecutive lines.

Fig. 8. Example of the curve of growth for CO2 transmission at 2000K

Equations (67, 70-73) show that SNB models lead to simple expression of the average transmissivity of uniform columns at equilibrium. Their extension to nonuniform columns, yet still at LTE, requires further approximations such as the Curtis-Godson approximation which are not detailed here (see Refs.51-54). The practical use of SNB models requires to rewrite the RTE in terms of transmissivities. It is easily shown for instance that for a medium at LTE, and if one neglects the incoming radiation at the boundaries, the RTE for the intensity averaged over Δσ may be written in the integral form

Radiation Phenomena behind Shock Waves

s

Iσ (s) = ∫ Iσ0 (s') 0

∂τσ (s',s) ds' . ∂s'

217

(74)

For a gaseous medium under NLTE, the source function should be replaced by the ratio ησ/κσ (s’) but this quantity may exhibit some strong variations with the wavenumber that could be spectrally correlated with the transmissivity. Figure 9 (left part) shows an example of the high resolution variations of ησ/κσ (s’) for the N2 Birge-Hopfield-2 system. These variations result from the mixing of rotational lines of different vibrational bands. The right part of this figure shows however that these variations are, to a good approximation, not correlated with the absorption coefficient or the transmissivity of the medium [53].

Fig. 9. Left part: Planck function at 10000K and 20000K, and the ratio ησ/κ σ in nonequilibrium conditions (Tv=Tel=10000K, T=Tr=20000K) for N2 Birge-Hopfield-2 system. Right part: net emission, averaged over spectral bands 1000 cm-1 wide, for the same N2 system and temperatures, and a uniform column of length 10 cm (from Ref.53).

The RTE, averaged over Δσ, may then be written in the form

Iσ (s) = ∫

ησ ∂τ (s') σ (s',s) ds' . κσ ∂s'

s

0

(75)

The above model has been implemented for the optically thick diatomic electronic systems of air plasmas in Ref.53 and was shown to predict accurately radiative transfer in Earth reentry flows. Model parameters

have been tabulated

versus two temperatures Ttr = Trot and Tv = Te, but the model can be applied to any nonequilibrium distribution of electronic levels in diatomic molecules. Note also that similar statistical models were applied to atomic lines in Ref. 8.

218

High Temperature Phenomena in Shock Waves

4.2.2 Global Models For media under LTE conditions, global radiation quantities, i.e. intensities or fluxes integrated over the whole spectrum, may be directly calculated from the distribution function of the absorption coefficient, weighted by the Planck function

F(k,Tp ,T) =

π σ SB Tp4

∫σ κ /

σ

(T )≤k

Iσ0 (Tp )dσ ,

(76)

where σSB is the Stefan-Boltzmann constant. This distribution function depends on the local medium temperature T and on another temperature Tp used to evaluate the Planck function. The integration of radiation is carried out over k values instead of σ in these models. where If the absorption spectrum can be assumed to be separable, i.e. designates the local state of the medium (temperature, pressure and concentrations) the set of elementary wavenumber intervals for which ξ(σ) belongs to [ξ, ξ +dξ] does not depend on the spatial position and the integration of the radiation intensity over k values, or more precisely over ξi values, is rigorous. This leads, after discretization of ξ space into N intervals [ξ j-, ξ j+], each one characterized by a representative value ξ j, to Itot =

∑I

j =1,...,N

j

∂ Ij ⎡ ⎤ σ T 4 (s) = k j ( χ ) ⎢a j ( χ ) SB − I j ⎥, ∂s π ⎣ ⎦

j = 1,..,N

(77)

where T is the temperature of the state χ kj(χ)=ξ jΦ(χ), and the weights aj(χ) are given by a j (χ ) =

π σ SBT 4

∫σ ξ /

− Φ ( χ ) ≤ κσ j

≤ ξ +j Φ ( χ )

Iσ0 (T ) d σ = F ( ξ +j Φ ( χ ), T , T ) − F ( ξ −j Φ ( χ ), T , T )

and satisfy the normalization relation

∑

N j =1

(78)

a j ( χ ) = 1.

In practice, and even at LTE conditions, the scaling approximation κσ (T)=ξ(σ)Φ(χ) is not satisfied in shock wave applications for two reasons. First, species concentrations strongly vary in the shock layer and the absorbing and emitting species are not the same everywhere. The second reason comes from the variations with temperature of line intensities. Even for a single species, the populations Nu of the excited emitting states vary strongly with T, which leads to a non-scaling behavior in the case of important temperature gradients. One remedy to this problem was proposed in Ref.55. It consists in splitting the actual spectra into fictitious spectra, each one gathering the radiation mechanisms and/or species which have similar variations with temperature. A correlation approximation remains however necessary to take into account the actual variations of the absorption coefficients. The α discretization of k space was first carried out in a reference condition denoted χ ref

Radiation Phenomena behind Shock Waves

219

α with a temperature Tref for a given fictitious species α, and the absorption coefficient in a current condition χ was deduced from the implicit relation

α α Fα (kαj ( χ ),Tref , χ )= Fα (kαj ( χαref ),Tref , χαref ) ,

(79)

where Fα is defined in a similar manner as in Eq.(76) for the spectrum of species α. α Fα (k,Tref , χ) =

π α4 σ SB Tref

∫σ κ /

α σ

( χ )≤k

Iσ0 (Tref )dσ

(80)

This led to the Mixture Absorption Distribution Function (MADF) model where the RTE has to be solved for instance 63=216 times (N=6 and 3 fictitious species)[56]. A similar approach was developed by Zhang and Modest [57] and called Multi-Scale Full-Spectrum Correlated-k Distribution. These authors introduced an overlap factor between the different spectra in order to reduce the number of RTEs to NM, where M is the number of fictitious species (or scales), instead of NM. This last approach was recently applied to CN radiation in nonequilibrium flows [58]. 4.3 Geometrical Treatment of Radiative Transfer

The Ray Tracing (RT) method, either in its deterministic or statistical form (given in the following subsection), is the most rigorous method for the geometrical integration of radiative transfer. It consists simply in discretizing the 4π total solid angle into M elementary solid angles ΔΩm, centered around the directions um, and performing angular integration of a given radiation quantity G according to M

∫ G(u)dΩ= ∑ω

4π sr

m=1

m

M

M

m=1

m=1

G(um ) = ∑ ΔΩm G(um ) = ∑ ∫

∫

Δϕ m Δθ m

sin(θ)dθ dϕ G(um ) ,

(81)

where θ and ϕ are the spherical angles. Rays are launched from each volume element and from each surface element of the boundaries, and the radiative transfer equation is solved along each ray according to

(

X

)

X

(

X

)

Iσ ( X , um ) = Iσ (0)exp −∫ κσ (s) ds + ∫ ησ exp −∫ κσ (s ') ds ' ds , 0

0

s

(82)

where s is the abscissa along the ray characterized by the unit vector um and s=0 designates the intersection point of this ray with the boundaries of the calculation domain. Such detailed geometrical calculations are however often prohibitive and approximate methods have been developed. A very brief overview of these methods is given below and the details may be found in classical textbooks dedicated to radiative transfer (see e.g., Refs.59-61).

220

High Temperature Phenomena in Shock Waves

4.3.1 The Discrete Ordinate Method (DOM) In this method, the discrete directions um and weights ωm in Eq.(81) are determined from a quadrature formula. In particular, ωm has no more the meaning of an elementary solid angle. Among the different available quadratures, the more popular is the SN quadrature [62] in which the directions are chosen from symmetry considerations. If a direction um, characterized by its direction cosines (μm, ηm, ξm) belongs to the set of quadrature directions, the 8 symmetric directions (±μm, ±ηm, ±ξm) must also belong to this set. The directions are thus defined on only one octant of the sphere. If n designates the number of possible latitudes of a direction cosine in an octant, the order of the quadrature is N=2n and the total number of directions is M=N(N+2). The weights are chosen in the SN quadrature in such a way that some moments are conserved. This leads for instance to

∫π

dΩ = 4π

∫π

μ 2 dΩ =

∫π

μ dΩ = π ⇒

4 sr

4 sr

2 sr

M

∑ω

⇒

m =1

4π 3

⇒

m

= 4π

M

∑ω m =1

m

M

∑ω

m m =1/ μ m >0

μ m2 =

4π 3

(83)

μ m = π , ....

These considerations are generally sufficient to determine the set of directions and associated weights. Another important difference with the ray tracing method is that the RTE, for a given direction, is solved by a finite volume type method instead of direct integration along a ray. This leads naturally to some numerical difficulties such as numerical diffusion or instabilities, depending on the used interpolation schemes. 4.3.2 Spherical Harmonics (PN) and Related Methods A decomposition of the radiation intensity on the spherical harmonics basis is used in these methods in order to simplify its directional dependence: N

Iσ (r,u) = ∑

l

∑

l = 0 m =− l

Al,mσ (r)Ylm (u) .

(84)

For a given wavenumber σ, the decomposition coefficients Alm only depend on the spatial position r and are governed by equations obtained by replacing this decomposition in the RTE and taking different moments of the result. The resulting equations are coupled partial differential equations that may be solved by classical finite volume or finite element methods. The most popular P1 method corresponds to a truncation at the first order N=1 in the decomposition. This yields the following equation for the zero order moment Gσ :

Radiation Phenomena behind Shock Waves

⎛1 ⎞ −∇⋅ ⎜ ∇Gσ (r)⎟ + 3κσ Gσ (r) =12π ησ (r), ⎝κσ ⎠

Gσ (r) = ∫

4π st

Iσ (r,u) dΩ.

221

(85)

The directional intensity, and then the radiative flux and its divergence, may be calculated from Iσ (r,u) =

1 4π

⎞ ⎛ 1 u ⋅ ∇Gσ (r)⎟ , ⎜ Gσ (r) − κσ ⎠ ⎝

(86)

once the above second order partial differential equation is solved for Gσ with appropriate boundary conditions. Let notice that Eqs.(84-86) are valid both for local equilibrium and nonequilibrium conditions. The source term ησ(r) can be replaced by κσ(r)Iσ0(r) under LTE conditions. The relative simplicity of the P1 approximation makes it very popular. The simple angular decomposition and truncation of the intensity to the first order leads generally to some discrepancies, in particular near the domain boundaries. Several studies show that the P1 approximation provides more accurate results for optically thick media than for optically thin ones (see e.g. Ref.56). The method can thus be used in combined manner with other methods to treat different spectral regions. It can be used for instance for wavenumbers close to the optically thick line centers while the ray tracing method, or Monte Carlo method, is used for the other non-thick parts of the spectrum. The use of higher order (P3) approximation leads of course to higher accuracy. Some attempts have been recently proposed to improve the efficiency of the PN approximation. In the SPN method developed by Larsen et al [63], the operator 1/ (1 + ε / κ σ u ⋅∇ ) , where ε = 1/(k ref Lref ) is a representative optical thickness, is developed for an optically thick wavenumber σ, in a power series. The resulting equations, for a truncation at the third order, are partial differential equations for two quantities (φ1σ and φ2σ) which can be written in non-dimensional form

−∇ ⋅

ε 2μ12 ∇φ1σ + κ σ φ1σ = 4πησ κσ

−∇ ⋅

ε 2μ 22 ∇φ 2σ + κ σ φ 2σ = 4πησ , κσ

(87)

with μi = (3 ± 6 /5) /7, i = 1,2 . This system of two partial differential equations is in fact coupled by the boundary conditions for φ1σ and φ2σ . These functions allow to calculate the radiative source term (in dimensional form) according to

222

High Temperature Phenomena in Shock Waves

∞ ⎛ 1 ⎞ PR = − ∫ ∇ ⋅ ⎜ ∇ ( a1φ1σ + a2φ2σ ) ⎟ dσ 0 κ ⎝ σ ⎠

(

(88)

)

where the constants ai are given by ai = 5 ∓ 5 6 30 , i = 1, 2 . The use of this method was shown to improve the P1 approximation for the treatment of photoionization in the problem of streamer propagation [64]. 4.3.3 The Tangent Slab Approximation When shock waves may be approximated as planar 1D media, the radiation transport can be simplified using analytical integrations over the two dimensions where the medium is assumed to be infinite. if μ denotes the direction cosine μ=cosθ, and θ is the angle between a propagation direction and the normal to the 1D layer, the radiative intensity at an abscissa z between the two boundaries (z=0 and z=E) is given by c − c'σ dc'σ ησ (c' )exp(− σ ) ,μ ≥ 0 κσ σ μ μ , c c −c c' −c dc' ησ σ Iσ (z, μ ) = Iσ ,2 exp(− σ ,E σ ) + ∫ (c' )exp(− σ σ ) σ , μ ≤ 0 cσ μ κσ σ μ μ

Iσ (z, μ ) = Iσ ,1 exp(−

cσ

μ

)+∫

cσ

0

(89)

,E

where Iσ,1 and Iσ,2 are the intensities leaving the boundaries 1 (z=0) and 2 (z=E), and z

cσ is the optical path given by cσ = ∫0κσ (z') dz' . Taking the moments of these intensities (multiplying by μ0 and μ1 and integrating over 4π steradians) leads respectively to the incoming intensity uσ,z and to the local radiative flux qσ,z along the z direction 1

uσ , z = 2π ∫ Iσ d μ = 2π ⎡⎣ Iσ ,1 E2 (cσ ) + Iσ ,2 E2 (cσ , E − cσ ) ⎤⎦ −1 +2π ∫

cσ

0

cσ η ησ σ (c 'σ ) E1 (cσ − c 'σ )dc 'σ + 2π ∫ (c 'σ ) E1 (c 'σ − cσ )dc 'σ cσ κσ κσ ,E

(90)

1

qσ , z = 2π ∫ Iσ μ d μ =2π ⎡⎣ Iσ ,1E3 (cσ ) − I σ ,2 E3 (cσ ,E − cσ ) ⎤⎦ −1

+2π ∫

cσ

0

cσ η ησ σ (c ' ) E2 (cσ − c 'σ ) dc 'σ − 2π ∫ (c ' ) E2 (c 'σ − cσ )dc 'σ cσ κσ σ κσ σ ,E

(91)

where the exponential-integral functions En are defined by 1 ⎛ c⎞ E n (c) = ∫ μ n −2 exp⎜ − ⎟ dμ 0 ⎝ μ⎠

(92)

Radiation Phenomena behind Shock Waves

223

It is worth noticing that Eq.(90) provides the incoming intensity that must be used to compute the absorption term in the balance equations of population densities of energy levels. The radiative source term may also be calculated from the intensity following

dqσ , z dz

= −2π κ σ ⎡⎣ Iσ ,1 E2 (cσ ) + Iσ ,2 E2 (cσ , E − cσ ) ⎤⎦ − 2πκσ ∫

cσ , E

0

ησ (c ' ) E ( c − c 'σ ) dc 'σ + 4πησ (cσ ) . κσ σ 1 σ

(93)

In practice the first and second order exponential-integral functions can be calculated from the third order function using the general formula dE n (c) = − E n −1 (c) n = 1, 2, 3, ... dc

(94)

Several studies have addressed the accuracy of the tangent slab approximation in the case of shock waves in front of atmospheric entry vehicles. The discrepancies are generally smaller in the region close to the stagnation line and higher in the detachment regions. The typical accuracy is about 10 to 30% in re-entry applications in the peak-heating region. 4.4 The Monte Carlo Method

The Monte Carlo method (MCM), applied to radiative transfer, can accommodate spatial, directional, and spectral aspects of the problem. Radiation transport is simulated stochastically in MCM by tracing energy bundles from emitting points until ending points. Bundle characteristics such as direction of emission, wavenumber of the bundle, etc, are chosen from random numbers and absorption is either considered as a local phenomenon with a randomly chosen point along the propagation direction, or calculated in a deterministic way from the length crossed by the bundle in a given cell. The random choice of each event follows of course probability densities in order to be physically consistent. The power emitted by a cell volume Vi is given by

Qie = ∫ dV ∫ Vi

π

θ =0

sinθ dθ ∫

2π

ϕ =0

dϕ ∫

∞

σ =0

ησ dσ ,

(95)

where θ and ϕ are the polar and azimuthal angles. The probability density P(M, θ, ϕ, σ) for a bundle to be emitted at point M, in a direction (θ, ϕ), with a wavenumber σ, is simply given by P( M,θ ,ϕ ,σ ) dV d θ d ϕ dσ =

dV sin θ dθ dϕ ησ dσ V i πsin θ d θ 2πdϕ ∞η d σ ∫ ∫ ∫ σ 0

0

0

(96)

224

High Temperature Phenomena in Shock Waves

and appears to be the product of independent probabilities related to each variable. As a general rule, the cumulated probability density function of any random variable may be seen as a random variable itself, with a uniform distribution in the range [0, 1]. Thus, the variables θ, ϕ and σ may be chosen from uniform random numbers Rθ , Rϕ , Rσ in this range according to

1 − cosθ ϕ Rθ = , Rϕ = , Rσ = 2 2π

∫ ∫

σ

0 ∞ 0

η (σ ') dσ ' η (σ ') dσ '

.

(97)

The choice of a point M inside Vi depends on the type of the used mesh. While the determination of θ and ϕ is performed analytically, the choice of σ is much more complex and may consume important CPU times if spectrally resolved calculations are performed with line structures. The inversion of the third relation in Eq.(97) can be drastically accelerated if one uses intermediate pre-tabulated values of Rσ . When absorption of the energy bundle is treated as a local phenomenon, advantage is taken from the fact that the transmissivity of the gaseous column between the emitting and absorbing points represents also the cumulated probability that the bundle escapes from the column. The transmissivity τσ is then considered as a uniform random variable in the range [0, 1] and the length labs traveled by the bundle is deduced from − ln(τσ ) = ∫

labs

0

κσ (s) ds ,

(98)

where the abscissa s = 0 designates the location of the emitting point. As a matter of fact, the bundle may be outside the calculation domain depending on the value of labs. In that case, miscellaneous surface events may occur such as absorption or reflection. It is found generally that the deterministic treatment of absorption is to be preferred to local deposition since it reduces significantly the statistical noise. When a bundle crosses a cell of volume Vj, entering it at length lin and leaving at length lout, its power is reduced by the fraction (τ σ (0, lin ) − τ σ (0, lout )) which is added to the power absorbed by cell Vj. The inspection of a given bundle is followed in this approach until the power carried by it becomes smaller than a cutoff criterion. In the most common MC methods, the number of bundles issued from each cell Vi or each surface element Sj is taken proportional to the power Qei or Qej emitted by this element and the number of bundles launched from Vi is calculated according to N i = NQie /(

Nv +Ns

∑ Q ) where N is the total number of bundles chosen for the simulation k =1

e k

and Nv and Ns stand respectively for the total number of volume and surface elements. The initial power carried by the different bundles is in this way almost the same. The net radiative power for a volume element is simply calculated at the end of the simulation as Pi = (Qia − Qie ) /Vi , where Qia is the total power absorbed by Vi, originating from all volume and surface elements.

Radiation Phenomena behind Shock Waves

225

When statistical narrow-band models are employed for the spectral behavior, the spectral correlation between emitted, transmitted and absorbed radiation must be treated carefully [65-66]. Taking the derivative of Eq. (74) or (75) with respect to s shows that absorption at abscissa s of radiation emitted at abscissa s’ requires the use of the second derivative ∂ 2τ σ / ∂s ∂s ' . In a discretized form, and assuming as discussed in Section 4.2.1 that η/κΔσ ≈ηΔσ /κΔσ in nonequilibrium conditions, the power of a bundle emitted at point si of the volume Vi and absorbed between the abscissa sjand sj+ of volume Vj is given by em,i Pabs ,j =

Qie Ni κ i

Δσ

)(

(

)

⎡ τ Δσ (s + δ s, s− ) − τ Δσ (s , s− ) − τ Δσ (s + δ s, s + ) − τ Δσ (s , s + ) ⎤ i j i j i j i j ⎥⎦ δ s ⎢⎣

(99)

In this expression, δs is the length of an elementary column in the source volume Vi. This length must be chosen sufficiently small so that the elementary column is optically thin. More details on the application of SNB models and MC methods in the framework of atmospheric entry applications may be found in Refs.65-66. Monte Carlo radiation simulations are known to possess a slow convergence rate scaling as 1/ N , where N is the total number of bundles. In order to control the convergence of MC simulations, N is generally subdivided into a number of M samples. Each sample m provides an estimation Pi,m of the radiative power at cell i and, at the limit of large numbers, an estimation of the variance σ 2 of the mean value Pi is given by

σ2 =

M 1 ⎡ Pi, m − Pi ⎤ ∑ ⎦ M ( M − 1) m=1 ⎣

2

(100)

5 Radiation and Flow-Field Coupling As stated above, radiative field strongly depend on the thermochemical state of the flow field, through the radiative properties κσ and ησ. Various models have been used to describe nonequilibrium thermochemical flows, from direct simulation Monte Carlo (DSMC) methods used to simulate high Mach numbers re-entry flow field at high altitude [67], to continuum fluid flow multi-temperaure [68] approaches, where the different energy modes (i.e. translation-rotation, vibration, electronic,…) are assumed to be characterized by Boltzmannian populations at distinct temperatures (Ttr, Tv, Tel, …). More detailed vibrational-specific or electronic-specific approaches [69] have been developed to account for non-Boltzmannian populations of levels of some specific modes. In such generally called collisional radiative description, these specific populations become state variables which can be determined in a full coupled manner with other aerothermal variables [70] or more usually as post-processing of the aerothermal fields (densities, velocities, temperatures) using additional approximations such as the QSS one [71], or the Lagrangian approach [72].

226

High Temperature Phenomena in Shock Waves

Once the populations of the energy levels of the radiating particle are determined from the thermochemical state of the fluid, emission and absorption coefficient can be evaluated. The radiative transfer equation may then be solved in order to obtain the field of radiative power PR and the distribution of the radiative flux qR on the boundaries of the medium. In return the radiative field affects the thermochemical state of the flow through source terms in energy balance and population balance equations. First, the radiative power - PR acts as a source term in the total energy balance equation. In multi-temperature approach, the radiative power - PR has moreover to be split into various relevant contributions which should appear as source terms in the energy balance equations of the various modes. In many approaches, where a unique temperature is used to describe vibrational and electronic energy modes [73-75] the total radiative power has been used as source term in both the total energy and the electro-vibrational energy balance equations. It has been generally observed that in near-equilibrium situations, radiation decreases the “equilibrium” temperature in the shock layer, increases the density, and thus reduces the shock standoff distance [73,74,76]. On the contrary, for strong non-equilibrium shock layer with significant ionization, radiation has been shown to reduce the electro-vibrational temperature and to increase the heavy particle translational one and thus the shock standoff distance [74]. The wall radiative fluxes are generally reduced in all situations since the electrovibrational temperatures are lowered. As pointed out in Ref.77, the coupling of radiative transfer to aerothermal fields through source terms in energy equations is usually achieved using an iteration procedure and a slight under-relaxation of the variations of the radiative source terms. In this last reference, wall ablation effects and their coupling to the flow simulations are investigated for various entry conditions (Fire II …). An additional coupling mechanism is thus considered, since the wall temperature in this model is strongly determined by the radiative flux. Here again, the coupling of the ablation model to the radiative transfer is obtained through a strongly under-relaxed iteration procedure. In more detailed approaches, radiative transfer should be accounted for as populating and depleting mechanisms of energy levels. Bound-free transitions such as photoionization or photodissociation even contribute as source terms in the balance equation of involved radiating species. In the frame of collisional radiative models, specific population equations such as Eqs.(62,64) have to be solved in a selfconsistent manner with the radiative transfer equation. Such studies remain quite sparse. In the case of calculations devoted to the Huygens probe entry into Titan, Johnston [75] evaluated escape factors related to the CN (B-X) violet system from a tangent slab Radiative Transfer approach iteratively coupled to a QSS approach. The B state populations obtained are slightly larger than those obtained by neglecting absorption (escape factor equal to 1) providing an increase of the radiative wall fluxes of about 16 %. More recently, in calculation cases representative of the Stardust Sample Return capsule, Sohn et al [47] have evaluated escape factors associated to the main resonant O and N transitions using a Monte Carlo RTE solver coupled to the NEQAIR QSS approach. Aerothermal fields were obtained from a DSMC simulation. Starting from escape factor equal to one, convergence of the relevant atomic populations was obtained at the first iteration; these populations was shown to be increased by a factor of two in comparison with the optically thin approach. In a similar way, the radiative wall fluxes also increased by a factor of two.

Radiation Phenomena behind Shock Waves

227

6 Conclusion and Perspectives This brief overview shows the complexity of radiation phenomena in strong shock waves, especially under non-equilibrium conditions. Several experimental, theoretical and numerical studies were devoted to this problem during the last two decades to support engineering aspects in the planned spatial missions. Modeling of nonequilibrium radiation remains however a challenging issue and further work is needed in several directions: - Uncertainties remain on the spectroscopic data in the UV and VUV regions where measurements are scarce or even lacking. - As the radiation is closely linked to the thermochemical state of the gases, a better understanding of the excitation processes in the flow is required to enable reliable radiation transport predictions. - The coupling between radiation and aerothermal fields, on one hand, and with energy level populations, on the other hand, is far to be fully understood. Selfconsistent modeling of these coupled phenomena is a formidable task and has not been fully undertaken yet. - The development of approximate and reliable spectral models should enable accurate prediction of radiation and aerothermal fields in complex 3D geometries. - The use of ablative thermal protection system, which produces pyrolysis gases and other ablative products, gives rise to additional difficulties. The radiative properties of products have to be determined. These products may absorb radiation from the shock layer and reduce the radiative heat flux to the surface through radiation blockage.

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66. Lamet, J.M., Perrin, M.-Y., Soufiani, A., Rivière, P., Tessé, L.: In: Proc. Third Int. Workshop on Radiation of High Temperature Gases in Atmospheric Entry. ESA, Heraklion (2008) 67. Ozawa, T., Zhong, J., Levin, D.A.: Phys. Fluids 20, 046102 (2008) 68. Gnoffo, P.A., Gupta, R.N., Shinn, J.L.: Conservation Equations and Physical Models for Hypersonic Air Flows in Thermal and Chemical Nonequilibrium, NASA TP-2867, NASA Langley Research Center, Hampton, VA 23665-5225 (1989) 69. Capitelli, M. (ed.): Non-equilibrium vibrational kinetics, Topics in Current Physics, vol. 39. Springer, Heidelberg (1986) 70. Panesi, M., Magin, T., Bourdon, A., Bultel, A., Chazot, O.: Journal of Thermophysics and Heat Transfer 23, 236 (2009) 71. Park, C.: AIAA Paper 84-0306 (1984) 72. Magin, T.E., Caillault, L., Bourdon, A., Laux, C.O.: J. Geophys. Research 111, E07S12 (2006) 73. Gökçen, T., Park, C.: AIAA paper 91-0570 (1991) 74. Hartung, L.C., Mitcheltree, R.A., Gnoffo, P.A.: J. Thermophys. Heat Transfer 8(2), 244 (1994) 75. Johnston, C.: Nonequilibrium Shock-Layer Radiative Heating for Earth and Titan Entry. PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia (November 17, 2006) 76. Kay, R.D., Gogel, T.H.: AIAA Paper 94-2091 (1994) 77. Gnoffo, P.A., Johnston, C.O., Thompson, R.A.: AIAA Paper 2009-1399 (2009)

Chapter 6

Structure of Shock Waves A.A. Raines1 and F.G. Tcheremissine2 1

2

Feza Gursey Institute, Istanbul, Turkey Dorodnicyn Computing Center of RAS, Moscow, Russia

1 Introduction Shock wave that appears from the solution of gas dynamic Euler equations as a discontinuity of gas parameters related by Rankine-Hugoniot conditions, in reality presents a narrow layer inside which the dissipative processes form the transition from one thermodynamically equilibrium state to another one. The fastest dissipative process is due to the elastic collisions between gas molecules. It imposes the characteristic length scale of the molecular mean free path and makes the hydrodynamic equations inapplicable for rigorous analysis. Hence, the application of computational methods of the kinetic theory is required. For gas mixtures and polyatomic gases additional dissipative processes of lower rate should be considered as well. For inert mixtures it is a collisional momentum and energy exchange between species of different masses, and for polyatomic gases the energy transfer between translational and internal degrees of freedom. At the microscopic level, the high nonequilibrium shock wave layer is described by velocity distribution functions of the components of the gas, and by internal energy spectra. At the macroscopic level, the shock wave structure can be represented by the densities of gas components, their velocities, kinetic temperature tensors of the gas as a whole and of the gas components, by the temperatures of the internal energy modes, and by the different energy fluxes. Besides its physical interest as an example of high non-equilibrium layer, the problem of the shock wave structure presents an important benchmark for computational methods of the kinetic theory. Its advantage consists of its onedimensional structure in the physical space, simple boundary conditions, and property of conservation of mass, momentum, and energy fluxes that can be used for verification of computations. It is also important that there exists good experimental data [1,2]. Historically, the problem has been used for the development of a number of numerical methods and approximate kinetic theories. In the case of a pure gas, various methods were applied: moment methods [3], solution of relaxation equations [4-7], direct simulation Monte Carlo (DSMC) method (in [8,9], later in [10-13], and their references), and in [14-15]. The first numerical solutions of the Boltzmann equation [16] were obtained by Nordsieck, Hicks, and Yen [17-19] and then in [20]. Later, the problem was solved by discrete-ordinate methods for the Boltzmann equation with different techniques of evaluation of the

232

High Temperature Phenomena in Shock Waves

collision integral: use of high order polynomial approximation of the distribution function in the velocity space [21,22], with application of the polynomial correction in the course of computations for complying with the conservation laws [23], and use of the conservative projection method [24-27]. Analysis of the velocity distribution function in an infinitely strong shock wave was done in [28]. For polyatomic gases, experimental measurements were made in [1,29-33] and computations in [1,34,35] by DSMC, in [36] with model equation, and in [37-39] by solving the generalized Boltzmann equation. Review of DSMC phenomenological models for rotational relaxation is presented in [40,41]. For binary gas mixtures, the shock wave structure has been investigated experimentally [42,44] and theoretically by different approaches: with moment methods [45,46], with fluid-dynamics models [47,48], by solving model equations [49,50], by DSMC method [51,52], (see [12] and references therein.). The above cited method [22] was extended to binary gas mixtures [53], the method with polynomial correction [23] was also applied to binary gas mixtures [54,55], the method [25,26] was first extended to binary gas mixtures [56,59], then to 3-4-component mixtures [60], later the computations for 3-component mixtures were repeated by the same method [61]. Here, the shock wave structure is computed by the Conservative Projection Method (CPM) [24-26]. The method is based on a special projection technique for evaluation of the collision operator that provides its explicit discrete form which approximates the continuum operator at a set of nodes in the phase space. The computed discrete collision operator is conservative for density, momentum and energy, and is equal to zero when the solution has a form of Maxwellian distribution function. The latter feature strongly increases the efficiency of calculations, especially in near-equilibrium parts of the flow. The integration grid for evaluation of the collision integrals are given by the method [62] which provides more uniform and more efficient distribution of the grid nodes than obtained by random number generators. The differential part of the Boltzmann equation is approximated by the flux conservative finite-difference scheme of the 2nd order [63]. In this scheme the transport of mass, momentum and energy between the nodes of the configuration space is realized in a conservative way.The method was extended to gas mixtures [57] and to gases with internal degrees of freedom, where it can incorporate real physical parameters of molecular potential and internal energy spectrum [64]. It was also applied for joint solution of the Boltzmann equation and Navier-Stokes equations [65]. A simple model for the rotational relaxation in the framework of the CPM was proposed [66]. The discrete-ordinate method introduced in [22] for a single gas was extended in [53] to the case of a binary gas mixture. In this approach the collision integral is approximated by a decomposition of the distribution functions on piecewise quadratic functions by the longitudinal molecular velocity ξ x , and a system of Laguerre polynomials by the transversal velocity ξr . The conservation laws are not satisfied exactly but with an accuracy sufficient for stable calculations. In [53] the authors have analysed the accuracy of computations, defined a convergence criterion and obtained high precision results by using powerful computers.

Structure of Shock Waves

233

2 Methodology of Computations The SW structure is studied on the basis of the Boltzmann kinetic equation for a monatomic gas and a mixture of monatomic gases, and with the use of the Generalized Boltzmann Equation (GBE) for a polyatomic gas having internal degrees of freedom. The kinetic equation is solved by a deterministic finite-difference method on fixed grids in velocity and configuration spaces. For a long time the main deficiency of the methods of direct solution of the Boltzmann equation has consisted in non-conservative evaluation of the multidimensional collision integral that produced spurious sources of mass, impulse, and energy. The problem has been solved [24] by application of a special projection technique for conservative evaluation of the collision integral. The method was considerably improved in [26] by inclusion of “inverse collisions” that strongly increases the accuracy of computations in near equilibrium parts of the flow. Further development consisted of improvements of some numerical techniques for accelerating the calculations and of extension of the method to gas mixtures and molecular gases with internal degrees of freedom [27,57,64]. The detailed description of the method for a simple gas can be found in [27]. A short description of the method is given below. 2.1 Solution of the Boltzmann Equation for a Pure Monatomic Gas The Boltzmann kinetic equation may be written in the following form ∂f ∂f +ξ =I ∂t ∂x

(2.1.1)

The distribution function f (ξ, x, t ) and the collision integral I (ξ , x , t ) are defined in 6dimension phase space (ξ, x ) and depend on time t . The collision integral, omitting variables x and t , can be presented in the standard form I (ξ) =

2π

bm

0

0

∫ ∫ ∫

R3

( f ' f *' − ff * ) gbdbd ϕ dξ *

The functions f ' and f *' contain post-collision velocities

(2.1.2)

ξ ' , ξ*'

which are

determined for a given molecular potential by velocities before collision ξ , ξ* an impact parameter b , g =| ξ* − ξ | and an angular parameter of the collision ϕ . The equation (2.1.1) is solved in a domain Ω of a volume V of the velocity space on a grid of N 0 equidistant nodes ξ γ with a step h. In configuration space an arbitrary discrete grid x i is applied. In the basis of Dirac δ–functions the distribution function and the collision integral can be presented in the following form

234

High Temperature Phenomena in Shock Waves N0

f (ξ , x, t ) = ∑ fγ ( x, t )δ (ξ − ξ γ ) γ =1

,

N0

I (ξ, x, t ) = ∑ Iγ (x, t )δ (ξ − ξγ ) γ =1

After the evaluation of the collision integral, the problem is reduced to a system of linear equations ∂fγ ∂t

+ ξγ

∂fγ ∂x

= Iγ

(2.1.3)

For construction of the projection method the integral (2.1.2) at a point ξγ is written in the form I γ ≡ I ( ξγ ) =

2π

*

R3

Using

the

bm

∫ ∫ ∫ ∫ δ (ξγ − ξ )( f R3

0

'

f*' − ff* ) gbdbdϕ dξ*dξ

0

φ (ξγ ) = δ (ξ − ξγ ) + δ (ξ* − ξγ ) − δ (ξ' − ξγ ) − δ (ξ*' − ξγ )

notation

and

knowing the properties of the collision integral, one can write this integral in the symmetric form Iγ =

1 4 R∫3

2π

bm

∫ ∫ ∫ φ (ξγ )( f R

3

0

'

f*' − ff* ) gbdbdϕ dξ*dξ

(2.1.4)

0

To evaluate (2.1.4), we define the domain Ω × Ω × 2π × bm , in which a uniform integration grid ξ α , ξ β , bν , ϕν of Nν nodes is built, so that ξα and ξ β belong to the ν ν ν

ν

velocity grid, excluding the values of variables bν , ϕν , for which the post-collision

velocities ξ α' , ξ 'β are outside of Ω . As the points ξα' , ξ 'β in general, do not coincide ν ν ν

ν

with the velocity grid, a regularization of the sum (2.1.4) is needed. Let ξ λ and ξ μ ν ν be the nearest vertices of the cells, in which we find the points ξ α' , ξ 'β and let ξ λ + s , ν ν ν

ξ μν − s be a pair of other symmetrically situated vertices. Then, the last two

δ – functions in φ (ξ γ ) are replaced by the expressions

δ (ξα' ν − ξγ ) = (1 − rν )δ (ξ λν − ξγ ) + rν δ (ξ λν + s − ξγ )

(2.1.5)

δ (ξ'βν − ξγ ) = (1 − rν )δ (ξμν − ξγ ) + rνδ (ξμν −s − ξγ ) This means that the contributions to the points ξα' , ξ 'β are distributed among the ν ν nearest

nodes.

Defining

E0 = (ξα' ν )2 + (ξ 'βν )2 ,

E1 = (ξ λν )2 + (ξ μν )2 ,

Structure of Shock Waves

235

E 2 = ( ξ λν + s ) 2 + ( ξ μν − s ) 2 , one of the following conditions is true: E1 ≤ E 0 < E 2 , or

E 2 < E0 ≤ E1 . The coefficient rν can be defined from the energy conservation law E 0 = (1 − rν ) E1 + rν E 2 , from which it results 0 ≤ rν < 1 . Using the obtained value of rν one finds the interpolation formula that is exact for the Maxwellian function f β = f M (ξ β )

fα'ν f β'ν = ( fλν f μν )1−rν ⋅ ( fλν +s f μν −s )rν

(2.1.6)

The integral sum (2.1.4) is calculated simultaneously in all nodes ξ γ of the velocity grid. Defining B = V π bm2 N 0 / (4 Nν ),

Δν = ( fαν f βν − fα'ν f β'ν ) gν , and introducing

Kroneker symbol δ γ ,β , one obtains an explicit discrete form of the Boltzmann collision integral in Cartesian velocity space, i.e. Nν

Iγ = B∑[−(δγ ,αν + δγ ,βν ) + (1 − rν )(δγ ,λν + δγ ,μν ) + rν (δγ ,λν +s + δγ ,μν −s )]Δν ν =1

The interpolation (2.1.6) annuls the collision integral from the Maxwellian distribution function

I ( fM , fM ) = 0

(2.1.7)

Assuming that the solution is close to the Maxwellian distribution function

f = f M + ε f (1) , ε << 1 This form makes the kinetic equation stiff with a factor ε −1 in the collision integral. The property (2.1.7) eliminates the stiffness because one has

I ( f , f ) = I ( f M , f M ) + 2ε I ( f M , f (1) ) Hence, the major part of the collision integral, which is theoretically equal to 0, is evaluated exactly, independently of the numbers of velocity nodes N 0 and integration grid nodes Nν . The method is economic by the number of arithmetic operations, because the values rν , gν , bν and velocity coordinates of functions f αν , f βν , f λν , f μν , f λν + s , f μν − s in arrays of the distribution function can be prepared in advance for all the nodes of the configuration space. The 8-dimension integration grid is generated by Korobov method [62] that gives the estimation O ( Nν−1 ) for the residual of integration in comparison with O ( Nν−0.5 )

for the Monte Carlo integration method.

236

High Temperature Phenomena in Shock Waves

After the collision integral is evaluated, the system of N 0 equations (2.1.3) is solved by the splitting method with a time step τ = t j +1 − t j ∂fγ* ∂f * + ξγ γ = 0 , ∂t ∂x

f γ*, j = f γ j

(2.1.8)

∂fγ j *, j +1 (2.1.9) = I γ , fγ = fγ ∂t The equation (2.1.8) is approximated by an explicit flux conservative scheme. The equation (2.1.9) is conveniently presented in the integral form fγ j +1 = f γ j +

t j +1

∫

I (t )dt

tj

Introducing the discrete variable tν = τν / Nν and an intermediate solution f γ j +ν / Nν , one obtains the scheme

fγ j+ν / Nν = fγ j+(ν −1)/ Nν +τ ⋅ Δγj+,ν(ν −1)/ Nν Here Δγj+,ν(ν −1) / Nν is the ν -h term of the sum (2.1.7). Therefore, the step (2.1.9) is computed as a continuous updating of the distribution function by successive contributions to the sum (2.1.7). The considered scheme is universal and can be applied for steady and unsteady flows as well. For computing the SW structure it is convenient to use a cylindrical coordinate system ξ x , ξ r in the velocity space in which the distribution function f ( ξ, x, t ) is replaced by the function f (ξ x , ξ r , x , t ) . The equation (2.1.3) is reduced to the equation ∂fγ ∂t

+ ξ x ,γ

∂fγ ∂x

= I γc

(2.1.10)

The discrete approximation of the collision operator is similar to the form (2.1.7), i.e. Nν

I γc = B c ∑ [−(δγ ,αν + δγ ,βν ) + (1 − pν )(δ γ ,λν + δγ ,μν ) + pν (δγ ,λν + s + δ γ ,μν − s )]Δνc

(2.1.11)

ν =1

Here pν

denotes

the

splitting

coefficient,

Bc =

π 2 N 0Vbm2

Δνc = ( fαν f βν − fα'ν f β'ν ) gν ξ r ,αν ξ r ,βν , and fα'ν f β'ν = (ξr ,λν ξ r ,μν f λν f μν ) pν (ξ r ,λν + sξ r ,μν − s f λν + s f μν − s )(1− pν ) / ξ rν ,αν ξ rν ,βν

2 Nν

,

Structure of Shock Waves

237

The equation (2.1.10) is solved by the stabilization method until the solution becomes independent of time. After the steady state distribution function is found, the gas dynamic parameters: density n( x ) , bulk velocity u( x ) , temperature T ( x) , longitudinal temperature Txx , and heat flux q( x ) can be computed as the corresponding sums

n( x ) =

mV N 1 V N V N0 , , T x ( ) = = ξ ξ u ( x ) f f ξ ∑ [(ξ x ,γ − u )2 + ξ r2,γ ]ξ r ,γ fγ , ∑ , γ γ x r ∑ r,γ γ 3knN 0 γ =1 n N 0 γ =1 N0 γ =1 0

0

Txx ( x) =

mV N0 ∑(ξx,γ − u)2 ξr,γ fγ , knN0 γ =1

q( x ) =

mV N 0 ∑ [(ξ x ,γ − u )2 + ξ r2,γ ](ξ x ,γ − u )ξ r ,γ fγ 3nkN 0 γ =1

2.2 Solution of the Generalized Boltzmann Equation

The Generalized Boltzmann Equation (GBE) has been obtained in [36] by considering classical interactions of rigid rotators and application of the quantum mechanics discretization rules to the obtained continuous energy spectrum. It replaces the Wang Chang–Uhlenbeck Equation (WC-UE) [67] for the case when the internal energy levels of molecules are degenerated. Although the GBE is similar to the WC-UE, it differs from the latter by a factor related to statistical weights of the levels and gives a different equilibrium spectrum. The GBE has the form

∂f i ∂f + ξ i = Ri ∂t ∂x

(2.2.1)

The collision operator is given by the expression

Ri = ∑

∞ 2π bm

∫ ∫ ∫(f

k

fl ω ijkl − fi f j ) Pijkl gbdbdϕ d ξ j

(2.2.2)

jkl −∞ 0 0

Here f i is the distribution function for the level i , Pijkl is the probability of the transfer from levels

i, j to the levels k , l , and the factor ωijkl = (qk ql ) /(qi q j ) , qi being the

degeneration of the energy level. For simple levels, the GBE is reduced to the WCUE, therefore it can be considered as a more general form of the kinetic equation for molecular gases with internal degrees of freedom. In [64] the conservative projection method for evaluation of the collision operator of WC-UE and GBE has been developed. A brief description of the method is given below. With the use of Dirac delta-function δ ( ξ i − ξ *n ) and Kroneker symbol δ n ,i the collision operator for the level n and the velocity vector ξ *n can be turned to the form Rn (ξ*n ) =

2π bm

∑ ∫ ∫ ∫δ

i , j ,k ,l R3 ×R 3 0 0

n ,i

δ (ξi − ξ*n )(ωijkl f k f l − fi f j ) gij Pijkl bdbdϕ dξi dξ j

(2.2.3)

238

High Temperature Phenomena in Shock Waves

Using the condition of the detailed balance

qi q j Pijkl gij bdbdϕ dξ i dξ j = qk ql Pklij g kl b ' db ' dϕ ' dξ k dξ l , one obtains the symmetric form of (2.2.3)

1 ∑ 4 i , j ,k ,l R3∫×R3

2π bm

∫ ∫ Ψ(ω

f k f l − f i f j ) gij Pijkl bdbdϕ dξ i dξ j

(2.2.4)

Ψ = (δn,iδ (ξi − ξ*n ) + δ n, jδ (ξ j − ξ*n ) − δn,kδ (ξk − ξ*n ) + δn,lδ (ξl − ξ*n ))

(2.2.5)

Rn (ξ*n ) =

kl ij

0 0

where

A limited domain Ω with the volume V in the velocity space is introduced and a uniform Cartesian grid S 0 with N 0 nodes is built. Let J be the grid of the internal energy levels, limited by the upper level jm . For evaluating the operators (2.2.4) in the nodes of the grid S = S 0 × J a uniform integration grid

Sν = {i, j, k , l , ξ i* , ξ j* , σ , ϕ}ν with Nν nodes is generated, in which the nodes i * , j * coincide with the nodes of the grid S0 , and the nodes {i , j , k , l }ν coincide with the nodes of the grid J . For the grid Sν , one defines the post-collision velocities ( ξ k )ν and ( ξ l )ν , which, in general case, do not belong to the grid S0 . The evaluation of the operators (2.2.4) will be made in the nodes ξ γ of the grid S0 . The continuous functions in (2.2.4) should be expressed through the discrete functions defined at the grid S f i ( ξ i , x , t ) = v 0 ∑ f i ,γ ( x , t )δ ( ξ i − ξ γ ) γ

Here γ is the index of a node of the grid S0 , ξγ is the velocity vector of this node, f i ,γ (x, t ) is the grid function for the level

i in the node γ , and v0 = V / N 0 . The

similar presentation is applied for the operator Rn (ξ n* ) . The above considerations are similar to those used for the classical Boltzmann equation. In (2.2.5), one replaces the δ -functions that contains non-grid vectors ξ k ξ l by the combinations of the δ functions at the nearest nodes, the nodes ξ λ and ξ λ + s for the vector ξ k , and the nodes

ξ μ and ξ μ − s for the vector ξ l (the index ν is omitted) ; the splitting coefficient r is determined from the same energy condition, and the interpolation (2.1.6.) is applied. Finally, one obtains the operator for the level n at the node γ in the form Nν

Rn ,γ = B ∑ nν (Φν(1) − Φν(2) )( Δν(2) − Δν(1) ) ν =1

(2.2.6)

Structure of Shock Waves

239

where the following notations have been used: B=

π bm2Vjm2 , Δ (1) = [ P kl g f f )] , ν ij ij i ,i* j , j* ν 4 Nν

Φν(1) = [δ n ,iδ γ ,i* + δ n , jδ γ , j* )]ν

,

Δν( 2) = [ Pijklωijkl gij f k (ξ k ) f l (ξ l )]ν

,

Φν(2) = [δ n ,k ((1 − r )δγ ,λ + rδγ ,λ + s ) + δ n ,l ((1 − r )δγ ,μ + rδγ ,μ −s )]ν

Here the indices i * and j * designate the velocity nodes for the levels i and j , respectively. The value nν denotes the number of permitted transitions to the levels

k , l from the levels i, j which depends on the relative velocity g ij . The formula (2.2.6) gives the explicit discrete approximation of the collision operator of GBE in the Cartesian velocity coordinates. The change to cylindrical coordinate system is similar to the procedure used for the Boltzmann equation. After the collision operator is evaluated, the kinetic equation is solved by a standard technique described in 2.1 for the classical Boltzmann equation. The computed discrete distribution function defines the macroscopic parameters. First, one computes the partial parameters, and then the gas parameters as the corresponding sums. In cylindrical coordinates one has

ni = ∑ξri,γ fi,γ , ui = γ

Trr ,i =

1 ∑ξri,γ ξxi,γ fi,γ ni γ

m 2 kni

∑γ ξ

ri ,γ

, Txx,i =

2 m ξri,γ (ξxi,γ − uxi ) fi ,γ , ∑ kni γ

1 ξ ri ,γ 2 f i ,γ , Ti = (2Trr ,i + Txx ,i ) 3

(2.2.7)

1 1 1 1 n = ∑ ni , u = ∑ ui ni , Txx = ∑ Txx ,i ni , Trr = ∑ Trr ,i ni , T = ∑ Ti ni n i n i n i n i i The normalized number densities ni / n define the spectrum of internal energy. 2.3 Two Levels Kinetic Model of RT Relaxation

In order to simplify the simulation of RT energy exchange in a gas, the 2 levels model, called “2LRT” model, was developed [66] Such simplification is strongly required for complex processes in which rotational excitation is accompanied by VT energy transfer. The model consists of 2 levels: the ground level with the rotational energy ε1 = 0 and the excited level with some energy ε 2 > kTmax , where Tmax is the maximum temperature in the problem under consideration. The distribution function is also composed of two parts, f1 and f 2 with corresponding populations of the levels

n1 and n2 . The gas density is n = n1 + n2 and the rotational energy is Erot = ε 2 n2 . Let the density of the gas at some point be n , the kinetic energy Ekin , and the rotational energy Erot . One can then determine the populations of the levels by the simple formulas n2 = Erot / ε 2 and n1 = n − n2 . The maximum value of Erot is given

240

High Temperature Phenomena in Shock Waves

by E rot = nkTmax , therefore n2 < nkTmax / ε 2 , and one obtains 0 < n2 < n and n1 > 0 . one can determine the equilibrium temperature, Knowing Ekin , Teq = 2( E kin + E rot ) / 5nk

and

n2,eq < nkTeq / ε 2 , n1,eq = n − n2,eq .

the These

equilibrium parameters

rotational

determine

the

populations equilibrium

distribution functions f1,M and f 2,M . For construction of the model equation we start from the Wang Chang–Uhlenbeck equation for the considered 2 levels system. ∂ f i / ∂t =

∑∫p

k ,l i, j

( f k f l − f i f j ) g i , j bdbd ϕ d ξ j

j , k ,l

(2.3.1)

In equation (2.3.1), we replace the collision operator by an elastic collision operator Qel and the non-elastic operator Qr . This replacement cannot be strictly justified, because for RT exchange the purely elastic collisions present an exception, and almost all collisions are accompanied with the transfer of relatively small part of kinetic energy to/from the rotational energy. On the other hand, because of small inelasticity of interactions the main collision relaxation process is close to the case of elastic collisions, except that one should take into account the inelastic transfer of the energy. The elastic operator is the same as the Boltzmann collision integral for a twocomponent gas mixture:

Qi ,el = ∑ ∫ ( f i ' f ' j − f i f j ) g i , j bdbd ϕ d ξ j j

(2.3.2)

Here the functions f i ' and f j ' contain post collision velocities. The non-elastic operator is taken in a relaxation form

Qr ,i = −ν r ( fi − fi*, M )

(2.3.3)

It was found from a number of numerical experiments that, for SW problem, the choice for fi*,M in equation (2.3.3) as the Maxwellian distribution functions fi , M is possible, but it is not the best choice. The function fi*,M represents the elliptic distribution defined by the diagonal elements of the temperature tensor. f i *, M = ni ,eq (

m 3/ 2 * * * −1/ 2 ) (TxxTyyTzz ) exp( − mc x2 / 2kTxx* − mc 2y / 2kTyy* − mc z2 / 2kTzz* ) (2.3.4) 2π k

where cx = ξ x − u , c y = ξ y − v, c z = ξ z − w , and

u, v, w are the components of the

bulk velocity vector. The components Taa* of the temperature tensor are defined by self-similar transformation of the initial components as

Taa* = Taa (Teq / Tkin )

(2.3.5)

Structure of Shock Waves

241

The use of the function given in equation (2.3.4), instead of the Maxwellian, means that the inelastic operator Qr preserves, to some extent, the shape of the distribution function in the velocity space. The RT relaxation frequency can be defined as a part of the relaxation frequencyν of the BGK model equation

ν r = a1v .

(2.3.6)

The non-elastic operator contributes to the evolution of the velocity distribution function towards the equilibrium state. To take into account its influence one should multiply the elastic collision operator by a factor (1 − a2ν r ), 0 < a2 < 1 . Finally, the proposed RT relaxation model contains two operators, the inelastic operator given by equation (2.3.3) with the frequency given by the equation (2.3.6), and the elastic operator Q i*, el = (1 − a 2ν r ) Q i , el . The coefficients a1 and a2 can be determined from comparisons of the solutions of the proposed model with solutions of the GBE. Some additional fitting parameters may also be applied in the model to obtain better agreement. The comparisons of SW computations with the use of 2LRT model with those by GBE are reported in [66,39]. 2.4 Solution of the Boltzmann Equation for a Gas Mixture

The system of Boltzmann kinetic equations for a mixture of monatomic gases containing K components is commonly written in a form

∂Fi ∂F + ξi i = I i , ∂t ∂x

i = 1, ..., K

The collision integral has the form Ii = ∑ ∫

2 π bm

∫ ∫ ( F ′F ′ − F F )gbdbd ε dξ i

j

i

j

j

,

j R3 0 0

(

)

where Fi = Fi ( ξi , x, t ) , Fi′ = Fi ξi′ , x, t , g = ξ j − ξi ,

bm is the maximum interaction distance, b and ε are the impact parameters of a binary collision. To extend the conservative method of evaluation of the collision integral [27] to the gas mixtures, it is sufficient to transform the equation from the velocity variables to the momentum variables (ξ i , x, t ) → (pi , x, t ),

Fi (ξ i , x, t ) → f (pi , x, t )

From the condition of the normalization ∫ Fd i ξi = ∫ f i dpi = ni , one obtains Fi ( ξ i , x , t ) = f i ( p i , x , t ) mi 3

242

High Temperature Phenomena in Shock Waves

The system of Boltzmann equations in the momentum space turns to the form ∂f i pi ∂f i + = Ii , ∂t mi ∂x

(2.4.1)

where the collision integral has the form:

Ii = ∑ ∫

2π bm

∫ ∫ ( f ′f ′ − f f )gbdbd ε dp i

j

i

j

j

, g=

j R3 0 0

pj

mj

−

pi

(2.4.2)

mi

The following properties should be conserved in the discrete form of the integral :

⎛ pi2 ⎞ p p p p p I ( ) ψ ( ) d = 0, ψ ( ) = 1, , ⎜ ⎟ i i ∫ i i i i mi ⎠ ⎝ R3

⎛ ( pi − pi 0 ) 2 ⎞ ⎛ ⎞ 2 1 exp I i ⎣⎡ f i ,M ⎦⎤ = 0 , f i , M = ni ⎜ ⎜− ⎟ ⎟ ⎜ mi 2kT ⎟⎠ ⎝ 2π kTmi ⎠ ⎝ The system (2.4.1) with collision integral (2.4.2) is solved on a uniform 3dimensional grid S0 in Cartesian momentum space Ω of the volume V, or a uniform 2-dimensional grid in cylindrical coordinate system. The use of a constant step in the coordinate space is required for the conservation of the total momentum in the projection method. The evaluation of the collision integrals is performed with the use of the 8dimentional uniform integration grid Sν = (piν ,p jν , bν , εν ) in a domain Ω× Ω× 2π × bm 3

with Nν nodes. The nodes pi ,ν and p j ,ν coincide with the momentum grid nodes, and all values of variables bν , εν for which the post-collision momentums p 'i ,ν or p ' j ,ν fall outside Ω are excluded. Proceeding in the same way as made for a polyatomic gas in 2.2, one obtains the collision integral for a species n in a momentum node γ in the form

I n ,γ =

1 ∑∑ ∫ 4 i j Ω×Ω

2 π bm

∫ ∫ φ γ ( f ′f ′ − f f ) gbdbd ε dp dp n,

i

j

i

j

j

i

,

0 0

with

(

)

(

φn , γ = δ n , i δ ( p i − p i , γ ) + δ n , j δ ( p j − p j ,γ ) − δ n ,iδ p i′ − p i ,γ − δ n , j δ p j ′ − p j ,γ

)

Finally, one obtains the discrete approximation of the collision integral in Cartesian coordinates in the explicit form Nν

I n ,γ = B ∑ (Φν(1) − Φν( 2) )Δν , ν =1

(2.4.3)

Structure of Shock Waves

243

with B=

π bm2VK 2 , Δ = [ g ( f ( ξ ) f ( ξ ) − f f )] , f (ξ ) f ( ξ ) = ( f f )1− r ( f f ) r ν ij k k l l i ,i * j , j * ν k k l l k ,λ l , μ k ,λ + s l , μ − s 4 Nν

Φν(1) =[δni, δγ,i* +δn, jδγ, j*)]ν , Φν(2) = [δn,k ((1 − r)δγ ,λ + rδγ ,λ+s ) + δn,l ((1 − r)δγ ,μ + rδγ ,μ−s )]ν , where the indices i * and j * denote the momentum nodes of the species i and j , respectively. The coefficient r (the index ν is omitted) is defined from the energy conservation law p ∗2 p ∗2 p2 p2 p2 p2 E0 = (1 − r ) E1 + rE2 , E0 = ij + jj , E1 = i ,λ + j , μ , E2 = i ,λ + s + j ,μ −s 2mi 2m j 2mi 2m j 2mi 2m j In the cylindrical coordinate system the integral (2.4.3) is changed in the same way as the integral for a pure monatomic gas. After the collision integral is calculated, the equation (2.4.1) is transformed to the system of discrete ordinate equations

∂fi ,γ ∂t

+

pi ,γ ∂fi ,γ G = I i ,γ , i = 1,....., K , γ = 1,....., N 0 mi ∂x

(2.4.4)

The system (2.4.4) is solved by the standard procedure with application of the splitting method until stabilization of the solution. The obtained distribution function defines the following gas dynamic parameters: number density ni , flow velocity for a species ui , temperature Ti , parallel temperature Txx ,i , and transversal temperature Trr ,i . For the whole gas one obtains the molecular number density n , the density ρ , the flow velocity u , and the temperature T . The listed macroscopic variables are defined as the sums of the momentum distribution functions. In cylindrical coordinates, one has

ni = ∑ pri ,γ fi ,γ , ui = 1

ni mi

γ

Txx ,i =

1 kni mi

∑γ p

ri ,γ

∑γ p

ri ,γ

p xi ,γ f i ,γ , Trr ,i =

1 2 kni mi

∑γ p

3 ri ,γ

( p xi ,γ − u xi mi ) 2 f i ,γ , Ti = 1 (2Trr ,i + Txx ,i ) 3

The parameters of the mixture are defined as the combinations of (2.4.5)

1 n = ∑ ni , ρ = ∑mi ni , u = ρ i i T =

1 ∑ [kniTi + mi ni (ui − u )2 / 3] kn i

∑mnu i

i

i

i

,

f i ,γ

(2.4.5)

244

High Temperature Phenomena in Shock Waves

2.5 Statement of the Boundary Problem and Presentation of the Computed Data Consider a plane shock wave traveling in the x direction with a Mach number M . Denote gas parameters ahead the SW as n1 , u1 , T1 , and those behind the SW as

n2 , u2 , T2 . The parameters on both sides are related by Rankine-Hugoniot conditions (γ + 1) M 2 n2 u1 T2 (2γ M 2 − (γ − 1))((γ − 1) M 2 + 2) = = = , (γ + 1)2 M 2 n1 u2 (γ − 1) M 2 + 2 T1 For a monatomic gas , γ = 5 / 3 , u1 = Mc1 and c1 = γ kT1 / m The problem is solved in the coordinate system attached with the SW. The steady SW structure is obtained as the evolution of initial discontinuity of gas parameters posed at x = 0 . The boundary conditions are imposed at sufficiently large distances from the discontinuity at x = − L1 and x = L2 where the gas can be considered as being in thermodynamic equilibrium with the corresponding Maxwellian distribution functions. In the case of a pure monatomic gas, the kinetic equation and boundary conditions may be written in a dimensionless form by introducing the characteristic velocity v1 = kT1 / m , the characteristic length as a molecular mean free path ahead 2 −1 the shock wave λ1 = ( 2π n1σ eff ) , and the characteristic time τ 1 = λ1 / v1 . Here σ eff is the efficient molecular diameter. Thus, in dimensionless form, the boundary conditions are

f (t = 0, x < 0, ξ x , ξ r ) = f (t , x = − L1 , ξ x , ξ r ) = n1 / (2π T1 )3/2 exp( − f ( t = 0, x > 0, ξ x , ξ r ) = f ( t , x = L 2 , ξ x , ξ r ) = n 2 / (2π T 2 ) 3/ 2 exp( −

(ξ x − u1 ) 2 + ξ r2 ), 2T1

( ξ x − u 2 ) 2 + ξ r2 ) 2T2

The function f ( t > 0, − L1 < x < L2 , ξ x , ξ r ) is searched as the solution of the Boltzmann kinetic equation. After the solution is found at the velocity grid S0 = (ξ xγ , ξ rγ ) , the gas parameters are computed as the corresponding sums, displayed in the paragraph 2.1. For presentation of the macroscopic gas parameters two forms are commonly used. In the first form the parameters normalized by their values ahead the shock wave, n ( x ) / n1 , T ( x ) / T1 , and so on, are presented. In the second form the computed results are presented by the following reduced paramers (with the asterics further removed)

n* = (n − n1) / (n2 − n1 ), T * = (T − T1) / (T2 − T1 ), u* = (u − u1 ) / (u1 − u2 )

Structure of Shock Waves

245

Similar presentations are used for the case of polyatomic gases and gas mixtures. The inverse shock thickness is defined at a sufficiently large stabilization time t∞ as

δ=

λ1 Lsw

=

λ1

⎛ dn(t∞ , x ) ⎞ ⎜ ⎟ n2 − n1 ⎝ dx ⎠ max

For polyatomic gases, the boundary conditions become

f i ,1 (ξ , x, t ) = n1[1 / (2π T1 )]3/2 exp[ −

x < 0,t = 0 fi ,2 (ξ , x, t ) = n2[1/ (2π T2 )]3/2 exp[−

(ξ − u1 ) 2 2i + 1 e ] exp( − i ) , for x = − L1 and 2T1 Qr T1

(ξ − u2 )2 2i + 1 e ] exp(− i ) , for x = L2 and x > 0, t = 0 , 2T2 Qr T2

where Qr is the statistical sum, and parameters (n, T , u)1,2 are defined by the RankineHugoniot relations with γ = 7 / 5 . For mixtures of monatomic gases, the boundary conditions are the same as for a pure monatomic gas, but written for each component having the same temperature and velocity. The characteristic length is defined as λ1 = ( 2π n1d12 )−1 , where d 1 is molecular diameter of the selected mixture component, and n1 is the number density of the mixture before the shock wave.

3 Shock Wave Structure in a Pure Monatomic Gas The analysis of the shock wave structure in the case of a pure monatomic gas is the oldest and the most numerically studied problem. It is also the most important benchmark for testing the different numerical approaches.

3.1 Gas of Hard Sphere Molecules The hard sphere (HS) molecule model represents the simplest model of the real molecular interaction law, although not very realistic. The model is often used for testing different approximate theories intended to replace the Boltzmann equation by something that could be more economic for calculations. Though we do not believe in the success of such attempts, the data on SW structure in the hard sphere gas that are computed with controlled high accuracy are presented below. Our aim is to obtain the accuracy of determination of the inverse SW thickness not less than 3%. To check the convergence of computed results by parameters of discretization a number of runs has been performed. It is worth to notice, that for HS model, the shock wave structure depends on a single dimensionless parameter, the Mach number.

246

High Temperature Phenomena in Shock Waves

In Table 1 a few results for M = 1.2 obtained with variation of the mesh hx along the coordinate x , with variation of the velocity mesh Δ ξ and with a number of nodes Nν of the integration grid are presented. In Table 2. similar data are presented for M = 10 . Table 1. Dependence of the inverse shock wave thickness for 1.2: а) at the coordinate and the number of integration nodes Nν mesh , b) at the velocity mesh

(a) 0.1 0.0677

0.05 0.0685

(b) Δ

0.4

0.3

5 10 0.0685

5 10 0.0677

Table 2. Dependence of the inverse shock wave thikness mesh , b) at the number of integration nodes

0.1 0.4253

10 0.4326

(a) 0.05 0.4346 (b) 2 10 0.4333

0.03 0.4379

0.2 10 0.0665

for

10: a) at the coordinate

0.025 0.4387

3 10 0.4346

5 10 0.4352

In Figs. 1, 2 and 3 the SW structures for M = 1.2,1.59, 2, 2.5, 3 are presented in comparison with data [21,22]. The computed inverse shock thickness and that from [21,22] are shown as δ and δ ' , respectively. The difference in the values of about 10% might be explained by the use of a rather crude coordinate mesh in the cited papers.

Structure of Shock Waves

247

(a)

b b)

d)

c)

e)

d temperature inside the shock wave: a) Fig. 1. Density, velocity, and c) 2, d) 2.5, e) 3., -- -- data [21]

1.2, b)

1 1.59,

248

High Temperature Pheenomena in Shock Waves

Fig. 2. Heat flux

inside the shock wave for

The stress tensor is defin ned as

1.2,

-- data [21]

Pxx = pxx − p ,

where p is the pressure and d p xx is a component of the pressure tensor. The plot off the inverse shock thickness δ versus Mach numder is shown in Fig.4. For increassing Mach numbers, the curve ap pproaches the limit 0.44 0.1.

Fig. 3. Compon nent of the stress tensor for

1.2, -- data [21]

The temperature profilee may have a known small “overshoot” shown in Figg.5. Defining this overshoot witth the value ϑ = (Tmax − T2 ) / (T2 − T1 ) , the dependence off ϑ versus M is given in Fig.6. In Fig.7 the cross sections of the distribution functionn for M = 3 are presented in Carrtesian coordinates when one of the transversal velocitiees is taken equal to 0. In Fig.8 the plot of the distribution function for a hypersonic S SW with M = 20 at x = 2λ1 is shown. The SW propagates from the left side. The deeltalike part of the distribution function that belongs to the cold undisturbed gas ahead the d by the computation. The results are in agreement with the shock wave is well resolved analysis [13] of the infinitelly strong shock wave.

Structure of Shock Waves

Fig. 4. Inveerse shock wave thickness for a hard sphere gas

Fig. 5. Details of the shock wave structure for a hypersonic wave

Fig. 6. Temperature T overshoot for a hard sphere gas

249

250

High Temperature Pheenomena in Shock Waves

Fig. 7. Cross sections of thee distribution function for

3; left:

Fig. 8. Distribution fu unction in Cartesian coordinates at

0.6, right: x

0.6 6

x = 2λ1 for M = 20

3.2 The Lennard-Jones Gas G We consider the Lennard-Jones potential (6,12) that contains two parameterss, a ε , that is diameter σ and an energy parameter p ⎛ ⎛ σ ⎞12 ⎛ σ ⎞ 6 ⎞ U ( r ) = 4ε ⎜ ⎜ ⎟ − ⎜ ⎟ ⎟ ⎜⎝ r ⎠ ⎝ r ⎠ ⎟⎠ ⎝

Because of the second parameter p the SW problem looses its unique dependencee on the Mach number, and depeends on the gas temperature as well. For comparisons w with the experimental data [1] we w consider an Argon gas at the room temperature 3000K. Accordingly to [68], for Argon, A one has ε = 120K that gives the tabulated valuee of the integral Ω ( 2,2) = 1.089 . Then, the efficient diameter σ eff = σ Ω( 2,2) that enterss in the definition of λ1 may bee computed.

Structure of Shock Waves

a)

251

b)

c) Fig. 9. Density profiles p for a shock wave in Argon; ,

d) -- data [1].

a)

c)

b)

d)

Fig. 10. Comparison of the sho ock wave structure in Argon with Lennard Jones potential andd the corresponding HS gas. Solid lines - HS gas, dashed lines - L-J gas; a)M 1.55, b)M 2.31, c)M 3.8, d)M 6.5.

252

High Temperature Phenomena in Shock Waves

In Fig.9 the computed density profiles of the SW are compared with the experimental ones, and in Fig.10 the comparison of the SW structure for a hard sphere (HS) gas and the gas with Lennard-Jones (LJ) potential is presented. In Fig.11 the dependence of the inverse SW thickness on Mach number for Argon is compared with the experimental data and the corresponding results for hard sphere gas. The curve for Argon is not monotonous and presents a maximum for M = 3.8 ± 0.3 . The comparison with experimental data [1] shows very good agreement for M ≥ 2.3 , and some disagreement for small Mach numbers. In this range, there is a better agreement with the data [2] obtained in a low Mach number shock tube.

Fig. 11. Inverse shock wave thickness as a function of Mach number for Argon (solid line) and for hard sphere gas (dashed line); -- experimental data [1], □ – data [2].

In Fig.12 the assymetry of the SW density profile as a function of the Mach number is presented for the case of Argon and for the HS gas. This assymetry is defined as follows: 1

The computed results are compared with experimental data [1]. The curve corresponding to the solution of the Navier-Stokes equations is also shown. One can notice that the N-S solution gives Q > 1 for all Mach numbers, when, for the above considered two cases, the curves cross the level Q = 1 at M ≈ 2.4 , in accordance with the experimental data.

Structure of Shock Waves

253

nsity profile as a function of Mach number. Solid line - LJ Arrgon Fig. 12. Assymetry of the den gas, dashed line - HS gas, and d dash-dotted line- solution of the N.-S. equations; -- data [11].

4 Shock Wave Struccture in a Polyatomic Gas In general case, the shock k wave structure in a polyatomic gas is formed by thhree dissipative processes: elastic collisional relaxation, rotational-translational (R RT) energy transfers, and vibrattional-translational (VT) energy transfers. The first proccess has the highest rate, it is folllowed by the second one, and the rate of the VT processs is usually some order of magn nitude less compared to the RT transfers. For calculating the non-eequilibrium flow of molecular gases, the most widely uused technique is the DSMC meethod. Here, we apply the computational methodology for computing the shock wav ve structure in a polyatomic gas using the Generaliized Boltzmann Equation (GBE)). The whole problem that includes both (VT) and (RT) energy transfers is solvedd by applying a three-stage splittting procedure to the GBE. The three stages consist of ffree molecular transport, VT rellaxation, and RT relaxation. For the VT relaxation, GBE E is always solved. For the RT relaxation, r two approaches are theoretically possible. In the first approach, for the RT reelaxation GBE is solved. This approach is computationaally very intensive since the totaal number of excited levels is the product of rotational and vibrational levels. In the seecond approach, a two-level model of RT relaxation tthat equilibrates rotational and translational t energies is employed. The second approacch is computationally much less intensive than the first and therefore is much m more efficient. Using this approach the computations are performed for SW structuree at high Mach numbers accoun nting for both vibrational and rotational excitations. 4.1 Shock Wave Structurre with Frozen Vibrational Levels The vibrational and rotation nal quanta have very different magnitudes. As an examp mple, the rotational quantum of Nitrogen N is equal to 2.9K, when the vibrational quantum m is 3340K. For Oxygen the corrresponding values are 2.1K and 2230K [69]. The VT crross

254

High Temperature Phenomena in Shock Waves

sections for both gases are by some orders of magnitude less the RT cross sections. Because of such a difference, in many cases the VT process could be neglected and only RT relaxation should be taken into account. On the other hand, the small value of the rotational quantum and rather high value of the cross section of RT process, which may be of the order of 0.25 - 0.1 of the elastic cross section [1] requires the replacement for molecular gases of the classic Boltzmann equation by GBE at all temperatures. When the temperature in SW does not approach the value of the vibrational quantum or when the cross section of the VT transfer at that temperature is negligibly small, the SW structure can be computed by solving the GBE equation for RT process only. Numerical study of the shock wave structure in a polyatomic gas with rotational degrees of freedom presents interest for two main reasons. First, it provides additional data about the process of rotational-translational (R-T) energy exchange that cannot be obtained by a physical experiment. Second, it can serve as a test and verification of numerical methods by comparison with existing experimental data. The experimental data for Nitrogen shock structure are obtained by registration of electron beam induced fluorescence [29-31], absorption of an electron beam [1], and by Raman spectroscopy[32], with the use of jets, wind tunnels, and shock tubes. The most definite experimental conditions were realized for moving shock waves in shock-tube experiments [1], but the applied method of measurements provided only density profiles. In other experiments with steady shocks formed in expanding free jets and wind tunnels, the thermodynamic equilibrium between rotational and translational modes before the front may be distorted, influencing not only the shock wave structure, but also rotational spectrum and rotational temperature as well. Most computations of the shock wave structure by DSMC method were carried out with application of different phenomenological relaxation models for the internal energy that involve numerous assumptions, which are not always physically justified [40,41]. More rigorous Monte Carlo approach that uses classical trajectory calculations of the interactions of rotating molecules [34] requires enormous amount of calculations. A more economic DSMC relaxation model based on the trajectory calculations was recently proposed [35]. In the present computations the molecular collisions are described by the LennardJones interaction potential with parameters and rotational spectrum data taken from [68]. Thus, for molecular Nitrogen, the depth of the energy hole ε = 91K , the degeneration of rotational level qi = 2i + 1, i = 0,1,..∞ , and the rotational energy of the level eri = ε 0 i (i + 1), ε 0 = 2.9 K . The molecular interaction during the collision consists of two phases. In the first phase, the molecules interact in an elastic manner according to the molecular potential. This stage determines the deviation angle of the relative velocity. In the second stage, the modulus of the relative velocity changes according to the energy conservation equation. For the transition probabilities Pijkl we apply the formulas [36] that are obtained by fitting the experimental data of molecular dynamics simulations of interactions of rigid rotors that model N 2 molecules

Structure of Shock Waves

P i jk l = P 0 ω

kl ij

[α

0

exp(− Δ

− Δ

1

− Δ

2

3

− Δ

4

) +

1

α

exp(− Δ

3

− Δ

255

4

)] ,

0

where

Δ1 =| Δe1 + Δe2 | / etr 0 ,

Δ 2 = 2 | Δe2 − Δe1 | / etot

Δ 3 = 4 | Δe1 | /(etr 0 + eri ) ,

Δ e1 = e r i − e r k , e tr 0 = m g

2

/4,

Δ 4 = 4 | Δe2 | /(etr 0 + erj )

Δ e 2 = e rj − e rl ,

α

0

= 0 . 4 e to t / e tr 0

etot = etr 0 + eri + erj .

The energy conservation law in a collision selects virtual collisions with non-zero probability. From the equation

m g ij2 / 4 + e ri + e rj = m g kl2 / 4 + e rk + e rl , it follows that Pijkl > 0 , if g kl2 ≥ 0 , otherwise Pijkl = 0 . The elastic collision is a particular case of the collision. The probabilities obey the normalization condition:

∑P

kl ij

k ,l

=1

that should be strictly satisfied in the

computations. The formula for transition probabilities is averaged over all interactions and does not depend on the impact parameter. In [37] the inelastic collisions were limited by some impact parameter common for all the interactions. In the presented computations, the deviation angle is limited to a value 0.13 below which the R-T transition is prohibited. The number of levels is selected according to the temperature range of the considered problem. For moderate Mach numbers, the SW structure in Nitrogen can be computed with real value of the spectral energy gap

ε 0 , but for the hypersonic

case the required number of levels becomes too high (up to 50-70 levels) and therefore the computations become very time consuming. To facilitate this problem it is possible to increase this energy gap and thereby reduce the number of levels, keeping the condition

ε 0* << kT0 where

T0 is the temperature ahead the SW, and

ε 0*

is the modified value of the gap. By conducting the numerical experiments, it was determined that this increase in spectral energy gap does not influence the results of calculations in any significant way. The details of the method and examples of computation of SW structure in the range of Mach numbers 1.5 – 15 are reported [38,39]. The numerical experiments have shown that this replacement remains reasonable good approximation until ε 0* < 0.25kT0 . The use of the “efficient” levels allows a considerable reduction of the CPU time.

256

High Temperature Phenomena in Shock Waves

In [38], the shock wave structure in Nitrogen is simulated for a wide range of Mach numbers and compared with experimental cases [1] for M= 1.53, 1.7, 2, 2.4, 3.2, 3.8, 6.1, 8.4, 10 and in [29-31] for M=7 and M=12.9. For the first 6 Mach numbers the obtained density profiles practically coincide with the experimental ones, and for other cases the results are very close. In Fig.13, the M=3.2 SW structure in Nitrogen at room temperature is computed with 44 rotational energy levels and compared in a density plot with experimental data [1]. For this strength of the SW the temperature does not reach the level of excitation of the vibrational degrees of freedom. Reduced gas parameters are shown in the left part of the figure: thus, density n , translational temperature T , rotational temperature Trot , and longitudinal temperature Txx , are normalized by the difference of their values behind and ahead the SW, and the distance is done in molecular mean free paths before the SW. The density and temperatures vary monotonously, but the longitudinal temperature tensor component has a maximum value before the center of the SW which is defined as the point at which the density has half of its maximum value. The rotational temperature rises with some delay with respect to the translational temperature which may be explained by the time required to transfer the increased kinetic energy into the rotational energy by inelastic collisions.

Fig. 13. Shock wave structure (left) and rotational spectra at different points along the shock wave (right) in Nitrogen at M=3.2

In the right part of the figure, the rotational spectrum calculated at several points along the wave front: x = −∞, x = xc − λ , x = xc , x = xc + λ , x = ∞ is shown. Here

xc denotes the SW center. The abscissas represent the number of the rotational levels and the ordinates the populations of the rotational levels. The spectrum in all

Structure of Shock Waves

257

positions looks close to the equilibrium one, but some peculiarity around the 7th-8th levels may be observed at the second spatial position. This deviation from equilibrium spectrum is better seen in Fig.14, where the populations of the levels are plotted in coordinates Z 2 = ln where

nr is the population of the rth level,

εr

∞

n = ∑ nr

nr , Z 1 = ε r = r ( r + 1)ε r 0 , (2 r + 1) n

the gas density at the given point,

r =0

the energy of the level, and ε r 0 the rotational energy quantum. The density and temperature are normalized by their left side values. The equilibrium degenerated distributions nr =

2r + 1 exp( −ε r / T ) at the SW boundaries are represented by straight n

lines in these coordinates. The rotational spectrum deviates from the equilibrium spectrum inside the SW, in agreement with the experimental data [31].

Fig. 14. Deviation of the rotational spectra from equilibrium for M=3.2

The most noticeable deviation occurs at the second point. In the center of the wave front and on the right part of the SW, the deviation is considerably smaller. It may be noted that, in the middle part the spectrum does not present a “mixture” of the left side and the right side spectral distributions which is confirmed by the experimental observations [32]. Fig.15 (left part) shows distributions of gas density, translational and rotational temperatures obtained in the present simulations for M=12.9 corresponding to the

258

High Temperature Phenomena in Shock Waves

experimental conditions [29] that were performed at low temperature, i.e. T1 = 9.15K , which is comparable to the rotational quantum. The comparison with the computations [34], which are close to the experimental data for density and rotational temperature profiles, are shown. The humps in curves of the translational temperature can be explained by slow R-T energy transfer because of low temperature ahead the SW. Right part of Fig.15 shows rotational spectrum for 26 levels at several points along the wave front. The center of SW is located at x=0. The rotational level number is represented on the x-axis and the population of the rotational levels on the y-axis. It is clearly seen that the rotational equilibrium inside the SW does not exist for this high Mach number. These results are in good agreement with experimental data [29] and computations [34]as well. The computations of high-speed flows require the use of a reduced number of “effective” rotational levels. The reduction of the number of levels is obtained by increasing the rotational quantum ε 0 . The results of computations of SW structure at M=10 and room temperature with 16 rotational levels and comparison with the results [1] are shown in Fig.16, left. It is seen that the longitudinal temperature Txx is about twice larger the translational temperature, therefore the R-T transitions may occur with much higher kinetic energies that are estimated in some phenomenological models based on the translational temperature.

Fig. 15. Shock wave structure (left) and rotational spectra at different points inside the shock wave (right) in Nitrogen at M=12.9. The data [34] are marked by the letter “K”.

In the right part of the figure the variation of relative populations n ( J ) / n of some “effective” levels J inside the shock wave is presented. One may notice decreasing populations for the ground and the first excited levels. The populations for other levels rise, but the rise of higher level populations begins with some delay. This may characterize the cascade character of the R-T process in which the rotational quantum passes from the low energy levels to the high energy ones.

Structure of Shock Waves

259

Fig. 16. Shock wave structure (left) and variation of spectral populations inside the shock wave (right) in Nitrogen at M=10 for a reduced number of levels

4.2 Shock Wave Structure with Excited Rotational and Vibrational Levels For the hypersonic SW, both rotational and vibrational degrees of freedom should be considered. Our aim is to obtain a qualitative picture of the SW structure. The interaction potential is now taken in the form of inverse 12-th power of the molecular interaction distance that gives a good approximation for the Lennard-Jones potential, but speeds up computations somewhat. The vibrational spectrum corresponds to Nitrogen with energy quantum ε vib = 3340 K . The vibration levels are not degenerated and the GBE transfers to the WC-UE. The probabilities of V-T transfers are generally accepted to be much smaller than those of the R-T transfers and are in the range of 0.0001 – 0.001 (or less) of the elastic collision probability. In the computation the probabilities of VT transfers between any 2 pairs of levels are taken equal to 0.001 of the elastic probability. The VT transfers are computed using WCUE and the RT transfers are computed by 2LRT kinetic model to simplify the computations. Computations are performed for the room temperature T1 = 300 K and Mach numbers 6 and 10. For the first Mach number, it was sufficient to take 4 vibrational levels, and for the second case the number of levels was 8. In computation of the shock structure, the vibration energy is calculated, i. e.

E v ib =

j = jm

∑

j ε v ib n j

j=0

Assuming that the vibration possesses two degrees of freedom, one can associate the vibration energy with the classic vibrational temperature T vib ,cl = E vib / k . This temperature measures the amount of energy stored in the vibrations. From the viewpoint of quantum mechanics, the vibrations form the Bose gas in which the temperature in the thermodynamic equilibrium is related to the vibrational energy by the formula [69]

260

High Temperature Phenomena in Shock Waves

E vib =

ε vib

exp(ε vib / kTvib , q ) − 1

Knowing Evib , the quantum vibration temperature Tvib ,q may be determined. In the classical limit, one obtains Tvib , q → Tvib ,cl . In Fig.17, both vibrational temperatures are represented as well as density, kinetic translational, longitudinal translational and rotational temperatures for M = 10 . All gas parameters here are normalized by their values ahead the SW. The SW structure differs very much from that presented in Fig.16. The variation of density remains monotonous, but it is not the case for the translational and rotational temperatures. This may be explained by the sharp increase of the kinetic energy and the slow transfer of this energy and the rotational one to the vibration mode. Because of this slow energy transfer, the first part of the SW layer is practically independent of the vibration and approximately corresponds to the adiabatic coefficient γ = 7 / 5 . Then the energy transfer to the vibrational levels gradually lowered the translational and rotational temperatures, which is followed by some increase of the gas density. It should be noted that the quantum vibrational temperature reaches the thermodynamic equilibrium value, but the classical vibrational temperature is below this limit. The vibrational spectrum is quite far from the equilibrium shape at the center of the SW, and then gradually tends to the equilibrium spectrum. In Fig.17, only the part of the x axis where the computed shock wave is located is shown, when the computations were made for x = ( − 120, 250) .

Fig. 17. Shock wave structure with excited rotational and vibrational levels for M=10. The flow parameters, normalized by their values ahead the shock (left), and the vibrational spectra at different abscissas (right) are shown.

The proposed methodology permits to model shock wave structure in polyatomic gases when the transitional probabilities for R-T and V-T processes are assigned. In case the vibrational levels are frozen, the detailed SW structure can be computed by strict approach solving the Generalized Boltzmann equation. When this assumption is

Structure of Shock Waves

261

not valid, the approximate approach with the use of the 2LRT model equation for R-T energy transfer can be recommended.

5 Shock Wave Structure in a Mixture of Monatomic Gases Here, we consider the shock wave structure in a mixture of monatomic gases assuming the hard sphere model. Compared to the case of a single component gas, the SW structure in the mixture is defined by a number of dimensionless parameters: Mach number M , concentrations of gas components before the shock wave χ i (1) = (ni / n)(1) , molecular mass ratio mi / m0 , and the ratio of molecular diameters

di / d0 . The parameters m0 and d 0 can be chosen arbitrary and thus, they are taken equal to the parameters of one of the components. Correspondingly, the characteristic molecular mean free path and the characteristic molecular velocity (or the characteristic molecular momentum p0 ) are defined with the parameters of the selected component m1 , d1 , and n(1) , T (1) are the gas density and the gas temperature respectively ahead the shock wave. The following parameters are also defined

λ0 = ( 2π n(1) d12 )−1 , v0 = 2kT (1) / m1 , p0 = m1v0

(1) Fig. 18. Shock wave structure for two components mixture at M = 1.5 , χ 2 = 0.9 with

different mass ratios: m2 m1 = 0.5 (left), and m2 m1 = 0.25 (right).

The ratio of diameters plays a relatively small role, because the real molecular diameters are close to each other. The mass ratios and concentrations influence the relaxation processes inside the shock wave and constitute it structure. The presentation of macroscopic values has two forms, the normalized form and the reduced form. The same notation is used for the variables in both forms: number

262

High Temperature Phenomena in Shock Waves

densities ni and n , flow velocities ui and u , temperatures Ti and T , parallel Txx.i and transversal Trr ,i temperatures. Notations inside the figures correspond to the curves listed from the top left to the top right. In Figs. 18-21, the results of calculations for a binary mixture with di d1 = 1 are presented. The heavy component is chosen as the first one. One can see that differences between the profiles of densities and temperatures of the components increase when the mass ratio decreases. For both cases, the temperature curves of the light component have a less steep slope than those of the heavy component. This feature can be explained by more intensive molecular diffusion of the light gas. The temperature of the heavy component T1 rises more quickly than the temperature of the light component T2 and exceeds it at some point inside the shock wave. Then T1 either approaches the downstream equilibrium temperature monotonously or becomes higher than the downstream temperature and then decreases. Monotonous behavior is seen in Fig.18 and at the left part of the Fig.19, in which the influence of concentration of the components is shown. At the right part of Fig.19, the non-monotonous behavior of the temperature, which appears at low concentration of the heavy component and when the Mach number is not too small is shown. This phenomenon has already been found by computations in early studies [47,51] and has been known as the temperature overshoot [12,52] . In Fig.20 (left), the components of the temperature tensors for two constituents of the mixture are shown. The higher hump at the heavy gas longitudinal temperature can be explained by the inertia of the heavy molecules penetrating more easily in the depth of the shock wave layer when the collisions with the light molecules prevail. The contribution of the longitudinal component with the big hump in T1 gives the overshoot of this temperature. The comparison of curves in Fig.20 (right) and Fig.19 shows that the temperature overshoot increases with the rise of the concentration of the light gas. In Fig.21, the shock wave structure for low mass ratio m2 m1 = 0.1 is presented. One can see a big difference of the densities and temperatures of the components. The temperature profile of the light component is much less steep than that of the heavy one. The curves of the total density and total velocity of the mixture are located between the curves of the components. From the computational point of view, small ratio of the molecular masses makes the calculations more intensive.

Structure of Shock Waves

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Fig. 19. Shock wave structure for two components mixture at M = 2, m 2 m1 = 0.25 with (1) different concentrations, χ 2 = 0.5 (left), and χ 2 (1) = 0.9 (right).

(1) Fig. 20. Temperature tensors (left) and gas parameters for M = 2, m2 m1 = 0.25 , χ 2 = 0.95

In Fig.22, the comparison of these results with the computations [53] is presented and shows a good agreement between them. For more exact comparison the parameters are normalized by their values ahead the shock wave. The details of the present calculations and the analysis of their accuracy can be found in the papers [57,58]. The computation of the binary mixture at high Mach number is presented in Fig.23. In Fig.24, the results of computations[60] of the 3 component mixture of monatomic gases Argon, Neon and Helium with real masses and molecular diameters are presented. Argon is numbered as the first component, Neon as the second, and Helium as the third. The computations are made

264

High Temperature Phenomena in Shock Waves

for the following parameters of the shock wave and of the mixture: M = 3, m2 m1 = 0.5, m3 m1 = 0.1, d 2 d1 = 0.7, d 3 d1 = 0.6, χ1(1) = 0.2, χ 2 (1) = 0.3, χ 3 (1) = 0.5

One can see the overshoot phenomenon for Argon and strong differences between the curves of all mixture components.

Fig. 21. Shock wave structure for M = 3, χ 2(1) = 0.5 , m2 m1 = 0.1. Parameters of the components (left), comparison of the velocity and density of components and of the mixture (right).

Fig. 22. Comparison of computations of the binary mixture for M = 3 , m2 m1 = 0.5 with

data [53], marked by squares, at χ 2 (1) = 0.1 (right), and χ 2 (1) = 0.9 (left).

In Fig.25, the results of computations for the 4-component mixture of Argon (1), Nitrogen (2), Methane (3) and Helium (4) are shown. Molecular masses and

Structure of Shock Waves

Fig.

23.

Temperature

m2 m1 = 0.5, χ 2

(1)

tensors

(left)

and

gas

parameters

(right)

for

265

M =6

= 0.5.

Fig. 24. Shock wave structure in a mixture of Argon (1) , Neon (2), and Helium (3)

diameters of the components are taken real, but the internal energy of Nitrogen and Methane is not taken into account. The computations are made for M = 3, and for the following parameters of the mixture: m2 m1 = 0.7, m3 m1 = 0.4, m4 m1 = 0.1, d 2 d 1 = 1 .0 3 4 , d 3 d 1 = 1 .1 4 4 , d 4 d1 = 0.6, χ1(1) = 0.1, χ 2 (1) = 0.2, χ 3(1) = 0.3, χ 4(1) = 0.4.

266

High Temperature Phenomena in Shock Waves

Fig. 25. Velocity and density for Argon (1), Nitrogen (2), Methane (3) and Helium (4) and for the mixture at M = 3

6 Conclusion The presented computations of the shock wave structure have been made by the unique approach based on the application of the Conservative Projection Method (CPM) for solving the classical Boltzmann equation for the monatomic gases and the Generalized Boltzmann equation for the molecular gases having internal energy levels. All computations were performed with desk computers without application of parallel processing. For the monatomic gases the computations can be made for real molecular potentials and with controlled high accuracy. Typical CPU time was of the order of 1 hour for precise computations. The computations of the mixtures of monatomic gases do not make the task much more difficult. The analysis of the shock wave structure in molecular gases with rotational energy levels requires considerable more efforts and strongly depends on the strength of the shock wave and on the number of the active levels. Some approximate methods making the computations less intensive have been proposed, but accurate calculations require today the use of powerful computers.

References 1. Alsmeyer, H.: J. Fluid Mech. 74, 495 (1976) 2. Garen, W., Synofzik, R., Frohn, A.: AIAA- Journal 12, 1132 (1974) 3. Mott-Smith, H.M.: Phys. Rev, 2nd Ser. 82, 885 (1951)

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4. Liepmann, H.W., Narashimha, R., Chahine, M.T.: Phys. Fluids 5, 1313 (1962) 5. Chu, C.K.: Phys. Fluids 8, 1, 12 (1965) 6. Holway Jr., H.: Kinetic theory of shock structure using an ellipsoidal distribution function. In: de Leeuw, J.H. (ed.) Rarefied Gas Dynamics, vol. 1, p. 193 (1965) 7. Anderson, D.: J. Fluid Mech. 25, 271 (1966) 8. Bird, G.A.: Shock wave structure in a rigid sphere gas. In: Leeuw, J.H. (ed.) Rarefied Gas Dynamics, vol. 1, p. 216 (1965) 9. Bird, G.A.: Phys. Fluids 13, 1172 (1970) 10. Cercignani, C.: The Boltzmann Equation and its Applications. Springer, Berlin (1988) 11. Fiszdon, W.: The structure of plane shock waves. In: Fiszdon, W. (ed.) Rarefied Gas Flows: Theory and Experiment, vol. 447. Springer, Vienna (1981) 12. Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford Univ. Press, Oxford (1994) 13. Cercignani, C., Frezzotti, A., Grosfils, P.: Phys. Fluids 11, 2757 (1999) 14. Hdjiconstantinou, N.G., Garcia, A.L.: Phys. Fluids 13, 4, 1040 (2001) 15. Kowalczyk, P., Palczewski, A., Russo, G., Walenta, Z.: Eur. J. Mech. B Fluids 27, 62 (2008) 16. Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases. Cambridge Univ.Press, Cambridge (1970) 17. Nordsieck, A., Hicks, B.I.: Monte Carlo evaluation of the Boltzmann collision integral. In: Brundin, C.L. (ed.) Rarefied Gas Dynamics, Proc.5th Intern. Symposium on RGD, vol. 1, p. 695. Plenum Press, N.Y (1967) 18. Yen, S.M.: Phys. Fluids 9, 1417 (1966) 19. Hicks, B.L., Yen, S.M.: Solution of the non-linear Boltzmann equation for plane shock waves. In: Trilling, L., Wachman, H.Y. (eds.) Rarefied Gas Dynamics, vol. 1, p. 313. Academic, New York (1969) 20. Tcheremissine, F.G.: Russian J. Comp. Math. Math Phys. 10, 654 (1970) 21. Ohvada, T.: Phys. Fluids A 5, 217 (1993) 22. Ohvada, T.: Numerical analysis of normal shock waves on the basis of the Boltzmann equation for hard-sphere molecules. In: Shizgal, B.D., Weaver, D.P. (eds.) Rarefied Gas Dynamics: Theory and Simulations, p. 482. IAA, Washington (1994) 23. Aristov, V.V., Tcheremissine, F.G.: Russian, J. Comp. Math. Math. Phys. 20, 190 (1980) 24. Cheremisin, F.G.: Dokl. Phys. 42, 607 (1997) 25. Tcheremissine, F.G.: Comput. Math. Appl. 35, 215 (1998) 26. Cheremisin, F.G.: Dokl. Phys. 45, 401 (2000) 27. Tcheremissine, F.G.: Comp. Math. Math. Phys. 46, 315 (2006) 28. Takata, S., Aoki, K.: Phys. Fluids 12, 2116 (2000) 29. Robben, F., Talbot, L.: Phys. Fluids 9, 633 (1966) 30. Robben, F., Talbot, L.: Phys. Fluids 9, 644 (1966) 31. Robben, F., Talbot, L.: Phys. Fluids 9, 653 (1966) 32. Smith, R.B.: Phys. Fluids 13, 1010 (1972) 33. Ramos, A., Mate, B., Tejeda, G., Fernandes, J.M., Montero, S.: Phys. Rev. 62, 4940 (2000) 34. Koura, K.: Phys. Fluids 14, 1689 (2002) 35. Tokumasu, T., Matsumoto, Y.: Phys. Fluids 11, 1907 (1999) 36. Beylich, A.A.: An Interlaced System for Nitrogen Gas. In: Proc. CECAM Workshop, ENS de Lyon, France (2000)

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37. Tcheremissine, F.: Direct Numerical Solution of the Boltzmann Equation. In: Capitelli, M. (ed.) 24th Intern. Symp. on Rarefied Gas Dynamics, RGD, AIP Conf. Proc., vol. 762, p. 677. Melville, New York (2005) 38. Tcheremissine, F.G., Kolobov, V.I., Arslanbekov, R.R.: Simulation of Shock Wave Structure in Nitrogen with Realistic Rotational Spectrum and Molecular Interaction Potential. In: Ivanov, R.,M., Rebrov, A. (eds.) 25th Intern. Symp. on Rarefied Gas Dynamics, p. 203. Novosibirsk Publishing House of the Siberian Branch of RAS (2007) 39. Tcheremissine, F.G., Agarwal, R.K.: Computation of Hypersonic Shock Waves in Diatomic Gases Using the Generalized Boltzmann Equation, Rarefied Gas Dynamics. In: Abbe, T. (ed.) 26th Intern. Symp. RGD, AIP Conference Proc., vol. 1084, p. 427. Melville, N-Y (2009) 40. Oran, E.S., Oh, C.K., Cybyk, B.Z.: Annul. Rev. Fluid Mech. 30, 403 (1998) 41. Wysong, I.G., Wadsworth, D.C.: Phys. Fluids 10, 2983 (1998) 42. Center, R.E.: Phys.Fluids 10, 1777 (1967) 43. Harnet, L.N., Munz, E.M.: Phys. Fluids 10, 565 (1972) 44. Gmurczyk, A.S., Tarczynski, M., Walenta, Z.A.: Shock wave structure in the binary mixtures of gases with disparate molecular masses. In: Campargue, R. (ed.) Rarefied Gas Dynamics, Commissariat à l’Energie Atomique, Paris, vol. 1, p. 333 (1979) 45. Oberai, M.M.: Phys.Fluids 9, 1634 (1966) 46. Oberai, M.M., Sinha, U.N.: Shock wave structure in binary gas mixture. In: Becker, M., Fiebig, M. (eds.) Rarefied Gas Dynamics, DFVLR, Porz-Wahn, vol. 1, p. B. 25 (1974) 47. Beylich, A.E.: Phys. Fluids 11, 2764 (1968) 48. Fernandez-Feria, R., Fernandez de la Mora, J.J.: Fluid Mech. 179, 21 (1987) 49. Abe, K., Oguchi, H.: Phys. Fluids 17, 1333 (1974) 50. Hamel, B.B.: Disparate mass mixture flows. In: Potter, J.L. (ed.) Rarefied Gas Dynamics, vol. 1, p. 171. AIAA, New York (1977) 51. Bird, G.A.: J.Fluid Mech. 31, 657 (1968) 52. Bird, G.A.: Shock wave structure in a gas mixtures. In: Oguchi, H. (ed.) Rarefied Gas Dynamics, vol. 1, p. 175. University of Tokyo Press, Tokyo (1984) 53. Kosuge, S., Aoki, K., Takata, S.: Eur. J. Mech. B Fluids 20, 87 (2001) 54. Mausbach, P., Beylich, A.E.: Numerical solution of the Boltzmann equation for onedimensional problems in binary mixtures. In: Proc. 13th Internat. Symp. Rarefied Gas Dynamics, vol. 1, p. 285. Plenum, New York (1985) 55. Raines, A.A.: Numerical solution of the Boltzmann equation for one-dimensional problem in a binary gas mixture. In: Beylich, A.E. (ed.) Proc. 17th Internat. RGD Symp., p. 328. Weinheim, New York (1991) 56. Raines, A.A.: Conservative method of evaluation of Boltzmann collision integrals for cylindrical symmetry. In: Brun, R., et al. (eds.) Rarefied Gas Dynamics, vol. 2, p. 173. Cepadues-Editions, Toulouse (1999) 57. Raines, A.A.: Eur. J. Mech. B Fluids 21, 599 (2002) 58. Raines, A.A.: Comp. Math. Math. Phys. 42, 1212 (2002) 59. Raines, A.A.: Fluid Dynamics 38, 132 (2003) 60. Raines, A.A.: Numerical solution of the Boltzmann equation for the shock wave in a gas mixture. In: 27th Intern. Symp. on Shock Waves, St. Petersburg, Russia, p. 213 (2009) 61. Josyua, E., Vedula, P., Bailey, W.F.: Kinetic solution of shock structure in a non-reactive gas mixture. In: AIAA 2010-817. Amer. Inst. Aeronaut. Astronaut., Orlando (2010) 62. Korobov, N.M.: Trigonometric Sums and their applications, Mir, Moscow (1989) 63. Boris, J.P., Book, D.L.: J. Comp. Phys. 11, 38 (1973) 64. Cheremisin, F.G.: Doklady Physics 47, 872 (2002)

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65. Popov, S.P., Tcheremissine, F.G.: A method of joint solution of the Boltzmann equation and Navier-Stokes equations Rarefied Gas Dynamics. In: 24th Internat. Symp. Rarefied Gas Dynamics, AIP Conf. Proc., vol. 82, p. 762. Melville, N.-Y (2005) 66. Tcheremissine, F.G.: Two levels kinetic model for rotational-translational transfers in a rarefied gas, http://www.chemphys.edu.ru/pdf/2007-10-22-001.pdf 67. Ferziger, J.H., Kaper, H.G.: Mathematical Theory of Transport Processes in Gases. North Holland, New-York (1972) 68. Hirshfelder, J.O., Curtiss, C.F., Bird, R.B.: Molecular theory of gases and liquids, N.Y., London (1954) 69. Landau, L.D., Lifshitz, E.M.: Theoretical Physics: Statistical Physics, Nauka-Fizmathlit, Moscow (1995)

Chapter 7

Shock Waves in Hypersonic Rarefied Flows V. Lago1, A. Chpoun2, and B. Chanetz3 1 2

Icare, CNRS, Orléans, France Université d’Evry, Evry, France 3 ONERA, Meudon, France

1 Introduction At very high altitudes, the atmosphere becomes so rarefied that it no longer behaves like a continuous medium when an object is flying in it. As illustrated in Fig.1[1], during re-entry, space vehicles experience rarefied flows in the high-altitude part of their trajectory.

Fig. 1. Space vehicles re-entry trajectories

272

High Temperature Phenomena in Shock Waves

The molecular character of the flow changes the thermal and aerodynamic properties of the flow. In the same time, due to the extreme high velocities, gas compression by a re-entry vehicle generates high enthalpy flows in its shock layer. The theoretical approach developed for continuous gas medium is no longer valid. Depending on the dominant physical phenomena present in the flow and the theoretical approaches necessary to tackle the problem, the rarefied hypersonic flows can be categorized in three regimes: •

The free molecular regime where the mean free path λ∞ for the molecular state is large compared with the characteristic dimension L of the model/vehicle ( λ∞ 1 ). The aerodynamic and thermal characteristics L

•

depend only on the incident flow and on the interaction between the incident molecules and the surface of the model which are described with the Boltzmann equation. The intermediate or transitional state which corresponds to mass densities for which the mean free path becomes comparable to the model characteristic dimension ( λ∞ ∼ 1 ). Direct Simulation Monte-Carlo (DSMC) L

•

method is the most convenient tool to describe the flows in this regime. The continuous flow where the model dimension is larger than the flowing gas mean free path ( λ∞ 1 ), and the classical fluid dynamic equations, L e.g. Navier-Stokes (NS) equations, are valid to describe the flow.

In terms of the rarefaction parameter

M∞ ReL , where M ∞ and ReL are the free flow

Mach number and Reynolds number respectively, the generally accepted boundary between the intermediate and free molecular regime lies at

M∞ = 10 ,and the ReL

boundary between transitional and continuous regime corresponds to

M∞ = 10−1 ReL

(Fig.2). In the continuous regime, there is a “slip state” where, as has been experimentally found, there exists a slip of the gas layer along the model surface accompanied by a temperature jump between the surface and this gas layer. The slip appears increasingly when the rarefaction parameter

M∞ becomes greater than 10 −2 . ReL

The flow can be still described using the continuous regime approach, e.g. NS equations with appropriate surface boundary conditions which take into account the temperature and velocity jumps.

Shock Waves in Hypersonic Rarefied Flows

273

Fig. 2. Flow regimes

In rarefied flow conditions, i.e. for small Reynolds numbers, the shock waves and viscous layers are very thick (Figs.3, 4, 5) [2]. Thus, classical gas dynamics approach for the analysis of flows and shock waves may be invalid, for example shock polar analysis or one-dimensional flow assumption. The flow surrounding the vehicle is dominated by the strong viscous shock wave-boundary layer interaction. Thermal non equilibrium is another aspect of rarefied flows. From the experimental point of view, while there is some similarity between conventional high density supersonic wind-tunnels and low density wind-tunnels regarding operational mode, the instrumentation used in low density wind-tunnel tests is fundamentally different. The rarefaction limits the magnitude of force, pressure and mass density to be measured and specific instrumentation is developed. Alternatively, the molecular character of the flow allows specific techniques based on light emission to be developed, e.g. electron beam technique for the measurement of local densities.

Fig. 3. Rarefied supersonic flow over a wedge at M=4 and ReL=3000 (CNRS/Orleans)

Fig. 4. Rarefied supersonic flow over a ramp at M=21.6 and ReL=39000 (DLR/Göttingen)

274

High Temperature Phenomena in Shock Waves

Fig. 5. Flow interaction between a rocket plume and an external surface in rarefied conditions (CNRS/Orléans

2 General Phenomena in Rarefied Flows 2.1 Flow Regime Classification The choice of the approach to describe an environment depends on the context in which it is located. This criterion will depend on the scale of the gradients of the problem studied in relation to the distance traveled by molecules between collisions. In classical fluid mechanics, the fluid is considered as a set of fluid particles. These particles are subject to physical laws of conservation of mass, momentum and energy that can be described by mathematical equations. These equations involve physical quantities characteristic of the flow such as velocity and temperature, but also physical properties of the fluid introduced into these equations in the form of transport coefficients as coefficient of viscosity or thermal conductivity depending on local fluid conditions. But this classical approach is valid only when the local population of molecules of the fluid is close to equilibrium; in other words this is valid if the variation of the local scale length of flow gradients dQ /Q is small (where Q is any macroscopic flow parameter) . If the flow is subsonic, i.e. the Mach number is smaller than one (Ma <<1), the speed is essentially the randomly molecular thermal velocity c, then the mean free path L can be defined as the ratio of the thermal velocity c to the collision frequency ν, L = c / ν . The criterium of thermal equilibrium is defined by the relation: Knlocal << 1

,

where the local Knudsen number is Knlocal = 1 cν

If the flow is supersonic, then Ma >>1, the speed becomes the stream velocity c0 and the mean free path L = c0 ν . The criterium to define the continuous regime can be expressed as follows:

Shock Waves in Hypersonic Rarefied Flows

275

dQ c0 dQ dQ L = = cos θ Q Q Q ν

Introducing the Knudsen number and the Mach number, the criterium can be written as:

P = Knlocal Ma

γπ 8

cos θ << 1

The parameter P was first introduced by Bird [3,4], who found a breakdown of tranlation equilibrium for P=0.02. Then the limit value for the continuum regime is P<<0.02. On the other hand, the validity of the local parameter is related to the concept of steady-state flow, but all processes are not in thermodynamic equilibrium. Indeed, the test is given below for the balance of translation, but other processes such as rotation, vibration and chemical processes require a greater number of collisions to reach equilibrium thermodynamics. In this case, the parameter Z = τ τ c can be introduced where τ is the relaxation time for the considered process and τ c is the characteristic time between two collisions. Then, the flow regime can be divided into three kinds: • • •

P<0.002/Z, the flow is in thermodynamic equilibrium 0.02/Z0.02, the flow is considered to be a rarefied flow and the classical mechanical laws cannot be applied.

The manifestation of rarefaction is the existence of local non-equilibrium in the gas. Different quantities require more or less collisions to reach equilibrium. The number of collisions required is usually characterized by the collisional number Z ranging from a few units for translation and rotation, to a few thousands for vibration and chemistry. For a considered specific process, the equilibrium is reached when P<<1/Z. In a shock layer, the thermodynamic conditions are such that the local rarefaction parameters are different compared to the global parameters characterizing the free jet, and there is not a simple relation between the local and global parameters. However, we can provide some elements for discussion regarding the parameters of rarefaction for certain areas of the flow. The degree of rarefaction is maximum within the shock wave [5,6]. This particularity involves that only numerical methods able to simulate this region are of molecular approach type. Another particular area of the flow is the boundary layer where the degree of rarefaction is important because of the strong gradients that exist there. Qualitatively, in the shock, the regime can be considered as molecular when the particles travel, after impact with the wall of the model, a distance much greater than the characteristic length of the model. This mean free path is connected to the free flow through a formula that involves the Mach number and ratio of wall temperature and free flow temperature, giving the simplified formula for the molecular regime application:

276

High Temperature Phenomena in Shock Waves

Kn∞ > 30 M∞ Hayes and Probstein [7] proposed a more detailed classification with seven flow stream regimes from the less rarefied to the most rarefied. 2.2 Shock Waves Thickness and Stand-Off Distance 2.2.1 Mixing Reynolds Number During a spatial mission, the flow becomes sufficiently rarefied so that it is necessary to consider the interaction between viscosity and rarefaction, which leads first to boundary layer thickening then shock thickening, these two regions constituting – when a high rarefaction level is reached - a large viscous compression zone. Lengrand[8] proposed an interesting rarefied parameter allowing a classification in five regimes. Thus, he defined a criterion based on a “mixing” Reynolds number Re* significant to qualify the shock wave thickness and the stand-off distance over cylinders and spheres in hypersonic flow. This “mixing” Reynolds number Re*correlates his experiments and the results existing in the open literature. It is defined as follows:

Re* = ρ1U1 R / μ (Tw ) , where the density ρ1 and velocity U1 are relative to the flow stream ahead of the shock, μ is calculated at the temperature at the wall Tw and R is the radius of the cylinder cross-section or of the sphere). The classification is then : - boundary layer regime for Re* > 1400 - mixing regime for 30 < Re* < 1400 - transitional layer regime 3 < Re* < 30 - first-collision regime for 0,3 < Re* < 3 - free-molecule regime Re* < 0.3 2.2.2 Shock-Wave Stand-Off Distance Ahead of Blunt Bodies In the non-viscous case, i.e. when viscosity effects are concentrated in the boundary layer and when the shock is not embedded in the boundary layer, Ambrosio and Wortman9 proposed two correlation formula available in a perfect gas where the ratio of specific heats γ is equal to 1.4 :

- one for a spherical body Δnv = 0.143 e3.24/M.M - one for a cylindrical body Δnv = 0.386 e4.67/M.M For a sphere, by considering a lot of experimental results, Lengrand notices that the real shock stand-off distance Δ differs from the viscous one Δnv given by Ambrosio and Wortman since Re* < 2000. Furthermore Δ/Δnv reaches 3 for Re*= 10. For a cylinder, Lengrand notices that the real shock stand-off distance Δ differs from the viscous one Δnv given by Ambrosio and Wortman since Re* = 100. Besides, Δ/Δnv reaches 3 for Re*= 5 and Δ/Δnv reaches 6 for Re*= 2. Lengrand establishes that the corresponding value Re* when Δ/Δnv differs from one is 20 times higher for the spheres than for the cylinders, which means that the reference length in the cylinder case would have been taken longer than the radius.

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277

2.2.3 Shock Wave Thickness Ahead of Blunt Bodies It is generally admitted that the shock-wave thickness δ is of the order of five mean free paths λ calculated with the conditions of the free stream [10]. It is considered by Lengrand as – infinitely thin – if his thickness is lower than 10% of the shock standoff distance Δ, i.e. δ/Δ < 0.1. In this situation, one considers that the shock is a discontinuity. The work performed by Lengrand shows that this condition is obtained for cylinders and spheres for Re* > 1000 – 2000. These results are based on the consideration of the density profile. The condition δ/Δ < 0.1 must be also respected by the other profiles. He also considers that shock thickness based on the rotational temperature Tr is of the order of Zr λ , where λ is the mean free-path ; Zr = 4 if Tr = 300 K and Zr = 10 if Tr = 1000 K, the temperature Tr being considered just behind the shock. For the sphere, the condition δ/Δ < 0.1 coincides with the condition Δ/Δnv = 1, but for the cylinders, the shock is thickening even if Δ/Δnv = 1. In this last case, the shock thickening appears although the shock stand-off distance is still not affected by the rarefaction. 2.2.4 Impact of Flat-Faced Leading-Edge Effects on Shock Stand-Off Distance and Shock Wave Thickness As it is well-known, the stand-off distance and the shock thickness increase when the flat-face thickness increases, but with a numerical study carried out in a twodimensional approach with a Direct Simulation Monte-Carlo (DMSC) solver, Santos[11] quantified the impact of the blunting effects on the shock-wave structure. He analyses the sensitivity of the shock stand-off distance and shock-wave thickness to shape variations of flat-faced leading edges. The calculations he performed correspond to the flow conditions experienced by a spacecraft at an altitude of 70 km. This altitude is associated with the transitional regime, corresponding to a mixing Reynolds number comprised between 3 and 30, and also characterized by a Knudsen number Kn = λ/L of the order of or larger than 10-2 . The dimensionless shock-wave stand-off distance Δ/λ (where λ corresponds to the upstream free flow) according the local Knudsen number Knt = λ /t (where t is the thickness of the sharp leading edge) is given in Table 1 which gives also a few shockwave thicknesses δ, normalized by the free stream mean free path λ : Table 1. Stand-off distance and shock thickness in function of the Knudsen number

Knt = λ /t

infinite

100

10

1

Δ/λ

0.0

0.096

0.209

0.614

δ/ λ

0.0

0.385

0.528

1.342

The shock wave thickness, for the bluntest leading edge case, corresponding to Knt = 1, is about 3.5 and 2.5 times larger than those for Knt of 100 and 10 respectively. In these last examples, one considers various leading edges, from very sharp ( Knt → ∞ ) to less sharp ones ( Knt = 1 ). But if one considers, for the same flow

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High Temperature Phenomena in Shock Waves

conditions, the case of the round leading edge Santos observes the largest shock thickness (δ/ λ = 3.350). Compared to the truncated wedges, this value is about 8.7, 6.3 and 2.5 times larger than the cases corresponding to Knt of 100, 10 and 1, respectively. 2.2.5 Characterisation of Shock Waves in Rarefied Regime over a Flat Plate In order to characterize shock wave over a flat plate in rarefied regime, Pitot pressures have been measured in the flow field along a flat plate for a Mach number of 26.2 and a unit Reynolds number of 640 /cm 1. Fig 6 shows plots of ratios of the local Pitot pressure and that measured in the non-disturbed flow. These curves correspond to different x-stations along the plate. They exhibit maxima which decrease when approaching the leading edge as the rarefaction parameter increases. The curve

corresponding to a rarefaction parameter V =

M∞ Re∞

equal to 1.13, which is fully

situated in the merged layer regime, does not exhibit any maximum. A schematic view of Pitot pressure profile is also given in Fig.7 which also shows a few characteristic points along the profile. The evolution of the characteristic points along the plate is also plotted in the same figure. The method of “maximum slope” defines the shock wave thickness as the interval between points 2 and 4. However, in rarefied cases, due to the diffuse structure of the shock wave, a better definition for the shock wave thickness is the interval between points 1 and 5 corresponding to the first point where the non-disturbed flow Pitot pressure is found and the point of maximum Pitot pressure. The shock thickness increases slightly along the flat plate from approximately 20 λ∞ up to 30 λ∞ .

Fig. 6. Pitot pressure profiles over a flat plate in rarefied regime ( V : Rarefaction parameter)

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279

Fig. 7. Shock wave characterization over a flat plate from Pitot pressure measurement in rarefied regime

2.3 Heat Flux in Rarefied Conditions

In a free molecular regime, an analytical approach proposed by Bird can be used to determine the heat flux at a wall. The starting point is the flux calculation of a quantity Q through an unity surface s given by

q=

ρ G ( s, s ' ) , β3 q

where 1 ⎡⎛ ⎞⎛ γ ⎞⎤ 1 γ +1 Gq ( s, s ') = ⎢⎜ 2s 2 + + s 2 ⎟⎥ 1 exp(− s '2 ) + 2π 2 s '(1 + erf s ' ⎟⎜ γ −1 ⎠⎝ γ − 1 ⎠ ⎦ 8π 2 ⎣⎝

s ' = s cos θ , where θ is the angle between the velocity direction and the vector normal to the surface. s=

1 c0 , where c0 is the macroscopic velocity, and β = 2RTtr 2 RTtr

The formula can be applied considering the balance between the incident and the reflected flux to the wall.

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High Temperature Phenomena in Shock Waves

The incident flux density can be expressed as follows

⎛ c2 γ ⎞ , qi = ρi RTtri c0i ⎜ 0i + ⎟ −1 ⎠ 2 γ RT tri ⎝ where the indices i and r correspond to incident and reflected quantities respectively, and the reflected flux density is given by qr =

ρr G ( s, s ' ) β r3 q

The diffused reflection model with complete accommodation gives

Ttri = Twall , β =

1 , c0r = 0 2 RTwall

sr = sr' = β c0r , ρr ≠ ρi Finally the heat flux density exchanged can be written as follows ⎛ c0 2 γ ⎞ γ +1 1 qexch = ρ i × R Ttri × c0i × ⎜ i + ⎟ − ρ i × 2 π × c0i × 2 RTwall ⎜ 2 RTtr γ − 1 ⎟ γ −1 8 π i ⎝ ⎠

2.4 Leading Edge Flow and Viscous Interaction in Supersonic Rarefied Flow

Shock wave-boundary layer interactions are among dominant aspects of supersonic and hypersonic rarefied aerodynamics. This phenomenon is very well illustrated by examining the flow over a flat plate with a sharp leading edge. The leading edge hypersonic flows were first analysed by Lees and Probstein (1952). As illustrated in Fig 8, a flat plate with a sharp leading edge placed parallel to the incoming flow in a supersonic rarefied flow produces different flow regimes and includes many aspects of high speed rarefied flows: First, starting from the leading edge, free molecular regime is established extending for a few mean free paths. Then after some transitional regime the developing shock wave and boundary layer merge to form the merged layer. Downstream, the boundary layer and shock wave develop separately. The region between the boundary layer and the shock wave consists of non-viscous potential flow. The flow in this region is dominated by the well-known shock wave/boundary layer interaction. First, the strong interaction regime where the boundary layer displacement thickness grows rapidly and affects the shock wave curvature. In turn, the induced pressure gradient affects the boundary layer development. In the following weak interaction regime, the influence of the shock wave on the boundary layer development can be neglected. The chart shown in Fig. 7.2 illustrates domains of existence of different flow regimes as a function of incoming Mach number and the Reynolds number based on local abscissa. On the flat plate surface, the region extending from the leading edge up to merging layer is

Shock Waves in Hypersonic Rarefied Flows

281

characterized by velocity and temperature jumps on the wall whereas, downstream, full accommodation of velocity and temperature between the wall and the flow exists. These regimes are called slip and non-slip flow regimes, respectively. The free molecular flow is also called kinetic regime. The transitional flow corresponds to the mixed regime and downstream, flows belong to the continuous regime.

Fig. 8. Leading edge flow regimes

2.5 Wall Pressure in Free Molecular Flow Regime

The pressure on a wedge surface placed in a free molecular supersonic flow of initial temperature, pressure and Mach number T1, p1 and M1 respectively, is given by ⎡⎛ 1 p γ M 12 1 sin 2 α ⎢⎜ = ⎜ s π + 2s 2 p1 2 ⎣⎢⎝

Tw T1

⎞ ⎛ 1 π exp( − s 2 ) + ⎜1 + 2 + ⎟⎟ ⎜ 2s 2 s ⎠ ⎝

⎤ Tw ⎞ ⎟ (1 + erf s )⎥ ⎟ T1 ⎠ ⎦⎥

,

where p is the pressure on the wedge, Tw is the wall temperature and s is defined as

s = M1

γ 2

sin α

A full thermal accommodation and diffuse reflection is assumed at the wall.

3 Experimental Approach 3.1 Hypersonic Rarefied Wind Tunnel

For reproducing rarefied flows at the ground level, a number of facilities has been built in many laboratories in the world, mainly during fifties and sixties. As for conventional wind tunnels, these facilities include a gas supplying system, a

282

High Temperature Phenomena in Shock Waves

convergent-divergent nozzle that expands gas from reservoir conditions to test conditions, the test section and finally a diffuser connected to an exhaust system including high-vacuum pumps. In general, these facilities are characterized by low mass-flow rates and also very low pressures in the test chamber. In these facilities, the Mach number range can extend from moderate supersonic up to Mach 30 corresponding to re-entry flows. Due to low mass-flow rates, most of these facilities require much less power than conventional high density hypersonic and supersonic wind tunnels and thus can operate in permanent mode. However, to simulate highenthalpy flows, blow-down type facilities may be required. In these cases, a large amount of thermal energy is accumulated in a reservoir and then released in a short time. In some cases, the test section can also operate as a vacuum chamber for satellite test purpose or jet-wall interaction studies. The nozzle may be either planar or axisymmetric and its exit diameter size may range from a few centimetres up to a metric scale. Low density gas flows leading to very low Reynolds numbers produce very thick boundary layers and thick shock waves. Thus, in rarefied wind tunnels, due to smooth density gradients, the conventional Schlieren system hardly operates and specific techniques of flow visualization and measurement are developed. In the same way, due to the very thick boundary layers produced along the inner surfaces of the wind tunnels, most of low density wind tunnels operate in free jet mode and necessitate an Eiffel type vacuum chamber. Low density wind tunnels facilities are used to investigate either basic physical phenomena (shock wave/ boundary later interaction, shock wave standing distance, chemical reaction, gas molecular dissociation, plasma…..) or to study full aerodynamic behaviour of a given model. The main parameters characterising low-density wind-tunnels are Mach number, Reynolds number and_ the most significant_ the Knudsen number. This last number is the ratio of test flow molecular mean free path and a characteristic length of the model or the nozzle exit section dimension. It characterizes the flow regime ranging from transitional to free molecular. As an example, the CNRS–SR3 facility located at Orleans, France, is a typical lowdensity wind tunnel built in the late sixties. It operates in a wide range of Mach numbers starting from 0.6 up to 22 by changing the nozzle. The facility covers Reynolds numbers from 102 up to 105. A schematic view of the facility is presented in Fig 9. For such facilities, one of the critical part of the installation is the pumping system. In this particular case, depending both on the required flow conditions and on the rarefaction level, two distinct pumping groups are used. For the highest flow densities, the chosen pumping group is composed of Roots pumps associated with two stages of rotative vacuum pumps, with an installed electrical power of about 1000 kW. At the lowest flow density levels, the pumping group, as shown in Fig 9 is composed of three stages including six rotative, vacuum pumps, two Roots pumps and two oil diffusion booster pumps. The cumulative pumping capacity of the whole system approaches 40 m3/s under pumping pressures up to about 10-4 atmosphere. One advantage of the facility is its almost unlimited running time. This is particularly useful to get rid of the instrumentation response time and to get a good stabilization of the flow conditions prior to tests. There exits also a facility adapted to study the transitional flow regime, like the ONERA R5CH , where a number of studies have been carried out to investigate the shock-shock and the shock-boundary layer interactions and regime transitions.

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283

Fig. 9: Schematic view of a supersonic low density wind-tunnel

3.2 Shock Wave-Boundary Layer Interactions in Low Density Flow

Viscous interaction phenomena at high Mach numbers are crucial especially in presence of thick boundary layers which considerably modify the external structure of the aerodynamic field due to the local variation in direction induced in the external flow. This modification then affects the structure of the boundary layer, giving rise to the conventional strong viscous interaction process7 and to shock wave-boundary layer interactions due to the impact of a shock wave on a boundary layer. This last phenomenon has been investigated in the low density R5Ch wind tunnel of ONERA[14,15] using advanced non-intrusive probing methods. The R5Ch wind tunnel is a cold blow down wind tunnel, the stagnation temperature Tst of nearly 1050 K being just sufficient to prevent the air from liquefying when expanded in the nozzle. The stagnation pressure pst was equal to 2.5 105 Pa leading to an unit Reynolds number equal to 168 000 m-1 for a nominal Mach number equal to M0 = 9.92. The unit Reynolds number was sufficiently low to insure a laminar regime throughout the interaction domain. The long duration run (90 s) allowed to make detailed measurements. The model used here is constituted by a hollow cylinder, with a sharp leading edge, followed by a flare terminated by a cylindrical part. To visualize the flow stream in rarefied flow, a well-adapted technique is the Electron Beam Fluorescence[16] (EBF). This technique is based on the formation of N2+ excited ions by an energetic electron beam (typically 25keV energy) crossing the flow. The almost immediate drop to a lower energy state gives rise to fluorescence whose intensity is proportional to the density. At high densities, quenching destroys the linearity of the response. Tomographic imaging with a sweeping electron beam creating a visualization plane is a classical application of EBF. The photograph of Fig.10 shows the flow stream around the cylinder and the attached shock wave at the sharp leading edge as well as the separation shock wave. These two shocks converge above the end of the flare.

284

High Temperature Ph henomena in Shock Waves

loc cation of explorations

Fig. 10. EBF visualization

In order to obtain quan ntitative density results, even when quenching occurss, a variant based on the detecction of brehmstrahlung and characteristic X-rays cann be employed[17]. These X-raays are emitted by electrons which are decelerated w when passing close to an atom. The method has the advantage that the signal is emittted ollisional quenching. The X-ray radiation at the pointt of instantly and shows no co measurement is collimateed and detected with X-ray counters equipped w with preamplifiers. As shown in n Fig.11, the beam crosses the model (though a small tuube) to avoid the intense X-ray ys production which would result from the impact of the beam on the surface.

Fig. 11. X-rays measurementts over a cylinder-flare model at Mach 9.92 EBF visualizaation showing the electron beam trav versing the model

The density measurem ments using electron-beam-excited X-ray detection tthus obtained for three profiles X/L X = 0.3 , X/L = 0.6 and X/L = 0.76 (shown in Fig.10) are

Shock Waves in Hypersonic Rarefied Flows

285

given in Figure 12, where they are compared to the results given by two NavierStokes codes (FLOW and NASCA) and a DSMC code[18].

25 Y (mm) 20

EXP DSMC FLOW NASCA

X / L = 0.3

15 10 5 0 0

0.5

1

1.5

2

2.5 ρ/ρ

25 Y (mm)

0

EXP DSMC FLOW NASCA

20 15

X / L = 0.6

10 5 0

0

0.5

1

1.5

2

2.5 ρ/ρ0 EXP DSMC FLOW NASCA

25 Y (mm) 20

X / L = 0.76

15 10 5 0

0

0.5

1

1.5

2

ρ/ρ0

2.5

Fig. 12. Density profile measurements by X-rays detection on the cylinder-flare model (see location in Fig.10)

Such experimental investigations are very useful to check numerical solvers in continuum or transitional regime.

286

High Temperature Phenomena in Shock Waves

3.3 Shock-Shock Interferences in Low Density

The most spectacular effect resulting from shock-shock interactions in high Mach number flows is the existence of pressure and heat flux peaks at the impingement on a nearby surface of a shear layer (Type III interference) or a supersonic jet (Type IV interference) originating from the shock intersection region[19,20]. These pressure and heat flux peaks are emphasized in rarefied flow and may reach values twenty times higher than those obtained at the stagnation point without interaction. The problem is crucial for air intakes of a future hypersonic vehicle. Thus, an in depth experimental investigation has been performed in the R5Ch wind tunnel already described. Temperature and density were probed using stimulated Raman scattering.

Fig. 13. EBF visualization of a Type IV shock-shock interference at Mach 9.92

The experimental arrangement includes a shock generator made of a 10° wedge located in front of a 16mm diameter cylinder perpendicular to the upstream flow field. The EBF visualization in Fig.13 shows the shock pattern which forms in the case of a type IV interaction. The supersonic jet issuing from the interference point is well visible. This configuration has been computed at NASA Langley by using a DSMC approach[20]. The computed iso-density lines are shown in Fig.14. The interference region has been explored in detail by the Coherent Raman Anti-Stokes Coherent Spectroscopy (CARS) along the horizontal lines indicated in Fig.14.

Fig. 14. Computed iso-density line in the Type IV shock/shock interference and location of CARS exploration lines. NASA DSMC calculation[13]

Shock Waves in Hypersonic Rarefied Flows

287

This method is based on fundamental physical processes related to the interaction between light and matter. In the Raman effect, when a photon strikes a molecule, it leaves a fraction of its energy to the molecule which is then raised to a higher energy state. When de-excitation occurs, a photon is released which has a longer wavelength, or lower energy, than the incident photon. In the present experiment scattering is produced by a first laser called the pump laser. The system includes a second laser, the probe laser, whose frequency was shifted so that the differences in wavelength with the pump laser matches a resonant frequency of the molecule. This arrangement is used in Coherent Anti-Stokes Raman Scattering (CARS) in which measurements are made with the Anti-Stokes radiation[21]. There are several variants of CARS, for example in Dual Line CARS (DLCARS) four beams are used to excite two energy levels of the studied molecule which allows a more direct determination of the density and temperature of the gas. The results thus obtained are reported in Fig.15. Rotational temperature Tr and density ρ/ρinf (where ρinf is the density in the free flow stream far upstream of the first shock emanating from the wedge) were probed along the three lines defined in Fig.14. The results are very useful to know the structure of the phenomena and to validate DSMC solvers. y = − 0,002 m

y = −0,004m

y = − 0,005 m

Fig. 15. Comparison with CARS measurements of computed temperatures and density distributions in the Type IV shock-shock interference. NASA DSMC calculation

288

High Temperature Phenomena in Shock Waves

3.4 Pressure Measurements in Rarefied Flow Regimes 3.4.1 Pressure Measurements and Orifice Diameter Effects For pressure measurements in rarefied flow regimes, because of extremely low pressure levels (pressure as low as 0.01 Pa) and the wide range of pressures encountered, specific techniques are developed. Some difficulties also arise in interpreting measurement results when the levels of rarefaction are high. Orifice effects must then also be taken into account as well as the thermo-molecular nonequilibrium existing along the tubes connecting pressure holes to the transducers. Moreover, density measurements also require prior outgassing of the facility. Finally, low density pressure measurements involve response times significantly longer than those of conventional wind-tunnels. As a consequence, low density facilities are usually of continuous running type. In general to minimize the probe response time, the sensitive element of the transducer is placed in the vicinity of the measurement hole. In addition, in rarefied flow regimes the hole diameter may be of the same order of magnitude as the gas local mean free path. Thus, the measured pressure may differ from the expected or real value. As a consequence, a number of studies have been carried out to calibrate the hole effects. 3.4.2 Pitot Pressure Measurement In the transition regime, stagnation pressure and then flow Mach number can still be measured using Pitot probe (Fig. 16) using an orifice effect correction procedure. In the case of Pitot pressure measurement, Bailly22 introduced the correlation ρ parameter Re2 2 where Re2 is the Reynolds number based on the orifice diameter ρ∞ and calculated with flow conditions behind the normal shock standing in front of the Pitot tube.

Fig. 16. Pitot pressure device

Then he showed that for values of the correlation parameters above 800, the measured pressure coincides with the real pressure. For the values of correlation parameter between 15 and 800 the measured pressures are slightly lower than the theoretical value. For the values below 15 the measured and theoretical values diverge

Shock Waves in Hypersonic Rarefied Flows

289

significantly. In the same manner, Potter et al. show[23] that the stagnation pressure measured by a Pitot probe is significantly different from the theoretical value when the Knudsen number based on the hole diameter and calculated from the conditions behind the standing normal shock wave is greater than 0.2 (Fig.17).

Fig. 17. Knudsen number effects on Pitot pressure measurement (Potter et al.[24])

Fig. 18. Rarefaction effects on Pitot pressure measurement (Allègre et al.[1])

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High Temperature Phenomena in Shock Waves

This condition corresponds to a Bailly parameter lower than 10. In the same manner, Allègre et al.[1] confirmed this trend by measuring stagnation pressure by Pitot device over a wide range of Mach numbers (Fig.18) 3.4.3 Static Pressure Measurement Concerning wall pressure measurements, measured and real pressures may diverge in the case of rarefied gas due to thermal non equilibrium between outer (flow field) and inner gas (gage volume). This effect is more important in the case of cooled wall than that of an adiabatic wall. Fig.19 shows for different ratios of the hole diameter to the gas local mean free path, the ratio of measured and real pressures as a function of the parameter Kw which takes into account the wall heat transfer [24].

Fig. 19. Rarefaction effects on isothermal wall pressure measurement (Potter et al.[23])

Fig. 20. Rarefaction effects on adiabatic wall pressure measurement (Allègre et al.[1])

Shock Waves in Hypersonic Rarefied Flows

291

For adiabatic wall pressure measurement, Allègre et al noted (Fig 20) very little effects of the orifice diameter on static pressure measurement for a wide range of Mach numbers. 3.4.4 Pressure Transducers The most commonly used pressure transducers in both low density and high density testing facilities are based on resistance, reluctance and capacitance variations. However, some specific devices designed for low pressure in low density facilities are the following: Pirani gage: It consists of very fine helical metal wire heated by Joule effect and placed in the centerline of a tube containing the gas whose pressure is to be determined. The heat conductivity of the surrounding gas is related in some extent to its pressure and density. Thus, the wire temperature at equilibrium depends on the gas pressure. This dependence is used for processing the measurements. The pressure range for Pirani gage lies between 0.1 Pa and 50 Pa. Thermistors: This type of device allows the sensitive element to be placed in line with the pressure orifice to avoid too long a response time. The principle of thermistor functioning is similar to that of Pirani gage. Heated electrically, its equilibrium temperature and therefore its resistance changes as a function of thermal conductivity of the surrounding gas pressure. The pressure range for thermistor gage is between 0.1 Pa and 100 Pa. Penning gage and liquid manometers: This type of gage is usually used to measure ultra-high vacuum pressures of less than 0.1 Pa. The gage is constituted of two unheated electrodes between which a cooled discharge is produced. The discharge current is then related to the surrounding gas pressure through a calibration procedure. For pressures higher than 0.1 Pa, the Penning gage becomes irrelevant when an intense luminescence for higher gas densities occurs. The measurement accuracy is particularly poor in the case of Penning gage. McLeod gage and liquid manometers: This type of gage is employed as a calibration device rather than a direct pressure measurement tool. The gage operating principle is very simple: McLeod gage take a known volume of gas at an unknown pressure. Then the initial volume is compressed to a known pressure and volume. The unknown pressure is then obtained through Boyle-Mariotte law. This pressure measurement is very precise and independent of the gas nature. 3.5 Heat Flux Measurements

Aerothermodynamics for space vehicles includes the science and technology of classical aerodynamics including hypersonic speed flow physics and chemistry reacting and dissociated flows. The research field covers as well external flows around aerospace vehicles and internal flows through vehicle propulsion systems. The external aerodynamics of aerospace vehicles deals with the transition from high altitude free molecular flow to continuum flow occurring when vehicles enter planetary atmospheres. Some of the outputs of these studies are the aerodynamic loads and kinetic heating rates, used for the structural, thermal and flight-control

292

High Temperature Phenomena in Shock Waves

design of the vehicles. One of the means to address aerodynamic design issues and to quantify their effects is the test in ground-based facilities such as classical wind tunnels, shock tunnels, plasma facilities and their instrumentation. For conventional wind tunnels, characterized by rather high flows densities, several instruments and techniques are available for aerodynamic heat rate measurements, such thin-film gages, coaxial surface thermocouples, thin-skin techniques, infrared thermography, phase-change paint technique, liquid crystals, and thermographic phosphor technique. Under rarefied flow conditions, heat rate measurements required a particular care and the choice between the available techniques becomes limited [25,26] The thin-skin technique is considered as the most accurate and reliable method, but limitations arise when analyses have to be applied on complicated shape models. For such configurations, the phase –change technique or liquid crystal allows a continuous thermal mapping over the coated surface but becomes inaccurate when the convective heat rate is lower than 6 kW/ m2. In such considerations the infrared technique is more appropriated because this method is more sensitive than the phasechange paint and especially, may provide the thermal mapping over complex models[27,28]. The main difficulty of temperature measurement by IR thermography is the accurate measurement of the model surface emission. As an example, the presence of graphite paint on a steel surface slightly increases the emissivity coefficient, but for a larger number of layers, the emissivity remains almost constant.

dt= 0

dt=2. 07

dt=0. 95

dt=3.02

Fig. 21. IR thermographic measurements of heat flux

Shock Waves in Hypersonic Rarefied Flows

293

Thus, the calibration of the IR camera with a simple model is a necessary step before using this method for more geometric complex models. The principle of the measurement is the same for the IR thermography than for the thin skin technique. The thin-skin technique for heat measurement in wind tunnels has been widely used during the Hermes program. A great number of tests have been carried out in the SR3 facility, nowadays renamed MARHy (Mach Adaptable Rarefied Hypersonic facility) and located in the Icare laboratory located at Orleans[29]. This facility is an open-jet wind tunnel producing continuous low-density flows at supersonic and hypersonic speed. In these conditions the reduction of thin-skin temperatures T to the heat rate quantity q involves the calorimetric heat balance for the thin-skin, which can be written as

q = ρ cb

dT , dt

where ρ and c are the density and specific heat of the model material respectively and b is the model skin thickness. The thermal radiation and heat conduction effects on the thin-skin element are neglected in the preceding relationship, and the skin temperature response is assumed to be due to convective heating only. The model is equipped with a set of thermocouples placed in the intrados of the model as presented in Fig.21, allowing to determine the heat flux from the thin-skin method. To illustrate this purpose, in Fig.21 one can see the over-heating of the Hermes model in a Mach 20 flow with a Reynolds number Re=824 cm-1. measured with an IR camera . 3.6 Shock Wave Control

In recent years there has been a growing interest in using weakly ionized gases for various different aerodynamic applications. Indeed, a weakly ionized gas appears like a new kind of flow control method when the abilities of traditional methods are close to natural limitations due to a strict localisation and slow response. Electroaerodynamic devices seems to be an appropriate means to increase abilities in flow control thanks to total electric control, no moving part and fast response time. The electro-hydrodynamic (EHD) technologies have been considered as good candidates to reduce the wave and viscous drag, heat fluxes, to produce sonic boom mitigation and to control boundary layer, turbulent transition or shock wave. In subsonic air flow, DC discharge or Dielectric Barrier Discharge (DBD) are used to produce “ionic wind”. Under Coulombian forces ions, drift from the first electrode to the second one and induces a secondary flow by collision with neutral species. This momentum is used to modify characteristics of the air flow. The first experiments of boundary layer control were made for low velocities (up to 25 m/s) but now efficiency has been extended for flow velocities up to 75 m/s [30]. In supersonic airflow the main problem is associated with generation of shock waves resulting in high mechanical and thermal load on elements of an aircraft construction, sharp growth of drag force, and reduction of ramjet efficiency. It has been found that gas discharge plasma can modify propagation of shock waves, reduce aerodynamic drag, increase lift but the nature of the observed effect was not clear and

294

High Temperature Phenomena in Shock Waves

from the very beginning there were a lot of controversial speculations around the problem[31] .It appears that plasma discharges induces gas heating or non-uniform gas heating. This energy addition to the flow results in an increase in the local sound speed and a reduction of gas density, that leads to expected modifications of the flow. For example some authors proposed that the local increase in pressure produced by the gas heating acts in a similar manner to a solid obstacle such as tab suddenly placed in the flow. Nevertheless some experimental observations are difficult to be explained only by heat release effects. For some authors it seems that plasma generation is not equivalent to conventional heating because the plasma structure is self-sustained with the flow structure and that the plasma influence on the flow leads to non-evident consequences. One example about this supposed non thermal effect is the non-symmetrical effect of AC and DC discharges on the drag reduction when the polarity is changed. Among the advanced reasons we can find a V-T relaxation (release of stored vibrational energy) or the similar explanation that in the subsonic plasma flow control: tangential momentum transfer due to ions accelerated by electrostatic forces (“ionic wind”). This non thermal mechanism should be responsible for this anomalous effect. A number of papers report observations of drag modification using different types of plasma generation in different aerodynamic conditions. For instance, a decrease of the tangential force on a plate in a subsonic flow is described by Leonov et al [32] or the modification of the shock wave in front of a cone is presented by Bivolaru et al[33]. The flow control of supersonic rarefied flow around a body by means of plasma discharge is a challenging work because low density flows presents experimental disadvantages: for example the low density does not allow the use of PIV or Schlieren methods to visualize the flow, and only the Pitot tube can be used to determine the localisation of the shock; moreover some care must be taken about first the material in which the Pitot tube is made off to avoid electrical perturbations, and second the corrections to be done for the interpretation of the measured pressure due to flow supersonic speed. The model presented here is a Plexiglas flat plate with a sharp leading edge (Fig.22). It is 5 mm thick, 100 mm long and 80 mm large. This plate is located in a Mach 2 air flow whose free stream pressure and temperature are 8 Pa and 167 K, respectively. The Knudsen number based on the plate length is 0.0026. The wind tunnel MARHy offers a wide range of Mach numbers (from 0.8 to 20) and Reynolds numbers, in a circular test section approximately 100 mm in diameter [34]. The rarefied air flow may be used in continuous mode with help of pumping group which can pump up to 4 g.s-1. For the present study, it runs with air and the nominal flow conditions are presented in Table 1. The Mach number is equal to 2 and the mean free path is equal to 0.375 mm. The plasma generation is ensured by means of an electrical discharge. Two aluminum strips of 5 mm thick are glued on the plate. Both of them are 80 mm large, one is 60 mm in the longitudinal direction and is glued at 3 mm from the leading edge of the plate, the other one is 15 mm long and is glued at 20 mm downstream of the first one Two different sources of electrical power are used: a Spellman high voltage power supply delivering a continuous (so called DC) signal up to 20 kV and 400 mA and a Trek amplifier. This device is able to amplify any input voltage by a factor of 1000 up

Shock Waves in Hypersonic Rarefied Flows

295

to 20 kV and 40 mA. This enables us to try out different kinds of signals from a continuous discharge at different input powers to a sinus signal at several frequencies. A negative potential was applied to the active electrode. (First experiments with a positive potential resulted in electrical mini-discharges in the whole chamber). In a typical experiment the input voltage is -1 kV, resulting in a 40mA current issued by the electrical source. More than 80% of this current is collected on the grounded electrode (33 mA). Table 2. Flow conditions for a 7.9 Pa, Mach 2 nozzle Stagnation Conditions

Flow Conditions

Pe = 7.9 Pa Te = 163 K po = 63 Pa To = 300 K ρo = 7.44×10-4 kg.m-3

ρe = 1.71×10-4 kg.m-3

Ve = 511 m.s-1 Mae = 2 μe = 1.1×10-5 Pa.s λe = 0.375 mm

qm = 3.34×10-3 kg.s-1

Fig. 22. Detailed view of the flat plate and its insertion into the test section

A home-made glass Pitot tube of 4 mm inner diameter is used to measure transverse profiles of the stagnation pressure at several locations from the leading edge. As illustrated in Fig.23, the discharge has an influence on the profile.

296

High Temperature Phenomena in Shock Waves

Fig. 23. DC discharge on supersonic rarefied air flow over flat plate

Mach 2

A stable DC discharge has been obtained on the upper face of a plate in a Mach 2, low density flow. As it can be observed, the shape of the plate disrupts the shock below the plate, because at the beginning there is a bow shock due to the shape, then the shock becomes oblique. However above the plate, the shock wave is oblique and the boundary layer can be distinguished. The shock on the lower part is stronger than on the upper one, due to the bevelling, so even without any discharge the lift is non-zero and is positive. It can also be observed the Pitot tube and the bow shock formed around it, and a very bright region where the active electrode is localized. Above the passive electrode, the region is dark, meaning that there is no plasma and in terms of surface temperature the contribution will come only from the active electrode zone. This temperature has been determined from the IR camera acquisitions (Fig.24). Experiments were carried out for three electrical powers and the temperatures of the active electrode surface are also given in Fig.24. This heating, due to ionic bombarding and recombination at the surface increases as the electrical potential increases. Pitot probe measurements demonstrate a thickening of the boundary layer when the upstream electrode is the cathode. This effect is weak or even inexistent when the downstream electrode is active. Thus one may think that this effect is directly connected to the heating of the plate. When the upstream part of the plate is hot, the boundary layer is more influenced by the heating.

Temperature (K)

473 537 670

Electrical power (W)

30 60 90

Fig. 24. Temperature distribution on the upper face of a plate.( -1.63 kV is applied on the downstream electrode)

Shock Waves in Hypersonic Rarefied Flows

70

70 W ithou t dis c harge W ith dis c harge 30 W W ith dis c harge 60 W W ith dis c harge 90 W

W ithout disc harge

50

50

40

40

30

20

10

20

25

30 35 40 P res s ion P it ot (P a)

45

50

0 20

55

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35 40 45 P ress ion P itot (P a)

50

55

60

70 W ith out dis c ha rge

W ithout dis c harge W ith dis c harge 30 W

60

60

W ith dis c harge 60 W W ith dis c harge 90 W

50

50

40

40 z (mm)

z (mm)

30

20

10

30

20

W ith dis c harge 30 W W ith dis c harge 60 W W ith dis c harge 90 W

30

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0 20

W ith dis charge 30 W W ith dis charge 60 W W ith dis charge 90 W

60

z (mm)

z (mm)

60

0 15

297

10

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30

35 40 45 P itot pres s ure (P a)

50

55

60

0 25

30

35

40 45 P res sion P itot (P a)

50

55

60

Fig. 25. Effect of plasma discharge on the total pressure profile measured above the upstream electrode with the potential applied downstream (left), and upstream (right)

References 1. Allègre, J.: Problèmes d’interactions liées aux régimes d’écoulements supersoniques et hypersoniques raréfiés, Ph.D. Thesis, Univ.Paris VI (1979) 2. Allègre, J., Raffin, M.: Experimental techniques in the field of low density aerodynamics, AGARD-AG-318(E) (April 1991) 3. Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flow. Clarendon Press, Oxford (1994) 4. Bird, G.A.: AIAA J. AIAA J. 8, 11, 1998 (1970) 5. Boyd, I.D., Chen, G., Candler, G.V.: Phys. Fluids A. 1, 210 (1995) 6. Boyd, I.D., Chen, G., Candler, G.V.: In: Proc. 23rd Intern. Symp. Rarefied Gas Dynamics, Whistler, Canada, July 20-25 (2002) 7. Hayes, W.D., Probstein, R.F.: Hypersonic Flow Theory. Academic Press, New York (1959) 8. Lengrand, J.C.: Le problème du corps émoussé dans un écoulement supersonique raréfié, Lab. Aérothermique, CNRS rapport 75, 505 (1975) 9. Ambrosoio, D., Wortman, A.: ARS J. 32, 281 (1962) 10. Candel, S.: Mécanique des Fluides, Dunod (1990) 11. Santos, W.: Flat-faced leading-edge effects on shock-detachment distance in hypersonic wedge-flow. Combustion and Propulsion Laboratory, Cachoiera Paulista, SP 12630-000 Brazil 12. Délery, J., Chanetz, B.: Experimental Aspects of Code Verification/Validation: Application to Internal Aerodynamics, VKI Lecture Series 2000-08 (2000) 13. Chanetz, B., Bur, R., Dussillols, L., Joly, V., Larigaldie, S., Lefèbvre, M., Marmignon, C., Mohamed, A.K., Oswald, J., Pot, T., Sagnier, P., Vérant, J.L., William, J.: Aerospace Science and Technology, vol. 4, 5, p. 347 (2000) 14. Mohamed, A.K., Pot, T., Chanetz, B.: In:16th Intern. Congress on Instrumentation in Aerospace Facilities, Dayton, OH (July 1995)

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15. Gorchakova, N., Kuznetsov, L., Yarigin, V., Chanetz, B., Pot, T., Bur, R., Taran, J.-P., Pigache, D., Schulte, D.: J. Moss, AIAA J. 40, 593 (2002) 16. Chanetz, B., Benay, R., Bousquet, J.-M., Bur, R., Pot, T., Grasso, F., Moss, J.: Aerospace Science and Technology 3, 205 (1998) 17. Edney, B.: Aero. Research Institute of Sweden, Rep.115, Stockholm (1968) 18. Moss, J.N., Pot, T., Chanetz, B., Lefebvre, M.: In: 22nd Intern. Symp. on Shock-Waves, London, UK, Paper No. 3570 (1999) 19. Lefebvre, M., Chanetz, B., Pot, T., Bouchardy, P., Varghese, P.: Aero. Research, 1994-4, 295 (1994) 20. Bailly, A.: Further experiments on impact pressure probe in a low density hypervelocity flow, AEDC-TDR-62-208 (1962) 21. Potter, J., Kinslow, M., Boylan, D.: In: 7th RGD Symposium (1970) 22. Potter, J., Kinslow, M.: In: 7th RGD Symp. (1970) 23. Délery, J., Chanetz, B.: Experimental aspects of code verification/validation: application to internal aerodynamics, VKI LS 2000-08 (2000) 24. Matthews, R.: In: 1st GAMNI-SMAI Meeting, Paris (1987) 25. Carlomagno, G., Luca, L.: In: 4th Intern. Symp. on Flow Visualization, Paris (1986) 26. Luca, L., Carlomagno, G., Buresi, G.: Experiments in Fluids 9, 121 (1990) 27. Allégre, J., Dubreuilh, X., Raffin, M.: In: Rarefied Gas Dynamics. Progress in Astro. and Aero., vol. 117 (1989) 28. Corke, T.C., Post, M.L.: AIAA Paper 2005-563 (2005) 29. Menart, J., Shang, J., Atzbach, C., Magoteaux, C., Slagel, M., Bilheimer, C.: AIAA Paper 2005-947 (2005) 30. Parisse, J.-D., Léger, L., Depussay, E., Lago, V., Burtschell, Y.: Phys. Fluids 21 (2009) 31. Lago, V., Lengrand, J.-C., Menier, E., Elizarova, T.G., Khokholov, A.A.: In: Abe, T. (ed.) Rarefied Gas Dynamics, vol. 1084, p. 901 (2009)

Chapter 8

High Enthalpy Non-equilibrium Shock Layer Flows: Selected Practical Applications S. Karl, J. Martinez Schramm, and K. Hannemann German Aerospace Center, DLR, Institute of Aerodynamics and Flow Technology, Spacecraft Department, Göttingen, Germany

1 Introduction A space vehicle (re)-entering the atmosphere of Earth or a different planet is subjected to flows that place the vehicle under extreme physical conditions. High temperature effects such as dissociation, vibrational excitation, electronic excitation or gas radiation within the shock layer in front of the vehicle will have to be correctly modelled by computational fluid dynamics (CFD) tools to be used in the framework of the design process of future entry or re-entry configurations. One important step during the development of a CFD code is the validation of the physical-chemical models used to describe the high temperature effects. The strategy generally pursued is to validate these models with data obtained in ground based testing facilities and / or flight tests. The validated CFD tool can subsequently be used for ground-to-flight extrapolation and for the computation of the flow field past (re)-entry vehicles at free flight conditions. Due to the complexity of the physical and chemical phenomena which are observed during (re)-entry, this validation can only be performed for a certain range of the flight trajectory or for a certain range of conditions in ground testing facilities. Even here, due to the large number of phenomena which need to be considered and the interactions between them, the validation of each aspect of the physical-chemical model is very difficult. Due to the fact that the development of in-flight measurement techniques as well as the measurement techniques applied in ground based testing to provide additional and more detailed information about the considered high enthalpy flows is a continuously progressing research field, code validation is not a one step process. It is rather an ongoing interaction between experiment and CFD. The aim of this combined effort is to improve both the knowledge of the flight environment and the facility performance on one hand and the performance of the physical-chemical modelling on the other hand, ultimately leading to a reduction in the uncertainty of predicted flow quantities. Related to this combined experimental / numerical effort, two selected examples are discussed here. The first describes the investigation of the chemical relaxation process in a cylinder shock layer flow in the High Enthalpy Shock Tunnel Göttingen (HEG), and the second is related to the numerical prediction of the radiative heat flux on the Huygens spacecraft during its entry into Titan’s atmosphere.

300

High Temperature Phenomena in Shock Waves

2 Chemical Relaxation in High Enthalpy Cylinder Shock Layer Flow When considering the re-entry trajectory of winged space vehicles returning from low earth orbit into the atmosphere, the most critical point concerning the heating loads on the vehicle is found in the continuum flow region in approximately 70 km altitude. In this region of the re-entry path, the velocity of the vehicle is approximately 6 km/s and the flow past the vehicle is accompanied by strong shock waves, leading to high temperatures ensuing dissociation reactions. The fundamental influence of the thermal and chemical relaxation processes caused by these high temperature effects on the external aerodynamics, i.e. the pressure distribution, flap efficiency, shock/shock and shock/boundary layer interactions and on the heating loads can be investigated by looking at the flow past basic generic flow configurations which are especially designed in order to focus on one of these effects. Additionally, these studies are well suited to validate the ground based facility performance, measurement techniques and computational fluid dynamics (CFD) codes. A test campaign was performed in the High Enthalpy Shock Tunnel Göttingen (HEG) of the German Aerospace Center to study the relaxation processes in the shock layer of a cylinder placed with its axis transverse to the flow[16]. This configuration was chosen because of the large shock stand-off distance that permits optical measurement techniques to investigate gas properties in the shock layer. Surface pressure and surface heat flux measurements and the determination of two dimensional shock layer density distributions by phase step holographic interferometry were performed. 2.1 High Enthalpy Shock Tunnel Göttingen (HEG) The HEG is a free piston driven shock tunnel[9,11,13] which was developed and constructed in the framework of the European HERMES program over the period 1989 – 1991. It was commissioned for use in 1991, at that time being the largest facility of its type worldwide. Since then it was extensively used in a large number of national and international space and hypersonic flight projects. In a free piston driven shock tunnel, the conventional driver of a shock tunnel is replaced by a free piston driver. This concept was proposed by Stalker[31]. A schematic and wave diagram of this type of facility is shown in Fig.1.

Fig. 1. Schematic and wave (x-t) diagram of a free piston driven shock tunnel

High Enthalpy Non-equilibrium Shock Layer Flows. Selected Practical Applications

301

Fig. 2. Schematic of the High Enthalpy Shock Tunnel Göttingen (HEG)

Free piston driven shock tunnels consist of a secondary reservoir, a compression tube, separated from an adjoining shock tube via the primary diaphragm, and a subsequent nozzle, test section and dump tank. A schematic and photos of HEG are given in Fig.2 and Fig.3, respectively. The high pressure air stored in the secondary reservoir is utilised to accelerate a heavy piston down the compression tube. During this quasi-adiabatic compression and heating of the light driver gas (typically helium or a helium argon mixture), the piston reaches a maximum velocity in the order of 300 m/s. The driver gas temperature increases with the driver gas volumetric compression ratio. When the main diaphragm burst pressure is reached it ruptures and the wave process as in a conventional reflected shock tunnel is initiated (see Fig.1). A shock wave is moving into the driven section and the head of a centred expansion wave is moving into the high pressure region. The numbers used in Fig.1 denote distinct regions of the flow. Region 1 contains the test gas at the initial shock tube filling conditions and region 4 contains the hot, compressed driver gas after piston compression. Region 2 contains the shock compressed test gas, while in region 3, the driver gas processed by the unsteady expansion wave is contained. The test and driver gas are separated by a contact surface.

Fig. 3. Photographic views of the High Enthalpy Shock Tunnel Göttingen (HEG)

302

High Temperature Phenomena in Shock Waves

After reflection of the incident shock wave at the right end wall of the shock tube, the test gas is brought to rest in region 0. Subsequently, the reflected shock wave penetrates the contact surface. Depending on the local conditions, three types of shock wave / contact surface interaction can be differentiated. Due to the fact that the shock compressed and heated slug of gas in region 0 is used in reflected shock tunnel operation as the reservoir driving the flow in the nozzle and test section, shock tube operation in tailored interface mode is most desirable (see, e.g.,[12]). Reflected shock tunnels are characterised by a convergent - divergent nozzle which is attached to the end of the shock tube. A thin secondary diaphragm is placed at the nozzle entrance in order to allow evacuation of the nozzle, test section and dump tank before the run. The nozzle entrance diameter is chosen sufficiently small such that the incident shock wave is almost completely reflected. The stagnant slug of test gas, generated by the shock reflection in region 0 is subsequently expanded through the hypersonic nozzle. The nozzle flow starting process is characterised by a wave system which passes through the nozzle before a steady flow is established (see Fig.1). The incident shock wave (a) is followed by a contact surface (b), an upstream facing secondary shock wave (c) and the upstream head of an unsteady expansion (d). The trajectory of the piston is chosen in a way that after main diaphragm rupture, the pressure and temperature of the driver gas in region 4 is maintained approximately constant. This is achieved by selecting the velocity of the piston at diaphragm rupture, and therefore the subsequent movement of the piston such that it compensates for the loss of the driver gas flowing into the shock tube. For that reason, in contrast to the constant volume driver of conventional shock tunnels, the free piston driver is a constant pressure driver. Due to the large forces occurring during the operation of the free piston driver, the compression tube, shock tube, nozzle assembly is allowed to move freely in axial direction. An inert mass placed at the compression tube / shock tube junction can significantly reduce the recoil motion of the facility during operation. The test section and the dump tank remain stationary. A sliding seal is used at the nozzle / test section interface. The overall length of HEG is 62 m and it weighs 280 t. Approximately, a third of its weight is contributed by the inert mass (see Fig.2 and left picture of Fig.3). The compression tube is closed by a hydraulic oil system (quick disk connect) at the main diaphragm station. The shock tube is connected to the nozzle of the tunnel at the downstream closure, which is also driven by oil hydraulics to close and seal the tunnel. The compression tube has a length of 33 m and a diameter of 0.55 m. The shock tube is 17 m long with a diameter of 0.15 m. The HEG was designed to provide a pulse of gas to a hypersonic convergent - divergent nozzle at stagnation pressures of up to 200 MPa, and stagnation enthalpies of up to 23 MJ/kg. Regarding the test gas, no basic limitations exist. The operating conditions presented in the present chapter are related to the test gas air. Additionally, operating conditions using nitrogen and carbon dioxide exist.

High Enthalpy Non-equilibrium Shock Layer Flows. Selected Practical Applications

free molecular regime

-7

-1

XIII

I

-4

II

IV

1

2

10

ent turbul

chem. nonequilibrium therm. equilibrium

XXXI

chem. equilibrium therm. equilibrium

~90%

0

ioni satio n

-3

10 6

Retr=10 [IXV flight]

XXI XXII

10

* * IXV-noseradius = 0.89 m

M=17

M=10

XIV M=6

IXV

N2 dissociation

SHEFEX II

vib. excitation

ρ⋅L [kg/m ]

2

ReHEG [1/m]

45⋅10 6 100⋅10

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-3

-2

-2

10

M=25

10

10

chem. / therm. nonequilibrium

laminar

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KnIXV-noseradius* *

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O2 disso ciati on

SHEFEX I

-5

10

0

10

transition regime

-6

10

~10%

~10%

~10%

~10%

Apollo 11

10

0.22⋅106 0.43⋅106 0.67⋅106 1.6⋅106 2.8⋅106 3.7⋅10

303

thermosphere

~90km

mesosphere ~50km

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~10km troposphere

100 150 200 250 300

T [K]

Fig. 4. HEG operating conditions in terms of the binary scaling parameter ρL and the flow velocity u

Originally, HEG was designed for the investigation of the influence of high temperature effects such as chemical and thermal relaxation on the aerothermodynamics of entry or re-entry space vehicles. In order to correctly model the chemical relaxation occurring behind the bow shock of a re-entry vehicle, the flight binary scaling parameter must be reproduced in ground based testing. Further, for high enthalpy testing an additional driving parameter which must be reproduced is the flow velocity. Therefore, the operating conditions of HEG are discussed in Fig.4 in terms of the binary scaling parameter ρL and the flow velocity u. Here L represents a characteristic length of the considered configuration. In addition to the HEG operating conditions, the most important fluid mechanical and chemical processes occurring during re-entry of a spacecraft in the Earth’s atmosphere are depicted in Fig.4. Further, as reference, the flight trajectories of a lifting body re-entry from low Earth orbit (ESA intermediate experimental vehicle (IXV)), a ballistic super orbital re-entry (Apollo 11) and two hypersonic flight experiments (DLR flight test SHEFEX I and SHEFEX II) are provided. An indication of the corresponding flight altitudes is given in the right diagram of Fig.4 showing the temperature variation of the Earth’s atmosphere. The transitions between regimes of different physical and chemical properties shown in Fig.4 depend on the chosen reference length and vary when different configurations are considered. Further, the boundaries shown have only symbolic character. In reality, no clear-cut dividing lines between the different regimes exist. The Knudsen number given in Fig.4 shows that the HEG operating conditions are located in the continuum flow regime. The high energy content of reentry flows leads to strong heating of the air in the vicinity of a spacecraft. Depending on the temperature level behind the shock wave (i.e. the flight velocity), the vibrational degrees of freedom of the air molecules are excited and dissociation reactions of oxygen- and nitrogen molecules may occur. Further, ionisation of the air constituents occurs. The high temperature effects described here are enabled by

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energy transfer from the translational energy stored in the random motion of the air particles, which is increased by the gas heating, to other forms of energy. Because this energy transfer is realised by air particle collisions, it requires a certain time period to develop. The time required to reach an equilibrium condition, is e.g. defined by the local temperature and density. Therefore, depending on the ratio of the relaxation time to a characteristic timescale of the flow, the chemical and thermal relaxation processes can be either in non-equilibrium or in equilibrium. Further, along a re-entry trajectory, the Reynolds number varies over several orders of magnitude. In high altitude flight the wall boundary layer of a re-entry vehicle is initially laminar. After exceeding a critical Reynolds number (shown exemplarily for the IXV configuration in Fig.4), the transition from a laminar to a turbulent boundary layer takes place. This process is linked with an increase of the skin friction and the wall heat flux. The HEG operating conditions I – IV are the high enthalpy conditions covering a total specific enthalpy range from 12 – 23 MJ/kg. Over the last years the HEG operating range was subsequently extended. In this framework the main emphasis was to generate test section conditions which allow investigating the flow past hypersonic configurations at low altitude Mach 6 up to Mach 10 flight conditions in approximately 33 km altitude. These low enthalpy conditions cover the range of total specific enthalpies from 1.5 – 6 MJ/kg. Details of the high enthalpy HEG operating conditions are provided in Table 1. For these conditions the test time amounts to approximately 1 ms. Table 1. Summary of HEG high enthalpy nozzle reservoir and test section flow conditions Condition

I

II

III

IV

p0 [MPa]

35

85

44

90

9100

9900

7000

8100

h0 [MJ/kg]

22

23

12

15

M∞ [-]

8.2

7.8

8.1

7.9

Rem [1/m· 10 ]

0.20

0.42

0.39

0.67

p∞ [Pa]

660

1700

790

1680

T∞ [K]

1140

1450

800

1060

ρ∞ [g/m ]

1.7

3.5

3.3

5.3

u∞ [m/s]

5900

6200

4700

5200

T0 [K]

6

3

A conical nozzle with a half cone angle of 6.5° was utilized for the cylinder shock layer flow experiments. The total length of the nozzle is 3.75 m, the throat radius is 0.011 m, and the exit radius is 0.44 m.

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2.2 Phase Step Holographic Interferometry Interferometry may be used as a technique to measure the variation of the refractive index in a gaseous flow in the test section of a short duration facility[12]. This information can be used to evaluate the density distribution of the investigated flow field. In particular, holographic interferometry does not require machining or manufacturing of test section windows, mirrors or lenses with high precision, because parasitic effects caused by imperfections of these components cancel out when applying the holographic two step procedure. Therefore, related to the application in short duration ground based test facilities, this technique has replaced the classical and labour intensive Mach-Zehnder interferometry. In the subsequent paragraphs, a brief introduction of the phase step holographic interferometry technique is given and the set up used at HEG is described. The absolute speed, c0 , at which light propagates in vacuum is a constant. In any kind of gaseous media the speed of light, c , will be lower. The ratio of the two speeds defines the refractive index:

n( ρ ) =

c0 = 1+ K λ ρ . c

(1)

The Gladstone-Dale relation describes that in a gaseous media, consisting of one species, the refractive index depends on the density, ρ , and the Gladstone-Dale constant K λ (see, e.g.,[20]). The latter is weakly dependent on the wavelength and is specific for each gas. For gas mixtures the refractive index is given by S

n( ρ ) = 1 + ρ ∑ K iλ ξi ,

(2)

i =1

where K iλ are the Gladstone-Dale coefficients for the single gas species, ξi are the species mass fractions, S is the number of species and λ is the wavelength of the laser light source. The definition of a linearly composed Gladstone-Dale constant applies not only to air and other neutral gas mixtures but also, in high temperature gas dynamics, to chemically homogenous gases where the molecules are either in different excited states, dissociated, or even ionized.

Fig. 5. Schematic of interference experiment

306

High Temperature Phenomena in Shock Waves

The basic principle of interferometry is sketched in Fig.5. Two rays of coherent light interfere in point P. Each ray passes a zone with different refractive index, which leads to a time shift, Δ t , due to the different propagation speeds of the light in the two zones:

Δt =

L L L − = ( n2 − n1 ) . c2 c1 co

(3)

If the resulting difference in the optical path length equals the wavelength, the phase shift φ between both rays is equal to 2π . Therefore, the following relationship can be derived:

K L φ L = ( n2 − n1 ) = λ ( ρ 2 − ρ1 ) . 2π λ λ

(4)

The light intensity measured in point P is proportional to the phase shift between the two rays. Consequently, the measurement of the intensity or phase shift in point P is directly related to the density difference in regions 1 and 2 in Fig.5. Using a Mach-Zehnder interferometer, which is shown schematically on the left side of Fig.6, the density in the test section can be evaluated if the density distribution is known at a reference point. The schematic emphasises that any imperfection which leads to a modification of the light path, disturbs the interference measurement in plane F. To avoid this, the measurement can be performed in two steps (see middle sketch of Fig.6). In the first step, the light beam (green) passing through the evacuated test section is recorded and in the second step the light beam (red) passing through the test section with flow is recorded. Subsequently, both beams are reconstructed resulting in an interference pattern in the measurement plane, P (see right sketch of Fig.6). The advantage of this technique is that any imperfection of the optical setup modifies the path of both light beams. Therefore, they cancel out and do not influence the reconstructed interference measurement. To store and reconstruct both beams, a holographic storage technique has to be used.

Fig. 6. Principle of Mach-Zehnder (left) and holographic interferometry (middle, right)

High Enthalpy Non-equilibrium Shock Layer Flows. Selected Practical Applications

307

Fig. 7. Schematic setup of the HEG holographic interferometry system

A schematic of the holographic interferometry setup used at HEG is shown in Fig.7. The green line shows the light path of the reference beam and the red line shows the light path of the object beam. The object light beam passes through the test section and is brought to interference with the reference beam on the holographic plate. Here the interference pattern between both beams is recorded. To achieve interference between the object and the reference beam, a light source with sufficient coherence length is needed. A seeded Innolas Nd:YAG Laser (Model Spitlight 300), emitting light at 532 nm is used because it has a coherence length larger than 1 m. The optical path length of one of the beams is approximately 15 m and the successful set up of the optical system is facilitated by aligning the path length difference between the two beams to the coherence length of the laser. The first exposure of the holographic plate with one reference beam is performed prior to a run in HEG and a second exposure with the second reference beam is obtained during the test time. After the chemical treatment of the holographic plate, two reconstruction waves are created in a separate reconstruction unit. These reference waves used in the reconstruction unit are identical to the reference waves used for both exposures. 2.3 CFD Code Steady state nozzle flow and cylinder flow computations were performed using the DLR CEVCATS-N code[11]. This block-structured, three dimensional finite-volume scheme solves the Navier-Stokes equations and uses multigrid strategies with residual averaging and local time-stepping to accelerate convergence to a steady state. The test gas air is modelled as a chemically reacting mixture of perfect gases and is assumed to consist of five species, namely molecular and atomic nitrogen and oxygen and nitric oxide. Reaction rate models by Park[25], Dunn and Kang[5] and Gupta et al.[8] were used. For the latter the third body efficiencies were set to unity. Hereby the dissociation reactions are delayed. The determination of the vibrational energy of the molecular species is based on the assumption of a harmonic oscillator. The transition of energy between the vibrational modes and the translational modes is approximated using the Landau-Teller formulation. The vibrational relaxation times were obtained from correlations of Millikan and White and Park (see, e.g.,[26]). The viscous fluxes

308

High Temperature Phenomena in Shock Waves

such as diffusion, shear stress and heat flux include the assumptions of Fick, Stokes and Fourier, respectively (see, e.g.,[1]). The species viscosities are provided by curve fits published by Blottner et al.[3]. The species thermal conductivities are given by the Eucken correction and the mixture viscosity and thermal conductivity are calculated by Wilke’s mixture rule (see, e.g.,[1]). At the subsonic inflow boundary of the nozzle both velocity components are extrapolated from the integration domain. Due to the low velocities and high densities it is assumed that the flow here is in chemical and thermal equilibrium. Therefore, the full state of the gas entering the nozzle can be calculated by using the total enthalpy and the entropy of the nozzle reservoir. At the inflow boundary of the cylinder calculations, uniform as well as spatially varying free stream conditions derived from the nozzle flow computations were used. The outflow boundaries for both types of computations were located in regions of the flow which are dominated by supersonic/hypersonic flow. Therefore, all conservative variables are extrapolated. At the solid walls no-slip conditions apply. It is assumed that the walls are isothermal and fully catalytic. The algebraic turbulence model of Baldwin and Lomax[2] was applied for the computation of the nozzle flow. 2.4 Experimental Setup and Results The diameter of the cylinder was 90 mm and its length was 380 mm. The cylinder model was equipped with 17 pressure transducers to measure surface pressure distributions and 17 thermocouples to measure surface heat flux distributions. These transducers were distributed along six rows located in the plane of symmetry at mid span location and 10, 20 and 30 mm to the left and the right of the plane of symmetry. The model was mounted on the nozzle centreline (Fig.8).

pressure gauges

thermocouples

Fig. 8. Cylinder model in the HEG test section including the grid used for the threedimensional flow field computations (left) and model dimensions and gauge positions (right)

High Enthalpy Non-equilibrium Shock Layer Flows. Selected Practical Applications

309

The results which will be discussed were obtained using HEG operating condition III (h0 = 12 MJ/kg and condition I (h0 = 22 MJ/kg, see Table 1). Two and three-dimensional computations of the flow past the cylinder model were performed assuming laminar flow, chemical non-equilibrium, and thermal equilibrium. The computational grid used for the 3D simulations is shown in the left part of Fig.8. It consists of 33x81x101 grid points. The symmetry plane of the 3D grid was also used for the 2D computations in order to avoid discrepancies caused by different resolutions. Two dimensional simulations were carried out using the reaction rate models by Park [25], Dunn and Kang [5] and Gupta et al. [8]. The different free stream conditions used for this study resulted from nozzle computations, each performed with one of the reaction rate models mentioned above. The resulting shock standoff distances were compared with the experimental values and a summary of the results is given in Fig.9.

Fig. 9. Comparison of the computed shock stand-off distance resulting from 2D flow field analyses with the values obtained in HEG

It can be seen that using the modified reaction rate model of Gupta et al. results in the smallest deviation between computed and measured shock stand-off distance. For that reason this chemistry model was also chosen for further 2D and 3D cylinder flow field computations. In order to investigate the influence of the flow past the edges of the cylinder on the flow in the central part of the model, 3D flow field computations were performed. A comparison between 2D and 3D results for surface pressure and heat flux along the stagnation line in span wise direction is shown in Fig.10. The centre plane of the cylinder is located at z/R=0. Two different sets of inflow conditions were used for the 3D simulations. The first set is referred to as "parallel" and represents a parallel inflow with averaged free stream conditions in the plane of the cylinder position. The second set is referred to as "conical". Here the diverging flow resulting from the conical nozzle computation is used to interpolate inflow conditions on the CFD grid around the cylinder. A comparison of the computed

310

High Temperature Phenomena in Shock Waves

surface pressure and heat flux distributions in the centre plane is given in Fig.11. It is clear from these plots that the surface data in the centre plane resulting from 3D computations compares well with the data obtained from the 2D computation. The heat flux and pressure distributions along the stagnation line of the cylinder obtained with the 3D computations show the influence of the cylinder edge effects on the surface pressure and heat flux. From the centre plane up to about z/R=2.5, no significant impact of the edge effects exists. It can also be seen from Fig.10 that compared to the “parallel” inflow condition, the utilisation of the “conical” condition results in only a small difference of the obtained surface pressure and heat flux in the centre plane.

Fig. 10. Comparison of surface pressure and heat flux along the stagnation line in span wise direction resulting from 2D and 3D computations

Fig. 11. Comparison of surface pressure and heat flux in the centre-plane resulting from 2D and 3D computations

High Enthalpy Non-equilibrium Shock Layer Flows. Selected Practical Applications

311

The comparison between computed and measured surface pressure and surface heat flux measurements for HEG run 627 and HEG run 619 (HEG condition III and condition I) is shown in Fig.12. Good agreement between the 2D predictions and the measurements was obtained (4% deviation for stagnation pressure and 1% deviation for stagnation heat flux). It is obvious that the surface pressure is not very sensitive with respect to the physical-chemical modelling in CFD tools. Therefore the potential of this quantity for code validation can be regarded as low. For the present flow and surface property conditions, this insensitivity was also observed for the surface heat flux. However, shock stand-off distance, density distribution in the shock layer and static free stream pressure are sensitive to variations in the physical-chemical modelling and are therefore well suited for validation purposes.

Fig. 12. Comparison between measured and computed surface pressure and heat flux

312

High Temperature Phenomena in Shock Waves

Fig. 13. Sketch of the line of sight phase shift distribution reconstruction based on the computed 3D flow fields (left) and comparison of computed and measured phase shift distribution (right); HEG operating condition III (run 627) and operating condition I (run 619)

It was shown through the comparison of the results of the two and three dimensional computations, that the flow in the symmetry plane of the cylinder can be regarded as two dimensional. However 3D effects become important for optical line of sight methods. Therefore, numerical phase shift distributions were obtained from the CFD solutions by using a procedure as described [10]. A computational ray tracing algorithm was implemented (see Fig.13) and numerical phase shifts were obtained for the 2D computations and the 3D computations with both “parallel” and “conical” free stream conditions. The comparison of the computed and measured phase shift along the stagnation line is given in Fig.14. These figures show clearly that the result obtained from the 3D computations by ray tracing along a set of lines of sight differ from the 2D CFD results. It should be emphasized that these differences are only due to the contributions of the outer flow regions past the cylinder edges. The use of the “conical” free stream condition for the 3D computation results in a shock stand-off distance which is reduced by about 3%.

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313

Fig. 14. Comparison of computed phase shift distributions along the stagnation line. The phase shift values are normalized by 2 π 5

phase shift

4 3

CFD, 3D with conical free stream C III CI Experiment Run 627, C III Run 619, C I

2 1 0 -1 -0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

x/ R

Fig. 15. Comparison of computed and measured phase shift distributions along the stagnation streamline for HEG operating condition III and condition I. [19]

Fig.15 shows the computed and measured phase shift distributions along the stagnation streamline. Using the result of the 3D computation with “conical” free stream condition, very good agreement was obtained with the measured phase shift distribution for both conditions [19]. The corresponding two-dimensional phase shift distributions are plotted in the right part of Fig.13. 2.5 Summary and Conclusions Experiments in HEG were performed using a cylinder as a basic configuration for CFD validation. This configuration has the advantage of a large shock stand-off distance that permits optical measurement techniques to investigate gas properties in the shock layer. The experiments utilizing HEG operation conditions I and III are well suited to investigate chemical relaxation phenomena. Both the nozzle flow (dominated by

314

High Temperature Phenomena in Shock Waves

recombination reactions) and the flow in the test section (dominated by dissociation reactions) are subjected to strong chemical non-equilibrium effects. The static pressure and the phase shift measurements proved to be a useful indicator to judge the suitability of different chemical relaxation models. The interpretation of optical line of sight measurements methods such as holographic interferometry requires the knowledge of the entire flow field in order to quantify three-dimensional effects. In general it can be concluded that the engineering numerical analysis of chemically reacting hypersonic flows is still subjected to large uncertainties. Available experimental data can be used to tune the chemistry model such that for a range of flow conditions good numerical predictions of the chemical relaxation process in the shock layer can be obtained. For the present flow conditions with a fully catalytic and cold model wall, surface heat flux and pressure measurements are not useful for the validation of relaxation models. Nevertheless, they are a good measure for the total enthalpy and the total pressure of the flow.

3 CFD Modeling of Radiation Phenomena in Shock Layers 3.1 Introduction, Definitions and Nomenclature The design framework of future space exploration missions requires the accurate characterization of the aerothermal environments of potential atmospheric (re)-entry vehicles. Reliable thermal and mechanical load predictions form the basis of the design of effective thermal protection systems as well as the prediction of vehicle flight characteristics and performance. From an aerothermodynamics point of view, the additional effect of gas radiation in the shock layer has to be considered for certain atmospheric (re)-entry missions. This is the case for planned future Lunar and Martian return missions, general sample return missions (e.g., Stardust), high speed Jupiter entry (like Galileo) but also for entry missions into atmospheres with constituents forming strongly radiating molecules (e.g., Titan atmosphere). The radiative heat flux computation relies on the implementation of adequate models for the prediction of the amount of energy released or absorbed in all regions of the flow field and its transport within the shock layer and to the surface of the space vehicle. The terms radiative heat transfer or thermal radiation are commonly used to describe the energy transport caused by electromagnetic waves or photons. Broad introductory discussions of the fundamentals of thermal radiation are given in many textbooks (e.g. Modest [21]). This section briefly reviews the basic equations to be solved and the nomenclature used in the following sections. All materials continuously emit and absorb radiation by lowering or raising their atomic or molecular energy excitation states. Radiative energy can be viewed as consisting of electromagnetic waves or mass less energy particles called photons. These waves or photons travel through gases with the speed c which is related to the speed of light in vacuum by c=c0/n, where n is the index of refraction. Each wave may be identified either by its frequency (ν in [1/s]), its wavelength (λ in [m]), its wavenumber (η in [1/m]) or its angular frequency ( ω in [rad/sec]). All four quantities are related to each other by

High Enthalpy Non-equilibrium Shock Layer Flows. Selected Practical Applications

ν=

ω 2π

=

c

λ

= cη

315

(5)

Each wave or photon carries an amount of energy which is determined as: E photon = hν

(6)

Since electromagnetic waves of vastly different wavelengths carry vastly different amounts of energy, their behaviour is often quite different. Therefore, they have been grouped into a number of different categories with approximate boundaries as shown in Table 2. Thermal radiation covers the regimes from infrared to ultraviolet. Table 2. Different wavelength regimes

Category radio waves microwaves infrared visible light ultraviolet X-rays gamma radiation

from wavelength --1m 1 mm 750 nm 400 nm 1 nm 1E-12 m

to wavelength 1m 1 mm 750 nm 400 nm 1 nm 1E-12 m ---

Every medium continuously emits radiation randomly into all directions at a rate depending on the temperature and the properties of the material. The radiative heat flux emitted from a surface element in all directions is called emissive power, E. spectral emissive power, Eλ [W/m2/m] total emissive power, E [W/m2]

emitted energy / time / area / wavelength emitted energy / time / area

While emissive power appears to be the natural choice to describe radiative heat flux leaving a surface, it is inadequate to model the directional dependence of the radiation field. This is particularly the case inside a participating (absorbing/emitting) medium. Therefore, very similar to the emissive power, the radiative intensity, I, is defined as heat flux per unit solid angle and unit area normal to the rays representing the solid angle. Again, spectral and total intensity are distinguished: spectral Intensity, Iλ [W/m2/sr/m] total Intensity, I [W/m2/sr]

heat flux / time / area / solid angle / wavelength heat flux / time / area / solid angle

316 High Temperature Phenomena in Shock Waves

The geometrical definition of an element of solid angle dω on a unit (hemi-) sphere is given in Fig.16. It is customary to describe the direction vector Ω in terms of a polar coordinate system using an azimuthal angle (here: θ) and an elevation angle (here: φ).

Length of sides 2 and 4: dφ Length of sides 1 and 3: cos(φ)dθ Therefore: d ω = cos(φ)dφdθ

Fig. 16. Definition of the solid angle dω

The radiative heat flux in the y-direction passing through a surface element dA in the x-z plane is obtained by integrating the intensity field over all directions of the complete sphere: q y = ∫ I (φ , θ ) sin (φ ) dw = ∫ N ∩

2π

0

π /2

∫π

− /2

I (φ , θ ) sin (φ )cos (φ )d φ dθ

(7)

projection

The projection of the surface elements normal to the rays from the definition of the intensity to the surface element in the x-z plane introduces the additional term sin(φ). 3.2 The Radiative Transfer Equation in Participating Media In order to describe the radiation field in a participating medium, the definitions of additional quantities are needed. The absorption coefficient αλ [1/m] is defined as the inverse mean free path of the photons or electromagnetic waves. It is a measure of the opacity of a medium. Furthermore, the power density, e, is defined here as the emitted radiative power per unit volume and solid angle. Once more it is distinguished between: spectral power density, eλ [W/m3/sr/m]

≡

emitted power / volume / solid angle / wavelength

total power density, e [W/m3/sr]

≡

emitted power / volume / solid angle

High Enthalpy Non-equilibrium Shock Layer Flows. Selected Practical Applications

317

The radiative transfer equation (8) describes the intensity propagation of electromagnetic radiation in a medium along a line of sight:

dI λ ( s) + α λ ( s) I λ ( s) = eλ ( s) ds

(8)

The solution of this ordinary linear differential equation for a given initial intensity I0 is given in equation (9): s α ( w ) dw ⎛ ⎞ − α ( r ) dr I λ (s ) = ⎜ I λ (0) + ∫ eλ (r )e ∫0 dr ⎟ e ∫0 0 ⎝ ⎠ r

s

(9)

To compute the radiative heat flux vector at a certain point in a three dimensional domain, equation (8) is needed to obtain the intensity field around this point which can be integrated over all directions using equation (7) yielding the heat flux vector: q x = ∫

∞

∫

4π

λ =0 0

s ⎛ ∫0 α ( w ) dw dr ⎞ e − ∫0 α ( r ) dr cos (φ ) cos (θ ) d ω d λ ⎜ I λ (0) + ∫0 eλ ( r )e ⎟ ⎝ ⎠ r

s

s α ( w ) dw ⎛ ⎞ − α ( r ) dr q y = ∫ ∫ ⎜ I λ (0) + ∫ eλ ( r ) e ∫0 dr ⎟ e ∫0 sin (φ ) d ω d λ λ =0 0 0 ⎝ ⎠ r s ∞ 4π ⎛ s α ( w ) dw ⎞ − α ( r ) dr q z = ∫ ∫ ⎜ I λ (0) + ∫ eλ ( r ) e ∫0 dr ⎟ e ∫0 cos (φ ) sin (θ ) d ω d λ λ =0 0 0 ⎝ ⎠ ∞

4π

r

s

(10)

The quantity s is the distance from the point under investigation in a direction defined by φ and θ in spherical coordinates. The aim of the different numerical models which are introduced in the following sections is to find an approximate solution for the set of equations (9) and (10). 3.3 One-Dimendional Approximations for the Solution of the Radiative Transfer Equation 3.3.1 The Infinite Slab Model The radiative heat flux in an infinite isothermal slab can be computed analytically. A combination of particular solutions for isothermal slabs was used by Lee [17] to compute heat fluxes for arbitrary property distributions. The solution for the radiative heat flux qi+1 at the boundary of an infinite isothermal slab is:

qi +1 = qi 2 E3 (κ ) + σ Tm4 (1 − 2 E3 (κ ) )

(11)

318 High Temperature Phenomena in Shock Waves

The optical thickness, κ, is defined as the product of the absorption coefficient, α, and the slab height, y: κ = α y (see Fig.17).

Fig. 17. Geometry of the infinite slab case

E3 is an exponential integral function and is defined as: 1

E3 (κ ) = ∫ μ exp(−κ / μ ) d μ 0

(12)

For moderate optical thicknesses, κ, the exponential integral function, E3, can be approximated [15] using the expression in Equation (13). A comparison of this approximation and the exact values of E3 is shown in Fig.18. E3 (κ ) ≈

1 1 κ3 − κ + ( 0.9228 − ln κ ) κ 2 + 2 2 3!

(13)

Fig. 18. Approximation for the exponential integral function E3

Infinite slabs with arbitrary stepwise property distributions can be computed by combining several isothermal slabs and using equation (12) to evaluate the heat fluxes at the cell boundaries as illustrated in Fig.19.

Fig. 19. Infinite slab multilayer model

High Enthalpy Non-equilibrium Shock Layer Flows. Selected Practical Applications

319

The overall radiative heat flux at a cell boundary n is:

qn = qn+ − qn−

(14)

The infinite slab model is a computationally very efficient 1D approximation for the radiative transfer equation. For the application to basic capsule geometries a maximum error of about 30% compared to the solution of the 3D radiative transfer equation including the back flow region was observed [22,23.] 3.3.2 Infinite Cylinder An analytical solution for radiative heat transfer in axisymmetric geometries is given by Sakai, Sawada and Mitsuda [23]. It is valid for the one-dimensional radiative heat flux in infinitely long cylinders. Equation (15) is valid for non-emitting black boundaries: q + (r ) = 4∫

π /2

0

+∫

⎡ rcosγ α ( r ′)e( r ′) D 2 ⎢⎣ ∫0

R 2 − r 2 sin 2γ

0

q − ( r ) = −4 ∫

π /2

0

α ( r ′)e( r ′) D2

(∫

(∫

rcosγ y

rcosγ

0

)

y

⎤

α (r ′′) dy ′ + ∫ α (r ′′) dy ′ dy ⎥ cosγ d γ 0

⎡ R 2 − r 2sin 2γ α ( r ′)e( r ′) D2 ⎢ ∫rcosγ ⎣

y = r ′2 − r 2 sin 2γ

)

α (r ′′) dy′ dy

(∫

y

rcosγ

y′ = r ′′2 − r 2 sin 2γ

)⎤⎦

⎦

(15)

α (r ′′)dy ′ ⎥cosγ d γ D2 ( z ) = ∫

1

0

μ

⎛ z⎞ exp ⎜ − ⎟ d μ ⎝ μ⎠ 1− μ 2

The overall heat flux in radial direction is obtained by combining the positive and negative contributions in equation (15): qrad ( r ) = q + ( r ) + q − ( r )

(16)

The result of this set of equations is the radial heat flux qrad at different radial positions, r (as illustrated in Fig.20).

Fig. 20. Schematic of 1D radiative heat flux in an infinitely long cylinder

320 High Temperature Phenomena in Shock Waves

3.4 Approximate Solution Methods of the Radiative Transfer Equation in Three Dimensions 3.4.1 The Discrete Transfer Model A popular scheme to solve the 3D radiation transfer problem is the Discrete Transfer Method. It was first introduced 1979 by Shah [29]. Later an improved version with special application to hypersonic aerothermodynamics problems was presented by Gregory and Cinnella [7]. In this model, the double integration over geometric parameters (angle and length) is achieved by superimposing a radiation subgrid on the considered domain (For simplicity, also the CFD grid can be used as the radiation grid). For every radiation cell in the calculation it is necessary to define a set of sample rays that will be considered for the integration over the angle (equation (10)). On each ray a number of representative points is defined which serves for the integration over the distance along each ray (equation (8)). A typical example of a (very coarse) radiative subgrid in a 2D plane is shown in Fig.21. The numerical solution procedure to obtain the radiative heat flux vector in one computational cell consists of five steps: 1. Choice of a distribution of sample rays (or sample directions) originating at the cell centre and ending at the boundaries of the domain. 2. Generation of a distribution of representative points for the integration of the radiative transfer equation along each ray. 3. Allocation of emission and absorption properties to each integration point (ray-tracing algorithm). 4. Integration of the transfer equation along each ray. 5. Angular integration by summing up the contributions coming from each direction weighted by the respective fraction of solid angle represented by a ray.

Fig. 21. Radiative grid and sample ray distribution for the discrete transfer model in a 2D plane

High Enthalpy Non-equilibrium Shock Layer Flows. Selected Practical Applications

321

In order to be able to compute the radiative heat flux vector at a point in a flowfield the sample ray distribution around it has to be specified. A possible choice is to divide the range of the azimuth and the elevation angles into equal intervals. This would result in a distribution as shown in Fig.22a. The disadvantage of this approach is clearly that rays near the "equator" are associated to a large fraction of solid angle whereas rays near the "north- and south-pole" are of tiny importance. Since all rays require the same computational effort it is desirable to assign equal fractions of solid angle and therefore equal importance to them. This can be done in an approximate way using an icosahedron as the basis for the direction distribution (Fig.22b). An icosahedron consists of 20 equilateral triangles of exactly the same size. These triangles can be further more subdivided into equilateral triangles. Rays that are associated to the corner points of those triangles represent approximately the same fraction of solid angle.

Fig. 22. Possible choices of sample directions for the Discrete Transfer Model

The main disadvantages associated with the Discrete Transfer Method are that the specification of a meaningful maximum length of a sample ray is difficult (especially critical if (multiple) surface reflections have to be taken into account). Further, it is generally difficult to design an adaptive algorithm which excludes regions of little interest for the radiation transport. 3.4.2 Solution of the Radiative Transfer Equation Using a Monte Carlo Method The Monte Carlo Method [36] directly simulates the physical process of radiative heat transport. It traces and collects the absorption and scattering behaviour of a large number of independent radiative energy particles (photons) which are emitted from each point of the system. Therefore, the radiative energy is not treated as a continuous and variable entitity, but is considered as a distributed collection of photons each with a fixed amount of energy. The main advantages of the Monte Carlo model are its applicability to complex geometries, the high adaptivity of the algorithm (e.g. many photos emitted from hot

322 High Temperature Phenomena in Shock Waves

regions, less photons elsewhere) and the easy implementation of scattering or reflection at surfaces. The disadvantage is the statistical noise in the numerical results. The schematic outline of the Monte-Carlo solution procedure is: 1. Assign the total emitted radiative power density in each control volume to a given number of virtual “photons” 2. Send each photon to a random direction 3. Trace the path of the photon through the computational domain 4. Integrate the optical distance (L=∫αds) 5. If the optical distance exceeds a random absorption limit then assign the energy carried by the virtual photon to the respective control volume A more detailed description of steps 1 to 5 of the schematic of the solution procedure is given below: Step 1 A uniform distribution (same number of photons for each control volume) or an adaptive distribution (number of photons depends on the emitted energy) can be chosen. For the adaptive option, a maximum number of sent particles per cell, Nmax, has to be specified. This number is then weighted for each volume by the maximum of sent energy over all volume elements according to: sent Qmax =

max(eV i i) , N max

(17)

where ei is the emitted power density of a cell i with volume Vi. The actual number of emitted photons per cell, Ni, is then easily obtained by:

⎛ eV ⎞ Ni = int ⎜ isenti ⎟ ⎝ Qmax ⎠

(18)

The result is that for the uniform case the same number of photons with different energy portions is emitted from each cell whereas for the adaptive case a different number of photons is emitted from each cell but each carrying the same amount of energy. Step 2 After the number of emitted photons and the amount of energy carried by each photon has been determined in step 1, all photons are emitted to a random direction for each computational cell successively. No directional weighting is applied for volume radiation. The direction vector d is computed from 3 evenly distributed random numbers R1 to R3 according to:

G ⎡ −1 + 2 R1 ⎤ G ⎢ G d1 ⎥ d1 = −1 + 2 R2 d = G ⎢ ⎥ d1 ⎣⎢ −1 + 2 R3 ⎦⎥

(19)

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323

The emission from surfaces is weighted according to Lambert’s cosine law. If Rθ and Rφ are evenly distributed random numbers representing the elevation and azimuthal directions respectively, the direction of the emitted photons is chosen according to:

φ = 2π Rφ θ = arccos 1 − Rθ

(20)

Steps 3 and 4 Having specified the start point of the photon (centre of mass of the respective computational cell) and its direction (d) the next problem is to trace the path of the photons through the computational domain. This is can be done using ray tracing algorithms such as described in detail by Widhalm [34] and Löhner [18]. The choice of the ray tracing algorithm strongly depends on the applied grid topology and is the most critical element for computational efficiency of the Monte-Carlo method. At surface elements the photon is either lost (free stream boundary), reflected or absorbed (walls). Two different wall treatments are implemented for wall with a given absorption coefficient αW: Diffuse reflection: The photon is absorbed and its energy, QP, is associated to the wall element. Another photon is emitted by the wall in a random direction according to equation (19) which carries the energy QP(1-αW). Probabilistic reflection: The photon is absorbed with a probability of αW (compared to an evenly distributed random number). Its entire energy is absorbed by the wall. If the photon is not absorbed (probability (1-αW)) the photon is specularly reflected. Step 5 As described in the introductory part of this section, the radiatve energy is treated as a distributed quantity in the Monte Carlo Method. It is postulated that all energy of each particle will eventually be absorbed by gas molecules after a certain optical flight distance ∫α(s)ds and that the energy possessed by each photon remains unchanged during its flight. This is precisely the same concept as the physical transport of radiative energy by “real” photons. When a photon is absorbed somewhere in the computational domain, its entire energy is assigned to the respective computational cell and the photon is lost. To decide whether a photon is absorbed after a certain flight distance or not, a uniformly distributed random number Rs is assigned to each photon. The absorption criterion is then:

∫ α (s)ds ≥ − ln(1 − R )

(21)

s

The left hand side of equation (21) is provided by the ray tracing algorithm described above. The right hand side transforms the uniformly distributed random number Rs to the physical absorption distribution given by Beer’s law:

(

I ( s ) = I 0 exp − ∫ α ( s ) ds

)

(22)

324

High Temperature Phenomena in Shock Waves

Fig. 23. Radiative intensity disstribution along a line of sight for Beer’s law: absorption of innitial radiative intensity I0; the erro or bars indicate the standard deviation of the Monte Carlo eerror based on 1000 computations

Fig.23 shows the comp puted intensity distributions along a single line of siight (Monte Carlo Method, equation (21)) compared to Beer’s law (equation (22)) for the pure absorption of an initiial radiative intensity I0. The quadratic error norm of the Monte Carlo solution comp pared to the exact value is shown in the right part of this figure. Clearly, for a large number of photons the Monte Carlo solution converges to c rate is of 1st order with respect to the num mber the exact distribution. The convergence of sample particles or photo ons. 3.4.3 Isothermal Cylindeer The Monte Carlo method was w tested against the canonical test case of an isotherm mal infinitely long cylinder with h constant absorption coefficient. The analytical solution is given in the set of equattions (15)–(16). The assumption of constant properrties (temperature and absorptiion coefficient) allows simplifications in the analyttical solution to avoid the oth herwise very tedious numerical integration process.. A summary of the test case co onditions is given in Table 3. A schematic of a possible solution procedure for 2D-axisymmetric problems is given in Fig.24. The initiaal 2D CFD grid is rotated around the axis of symmeetry. Hence, a 3D-toroid is creatted from each initial 2D CFD cell. The angular resoluttion was chosen for the present test case according to the 2D discretisation (8, 16, 32 and y). After the Monte Carlo simulation was performed on the 64 “pie slices”, respectively artificial 3D domain, the absorbed and emitted energy particles (photons) w were summed up on each toroiid and were assigned to the initial 2D cell. Hence, no statistical information is losst.

High Enthalpy Non-equilibrium Shock Layer Flows. Selected Practical Applications

325

Table 3. Properties and boundary conditions for the isothermal cylinder test case

Test case:

isothermal cylinder, constant absorption coefficient

Geometry:

Radius R=1.0 m

Properties:

Temperature T = 10000 K, gray radiation Absorption coefficient: α = 1.0 m-1 Emission power density: e = ασT4/π

Discretisation:

Structured CFD grids of resolution n × n with n = 8, 16, 32, 64 32, 64, 128 or 256 emitted photons / cell Length of the CFD cylinder L = 10 m, reflecting boundaries at L = 0 m and L = 10 m

Fig. 24. Schematic of the solution procedure for axi-symmetric radiative transfer problems using the Monte Carlo Method

Results from the Monte Carlo simulation (radial radiative heat flux, see Fig.20 and divergence of radial radiative heat flux) compared to the exact values from equations (15) are shown in Fig.25 and Fig.26. The error bars in these Figures represent the standard deviation of the divergence of the radiative heat flux for the same radius at different axial locations (The theoretical result depends only on the radius).

326

High Temperature Phenomena in Shock Waves

Fig. 25. Radial radiative heat flux f and its divergence for the isothermal cylinder test case. L Left: computational grid size of 8x8 8 cells, 8 slices in circumferential direction, 32 emitted photoons / cell; Right: computational griid size of 16x16 cells, 16 slices in circumferential direction, 64 emitted photons / cell

Fig. 26. Radial radiative heat flux f and its divergence for the isothermal cylinder test case. L Left: computational grid size of 32x32 cells, 32 slices in circumferential direction, 128 emiitted photons / cell; Right: compu utational grid size of 64x64 cells, 64 slices in circumferenntial direction, 256 emitted photonss / cell

High Enthalpy Non-equilibrium Shock Layer Flows. Selected Practical Applications

327

A quantitative comparison of the different Monte Carlo results is given in Table 4. The CPU time was measured on a AMD Opteron 2.4GHz PC-system. The conclusions which can be drawn from inspecting this Table 3 are: • •

•

•

With increasing resolution the Monte Carlo result convergences to the analytical solution, The required CPU time scales approximately with [number of cells][2,5] and a convergence rate of 1st order with respect to the spatial discretisation cell size Δx is observed, which means halving the error of the numerical model requires an increase of CPU time of a factor of 32, This factor is consistent with the 2D expectation for 1st order accuracy and n2 complexity which is a factor of: 4 [cells] × 4 [rays / cell] × 2 [cells crossed / ray] = 32) The CPU time / traced photon shows the efficiency of the implemented ray tracer. The required computational time scales (almost) linearly with the average number of crossed cells. Table 4. Some statics of the Monte Carlo solutions of the isothermal cylinder test case

Configuration

total CPU time

CPU time / traced photon

83 points × 32 rays

0.25 sec

1.5 × 10-5 sec

163 points × 64 rays

6.7 sec

323 points × 128 rays 643 points × 256 rays

Error of total heat flux at R = 1m 18.8 %

Std. deviation of rad. source term at R = 1 m 4.1 %

2.5 × 10-5 sec

10.1 %

1.4 %

204 sec

4.9 × 10-5 sec

4.9 %

0.8 %

6280 sec

9.4 × 10-5 sec

2.4 %

0.6 %

3.5 Huygens Entry Peak Heating Prediction

The Monte Carlo Method was applied to study the aerothermodynamics of the Huygens probe [14] at peak heating conditions during its entry into the Titan atmosphere. Comprehensive parametric studies concerning the characterization of the aerothermal environment using different CFD tools are available in the literature [24,35,33]. The peak heating point of the flight trajectory was chosen for the present investigation using the DLR TAU code [27].

328 High Temperature Phenomena in Shock Waves

The computational grid consisted of 120 × 80 cells in the tangential and normal directions, respectively, and its shape was adapted to the bow shock wave. The thermochemistry modelling was based on the detailed 13 species chemical nonequilibrium model described by Gökcen [6]. Thermal equilibrium was assumed. The free stream conditions for the peak heating point are summarized in Table 5. The viscous wall of the capsule configuration was modelled using a fully catalytic (local chemical equilibrium) radiative equilibrium boundary condition employing a wall emissivity of 0.9. The entire flow field was assumed to be laminar. Table 5. Free stream conditions applied to the Huygens peak-heating test case

Density, ρ∞

0.296 g/m3

Temperature, T∞

176.6 K

Velocity, u∞

5126.3 m/s

CH4 mole fraction

2.3 %

The applied computational grids and details from the initial (obtained without radiation effects) CFD solution are shown in Fig.27 and Fig.28. The results in these Figures show that the flow in the shock layer is affected by chemical non-equilibrium. The temperature on the stagnation stream line in the shock layer drops from about 9000 K downstream of the shock to 6500 K at the boundary layer edge. Excellent agreement with reference data [24] was achieved concerning the species concentration distributions in the shock layer.

Fig. 27. Initial CFD solution for the Huygens peak-heating case: Left: Computational grid and pressure contours; Right: Temperature distribution along the stagnation line

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Fig. 28. Species concentrations along the stagnation line

Based on this initial CFD solution, a preparatory study of the impact of the applied spectral modelling parameters on the radiative stagnation point heat flux was undertaken using the PARADE 1D radiation transport tool [30] on the flow field property distribution along the stagnation streamline. Only the radiation of the CN molecule was considered in the present study. An exemplary uncoupled radiative heat flux spectrum is shown in Fig.29.

Fig. 29. Radiative heat flux spectrum on the Huygens stagnation point resulting from 1D analysis

330 High Temperature Phenomena in Shock Waves

Fig.30 shows the influence of the considered spectral range on the integrated radiative heat flux at the stagnation point. The radiative surface heat flux below a wavelength of 300 nm is negligible. The integration of the spectrum from this figure reveals that 98% of the total radiative heat flux is contained in the spectral range from 300 to 1500 nm. This range was applied for the further investigations. The error which is introduced by coarse the spectral resolutions is shown in the right part of Fig.30. The error is significantly less for constant frequency increments (df = const.) due to the clustering of points in the high frequency region. A spectral resolution of 20000 points at constant frequency increments was chosen for the further investigation. The error introduced by this choice is about 10%, compared to a resolution of 400,000 points which is needed for spectral convergence of the radiative heat flux. Because of the high numerical complexity of the Monte Carlo algorithm the required CPU-time for the numerical solution of the radiative transfer equation increases strongly with the applied spatial resolution. Therefore, a coarser “radiative subgrid” was used for the Monte Carlo algorithm and an inverse-distance interpolation algorithm [32] was applied to transfer the input and solution data between the flow-solver and Monte Carlo grids.

Fig. 30. Influence of the spectral range (left) and the spectral resolution (right) on the integrated radiative stagnation point heat flux

A parametric study was carried out to assess and optimize the numerical parameters of the Monte Carlo scheme. A summary of the considered cases is given in Table. The CPU time is measured in [h] on a single core of a 2.5 GHz Intel QuadProcessor. Spatial adaptation was applied for all computations concerning the number of sample photons sent from each computational cell. The maximum number of sent particles per cell, Nmax, was specified either as a constant or as a function of the considered wavelength. This number is then weighted for each volume by the maximum of sent energy over all volume elements according to equation (18).

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Additional spectral adaptivity was applied to the cases marked with the label “adaptive” in Table 6. The maximum number of sample photons per cell, Nmax, was chosen to depend on the spectral radiative heat flux distribution at the stagnation point (see Fig.29). ⎡ ⎤ q (λ ) Nmax = int ⎢ Nmax + 1⎥ max ⎡⎣q ( λ ) ⎤⎦ ⎥ ⎢⎣ ⎦

(23)

Two different radiative subgrids (left plot of Fig.31) were applied in the present study. The resolutions of the standard and fine grid levels were 30×20 points and 60×40 points in the tangential and wall normal directions, respectively. Both grids were derived from the flow-solver grid and are adapted to the bow shock shape with refinement in the vicinity of the shock and the boundary layer close to the vehicle surface. An exemplary distribution of the radiative source term for the uncoupled case is shown in the right part of Fig.31. The distributions of surface heat flux and of the radiative source term along the stagnation streamline resulting from the different numerical parameter settings are shown in Fig.32. Table 6. Numerical parameters for the Monte Carlo algorithm

Case Non-adaptive Adaptive, less photons Adaptive, standard

Adaptive, fine subgrid

Sub-grid Sample photons / cell 30 × 20 Nmax = const = 2000 30 × 20 Nmax = 1 – 4000 (dep. on wavelength) 30 × 20 Nmax = 2 – 8000 (dep. on wavelength) 60 × 40 Nmax = 1 – 4000 (dep. on wavelength)

CPU-time 296 h 34 h

68 h 290 h

Fig. 31. Radiative subgrids (left) and exemplary distribution of the uncoupled radiative source term (right)

332 High Temperature Phenomena in Shock Waves

The conclusion that can be drawn from inspecting Fig.32 and Table 6 are: (1): the results for the adaptive and non-adaptive (in the spectral domain) computations are identical; (2): the chosen number of sample photons is sufficient to adequately model the emission and self-absorption phenomena in the flow field; (3): the introduction of spectral adaptation results in a significant decrease of required CPU-time, and, (4): reasonable grid convergence concerning the spatial discretization of the radiative transport problem was achieved. Based on these considerations, the “adaptive-standard” configuration was chosen for the coupled simulation of the Huygens peak heating case. A fully converged updated CFD solution was obtained using the constant radiative source term distribution from each coupling step. The radiative transfer calculation was then repeated using the updated CFD solution. To enhance convergence of this loosely coupled approach under-relaxation of the source term was applied for the initial coupling step, with a relaxation factor of 0.7. All successive coupling steps were performed using a relaxation factor of 1. The iterative loose-coupling procedure quickly converges after 4 coupling steps. The convergence history is shown in Fig.33.

Fig. 32. Uncoupled radiative surface heat fluxes (left) and distribution of the radiative source term along the stagnation streamline (right) for different radiative subgrids and numerical parameters of the Monte Carlo algorithm

The resulting coupled and uncoupled surface heat flux distributions are shown in Fig.34. The radiative surface heat fluxes are weighted with a wall absorption coefficient of αwall=0.9. Good agreement with available reference data from literature [24] was achieved. Due to the cooling effect of the flow-radiation coupling the radiative heat flux at the stagnation point is decreased by 22% compared to the uncoupled case. The convective heat flux decreases by 23%.

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Fig. 33. Convergence history of the total convective and radiative heat loads for the loose radiation-flow coupling approach as applied to the Huygens peak-heating case

The radiative source term is only affected by the flow-radiation coupling in the vicinity of the shock (Fig.36). The reduction of the radiative wall heat flux is mainly driven from this region. This is because the reduction of static temperature in the shock layer (right part of Fig.35) is compensated by an increase in CN partial density (left part of Fig.35).

Fig. 34. Coupled and uncoupled surface heat flux distributions including comparison with reference data [24] (the hollow symbols correlate to uncoupled data, and the solid symbols correlate to coupled data)

334 High Temperature Phenomena in Shock Waves

Fig. 35. Coupled and uncoupled distributions of CN partial density and temperature along the stagnation streamline

Fig. 36. Coupled and uncoupled distributions of the radiative source term along the stagnation streamline.

4 Summary and Conclusions Radiative heat transfer can become an important energy transport mechanism in high enthalpy shock layers. Radiative effects have to be considered in air flows at temperatures above approximately 10 000 K or in flows with strongly radiating constituents such as CN. The numerical modelling of gas radiation requires accurate prediction of the aerothermochemistry in the shock layer including chemical and thermal non-equilibrium effects that strongly influence the emission and absorption properties of the flow medium. A significant part of the radiation is emitted from the relaxation zone downstream of the bow shock which might be strongly affected by

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non-equilibrium effects. Then, appropriate models for the computation of local absorption and emission spectra from the local flow properties have to be developed and applied. Knowing the local spectral distribution of radiative properties in the flow field, the next step is to numerically solve the radiative transfer equation to compute the radiation heat flux distribution. Due to the mathematical nature of the radiative transfer equation, this can become a numerically expensive procedure. Because no general averaging procedures for spectral properties exist, the transfer equation has to be solved in a line-by-line manner for a large number of different wavelengths in the spectrum which additional complicates the numerical solution process. Averaged local absorption and emission coefficients can be applied for optically thin (Planckaverage) or thick (Rosseland-average) cases. Further, different approximate averaging methods for intermediate cases exist such as smeared band or multi-bin models. Having solved the radiative transfer equation, the divergence of the radiative heat flux can be coupled to the energy equation of the CFD solver to account for local heating (absorption) or cooling (emission) effects in the flow field. Because of the large computational cost for the radiative transfer prediction, a loose coupling strategy is beneficial; after a converged solution of the flow field is obtained, the radiation field is subsequently predicted and the radiative source terms are coupled to the conservation equation of energy to obtain an updated converged CFD solution. The application of under-relaxation of the radiative source term during the first coupling steps can significantly improve the convergence characteristics of this coupling approach.

References 1. Anderson, J.D.: Hypersonic and High Temperature Gas Dynamics. McGraw-Hill (1989) 2. Baldwin, B.S., Lomax, H.: AIAA Paper 78–257 (1978) 3. Blottner, F.G., Johnson, M., Ellis, M.: Chemically reacting viscous flow program for multi-component gas mixtures. Sandia Laboratories, SC-RR-70-745 (1971) 4. Brück, S., Radespiel, R., Longo, J.M.A.: AIAA Paper 97–0275 (1997) 5. Dunn, M.G., Kang, S.W.: Theoretical and experimental studies of reentry plasmas, NASA CR-2232 (1973) 6. Gökcen, T.: AIAA Paper 2004-2469 (2004) 7. Gregory, J.E., Cinella, P.: Computers and Fluids 24 (5), 523 (1995) 8. Gupta, R.N., Yos, J.M., Thompson, R.A., Lee, K.P.: A Review of Reaction Rates and Thermodynamic and Transport Properties for an 11-Species Air Model for Chemical and Thermal Non-Equilibrium Calculations to 30000 K. In: NASA Reference Publication, vol. (1232) (1990) 9. Hannemann, K., Schnieder, M., Reimann, B., Martinez Schramm, J.: AIAA Paper 20002593 (2000) 10. Hannemann, K., Martinez Schramm, J., Karl, S., Beck, W.H.: AIAA Paper 2002-2913 (2002) 11. Hannemann, K.: AIAA Paper 2003-0978 (2003) 12. Hannemann, K., Martinez Schramm, J.: High Enthalpy, High Pressure Short Duration Testing of Hypersonic Flows. In: Tropea, C., Foss, J., Yarin, A. (eds.) Springer Handbook of Experimental Fluid Mechanics, vol. 1081. Springer, Heidelberg (2007)

336 High Temperature Phenomena in Shock Waves 13. Hannemann, K., Martinez Schramm, J., Karl, S.: Recent extensions to the High Enthalpy Shock Tunnel Göttingen (HEG). In: Proceedings of the 2nd International ARA Days Ten Years after ARD, Arcachon, France (2008) 14. Harland, D.M.: Mission to Saturn: Cassini and the Huygens Probe, 1st edn. Springer, Berlin (2002) 15. Jarms, K.: Chemical non-equilibrium boundary layer flow including radiation, Stagiare report 1991-18/AR, VKI (1991) 16. Karl, S., Martinez Schramm, J., Hannemann, K.: High enthalpy shock tunnel flow past a cylinder. In: A basis for CFD validation New Results in Numerical and Experimental Fluid Mechanics, IV, vol. 87. Springer, Heidelberg (2004) 17. Lee, H., Buckius, R.O.: International Journal of Heat and Mass Transfer 26 (7), 1055 (1983) 18. Lohner, R., Ambrosiano, J.: Journal of Computational Physics 91 (1990) 19. Martinez Schramm, J.: Aerothermodynamische Untersuchung einer Wiedereintrittskonfiguration und ihrer Komponenten in einem impulsbetriebenen Hochenthalpie-Stoßkanal, Dissertation Universität Göttingen (2008) 20. Merzkirch, W.: Flow Visualization. Academic Press (1974) 21. Modest, M.F.: Radiative Heat Transfer. McGraw Hill (1992) 22. Olejniczak, J., Grinstead, J., Bose, D.: An Overview of Radiation Modeling Work for Shock Heated Gas for RTO AVT-136. In: 6th European Symposium on Aerothermodynamics for Space Vehicles, Versailles (2008) 23. Osawa, H., Matsuyama, S., Ohnishi, N., Sawada, K.: AIAA Paper 2006-3772 (2006) 24. Osawa, H., Matsuyama, S., Ohnishi, N., Furudate, M., Sawada, K.: J. Thermophysics and Heat Transfer 22 (2) (2008) 25. Park, C.: AIAA Paper 85-0247 (1985) 26. Park, C.: Nonequilibrium Hypersonic Aerothermodynamics. John Wiley & Sons, New York (1989) 27. Reimann, B., Johnston, I., Hannemann, V.: The DLR TAU- Code for High Enthalpy Flows, Notes on Num. Fluid Mech. and Multidisc. Design, vol. 87. Springer, Heidelberg (2004) 28. Sawada, K., Sakai, T., Mitsuda, M.: AIAA Paper 98-0861 (1998) 29. Shah, N.G.: New Method of Computation of Radiation Heat transfer in Combustion Chambers, PhD Thesis, London Imperial College of Science and Technology (1979) 30. Smith, A.J.: Plasma Radiation Database Parade, Final Report I3, ESA TR28/96 (2006) 31. Stalker, R.J.: AIAA Journal 5, 12 (1967) 32. Tecplot 360 User Manual, Tecplot Inc. (2010) 33. Walpot, L., Caillault, L., Molina, R.C., Laux, C.O., Blanquaert, T.: J. Thermophysics Heat Transfer 20(4) (2006) 34. Widhalm, M., Bartels, C., Meyer, J., Kroll, N.: AIAA Paper 2008-472 (2008) 35. Wright, M.J., Oleiniczak, J., Walpot, L., Raynaud, E., Magin, T., Caillault, L., Hollis, B.R.: AIAA Paper 2006-382 (2006) 36. Yang, W.J., Taniguchi, H., Kudo, K.: Radiative Heat Transfer by the Monte Carlo Method. In: Advances in Heat Transfer, vol. 27. Academic Press (1995)

Author Index

Lago, V. 271 Laricchiuta, A. Longo, S. 11

Brun, R. 1 Bruno, D. 11 Capitelli, M. 11 Chanetz, B. 271 Chpoun, A. 271 Colonna, G. 11

Magin, T.E. 149 Martinez Schramm, J.

d’Ammando, G. 11 d’Angola, A. 11 Giordano, D. 11 Gorse, C. 11 Hannemann, K. 299 Huo, W.M. 149 Ibraguimova, L.

11

99

Karl, S. 299 Kustova, E.V. 59

Nagnibeda, E.A.

299

59

Panesi, M. 149 Perrin, M.Y. 193 Raines, A.A. 231 Rivi`ere, Ph. 193 Shatalov, O. Soufiani, A.

99 193

Tcheremissine, F.G.

231